REVIEWS OF MODERN PHYSICS, VOLUME 78, OCTOBER–DECEMBER 2006 Dynamical principles in neuroscience Mikhail I. Rabinovich* Institute for Nonlinear Science, University of California, San Diego, 9500 Gilman Drive 0402, La Jolla, California 92093-0402, USA Pablo Varona GNB, Departamento de Ingeniería Informática, Universidad Autónoma de Madrid, 28049 Madrid, Spain and Institute for Nonlinear Science, University of California, San Diego, 9500 Gilman Drive 0402, La Jolla, California 92093-0402, USA Allen I. Selverston Institute for Nonlinear Science, University of California, San Diego, 9500 Gilman Drive 0402, La Jolla, California 92093-0402, USA Henry D. I. Abarbanel Department of Physics and Marine Physical Laboratory (Scripps Institution of Oceanography) and Institute for Nonlinear Science, University of California, San Diego, 9500 Gilman Drive 0402, La Jolla, California 92093-0402, USA 共Published 14 November 2006兲 Dynamical modeling of neural systems and brain functions has a history of success over the last half century. This includes, for example, the explanation and prediction of some features of neural rhythmic behaviors. Many interesting dynamical models of learning and memory based on physiological experiments have been suggested over the last two decades. Dynamical models even of consciousness now exist. Usually these models and results are based on traditional approaches and paradigms of nonlinear dynamics including dynamical chaos. Neural systems are, however, an unusual subject for nonlinear dynamics for several reasons: 共i兲 Even the simplest neural network, with only a few neurons and synaptic connections, has an enormous number of variables and control parameters. These make neural systems adaptive and flexible, and are critical to their biological function. 共ii兲 In contrast to traditional physical systems described by well-known basic principles, first principles governing the dynamics of neural systems are unknown. 共iii兲 Many different neural systems exhibit similar dynamics despite having different architectures and different levels of complexity. 共iv兲 The network architecture and connection strengths are usually not known in detail and therefore the dynamical analysis must, in some sense, be probabilistic. 共v兲 Since nervous systems are able to organize behavior based on sensory inputs, the dynamical modeling of these systems has to explain the transformation of temporal information into combinatorial or combinatorial-temporal codes, and vice versa, for memory and recognition. In this review these problems are discussed in the context of addressing the stimulating questions: What can neuroscience learn from nonlinear dynamics, and what can nonlinear dynamics learn from neuroscience? DOI: 10.1103/RevModPhys.78.1213 PACS number共s兲: 87.19.La, 05.45.⫺a, 84.35.⫹i, 87.18.Sn CONTENTS 3. Synaptic plasticity 1222 4. Examples of the cooperative dynamics of I. What are the Principles? A. Introduction individual neurons and synapses 1214 B. Robustness and adaptability in small microcircuits 1214 C. Intercircuit coordination D. Chaos and adaptability III. Informational Neurodynamics A. Time and neural codes 1. Temporal codes 2. Spatiotemporal codes 3. Coexistence of codes 4. Temporal-to-temporal information transformation: Working memory B. Information production and chaos 1. Stimulus-dependent motor dynamics 2. Chaos and information transmission C. Synaptic dynamics and information processing B. Classical nonlinear dynamics approach for neural systems C. New paradigms for contradictory issues 1215 1217 II. Dynamical Features of Microcircuits: Adaptability and Robustness 1218 A. Dynamical properties of individual neurons and synapses 1218 1. Neuron models 1218 2. Neuron adaptability and multistability 1219 *Electronic address: [email protected] 0034-6861/2006/78共4兲/1213共53兲 1213 1223 1224 1228 1229 1231 1231 1231 1232 1233 1234 1237 1237 1239 1240 ©2006 The American Physical Society 1214 Rabinovich et al.: Dynamical principles in neuroscience D. Binding and synchronization IV. Transient Dynamics: Generation and Processing of Sequences A. Why sequences? B. Spatially ordered networks 1. Stimulus-dependent modes 2. Localized synfire waves C. Winnerless competition principle 1. Stimulus-dependent competition 2. Self-organized WLC networks 3. Stable heteroclinic sequence 4. Relation to experiments D. Sequence learning E. Sequences in complex systems with random connections F. Coordination of sequential activity V. Conclusion Acknowledgments Glossary References 1242 1244 1244 1244 1244 1247 1248 1248 1249 1250 1251 1252 1254 1256 1258 1259 1259 1260 “Will it ever happen that mathematicians will know enough about the physiology of the brain, and neurophysiologists enough of mathematical discovery, for efficient cooperation to be possible?” —Jacques Hadamard I. WHAT ARE THE PRINCIPLES? A. Introduction Building dynamical models to study the neural basis of behavior has a long tradition 共Ashby, 1960; Block, 1962; Rosenblatt, 1962; Freeman, 1972, 2000兲. The underlying idea governing neural control of behavior is the three-step structure of nervous systems that have evolved over billions of years, which can be stated in its simplest form as follows: Specialized neurons transform environmental stimuli into a neural code. This encoded information travels along specific pathways to the brain or central nervous system composed of billions of nerve cells, where it is combined with other information. A decision to act on the incoming information then requires the generation of a different motor instruction set to produce the properly timed muscle activity we recognize as behavior. Success in these steps is the essence of survival. Given the present state of knowledge about the brain, it is impossible to apply a rigorous mathematical analysis to its functions such as one can apply to other physical systems like electronic circuits, for example. We can, however, construct mathematical models of the phenomena in which we are interested, taking account of what is known about the nervous system and using this information to inform and constrain the model. Current knowledge allows us to make many assumptions and put them into a mathematical form. A large part of this review will discuss nonlinear dynamical modeling as a particularly appropriate and useful mathematical framework that can be applied to these assumptions in order to Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 FIG. 1. 共Color online兲 Illustration of the functional parts and electrical properties of neurons. 共a兲 The neuron receives inputs through synapses on its dendritic tree. These inputs may or may not lead to the generation of a spike at the spike generation zone of the cell body that travels down the axon and triggers chemical transmitter release in the synapses of the axonal tree. If there is a spike, it leads to transmitter release and activates the synapses of a postsynaptic neuron and the process is repeated. 共b兲 Simplified electrical circuit for a membrane patch of a neuron. The nonlinear ionic conductances are voltage dependent and correspond to different ion channels. This type of electrical circuit can be used to model isopotential single neurons. Detailed models that describe the morphology of the cells use several isopotential compartments implemented by these circuits coupled by a longitudinal resistance; these are called compartmental models. 共c兲 A typical spike event is of the order of 100 mV in amplitude and 1 – 2 ms in duration, and is followed by a longer after-hyperpolarization period during which the neuron is less likely to generate another spike; this is called a refractory period. simulate the functioning of the different components of the nervous system, to compare simulations with experimental results, and to show how they can be used for predictive purposes. Generally there are two main modeling approaches taken in neuroscience: bottom-up and top-down models. • Bottom-up dynamical models start from a description of individual neurons and their synaptic connections, that is, from acknowledged facts about the details resulting from experimental data that are essentially reductionistic 共Fig. 1兲. Using these anatomical and physiological data, the particular pattern of connectivity in a circuit is reconstructed, taking into account the strength and polarity 共excitatory or inhibitory兲 of the synaptic action. Using the wiring diagram thus obtained along with the dynamical features of the neurons and synapses, bottom-up models have been able to predict functional properties of Rabinovich et al.: Dynamical principles in neuroscience neural circuits and their role in animal behavior. • Top-down dynamical models start with the analysis of those aspects of an animal’s behavior that are robust, reproducible, and important for survival. The top-down approach is a more speculative big-picture view that has historically led to different levels of analysis in brain research. While this hierarchical division has put the different levels on an equal footing, the uncertainty implicit in the top-down approach should not be minimized. The first step in building such large-scale models is to determine the type of stimuli that elicit specific behaviors; this knowledge is then used to construct hypotheses about the dynamical principles that might be responsible for their organization. The model should predict how the behavior evolves with a changing environment represented by changing stimuli. It is possible to build a sufficiently realistic neural circuit model that expresses dynamical principles even without knowledge of the details of the neuroanatomy and neurophysiology of the corresponding neural system. The success of such models depends on the universality of the underlying dynamical principles. Fortunately, there is a surprisingly large amount of similarity in the basic dynamical mechanisms used by neural systems, from sensory to central and motor processing. Neural systems utilize phenomena such as synchronization, competition, intermittency, and resonance in quite nontraditional ways with regard to classical nonlinear dynamics theory. One reason is that the nonlinear dynamics of neural modules or microcircuits is usually not autonomous. These circuits are continuously or sporadically forced by different kinds of signals, such as sensory inputs from the changing environment or signals from other parts of the brain. This means that when we deal with neural systems we have to consider stimulusdependent synchronization, stimulus-dependent competition, etc. This is a departure from the considerations of classical nonlinear dynamics. Another very important feature of neuronal dynamics is the coordination of neural activities with very different time scales, for example, theta rhythms 共4 – 8 Hz兲 and gamma rhythms 共40– 80 Hz兲 in the brain. One of our goals in this review is to understand why neural systems are very specific from the nonlinear dynamics point of view and to discuss the importance of such specificities for the functionality of neural circuits. We will talk about the relationship between neuroscience and nonlinear dynamics using specific subjects as examples. We do not intend to review here the methods or the nonlinear dynamical tools that are important for the analysis of neural systems as they have been discussed extensively in many reviews and books 共e.g., Guckenheimer and Holmes, 1986; Crawford, 1991; Abarbanel et al., 1993; Ott, 1993; Kaplan and Glass, 1995; Abarbanel, 1997; Kuznetsov, 1998; Arnold et al., 1999; Strogatz, 2001; Izhikevich, 2006兲. Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 1215 B. Classical nonlinear dynamics approach for neural systems Let us say a few words about the role of classical dynamical theory. It might seem at first sight that the apparently infinite diversity of neural activity makes its dynamical description a hopeless, even meaningless, task. However, here one can exploit the knowledge accumulated in classical dynamical theory, in particular, the ideas put forth by Andronov in 1931 concerning the structural stability of dynamical models and the investigation of their bifurcations 共Andronov, 1933; Andronov and Pontryagin, 1937; Andronov et al., 1949兲. The essential points of these ideas can be traced back to Poincaré 共Poincaré, 1892; Goroff, 1992兲. In his book La Valeur de la Science, Poincaré 共1905兲 wrote that “the main thing for us to do with the equations of mathematical physics is to investigate what may and should be changed in them.” Andronov’s remarkable approach toward understanding dynamical systems contained three key points: • Only models exhibiting activity that does not vary with small changes of parameters can be regarded as really suitable to describe experiments. He referred to them as models or dynamical systems that are structurally stable. • To obtain insight into the dynamics of a system it is necessary to characterize all its principal types of behavior under all possible initial conditions. This led to Andronov’s fondness for the methods of phasespace 共state-space兲 analysis. • Considering the behavior of the system as a whole allows one to introduce the concept of topological equivalence of dynamical systems and requires an understanding of local and global changes of the dynamics, for example, bifurcations, as control parameters are varied. Conserving the topology of a phase portrait for a dynamical system corresponds to a stable motion of the system with small variation of the governing parameters. Partitioning parameter space for the dynamical system into regions with different phase-space behavior, i.e., finding the bifurcation boundaries, then furnishes a complete picture of the potential behaviors of a dynamical model. Is it possible to apply such a beautiful approach to biological neural network analysis? The answer is yes, at least for small, autonomous neural systems. However, even in these simple cases we face some important restrictions. Neural dynamics is strongly dissipative. Energy derived from biochemical sources is used to drive neural activity with substantial energy loss in action-potential generation and propagation. Nearly all trajectories in the phase space of a dissipative system are attracted by some trajectories or sets of trajectories called attractors. These can be fixed points 共corresponding to steady-state activity兲, limit cycles 共periodic activity兲, or strange attractors 共chaotic dynamics兲. The behavior of dynamical systems with attractors is usually structurally stable. Strictly speaking a strange attractor is itself structurally 1216 Rabinovich et al.: Dynamical principles in neuroscience FIG. 2. Six examples of limit cycle bifurcations observed in living and model neural systems 关see Chay 共1985兲; Canavier et al. 共1990兲; Guckenheimer et al. 共1993兲; Huerta et al. 共1997兲; Crevier and Meister 共1998兲; Maeda et al. 共1998兲; Coombes and Osbaldestin 共2000兲; Feudel et al. 共2000兲; Gavrilov and Shilnikov 共2000兲; Maeda and Makino 共2000兲; Mandelblat et al. 共2001兲; Bondarenko et al. 共2003兲; Gu et al. 共2003兲; Shilnikov and Cymbalyuk 共2005兲; Soto-Trevino et al. 共2005兲兴. unstable, but its existence in the system state space is a structurally stable phenomenon. This is a very important point for the implementation of Andronov’s ideas. The study of bifurcations in neural models and in in vitro experiments is a keystone for understanding the dynamical origin of many single-neuron and circuit phenomena involved in neural information processing and the organization of behavior. Figure 2 illustrates some typical local bifurcations 关their support consists of an equilibrium point or a periodic trajectory—see the detailed definition by Arnold et al. 共1999兲兴 and some global bifurcations 共their support contains an infinite set of orbits兲 of periodic regimes observed in neural systems. Many of these bifurcations are observed both in experiments and in models, in particular in the conductancebased Hodgkin-Huxley–type equations 共Hodgkin and Huxley, 1952兲, considered the traditional framework for modeling neurons, and in the analysis of network stability and plasticity. The most striking results in neuroscience based on classical dynamical system theory have come from bottom-up models. These results include the description Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 of the diversity of dynamics in single neurons and synapses 共Koch, 1999; Vogels et al., 2005兲, the spatiotemporal cooperative dynamics of small groups of neurons with different types of connections 共Selverston et al., 2000; Selverston, 2005兲, and the principles of synchronization in networks with dynamical synapses 共Loebel and Tsodyks, 2002; Elhilali et al., 2004; Persi et al., 2004兲. Some top-down models also have attempted a classical nonlinear dynamics approach. Many of these models are related to the understanding and description of cognitive functions. Nearly half a century ago, Ashby hypothesized that cognition could be modeled as a dynamical process 共Ashby, 1960兲. Neuroscientists have spent considerable effort implementing the dynamical approach in a practical way. The most widely studied examples of cognitive-type dynamical models are multiattractor networks: models of associative memory that are based on the concept of an energy function or Lyapunov function for a dynamical system with many attractors 共Hopfield, 1982兲 关see also Cohen and Grossberg 共1983兲; Waugh et al. 共1990兲; Doboli et al. 共2000兲兴. The dynamical process in such networks is often called Rabinovich et al.: Dynamical principles in neuroscience 1217 “computation with attractors.” The idea is to design during the learning stage, in a memory network phase space, a set of attractors, each of which corresponds to a specific output. Neural computation with attractors involves the transformation of a given input stimulus, which defines an initial state inside the basin of attraction of one attractor, leading to a fixed desired output. The idea that computation or information processing in neural systems is a dynamical process is broadly accepted today. Many dynamical models of both bottom-up and top-down type that address the encoding and decoding of neural information as the inputdependent dynamics of a nonautonomous network have been published in the last few years. However, there are still huge gaps in our knowledge of the actual biological processes underlying learning and memory, making accurate modeling of these mechanisms a distant goal. For reviews see Arbib et al. 共1997兲 and Wilson 共1999兲. Classical nonlinear dynamics has provided some basis for the analysis of neural ensembles even with large numbers of neurons in networks organized as layers of nearly identical neurons. One of the elements of this formulation is the discovery of stable low-dimensional manifolds in a very high-dimensional phase space. These manifolds are mathematical images of cooperative modes of activity, for example, propagating waves in nonequilibrium media 共Rinzel et al., 1998兲. Models of this sort are also interesting for the analysis of spiral waves in cortical activity as experimentally observed in vivo and in vitro 共Huang et al., 2004兲. Many interesting questions have been approached by using the phase portrait and bifurcation analysis of models and by considering attractors and other asymptotic solutions. Nevertheless, new directions may be required to address the important complexity of nervous system functions. C. New paradigms for contradictory issues The human brain contains approximately 1011 neurons and a typical neuron connects with ⬇104 other neurons. Neurons show a wide diversity in terms of their morphology and physiology 共see Fig. 3兲. A wide variety of intracellular and network mechanisms influence the activity of living neural circuits. If we take into account that even a single neuron often behaves chaotically, we might argue that such a complex system most likely behaves as if it were a turbulent hydrodynamic flow. However, this is not what is observed. Brain dynamics are more or less regular and stable despite the presence of intrinsic and external noise. What principles does nature use to organize such behavior, and what mathematical approaches can be utilized for their description? These are the very difficult questions we need to address. Several important features differentiate the nervous system from traditional dynamical systems: • The architecture of the system, the individual neural units, the details of the dynamics of specific neurons, as well as the connections among neurons are not Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 FIG. 3. Examples of 共a兲 the anatomical diversity of neurons, and 共b兲 the single-neuron membrane voltage activity associated with them. 共1兲 Lobster pyloric neuron; 共2兲 neuron in rat midbrain; 共3兲 cat thalamocortical relay neuron; 共4兲 guinea pig inferior olivary neuron; 共5兲 aplysia R15 neuron; 共6兲 cat thalamic reticular neuron; 共7兲 sepia giant axon; 共8兲 rat thalamic reticular neuron; 共9兲 mouse neocortical pyramidal neuron; 共10兲 rat pituitary gonadotropin-releasing cell. In many cases, the behavior depends on the level of current injected into the cell as shown in 共b兲. Modified from Wang and Rinzel, 1995. usually known in detail, so we can describe them only in a probabilistic manner. • Despite the fact that many units within a complex neural system work in parallel, many of them have different time scales and react differently to the same nonstationary events from outside. However, for the whole system, time is unified and coherent. This means that the neural system is organized hierarchically, not only in space 共architecture兲 but also in time: each behavioral event is the initial condition for the next window of time. The most interesting phenomenon for a neural system is the presence not of at- 1218 Rabinovich et al.: Dynamical principles in neuroscience tractor dynamics but of nonstationary behavior. Attractor dynamics assumes long-time evolution from initial conditions; we must consider transient responses instead. • The structure of neural circuits is—in principle— genetically determined; however, it is nevertheless not fixed and can change with experience 共learning兲 and through neuromodulation. We could expand this list, but the facts mentioned already make the point that the nervous system is a very special field for the application of classical nonlinear dynamics, and it is clear now why neurodynamics needs new approaches and a fresh view. We use the following arguments to support an optimistic view about finding dynamical principles in neuroscience: • Complex neural systems are the result of evolution, and thus their complexity is not arbitrary but follows some universal rules. One such rule is that the organization of the central nervous system 共CNS兲 is hierarchical and based on neural modules. • It is important to note that many modules are organized in a very similar way across different species. Such units can be small, like central pattern generators 共CPGs兲, or much more complex, like sensory systems. In particular, the structure of one of the oldest sensory systems, the olfactory system, is more or less the same in invertebrates and vertebrates and can be described by similar dynamical models. • The possibility of considering the nervous system as an ensemble of interconnected units is a result of the high level of autonomy of its subsystems. The level of autonomy depends on the degree of selfregulation. Self-regulation of neural units on each level of the nervous system, including individual neurons, is a key principle determining hierarchical neural network dynamics. • The following conjecture seems reasonable: Each specific dynamical behavior of the network 共e.g., traveling waves兲 is controlled by only a few of the many parameters of a system 共like neuromodulators, for example兲, and these relevant parameters influence the specific cell or network dynamics independently—at least in a first approximation. This idea can be useful for the mathematical analysis of network dynamics and can help to build an approximate bifurcation theory. The goal of this theory is to predict the transformation of specific dynamics based on bifurcation analysis in a low-dimensional control subspace of parameters. • For the understanding of the main principles of neurodynamics, phenomenological top-down models are very useful because even different neural systems with different architectures and different levels of complexity demonstrate similar dynamics if they execute similar functions. Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 In the main part of this review we discuss two critical functional properties of neural systems that at first glance appear incompatible: robustness and sensitivity. Finding solutions to such apparent contradictions will help us formulate some general dynamical principles of biological neural network organization. We note two examples. Many neural systems, especially sensory systems, must be robust against noise and at the same time must be very sensitive to incoming inputs. A new paradigm that can deal with the existence of this fundamental contradiction is the winnerless competition 共WLC兲 principle 共Rabinovich et al., 2001兲. According to this principle, a neural network with nonsymmetric inhibitory connections is able to exhibit structurally stable dynamics if the stimulus is fixed, and qualitatively change its dynamics if the stimulus is changed. This ability is based on different features of the signal and the noise, and the different ways they influence the dynamics of the system. Another example is the remarkable reproducibility of transient behavior. Because transient behavior, in contrast to the long-term stable stationary activity of attractors, depends on initial conditions, it is difficult to imagine how such behavior can be reproducible from experiment to experiment. The solution to this paradox is related to the special role of global and local inhibition, which sets up the initial conditions. The logic of this review is related to the specificity of neural systems from the dynamical point of view. In Sec. II we consider the possible dynamical origin of robustness and sensitivity in neural microcircuits. The dynamics of information processing in neural systems is considered in Sec. III. In Sec. IV, together with other dynamical concepts, we focus on a new paradigm of neurodynamics: the winnerless competition principle in the context of sequence generation, sensory coding, and learning. II. DYNAMICAL FEATURES OF MICROCIRCUITS: ADAPTABILITY AND ROBUSTNESS A. Dynamical properties of individual neurons and synapses 1. Neuron models Neurons receive patterned synaptic input and compute and communicate by transforming these synaptic input patterns into an output sequence of spikes. Why spikes? As spike wave forms are similar, information encoded in spike trains mainly relies on the interspike intervals. Relying on timing rather than on the details of action-potential wave forms increases the reliability and reproducibility in interneural communication. Dispersion and attenuation in transmission of neural signals from one neuron to others changes the wave form of the action potentials but preserves their timing information, again allowing for reliability when depending on interspike intervals. The nature of spike train generation and transformation depends crucially on the properties of many voltage-gated ionic channels in neuron cell membranes. Rabinovich et al.: Dynamical principles in neuroscience The cell body 共or soma兲 of the neuron gives rise to two kinds of processes: short dendrites and one or more long, tubular axons. Dendrites branch out like trees and receive incoming signals from other neurons. In some cases the synaptic input sites are on dendritic spines, thousands of which can cover the dendritic arbor. The output process, the axon, transmits the signals generated by the neuron to other neurons in the network or to an effector organ. The spikes are rapid, transient, all-ornone 共binary兲 impulses, with a duration of about 1 ms 共see Fig. 1兲. In most cases, they are initiated at a specialized region at the origin of the axon and propagate along the axon without distortion. Near its end, the tubular axon divides into branches that connect to other neurons through synapses. When the spike emitted by a presynaptic neuron reaches the terminal of its axon, it triggers the emission of chemical transmitters in the synaptic cleft 共the small gap, of order a few tens of nanometers, separating the two neurons at a synapse兲. These transmitters bind to receptors in the postsynaptic neuron, causing a depolarization or hyperpolarization in its membrane, exciting or inhibiting the postsynaptic neuron, respectively. These changes in the polarization of the membrane relative to the extracellular space spread passively from the synapses on the dendrites across the cell body. Their effects are integrated, and, when there is a large enough depolarization, a new action potential is generated 共Kandel et al., 2000兲. Other types of synapses called gap junctions function as Ohmic electrical connections between the membranes of two cells. A spike is typically followed by a brief refractory period, during which no further spikes can be fired by the same neuron. Neurons are quite complex biophysical and biochemical entities. In order to understand the dynamics of neurons and neural networks, phenomenological models have to be developed. The Hodgkin-Huxley model is foremost among such phenomenological descriptions of neural activity. There are several classes of neural models possessing various degrees of sophistication. We summarize the neural models most often considered in biological network development in Table I. For a more detailed description of these models see, for example, Koch 共1999兲, Gerstner and Kistler 共2002兲, and Izhikevich 共2004兲. Detailed conductance-based neuron models take into account ionic currents flowing across the membrane 共Koch, 1994兲. The neural membrane may contain several types of voltage-dependent sodium, potassium, and calcium channels. The dynamics of these channels can also depend on the concentration of specific ions. In addition, there is a leakage current of chloride ions. The flow of these currents results in changes in the voltage across the membrane. The probability that a type of ionic channel is open depends nonlinearly on the membrane voltage and the current state of the channel. These dependencies result in a set of several coupled nonlinear differential equations describing the electrical activity of the cell. The intrinsic membrane conductances can enable neurons to generate different spike patterns, inRev. Mod. Phys., Vol. 78, No. 4, October–December 2006 1219 cluding high-frequency bursts of different durations which are commonly observed in a variety of motor neural circuits and brain regions 关see Fig. 3共b2兲兴. The biophysical mechanisms of spike generation enable individual neurons to encode different stimulus features into distinct spike patterns. Spikes, and bursts of spikes of different durations, code for different stimulus features, which can be quantified without a priori assumptions about those features 共Kepecs and Lisman, 2003兲. How detailed does the description of neurons or synapses have to be to make a model of neural dynamics biologically realistic while still remaining computationally tractable? It is reasonable to separate neuron models into two classes depending on the general goal of the modeling. If we wish to understand, for example, how the ratio of inhibitory to excitatory synapses in a neural ensemble with random connections influences the activity of the whole network, it is reasonable to use a simple model that keeps only the main features of neuron behavior. The existence of a spike threshold and the increase of the output spike rate with an increase in the input may be sufficient. On the other hand, if our goal is to explain the flexibility and adaptability of a small network like a CPG to a changing environment, the details of the ionic channel dynamics can be of critical importance 共Prinz et al., 2004b兲. In many cases neural models built on simplified paradigms lead to more detailed conductance-based models based on the same dynamical principles but implemented with more biophysically realistic mechanisms. A good indication that the level of the description was chosen wisely comes if the model can reproduce with the same parameters the main bifurcations observed in the experiments. 2. Neuron adaptability and multistability Multistability in a dynamical system means the coexistence of multiple attractors separated in phase space at the same value of the system’s parameters. In such a system qualitative changes in dynamics can result from changes in the initial conditions. A well-studied case is the bistability associated with a subcritical AndronovHopf bifurcation 共Kuznetsov, 1998兲. Multistable modes of oscillation can arise in delayed-feedback systems when the delay is larger than the response time of the system. In neural systems multistability could be a mechanism for memory storage and temporal pattern recognition in both artificial 共Sompolinsky and Kanter, 1986兲 and living 共Canavier et al., 1993兲 neural circuits. In a biological nervous system recurrent loops involving two or more neurons are found quite often and are particularly prevalent in cortical regions important for memory 共Traub and Miles, 1991兲. Multistability emerges easily in these loops. For example, the conditions under which time-delayed recurrent loops of spiking neurons exhibit multistability were derived by Foss et al. 共1996兲. The study used both a simple integrate-and-fire neuron and a Hodgkin-Huxley 共HH兲 neuron whose recurrent inputs are delayed versions of their output spike trains. The authors showed that two kinds of multistability with 1220 Rabinovich et al.: Dynamical principles in neuroscience TABLE I. Summary of many frequently used neuronal models. Model Integrate-andfire neurons Example dv共t兲 dt = 冦 − v共t兲 + Iext + Isyn共t兲, 0 ⬍ v共t兲 ⬍ v共t−0 兲 = v共t+0 兲 = 0, Isyn共t兲 = g 兺 f共t − t spike兲 spikes and f共t兲=A关exp共−t / 1兲 − exp共−t / 2兲兴 冧 Variables Remarks References v共t兲 is the neuron membrane potential; is the threshold for spike generation. Iext is an external stimulus current; Isyn is the sum of the synaptic currents; and 1 and 2 are time constants characterizing the synaptic currents. A spike occurs when the neuron reaches the threshold in v共t兲 after which the cell is reset to the resting state. Lapicque, 1907 Rate models ȧi共t兲 = Fi共ai共t兲兲关Gi„ai共t兲… − 兺jijQj„aj共t兲…兴 ai共t兲 ⬎ 0 is the spiking rate of the ith neuron or cluster; ij is the connection matrix; and F , G , Q are polynomial functions. This is a generalization of the LotkaVolterra model 关see Eq. 共9兲兴. Fukai and Tanaka, 1997; Lotka, 1925; Volterra, 1931 McCulloch and Pitts xi共n + 1兲 = ⌰共兺jgijxj共n兲 − 兲 1, x ⬎ 0 ⌰共x兲 = 0, x 艋 0 再 is the firing threshold; xj共n兲 are synaptic inputs at the discrete “time” n; xi共n + 1兲 is the output. Inputs and outputs are binary 共one or zero兲; the synaptic connections gij are 1, −1, or 0. The first computational model for an artificial neuron; it is also known as a linear threshold device model. This model neglects the relative timing of neural spikes. McCulloch and Pitts, 1943 v共t兲 is the membrane potential, m共t兲, and h共t兲, and n共t兲 represent empirical variables describing the activation and inactivation of the ionic conductances; I is an external current. The steady-state values of the conductance variables m⬁ , h⬁ , n⬁ have a nonlinear voltage dependence, typically through sigmoidal or exponential functions. These ODEs represent point neurons. There is a large list of models derived from this one, and it has become the principal tool in computational neuroscience. Other ionic currents can be added to the right-hand side of the voltage equation to better reproduce the dynamics and bifurcations observed in the experiments. Hodgkin and Huxley, 1952 x共t兲 is the membrane potential, and y共t兲 describes the dynamics of fast currents; I is an external current. The parameter values a, b, and c are constants chosen to allow spiking. A reduced model describing oscillatory spiking neural dynamics including bistability. FitzHugh, 1961; Nagumo et al., 1962 冎 ˙ = g 关v − v共t兲兴 Hodgkin-Huxley Cv共t兲 L L + gNam共t兲3h共t兲关vNa − v共t兲兴 + gKn共t兲4共vK兲 − v共t兲 + I, ˙ = m⬁„v共t兲… − m共t兲 m共t兲 m„v共t兲… h⬁„v共t兲… − h共t兲 ˙ = h共t兲 h„v共t兲… n⬁„v共t兲… − n共t兲 ˙ = n共t兲 n„v共t兲… FitzHugh-Nagumo ẋ x − cx3 − y + I, ẏ = x + by − a Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 1221 Rabinovich et al.: Dynamical principles in neuroscience TABLE I. 共Continued.兲 Model Variables Remarks 兵E共x , t兲 , I共x , t兲其 are the number density of active excitatory and inhibitory neurons at location x of the continuous neural media. „wee共x兲 , wie共x兲 , wei共x兲 , wii共x兲… are connectivity distributions among the populations of cells. 兵Le , Li其 are nonlinear responses reflecting different populations of thresholds. The operator 丢 is a convolution involving the connectivity distributions. The first “mean-field” model. It is an attempt to describe a cluster of neurons, to avoid the inherent noisy dynamical behavior of individual neurons; by averaging to a distribution noise is reduced. Wilson and Cowan, 1973 v共t兲 is the membrane potential; n共t兲 describes the recovery activity of a calcium current; I is an external current. Simplified model that reduces the number of dynamical variables of the HH model. It displays action potential generation when changing I leads to a saddle-node bifurcation to a limit cycle. Morris and Lecar, 1981 ˙ = y共t兲 + ax共t兲2 − bx共t兲3 − z共t兲 + I Hindmarsh-Rose x共t兲 ˙ y共t兲 = C − xx共t兲2 − y共t兲 ˙ = rˆs关x共t兲 − x 兴 − z共t兲‰ z共t兲 x共t兲 is the membrane potential; y共t兲 describes fast currents; z共t兲 describes slow currents; and I is an external current. Simplified model that uses a polynomial approximation to the right-hand side of a Hodgkin-Huxley model. This model fails to describe the hyperpolarized periods after spiking of biological neurons. Hindmarsh and Rose, 1984 di共t兲 =+ dt 共t兲 is the phase of the ith neuron with approximately periodic behavior; and Hij is the connectivity function determining how neuron i and j interact. First introduced for chemical oscillators; good for describing strongly dissipative oscillating systems in which the neurons are intrinsic periodic oscillators. Cohen et al., 1982; Ermentrout and Kopell, 1984; Kuramoto, 1984 xt represents the spiking activity and yt represents a slow variable. A discrete time map. One of a class of simplephenomenological models for spiking, bursting neurons. This kind of model can be computationally very fast, but has little biophysical foundation. Cazelles et al., 2001; Rulkov, 2002 Wilson-Cowan Example Morris-Lecar E共x , t兲 t I共x , t兲 t = −E共x , t兲 + 关1 − rE共x , t兲兴 ⫻ Le关E共x , t兲 丢 wee共x兲 − I共x , t兲 丢 wei共x兲 + Ie共x , t兲兴 = −I共x , t兲 + 关1 − rI共x , t兲兴 ⫻ Li关E共x , t兲 丢 wie共x兲 − I共x , t兲 丢 wii共x兲 + Ii共x , t兲兴 ˙ = g 关v − v共t兲兴 + n共t兲g v共t兲 L L n ⫻ 关vn − v共t兲兴 + gmm⬁„v̇共t兲…关vm − v共t兲兴 + I, ˙ = „v共t兲…关n „v共t兲… − n共t兲兴 n共t兲 ⬁ 1 v − vm m⬁共v兲 = 1 + tanh 0 2 vm 1 v − vn n⬁共v兲 = 1 + tanh 0 2 vn v − vn 共v兲 = n cosh 2vn0 共 共 兲 兲 0 Phase oscillator models Map models xt+1共i兲 = 兺 H 共 共t兲 − 共t兲兲 ij i j j ␣ + yt共i兲 1 + xt共i兲2 ⑀ + xt共j兲 N j 兺 yt+1共i兲 = yt共i兲 − xt共i兲 − ␤ Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 References 1222 Rabinovich et al.: Dynamical principles in neuroscience respect to initial spiking functions exist, depending on whether the neuron is excitable or repetitively firing in the absence of feedback. Following Hebb’s 共1949兲 ideas most studies of the mechanisms underlying learning and memory focus on changing synaptic efficacy. Learning is associated with changing connectivity in a network. However, the network dynamics also depends on complex interactions among intrinsic membrane properties, synaptic strengths, and membrane-voltage time variation. Furthermore, neuronal activity itself modifies not only synaptic efficacy but also the intrinsic membrane properties of neurons. Papers by Marder et al. 共1996兲 and Turrigiano et al. 共1996兲 present examples showing that bistable neurons can provide short-term memory mechanisms that rely solely on intrinsic neuronal properties. While not replacing synaptic plasticity as a powerful learning mechanism, these examples suggest that memory in networks could result from an ongoing interplay between changes in synaptic efficacy and intrinsic neuron properties. To understand the biological basis for such computational properties we must examine both the dynamics of the ionic currents and the geometry of neuronal morphology. 3. Synaptic plasticity Synapses as well as neurons are dynamical nonlinear devices. Although synapses throughout the CNS share many features, they also have distinct properties. They operate with the following sequences of events: A spike is initiated in the axon near the cell body, it propagates down the axon, and arrives at the presynaptic terminal, where voltage-gated calcium channels admit calcium, which triggers vesicle fusion and neurotransmitter release. The released neurotransmitter then binds to receptors on the postsynaptic neuron and changes their conductance 共Nicholls et al., 1992; Kandel et al., 2000兲. This series of events is regulated in many ways, making synapses adaptive and plastic. In particular, the strength of synaptic conductivity changes in real time depending on their activity, as Katz observed many years ago 共Fatt and Katz, 1952; Katz, 1969兲. A description of such plasticity was made in 1949 by Hebb 共1949兲. He proposed that “When an axon of cell A is near enough to excite a cell B and repeatedly or persistently takes part in firing it, some growth process or metabolic change takes place in one or both cells such that A’s efficiency, as one of the cells firing B, is increased.” This neurophysiological postulate has since become a central concept in neuroscience through a series of classic experiments demonstrating Hebbian-like synaptic plasticity. These experiments show that the efficacy of synaptic transmission in the nervous system is activity dependent and continuously modified. Examples of such modification are long-term potentiation and depression 共LTP and LTD兲, which involve increased or decreased conductivity, respectively, of synaptic connections between two neurons, leading to increased or Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 decreased activity over time. Long-term potentiation and depression are presumed to produce learning by differentially facilitating the association between stimulus and response. The role of LTP and LTD, if any, in producing more complex behaviors is less closely tied to specific stimuli and more indicative of cognition, and is not well understood. Long-term potentiation was first reported in the hippocampal formation 共Bliss and Lomo, 1973兲. Changes induced by LTP can last for many days. Long-term potentiation has long been regarded, along with its counterpart LTD, as a potential mechanism for short-termmemory formation and learning. In fact, the hypothesis is widely accepted in learning and memory research that activity-dependent synaptic plasticity is induced at appropriate synapses during memory formation and is both necessary and sufficient for the information storage underlying the type of memory mediated by the brain area in which that plasticity is observed 关see for a review Martin et al. 共2000兲兴. Hebb did not anticipate LTD in 1949, but along with LTP it is thought to play a critical role in “rewiring” biological networks. The notion of a coincidence requirement for Hebbian plasticity has been supported by classic studies of LTP and LTD using presynaptic stimulation coupled with prolonged postsynaptic depolarization 关see, for example, Malenka and Nicoll 共1999兲兴. However, coincidence there was loosely defined with a temporal resolution of hundreds of milliseconds to tens of seconds, much larger than the time scale of typical neuronal activity characterized by spikes that last for a couple of milliseconds. In a natural setting, presynaptic and postsynaptic neurons fire spikes as their functional outputs. How precisely must such spiking activities coincide in order to induce synaptic modifications? Experiments addressing this critical issue led to the discovery of spike-timingdependent synaptic plasticity 共STDP兲. Spikes initiate a sequence of complex biochemical processes in the postsynaptic neuron during the short time window following synaptic activation. Identifying detailed molecular processes underlying LTP and LTD remains a complex and challenging problem. There is good evidence that it consists of a competition between processes removing 共LTD兲 and processes placing 共LTP兲 phosphate groups from on postsynaptic receptors, or increasing 共LTP兲 or decreasing 共LTD兲 the number of such receptors in a dendritic spine. It is also widely accepted that N-methyl-D-aspartate 共NMDA兲 receptors are crucial for the development of LTP or LTD and that it is calcium influx onto the postsynaptic cell that is critical for both LTP and LTD. Experiments on synaptic modifications of excitatory synapses between hippocampal glutamatergic neurons in culture 共Bi and Poo, 1998, 2001兲 共see Fig. 4兲 indicate that if a presynaptic spike arrives at time tpre and a postsynaptic spike is observed or induced at tpost, then when = tpost − tpre is positive the incremental percentage increase in synaptic strength behaves as Rabinovich et al.: Dynamical principles in neuroscience FIG. 4. Spike-timing-dependent synaptic plasticity observed in hippocampal neurons. Each data point represents the relative change in the amplitude of evoked postsynaptic current after repetitive application of presynaptic and postsynaptic spiking pairs 共1 Hz for 60 s兲 with fixed spike timing ⌬t, which is defined as the time interval between postsynaptic and presynaptic spiking within each pair. Long-term potentiation 共LTP兲 and depression 共LTD兲 windows are each fitted with an exponential function. Modified from Bi, 2002. ⌬g ⬇ a Pe −␤P , g 共1兲 with ␤P ⬇ 1 / 16.8 ms. When ⬍ 0, the percentage decrease in synaptic strength behaves as ⌬g ⬇ − a De ␤D , g 共2兲 with ␤D ⬇ 1 / 33.7 ms. aP and aD are constants. This is illustrated in Fig. 4. Many biochemical factors contribute differently to LTP and LTD in different synapses. Here we discuss a phenomenological dynamical model of synaptic plasticity 共Abarbanel et al., 2002兲 which is very useful for modeling neural plasticity; its predictions agree with several experimental results. The model introduces two dynamical variables P共t兲 and D共t兲 that do not have a direct relationship with the concentration of any biochemical components. Nonlinear competition between these variables imitates the known competition in the postsynaptic cell. These variables satisfy the following simple firstorder kinetic equations: dP共t兲 = f„Vpre共t兲…关1 − P共t兲兴 − ␤PP共t兲, dt dD共t兲 = g„Vpost共t兲…关1 − D共t兲兴 − ␤DD共t兲, dt 共3兲 where the functions f共V兲 and g共V兲 are typical logistic or sigmoidal functions which rise from zero to the order of unity when their argument exceeds some threshold. These driving or input functions are a simplification of Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 1223 the detailed way in which each dynamical process is forced. The P共t兲 process is associated with a particular time constant 1 / ␤P while the D共t兲 process is associated with a different time constant 1 / ␤D. Experiments show that ␤P ⫽ ␤D, and this is the primary embodiment of the two different time scales seen in many observations. The two time constants are a coarse-grained representation of the diffusion and leakage processes which dampen and terminate activities. Presynaptic voltage activity serves to release neurotransmitters in the usual manner and this in turn induces the postsynaptic action of P共t兲, which has a time course determined by the time constant ␤P−1. Similarly, the postsynaptic voltage, constant or time varying, can be associated with the induction of the D共t兲 process. P共t兲 and D共t兲 compete to produce a change in synaptic strength ⌬g共t兲 as d⌬g共t兲 = ␥关P共t兲D共t兲 − D共t兲P共t兲兴, dt 共4兲 where ⬎ 1 and ␥ ⬎ 0. This dynamical model reproduces some of the key STDP experimental results like, for example, those shown in Fig. 4. It also accounts for the case where the postsynaptic cell is depolarized while a presynaptic spike train is presented to it. 4. Examples of the cooperative dynamics of individual neurons and synapses To illustrate the dynamical significance of plastic synapses we consider the synchronization of two neurons: a living neuron and an electronic model neuron coupled through a STDP or inverse STDP electronic synapse. Using hybrid circuits of model electronic neurons and biological neurons is a powerful method for analyzing neural dynamics 共Pinto et al., 2000; Szücs et al., 2000; LeMasson et al., 2002; Prinz et al., 2004a兲. The representation of synaptic input to a cell using a computer to calculate the response of the synapse to specified presynaptic input goes under the name “dynamic clamp” 共Robinson and Kawai, 1993; Sharp et al., 1993兲. It has been shown in modeling and in experiments 共Nowotny, Zhigulin, et al., 2003; Zhigulin et al., 2003兲 that coupling through plastic electronic synapses leads to neural synchronization or, more correctly, entrainment that is more rapid, more flexible, and much more robust against noise than synchronization mediated by connections of constant strength. In these experiments the neural circuit consists of a specified presynaptic signal, a simulated synapse 共via the dynamic clamp兲, and a postsynaptic biological neuron from the Aplysia abdominal ganglion. The presynaptic neuron is a spike generator producing spikes of predetermined form at predetermined times. The synapse and its plasticity are simulated by dynamic clamp software 共Nowotny, 2003兲. In each update cycle of ⬃100 s the presynaptic voltage is acquired, the spike generator voltage is updated, the synaptic strength is determined according to the learning rule, and the resulting synaptic current is calculated and injected into the living neuron through a current injection electrode. As 1224 Rabinovich et al.: Dynamical principles in neuroscience one presents the presynaptic signal many times, the synaptic conductance changes from one fixed value to another depending on the properties of the presynaptic signal. The calculated synaptic current is a function of the presynaptic and postsynaptic potentials of the spike generator Vpre共t兲 and the biological neuron Vpost共t兲, respectively. It is calculated according to the following model. The synaptic current depends linearly on the difference between the postsynaptic potential Vpost and its reversal potential Vrev, on an activation variable S共t兲, and on its maximal conductance g共t兲: Isyn共t兲 = g共t兲S共t兲关Vpost共t兲 − Vrev兴. 共5兲 The activation variable S共t兲 is a nonlinear function of the presynaptic membrane potential Vpre and represents the percentage of neurotransmitter docked on the postsynaptic cell relative to the maximum that can dock. It has two time scales: a docking time and an undocking time. We take it to satisfy the dynamical equation S⬁„Vpre共t兲… − S共t兲 dS共t兲 = . dt syn关S1 − S⬁„V1共t兲…兴 共6兲 S⬁共V兲 is a sigmoid function which we take to be S⬁共V兲 = 再 tanh关共V − Vth兲/Vslope兴 for V ⬎ Vth 0 otherwise. 冎 共7兲 The time scale is syn共S1 − 1兲 for neurotransmitter docking and synS1 for undocking. For AMPA excitatory receptors, the docking time is about 0.5 ms, and the undocking time is about 1.5 ms. The maximal conductance g共t兲 is determined by the learning rule discussed below. In the experiments, the synaptic current is updated at ⬃10 kHz. To determine the maximal synaptic conductance g共t兲 of the simulated STDP synapse, an additive STDP learning rule was used. This is accurate if the time between presented spike pairs is long compared to the time between spikes in the pair. To avoid runaway behavior, the additive rule was applied to an intermediate graw that was then filtered through a sigmoid function. In particular, the change ⌬graw in synaptic strength is given by ⌬graw共⌬t兲 = 冦 ⌬t − 0 −共⌬t− 兲/ 0 + for ⌬t ⬎ 0 e + ⌬t − 0 共⌬t− 兲/ 0 − for ⌬t ⬍ 0 , A− e − A+ 冧 共8兲 where ⌬t = tpost − tpre is the difference between postsynaptic and presynaptic spike times. The parameters + and − determine the widths of the learning windows for potentiation and depression, respectively, and the amplitudes A+ and A− determine the magnitude of synaptic change per spike pair. The shift 0 reflects the finite time of information transport through the synapse. As one can see in Fig. 5, the postsynaptic neuron quickly synchronizes to the presynaptic spike generator which presents spikes with an interspike interval 共ISI兲 of 255 ms 共top panel兲. The synaptic strength continuously Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 FIG. 5. Example of a synchronization experiment. Top: The interspike intervals 共ISIs兲 of the postsynaptic biological neuron. Bottom: The synaptic strength g共t兲. Presynaptic spikes with ISI of 255 ms were presented to a postsynaptic neuron with periodic oscillations at an ISI of 330 ms. Before coupling with the presynaptic spike generator, the biological neuron spikes tonically at its intrinsic ISI of 330 ms. Coupling was switched on with g共t = 0兲 = 15 nS at time 6100 s. As one can see the postsynaptic neuron quickly synchronizes to the presynaptic spike generator 共top panel, dashed line兲. The synaptic strength continuously adapts to the state of the postsynaptic neuron, effectively counteracting adaptation and other modulations of the system. This leads to a very precise and robust synchronization at a nonzero phase lag. The precision of the synchronization manifests itself in small fluctuations of the postsynaptic ISIs in the synchronized state. Robustness and phase lag cannot be seen directly. Modified from Nowotny, Zhigulin, et al., 2003. adapts to the state of the postsynaptic neuron, effectively counteracting adaptation and other modulations of the system 共bottom panel兲. This leads to a very precise and robust synchronization at a nonzero phase lag. The precision of the synchronization manifests itself in small fluctuations of the postsynaptic ISIs in the synchronized state. Robustness and phase lag cannot be seen directly in Fig. 5. Spike-timing-dependent plasticity is a mechanism that enables synchronization of neurons with significantly different intrinsic frequencies as one can see in Fig. 6. The significant increase in the regime of synchronization associated with synaptic plasticity is a welcome, perhaps surprising, result and addresses the issue raised above about robustness of synchronization in neural circuits. B. Robustness and adaptability in small microcircuits The precise relationship between the dynamics of individual neurons and the mammalian brain as a whole remains extremely complex and obscure. An important reason for this is a lack of knowledge on the detailed cell-to-cell connectivity patterns as well as a lack of knowledge on the properties of the individual cells. Although large-scale modeling of this situation is attempted frequently, parameters such as the number and kind of synaptic connections can only be estimated. By Rabinovich et al.: Dynamical principles in neuroscience FIG. 6. 共Color online兲 The presynaptic signal generator presents a periodic spike train with ISI of T1 to a postsynaptic neuron with ISI of T02, before coupling. When neurons are coupled, T02 → T2. We plot the ratio of these periods after coupling as a function of the ratio before coupling 共a兲, for a synapse with constant g and 共b兲 for a synaptic connection g共t兲 following the rule in the text. The enlarged domain of one-toone synchronization in the latter case is quite clear and, as shown by the change in the error bar sizes, the synchronization is much better. This result persists when noise is added to the presynaptic signal and to the synaptic action 共not shown兲. Modified from Nowotny, Zhigulin, et al., 2003. using the less complex microcircuits 共MCs兲 of invertebrates, a more detailed understanding of neural circuit dynamics is possible. Central pattern generators are small MCs that can produce stereotyped cyclic outputs without rhythmic sensory or central input 共Marder and Calabrese, 1996; 1225 Stein et al., 1997兲. Thus CPGs are oscillators, and the image of their activity in the corresponding system state space is a limit cycle when oscillations are periodic and a strange attractor in more complex cases. Central pattern generators underlie the production of most motor commands for muscles that execute rhythmic animal activity such as locomotion, breathing, heartbeat, etc. The CPG output is a spatiotemporal pattern with specific phase lags between the temporal sequences corresponding to the different motor units 共see below兲. The network architecture and the main features of CPG neurons and synapses are known much better than any other brain circuits. Examples of typical invertebrate CPG networks are shown in Fig. 7. Common to many CPG circuits are electrical and inhibitory connections and the spiking-bursting activity of their neurons. The characteristics of the spatiotemporal patterns generated by the CPG, such as burst frequency, phase, length, etc., are determined by the intrinsic properties of each individual neuron, the properties of the synapses, and the architecture of the circuit. The motor patterns produced by CPGs fall into two categories: those that operate continuously such as respiration 共Ramirez et al., 2004兲 or heartbeat 共Cymbalyuk et al., 2002兲, and those that are produced intermittently such as locomotion 共Getting, 1989兲 or chewing 共Selverston, 2005兲. Although CPGs autonomously establish correct rhythmic firing patterns, they are under constant supervision by descending fibers from higher centers and by local reflex pathways. These inputs allow the animal to constantly adapt its behavior to the immediate environment, which suggests that there is considerable flexibility in the dynamics of motor systems. In addition there is now a considerable body of information showing that anatomically defined small neural circuits can be reconfigured in a more general way by neuromodulatory substances in the blood, or released synaptically so that they are functionally altered to produce different stable spatiotemporal patterns, which must also be flexible in response to sensory inputs on a cycle-by-cycle basis; see FIG. 7. Examples of invertebrate CPG microcircuits from arthropod, mollusk, and annelid preparations. All produce rhythmic spatiotemporal motor patterns when activated by nonpatterned input. The black dots represent chemical inhibitory synapses. Resistors represent electrical connections. Triangles are chemical excitatory synapses, and diodes are rectifying synapses 共electrical synapses in which the current flows only in one direction兲. Individual neurons are identifiable from one preparation to another. Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 1226 Rabinovich et al.: Dynamical principles in neuroscience Simmers and Moulins 共1988兲, for example. Central pattern generators have similarity with neural MCs in the brain 共Silberberg et al., 2005; Solis and Perkel, 2005; Yuste et al., 2005兲 and are often studied as models of neural network function. In particular, there are important similarities between vertebrate spinal cord CPGs and neocortical microcircuits which have been emphasized by Yuste et al. 共2005兲: 共i兲 CPG interactions, which are fundamentally inhibitory, dynamically regulate the oscillations. Furthermore, subthresholdactivated voltage-dependent cellular conductances that promote bistability and oscillations also promote synchronization with specific phase lags. The same cellular properties are also present in neocortical neurons, and underlie the observed oscillatory synchronization in the cortex. 共ii兲 Neurons in spinal cord CPGs show bistable membrane dynamics, which are commonly referred to as plateau potentials. A correlate of bistable membrane behavior, in this case termed “up” and “down” states, has also been described in the striatum and neocortex both in vivo and in vitro 共Sanchez-Vives and McCormick, 2000; Cossart et al., 2003兲. It is still unclear whether this bistability arises from intrinsic or circuit mechanisms or a combination of the two 共Egorov et al., 2002; Shu et al., 2003兲. 共iii兲 Both CPGs and cortical microcircuits demonstrate attractor dynamics and transient dynamics 关see, for example, Abeles et al. 共1993兲; Ikegaya et al. 共2004兲兴. 共iv兲 Modulations by sensory inputs and neuromodulators are also a common characteristic that is shared between CPGs and cortical circuits. Examples in CPGs include the modulation of oscillatory frequency, of temporal coordination among different populations of neurons, of the amplitude of network activity, and of the gating of CPG input and output 共Grillner, 2003兲. 共v兲 Switching between different states of CPG operation 共for example, switching coordinated motor patterns for different modes of locomotion兲 is under sensory afferent and neurochemical modulatory control. This makes CPGs multifunctional and dynamically plastic. Switching between cortical activity states is also under modulatory control, as shown, for example, by the role of the neurotransmitter dopamine in working memory in monkeys 共Goldman-Rakic, 1995兲. Thus modulation reconfigures microcircuit dynamics and transforms activity states to modify behavior. The CPG concept was built around the idea that behaviorally relevant spatiotemporal cyclic patterns are generated by groups of nerve cells without the need for rhythmic inputs from higher centers or feedback from structures that are moving. If activated, isolated invertebrate preparations can generate such rhythms for many hours and as a result have been extremely important in trying to understand how simultaneous cooperative interactions between many cellular and synaptic parameters can produce robust and stable spatiotemporal patterns 关see Fig. 8共d兲兴. An example of a three-neuron CPG phase portrait is shown in Figs. 8共a兲–8共c兲. The effect of a hyperpolarizing current leads to changes in the pattern as reflected by the phase portrait in Figs. 8共b兲 and 8共c兲. Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 FIG. 8. 共Color兲 Phase portrait of typical CPG output. The data were recorded in the pyloric CPG of the lobster stomatogastric ganglion. Each axis represents the firing rate of one of three pyloric neurons: LP, VD, and PD 共see Fig. 7兲. 共a兲 The orbit of the oscillating pyloric system is shown in blue and the average orbit is shown in red; 共b兲 the same but with a hyperpolarizing dc current injected into the PD; 共c兲 the difference between the averaged orbits; 共d兲 time series of the membrane potentials of the three neurons. Figure provided by T. Nowotny, R. Levi, and A. Szücs. Neural oscillations arise either through interactions among neurons 共network-based mechanism兲 or through interactions among currents in individual neurons 共pacemaker mechanism兲. Some CPGs use both mechanisms. In the simplest case, one or more neurons with intrinsic bursting activity acts as the pacemaker for the entire CPG circuit. The intrinsic currents may be constitutively active or they may require activation by neuromodulators, so-called conditional bursters. Synaptic connections act to determine the pattern by exciting or inhibiting other neurons at the appropriate time. Such networks are extremely robust and have generally been thought to be present in systems in which the rhythmic activity is active all or most of the time. In the second case, it is the synaptic interactions between nonbursty neurons that generate the rhythmic activity and many schemes for the types of connections necessary to do this have been proposed. Usually reciprocal inhibition serves as the basis for generating bursts in antagonistic neurons and there are many examples of cells in pattern-generating microcircuits connected in this way 共see Fig. 7兲. Circuits of this type are usually found for behaviors that are intermittent in nature and require a greater degree of flexibility than those based on pacemaker cells. Physiologists know that reciprocal inhibitory connections between oscillatory neurons can produce, as a result of the competition, sequential activity of neurons and rhythmic spatiotemporal patterns 共Szekely, 1965; Stent and Friesen, 1977; Rubin and Terman, 2004兲. However, even for a rather simple MC, consisting of just three neurons, there is no quantitative description. If the connections are symmetric, the MC can reach an attrac- 1227 Rabinovich et al.: Dynamical principles in neuroscience tor. It is reasonable to hypothesize that asymmetric inhibitory connections are necessary to preserve the order of patterns with more than two phases per cycle. The contradiction, noted earlier, between robustness and flexibility can then be resolved because external signals can modify the effective topology of connections so one can have functionally different networks for different stimuli. Theoretical analysis and computer experiments with MCs based on the winnerless competition principle 共discussed in Sec. IV.C兲 show that sufficient conditions for the generation of sequential activity do exist and the range of allowed nonsymmetric inhibitory connections is quite wide 共Rabinovich et al., 2001; Varona, Rabinovich, et al., 2002; Afraimovich, Rabinovich, et al., 2004兲 We illustrate this using a Lotka-Volterra rate description of neuron activity: 冉 N 冊 dai共t兲 = ai共t兲 1 − 兺 ij共Si兲aj共t兲 + Si, dt j=1 i = 1, . . . ,N, 共9兲 where the rate of each N neuron is ai共t兲, the connection matrix is ij, and the stimuli Si are constants here. This model can be justified as a rate neural model as follows 共Fukai and Tanaka, 1997兲. The firing rate ai共t兲 and membrane potential vi共t兲 of the ith neuron can be described by ai共t兲 = G共vi − 兲, 共10兲 dvi共t兲 = − vi共t兲 + Ii共t兲, dt 共11兲 where G共vi − 兲 is a gain function, and are constants, and the input current Ii共t兲 to neuron i is generated by the rates aj共t兲 of the other neurons: N Ii共t兲 = Si − 兺 ijaj共t兲. 共12兲 j Here Si is the input and ij is the strength of the inhibitory synapse from neuron j to neuron i. We suppose that G共x兲 is a sigmoidal function: G共x兲 = G0/关1 + exp共− ␤x兲兴. 共13兲 Let us then make two assumptions: 共i兲 the firing rate is always much smaller than its maximum value G0; and 共ii兲 the system is strongly dissipative 共this is reasonable because we are considering inhibitory networks兲. Based on these assumptions, after combining and rescaling Eqs. 共10兲–共13兲, we obtain the Lotka-Volterra rate description 共9兲 with an additional positive term on the right side that can be replaced by a constant 关see Fukai and Tanaka 共1997兲 for details兴. The tests of whether WLC is operating in a reduced pyloric CPG circuit are shown in Fig. 9. This study used estimates of the synaptic strengths shown in Fig. 9共a兲. Some of the key questions here are these. 共i兲 What is the minimal strength for the inhibitory synapse from the pyRev. Mod. Phys., Vol. 78, No. 4, October–December 2006 FIG. 9. 共Color online兲 Competition without winner in a model of the pyloric CPG. 共a兲 Schematic diagram of the three-neuron network used for rate modeling. Black dots represent chemical inhibitory synapses with strengths given in nanoseconds 共X ⬎ 160兲. 共b兲 Phase portrait of the model: The limit cycle corresponding to the rhythmic activity is in the 2D simplex 共Zeeman and Zeeman, 2002兲. 共c兲 Robustness in the presence of noise: Noise introduced into the model shows no effect on the order of activation for each type of neuron. Figure provided by R. Huerta. loric dilator 共PD兲 neuron or AB group to the VD neuron such that WLC exists? 共ii兲 Does the connectivity obtained from the competition without winner condition produce the order of activation observed in the pyloric CPG? 共iii兲 Is this dynamics robust against noise, in the sense that strong perturbations of the system do not alter the sequence? If the strengths of ij are taken as 冢 冣 1 1.25 0 1 1.25 , ij = 0.875 X/80 0.625 1 the WLC formulas imply that the sufficient conditions for a reliable and robust cyclic sequence are satisfied if X ⬎ 160. The activation sequence of the rate model with noise shown in Fig. 9共c兲 is similar to that observed experimentally in the pyloric CPG. When additive Gaussian noise is introduced into the rate equations, the activation order of neurons is not broken, but the period of the limit cycle depends on the level of perturbation. Therefore the cyclic competitive sequence is robust and can be related to the synaptic connectivity seen in real MCs. If individual neurons in a MC are not oscillating, one can consider small subgroups of neurons that may form oscillatory units and apply the WLC principle to these units. An important question about modeling the rhythmic activity of small inhibitory circuits is how the specific dynamics of individual neurons influences the network rhythm generation. Figure 10 represents the threedimensional 共3D兲 projection of the many-dimensional phase portrait of a circuit with the same architecture as 1228 Rabinovich et al.: Dynamical principles in neuroscience FIG. 10. 共Color online兲 Three-dimensional projection of the many-dimensional phase portrait of a circuit with the same architecture as the one shown in Fig. 9, using Hodgkin-Huxley spiking-bursting neuron models. shown in Fig. 9共a兲 but using Hodgkin-Huxley spikingbursting neuron models. The switching dynamics seen in the rate model is shown in Fig. 9共c兲, and this circuit is robust when noise is added to it. Pairs of neurons can interact via inhibitory, excitatory, or electrical 共gap junction兲 synapses to produce basic forms of neural activity which can serve as the foundation for MC dynamics. Perhaps the most common 共and well-studied兲 CPG interaction consists of reciprocal inhibition, an arrangement that generates a rhythmic bursting pattern in which neurons fire approximately out of phase with each other 共Wang and Rinzel, 1995兲. This is called a half-center oscillator. It occurs when there is some form of excitation to the two neurons sufficient to cause their firing and some form of decay mechanism to slow high firing frequencies. The dynamical range of the bursting activity varies with the synapse strength and in some instances can actually produce in-phase bursting. Usually reciprocal excitatory connections 共unstable if too large兲 or reciprocal excitatory-inhibitory connections are able to reduce the intrinsic irregularity of neurons 共Varona, Torres, Abarbanel, et al., 2001兲. Modeling studies with electrically coupled neurons have also produced nonintuitive results 共Abarbanel et al., 1996兲. While electrical coupling is generally thought to provide synchrony between neurons, under certain conditions the two neurons can burst out of phase with each other 共Sherman and Rinzel, 1992; Elson et al., 1998, 2002兲; see Fig. 11 and also Chow and Kopell 共2000兲 and Lewis and Rinzel 共2003兲. An interesting modeling study of three neurons 共Soto-Trevino et al., 2001兲 with synapses that are activity dependent found that the synaptic strengths self-adjusted in different combinations to produce the same three-phase rhythm. There are many examples of vertebrate MCs in which a collection of neurons can be conceptually isolated to perform a particular function or to represent the canonical or modular circuit for a particular brain region 关see Shepherd 共1998兲兴. Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 FIG. 11. Artificial electrical coupling between two living chaotic PD cells of the stomatogastric ganglion of a crustacean can demonstrate both synchronous and asynchronous regimes of activity. In this case the artificial electrical synapse was built on top of the existing natural coupling between two PD cells. This shows different synchronization levels 共a兲–共d兲 as a function of the artificial coupling ga and a dc current I injected in one of the cells. 共a兲 With their natural coupling ga = 0 the two cells are synchronized and display irregular spiking-bursting activity. 共b兲 With an artificial electrical coupling that changes the sign of the current ga = −200 nS, and thus compensates the natural coupling, the two neurons behave independently. 共c兲 Increasing the negative conductance leads to a regularized antiphase spiking activity 共by mimicking mutual inhibitory synapses兲. 共d兲 With no artificial coupling but adding a dc current two neurons are synchronized, displaying tonic spiking activity. Modified from Elson et al., 1988. C. Intercircuit coordination It is often the case that more or less independent MCs must synchronize in order to perform some coordinated function. There is a growing literature suggesting that large groups of neurons in the brain synchronize oscillatory activity in order to achieve coherence. This may be a mechanism for binding disparate aspects of cognitive function into a whole 共Singer, 2001兲, as we will discuss in Sec. III.D. However, it is more persuasive to examine intercircuit coordination in motor circuits where the phases of different segments or limbs actually control movements. For example, the pyloric and gastric circuits can be coordinated in the crustacean stomatogastric system by a higher-level modulatory neuron that channels the faster pyloric rhythm to a key cell in the gastric mill rhythm 共Bartos and Nushbaum, 1997; Bartos et al., 1999兲 共Fig. 12兲. In crab stomatogastric MCs, the gastric mill cycle has a period of approximately 10 s while the pyloric period is approximately 1 s. When an identified modulatory projection neuron 共MCN1兲 关Fig. 12共a兲兴 is activated, the gastric mill pattern is largely controlled by Rabinovich et al.: Dynamical principles in neuroscience FIG. 12. 共a兲 Schematic circuit diagram underlying MCN1 activation of the gastric mill rhythm of a crustacean. The circuit represents two phases of the rhythm, retraction 共left兲 and protraction 共right兲. Lighter lines represent inactive connections. LG, Int1, and DG are members of the gastric CPG and AB and PD are members of the pyloric CPG. Arrows represent functional transmission pathways from the MCN1 neuron. Bars are excitatory and dots are inhibitory. 共b兲 The gastric mill cycle period; the timing of each cycle is a function of the pyloric rhythm frequency. With the pyloric rhythm turned off, the gastric rhythm cycles slowly 共LG兲. Replacing the AB inhibition of Int1 with current into LG using a dynamic clamp reduces the gastric mill cycle period. Modified from Barots et al., 1999. interactions between MCN1 and gastric neurons LG and Int 1 共Bartos et al., 1999兲. When Int 1 is stimulated, the AB to LG synapse 关see Fig. 12共b兲兴 plays a major role in determining the gastric cycle period and coordination between the two rhythms. The two rhythms become coordinated because LG burst onset occurs with a constant latency after the onset of the triggering pyloric input. These results suggest that intercircuit synapses can enable an oscillatory circuit to control the speed of a slower oscillatory circuit as well as provide a mechanism for intercircuit coordination 共Bartos et al., 1999兲. Another type of intercircuit coupling occurs among segmental CPGs. In the crayfish, abdominal appendages Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 1229 called swimmerets beat in a metachronal rhythm from posterior to anterior with a frequency-independent phase lag of about 90°. Like most rhythms of this kind, the phase lag must remain constant over different frequencies. In theoretical and experimental studies by Jones et al. 共2003兲, it was shown that such phase constancy could be achieved by ascending and descending excitatory and inhibitory synapses, if the right connections were made. It appears realistic to look at rhythmic MCs as recurrent networks with many intrinsic feedback connections so that the information on a complete spatiotemporal pattern is contained in the long-term activity of just a few neurons in the circuit. The number of intercircuit connections necessary for coordination of the rhythms is therefore much smaller than the total number of neurons in the MC. To investigate coordinating two elements of a population of neurons, one may investigate how various couplings, implemented in a dynamical clamp, might operate in the cooperative behavior of two pyloric CPGs. This is a hybrid and simplified model of the more complex interplay between brain areas whose coordinated activity might be used to achieve various functions. We now describe such a set of experiments. Artificially connecting neurons from the pyloric CPG of two different animals using a dynamic clamp could lead to different kinds of coordination depending on which neurons are connected and what kind of synapses are used 共Szücs et al., 2004兲. Connecting the pacemaker group with electrical synapses could achieve same-phase synchrony; connecting them with inhibitory synapses provided much better coordination but out of phase. The two pyloric circuits 共Fig. 13兲 are representative of circuits driven by coupled pacemaker neurons that communicate with each other via both graded and conventional chemical interactions. But while the unit CPG pattern is formed in this way, coordinating fibers must use spike-mediated postsynaptic potentials only. It therefore becomes important to know where in the circuit to input these connections in order to achieve maximum effectiveness in terms of coordinating the entire circuit and ensuring phase constancy at different frequencies. Simply coupling the PDs together electrically is rather ineffective although the bursts 共not spikes兲 do synchronize completely even at high coupling strengths. The fact that the two PDs are usually running at slightly different frequencies leads to bouts of chaos in the two neurons, i.e., a reduction in regularity. More effective synchronization occurs when the pacemaker groups are linked together with moderately strong reciprocal inhibitory synapses in the classic half-center configuration. Bursts in two CPGs are of course 180° out of phase, but the frequencies are virtually identical. The best in-phase synchronization is obtained when the LPs are coupled to the contralateral PDs with inhibitory synapses 共Fig. 13兲. D. Chaos and adaptability Over the past decades there have been many reports of the observation of chaos in the analysis of various 1230 Rabinovich et al.: Dynamical principles in neuroscience FIG. 13. Coupling of two biological pyloric CPGs Pyl 1 and Pyl 2 by means of dynamic clamp artificial inhibitory synapses. The dynamic clamp is indicated by DCL. Reciprocal inhibitory coupling between the pacemaker groups AB and PD leads to antiphase synchronization while nonreciprocal coupling from the LPs produces in-phase synchronization. Figure provided by A. Szücs. time courses of data from a variety of neural systems ranging from the simple to the complex 共Glass, 1995; Korn and Faure, 2003兲. Perhaps the outstanding feature of these observations is not the presence of chaos but the appearance of low-dimensional dynamical systems as the origin of spectrally broadband, nonperiodic signals observed in many instances 共Rabinovich and Abarbanel, 1998兲. All chaotic oscillations occur in a bounded state-space region of the system. This state space is captured by the multivariate time course of the vector of dynamical degrees of freedom associated with neural spike generation. These degrees of freedom are comprised of the membrane voltage and the characteristics of the various ion currents in the cell. Using nonlinear dynamical tools one can reconstruct a mathematically faithful proxy state space for the neuron by using the membrane voltage and its time-delayed values as coordinates for the state space 共see Fig. 14兲. Chaos seems to be almost unavoidable in natural systems comprised of numerous simple or slightly complex subsystems. As long as there are three or more dimensions, chaotic motions are generic in the broad mathematical sense. So neurons are dealt a chaotic hand by nature and may have little choice but to work with it. Accepting that chaos is more or less the only choice, we can ask what benefits accrue to the robustness and adaptability of neural activity. Chaos itself may not be essential for living systems. However, the multitude of regular regimes of operation that can be accomplished in dynamical systems composed of elements which themselves can be chaotic gives rise to a basic principle that nature may use for the orRev. Mod. Phys., Vol. 78, No. 4, October–December 2006 FIG. 14. Chaotic spiking-bursting activity of isolated CPG neutrons. Top panel: Chaotic membrane potential time series of a synaptically isolated LP neuron from the pyloric CPG. Bottom panel: State-space attractor reconstructed from the voltage measurements of the LP neuron shown in the top panel using delayed coordinates 关x共t兲 , y共t兲 , z共t兲兴 = 关V共t兲 , V共t − T兲 , V共t − 2T兲兴. This attractor is characterized by two positive Lyapunov exponents. Modified from Rabinovich and Abarbanel, 1998. ganization of neural assemblies. In other words, chaos is not responsible for the work of various neural structures, but rather for the fact that those structures function at the edge of instability, and often beyond it. By recognizing chaotic motions in a system state space as unstable, but bounded, this geometric notion gives credence to the otherwise unappealing idea of system instability. The instability inherent in chaotic motions, or more precisely in nonlinear dynamics of systems with chaos, facilitates the extraordinary ability of neural systems to adapt, make transitions from one pattern of behavior to another when the environment is altered, and consequently create a rich variety of patterns. Thus chaos gives a means to explore the opportunities available to the system when the environment changes, and acts as a precursor to adaptive, reliable, and robust behavior for living systems. Throughout evolution neural systems have developed different methods of self-control or self-organization. On the one hand, such methods preserve all advantages of the complex behavior of individual neurons, such as allowing regulation of the time period of transitions between operating regimes, as well as regulation of the operation frequency in any given regime. They also preserve the possibility of a rich variety of periodic and nonperiodic regimes of behavior; see Fig. 11 and Elson et al. 共1988兲 and Varona, Torres, Huerta, et al. 共2001兲. On the other hand, these control or organizational techniques provide the needed predictability of behavioral patterns in neural assemblies. Organizing chaotic neurons through appropriate wiring associated with electrical, inhibitory, and excitatory connections appears to allow for essentially regular operation of such an assembly 共Huerta et al., 2001兲. As an example we mention the dynamics of an artificial micro- Rabinovich et al.: Dynamical principles in neuroscience 1231 III. INFORMATIONAL NEURODYNAMICS FIG. 15. Average bursting period of the model heartbeat CPG activity as a function of the inhibitory coupling ⑀. Modified from Malkov et al., 1996. circuit that mimics the leech heartbeat CPG 共Calabrese et al., 1995兲. This CPG model consists of six chaotic neurons implemented with Hindmarsh-Rose equations reciprocally coupled to their neighbors through inhibitory synapses. The modeling showed that in spite of chaotic oscillations of individual neurons the cooperative dynamics is regular and, most importantly, the period of bursting of the cooperative dynamics sensitively depends on the values of the inhibitory connections 共Malkov et al., 1996兲 共see Fig. 15兲. This example shows the high level of adaptability of a network consisting of chaotic elements. Chaotic signals have many of the traditional characteristics attributed to noise. In the present context we recognize that both chaos and noise are able to organize the irregular behavior of individual neurons or neural assemblies, but the principal difference is that dynamical chaos is a controllable irregularity, possessing structure in state space, while noise is an uncontrollable action of dynamical systems. This distinction is extremely important for information processing as discussed below 共see Sec. III.B.2 and its final remarks兲. There are several possible functions for noise 共Lindner et al., 2004兲, even seen as high-dimensional essentially unpredictable chaotic motion, in neural network studies. In high-dimensional systems composed here of many coupled nonlinear oscillators, there may be small basins of attraction where, in principle, the system could become trapped. Noise will blur the basin boundaries and remove the possibility that the main attractors could accidentally be missed and highly functional synchronized states lost to neuronal activity. Some noise may persist in the dynamics of neurons to smooth out the actions of the chaotic dynamics active in creating robust, adaptable networks. Chaos should not be mistaken for noise, as the former has phase-space structure which can be utilized for synchronization, transmission of information, and regularization of the network for performance of critical functions. In the next section we discuss the role of chaos in information processing and information creation. Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 The flow of information in the brain goes from sensory systems, where it is captured and encoded, to central nervous systems, where it is further processed to generate response signals. In the central nervous system command signals are generated and transported to the muscles to produce motor behavior. At all these stages learning and memory processes that need specific representations take place. Thus it is not surprising that the nervous system has to use different coding strategies at different levels of the transport, storage, and use of information. Different transformations of codes have been proposed for the analysis of spiking activity in the brain. The details depend on the particular system under study but some generalization is possible in the framework of analyzing the spatial, temporal, and spatiotemporal codes. There are many unknown factors related to the cooperation between these different forms of information coding. Some key questions are as follows: 共i兲 How can neural signals be transformed from one coding space to another without loss of information? 共ii兲 What dynamical mechanisms are responsible for storing time in memory? 共iii兲 Can neural systems generate new information based on their sensory inputs? In this section, we discuss some important experimental results and new paradigms that can help to address these questions. A. Time and neural codes Information from sensory systems arrives at sensory neurons as analog changes in light intensity or temperature, or chemical concentration of an odorant, or skin pressure, etc. These analog data are represented in internal neural circuit dynamics and computations by action-potential sequences passed from sensory receivers to higher-order brain processes. Neural codes guarantee the efficiency, reliability, and robustness of the required neural computations 共Machens, Gollisch, et al., 2005兲. 1. Temporal codes Two of the central questions in understanding the dynamics of information processing in the nervous system are how information is encoded and how the coding space depends on time-dependent stimuli. A code in the biophysical context of the nervous system is a specific representation of the information operated on or created by neurons. A code requires a unit of information. However, this is already a controversial issue since, as we have previously discussed, information is conveyed through chemical and electrical synapses, neuromodulators, hormones, etc., which makes it difficult to point out a single universal unit of information. A classical assumption at the cellular level, valid for many neural systems, is that a spike is an all-or-nothing event and thus a good candidate for a unit of information, at least in a computational sense. This is not the only sim- 1232 Rabinovich et al.: Dynamical principles in neuroscience FIG. 16. 共Color online兲 Two possible codes for the activity of a single neuron. In a rate code, different inputs 共A–D兲 are transformed into different output spiking rates. In a timing code, different inputs are transformed into different spiking sequences with precise timing. plification needed to analyze neural codes for a first approach. A coding scheme needs to determine a coding space and take into account time. A common hypothesis is to consider a universal time for all neural elements. Although this is the approach we discuss here, we remind the reader that this is also an arguable issue, since neurons can sense time in many different ways: by their intrinsic activity 共subcellular dynamics兲 or by external input 共synaptic and network dynamics兲. Internal and external 共network兲 clocks are not necessarily synchronized and can have different degrees of precision, time scales, and absolute references. Some dynamical mechanisms can contribute to make neural time unified and coherent. On the one hand, when we consider just a single neuron, a spike as the unit of information, and a universal time, we can talk about two different types of encoding: the frequency of firing can encode information about the stimulus in a rate code; on the other hand, the exact temporal occurrence of spikes can encode the stimulus and its response in a precise timing code. The two coding alternatives are schematically represented in Fig. 16. In this context, a precise timing or temporal code is a code in which relative spike timings 共rather than spike counts兲 are essential for information processing. Several experimental recordings have shown the presence of both types of single-cell coding in the nervous system 共Adrian and Zotterman, 1926; Barlow, 1972; Abeles, 1991; McClurkin et al., 1991; Softky, 1995; Shadlen and Newsome, 1998兲. In particular, fine temporal precision and reliability of spike dynamics are reported in many cell types 共Segundo and Perkel, 1969; Mainen and Sejnowski, 1995; deCharms and Merzenich, 1996; de Ryter van Steveninck et al., 1997; Segundo et al., 1998; Mehta et al., 2002; Reinagel and Reid, 2002兲. Single neurons can display these two codes in different situations. 2. Spatiotemporal codes A population of coupled neurons can have a coding scheme different from the sum of the individual coding mechanisms. Interactions among neurons through their synaptic connections, i.e., their cooperative dynamics, allow for more complex coding paradigms. There is much experimental evidence which shows the existence of socalled population codes that collectively express a complex stimulus better than the individual neurons 关see, Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 e.g., Georgopoulus et al. 共1986兲; Wilson and McNaughton 共1993兲; Fitzpatrick et al. 共1997兲; Pouget et al. 共2000兲兴. The efficacy of population coding has been assessed mainly using measures of mutual information in modeling efforts 共Seung and Sompolinsky, 1993; Panzeri et al., 1999; Sompolinsky et al., 2001兲. Two elements can be used to build population codes: neuronal identity 共i.e., neuronal space兲 and the time occurrence of neural events 共i.e., the spikes兲. Accordingly, information about the physical world can be encoded in temporal or spatial 共combinatorial兲 codes, or combinations of these two: spike time can represent physical time 共a pure temporal code兲, spike time can represent physical space, neuronal space can represent physical time 共a pure spatial code兲, and neuronal space can represent physical space 共Nádasdy, 2000兲. When we consider a population of neurons, information codes can be spatial, temporal, or spatiotemporal. Population coding can also be characterized as independent or correlated 共deCharms and Christopher, 1998兲. In an independent code, each neuron represents a separate signal: all information that is obtainable from a single neuron can be obtained from that neuron alone, without reference to the activities of other neurons. For a correlated or coordinated coding messages are carried at least in part by the relative timing of the signals from a population of neurons. The presence of network coding, i.e., a spatiotemporal dynamical representation of incoming messages, has been confirmed in several experiments. As an example, we discuss here the spatiotemporal representation of episodic experiences in the hippocampus 共Lin et al., 2005兲. Individual hippocampal neurons respond to a wide variety of external stimuli 共Wilson and McNaughton, 1994; Dragoi et al., 2003兲. The response variability at the level of individual neurons poses an obstacle to the understanding of how the brain achieves its robust real-time neural coding of the stimulus 共Lestienne, 2001兲. Reliable encoding of sensory or other network inputs by spatiotemporal patterns resulting from the dynamical interaction of many neurons under the action of the stimulus can solve this problem 共Hamilton and Kauer, 1985; Laurent, 1996; Vaadia et al., 1999兲. Lin et al. 共2005兲 showed that mnemonic short-time episodes 共a form of one-trial learning兲 can trigger firing changes in a set of CA1 hippocampal neurons with specific spatiotemporal relationships. To find such representations in the central nervous system of an animal is an extremely difficult experimental and computational problem. Because the individual neurons that participate in the representation of a specific stimulus and form a temporal neural cluster in different trials can be different, it is necessary to measure simultaneously the activity of a large number of neurons. In addition, because of the variability in the individual neuron responses, the spatiotemporal patterns of different trials may also look different. Thus, to show the functional importance of the spatiotemporal representation of the stimulus, the reader has to use sophisticated methods of data analysis. Lin et al. 共2005兲 developed a 96-channel array to record Rabinovich et al.: Dynamical principles in neuroscience 1233 FIG. 17. 共Color online兲 Temporal dynamics of individual CA1 neurons of the hippocampus in response to “startling” events. Spike raster plots 关共a兲–共d兲 upper, seven repetitions each兴 and corresponding perievent histogram 关共a兲–共d兲 lower, bin width 500 ms兴 for units exhibiting the four major types of firing changes observed: 共a兲 transient increase, 共b兲 prolonged increase, 共c兲 transient decrease, 共d兲 and prolonged decrease. From Lin et al., 2005. simultaneously the activity patterns of as many as 260 individual neurons in the mouse hippocampus during various startling episodes 共air blow, elevator drop, and earthquake shake兲. They used multiple-discriminant analysis 共Duda et al., 2001兲 and showed that, even though individual neurons express different temporal patterns in different trials 共see Fig. 17兲, it is possible to identify functional encoding units in the CA1 neuron assembly 共see Fig. 18兲. The representation of nonstationary sensory information, say, a visual stimulus, can use the transformation of a temporal to a spatial code. The recognition of a specific neural feature can be implemented through the transformation of a spatial code into a temporal one through coincidence detection of spikes. A spatial representation can be transformed into a spatiotemporal one to provide the system with higher capacity and robustness and sensitivity at the same time. Finally, a spatiotemporal code can be transformed into a spatial code in processes related to learning and memory. These possibilities are summarized in Fig. 19. Morphological constraints of neural connections in some cases impose a particular spatial or temporal code. For example, projection neurons transfer information between areas of the brain along parallel pathways by preserving the input topography as neuronal specificity at the output. In many cases the input topography is transformed to a different topography that is preserved; for example, the retinotopic map of the primary visual areas and somatotopic maps of the somatosensory and motor areas. Other transformations do not preserve topology. These include transformations in place cells in the hippocampus, and the tonotopic representation in the auditory cortex. There is a high degree of convergence and divergence of projections in some of these transformations that can be a computationally optimal design 共Garcia-Sanchez and Huerta, 2003兲. In most of these transformations, the temporal dimension of the stimulus is encoded by spike timing or by the onset of firing-rate transients. An example of transforming a spatiotemporal code to a pure spatial code was found in the olfactory system of Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 locusts, and has been modeled by Nowotny, Rabinovich, et al. 共2003兲 and Nowotny et al. 共2005兲. Figure 20 gives a graphical explanation of the connections involved. The complex spatiotemporal code of sequences of transiently synchronized groups of projection neurons in the antennal lobe 共Laurent et al., 2001兲 is sliced into temporal snapshots of activity by feedforward inhibition and coincidence detection in the next processing layer, the mushroom body 共Perez-Orive et al., 2002兲. This snapshot code is presumably integrated over time in the next stages of the mushroom lobes, completing the transformation of the spatiotemporal code in the antennal lobe to a purely spatial code. It was shown in simulations that the temporal information on the sequence of activity in the antennal lobe that could be lost in downstream temporal integration can be restored through slow lateral excitation in the mushroom body 共Nowotny, Rabinovich, et al., 2003兲. This has been reported experimentally 共Leitch and Laurent, 1996兲. With this extra feature the transformation from a spatiotemporal code to a pure spatial code becomes free of information loss. 3. Coexistence of codes Different stages of neural information processing are difficult to study in isolation. In many cases it is hard to distinguish between what is an encoding of an input and what is a static or dynamic, perhaps nonlinear, response to that input. This is a crucial observation that is often missed. Encoding and decoding may or may not be part of a dynamical process. However, the creation of information 共discussed in the next section兲 and the transformation of spatial codes to temporal or spatiotemporal codes are always dynamical processes. Another, but less frequently addressed, issue about coding is the presence of multiple encodings in singlecell signals 共Latorre et al., 2006兲. This may occur since multifunctional networks may need multiple coexisting codes. The neural signatures in interspike intervals of CPG neurons provide an example 共Szücs et al., 2003兲. Individual fingerprints characteristic of the activity of each neuron coexist with the encoding of information in 1234 Rabinovich et al.: Dynamical principles in neuroscience formulation of information theory 共Fano, 1961; Gallager, 1968兲. For example, cochlear afferents in birds bifurcate to two different areas of the brain with different decoding properties. One area extracts information about relative timing from a spike train, whereas the other extracts the average firing rate 共Konishi, 1990兲. 4. Temporal-to-temporal information transformation: Working memory FIG. 18. 共Color online兲 Classification, visualization, and dynamical decoding of CA1 ensemble representations of startle episodes by multiple-discriminant analysis 共MDA兲 methods. 共a兲 Firing patterns during rest, air blow, drop, and shake epochs are shown after being projected to a three-dimensional space obtained by using MDA for mouse A; MDA1–MDA3 denote the discriminant axes. Both training 共dark symbols兲 and test data are shown. After the identification of startle types, a subsequent MDA is further used to resolve contexts 共full vs empty symbols兲 in which the startle occurred for air-blow context 共b兲 and for elevator drop 共c兲. 共d兲 Dynamical monitoring of ensemble activity and the spontaneous reactivation of startle representations. Three-dimensional subspace trajectories of the population activity in the two minutes after an air-blow startle in mouse A are shown. The initial response to an air blow 共black line兲 is followed by two large spontaneous excursions 共blue/dark and red/light lines兲, characterized by coplanar, geometrically similar lower-amplitude trajectories 共directionality indicated by arrows兲. 共e兲 The same trajectories as in 共a兲 from a different 3D angle. 共f兲 The timing 共t1 = 31.6 s and t2 = 54.8 s兲 of the two reactivations 共marked in blue/dark and red/ light, respectively兲 after the actual startle 共in black兲 共t = 0 s兲. The vertical axis indicates the air-blow classification probability. From Lin et al., 2005. the frequency and phase of the spiking-bursting rhythms. This is an example that shows that codes can be nonexclusive. In bursting activity, coding can exist in slow waves, but also, and simultaneously, in the spiking activity. In the brain, specific neural populations often send messages through projections to several information “users.” It is difficult to imagine that all of them decode the incoming signals in the same way. In neuroscience the relationship between the encoder and decoder is not a one-to-one map but can be many simultaneous maps from the senders to different receivers, based on different dynamics. This departs from Shannon’s classical Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 There is another important code transformation of interest here: the transformation of a finite amount of temporal information to a slow temporal code lasting for seconds, minutes, or hours. We are able to remember a phone number from someone who just called us. Persistent dynamics is one of the mechanisms for this phenomenon, which is usually named short-term memory 共STM兲 or working memory; it is a basic function of the brain. Working memory, in contrast to long-term memory which most likely requires molecular 共membrane兲 or structural 共connection兲 changes in neural circuits, is a dynamical process. The dynamical origins of working memory can vary. One plausible idea is that STMs are the result of active reverberation in interconnected neural clusters that fire persistently. Since its conceptualization 共de Nó, 1938; Hebb, 1949兲, reverberating activity in microcircuits has been explored in many modeling papers 共Grossberg, 1973; Amit and Brunel, 1997a; Durstewitz et al., 2000; Seung et al., 2000; Wang, 2001兲. Experiments with cultured neuronal networks show that reverberatory activity can be evoked in circuits that have no preexisting anatomical specialization 共Lau and Bi, 2005兲. The reverberation is primarily driven by recurrent synaptic excitation rather than complex individual neuron dynamics such as bistability. The circuitry necessary for reverberating activity can be a result of network selforganization. Persistent reverberatory activity can exist even in the simplest circuit, i.e., an excitatory neuron with inhibitory self-feedback 共Connors, 2002; Egorov et al., 2002兲. In this case, reverberation depends on asynchronous transmitter release and intracellular calcium stores as shown in Fig. 21. Nature seems to use different dynamical mechanisms for persistent microcircuit activity: cooperation of many interconnected neurons, persistent dynamics of individual neurons, or both. These mechanisms each have distinct advantages. For example, network mechanisms can be turned on and off quickly 共McCormick et al., 2003兲 关see also Brunel and Wang 共2001兲兴. Most dynamical models with persistent activity are related to the analysis of microcircuits with local feedback excitation between principal neurons controlled by disynaptic feedback inhibition. Such basic circuits spontaneously generate two different modes: relative quiescence and persistent activity. The triggering between modes is controlled by incoming signals. The review by Brunel 共2003兲 considers several basic models of persistent dynamics, including bistable networks with excitation only and multistable models for working memory of a discrete set 1235 Rabinovich et al.: Dynamical principles in neuroscience FIG. 19. 共Color online兲 Summary of possible scenarios for the transformation of codes, their functional implications, and the dynamical mechanism involved. of pictures with structured excitation and global inhibition. Working memory is used for tasks such as planning, organizing, rehearsing, and movement preparation. Experiments with functional magnetic resonance imaging reveal some aspects of the dynamics of working memory 关see, for example, Diwadkar et al. 共2000兲 and Nystrom et al. 共2000兲兴. It is important to note that working memory has a limited capacity of around four to seven items 共Cowan, 2001; Vogel and Michizawa, 2004兲. An essential feature attributed to working memory is the labile and transient nature of its representations. Because such representations involve many coupled neurons from cortical areas 共Curts and D’Esposito, 2003兲, it is natural to model working memory as the spatiotemporal dynamics of large neural networks. A popular idea is to model working memory with attractors. Representation of items in working memory by attractors may guarantee its robustness. Although robustness is an important requisite for a working-memory system, its transient properties are also important. Consider a foraging task in which an animal uses visual input to catch prey 共Nakahara and Doya, 1998兲. It is helpful to store the location of the prey in the animal’s working memory if the prey goes behind a bush and the sensory cue becomes temporarily unavailable. However, the memory should not be retained forever because the prey may have actually gone away or may have been eaten by another animal. Furthermore, if more prey appears near the animal, the animal should quickly load the location of the new prey into its working memory without being disturbed by the old memory. This example illustrates that there are more requirements for a working-memory system than solely robust Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 maintenance. First, the activity should be maintained but not for too long. Second, the activity should be reset quickly when there is a novel sensory cue that needs to be stored. In other words, the neural dynamics involved in working memory for goal-directed behaviors should have the properties of long-term maintenance and quick switching. A corresponding model based on “nearsaddle-node” bifurcation dynamics has been suggested by Nakahara and Doya 共1998兲. The authors have analyzed the dynamics of a network of model neural units that are described by the following map 共see Fig. 22兲: 冉 冊 yi共tn+1兲 = F ayi共tn兲 + b + 兺 ijyi共tn兲 + ␥iIi共tn兲 , j⫽i 共14兲 where yi共tn兲 is the firing rate of the ith unit at time tn, F共z兲 = 1 / 关1 + exp共−z兲兴 is a sigmoid function, a is the selfconnection weight, ij are the lateral connection weights, Ii共t兲 are external inputs, b is the bias, and ␥i are constants used to scale the inputs Ii共t兲. As the bias b is increased, the number of fixed points changes sequentially from one to two, three, two, and then back to one. A saddle-node bifurcation occurs when the stable transition curve y共tn+1兲 = F共z兲 is tangent to the fixed point y共tn+1兲 = y共tn兲 共see Fig. 22兲. Just near the saddle-node bifurcation the system shows persistent activity. This means that it spends a long time in the narrow channel between the bisectrix and the sigmoid activation curve and then goes to the fixed point quickly. Such dynamical behavior reminds one of the well-known intermittency phenomenon in physics 共Landau and Lifshitz, 1987兲. Because the effect of the sum of the lateral and external inputs in Eq. 共14兲 is equivalent to a change in the bias, the mechanism may satisfy the requirements of the dy- 1236 Rabinovich et al.: Dynamical principles in neuroscience FIG. 20. 共Color兲 Illustration of the transformation of temporal into spatial information. If a coincidence detection occurs, the local excitatory connections activate the neighbors of the active neuron 共yellow neurons兲. Coincidence detection of input is now more probable in these activated neighborhoods than in other Kenyon cells 共KCs兲. Which of the neighbors might fire a spike, however, depends on the activity of the projection neurons 共PNs兲 in the next cycle. It might be a different neuron for active group B of PNs 共upper branch兲 than for active group C 共lower branch兲. In this way local sequences of active KCs form. These depend on the identity of active PNs 共coincidence detection兲 as well as on the temporal order of their activity 共activated neighborhoods兲. Modified from Nowotny, Rabinovich, et al., 2003. namics of working memory for goal-directed behavior: long-term maintenance and quick switching. Another reasonable model for working memory consists of competitive networks with stimulus-dependent inhibitory connections 关as in Eq. 共9兲兴. One of the advantages of such a model is the ability to have both working memory and stimulus discrimination. This idea was proposed by Machens, Romo, et al. 共2005兲 in relation to the frontal-lobe neural architecture. The network first perceives the stimulus, then holds it in the working memory, and finally makes a decision by comparing that stimulus with another one. The model integrates both working memory and decision making since the number of stable fixed points and the size of the basins of attractors are controlled by the connection matrix ij共S兲 which depends on the stimuli S. The working-memory phase corresponds to the bifurcation boundary, i.e., ij = ji = ii. In the state space of the dynamical model, this phase is represented by a stable manifold called a “continuous Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 FIG. 21. 共Color online兲 Reverberation can be the dynamical origin for working memory in minimal circuits. 共a兲 Most neurons respond to excitatory stimuli 关upward steps in the line below 共c兲兴 by spiking only as long as each stimulus lasts. 共b兲 Very rare neurons are bistable: brief excitation leads to persistent spiking, always at the same rate; brief inhibition 关downward steps in the line below 共c兲兴 can turn it off. 共c兲 Multistable neurons persistently increase or decrease their spiking across a range of rates in response to repeated brief stimuli. 共d兲 In the reverberatory network model of short-term memory discussed in the text, an excitatory stimulus 共left arrow兲 leads to recursive activity in interconnected neurons. Inhibitory stimuli 共bar on the right兲 can halt the activity. 共e兲 Egorov et al. 共2002兲 suggest that graded persistent activity in single neurons 关as in 共c兲兴 might be triggered by a pulse of internal Ca2+ ions that enter through voltage-gated channels; Ca2+ then activates calciumdependent nonspecific cation 共CAN兲 channels, through which an inward current 共largely comprising Na+ ions兲 enters, persistently exciting the neuron. The positive feedback loop 共broken arrows兲 may include the activity of many ionic channels. Modified from Connors, 2002. attractor.” This is an attractor that consists of continuous sets of fixed points 关see Amari 共1977兲 and Seung 共1998兲兴. Thus the stimulus creates a specific fixed point and, at the next stage, the working memory 共a continuous attractor兲 maintains it. During the comparison and decision phase, the second stimulus is mapped onto the same state space as another attractor. The criterion of the decision maker is reflected in the positions of the separatrices that separate the basins of attraction of different FIG. 22. Temporal responses of self-recurrent units: Nearsaddle-node bifurcation with a = 11.11, b = −7.9 共center panels兲. Increased bias, b = −3.0 共left panels兲. Decreased bias b = −9.0 共right panels兲. Modified from Nakahara and Doya, 1998. Rabinovich et al.: Dynamical principles in neuroscience 1237 FIG. 23. Hallucinations generated by LSD are an example of a dynamical representation of the internal activity of the visual cortex without an input stimulus. Figure shows examples of 共a兲 funnel and 共b兲 spiral hallucinations. Modified from Bressloff, et al. 2001. stimuli, i.e., fixed points 共see an alternative approach in Rabinovich and Huerta, 2006兲. We think that the intersection of the mechanisms responsible for persistent activity of single neurons with the activity of a network with local or nonlocal recurrence provides robustness against noise and perturbations, and at the same time makes working memory more flexible. B. Information production and chaos Information processing in the nervous system involves more than the encoding, transduction, and transformation of incoming information to generate a corresponding response. In many cases, neural information is created by the joint action of the stimulus and the individual neuron and network dynamics. A creative activity like improvisation on the piano or writing a new poem results in part from the production of new information. This information is generated by neural circuits in the brain and does not directly depend on the environment. Time-dependent visual hallucinations are one example of information produced by neural systems, in this case the visual cortex, themselves. Such hallucinations consist in seeing something that is not in the visual field. There are interesting models, beginning from the pioneering paper of Ermentrout and Cowan 共1979兲, that explain how the intrinsic circuitry of the brain’s visual cortex can generate the patterns of activity that underlie hallucinations. These hallucination patterns usually take the form of checkerboards, honeycombs, tunnels, spirals, and cobwebs 共see two examples in Fig. 23兲. Because the visual cortex is an excitable medium it is possible to use spatiotemporal amplitude equations to describe the dynamics of these patterns 共see the next section兲. These models are based on advances in brain anatomy and physiology that have revealed strong short-range connections and weaker long-range connections between neurons in the visual cortex. Hallucination patterns can be quasistatic, periodically repeatable, or chaotically repeatable as in low-dimensional convective turbulence; see for a review Rabinovich et al. 共2000兲. Unpredictability of the specific pattern in the hallucination sequences Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 FIG. 24. 共Color online兲 Dual sensory network dynamics. Top panels: Schematic representation of the dual role of a single statocyst, the gravity sensory organ of the mollusk Clione. During normal swimming, a stonelike structure, the statolith, hits the mechanoreceptor neurons that react to this excitation. In Clione’s hunting behavior, the statocyst receptors receive additional excitation from the cerebral hunting neuron 共H兲 which generates a winnerless competition among them. Bottom panels: Chaotic sequential switching displayed by the activity of the statocyst during hunting mode in a model of a six-receptor network. This panel displays the time intervals in which each neuron is active 共ai ⬎ 0.03兲. Each neuron is represented by a different color. The dotted rectangles indicate the activationsequence locks among units that are active at a given time interval within each network for time windows in which all six neurons are active. 共movie兲 means the generation of information that in principle can be characterized by the value of the Kolmogorov-Sinai entropy 共Scott, 2004兲. The creation or production of new information is a theme that has been neglected in theoretical neuroscience, but it is a provocative and challenging point that we discuss in this section. As mentioned before, information production or creation must be a dynamical process. Below we discuss an example that emphasizes the ability of neural systems to produce information-rich output from information-poor input. 1. Stimulus-dependent motor dynamics A simple network with which we can discuss the creation of new information is the gravity-sensing neural network of the marine mollusk Clione limacina. Clione is a blind planktonic animal, negatively buoyant, that has to maintain continuous motor activity in order to keep its preferred head-up orientation. Its motor activity is controlled by wing CPGs and tail motor neurons that use signals from its gravity-sensing organs, the statocysts 共Panchin et al., 1995兲. A six-receptor neural network model with synaptic inhibition has been built to describe a single statocyst 共Varona, Rabinovich, et al., 2002兲 共see Fig. 24兲. This is a small sphere in which a statolith, a stonelike structure, moves according to the gravitational 1238 Rabinovich et al.: Dynamical principles in neuroscience FIG. 25. 共Color online兲 Clione swimming trajectories in different situations. 共a兲 Three-dimensional trajectory of routine swimming. Here and in the following figures, different colors 共gray tones兲 are used to emphasize the three-dimensional perception of the trajectories and change according to the x axis. The indicated time t is the duration of the trajectory. 共b兲 Swimming trajectory of Clione with the statocysts surgically removed. 共c兲 Trajectory of swimming during hunting behavior evoked by the contact with the prey. 共d兲 Trajectory of swimming after immersion of Clione in a solution that pharmacologically evokes hunting. Modified from Levi et al., 2004. field. The statolith excites the neuroreceptors by pressing down on them. When excited, the receptors send signals to the neural systems responsible for wing beating and tail orientation. The statocysts have a dual role 共Levi et al., 2004, 2005兲. During normal swimming only neurons that are excited by the statolith are active, and this leads to a winner-take-all dynamical mode as a result of inhibitory connections in the network. 共Winner-take-all dynamics is essentially the same as the attractor-based computational ideas discussed earlier.兲 However, when Clione is searching for its food, a cerebral hunting neuron excites each neuron of the statocyst 共see Fig. 24兲. This triggers a competition between all statocyst neurons whose signals participate in the generation of a complex motion that the animal uses to scan the immediate space until it finds its prey 共Levi et al., 2004, 2005兲 共see Fig. 25兲. The following Lotka-Volterra-type dynamics can be used to describe the activity of this network: 冉 N 冊 dai共t兲 = ai共t兲 共H,S兲 − 兺 ijaj共t兲 + Hi共t兲 + Si共t兲, dt j=1 共15兲 where ai共t兲 艌 0 represents the instantaneous spiking rate of the statocyst neurons, Hi共t兲 represents the excitatory stimulus from the cerebral hunting interneuron to neuRev. Mod. Phys., Vol. 78, No. 4, October–December 2006 FIG. 26. 共Color online兲 Irregular switching in a network of six statocyst receptors. Traces represent the instantaneous spiking rate of each neuron ai 关neurons 1,2,3 are shown in 共a兲, neurons 4,5,6 in 共b兲兴. Note that after a neuron is silent for a while, its activity reappears with the same sequence relative to the others 共see arrows, and Fig. 24兲. 共c兲 A projection of the phase portrait of the strange attractor in 3D space; see model 共15兲. ron i, Si共t兲 represents the action of the statolith on the receptor that it is pressing, and ij is the nonsymmetric statocyst connection matrix. When there is no stimulus from the hunting neuron, Hi = 0, or the statolith, Si = 0, then 共H , S兲 = −1 and all neurons are silent. When the hunting neuron is active Hi ⫽ 0 and/or the statolith is pressing one of the receptors, Si ⫽ 0, 共H , S兲 = + 1. During hunting Hi ⫽ 0, and we assume that the action of the hunting neuron overrides the effect of the statolith and thus Si ⬇ 0. As a result of the competition, the receptors display a highly irregular, in fact chaotic, switching activity. The phase-space image of the chaotic dynamics of the statocyst model in this behavioral mode is a strange attractor 关the heteroclinic loops in the phase space of Eq. 共15兲 become unstable; see Sec. IV.C兴. For six receptors we have shown 共Varona, Rabinovich, et al., 2002兲 that the observed dynamical chaos is characterized by two positive Lyapunov exponents. The bottom panel in Fig. 24 is an illustration of the nonsteady switching activity of the receptors. An interesting phenomenon can be seen in this figure and is also pointed out in Fig. 26. Although the timing of each activity is irregular, the sequence of switching among the statocyst receptors is the same for those neurons that are active at a given time window. Dotted rectangles in Fig. 24 point out this fact. The activation-sequence lock Rabinovich et al.: Dynamical principles in neuroscience among the statocyst receptor neurons emerges in spite of the highly irregular timing of the switching dynamics and is a feature that can be used for motor coordination 共Venaille et al., 2005兲. In this example the winnerless competition is triggered by a constant excitation to all statocyst receptors 关Hi = ci; see details by Varona, Rabinovich, et al. 共2002兲兴. Thus the stimulus has low information content. Nonetheless, the network of statocyst receptors can use this activity to generate an information-rich signal with positive Kolmogorov-Sinai entropy. This entropy is equal to the value of the new information encoded in the dynamical motion. The statocyst sensory network is thus multifunctional and can generate a complex spatiotemporal pattern useful for motor coordination even when its dynamics are not evoked by gravity, as during hunting. 2. Chaos and information transmission To illustrate the role of chaos in information transmission, we use as an example the inferior olive 共IO兲, which is an input system to the cerebellum. Neurons of the IO may chaotically recode the high-frequency information carried by its inputs into chaotic, low-rate output 共Schweighofer et al., 2004兲. The IO has been proposed as a system that controls and coordinates different rhythms through the intrinsic oscillatory properties of its neurons and the nature of their electrical interconnections 共Llinás and Welsh, 1993; de Zeeuw et al., 1998兲. It has also been implicated in motor learning 共Ito, 1982兲 and in comparing tasks of intended and achieved movements as a generator of error signals 共Oscarsson, 1980兲. Experimental recordings show that IO cells are electrically coupled and display subthreshold oscillations and spiking activity. Subthreshold oscillations have a relevant role for information processing in the context of a system with extensive electrical coupling. In such systems the spiking activity can be propagated through the network, and, in addition, small differences in hyperpolarized membrane potentials propagate among neighboring cells. A modeling study suggests that electrical coupling in IO neurons may induce chaos, which would allow information-rich, but low-firing-rate, error signals to reach individual Purkinje cells in the cerebellar cortex. This would provide the cerebellar cortex with essential information for efficient learning without disturbing ongoing motor control. The chaotic firing leads to the generation of IO spikes with different timing. Because the IO has a low firing rate, an accurate error signal will be available for individual Purkinje cells only after repeated trials. Electrical coupling can provide the source of disorder that induces a chaotic resonance in the IO network 共Schweighofer et al., 2004兲. This resonance leads to an increase in information transmission by distributing the high-frequency components of the error inputs over the sporadic, irregular, and non-phase-locked spikes. Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 1239 The IO single-neuron model consists of two compartments that include a low-threshold calcium current 共ICal兲, an anomalous inward rectifier current 共Ih兲, a Hodgkin-Huxley-type sodium current 共INa兲, and a delayed rectifier potassium current 共IKd兲 in the somatic compartment 共see Table I兲. The dendritic compartment contains a calcium-activated potassium current 共IKCa兲 and a high-threshold calcium current 共ICah兲. This compartment also receives electrical connections from neighboring neurons. Fast ionic channels are located in the soma, and slow channels are located in the dendritic compartment. Some of the channel conductances depend on the calcium concentration. The equations for each compartment of a single neuron can be summarized as CM dV共t兲 = − 共Iion + Il + Iinj + Icomp兲, dt 共16兲 where CM is the membrane capacitance, Il is a leak current, Iinj is the injected stimulus current, Icomp connects the compartments, and Iion is the sum of the currents above for each compartment. In addition, the dendritic compartment has the electrical coupling current Iec = gc兺i关V共t兲 − Vi共t兲兴, where the index i runs over the neighbors of each neuron, and gc is the electrical coupling conductance. Each IO neuron is represented by a system of ordinary differential equations 共ODEs兲, and the network is a set of these systems coupled through the electrical coupling currents Iec. The networks examined consisted of 2 ⫻ 2, 3 ⫻ 3, and 9 ⫻ 3 neurons, where cells are connected to their two, three, or four neighbors depending on their positions in the grid. This is a complex network, even when it is only 2 ⫻ 2, and one must select an important feature of the dynamics to characterize its behavior. The largest Lyapunov exponent of the network is a good choice as it is independent of initial conditions and tells us about information flow in the network. Figure 27 displays the largest Lyapunov exponent for each network as a function of the electric coupling conductance gc. We also see in Fig. 27 that the gc producing the largest Lyapunov exponent yields the largest information transfer through the network, evaluated as the average mutual information per spike. In a more general framework than the IO, it is remarkable that the chaotic activity of individual neurons unexpectedly underlies higher flexibility and, at the same time, greater accuracy and precision in their neural dynamics. The origin of this phenomenon is the potential ability of coupled neurons with chaotic behavior to synchronize their activities and generate rhythms whose period depends on the strength of the coupling or other network parameters 关for a review see Rabinovich and Abarbanel 共1998兲 and Aihara 共2002兲兴. Networks with many chaotic neurons can generate interesting transient dynamics, i.e., chaotic itinerancy 共CI兲 共Tsuda, 1991; Rowe, 2002兲. Chaotic itinerancy results from weak instabilities in the attractors, i.e., attractor sets in whose 1240 Rabinovich et al.: Dynamical principles in neuroscience tally hinder the power of data analyses: 共i兲 finite statistical fluctuations, 共ii兲 external noise, and 共iii兲 nonstationarity of the neural circuit activity 关see, for example, Lai et al. 共2003兲兴. C. Synaptic dynamics and information processing FIG. 27. 共Color online兲 Chaotic dynamics increases information transmission in IO models. Top panel: Largest Lyapunov exponent as a function of the electrical coupling strength gc for different IO networks of nonidentical cells. Bottom panel: Network average mutual information per spike as a function of gc. Modified from Schweighofer et al., 2004. neighborhood there are trajectories that do not go to the attractors 共Milnor-type attractors兲. A developed CI motion needs both many neurons and a very high level of interconnections. This is in contrast to the traditional concept of computation with attractors 共Hopfield, 1982兲. Chaotic itinerancy yields computations with transient trajectories; in particular, there can be motion along separatrices as in winnerless competition dynamics 共Sec. IV.C兲. Although CI is an interesting phenomenon, applying it to explain and predict the activity of sensory systems 共Kay, 2003兲, and to any nonautonomous neural circuit dynamics, poses a question that has not been answered yet: How can CI be reproducible and robust against noise and at the same time sensitive to a stimulus? To conclude this section it is necessary to emphasize that the answer to the question of the functional role of chaos in real neural systems is still unclear. In spite of the attractiveness of such ideas as 共i兲 chaos makes neural circuits more flexible and adaptive, 共ii兲 chaotic dynamics create information and can help to store it 共see above兲, and 共iii兲 the nonlinear dynamical analyses of physiological data 共e.g., electroencephalogram time series兲 can be important for the prediction or control of pathological neural states, it is extremely difficult to confirm these ideas directly in in vivo or even in vitro experiments. In particular, there are three obstacles that can fundamenRev. Mod. Phys., Vol. 78, No. 4, October–December 2006 Synaptic transmission in many networks of the nervous system is dynamical, meaning that the magnitude of postsynaptic responses depends on the history of presynaptic activity 共Thompson and Deuchars, 1994; Fuhrmann et al., 2002兲. This phenomenon is independent of 共or in addition to兲 the plasticity mechanisms of the synapses 共discussed in Sec. II.A.3兲. The role of synapses is often considered to be the simple notification to the postsynaptic neuron of presynaptic cell activity. However, electrophysiological recordings show that synaptic transmission can imply activity-dependent changes in response to presynaptic spike trains. The magnitude of postsynaptic potentials can change rapidly from one spike to another, depending on the particular temporal distribution of the presynaptic signals. Thus each single postsynaptic response can encode information about the temporal properties of the presynaptic signals. The magnitude of the postsynaptic response is determined by the interspike intervals of the presynaptic activity and by the probabilistic nature of neurotransmitter release. In depressing synapses a short interval between presynaptic spikes is followed by small postsynaptic responses, while long presynaptic interspike intervals are followed by a large postsynaptic response. Facilitating synapses tend to generate responses that grow with successive presynaptic spikes. In this context, several theoretical efforts have tried to explore the capacity of single responses of dynamical synapses to encode temporal information about the timing of presynaptic events. Theoretical models for dynamical synapses are based on the time variation of the fraction of neurotransmitter released from the presynaptic terminal R共t兲, 0 艋 R共t兲 艋 1. When a presynaptic spike occurs at time tsp, the fraction U of available neurotransmitters and the recovery time constant rec determine the rate of return of resources R共t兲 to the available presynaptic pool. In a depressing synapse, U and rec are constant. A simple model describes the fraction of synaptic resources available for transmission as 共Fuhrmann et al., 2002兲 dR共t兲 1 − R共t兲 − UR共t兲␦共t − tsp兲, = dt rec 共17兲 and the amplitude of the postsynaptic response at time tsp is proportional to R共tsp兲. For a facilitating synapse, U becomes a function of time U共t兲 increasing at each presynaptic spike and decaying to the baseline level when there is no presynaptic activity: Rabinovich et al.: Dynamical principles in neuroscience FIG. 28. Dynamical synapses imply that synaptic transmission depends on previous presynaptic activity. This shows the average postsynaptic activity generated in response to a presynaptic spike train 共bottom trace兲 in a pyramidal neuron 共top trace兲 and in a model of a depressing synapse 共middle trace兲. Postsynaptic potential in the model is computed using a passive membrane mechanism m共dV / dt兲 = −V + RiIsyn共t兲, where Ri is the input resistance. Modified from Tsodyks and Markram, 1997. dU共t兲 U共t兲 + U1关1 − U共t兲兴␦共t − tsp兲, =− dt facil 共18兲 where U1 is a constant determining the step increase in U共t兲 and facil is the relaxation time constant of the facilitation. Other approaches to modeling dynamical synapses include probabilistic models to account for fluctuations in presynaptic release of neurotransmitters. At a synaptic connection with N release sites we can assume that at each site there can be, at most, one vesicle available for release, and that the release at each site is an independent event. When a presynaptic spike is produced at time tsp, each site containing a vesicle will release it with the same probability U共t兲. Once a release occurs, the site can be refilled during a time interval dt with probability dt / rec. The probabilistic release and recovery can be described by the probability Pv共t兲 for a vesicle to be available for release at any time t: dPv共t兲 1 − Pv共t兲 − U共t兲Pv共t兲␦共t − tsp兲. = dt rec 共19兲 Figure 28 shows how this formulation permits an accurate description of a depressing synapse in response to a specified presynaptic spike train. The transmission of sensory information from the environment to decision centers through neural communication channels requires a high degree of reliability and Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 1241 sensitivity from networks of heterogeneous, inaccurate, and sometimes unreliable components. The properties of the channel itself, assuming the sensor is accurate, must be richer than conventional channels studied in engineering applications. Those channels are passive and, when of high quality, can relay inputs accurately to a receiver. Neural communication channels are composed of dynamically active elements capable of complex autonomous oscillations. Individually, chaotic neurons can create information in a way similar to the study of nonlinear systems with unstable trajectories: Two states of the system, indistinguishable because only finiteresolution observations can occur, may through the action of the instabilities of the nonlinear dynamics find themselves in the future widely separated in state space, and thus distinguishable. Information about different states that was unavailable at one time may become available at a later time. Biological neural communication pathways are able to recover information from a hidden coding space and to transfer information from one time scale to another because of the intrinsic nonlinear dynamics of synapses. As an example, we discuss a very simple neural information channel composed of sensory input in the form of a spike train that arrives at a model neuron and then moves through a realistic dynamical synapse to a second neuron where the information in the initial sensory signal is read 共Eguia et al., 2000兲. The model neurons are four-dimensional generalizations of the HindmarshRose neuron, and a model of chemical synapse derived from first-order kinetics is used. The four-dimensional model neuron has a rich variety of dynamical behaviors, including periodic bursting, chaotic bursting, continuous spiking, and multistability. For many of these regimes, the parameters of the chemical synapse can be tuned so that the information about the stimulus, which is unreadable to the first neuron in the path, can be recovered by the dynamical activity of the synapse, and the second neuron can read it 共see Fig. 29兲. The quantitative description of this unexpected phenomenon was done by calculating the average mutual information I共S , N1兲 between the stimulus S and the response of the first neuron N1, and I共S , N2兲 between the stimulus and the response of the second neuron N2. The result in the example shown in Fig. 29 is I共S , N2兲 ⬎ I共S , N1兲. This result indicates how nonlinear synapses and neurons acting as input and output systems along a communication channel can recover information apparently hidden in earlier synaptic connections in the pathway. Here the measure of information transmission used is the average mutual information between elements, and because the channel is active and nonlinear, the average mutual information between the sensory source and the final neuron may be greater than the average mutual information found in an intermediate neuron in the channel 共but not greater than the original information兲. Another form of synaptic dynamics involved in information processing and especially in learning is STDP 共already discussed in Sec. II.A.3兲. Information transduc- 1242 Rabinovich et al.: Dynamical principles in neuroscience FIG. 29. Example of recovery of hidden information in neural channels. A presynaptic cell receives specified input and connects to a postsynaptic cell through a dynamical synapse. Top panel: the time series of synaptic input to the presynaptic cell J1共t兲; middle panel: the membrane potential of the first bursting neuron X1共t兲; bottom panel: the membrane potential of the second bursting neuron X2共t兲. Note that features of the input hidden in the response X1共t兲 are recovered in the response following a dynamical synapse X2共t兲 共note hyperpolarization regions for X2兲. Modified from Eguia et al., 2000. tion is influenced by STDP 共Chechik, 2003; Hopfield and Brody, 2004兲, which also plays an important role in binding and synchronization. D. Binding and synchronization We have discussed the diversity of neuron types and the variability of neural activity. Neural processing requires the fast interaction of many neurons in different neural subsystems. There are several dynamical mechanisms contributing to the complex integration of information that neural systems perform. Among them, the synchronization of neural activity is the one that has captured the most attention. Synchronization of neural activity is also one of the proposed solutions to a widely discussed question in neuroscience: the binding problem, which we describe briefly in this section. The binding problem was originally formulated as a theoretical problem by von der Malsburg in 1981 关see a review by von der Malsburg 共1999兲, and Roskies 共1999兲; Singer 共1999兲兴. However, examples of binding had already been proposed by Rosenblatt 共1962兲 for the visual system 关for a review of the binding problem in vision see Singer 共1999兲, and Wolfe and Cave 共1999兲兴. The binding problem is formulated as the need for a coherent representation of an object provided by the association of all Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 its features 共shape, color, location, speed, etc.兲. The association of all features or binding allows a unified perception of the object. The binding problem is a generalized task of the nervous system as it seeks to reconstruct any total perception from its components. There are also cognitive binding problems related to cognitive identification and memory. No doubt the binding problem, like many other problems in biology, has multiple solutions. These solutions are most likely implemented through the use of dynamical mechanisms for the control of neural activity. The most widely studied mechanism proposed to solve the binding problem is temporal synchrony 共or temporal correlation兲 共Singer and Gray, 1995兲. It has been suggested by von der Malsburg and Schneider 共1986兲 that synchronization is the basis for perceptual binding. However, there is still criticism of the temporal binding hypothesis 共Ghose and Maunsell, 1999; Riesenhuber and Poggio, 1999兲. Obviously, neural oscillations and synchronous signals are ubiquitous in the brain, and neural systems can make use of these phenomena to encode, learn, and create effective outputs. There are several lines of experimental evidence that reveal the use of synchronization and activity correlation for binding tasks. Figure 30 shows an example of how neural synchronization correlates with the perceptual segmentation of a complex visual pattern into distinct, spatially overlapping surfaces 共Castelo-Branco et al., 2000兲 共see the figure caption for details兲. Indeed, modeling studies show that involving time in these processes can lead to the binding of different features. The idea is to use the coincidence of certain events in the dynamics of different neural units for binding. Usually such dynamical binding is represented by synchronous neurons or neurons that are in phase with an external field. However, dynamical events such as phase or frequency variations usually are not very reproducible and robust. As discussed in the next section, it is reasonable to hypothesize that brain circuits displaying sequential switching of neural activity use the coincidence of this switching to implement dynamical binding of different WLC networks. Any spatiotemporal coding needs the temporal coordination of neural activity among different populations of neurons to provide 共i兲 better recognition of specific features, 共ii兲 faster processing, 共iii兲 higher information capacity, and 共iv兲 feature binding. Neural synchronization has been observed throughout the nervous system, particularly in sensory systems, for example, in the olfactory system 共Laurent and Davidowitz, 1994兲 and the visual system 共Gray et al., 1989兲. From the point of view of dynamical system theory, transient synchronization is an ideal mechanism for binding neurons into assemblies for several reasons: 共i兲 the synchronized neurons do not necessarily have to be neighbors; 共ii兲 a synchronization event depends on the state of the neuron and the stimulus and can be very selective, that is, neurons from the same network can be temporal members of different cell assemblies at different instants of time; 共iii兲 basic brain rhythms are able to synchronize neurons responsible for Rabinovich et al.: Dynamical principles in neuroscience FIG. 30. An example of binding showing dependence of synchrony on transparency conditions and receptive field 共RF兲 configuration in the cat visual cortex. 共a兲 Stimulus configuration. 共b兲 Synchronization between neurons with nonoverlapping RFs and similar directional preferences recorded from areas A18 and PMLS of the cat visual cortex. Left, RF constellation and tuning curves; right, cross correlograms for responses to a nontransparent 共left兲 and transparent plaid 共right兲 moving in the cells’ preferred direction. Grating luminance was asymmetric to enhance perceptual transparency. Small dark correlograms are shift predictors. 共c兲 Synchronization between neurons with different direction preferences recorded from A18 共polar and RF plots, left兲. Top, correlograms of responses evoked by a nontransparent 共left兲 and a transparent 共right兲 plaid moving in a direction intermediate to the cells’ preferences. Bottom, correlograms of responses evoked by a nontransparent plaid with reversed contrast conditions 共left兲, and by a surface defined by coherent motion of intersections 共right兲. Scale on polar plots: discharge rate in spikes per second. Scale on correlograms: abscissa, shift interval in ms, bin width 1 ms; ordinate, number of coincidences per trial, normalized. Modified from Castelo-Branco et al., 2000. the processing of information from different sensory inputs; and 共iv兲 the synchronization is possible even between neural oscillators with strongly different frequencies 共Rabinovich et al., 2006兲. In early visual processing neurons that encode features of a complex visual percept are associated in funcRev. Mod. Phys., Vol. 78, No. 4, October–December 2006 1243 tional assemblies through gamma-frequency synchronization 共Engel et al., 2001兲. When sensory stimuli are perceptually or attentionally selected, and the respective neurons are bound together to raise their saliency, then gamma-frequency synchronization among these neurons is also enhanced. Gamma-mediated coupling and its modulation by attention are not limited to the visual system: they are also found in the auditory 共Tiitinen et al., 1993兲 and somatosensory domains 共Desmedt and Tomberg, 1994兲. Gamma oscillations allow visiomotor binding between posterior and central brain regions 共Rodriguez et al., 1999兲 and are involved in short-term memory. As a means for dynamically binding neurons into assemblies, gamma-frequency synchronization appears to be the prime mechanism for stabilizing cortical connections among members of a neural assembly over time. On the other hand, neurons can increase or decrease the strength of their synaptic connections depending on the precise coincidence of their activation 共STDP兲, and gamma-frequency synchronization provides the required temporal precision. Hatsopoulos et al. 共2003兲 and Jackson et al. 共2003兲 revealed the functional significance of neural synchronization and correlations within the motor system. Preeminent among brain actions must be the aggregation of disparate spiking patterns to form spatially and temporally coherent neural codes that then drive alpha motor neurons and their associated muscles. Essentially, motor binding seems to describe exactly what motor structures of the mammalian brain do: provide high-level coordination of simple and complex voluntary movements. Neurons with similar functional output have an increased likelihood of exhibiting neural synchronization. In contrast to classical synchronization 共Pikovsky et al., 2001兲, synchronization in the CNS is always transient. The phase-space image of transient synchronization can be a saddle limit cycle in the vicinity of which the system spends finite time. Alternatively, it can be a limit cycle whose basin of attraction decreases in time. In both cases the system is able to leave the synchronization region after a specific stage of processing is completed and proceed with the next task. This is a broad area where the issues and approaches are not settled, and thus it provides an opportunity for innovative ideas to explain the phenomenon. To conclude this section, we note that the functional role of synchronization in the CNS and the importance of spike-timing coding in general are still a subject of debate. On the one hand, it is possible to build models that use dynamical patterns of spikes for neural computations, e.g., representation, recognition, and decision making. Examples of such spike-timing-based computational models have been discussed by Hopfield and Brody 共Hopfield and Brody, 2001; Brody and Hopfield, 2003兲. In this work the authors showed, in particular, that spike synchronization across many neurons can be achieved in the absence of direct synaptic interactions between neurons through phase locking to a common underlying oscillatory potential 共like gamma oscillation; see above兲. On the other hand, the real connections of 1244 Rabinovich et al.: Dynamical principles in neuroscience FIG. 31. 共Color online兲 Spontaneous spatiotemporal patterns observed in the neocortex in vitro under the action of carbachol. Images composed of optical signals recorded by eight detectors arranged horizontally. The optical signal from each detector was normalized to the maximum on that detector during that period and normalized values were assigned colors according to a linear color scale 共at the top right兲. The traces above images 2 and 5 are optical signals from two optical detectors labeled with this color scale. The x direction of the images represents time 共12 s兲 and the y direction of each image represents 2.6 mm of space in cortical tissue. Note also that the first spike had a high amplitude but propagated more slowly in the tissue. Modified from Bao and Wu, 2003. such theoretical models with experiments in vivo are not established 关see also Fell et al. 共2003兲 and O’Reilly et al. 共2003兲兴. IV. TRANSIENT DYNAMICS: GENERATION AND PROCESSING OF SEQUENCES A. Why sequences? The generation and control of sequences is of crucial importance in many aspects of animal life. Working memory, bird songs, finding food in a labyrinth, jumping from one stone to another on the shore—all these are the results of sequential activity generated by the nervous system. Lashley called the problem of coordination of constituent actions into organized sequential spatiotemporal patterns the action syntax problem 共Lashley, 1960兲. The generation of sequences is also important for intermediate information processing as we discuss below. The sequences can be cyclic, like many brain rhythms and spatiotemporal patterns generated by CPGs. They can also be irregular, like neocortical theta oscillations 共4 – 10 Hz兲 generated spontaneously in cortical networks 共Bao and Wu, 2003兲 共see Fig. 31兲. The sequences can be finite in time like those generated by a neural circuit under the action of external input as in sensory systems. From a physicist’s point of view, any reproducible finite sequence that is functionally meaningful results from the cooperative transient dynamics of the corresponding Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 neural ensemble or individual neurons. Even brain rhythms demonstrate transient dynamics because the circuit’s periodic activity is modulated by nonstationary sensory inputs or signals from the periphery. It is important to emphasize the fundamental role of inhibition in the generation and control of sequences in the nervous system. In this section we concentrate on the origin of sequence generation and the mechanisms of reproducibility, sensitivity, and functional reorganization of MCs. In the standard study of nonlinear dynamical systems, attention is focused on the long-time behavior of a system. This is typically not the relevant question in neuroscience. Here we must address the transient responses to a stimulus external to the neural system and must consider the short-term binding of a collection of responses, perhaps from different sensory inputs, to facilitate action commands directed to the motor system. If you attempt to swat a fly, it cannot ask you to perform this action many times so that it can average over your actions, allowing it to perform some standard optimal response. Few flies wanting this repetition would survive. B. Spatially ordered networks 1. Stimulus-dependent modes Many neural ensembles are anatomically organized as slightly inhomogeneous excitable media. Examples of such media are retina 共Tohya et al., 2003兲, IO network 共Leznik and Llinas, 2002兲, cortex 共Ichinohe et al., 2003兲, and thalamocortical layers 共Contreras et al., 1996兲. All these are neuronal lattices with chemical or electrical connections occurring primarily between neighbors. There are some general dynamical mechanisms of sequence generation in such spatially ordered networks. These mechanisms are usually related to the existence of wave modes such as those shown in Fig. 31 that are modulated by external inputs or stimuli. Many significant observational and modeling results for this subject are found in the visual system. Visual systems are organized differently for different classes of animals. For example, the mammalian visual cortex has several topographically organized representations of the visual field and neurons at adjacent points in the cortex are excited by stimuli presented at adjacent regions of the visual field. This indicates there is a continuous mapping of the coordinates of the visual field to the coordinates of the cortex 共van Essen, 1979兲. In contrast to such a mapping connections from the visual field to the visual cortex in the turtle, for example, are more complex: A local spot in the visual field activates many neurons in the cortex but in an ordered way. As a result the excitation of the turtle visual cortex is distributed and not localized, and this suggests the temporal dynamics of several interacting membrane modes 共see Fig. 32兲. In the mammalian cortex a moving stimulus evokes a localized wave or wave front, while in the turtle visual cortex a differentially moving stimulus modulates temporal interactions of the cortical modes differently and is represented by different sequential switchings between them. 1245 Rabinovich et al.: Dynamical principles in neuroscience FIG. 32. 共Color online兲 Sequential changing of cortical modes in the turtle visual cortex. Comparison between the spatial organization of the cortical activity in the turtle visual system and the normal modes of a rectangular membrane 共drum兲. From Senseman and Robbins, 1999. To understand the dynamics of the wave modes, i.e., stability, sensitivity to stimuli, dependence on neuromodulators, etc., one has to build a model that is based on the experimental information about the possibility of these modes maintaining the topological space structure observed in experiments. In many similar situations one can introduce cooperative or population variables that can be interpreted as the amplitude of such modes depending on time. The corresponding amplitude equations are essentially the widely studied evolution equations of the dynamical theory of pattern formation 共Cross and Hohenberg, 1993; Rabinovich et al., 2000兲. For an analysis of the wave mode dynamics of the turtle visual cortex Senseman and Robbins 共1999兲 used the Karhunen-Loeve decomposition and a snapshot of a spatiotemporal pattern at time t = t0 could be represented as a weighted sum of basic modes Mi共x , y兲 with coordinates 共x , y兲 on the image: N u共x,y,t 兲 = 兺 ai共t0兲Mi共x,y兲, 0 共20兲 i where u共x , y , t兲 represents the cooperative dynamics of these modes. The presentation of different visual stimuli, such as spots of light at different points in the visual field, produced spatiotemporal patterns represented by different trajectories in the phase space a1共t兲 , a2共t兲 , . . . , an共t兲. Du et al. 共2005兲 showed that it is possible to make a reduction in the dimensionality of the wave modes by a second Karhunen-Loeve decomposition, which maps in some time window the trajectory in 共ai兲 space into a point in a low-dimensional space 共see Fig. 33兲. The observed transient dynamics is similar to the experimental results on the representation of different odors in the invertebrate olfactory system 关see Fig. 46 and Galan et al. 共2004兲兴. Nenadic et al. 共2002兲 used a large-scale computer model of turtle visual cortex to reproduce qualitatively the features of the cortical mode dynamics seen in these experiments. It is remarkable that not only do spatiotemporal patterns evoked by a direct stimulus look like wave modes, but even spontaneous activity in the sensory cortex is well organized and very different from turbulent flow 共Arieli et al., 1996兲. This means that the common assumption about the stochastic and uncorrelated spontaRev. Mod. Phys., Vol. 78, No. 4, October–December 2006 FIG. 33. 共Color兲 Space representation of cortical responses in the turtle visual cortex to left, center, and right stimuli. From Du et al., 2005. neous activity of neighboring neurons in neural networks 关see, for example, van Vreeswijk and Sompolinsky 共1996兲; Amit and Brunel 共1997b兲兴 is not always correct. Local field potentials and recordings from single neurons indicate the presence of highly synchronous ongoing activity patterns or wave modes 共see Fig. 34兲. The spontaneous activity of a single neuron connected with others, in principle, can be reconstructed using the evoked patterns of network activity 共Tsodyks et al., 1999兲. There are some illustrative models of wave modes that we note here. In 1977 Amari 共1977兲 found spatially localized regions of high neural activity 共“bumps”兲 in network models consisting of a single layer of coupled excitatory and inhibitory rate neurons. Laing et al. 共2002兲 extended Amari’s results to a nonmonotonic connection function 共“Mexican hat” with oscillating tails兲 共shown in Fig. 35兲 and a neural layer in two spatial dimensions: u共x,y,t兲 = − u共x,y,t兲 t + 冕冕 ⍀ 共x − q,y − p兲f„u共q,p,t兲…dq dp, 共21兲 2 f共u兲 = 2e−/共u − th兲 ⌰共u − th兲, 共x,y兲 = e−b 冑x2+y2 共22兲 关b sin共冑x2 + y2兲 + cos共冑x2 + y2兲兴. 共23兲 An example of a typical localized mode in such neural media with local excitation and long-range inhibition is represented in Fig. 36. Different modes 共with different numbers of bumps兲 can be switched from one to another by transient external stimuli. Multiple items can be stored in this model because of the oscillating tails of the 1246 Rabinovich et al.: Dynamical principles in neuroscience FIG. 34. 共Color online兲 Relation between the spiking activity of a single neuron and the population state of cortical networks. 共a兲 From bottom to top: stimulus time course; correlation coefficient of the instantaneous snapshot of population activity with the spatial pattern obtained by averaging over all patterns observed at the times corresponding to spikes evoked by the optimal orientation of the stimulus called the neuron’s preferred cortical state 共PCS兲 pattern; observed spike train of evoked activity with the optimal orientation for that neuron; reconstructed spike train. The similarity between the reconstructed and observed spike trains is evident. Also, strong upswings in the values of correlation coefficients are evident each time the neuron emits bursts of action potentials. Every strong burst is followed by a marked downswing in the values of the correlation coefficients. 共b兲 The same as 共a兲, but for a spontaneous activity recording session from the same neuron 共eyes closed兲. 共c兲 The neuron’s PCS, calculated during evoked activity and used to obtain both 共a兲 and 共b兲. 共d兲 The cortical state corresponding to spontaneous action potentials. The two patterns are nearly identical 共correlation coefficient 0.81兲. 共e兲 and 共f兲 Another example of the similarity between the neuron’s PCS 共e兲 and the cortical state corresponding to spontaneous activity 共f兲 from a different cat obtained with the high-resolution imaging system 共correlation coefficient 0.74兲. Modified from Tsodyks et al., 1999. effective connection strength. This is the result of the common activity of the excitatory and inhibitory connections between neurons. Inhibition plays a crucial role for the stability of localized modes 共Laing et al., 2002兲. Localized modes with different numbers of bumps remind one of complex localized patterns in a dissipative nonequilibrium media 共Rabinovich et al., 2000兲. Based on this analogy, it is reasonable to hypothesize that different modes may coexist in a neural layer and their interaction and annihilation can explain the sequential effectiveness of the different events. This suggests they could be a model of sequential working memory 共see below兲. Many rhythms of the brain can take the form of waves: spindle waves 共7 – 14 Hz兲 seen at the onset of sleep 共Kim et al., 1995兲, slower delta rhythms of deeper sleep, the synchronous discharge during an epileptic sei- FIG. 35. Connection function 共x , y兲, centered at the center of the domain. Modified from Laing et al., 2002. FIG. 36. Six-bump stable solution of the model 共21兲–共23兲: b = 0.45, = 0.1, th= 1.5. Modified from Laing et al., 2002. Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 Rabinovich et al.: Dynamical principles in neuroscience zure 共Connors and Amitai, 1997兲, waves of excitation associated with sensory processing, 40-Hz oscillations, and others. In thalamocortical networks the same clusters of neurons are responsible for different modes of rhythmic activity. What is the dynamical origin of such multifunctionality? There is no unique answer to this question, and there are several different mechanisms that can be responsible for it 共we have already discussed this for small invertebrate networks; see Sec. II.B兲. Terman et al. 共1996兲 studied the transition between spindling and delta sleep rhythms. The authors showed that these two rhythms make different uses of the fast inhibition and slow inhibition generated by thalamic reticularis cells. These two types of inhibition are mediated in the cortex by GABA共A兲 and GABA共B兲 receptors, respectively 共Schutter, 2002; Tams et al., 2003兲. The wave mode equation discussed above is familiar to physicists and can be written both when interactions between neuron populations are homogeneous and isotropic 共Ermentrout, 1998兲 and when the neural layer is partitioned into domains or hypercolumns like the primary visual cortex 共V1兲 of cats and primates, which has a crystallinelike structure at the millimeter length scale 共Bressloff, 2002; Bressloff and Cowan, 2002兲. In the next section we discuss the propagation of patterns of synchronous activity along spatially ordered neural networks. 2. Localized synfire waves Auditory and visual sensory systems have a very high temporal resolution. For example, the retina is able to resolve sequential temporal patterns with a precision in the millisecond range. Does the transmission of sensory information from the periphery to the cortex maintain such high resolution? If the answer is yes, what are the dynamical mechanisms responsible for this? These questions are still open. There are several neurophysiological experiments that show the ability of neural systems to transmit temporarily modulated responses of sensory networks with high precision over several processing levels. For example, cross correlations between simultaneously recorded responses of retinal cells relay neurons within the thalamus, and cortical neurons show that the oscillatory patterning is reliably transmitted to the cortex with a resolution in the millisecond range 关see for reviews Singer 共1999兲 and Nase et al. 共2003兲兴. A similar phenomenon was observed by Kimpo et al. 共2003兲 who showed evidence for the preserved timing of spiking activity through multiple steps of a neural control loop in the bird brain. The dynamical origin of such precise message propagation, independent of the rate fluctuation, is often attributed to synchronization of the many neurons in the overall circuit 共Abeles, 1991; Diemann et al., 1999兲. We now discuss briefly the dynamics of waves of synchronous neural firing, i.e., synfire waves. One modeling study 共Diesmann et al., 1999兲 has shown that the stable propagation of localized synfire waves, short-lasting synchronous spiking activity, is possible along a sequence of Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 1247 FIG. 37. Sequence of pools of excitatory neurons, connected in a feedforward way by so-called divergent and convergent connections. The network is called a synfire chain if it supports the propagation of synchronous spike patterns. Modified from Gewaltig et al., 2001. layers or pools of neurons in a feedforward cortical network such as the one shown in Fig. 37, a synfire chain 共Abeles, 1991兲. The degree of temporal accuracy of spike times among the pools’ members determines whether subsequent pools can reproduce 共or even improve兲 this accuracy 关Fig. 38共a兲兴, or whether synchronous excitation disperses and eventually dies out as in Fig. 38共b兲 for a smaller number of spikes in the volley. Thus in the context of synfire network function the quality of timing is judged on whether synchronous spiking is sustained or whether it dies out. Diesmann et al. 共1999兲, Cateau and Fukai 共2001兲, Kistler and de Zeeuw 共2002兲, and Nowotny and Huerta 共2003兲 have shown that if the pool size is more than a critical value determined by the connectivity between layers, the wave activity initiated at the first pool propagates from one pool to the next, forming a synfire wave. Nowotny and Huerta 共2003兲 have theoretically proven that no other states exist beyond synchronized or unsynchronized volleys as shown in the experiments by Reyes 共2003兲. The synfire feedforward chain 共Fig. 37兲 is an oversimplified model for analyzing synfire waves because in reality any network with synfire chains is embedded in a larger cortical network that also has inhibitory neurons FIG. 38. Propagation of firing activity in synfire chains. 共a兲 Stable and 共b兲 unstable propagation of synchronous spiking in a model of cortical networks. Raster displays of propagating spike volley along fully connected synfire chain. Panels show the spikes in ten successive groups of 100 neurons each 共synaptic delays arbitrarily set to 5 ms兲. Initial spike volley 共not shown兲 was fully synchronized, containing 共a兲 50 or 共b兲 48 spikes. Modified from Diesmann et al., 1999. 1248 Rabinovich et al.: Dynamical principles in neuroscience and many recurrent connections. This problem is discussed in detail by Aviel et al. 共2003兲. C. Winnerless competition principle 1. Stimulus-dependent competition Here we consider a paradigm of sequence generation that does not depend on the geometrical structure of the neural ensemble in physical space. It can, for example, be a two-dimensional layer with connections between neighbors or a three-dimensional network with sparse random connections. This paradigm can be helpful for the explanation and prediction of many dynamical phenomena in neural networks with excitatory and inhibitory synaptic connections. The paradigm is called the winnerless competition principle. We have touched on aspects of WLC networks earlier, and here we expand on their properties and their possible use in neuroscience. “Survival of the fittest” is a cliché that is often associated with the term competition. However, competition is not merely a means of determining the winner, as in a winner-take-all network. It is also a multifunctional instrument that nature uses at all levels of the neuronal hierarchy. Competition is also a mechanism that maintains the highest level of variability and stability of neural dynamics, even if it is a transient behavior. Over two hundred years ago the mathematicians Borda and de Condorcet were interested in the process of plurality elections at the French Royal Academy of Sciences. They considered voting dynamics in a case of three candidates A, B, and C. If A beats B and B beats C in a head-to-head competition, we might reasonably expect A to beat C. Thus predicting the results of the election is easy. However, this is not always the case. It may happen that C beats A, resulting in a so-called Condorcet triangle, and there is no real winner in such a competitive process 共Borda, 1781; Saari, 1995兲. This example is also called a “voting paradox.” The dynamical image of this phenomenon is a robust heteroclinic cycle 共see Fig. 39兲. In some specific cases the heteroclinic cycle is even structurally stable 共Guckenheimer and Holmes, 1988; Krupa, 1997; Stone and Armbruster, 1999; Ashwin et al., 2003; Postlethwaite and Dawes, 2005兲. The competition without a winner is also known in hydrodynamics: Busse and Heikes discovered that convective roll patterns in a rotating plane layer exhibit sequential changes of the roll’s direction as a result of the competition between patterns with different roll orientations. No pattern becomes a winner and the system exhibits periodic or chaotic switching dynamics 共Busse and Heikes, 1980兲. For review see Rabinovich et al. 共2000兲. The same phenomenon has been discovered in a genetic system, i.e., in experiments with a synthetic network of three transcriptional regulators 共Elowitz and Leibler, 2000兲. Specifically, these authors described three repressor genes A, B, and C organized in a closed chain with unidirectional inhibitory connections such that A, B, and C beat each other. This network behaves like a Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 FIG. 39. 共Color online兲 Illustration of WLC dynamics. Top panel: Phase portrait corresponding to the autonomous WLC dynamics of a three-dimensional case. Bottom panel: Projection of a nine-dimensional heteroclinic orbit of three inhibitory coupled FitzHugh-Nagumo spiking neurons in a threedimensional space 共the variables 1, 3, 3 are linear combinations of the actual phase variables of the system兲. From Rabinovich et al., 2001. clock: it periodically induces synthesis of green fluorescent proteins as an indicator of the state of individual cells on a time scale of hours. In neural systems such clock competitive dynamics can result from the inhibitory connections among neurons. For example, Jefferys et al. 共1996兲 showed that hippocampal and neocortical networks of mutually inhibitory interneurons generate collective 40-Hz rhythms 共gamma oscillations兲 when excited tonically. Another example of neural competition without a winner was discussed by Ermentrout 共1992兲. The author studied the dynamics of a single inhibitory neuron connected to a small cluster of loosely coupled excitatory cells and observed the emergence of a limit cycle through a heteroclinic cycle. For autonomous dynamical systems competition without a winner is a well-known phenomenon. We use the term WLC principle for the nonautonomous transient dynamics of neural systems receiving external stimuli and exhibiting sequential switching among temporal winners. The main point of the WLC principle is the transformation of incoming inputs into spatiotemporal outputs based on the intrinsic switching dynamics of the neuronal ensemble 共see Fig. 40兲. In the phase space of the network, such switching dynamics are represented by a heteroclinic sequence whose architecture depends on the stimulus. Such a sequence consists of 1249 Rabinovich et al.: Dynamical principles in neuroscience FIG. 40. Transformation of the identity spatial input into spatiotemporal output based on the intrinsic sequential dynamics of a neural ensemble with WLC. many saddle equilibria or saddle cycles and many heteroclinic orbits connecting them, i.e., many separatrices. The sequence can serve as an attracting set if every semistable set has only one unstable direction 关see also Ashwin and Timme 共2005兲兴. The key points on which WLC networks are based are the following: 共i兲 the stimulus-dependent heteroclinic sequence corresponding to a specific order of switching has a large basin of attraction, i.e., the sequence is robust; and 共ii兲 the topology of the heteroclinic sequence sensitively depends on the incoming signals, i.e., WLC dynamics have a high resolution. In this manner stimulus-dependent sequential switching of neurons or groups of neurons 共clusters兲 is able to resolve the fundamental contradiction between sensitivity and robustness in sensory recognition. Any kind of sequential activity can be programmed, in principle, by a network with stimulus-dependent nonsymmetric inhibitory connections. It can be the creation of spatiotemporal patterns of motor activity, the transformation of the spatial information into spatiotemporal information for successful recognition 共see Fig. 40兲, and many other computations. The generation of sequences in inhibitory networks has already been discussed when we analyzed the dynamics of CPGs 共see Sec. II.B兲 focusing on rhythmic activity. The mathematical image in phase space of the rhythmic sequential switching shown in Figs. 8 and 9 is a limit cycle in the vicinity of the heteroclinic contour 关cf. Fig. 39共a兲兴. WLC dynamics can be described in the framework of neural models at different levels. These could be rate models, Hodgkin-Huxley-type models, or even simple map models 共see Table I兲. For spiking neurons or groups of synchronized spiking neurons in a network with nonsymmetrical lateral inhibition WLC may lead to switching between active and inactive states. The mathematical image of such switching activity is also a heteroclinic loop, but in this case the separatrices do not connect saddle equilibrium points 关Fig. 39共a兲兴 but saddle limit cycles as shown in Fig. 39共b兲. The WLC dynamics in a model network of nine spiking neurons with inhibitory connections is shown in Fig. 41. Similar results based on Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 FIG. 41. Spatiotemporal patterns generated by a network of nine FitzHugh-Nagumo neurons with inhibitory connections. The left and right panels correspond to two different stimuli. From Rabinovich et al., 2001. a map model of neurons have been reported by Casado 共2003兲. An important advantage of WLC networks is that they can produce different spatiotemporal patterns in response to different stimuli, and, remarkably, neurons spontaneously form synchronized clusters despite the absence of excitatory synaptic connections. For a discussion of synchronization with inhibition see also van Vreeswijk et al. 共1994兲 and Elson et al. 共2002兲. Finally WLC networks also possess a strikingly different capacity or ability to represent in a distinguishable manner a number of different patterns. In an attractor computation network of the Hopfield variety, a network with N attractors has been shown to have a capacity of approximately N / 7. In a simple WLC network with N nodes, this capacity has been shown 共Rabinovich et al., 2001兲 to be of order e共N − 1兲!, which is a remarkable gain in capacity. 2. Self-organized WLC networks It is generally accepted that there is insufficient genetic information available to account for all the synaptic connectivity in the brain. How then can the functional architecture of WLC circuits be generated in the process of development? One possible answer has been found by Huerta and Rabinovich. Starting with a model circuit consisting of 100 rate model neurons connected randomly with weak inhibitory synapses, new synaptic strengths are computed for the connections using Hebb learning rules in the presence of weak noise. The neuron rates ai共t兲 satisfy a Lotka-Volterra model familiar from our earlier discussion. In this case the matrix ij共t兲 is a dynamical variable: 冉 冊 dai共t兲 = ai共t兲 共S兲 − 兺 ij共t兲aj共t兲 + i共t兲. dt j 共24兲 共S兲 is a function dependent on the stimulus S, ij共t兲 are the strengths of the inhibitory connections determined 1250 Rabinovich et al.: Dynamical principles in neuroscience FIG. 42. 共Color兲 Result of simulating a network of 100 neurons subject to the learning rule g共ai , aj兲 = aiaj关10 tanh共aj − ai兲 + 1兴. The activity of representative neurons in this network is shown in different colors. The system starts from random initial conditions for the connections. The noise level is = 0.01. For simplicity, the switching activity of only four of the 100 neurons is shown. by some learning rules, and i共t兲 is Gaussian noise with 具i共t兲j共t⬘兲典 = ␦ij␦共t − t⬘兲. The learning is described by the equations dij共t兲 = ij共t兲g„ai共t兲,aj共t兲,S… − 关ij共t兲 − ␥兴, dt 共25兲 where g共ai , aj , S兲 represents the strengthening of interactions from neuron i to neuron j as a function of the external stimulus S. The parameter ␥ represents the lower bound of the coupling strengths among neurons. Figure 42 shows the activity of representative neurons in a network built with this model. After the selforganization phase, this network displays WLC switching dynamics. Winnerless competition dynamics can also be the result of local self-organization in networks of HH model neurons that display STDP with inhibitory synaptic connections as shown in Fig. 43. Such mechanisms of selforganization, as shown by Nowotny and Rabinovich, can be appropriate for networks that generate not only rhythmic activity but also transient heteroclinic sequences. 3. Stable heteroclinic sequence The phase-space image of nonrhythmic WLC dynamics is a trajectory in the vicinity of a stable heteroclinic sequence 共SHS兲 in the state space of the system. Such a sequence 共see Fig. 44兲 is an open chain of saddle fixed points connected by one-dimensional separatrices which retain nearby trajectories in its vicinity. The flexibility of WLC dynamics is provided by their dependence on the identity of participating neural clusters of stimuli. Sequence generation in chainlike or layerlike networks of neurons may result from a feedforward wavelike propagation of spikes like waves in synfire chains 共see above兲. In contrast, WLC dynamics does not need a specific spatial organization of the network. However, the image of a wave is a useful one, because in the case of WLC a wave of neural activity propagates in state space along the SHS. Such a wave is initiated by a stimulus. The Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 FIG. 43. 共Color online兲 Example of WLC dynamics entrained in a network by a local learning rule. In isolation, the four HH neurons in the network are rebound bursters, i.e., they fire a brief burst of spikes after being strongly inhibited. The all-toall inhibitory synapses in the small network are governed by a STDP learning rule which strengthens the synapse for positive time delays between postsynaptic and presynaptic activity and weakens it otherwise. Such STDP of inhibitory synapses has been observed in the entorhinal cortex of rats 共Haas et al., 2006兲. 共a兲 Before entrainment the neurons just follow the input signal of periodic current pulses. 共b兲 The resulting bursts strengthen the forward synapses corresponding to the input sequence making them eventually strong enough to cause rebound bursts. 共c兲 After entrainment activating any one of the neurons leads to an infinite repetition of the trained sequence carried by the successive rebound bursts of the neurons. speed of the sequential switching depends on the noise level . Noise controls the distance between trajectories realized by the system and the SHS. For trajectories that get closer to the SHS the time that the system spends near semistable states 共saddles兲, i.e., the interval between switching, becomes longer 共see Fig. 44兲. The mechanism of reproducing transient sequential neural activity has been analyzed by Aframovich, Zhigulin, et al. 共2004兲 共see Fig. 44兲. It is quite general and does not depend on the details of the neuronal model. Saddle points in the phase space of the neural network can be replaced by saddle limit cycles or even chaotic sets that describe neural activity in more detail, as in typical spiking or spiking-bursting models. This feature is important for neural modeling because it may help to build a bridge between the concepts of neural competition and synchronization of spikes. We can formulate the necessary conditions for the connectivity of a WLC network that must be satisfied in order for the network to exhibit reproducible sequential dynamics along the heteroclinic chain. As before, we base our discussion on the rate model Rabinovich et al.: Dynamical principles in neuroscience FIG. 44. A stable open heteroclinic sequence in a neural circuit with WLC. Wsi is a stable manifold of the ith saddle fixed point 共heavy dots兲. The trajectories in the vicinity of the SHS represent sequences with different timings. The time intervals between switches is proportional to T ⬃ 兩ln 兩 / u, where u is a positive Lyapunov exponent that characterizes the onedimensional unstable separatrices of the saddle points 共Stone and Holmes, 1990兲. Modified from Afraimovich, Zhigulin, et al., 2004. 冉 冊 Nl ai共t兲 = ai共t兲 i共Sជ l兲 − 兺 ij共Sជ l兲aj共t兲 + i共t兲, dt j 共26兲 where i共t兲 is an external Gaussian noise. In this model it is assumed that the stimulus Sជ l influences the matrix ij and increments i only in the subnetwork Nl. Each increment i controls the time constant of an initial exponential growth from the resting state ai共t兲 = 0. As shown by Aframovich, Zhigulin, et al. 共2004兲 to assure that the SHS is in the phase space of the system 共26兲 the following inequalities must be satisfied: ik−1 ik ik+1 ik ⬍ ik−1ik ⬍ ik−1 ik − 1 ⬍ ik+1ik ⬍ iik ⬎ ik−1ik + + 1, ik+1 ik i − ik−1 ik , . 共27兲 共28兲 共29兲 im is the increment of the mth saddle whose unstable manifold is one dimensional; ik±1ik is the strength of the inhibitory connection between neighboring saddles in the heteroclinic chain. The computer modeling result of a network with parameters that satisfy 共27兲–共29兲 is shown in Fig. 45. In the next section we discuss some experiments that support the SHS paradigm. 4. Relation to experiments The olfactory system may serve as one example of a neural system that generates transient, but trial-to-trial Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 1251 FIG. 45. 共Color online兲 Time series of the activity of a WLC network during ten trials 共only 20 neurons are shown兲: simulations of each trial were started from a different random initial condition. In this plot each neuron is represented by a different color and its level of activity by the saturation of the color. From Afraimovich, Zhigulin, et al., 2004. reproducible, sequences of neuronal activity which can be explained with the WLC principle. The complex intrinsic dynamics in the antennal lobe 共AL兲 of insects transform static sensory stimuli into spatiotemporal patterns of neural activity 共Laurent et al., 2001兲. Several experimental results about the reproducibility of the transient spatiotemporal AL dynamics have been published 共Stopfer et al., 2003; Galan et al., 2004; Mazor and Laurent, 2005兲 共see Fig. 46兲. In experiments described by Galan et al. 共2004兲 bees were presented with different odors, and neural activity in the AL was recorded using calcium imaging. The authors analyzed the transient trajectories in the projection neuron activity space and found that trajectories representing different trials of stimulation with the same odor were very similar. It was shown that after a time interval of about 800 ms different odors are represented in phase space by different static attractors, i.e., the transient spatiotemporal patterns converge to different spatial patterns of activity. However, the authors emphasize that due to the reproducibility of the transient dynamics some odors were recognized in the early transient stage as soon as 300 ms after the onset of the odor presentation. It is highly likely that the transient trajectories observed in these experiments represent realizations of a SHS. The generation of reproducible sequences plays also a key role in the high vocal center 共HVC兲 of the songbird system 共Hahnloser et al., 2002兲. Like a CPG, this neural system is able to generate sparse spatiotemporal patterns without any rhythmic stimuli in vitro 共Solis and Perkel, 2005兲. In its projections to the premotor nucleus RA, HVC in an awake singing bird sends sparse bursts of high-frequency signals once for each syllable of the song. These bursts have an interspike interval about 2 ms and last about 8 ms within a syllable time scale of 100– 200 ms. The bursts are shown for several HVC → RA projection neurons in Fig. 47. The HVC also contains many inhibitory interneurons 共Mooney and Prather, 2005兲. The interneurons burst densely throughout the vocalizations, in contrast to the bursting of the RA-projecting HVC neurons at single precise timings. A 1252 Rabinovich et al.: Dynamical principles in neuroscience FIG. 47. 共Color online兲 HVC songbird patterns. Spike raster plot of ten HVC共RA兲 neurons recorded in one bird during singing. Each row of tick marks shows spikes generated during one rendition of the song or call; roughly ten renditions are shown for each neuron. Modified from Hahnloser et al., 2002. FIG. 46. 共Color兲 Transient AL dynamics. Top panel: Trajectories of the antennal lobe activity during poststimulus relaxation in one bee. Modified from Galan et al., 2004. Bottom panel: Visualization of trajectories representing the response of a PN population in a locust AL over time. Time-slice points were calculated from 110 PN responses to four concentrations 共0.01, 0.05, 0.1, 1兲 of three odors, projected onto three dimensions using locally linear embedding, an algorithm that computes low-dimensional, neighborhood-preserving embeddings of high-dimensional inputs 共Roweis and Saul, 2000兲. Modified from Stopfer et al., 2003. plausible hypothesis is that HVC’s synaptic connections are nonsymmetric and WLC can be a mechanism of the neural spatiotemporal pattern generation of the song. This would provide a basis for the reproducible patterned output from the HVC when it receives a song command stimulus. D. Sequence learning Sequence learning and memory as sequence generation require temporal asymmetry in the system. Such asymmetry can result from specific properties of the network connections, in particular, asymmetry of the connections, or can result from temporal asymmetry in the dynamical features of individual neurons and synapses, or both. The specific dynamical mechanisms of sequence learning depend on the time scale of the sequence that Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 this neural system needs to learn. Learning of fast sequences, 20– 30 ms and faster, needs precise synchronization of the spikes or phases of neural waves. One possible mechanism for this can be the learning of synfire waves. For slow sequences, like autonomous repetitive behavior, it would be preferable to learn relevant behavioral events that typically occur on the time scale of hundreds of milliseconds or slower and the switching 共transitions兲 between them. Networks whose dynamics are based on WLC are able to do such a job. We consider here slow sequence learning and spatial sequential memory 共SSM兲. The idea is that sequential memory is encoded in a multidimensional dynamical system with a SHS. Each of the saddle points represents an event in a sequence to be remembered. Once the state of the system approaches one fixed point representing a certain event, it is drawn along an unstable separatrix toward the next fixed point, and the mechanism repeats itself. The necessary connections are formed in the learning phases by different sensory inputs originated by sequential events. Seliger et al. 共2003兲 have discussed a model of the SSM in the hippocampus. It is well accepted that the hippocampus plays the central role in acquisition and processing information related to representing motion in physical space. The most spectacular manifestation of this role is the existence of so-called place cells which repeatedly fire when an animal is in a certain spatial location 共O’Keefe and Dostrovsky, 1971兲. Experimental research also favors an alternative concept of spatial memory based on a linked collection of stored episodes 共Wilson and McNaughton, 1993兲. Each episode comprises a sequence of events, which, besides spatial locations, may include other features of the environment 共orientation, odor, sound, etc.兲. It is plausible to describe the corresponding learning with a population model that represents neural activity by rate coding. Seliger et al. 共2003兲 have proposed a two-layer dynamical model of SSM that can answer the following key questions: 共i兲 How is a certain event, e.g., an image of the environment, recorded in the structure of the synaptic connections be- Rabinovich et al.: Dynamical principles in neuroscience tween multiple sensory neurons 共SNs兲 and a single principal neuron 共PN兲 during learning? 共ii兲 What kind of cooperative dynamics forces individual PNs to fire sequentially, in a way that would correspond to a specific sequence of snapshots of the environment? 共iii兲 How complex should this network be in order to store a certain number of different episodes without mixing different events or storing spurious episodes? The two-layer structure of the SSM model is reminiscent of the projection network implementation of the normal form projection algorithm 共NFPA兲; see Baird and Eeckman 共1993兲. In the NFPA model, the dynamics of the network is cast in terms of normal form equations which are written for amplitudes of certain normal forms corresponding to different patterns stored in the system. The normal form dynamics can be chosen to follow certain dynamical rules. Baird and Eeckman 共1993兲 have shown that a Hopfield-type network with improved capacity can be built using this approach. Furthermore, it has been suggested 共Baird and Eeckman, 1993兲 that specific choices of the coupling matrix for the normal form dynamics can lead to multistability among more complex attracting sets than simple fixed points, such as limit cycles or even chaotic attractors. For example, quasiperiodic oscillations can be described by a normal form that corresponds to a multiple Hopf bifurcation 共Guckenheimer and Holmes, 1986兲. As shown below, a model of SSM after learning is completed can be viewed as a variant of the NFPA with a specific choice of normal form dynamics corresponding to winnerless competition among different patterns. To illustrate these ideas consider a two-level network of Ns SNs 关xi共t兲兴 and Np principal neurons 关ai共t兲兴. One can reasonably assume that sensory neurons do not have their own complex dynamics and are slaved either to external stimuli in the learning or storing regime or to the PNs in the retrieval regime. In the learning regime, xi共t兲 is a binary input pattern consisting of 0’s and 1’s. Np During the retrieval phase, xi共t兲 = 兺j=1 Pijaj共t兲, where Pij is the Ns ⫻ Np projection matrix of connections among SNs and PNs. The PNs are driven by SNs during the learning phase, but they also have their own dynamics controlled by inhibitory interconnections. When learning is complete, the direct driving from SNs is disconnected. The equations for the PN rates ai共t兲 read N N s p dai共t兲 = ai共t兲 − ai共t兲 兺 Vijaj共t兲 + ␣ai 兺 PTij xj共t兲 + 共t兲, dt j=1 j=1 共30兲 where ␣ ⫽ 0 in the learning phase and ␣ = 0 in the retrieval phase, and PTij is the projection matrix. The coupling between SNs and PNs is bidirectional. The last term on the right-hand side of Eq. 共30兲 represents small positive external perturbations which can input signals from other parts of the brain that control learning and retrieval dynamics. Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 1253 After a certain pattern is presented to the model, the sensory stimuli reset the state of the PN layer according NS T to the projection rule ai共t兲 = 兺j=1 Pij xj共t兲, but ai共t兲 change according to Eq. 共30兲. The dynamics of SNs and PNs during the learning and retrieval phases have two learning processes: 共i兲 forming the projection matrix Pij which is responsible for connecting a group of sensory neurons of the first layer corresponding to a certain stored pattern to a single PN which represents this pattern at the PN level; and 共ii兲 learning of the competition matrix Vij which is responsible for the temporal 共logical兲 ordering of the sequential memory. The slow learning dynamics of the projection matrix is controlled by the following equation: Ṗij = ⑀ai共␤xj − Pij兲 共31兲 with ⑀ Ⰶ 1. We assume that initially all Pij connections are nearly identical Pij = 1 + ij, where ij are small random perturbations, 兺jij = 0, 具2ij典 = 20 Ⰶ 1. Additionally, we assume that initially the matrix Vij is purely competitive: Vii = 1 and Vij = V0 ⬎ 1 for i ⫽ j. Suppose we want to memorize a certain pattern A in our projection matrix. We apply a set of inputs Ai corresponding to the pattern A of the SNs. As before, we assume that external stimuli render the SNs in one of two states: excited, Ai = 1, and quiescent, Ai = 0. The initial state of the PN layer is fully excited: ai共0兲 = 兺jPijAj. According to the competitive nature of interactions between PNs after a short transient, only one of them, the neuron A which corresponds to the maximum ai共0兲, remains excited and the others become quiescent. Which neuron becomes responsible for the pattern A is actually random, as it depends on the initial projection matrix Pij. It follows from Eq. 共31兲 that for small ⑀ synapses of suppressed PNs do not change, whereas synapses of the single excited neuron evolve such that connections between excited SNs and PNs neurons amplify toward ␤ ⬎ 1, and connections between excited PNs and quiescent SNs decay to zero. As a result, the first input pattern will be recorded in one of the matrix Pij rows, while other rows will remain almost unchanged. Now suppose that we want to record a second pattern different from the first one. We can repeat the procedure described above, namely, apply external stimuli associated with pattern B to the SNs, project them to the initial state of the PN layer, ai共0兲 = 兺jPijBj, and let the system evolve. Since synaptic connections from SNs suppressed by the first pattern to neuron A have been eliminated, a new set of stimuli corresponding to pattern B will excite neuron A more weakly than most of the others, and competition will lead to selection of one PN B different from neuron A. In this way we can record as many patterns as there are PNs. The sequential order of the patterns recorded in the projection network is determined by the competition matrix Vij, Eq. 共30兲. Initially it is set to Vij = V0 ⬎ 1 for i ⫽ j and Vii = 1 which corresponds to winner-take-all competition. The goal of sequential spatial learning is to 1254 Rabinovich et al.: Dynamical principles in neuroscience E. Sequences in complex systems with random connections FIG. 48. Amplitudes of principal neurons during the memory retrieval phase in a two-layer dynamical model of sequential spatial memory. 共a兲 Periodic retrieval, two different test patterns presented; 共b兲 aperiodic retrieval with modulated inhibition 共see text兲. Modified from Seliger et al., 2003. record the transition of pattern A to pattern B in the form of suppressing the competition matrix element VBA. We suppose that the slow dynamics of the nondiagonal elements of the competition matrix are controlled by the delay-differential equation V̇ij = ⑀ai共t兲aj共t − 兲共V1 − Vij兲, 共32兲 where is constant. Equation 共32兲 shows that only the matrix elements corresponding to ai共t兲 ⫽ 0 and aj共t − 兲 ⫽ 0 are changing toward the asymptotic value V1 ⬍ 1 corresponding to the desired transition. Since most of the time, except for short transients, only one PN is excited, only one of the connections Vij is changing at any time. As a result, an arbitrary, nonrepeating, sequence of patterns can be recorded. When a test pattern T is presented to the sensory layer, xi共0兲 = T共i兲 ai共0兲 = 兺iPijTTj, and T resembles one of the recorded patterns, this will initiate a periodic sequence of patterns corresponding to the previously recorded sequence in the network. Figure 48 shows the behavior of principal neurons after different initial patterns resembling different digits have been presented. In both cases, the system quickly settles onto a cyclic generation of patterns associated with a given test pattern. At any given time, except for a short transient time between patterns, only a single PN is on, corresponding to a particular pattern. Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 The level of cellular and network complexity in the nervous system leads one to ask: How do evolution and genetics build a complex brain? Comparative studies of the neocortex indicate that early mammalian neocortices were composed of only a few cortical fields and in primates the neocortex expanded dramatically; the number of cortical fields increased and the connectivity between them became very complex. The architecture of the microcircuitry of the mammalian neocortex remains largely unknown in terms of cell-to-cell connections; however, the connections of groups of neurons with other groups are becoming better understood thanks to new anatomical techniques and the use of slice techniques. Many parts of the neocortex developed under strict genetic control as precise networks with connections that appear similar from animal to animal. Kozloski et al. 共2001兲 discussed visual networks in this context. However, the local connectivity can be probabilistic or random as a consequence of experience-dependent plasticity and self-organization 共Chklovskii et al., 2004兲. In particular, the imaging of individual pyramidal neurons in the mouse barrel cortex over a period of weeks 共Maravall et al., 2004兲 showed that sensory experience drives the formation and elimination of synapses and that these changes might underlie adaptive remodeling of neural circuits. Thus the brain appears as a compromise between existing genetic constraints and the need to adapt, i.e., networks are formed by both genetics and activitydependent or self-organizing mechanisms. This makes it very difficult to determine the principles of network architecture and to build reasonable dynamical models that are able to predict the reactions of a complex neural system to changes in the environment; we have to take into account that even self-organized networks are under genetic control but in a different sense. For example, genetics can control the average balance between excitatory and inhibitory synaptic connections, sparseness of the connections, etc. The point of view that the infant cortex is not a completely organized machine is based on the supposition that there is insufficient storage capacity in the DNA to control every neuron and every synapse. This idea was formulated first by Alan Turing in 1948 共Ince, 1992兲. A simple calculation reveals that the total size of the human genome can specify the connectivity of about 105 neurons. The human brain actually contains around 1011 neurons. Let us say that we have N neurons. Each neuron requires Np log2N bits to completely specify its connections, where p is the average number of connections. Therefore we need at least N2p log2N bits to specify the entire on-off connectivity matrix of N neurons. If the connectivity degree p is not very sparse then we just need N2 bits. So, if we solve min共N2 , N2p log2N兲 = 3.3 ⫻ 109 base pairs in the human genome using a connectivity degree of 1%, we obtain a maximum of 105 neurons that can be completely specified. Since we do not know how much of the genome is used for brain connec- 1255 Rabinovich et al.: Dynamical principles in neuroscience tivity, it is not possible to narrow down the estimation. Nevertheless, it does not make sense to expect the whole genome to specify all connections in the brain. This simple estimate makes clear that learning and synaptic plasticity have a very important role in determining the connectivity of brain circuits. The dynamics of complex network models are difficult to dissect. The mapping of the corresponding local and global bifurcations in a low-dimensional system has been extensively studied. To perform such analysis in highdimensional systems is very demanding if not impossible. Average measures, such as mean firing rates, average membrane potential, correlations, etc., can help us to understand the dynamics of the network as a function of a few variables. One of the first models to use a meanfield approach was the Wilson-Cowan model 共Wilson and Cowan, 1973兲. Individual neurons in the model resemble integrate-and-fire neurons with a membrane integration time and a refractory period r. Wilson and Cowan’s main hypothesis is that the unreliable individual responses, when grouped together, can lead to more reliable operations. The Wilson-Cowan formalism can be reduced to the following equations: E共x,t兲 = − E共x,t兲 + 关1 − rE共x,t兲兴 t ⫻ Le − 冕 冋冕 E共y,t兲wee共y,x兲dy 册 I共y,t兲wei共y,x兲dy + Se共x,t兲 , − 冕 冋冕 dxi共t兲 =⌰ dt 共33兲 E共y,t兲wie共y,x兲dy 册 I共y,t兲wii共y,x兲dy + Si共x,t兲 , 共34兲 where E共x , t兲 and I共x , t兲 are the proportions of firing neurons in the excitatory and inhibitory population, the coordinate x is a continuous variable that represents the position in the cortical surface, wee, wei, wie, and wii are the connectivity weights, and Se and Si are external inputs to the excitatory and inhibitory populations, respectively. The gain functions Le and Li basically reflect the expected proportions of excitatory and inhibitory neurons receiving at least threshold excitation per unit of time. One subtle trick used in the derivation of this model is that the membrane integration time is introduced through synaptic connections. The model expressed in this form attempts to eliminate the uncertainty of single neurons by grouping them according to those with reliable common responses. We are still left with the problem of what to expect in a network of clusters connected randomly to each other. Here we will discuss it in more detail. Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 冉兺 NE j=1 冊 NI wEE ij xj共t兲 E − 兺 wEI − xi共t兲, ij yj共t兲 + Si j=1 共35兲 I共x,t兲 = − I共x,t兲 + 关1 − rI共x,t兲兴 t ⫻ Li In a random network of excitatory and inhibitory neurons, it is not uncommon to find oscillatory activity 共Jin, 2002; Huerta and Rabinovich, 2004兲. However, it is more interesting to study the transient behavior of neural recurrent networks. These are fast behaviors and important for sensory processing and for the control of motor commands. In studying this one needs to address two main issues: 共i兲 whether it is possible to consistently find networks with random connections, described by equations similar to Eqs. 共33兲 and 共34兲, behaving regularly, and 共ii兲 whether transient behavior in these networks is reproducible. Huerta and Rabinovich 共2004兲 showed, using the Wilson-Cowan formalism, periodic sequential activity 共limit cycles兲 is more likely to be found in regions of the control parameter space where inhibitory and excitatory synapses are slightly out of balance. However, reproducible transient dynamics is more likely found in the region of parameter space far from balanced excitation and inhibition. In particular, the authors investigated the model dyi共t兲 =⌰ dt 冉 NE NI j=1 j=1 冊 兺 wIEij xj共t兲 − 兺 wIIij yj共t兲 + SIi − yi共t兲, 共36兲 where x共t兲 and yi共t兲 represent the fractions of active neurons in cluster i of the excitatory and inhibitory populations, respectively. The numbers of excitatory and inhibitory clusters are NE and NI. The labels E and I are used to denote quantities associated with the excitatory or inhibitory populations, respectively. The external inputs SE,I are instantaneous kicks applied to a fraction of the total population at time zero. The gain function is ⌰共z兲 = 兵tanh关共z − b兲 / 兴 + 1其 / 2, with a threshold b = 0.1 below the excitatory and inhibitory synaptic strength of a single connection. Clusters are taken to have very sharp thresholds of excitability by choosing = 0.01. There is a wide range of values that generates similar results. The time scale is set as done by Wilson and Cowan 共1973兲, = 10 ms. The connectivity matrices wXY have entries ij drawn from a Bernoulli process 共Huerta and Rabinovich, 2004兲. The main control parameters in this problem are the probabilities of connections from population to population. Now we can answer the following question: What kind of activity can a network with many neurons and random connections produce? Intuition suggests that the answer has to be a complex multidimensional dynamics. However, this is not the case 共Fig. 49兲: most observable stimulus-dependent dynamics are more simple and reproducible; periodic, transient, or chaotic 共also low dimensional兲. 1256 Rabinovich et al.: Dynamical principles in neuroscience FIG. 50. 共Color兲 Spatiotemporal patterns of coordinated rhythms induced by stimuli in a model of the inferior olive. Several structures with different frequencies can coexist simultaneously in a commensurate representation of the spiking frequencies when several stimuli are present. Incommensurate stimuli are introduced in the form of current injections in different clusters of the network. 共Panels on the right show the positions of the input clusters.兲 These current injections induce different spiking frequencies in the neurons. Colors in these panels represent different current injections, and thus different spiking frequencies in the input clusters. Top row shows the activity of a network with two different input clusters. Bottom row shows the activity of a network with 25 different input clusters. Sequences develop in time from left to right. Regions with the same color have synchronous behavior. Color bar maps the membrane potential. Red corresponds to spiking neurons 共−45 mV is above the firing threshold in the model兲. Dark blue means hyperpolarized activity. Bottom panel shows the activity of a single neuron with subthreshold oscillations and spiking activity. Modified from Varona, Aguirre, et al., 2002. FIG. 49. 共Color online兲 Three-dimensional projections of simulations of random networks of 200 neurons. For illustrative purposes we show three types of dynamics that can be generated by a random network: 共top兲 chaos, 共middle兲 limit cycle 共both in the areas of parameter space that are close to balanced兲, and 共bottom兲 transient dynamics 共far from balanced兲. This is a very important point for understanding cortex dynamics that involves the cooperative activity of many complex networks 共units or microcircuits兲. From the functional point of view, the stimulus-dependent dynamics of the cortex can be considered as a coordinated behavior of many units with low-dimensional transient dynamics. This is the basis of a new approach to cortex modeling named the “liquid-state machine” 共Maass et al., 2002兲. F. Coordination of sequential activity Coordination of different sequential behaviors is crucially important for survival. From the modeling point of view it is a very complex problem. The IO 共a network Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 already discussed in Sec. III.B.2兲 has been suggested as a system that coordinates motor voluntary movements involving several simultaneous rhythms 共Llinás and Welsh, 1993兲. Here an example of how subthreshold oscillations coordinate different incommensurate rhythms in a commensurate fashion is shown. In the IO, neurons are electrically coupled to their close neighbors. Their activity is characterized by subthreshold oscillations and spiking activity 共see Fig. 50兲. The cooperative dynamics of the IO under the action of several incommensurate inputs has been modeled by Varona, Aguirre, et al. 共2002兲. The results of these large-network simulations show that the electrical coupling of IO neurons produces quasisynchronized subthreshold oscillations. Because spiking activity can happen only on top of these oscillations, incommensurate inputs can produce regions with different commensurate spiking frequencies. Several spiking frequencies are able to coexist in these networks. The coexistence of different rhythms is related to the different clusterization of the spatiotemporal patterns. Another important question related to coordination of several sequential behaviors concerns the dynamical principles that can be a basis for fast neuronal planning and reaction to a changing environment. One might Rabinovich et al.: Dynamical principles in neuroscience think that the WLC principle can be generalized in order to organize the sequential switching according to 共i兲 the learned skill and 共ii兲 the dynamical sensory inputs. The corresponding mathematical model might be similar to Eqs. 共26兲–共29兲 together with a learning rule similar to Eq. 共25兲. Stimuli Sl change sequentially and the timing of each step 共the time that the system spends moving from the vicinity of one saddle to the vicinity of the next one; see Fig. 51兲 should be coordinated with the time of change in the environment. In recurrent networks, as a result of learning, the stimulus can go sequentially to the specific goal of an optimal heteroclinic sequence among many such sequences that exist in the phase space of the model. What is important is that at the same time, i.e., in parallel with the choosing of the rest of the motor plan, the already existing part of the motor activity plan is executed. The two ideas just discussed can be applied to the cerebellar circuit, which is an example of a complex recurrent network 共see Fig. 52兲. To give an impression of the complexity of the cerebellar cortex we note that it is organized into three layers: the molecular layer, the Purkinje cell layer, and the granule cell layer. Only two significant inputs reach the cerebellar cortex: mossy fibers and climbing fibers. Mossy fibers are in the majority 共4:1兲 and carry a wealth of sensory and contextual information of multiple modalities. They make specialized excitatory synapses in structures called “glomeruli” with the dendrites of numerous granule cells. Granule cell axons form parallel fibers that run transversely in the molecular layer, making excitatory synapses with Purkinje cells. Each Purkinje cell receives ⬇150 000 synapses. These synapses are thought to be major storage sites for the information acquired during motor learning. The Purkinje cell axon provides the only output from the cerebellar cortex. This is via the deep cerebellar nuclei. Each Purkinje cell receives just one climbing fiber input from the inferior olive, but this input is very powerful because it involves several hundreds of synaptic contacts. The climbing fiber is thought to have a role in teaching in the cerebellum. The Golgi cell is excited by mossy fibers and granule cells and exercises an inhibitory feedback control upon granule cell activity. Stellate and basket cells are excited by parallel fibers in order to provide feedforward inhibition to Purkinje cells. The huge number of inhibitory neurons and the architecture of the cerebellar networks 共de Zeeuw et al., 1998兲 support the generalized WLC mechanism for coordination. A widely discussed hypothesis is that the specific circuitry of the IO, cerebellar cortex, and deep cerebellar nuclei called the slow loop 共see Fig. 52兲 can serve as a dynamical working memory or as a neuronal clock with ⬇100-ms cycle time which would make it easy to connect it to behavioral time scales 共Kistler and de Zeeuw, 2002; Melamed et al., 2004兲. Temporal coordination and, in particular, synchronization of neural activity is a robust phenomenon, frequently observed across populations of neurons with diverse membrane properties and intrinsic frequencies. In the light of such diversity the question of how precise Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 1257 FIG. 51. Illustration of the learned sequential switching in a recurrent network with WLC dynamics: Thin lines, possible learned sequences; thick line, sequential switching chosen online by the dynamical stimulus. synchronization can be achieved in heterogeneous networks is critical. Several mechanisms have been suggested and many of them require an unreasonably high degree of network homogeneity or very strong connectivity to achieve coherent neural activity. As discussed above 共Sec. II.A.4兲, in a network of two synaptically coupled neurons STDP at the synapse leads to the dynamical self-adaptation of the synaptic conductance to a value that is optimal for the entrainment of the postsynaptic neuron. It is interesting to note that just a few STDP synapses are able to make the entrainment of a FIG. 52. A schematic representation of the mammalian cerebellar circuit. Arrows indicate the direction of transmission across each synapse. Sources of mossy fibers: Ba, basket cell; BR, brush cell; cf, climbing fiber; CN, cerebellar nuclei; Go, Golgi cell; IO, inferior olive; mf, mossy fiber; pf, parallel fiber; PN, pontine nuclei; sb and smb, spiny and smooth branches of P cell dendrites, respectively; PC, Purkinje cell; bat, basket cell terminal; pcc, P cell collateral; no, nucleo-olivary pathway; nc, collateral of nuclear relay cell. Modified from Voogd and Glickstein, 1998. 1258 Rabinovich et al.: Dynamical principles in neuroscience heterogeneous network of electrically coupled neurons more effective 共Zhigulin and Rabinovich, 2004兲. It has been shown that such a network oscillates with a much higher degree of coherence than when it is subject to stimulation that is mediated by STDP synapses as compared with stimulation through static synapses. The observed phenomenon depends on the number of stimulated neurons, the strength of electrical coupling, and the degree of heterogeneity. In reality, long-term plasticity depends not only on spike timing 共STDP兲 but also on the firing rate and the cooperativity among different neuronal inputs 共Sjöström et al., 2001兲. This makes modeling self-organization and learning more challenging. Real behavior in nonstationary or complex environments, as already discussed, requires switching between different sequential activities. Jancke et al. 共2000兲 have identified distributed regions in different parts of the cortex that are involved in the switching among sequential movements. It is important for dynamical modeling that this differential pattern of activation is not seen for simple repetitive movements. Thus such movements are too simple to evoke additional activation. This means that a dynamical model that aims to describe the sequential behavior in general has to correctly describe the switching from a low-dimensional subspace to a high-dimensional state space, and vice versa. There are no general methods for describing multidimensional dissipative nonlinear systems with such transient but reproducible dynamics. We think that the WLC principle might be the first step in this direction. V. CONCLUSION Physicists, mathematicians, and physiologists all agree that an important attribute of any dynamical model of CNS activity is that not only should it be able to fit the available anatomical and physiological data, but it should also be capable of explaining function and predicting behavior. However, the ways in which physicists and mathematicians, on one hand, and physiologists, on the other hand, use modeling are based on their own experience and views and thus are different. In this review we tried to bring these different viewpoints closer together and, using many examples from the sensory, motor, and central nervous systems, discussed just a few principles like reproducibility, adaptability, robustness, and sensitivity. Let us return to the questions formulated at the beginning of the review: • What can nonlinear dynamics learn from neuroscience? • What can dynamics? neuroscience learn from nonlinear After reading this review, we hope the reader can join us in integrating the key messages in our presentation. Perhaps we may offer our compact formulation. Addressing the first question of what nonlinear dynamics can learn from neuroscience: Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 • The most important activities of neuronal systems are transient and cannot be understood by analyzing attractor dynamics alone. These need to be augmented by reliable descriptions of stimulusdependent transient motions in state space as this comprises the heart of most neurobiological activity. Nonetheless, because the dynamics of realistic neuronal models are strongly dissipative, their stimulusdependent transient behavior is strongly attracted to some low-dimensional manifolds embedded in the high-dimensional state space of the neural network. It is a strong stimulus to nonlinear dynamics to develop a theory of reasonably low-dimensional transient activity and, in particular, to consider the local and global bifurcations of such objects as homoclinic and heteroclinic trajectories. • For many dynamical problems of neuroscience, in contrast to traditional dynamical approaches, the initial conditions do matter crucially. Persistent neuronal activity 共i.e., dynamical memory兲, stimulusdependent transient competition, stimulusdependent transient synchronization, and stimulusdependent synaptic plasticity are all aspects of this. Clearly, addressing these important phenomena will require an expansion in our approaches to dynamical systems. Addressing the second question of what neuroscience can learn from nonlinear dynamics: • Dynamical models confirm the key role of inhibition in neuronal systems. The function of inhibition is not just to organize a balance with excitation in order to stabilize a network but much more: 共a兲 inhibitory networks can generate rhythms, such as reproducible and adaptive motor rhythms in CPGs, or gamma rhythms in the brain; 共b兲 they are responsible for the transformation of an identity sensory code to a spatiotemporal code important for better recognition in an acoustically cluttered environment; and 共c兲 thanks to inhibition, neural systems can be at the same time very sensitive to their input and robust against noise. • Dynamical chaos is not just a fundamental phenomenon but also important for the survival of living organisms. Neuronal systems may use chaos for the organization of nontrivial behavior such as the irregular hunting-swimming of Clione and for the organization of higher brain functions. • The improvement in yield, stability, and longevity of multielectrode recordings, new imaging techniques, combined with new data processing methods, have allowed neurophysiologists to describe brain activities as the dynamics of spatiotemporal patterns in some virtual space. We think this is a basis for building a bridge between transient large-scale brain activity and animal behavior. And finally as we pursue the investigation of dynamical principles in neuroscience, we hope that eventually not to see these two questions apart from one another Rabinovich et al.: Dynamical principles in neuroscience but as an integrated approach to deep and complex scientific problems. ACKNOWLEDGMENTS We thank Ramon Huerta and Thomas Nowotny for their help, and Rafael Levi and Valentin Zhigulin for useful comments. This work was supported by NSF Grant No. NSF/EIA-0130708, and Grant No. PHY 0414174; NIH Grant No. 1 R01 NS50945 and Grant No. NS40110; MEC BFI2003-07276, and Fundación BBVA. GLOSSARY AL AMPA receptors bumps CA1 carbachol Clione CNS CPG depolarization dynamic clamp GABA heteroclinic loop heteroclinic trajectory antennal lobe, the first site of sensory integration from the olfactory receptors of insects. transmembrane receptor for the neurotransmitter glutamate that mediates fast synaptic transmission. spatially localized regions of high neural activity. subsystem of the hippocampus with a very active role in general memory. chemical that induces oscillations in in vitro preparations. marine mollusk whose nervous system is frequently used in neurophysiology studies. central nervous system. central pattern generator, a small neural circuit that can produce stereotyped rhythmic outputs without rhythmic sensory or central input. any change in the neuron membrane potential that makes it more positive than when the cell is in its resting state. a computer setup to insert virtual conductances into a neural membrane typically used to add synaptic input to a cell by calculating the response current to a specific presynaptic input. neurotransmitter of typically inhibitory synapses; they can be mediated by fast GABA共A兲 or slow GABA共B兲 receptors. a closed chain of heteroclinic trajectories. trajectory that lies simultaneously on the stable manifold of one saddle point 共or limit cycle兲 and the unstable manifold of another saddle 共or limit cycle兲 connecting them. Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 HH 1259 Hodgkin-Huxley neuron model. HVC high vocal center in the brain of songbirds. hyperpolarization any change in the neuron membrane potential that makes it more negative than when the cell is in its resting state. interneurons neurons whose axons remain within a particular brain region as contrasted with projection neurons, which have axons projecting to other brain regions, or with motoneurons, which innervate muscles. IO inferior olive, a neural system that is an input to the cerebellar cortex presumably involved in motor coordination. KCs Kenyon cells, interneurons of the mushroom body of insects. Kolmogorov-Sinai a measure of the degree of preentropy dictability of further states visited by a chaotic trajectory started within a small region in a state space. LP lateral pyloric neuron of the crustacean stomatogastric CPG. LTD long-term depression, activitydependent decrease of synaptic efficacy transmission. LTM long-term memory. LTP long-term potentiation, activity-dependent reinforcement of synaptic efficacy transmission. Lyapunov exponents j the rate of exponential divergence from perturbed initial conditions in the jth direction of the state space. For trajectories belonging to a strange attractor the spectrum j is independent of initial conditions and characterizes the stable chaotic behavior. MCs microcircuits; circuits composed of a small number of neurons that perform specific operational tasks. mushroom body lobed subsystem of the insect brain involved in classification, learning, and memory of odors. mutual information a measure of the independence of two signals X and Y, i.e., the information of X that is shared by Y. In the discrete case, if the joint probability density function of X and Y is p共x , y兲 = P共X = x , Y = y兲, the probability 1260 neuromodulators neurotransmitters pacemaker PD phase synchronization 共locking兲 place cell plasticity PN Purkinje cell RA receptor receptor neuron SHS SN SSM statocyst STDP STM structural stability synapse Rabinovich et al.: Dynamical principles in neuroscience density function of X alone is f共x兲 = P共X = x兲, and the probability density function of Y alone is g共y兲 = P共Y = y兲, then the mutual information of X and Y is given by I共X , Y兲 = 兺x,yp共x , y兲log2关p共x , y兲 / f共x兲g共y兲兴. a substance other than a neurotransmitter, released by neurons that can affect the intrinsic and synaptic dynamics of other neurons. chemicals that are used to relay at the synapses the signals between neurons. neuron or circuit that has endogenous rhythmic activity. pyloric dilator neuron of the crustacean CPG. the onset of a certain relationship between the phases of coupled self-sustained oscillators. a type of neuron found in the hippocampus that fires strongly when an animal is in a specific location in an environment. changes that occur in the organization of synaptic connections or intracellular dynamics. projection or principal neurons. main cell type of the cerebellar cortex. premotor nucleus of the songbird brain. a protein on the cell membrane that binds to a neurotransmitter, neuromodulator, or other substance, and initiates the cellular response to the ligand. sensory neuron. stable heteroclinic sequence. sensory neuron. sequential spatial memory. balance organ in some invertebrates that consists of a spherelike structure containing a mineralized mass 共statolith兲 and several sensory neurons also called statocyst receptors. spike-timing-dependent plasticity. short-term memory. condition in which small changes in the parameters do not change the topology of the phase portrait in the state space. specialized junction through which neurons signal to one an- Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 synfire chain WLC other. There are at least three different types of synapses: excitatory and inhibitory chemical synapses and electrical synapses or gap junctions. propagation of synchronous spiking activity in a sequence of layers of neurons belonging to a feedforward network. winnerless-competition principle for the nonautonomous transient dynamics of neural systems receiving external stimuli and exhibiting sequential switching among temporal “winners.” REFERENCES Abarbanel, H., R. Huerta, and M. I. Rabinovich, 2002, Proc. Natl. Acad. Sci. U.S.A. 99, 10132. Abarbanel, H. D. I., 1997, Analysis of Observed Chaotic Data 共Springer, New York兲. Abarbanel, H. D. I., R. Huerta, M. I. Rabinovich, N. F. Rulkov, P. F. Rowat, and A. I. Selverston, 1996, Neural Comput. 8, 1567. Abarbanel, H. D. I., M. Rabinovich, and M. Sushchik, 1993, Introduction to Nonlinear Dynamics for Physicists 共World Scientific, Singapore兲. Abeles, M., 1991, Corticonics: Neural Circuits of the Cerebral Cortex 共Cambridge University Press, Cambridge, England兲. Abeles, M., H. Bergman, E. Margalit, and E. Vaadia, 1993, J. Neurophysiol. 79, 1629. Adrian, E. D., and Y. Zotterman, 1926, J. Physiol. 共London兲 61, 151. Afraimovich, V., V. Zhigulin, and M. Rabinovich, 2004, Chaos 14, 1123. Afraimovich, V. S., M. I. Rabinovich, and P. Varona, 2004, Int. J. Bifurcation Chaos Appl. Sci. Eng. 14, 1195. Aihara, K., 2002, Proc. IEEE 90, 919. Amari, S., 1977, Biol. Cybern. 27, 77. Amit, D., and N. Brunel, 1997a, Cereb. Cortex 7, 237. Amit, D. J., and N. Brunel, 1997b, Cereb. Cortex 7, 237. Andronov, A., 1933, in The First All-Union Conference on Auto-oscillations 共GTTI, Moscow Leningrad兲, pp. 32–71. Andronov, A., A. Vitt, and S. Khaikin, 1949, Theory of Oscillations 共Princeton University Press, Princeton, NJ兲. Andronov, A. A., and L. Pontryagin, 1937, Dokl. Akad. Nauk SSSR 14, 247. Arbib, M. A., P. Érdi, and J. Szentágothai, 1997, Neural Organization: Structure, Function, and Dynamics 共Bradford Book/ MIT, Cambridge, MA兲. Arieli, A., A. Sterkin, A. Grinvald, and A. Aertsen, 1996, Science 273, 1868. Arnold, V., V. Afrajmovich, Y. Ilyashenko, L. Shilnikov, and N. Kazarinoff, 1999, Bifurcation Theory and Catastrophe Theory 共Springer, New York兲. Ashby, W. R., 1960, Design for a Brain, 2nd ed. 共Wiley, New York兲. Ashwin, P., M. Field, A. Rucklidge, and R. Sturman, 2003, Chaos 13, 973. Aswin, P., and M. Timme, 2005, Nature 共London兲 436, 36. Rabinovich et al.: Dynamical principles in neuroscience Aviel, Y., C. Mehring, M. Abeles, and D. Horn, 2003, Neural Comput. 15, 1321. Baird, B., and F. Eeckman, 1993, in Associative Neural Memories: Theory and Implementation, edited by M. H. Hassoun 共Oxford University Press, New York兲, p. 135. Bao, W., and J.-Y. Wu, 2003, J. Neurophysiol. 90, 333. Barlow, H. B., 1972, Perception 1, 371. Bartos, M., Y. Manor, F. Nadim, E. Marder, and M. P. Nusbaum, 1999, J. Neurosci. 19, 6650. Bartos, M., and M. P. Nusbaum, 1997, J. Neurosci. 17, 2247. Bi, G., 2002, Biol. Cybern. 87, 319. Bi, G., and M. Poo, 1998, J. Neurosci. 18, 10464. Bi, G., and M. Poo, 2001, Annu. Rev. Neurosci. 24, 139. Bliss, T. V., and T. Lomo, 1973, J. Physiol. 共London兲 232, 331. Block, H. D., 1962, Rev. Mod. Phys. 34, 123. Bondarenko, V., G. S. Cymbalyuk, G. Patel, S. Deweerth, and R. Calabrese, 2003, Neurocomputing 52-54, 691. Borda, J. C., 1781, Memoire sur les elections au scrutin 共Histoire de l’Academie Royale des Sciences, Paris兲. Bressloff, P., 2002, Phys. Rev. Lett. 89, 088101. Bressloff, P., and J. D. Cowan, 2002, Neural Comput. 14, 493. Bressloff, P., J. Cowan, M. Golubitsky, P. Thomas, and M. Weiner, 2001, Philos. Trans. R. Soc. London, Ser. B 356, 299. Brody, C., and J. Hopfield, 2003, Neuron 37, 843. Brunel, N., 2003, Cereb. Cortex 13, 1151. Brunel, N., and X. Wang, 2001, J. Comput. Neurosci. 11, 63. Busse, F., and K. Heikes, 1980, Science 208, 173. Calabrese, R., F. Nadim, and O. Olsen, 1995, J. Neurobiol. 27, 390. Canavier, C., D. Baxter, J. Clark, and J. Byrne, 1993, J. Neurophysiol. 69, 2252. Canavier, C., J. Clark, and J. Byrne, 1990, Biophys. J. 57, 1245. Casado, J., 2003, Phys. Rev. Lett. 91, 208102. Castelo-Branco, M., R. Goebel, S. Neuenschwander, and W. Singer, 2000, Nature 共London兲 405, 685. Cateau, H., and T. Fukai, 2001, Neural Networks 14, 675. Cazelles, B., M. Courbage, and M. I. Rabinovich, 2001, Europhys. Lett. 56, 504. Chay, T., 1985, Physica D 16, 223. Chechik, G., 2003, Neural Comput. 15, 1481. Chklovskii, D. B., B. W. Mel, and K. Svoboda, 2004, Nature 共London兲 431, 78. Chow, C., and N. Kopell, 2000, Neural Comput. 12, 1643. Cohen, A., P. Holmes, and R. Rand, 1982, J. Math. Biol. 13, 345. Cohen, M. A., and S. Grossberg, 1983, IEEE Trans. Syst. Man Cybern. 13, 815. Connors, B., 2002, Nature 共London兲 420, 133. Connors, B. W., and Y. Amitai, 1997, Neuron 18, 347. Contreras, D., A. Destexhe, T. Sejnowski, and M. Steriade, 1996, Science 274, 771. Coombes, S., and A. Osbaldestin, 2000, Phys. Rev. E 62, 4057. Cossart, R., D. Aronov, and R. Yuste, 2003, Nature 共London兲 423, 283. Cowan, N., 2001, Behav. Brain Sci. 24, 87. Crawford, J., 1991, Rev. Mod. Phys. 63, 991. Crevier, D., and M. Meister, 1998, J. Neurophysiol. 79, 1869. Cross, M., and P. Hohenberg, 1993, Rev. Mod. Phys. 65, 851. Curts, C. E., and M. D’Esposito, 2003, Trends Cogn. Sci. 7, 415. Cymbalyuk, G., O. Gaudry, M. Masino, and R. Calabrese, 2002, J. Neurosci. 22, 10580. deCharms, R., and M. Merzenich, 1996, Nature 共London兲 361, Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 1261 610. deCharms, R. C., 1998, Proc. Natl. Acad. Sci. U.S.A. 95, 15166. de Nó, R. L., 1938, J. Neurophysiol. 1, 207. de Ryter van Steveninck, R., G. Lewen, S. Strong, R. Koberle, and W. Bialek, 1997, Science 275, 1805. Desmedt, J., and C. Tomberg, 1994, Neurosci. Lett. 168, 126. de Zeeuw, C. I., J. I. Simpson, C. C. Hoogenaraad, N. Galjart, S. K. E. Koekkoek, and T. J. H. Ruigrok, 1998, Trends Neurosci. 21, 391. Diesmann, M., M.-O. Gewaltig, and A. Aertsen, 1999, Nature 共London兲 402, 529. Diwadkar, V. A., P. A. Carpenter, and M. Just, 2000, Neuroimage 11, 85. Doboli, S., A. A. Minai, and P. Best, 2000, Neural Comput. 12, 1009. Dragoi, G., K. Harris, and G. Buzsaki, 2003, Neuron 39, 843. Du, X., B. K. Ghosh, and P. Ulinski, 2005, IEEE Trans. Biomed. Eng. 52, 566. Duda, R., E. Hart, and D. Stork, 2001, Pattern Classification 共Wiley, New York兲. Durstewitz, D., J. Seamans, and T. Sejnowski, 2000, Nat. Neurosci. 3, 1184. Egorov, A., B. Hamann, E. Fransen, M. Hasselmo, and A. Alonso, 2002, Nature 共London兲 420, 173. Eguia, M. C., M. I. Rabinovich, and H. D. I. Abarbanel, 2000, Phys. Rev. E 62, 7111. Elhilali, M., J. Fritz, D. Klein, J. Z. Simon, and S. Shamma, 2004, J. Neurosci. 24, 1159. Elowitz, M., and S. Leibler, 2000, Nature 共London兲 403, 335. Elson, R. C., A. I. Selverston, H. D. I. Abarbanel, and M. I. Rabinovich, 2002, J. Neurophysiol. 88, 1166. Elson, R. C., A. I. Selverston, R. Huerta, N. F. Rulkov, M. I. Rabinovich, and H. D. I. Abarbanel, 1998, Phys. Rev. Lett. 81, 5692. Engel, A. K., P. Fries, and W. Singer, 2001, Nat. Rev. Neurosci. 2, 704. Ermentrout, B., 1992, Neural Networks 5, 415. Ermentrout, G., and J. Cowan, 1979, Biol. Cybern. 34, 137. Ermentrout, G. B., 1998, Rep. Prog. Phys. 61, 353. Ermentrout, G. B., and N. Kopell, 1984, SIAM J. Math. Anal. 15, 215. Fano, R. M., 1961, Ed., Transmission of Information: A Statistical Theory of Communications 共MIT, New York兲. Fatt, P., and B. Katz, 1952, J. Physiol. 共London兲 117, 109. Fell, J., G. Fernandez, P. Klaver, C. Elger, and P. Fries, 2003, Brain Res. Rev. 42, 265. Feudel, U., A. Neiman, X. Pei, W. Wojtenek, H. Braun, M. Huber, and F. Moss, 2000, Chaos 10, 231. FitzHugh, R., 1961, Biophys. J. 1, 445. Fitzpatrick, D. C., R. Batra, T. R. Sanford, and S. Kuwada, 1997, Nature 共London兲 388, 871. Foss, J., A. Longtin, B. Mensour, and J. Milton, 1996, Phys. Rev. Lett. 76, 708. Freeman, W., 1972, Progress in Theoretical Biology 共Academic, New York兲, Vol. 2. Freeman, W., 2000, Neurodynamics: An Exploration in Mesoscopic Brain Dynamics 共Springer, New York兲. Fuhrmann, G., I. Segev, H. Markram, and M. Tsodyks, 2002, J. Neurophysiol. 87, 140. Fukai, T., and S. Tanaka, 1997, Neural Comput. 9, 77. Galan, R. F., S. Sachse, C. G. Galizia, and A. V. M. Herz, 2004, Neural Comput. 16, 999. 1262 Rabinovich et al.: Dynamical principles in neuroscience Gallager, R. G., 1968, Ed., Information Theory and Reliable Communication 共Wiley, New York兲. Garcia-Sanchez, M., and R. Huerta, 2003, J. Comput. Neurosci. 15, 5. Gavrilov, N., and A. Shilnikov, 2000, Am. Math. Soc. Transl. 200, 99. Georgopoulos, A. P., A. B. Schwartz, and R. E. Kettner, 1986, Science 233, 1416. Gerstner, W., and W. Kistler, 2002, Spiking Neuron Models 共Cambridge University Press, Cambridge, England兲. Getting, P., 1989, Annu. Rev. Neurosci. 12, 185. Gewaltig, M.-O., M. Diesmann, and A. Aertsen, 2001, Neural Networks 14, 657. Ghose, G., and J. Maunsell, 1999, Neuron 24, 79. Glass, L., 1995, The Handbook of Brain Theory and Neural Networks 共MIT, Cambridge兲, pp. 186–189. Goldman-Rakic, P., 1995, Neuron 14, 477. Goroff, D., 1992, Ed., New Methods of Celestial Mechanics 共AIP, New York兲. Gray, C., P. Konig, A. Engel, and W. Singer, 1989, Nature 共London兲 338, 334. Grillner, S., 2003, Nat. Rev. Neurosci. 4, 573. Grossberg, S., 1973, Stud. Appl. Math. 52, 213. Gu, H., M. Yang, L. Li, Z. Liu, and W. Ren, 2003, Phys. Lett. A 319, 89. Guckenheimer, J., S. Gueron, and R. Harris-Warrick, 1993, Philos. Trans. R. Soc. London, Ser. B 341, 345. Guckenheimer, J., and P. Holmes, 1986, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields 共Springer, New York兲. Guckenheimer, J., and P. Holmes, 1988, Math. Proc. Cambridge Philos. Soc. 103, 189. Hass, J., T. Nowotny, and H. D. I. Abarbanel, 2006, J. Neurophysiol., doi:10.1152/jn.00551.2006. Hahnloser, R. H., A. A. Kozhevnikov, and M. S. Fee, 2002, Nature 共London兲 419, 65. Hamilton, K., and J. Kauer, 1985, Brain Res. 338, 181. Hatsopoulos, N., L. Paninski, and J. Donoghue, 2003, Exp. Brain Res. 149, 478. Hebb, R., 1949, The Organization of Behavior 共Wiley, New York兲. Hindmarsh, J. L., and R. M. Rose, 1984, Proc. R. Soc. London, Ser. B 221, 87. Hodgkin, A. L., and A. F. Huxley, 1952, J. Physiol. 共London兲 117, 500. Hopfield, J., and C. Brody, 2001, Proc. Natl. Acad. Sci. U.S.A. 98, 1282. Hopfield, J., and C. Brody, 2004, Proc. Natl. Acad. Sci. U.S.A. 101, 337. Hopfield, J. J., 1982, Proc. Natl. Acad. Sci. U.S.A. 79, 2554. Huang, X., W. C. Troy, Q. Yang, H. Ma, C. R. Laing, S. J. Schiff, and J.-Y. Wu, 2004, J. Neurosci. 24, 9897. Huerta, R., and M. I. Rabinovich, 2004, Phys. Rev. Lett. 93, 238104. Huerta, R., M. Rabinovich, H. H. D. I. Abarbanel, and M. Bazhenov, 1997, Phys. Rev. E 55, R2108. Huerta, R., P. Varona, M. Rabinovich, and H. Abarbanel, 2001, Biol. Cybern. 84, L1. Ichinohe, N., F. Fujiyama, T. Kaneko, and K. S. Rockland, 2003, J. Neurosci. 23, 1372. Ikegaya, Y., G. Aaron, R. Cossart, D. Aronov, I. Lampl, D. Ferster, and R. Yuste, 2004, Science 304, 559. Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 Ince, D., 1992, Ed., Mechanical Intelligence: Collected Works of A. M. Turing 共North-Holland, Amsterdam兲. Ito, M., 1982, Annu. Rev. Neurosci. 5, 275. Izhikevich, E., 2004, IEEE Trans. Neural Netw. 15, 1063. Izhikevich, E., 2006, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting 共MIT, Cambridge, MA兲. Jackson, A., V. Gee, S. Baker, and R. Lemon, 2003, Neuron 38, 115. Jancke, L., M. Himmelbach, N. Shah, and K. Zilles, 2000, Neuroimage 12, 528. Jefferys, J., R. Traub, and M. Whittington, 1996, Trends Neurosci. 19, 202. Jin, D. Z., 2002, Phys. Rev. Lett. 89, 208102. Jones, S. R., B. Mulloney, T. J. Kaper, and N. Kopell, 2003, J. Neurosci. 60, 3457. Kandel, E. R., J. H. Schwartz, and T. M. Jessell, 2000, Principles of Neural Science 共McGraw-Hill, New York兲. Kaplan, D., and L. Glass, 1995, Understanding Nonlinear Dynamics 共Springer, New York兲. Katz, B., 1969, The Release of Neurotransmitter Substances 共Thomas, Springfield, IL兲. Kay, L., 2003, Chaos 13, 1057. Kepecs, A., and J. Lisman, 2003, Network Comput. Neural Syst. 14, 103. Kim, U., T. Bal, and D. A. McCormick, 1995, J. Neurophysiol. 74, 1301. Kimpo, R. R., F. E. Theunissen, and A. J. Doupe, 2003, J. Neurosci. 23, 5730. Kistler, W. M., and C. I. de Zeeuw, 2002, Neural Comput. 14, 2597. Koch, C., 1999, Biophysics of Computation 共Oxford University Press, New York兲. Konishi, M., 1990, Cold Spring Harbor Symp. Quant. Biol. 55, 575. Korn, H., and P. Faure, 2003, C R. Seances Soc. Biol. Fil 326, 787. Kozloski, J., F. Hamzei-Sichani, and R. Yuste, 2001, Science 293, 868. Krupa, P., 1997, J. Nonlinear Sci. 7, 129. Kuramoto, Y., 1984, Chemical Oscillations, Waves, and Turbulence 共Springer, New York兲. Kuznetsov, Y., 1998, Elements of Applied Bifurcation Theory, 2nd ed. 共Springer, New York兲. Lai, Y. C., M. A. F. Harrison, M. G. Frei, and I. Osorio, 2003, Phys. Rev. Lett. 91, 068102. Laing, C. R., W. C. Troy, B. Gutkin, and G. B. Ermentrout, 2002, SIAM J. Appl. Math. 63, 62. Landau, L. D., and E. M. Lifshitz, 1987, Fluid Mechanics 共Pergamon, New York兲. Lapicque, L., 1907, J. Physiol. Pathol. Gen. 9, 620. Lashley, K., 1960, in The Neuropsychology of Lashley, edited by F. A. Beach, D. O. Hebb, C. T. Morgan, and H. W. Nissen 共McGraw-Hill, New York兲, pp. 506–521. Latorre, R., F. B. Rodriquez, and P. Varona, 2006, Biol. Cybern. 95, 169. Lau, P., and G. Bi, 2005, Proc. Natl. Acad. Sci. U.S.A. 102, 10333. Laurent, G., 1996, Trends Neurosci. 19, 489. Laurent, G., and H. Davidowitz, 1994, Science 265, 1872. Laurent, G., M. Stopfer, R. W. Friedrich, M. I. Rabinovich, and H. D. I. Abarbanel, 2001, Annu. Rev. Neurosci. 24, 263. Leitch, B., and G. Laurent, 1996, J. Comp. Neurol. 372, 487. Rabinovich et al.: Dynamical principles in neuroscience LeMasson, G., S. R.-L. Masson, D. Debay, and T. Bal, 2002, Nature 共London兲 417, 854. Lestienne, R., 2001, Prog. Neurobiol. 共Oxford兲 65, 545. Levi, R., P. Varona, Y. I. Arshavsky, M. I. Rabinovich, and A. I. Selverston, 2004, J. Neurophysiol. 91, 336. Levi, R., P. Varona, Y. I. Arshavsky, M. I. Rabinovich, and A. I. Selverston, 2005, J. Neurosci. 25, 9807. Lewis, T., and J. Rinzel, 2003, J. Comput. Neurosci. 14, 283. Leznik, E., and R. Llinas, 2002, Ann. N.Y. Acad. Sci. 978, 529. Lin, L., R. Osan, S. Shoham, W. Jin, W. Zuo, and J. Tsien, 2005, Proc. Natl. Acad. Sci. U.S.A. 102, 6125. Lindner, B., J. Garca-Ojalvo, A. Neiman, and L. SchimanskyGeier, 2004, Phys. Rep. 392, 2004. Llinás, R., and J. P. Welsh., 1993, Curr. Opin. Neurobiol. 3, 958. Loebel, A., and M. Tsodyks, 2002, J. Comput. Neurosci. 13, 111. Lotka, A. J., 1925, Elements of Physical Biology 共Williams & Wilkins Co., Baltimore兲. Maass, W., T. Natschläger, and H. Markram, 2002, Neural Comput. 14, 2531. Machens, C., T. Gollisch, O. Kolesnikova, and A. Herz, 2005, Neuron 47, 447. Machens, C., R. Romo, and C. Brody, 2005, Science 307, 1121. Maeda, Y., and H. Makino, 2000, BioSystems 58, 93. Maeda, Y., K. Pakdaman, T. Nomura, S. Doi, and S. Sato, 1998, Biol. Cybern. 78, 265. Mainen, Z., and T. Sejnowski, 1995, Science 268, 1503. Malenka, R. C., and R. A. Nicoll, 1999, Science 285, 1870. Malkov, V., M. Rabinovich, and M. M. Sushchik, 1996, Proceedings of Nizhny Novgorod University, edited by V. D. Shalfeev 共Nizhny Novgorov Univerity, Nizhny Novgorod兲, p. 72. Mandelblat, Y., Y. Etzion, Y. Grossman, and D. Golomb, 2001, J. Comput. Neurosci. 11, 43. Maravall, M., I. Y. Y. Koh, W. Lindquist, and K. Svoboda, 2004, Cereb. Cortex 10, 1093. Marder, E., and R. L. Calabrese, 1996, Physiol. Rev. 76, 687. Marder, E., L. A. G. Turrigiano, Z. Liu, and J. Golowasch, 1996, Proc. Natl. Acad. Sci. U.S.A. 93, 13481. Martin, S. J., P. D. Grimwood, and R. G. M. Morris, 2000, Annu. Rev. Neurosci. 23, 649. Mazor, O., and G. Laurent, 2005, Neuron 48, 661. McClurkin, J. W., L. M. Optican, B. J. Richmond, and T. J. Gawne, 1991, Science 253, 675. McCormick, D., Y. Shu, A. Hasenstaub, M. Sanches-Vives, M. Badoual, and T. Bal, 2003, Cereb. Cortex 13, 1219. McCulloch, W. S., and W. Pitts, 1943, Bull. Math. Biophys. 5, 115. Mehta, M. R., A. K. Lee, and M. A. Wilson, 2002, Nature 共London兲 417, 741. Melamed, O., W. Gerstner, W. Maas, M. Tsodyks, and H. Markram, 2004, Trends Neurosci. 27, 11. Mooney, R., and J. Prather, 2005, J. Neurosci. 25, 1952. Morris, C., and H. Lecar, 1981, Biophys. J. 35, 193. Nádasdy, Z., 2000, J. Physiol. 共Paris兲 94, 505. Nagumo, J., S. Arimoto, and S. Yoshizawa, 1962, Proc. IRE 50, 2061. Nakahara, H., and K. Doya, 1998, Neural Comput. 10, 113. Nase, G., W. Singer, H. Monyer, and A. K. Engel, 2003, J. Neurophysiol. 90, 1115. Nenadic, Z., B. K. Ghosh, and P. Ulinski, 2002, IEEE Trans. Biomed. Eng. 49, 753. Nicholls, J. G., A. R. Martin, and B. G. Wallace, 1992, From Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 1263 Neuron to Brain: A Cellular and Molecular Approach to the Function of the Nervous System 共Sinauer Associates, Sunderland, MA兲. Nowotny, T., 2003, URL http://inls.ucsd.edu/$\sim$nowotny/ dynclamp.html Nowotny, T., and R. Huerta, 2003, Biol. Cybern. 89, 237. Nowotny, T., R. Huerta, H. Abarbanel, and M. Rabinovich, 2005, Biol. Cybern. 93, 436. Nowotny, T., M. I. Rabinovich, R. Huerta, and H. D. I. Abarbanel, 2003, J. Comput. Neurosci. 15, 271. Nowotny, T., V. P. Zhigulin, A. I. Selverston, H. D. I. Abarbanel, and M. I. Rabinovich, 2003, J. Neurosci. 23, 9776. Nystrom, L. E., T. S. Braver, F. W. Sabb, M. R. Delgado, D. C. Noll, and J. D. Cohen, 2000, Neuroimage 11, 424. O’Keefe, J., and J. Dostrovsky, 1971, Brain Res. 34, 171. O’Reilly, R., R. Busby, and R. Soto, 2003, The Unity of Consciousness—Binding, Integration and Dissociation 共Oxford University Press, Oxford兲, Chap. 3, p. 2.5. Oscarsson, O., 1980, in The Inferior Olivary Nucleus, edited by J. Courville, C. de Montigny, and Y. Lamarre 共Raven, New York兲, pp. 279–289. Ott, E., 1993, Chaos in Dynamical Systems 共Cambridge University Press, Cambridge, England兲. Panchin, Y., Y. Arshavsky, T. Deliagina, L. Popova, and G. Orlovsky, 1995, J. Neurophysiol. 73, 1924. Panzeri, S., S. Schultz, A. Treves, and E. Rolls, 1999, Proc. R. Soc. London, Ser. B 266, 1001. Perez-Orive, J., O. Mazor, G. C. Turner, S. Cassenaer, R. I. Wilson, and G. Laurent, 2002, Science 297, 359. Persi, E., D. Horn, V. Volman, R. Segev, and E. Ben-Jacob, 2004, Neural Comput. 16, 2577. Pikovsky, A., M. Rosenblum, and J. Kurths, 2001, Synchronization: A Universal Concept in Nonlinear Sciences 共Cambridge University Press, Cambridge, England兲. Pinto, R., P. Varona, A. Volkovskii, A. Szücs, H. Abarbanel, and M. Rabinovich, 2000, Phys. Rev. E 62, 2644. Poincaré, A., 1905, Le Valeur de la Science 共Flammarion, Paris兲. Poincaré, H., 1892, Méthodes Nouvells de la Mécanique Céleste 共Gauthier-Villars, Paris兲. Postlethwaite, C. M., and J. H. P. Dawes, 2005, Nonlinearity 18, 1477. Pouget, A., P. Dayan, and R. Zemel, 2000, Nat. Rev. Neurosci. 1, 125. Prinz, A., L. F. Abbott, and E. Marder, 2004a, Trends Neurosci. 27, 218. Prinz, A. A., D. Bucher, and E. Marder, 2004b, Nat. Neurosci. 7, 1345. Rabinovich, M., A. Ezersky, and P. Weidman, 2000, The Dynamics of Patterns 共World Scientific, Singapore兲. Rabinovich, M., R. Huerta, and P. Varona, 2006, Phys. Rev. Lett. 96, 014101. Rabinovich, M., A. Volkovskii, P. Lecanda, R. Huerta, H. D. I. Abarbanel, and G. Laurent, 2001, Phys. Rev. Lett. 87, 068102. Rabinovich, M. I., and H. D. I. Abarbanel, 1998, Neuroscience 87, 5. Rabinovich, M. I., and R. Huerta, 2006, Phys. Rev. Lett. 97, 188103. Ramirez, J., A. Tryba, and F. Pena, 2004, Curr. Opin. Neurobiol. 6, 665. Reinagel, P., and R. C. Reid, 2002, J. Neurosci. 22, 6837. Reyes, A. D., 2003, Nat. Neurosci. 6, 593. Riesenhuber, M., and T. Poggio, 1999, Neuron 24, 87. 1264 Rabinovich et al.: Dynamical principles in neuroscience Rinzel, J., D. Terman, X. Wang, and B. Ermentrout, 1998, Science 279, 1351. Robinson, H., and N. Kawai, 1993, J. Neurosci. Methods 49, 157. Rodriguez, E., N. George, J. Lachaux, J. Martinerie, B. Renault, and F. Varela, 1999, Nature 共London兲 397, 430. Rosenblatt, F., 1962, Principles of Neurodynamics: Perceptions and the Theory of Brain Mechanisms 共Spartan Books, New York兲. Roskies, A., 1999, Neuron 24, 7. Rowe, D., 2002, Behav. Brain Sci. 24, 5. Roweis, S., and L. Saul, 2000, Science 290, 2323. Rubin, J. E., and D. Terman, 2004, J. Comput. Neurosci. 16, 211. Rulkov, N. F., 2002, Phys. Rev. E 65, 041922. Saari, G., 1995, Basic Geometry of Voting 共Springer-Verlag, Berlin兲. Sanchez-Vives, M., and D. McCormick, 2000, Nat. Neurosci. 3, 1027. Schutter E. D., 2002, Curr. Biol. 12, R363. Schweighofer, N., K. Doya, H. Fukai, J. V. Chiron, T. Furukawa, and M. Kawato, 2004, Proc. Natl. Acad. Sci. U.S.A. 101, 4655. Scott, A., 2004, Ed., Encylopedia of Nonlinear Science 共Routledge, New York兲. Segundo, J. P., and D. H. Perkel, 1969, UCLA Forum Med. Sci. 11, 349. Segundo, J. P., G. Sugihara, P. Dixon, M. Stiber, and L. F. Bersier, 1998, Neuroscience 87, 741. Seliger, P., L. S. Tsimring, and M. I. Rabinovich, 2003, Phys. Rev. E 67, 011905. Selverston, A., 2005, Cell Mol. Neurobiol. 25, 223. Selverston, A., M. Rabinovich, H. Abarbanel, R. Elson, A. Szücs, R. Pinto, R. Huerta, and P. Varona, 2000, J. Physiol. 共Paris兲 94, 357. Senseman, D. M., and K. A. Robbins, 1999, J. Neurosci. 19, RC3 1. Seung, H., D. Lee, B. Reis, and D. Tank, 2000, J. Comput. Neurosci. 9, 171. Seung, H. S., 1998, in Advances in Neural Information Processing Systems, edited by M. I. Jordan, M. J. Kearns, and S. A. Solla 共MIT, Cambridge, MA兲, Vol. 10. Seung, H. S., and H. Sompolinsky, 1993, Proc. Natl. Acad. Sci. U.S.A. 90, 10749. Shadlen, M. N., and W. T. Newsome, 1998, J. Neurosci. 18, 3870. Sharp, A., M. O’Neil, L. Abbott, and E. Marder, 1993, Trends Neurosci. 16, 389. Shepherd, G., 1998, The Synaptic Organization of the Brain 共Oxford University Press, New York兲. Sherman, A., and J. Rinzel, 1992, Proc. Natl. Acad. Sci. U.S.A. 89, 2471. Shilnikov, A., and G. Cymbalyuk, 2005, Phys. Rev. Lett. 94, 048101. Shu, Y., A. Hasenstaub, and D. McCormick, 2003, Nature 共London兲 423, 288. Silberberg, G., S. Grillner, F. LeBeau, R. Maex, and H. Markram, 2005, Trends Neurosci. 28, 541. Simmers A. J., and M. Moulins, 1988, J. Neurophysiol. 59, 740. Singer, W., 1999, Neuron 24, 49. Singer, W., 2001, Ann. N.Y. Acad. Sci. 929, 123. Singer, W., and C. M. Gray, 1995, Annu. Rev. Neurosci. 18, 555. Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 Sjöström, P. J., G. G. Turrigiano, and S. B. Nelson, 2001, Neuron 32, 1149. Softky, W. R., 1995, Curr. Opin. Neurobiol. 5, 239. Solis, M., and D. Perkel, 2005, J. Neurosci. 25, 2811. Sompolinsky, H., and I. Kanter, 1986, Phys. Rev. Lett. 57, 2861. Sompolinsky, H., H. Yoon, K. Kang, and M. Shamir, 2001, Phys. Rev. E 64, 051904. Soto-Trevino, C., P. Rabbah, E. Marder, and F. Nadim, 2005, J. Neurophysiol. 94, 590. Soto-Trevino, C., K. A. Thoroughman, E. Marder, and L. E. Abbott, 2001, Nat. Neurosci. 4, 297. Stein, S. G., S. Grillner, A. I. Selverston, and G. S. Douglas, 1997, Eds., Neurons, Networks, and Motor Behavior 共MIT, Cambridge, MA兲. Stent, G. S., and W. O. Friesen, 1977, Biol. Cybern. 28, 27. Stone, E., and D. Armbruster, 1999, Chaos 9, 499. Stone, E., and P. Holmes, 1990, SIAM J. Appl. Math. 50, 726. Stopfer, M., V. Jayaraman, and G. Laurent, 2003, Neuron 39, 991. Strogatz, S. H., 2001, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering 共Perseus Books Group, Cambridge, MA兲. Szekely, G., 1965, Acta Physiol. Acad. Sci. Hung. 27, 285. Szücs, A., R. D. Pinto, M. I. Rabinovich, H. D. I. Abarbanel, and A. I. Selverston, 2003, J. Neurophysiol. 89, 1363. Szücs, A., P. Varona, A. Volkovskii, H. D. I. Abarbanel, M. Rabinovich, and A. Selverston, 2000, NeuroReport 11, 563. Szücs, A., A. I. Selverston, M. I. Rabinovich, and H. D. I. Abarbanel, 2004, Soc. Neurosci Abs. 420, 4. Tams, G., A. Lorincz, A. Simon, and J. Szabadics, 2003, Science 299, 1902. Terman, D., A. Bose, and N. Kopell, 1996, Proc. Natl. Acad. Sci. U.S.A. 93, 15417. Thompson, A. M., and J. Deuchars, 1994, Trends Neurosci. 17, 119. Tiitinen, H., J. Sinkkonen, K. Reinikainen, K. Alho, J. Lavikainen, and R. Naatanen, 1993, Nature 共London兲 364, 59. Tohya, S., A. Mochizuki, and Y. Iwasa, 2003, J. Theor. Biol. 221, 289. Traub, R. D., and R. Miles, 1991, Ann. N.Y. Acad. Sci. 627, 277. Tsodyks, M., T. Kenet, A. Grinvald, and A. Arieli, 1999, Science 286, 1943. Tsodyks, M., and H. Markram, 1997, Proc. Natl. Acad. Sci. U.S.A. 94, 719. Tsuda, I., 1991, World Futures 32, 167. Turrigiano, G. G., E. Marder, and L. Abbott, 1996, J. Neurophysiol. 75, 963. Vaadia, E., I. Haalman, M. Abeles, H. Bergman, Y. Prut, H. Slovin, and A. Aertsen, 1995, Nature 共London兲 373, 515. van Essen, D. C., 1979, Annu. Rev. Neurosci. 2, 227. van Vreeswijk, C., L. F. Abbott, and G. B. Ermentrout, 1994, J. Comput. Neurosci. 1, 313. van Vreeswijk, V., and H. Sompolinsky, 1996, Science 274, 1724. Varona, P., C. Aguirre, J. J. Torres, M. I. Rabinovich, and H. D. I. Abarbanel, 2002, Neurocomputing 44-46, 685. Varona, P., M. I. Rabinovich, A. I. Selverston, and Y. I. Arshavsky, 2002, Chaos 12, 672. Varona, P., J. J. Torres, H. D. I. Abarbanel, M. I. Rabinovich, and R. Elson, 2001, Biol. Cybern. 84, 91. Varona, P., J. J. Torres, R. Huerta, H. D. I. Abarbanel, and M. I. Rabinovich, 2001, Neural Networks 14, 865. Rabinovich et al.: Dynamical principles in neuroscience Venaille, A., P. Varona, and M. I. Rabinovich, 2005, Phys. Rev. E 71, 061909. Vogel, E. K., and M. G. Machizawa, 2004, Nature 共London兲 428, 748. Vogels, T., K. Rajan, and L. Abbott, 2005, Annu. Rev. Neurosci. 28, 357. Volterra, V., 1931, in Animal Ecology, edited by R. N. Chapman 共McGraw-Hill, New York兲, pp. 409–448. von der Malsburg, C., 1999, Neuron 24, 95. von der Malsburg, C., and W. Schneider, 1986, Biol. Cybern. 54, 29. Voogd, J., and M. Glickstein, 1998, Trends Neurosci. 21, 370. Wang, X., 2001, Trends Neurosci. 24, 455. Wang, X.-J., and J. Rinzel, 1995, The Handbook of Brain Theory and Neural Networks 共MIT, Cambridge, MA兲, p. 686. Rev. Mod. Phys., Vol. 78, No. 4, October–December 2006 1265 Waugh, F., C. Marcus, and R. Westervelt, 1990, Phys. Rev. Lett. 64, 1986. Wilson, H. R., 1999, Spikes, Decisions, and Actions 共Oxford University Press, New York兲. Wilson, H. R., and J. D. Cowan, 1973, Kybernetik 13, 55. Wilson, M. A., and B. L. McNaughton, 1993, Science 261, 1055. Wilson, M. A., and B. McNaughton, 1994, Science 265, 676. Wolfe, J., and K. Cave, 1999, Neuron 24, 11. Yuste, R., J. MacLean, J. Smith, and A. Lansner, 2005, Nat. Rev. Neurosci. 6, 477. Zeeman, E., and M. Zeeman, 2002, Nonlinearity 15, 2019. Zhigulin, V. P., and M. I. Rabinovich, 2004, Neurocomputing 58-60, 373. Zhigulin, V. P., M. I. Rabinovich, R. Huerta, and H. D. I. Abarbanel, 2003, Phys. Rev. E 67, 021901.
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