Existence and Control of Go/No-Go Decision Transition Threshold in the Striatum

Existence and Control of Go/No-Go Decision Transition Threshold in the Striatum
RESEARCH ARTICLE
Existence and Control of Go/No-Go Decision
Transition Threshold in the Striatum
Jyotika Bahuguna1,2*, Ad Aertsen1, Arvind Kumar1,2*
1 Bernstein Center Freiburg and Faculty of Biology, University of Freiburg, Freiburg, Germany,
2 Computational Biology, School of Computer Science and Communication, KTH Royal Institute of
Technology, Stockholm, Sweden
* [email protected] (JB); [email protected] (AK)
a11111
Abstract
OPEN ACCESS
Citation: Bahuguna J, Aertsen A, Kumar A (2015)
Existence and Control of Go/No-Go Decision
Transition Threshold in the Striatum. PLoS Comput
Biol 11(4): e1004233. doi:10.1371/journal.
pcbi.1004233
Editor: Kim T. Blackwell, The Krasnow Institute for
Advanced Studies, UNITED STATES
Received: June 23, 2014
Accepted: March 11, 2015
Published: April 24, 2015
Copyright: © 2015 Bahuguna et al. This is an open
access article distributed under the terms of the
Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any
medium, provided the original author and source are
credited.
Data Availability Statement: There is no data that
needs to be made available. The simulation code to
reproduce the key results is available at the github
https://github.com/arvkumar/BasalGanglia/tree/
master/Striatum/striatum_d1_d2_network_
bahuguna_2015 The execution of the code requires
the neuronal network simulator NEST (www.nestinitiative.org/) and Python.
Funding: This work was supported by the German
Federal Ministry of Education and Research (BMBF
01GQ0420 to BCCN Freiburg and 01GQ0830 to
BFNT Freiburg/Tübingen), the BrainLinks-BrainTools
Cluster of Excellence funded by the German
A typical Go/No-Go decision is suggested to be implemented in the brain via the activation
of the direct or indirect pathway in the basal ganglia. Medium spiny neurons (MSNs) in the
striatum, receiving input from cortex and projecting to the direct and indirect pathways express D1 and D2 type dopamine receptors, respectively. Recently, it has become clear that
the two types of MSNs markedly differ in their mutual and recurrent connectivities as well as
feedforward inhibition from FSIs. Therefore, to understand striatal function in action selection, it is of key importance to identify the role of the distinct connectivities within and between the two types of MSNs on the balance of their activity. Here, we used both a reduced
firing rate model and numerical simulations of a spiking network model of the striatum to analyze the dynamic balance of spiking activities in D1 and D2 MSNs. We show that the asymmetric connectivity of the two types of MSNs renders the striatum into a threshold device,
indicating the state of cortical input rates and correlations by the relative activity rates of D1
and D2 MSNs. Next, we describe how this striatal threshold can be effectively modulated by
the activity of fast spiking interneurons, by the dopamine level, and by the activity of the
GPe via pallidostriatal backprojections. We show that multiple mechanisms exist in the
basal ganglia for biasing striatal output in favour of either the `Go' or the `No-Go' pathway.
This new understanding of striatal network dynamics provides novel insights into the putative role of the striatum in various behavioral deficits in patients with Parkinson's disease, including increased reaction times, L-Dopa-induced dyskinesia, and deep brain stimulationinduced impulsivity.
Author Summary
The basal ganglia (BG) play a crucial role in a variety of cognitive and motor functions.
BG dysfunction leads to brain disorders such as Parkinson’s disease. At the main input
stage of the BG, the striatum, two competing pathways originate. Neurons projecting on
these pathways either express D1 or D2 type dopamine receptors. Because activity of D1
or D2 neurons facilitate go or no-go type decision, it is important to study the balance of
the D1 and D2 neuron activity. Contrary to the common assumption thus far, recent data
PLOS Computational Biology | DOI:10.1371/journal.pcbi.1004233 April 24, 2015
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Go/No-Go Decision Transition Threshold in the Striatum
Research Foundation (DFG #EXC 1086), the EU
(INTERREG-V Grant to Neurex: TriNeuron) and the
German-Israeli Foundation. AK and AA acknowledge
INTERREG IV Rhin supérieur program and european
funds for regional development (FEDER) through the
project TIGER A31. We also thank the Albert Ludwigs
University Freiburg in the funding programme Open
Access Publishing.The funders had no role in study
design, data collection and analysis, decision to
publish, or preparation of the manuscript.
Competing Interests: The authors have declared
that no competing interests exist.
shows an asymmetry in striatal circuit with D1 receiving higher inhibition from D2 and
fast-spiking neurons. Here, we studied the functional implications of the asymmetric connectivity between D1 and D2 neurons. Our analysis and simulations show that the asymmetric connectivity between these neurons gives rise to a decision transition threshold
(DTT), as a consequence D1 (D2) neurons have higher firing rate at lower (higher) average
cortical firing rates. Importantly, DTT can be modulated by input correlations, local connectivity, feedforward inhibition and dopamine. Our results suggest that abnormal
changes in the DTT could be a plausible mechanism underlying the cognitive and motor
deficits associated with brain diseases involving BG malfunction.
Introduction
The basal ganglia (BG) are a set of nuclei, located at the base of the forebrain, which play a crucial role in a variety of motor and cognitive functions. The striatum is the main input stage of
the basal ganglia, receiving inputs from widely distributed areas in the cortex [1], and projecting to the BG output nuclei Globus Pallidus Interna (GPi) and Substantia Nigra (SNr) via the
so-called direct and indirect pathways, respectively [2]. The integration of multi-modal sensory
signals with motor and/or cognitive inputs in the striatum sets the stage for action selection.
Therefore, to understand the computations performed by the basal ganglia, it is of key importance to characterize the dynamical properties of striatal network activity.
Nearly 95% of the neurons in the striatum are inhibitory medium spiny neurons (MSNs).
The remaining 5% are inhibitory interneurons, such as fast spiking interneurons (FSIs) and
tonically active neurons (TANs) [3]. The MSNs are classified as D1 and D2 type neurons, depending on their dopamine receptor expression. Interestingly, D1 MSNs project to the GPi and
SNr, forming the direct pathway, whereas D2 neurons project to Globus Pallidus Externa
(GPe), forming the indirect pathway [2]. Consistent with basal ganglia anatomy and previous
models [4–6], selective activation of D1 MSNs in the rat increases ambulation, whereas selective activation of D2 MSNs increases freezing behavior [7]. However, a complete shutdown of
activity in either of the two neuron subpopulations might not occur in awake behaving animals
[8]. In such a scenario,though, action selection could still be performed by a relative increase in
the activity of one subpopulation compared to the other.
Thus far, in computational models of the interactions between direct and indirect pathways
[4–6, 9], D1 and D2 MSNs have been considered as interchangeable inhibitory neuron subpopulations. In such single population models, the striatal output is controlled by the strength of
cortico-striatal synaptic weights. The recurrent inhibition is not strong enough to support winner-take-all dynamics as earlier speculated [10], but may be sufficient to allow for a winner-less
competition [11] and may enhance the saliency of the cortical input representation in the striatum [12]. This, however, is a highly simplistic view of the recurrent inhibition within the striatum and, as will be described below, is inconsistent with experimental data, especially given the
recent findings on the recurrent connectivity in the striatum.
Recent experiments have shown that D1 and D2 MSNs have quite different anatomical and
electrophysiological properties [13]. Moreover, the striatal circuit also shows a highly specific
connectivity in terms of the mutual inhibition between the MSN subpopulations [14, 15] and
the feedforward inhibition from FSIs [16]. Paired neuron recordings showed that D2 MSNs
make more and stronger connections to D1 MSNs, than vice versa. Furthermore, FSIs preferentially innervate D1 MSNs as compared to D2 MSNs (Fig 1A). The computational role of this
specific connectivity within the striatum is not clear and cannot be inferred from previous
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Go/No-Go Decision Transition Threshold in the Striatum
Fig 1. Asymmetry and decision transition threshold in the striatum. (A) Schematic of the striatal circuit (B) Steady state firing rates of the D1 (blue), D2
(red) MSNs and FSI (green) as a function of cortical inputs as estimated from the linearized mean-field dynamics model of the striatum. Here we consider the
‘multiplicative scenario’ for the extra input the the D1 MSNs. We refer to the crossover point (ltran
CTX ), where the bias of striatal activity (ΔMSN) changes from D1
to D2 MSNs, as the decision transition threshold 13 Hz here and marked with the dashed line. The grey and black traces show the firing rates of the D1 and
D2 MSNs when they received cortical inputs with the same strength, respectively. This shows that extra input to D1 MSNs is necessary to activate the ‘direct’
pathway. Otherwise, D1 MSNs cannot have higher firing rates than D2 MSNs.
doi:10.1371/journal.pcbi.1004233.g001
models, which assumed a single homogeneously connected MSN population in the striatum.
Specifically, it is important to identify the effect of the distinct D1 and D2 connectivities on the
competition between the direct and indirect pathways.
Here we describe the effect of the heterogenous connectivity of D1 and D2 neurons on their
mutual interactions using both a reduced firing rate model and numerical simulations of a
spiking striatal network model. We show that the firing rates of both D1 and D2 MSNs change
in a non-monotonic manner in response to cortical input rates and correlations. Interestingly,
higher output rates in D1 than in D2 MSNs, and vice versa, were observed for separate, nonoverlapping ranges of cortical input rate and correlation. Correlations in the input can further
change the range of cortical inputs for which either D1 or D2 MSNs have the higher firing rate.
Thus, we argue that the striatum acts as a threshold device for cortical input rates by changing
the magnitude of the difference between firing rates of D1 and D2 MSNs, depending on the
level of cortical input rates and correlations.
While the main determinant of the striatal threshold is the asymmetric connectivity among
the various striatal elements, the threshold is not fixed and can be dynamically adjusted by the
dopamine level, by the connectivity and firing rate of fast spiking interneurons (FSI), and by
the GPe activity. That is, changes in the striatal threshold could reflect changes in the operating
point of the striatum, behavioral context, and learning and reward history.
These novel insights concerning the interactions between direct and indirect pathways suggest putative mechanistic explanations for the role of striatum in cognitive deficits such as
L-Dopa-induced Dyskinesia (LID), deep brain stimulation (DBS)-induced impulsivity and increased reaction times in Parkinson’s disease (PD) patients.
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Go/No-Go Decision Transition Threshold in the Striatum
Results
The striatal MSNs expressing D1 and D2 type dopamine receptors initiate the direct (‘Go’) and
indirect (‘No-Go’) pathways of the basal ganglia, respectively. These two pathways converge in
the GPi/SNr, their relative activity balance lets the animal choose between a ‘Go’ or a ‘No-Go’
action [4, 5]. In this framework, the decision making process in the striatum is mediated by the
selective activation of one of the two MSN (D1 or D2) subpopulations. However, recent experimental data suggest that such complete shutdown of activity in either one of the two neuron
subpopulations may not occur in awake behaving animals [8]. Hence, we should consider the
alternative possibility that action selection is performed by a relative increase in the activity of
one subpopulation compared to the other.
Therefore, to understand the effect of the recurrent, mutual connectivity and feedforward
inhibition [14–16] on the relative balance of the activities in the direct and indirect pathways
we studied the dynamics of the striatal network. Specifically, we evaluated the firing rates of the
D1 and D2 MSNs, in response to cortical input rates and input correlations.
D1 MSNs require overall stronger input from cortex than D2 MSNs
Experimental measurements of the mutual connectivity between striatal neurons show that D2
MSNs make more and stronger inhibitory connections on D1 MSNs than vice versa [14, 15].
In Lhx6-GFP transgenic mice, FSIs preferentially target D1 MSNs as compared to D2 MSNs
[16] (however cf. [15]). This leads to the following two inequalities: J12 > J21, J1F J2F, with J12
denoting the connection from D2 MSN to D1 MSN, J21 the obverse, J1F the connection from
FSI to D1 MSN and J2F likewise to D2 MSN (cf. Table 1). These inequalities imply that if the
two MSN subpopulations receive the same amount of excitatory input, D2 MSNs will always
have a higher firing rate.
We confirmed this by evaluating the fixed points of the linearized dynamics of the D1 and
D2 MSNs in a mean field model (Eqs 25–27, Fig 1B—grey and black traces). In our spiking network simulations this corresponded to a lower mean firing rate of the D1 population compared
to the D2 population. This bias might be considered functionally useful, because for similar
input strengths (i.e. when the cortex does not impose any preference for either direct or indirect pathways), the indirect pathway will dominate the striatal network dynamics and the ‘NoGo’ would be the default state of the basal ganglia.
Thus, in order for D1 MSN activity (λD1) to exceed D2 MSN activity (λD2), D1 MSNs must
receive either stronger (JC1 > JC2), higher excitatory inputs (λctx_d1 > λctx_d2), or more excitatory synapses. Alternatively, D1 MSNs could be more excitable. Indeed, D1 MSNs have more
Table 1. Striatal network parameters.
Connections
Strengths
J11
-0.06
J12
-0.21
J21
-0.04
J22
-0.22
J1F
-0.09
J2F
-0.06
JC1
1.06
JC2
1.0
doi:10.1371/journal.pcbi.1004233.t001
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Go/No-Go Decision Transition Threshold in the Striatum
primary dendrites and a larger dendritic arborization [13], thereby potentially receiving more
inputs. Moreover, pyramidal tract neurons have stronger connections to D1 MSNs than to D2
MSNs, while intratelencephalic tract neurons innervate D1 and D2 MSNs equally. [1]. Despite
this evidence, current data are not sufficient to determine the exact strength and rate of excitatory inputs to D1 and D2 MSNs. To estimate how much additional excitation would be required for D1 MSNs to have their firing rates exceed over those of D2 MSNs, we systematically
varied the drive of cortical inputs to D1 and D2 MSNs and calculated the response firing rates
of the two subpopulations, for the firing rate model (Fig 2). In the following we discuss two different scenarios for systematically varying the cortical excitatory drive to D1 and D2 MSNs.
Scenario I: Increase the relative cortical input rate to D1 MSNs. One possibility is that
the cortical input rate to D1 MSNs is higher than to D2 MSNs while the strength of the cortico-striatal synapses to the two types of MSNs is the same. This results in an ‘additive’ scenario, in which we kept the strength of the cortical inputs to D1 and D2 MSNs equal (JC1 = JC2),
and systematically increased the firing rate of the cortical inputs to both D1 and D2 MSNs:
JC1
¼
JC2
lctx
d1
¼
lCTX JC1 þ DCTX ;
lctx
d2
¼
lCTX JC2 ;
DCTX
¼
lctx
DMSN
¼
lD1 lD2
d1
8ð0 Hz lCTX 20 HzÞ; 8ð0 Hz DCTX 13 HzÞ
8ð0 Hz < lCTX < 20 HzÞ
lctx
ð1Þ
d2
where λctx_d1 and λctx_d2 denote the total cortical drive to the D1 and D2 MSNs, respectively
and ΔCTX denotes the extra excitatory input to the D1 MSNs. This scenario is equivalent to a
state when no learning has happened and the cortex has no preference for choosing between
activating D1 or D2 MSNs. Here, we considered only those parameter ranges for which neither
of the two subpopulations was completely shut down by mutual and/or recurrent inhibition,
because that would be a trivial solution of the network dynamics, which is also not supported
by the experimental data [8].
When the two types of MSNs are driven with identical input rates (λctx_d1 = λctx_d2), the output firing rate of D2 MSNs always exceeded that of D1 MSNs, irrespective of the magnitude of
the input rate (Fig 2A: main diagonal, Fig 2B: points corresponding to ΔCTX = 0.0Hz). This
shows the inherent bias of the striatal network towards higher firing rates of D2 MSNs [17], for
the reasons explained above.
When the input rate to D1 MSNs exceeds that to D2 MSNs by a sufficiently large amount
(here ΔCTX 2Hz), D1 MSNs always fired at a higher rate than D2 MSNs, irrespective of the
magnitude of the cortical input rate (Fig 2A: area below the dashed line, Fig 2B: area corresponding to ΔCTX > 2.0Hz).
An interesting transitional region was observed when the input rate to D1 MSNs exceeded
that to D2 MSNs by only a small amount (here: 0Hz < ΔCTX < 2Hz). In this case, the ‘winner’
of the competition between D1 and D2 MSNs depends on the size of the cortical input rates.
For small input rates to the two MSNs, D1 MSNs have the relatively higher firing rates (i.e.
ΔMSN > 0 Hz). However, as both input rates were increased, while maintaining the small difference between the two, the recurrent inhibition from D2 to D1 MSNs increased such that, beyond a certain input rate ltran
CTX (here 10 Hz), it exceeded the extra small excitatory drive to
the D1 MSNs, resulting in the output rate of D2 MSNs surpassing that of D1 MSNs (i.e. ΔMSN
< 0 Hz). Hence, in this transitional region with small differences in the excitatory inputs to
both MSN types (0Hz < ΔCTX < 2Hz), it was possible to have higher firing rate for either of D1
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Go/No-Go Decision Transition Threshold in the Striatum
Fig 2. Balance of D1 and D2 MSNs firing rates as a function of cortical input rates. (A) ΔMSN as a function of λctx_d1 and λctx_d2 (cf. Eq 1), scenario-I.
Diagonal (marked with white line) represents the case that the two MSN populations receive equal cortical drive (λctx_d1 = λctx_d2, X = 0.0). With these inputs,
ΔMSN is always negative. This shows the inherent bias towards D2, due to the asymmetrical connectivity. The area below the diagonal represents the regime
of higher input drive to D1 λctx_d1 > λctx_d2, X > 0. Off-diagonal bands represent ΔMSN for a constant difference in cortical input rates (ΔCTX) to the two MSN
populations. The dashed line marks the desirable regime of operation for the striatum, in which a systematic increase in the cortical input can reverse the sign
of ΔMSN. (B) ΔMSN as a function of ΔCTX to the two MSN populations. For a constant value of ΔCTX, ΔMSN changes depending on λCTX (cf. Eq 1). The oval
marks the region where an increase in the cortical input rates for a constant ΔCTX changes the sign of ΔMSN from positive to negative, indicating a higher firing
rate of the D2 MSNs. (C) Same as in panel A for the scenario-II (cf. Eq 9). In this scenario we considered JC1 > JC2, therefore, the difference in the drive to D1
and D2 MSNs scales with the difference between JC1 and JC2. Since JC1 > JC2, the diagonal itself lies in a desirable regime, where ΔMSN changes from
positive to negative for increasing cortical input rate. (D) Same as in panel B for the scenario-II.
doi:10.1371/journal.pcbi.1004233.g002
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Go/No-Go Decision Transition Threshold in the Striatum
or D2 MSNs, depending on the size of the cortical input drive (Fig 2A, dashed line and 2B, set
of points marked by the oval).
Thus, as a consequence of the asymmetric connectivity between D1 and D2 MSNs, the striatum can generate higher firing rates for either D1 or D2 MSNs, depending on the cortical input
rates to the two, provided the difference between the two inputs is small enough. As a result,
the difference between the two output rates ΔMSN reverses as the cortical input rate is increased.
In other words: due to the asymmetric connectivity between D1 and D2 MSNs, the striatum
can flexibly choose between the direct and the indirect pathway, depending on the level of the
cortical input firing rates. By contrast, in a regime with a large difference between the excitatory
drives to D1 and D2 MSNs, the striatum would lose this ability to switch between the two, depending on the cortical input.
To understand the mechanism underlying the sign reversal we derived an expression for
ΔMSN. In this input scenario the dynamics of the two MSN populations are given by:
_ ¼ 0:01l þ Sð J l þ J l þ J l þ l þ D Þ
lD1
D1
11 D1
12 D2
1F FSI
CTX
CTX
ð2Þ
_ ¼ 0:01l þ Sð J l þ J l þ J l þ l Þ
lD2
D2
21 D1
22 D2
2F FSI
CTX
ð3Þ
To simplify the analysis, we ignored the leak term and considered a linear transfer function,
S(z) = z for which we calculated the fixed point rates λD1 and λD2. It should be noted that absolute values of parameters are used for the analysis, and to calculate λD1 and λD2 (Eqs 4, 5) correct sign of the various weights must be used, i.e. J11, J12, J22, J21, J1F, J2F are negative and
similarly, ΔCTX is positive.
lD1 ¼
lFSI ð J2F J12 J1F J22 Þ þ lCTX ð J22 J12 Þ þ J22 DCTX
ð J22 J11 J12 J21 Þ
ð4Þ
lD2 ¼
lFSI ð J2F J11 J1F J21 Þ þ lCTX ð J21 J11 Þ þ J21 DCTX
ð J12 J21 J22 J11 Þ
ð5Þ
where ΔCTX is again the extra excitatory input given to D1 MSNs. The response rate difference
ΔMSN = λD1 − λD2 now reduces to:
DMSN ¼
ð J2F J12 þ J2F J11 J1F J21 JF1 J22 ÞlFSI þ lCTX ð J22 þ J21 J11 J12 Þ þ DCTX ð J22 þ J21 Þ
ð J22 J11 J12 J21 Þ
ð6Þ
which can be divided into two terms, Inpstr (Eq 7) and Inpadd (Eq 8):
I
II
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
l ð J þ J21 J11 J12 Þ þ ð J2F ð J12 þ J11 Þ J1F ð J21 þ J22 ÞÞlFSI
Inpstr ¼ CTX 22
ð J22 J11 J12 J21 Þ
Inpadd ¼
DCTX ð J22 þ J21 Þ
ð J22 J11 J12 J21 Þ
ð7Þ
ð8Þ
Here, Inpstr is a measure of the effective external input, that is, cortical excitation and feedforward inhibition, each scaled by the recurrent dynamics of the striatum. The first term in
Inpstr(term I) has an effective negative contribution to ΔMSN, due to the higher effective recurrent inhibition to D1 ((J11 + J12) > (J22 + J21)). The contribution of FSI inhibition (term II)
could be effectively positive or negative, depending on the ratio of feedforward (J1F and J2F)
and recurrent ((J11 + J12) and (J22 + J21)) inhibition. In our case, it is negative because (J2F(J11 +
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Go/No-Go Decision Transition Threshold in the Striatum
Fig 3. Mechanism of decision transition threshold. The mechanism of decision transition threshold is explained analytically. (A) ΔMSN plotted for different
values of λCTX and λFSI. The dashed line refers to the case of increasing cortical excitation and feedforward FSI inhibition (as also discussed in Fig 2A).
However, ΔMSN changes from positive to negative along the rows (keeping λFSI constant and increasing λCTX) and along the columns (keeping λCTX constant
and increasing λFSI). (B) Mechanism for scenario I, ‘additive’ input, Refer to Eqs 4–8. ΔMSN = Inpstr + Inpadd plotted for increasing values of λCTX. (C) ΔMSN =
λD1 − λD2 as calculated from Eqs 2, 3. (D) ΔMSN plotted for different values of λCTX and λFSI. The dashed line refers to the case of increasing cortical excitation
and feedforward FSI inhibition (as also discussed in Fig 2C). However, ΔMSN changes from positive to negative by increasing λFSI for a constant value of λCTX.
(E) ΔMSN = D1eff + D2eff plotted for increasing values of λCTX. Mechanism for scenario II, ‘multiplicative’ input. Refer to Eqs 13–14. (F) ΔMSN = λD1 − λD2 as
calculated from Eqs 21, 22.
doi:10.1371/journal.pcbi.1004233.g003
J12) < J1F(J22 + J21)). On the other hand, the additional input (Inpadd) is a positive constant proportional to ΔCTX. That is, ΔMSN shows a competition between Inpadd and Inpstr.
When λCTX is increased, Inpstr increases linearly and exceeds the Inpadd at the value ltran
CTX ,
thereby switching the sign of ΔMSN from positive to negative (Fig 3B). This approximative result of the linearized equation without leak was confirmed when we numerically calculated
ΔMSN from the full Eqs. 2–3 (Fig 3C).
Inpstr (Eq 7) is determined by both λCTX and λFSI. Therefore, to understand the contribution
of the activity of fast spiking neurons (assuming their overall contribution to be negative), we
estimated ΔMSN while independently varying λCTX and λFSI (Fig 3A). We found that for a fixed
value of λCTX the sign of ΔMSN could be reversed by an increase in λFSI. Similarly, for a constant
value of λFSI the sign of ΔMSN could be reversed by an increase in λCTX. In fact, in this scenario,
ΔMSN reverses its sign with respect to λCTX even in the absence of feedforward inhibition.
Hence we conclude that the recurrent connectivity between the two MSN populations is the
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Go/No-Go Decision Transition Threshold in the Striatum
main determinant of the ΔMSN sign reversal. Note that in Fig 3C and 3B we showed the sign reversal of ΔMSN when both λFSI and λCTX increased in proportion (cf. black lines in Fig 3A).
Scenario II: Increase the relative strength of cortical projection to D1 MSNs. Instead of
increasing the rate of cortical excitation, we could also increase the strength of the cortical synapses to the D1 MSNs to provide these neurons with more input. This gives rise to a second, in
this case a ‘multiplicative’ scenario (Fig 2C and 2D):
JC1
>
JC2
lctx
d1
¼
lCTX JC1 ;
8ð0 Hz < lCTX < 20 HzÞ
lctx
d2
¼
lCTX JC2 ;
8ð0 Hz < lCTX < 20 HzÞ
DCTX
¼
lCTX JC1 lCTX JC2
ð9Þ
This scenario refers to a state in which cortico-striatal projection strengths have been modified
in favour of D1 (and possibly also D2, but less) MSNs, due to synaptic learning. As in scenario
I, we estimated ΔMSN by systematically varying the input rate (λCTX) and the strength of the
cortico-striatal projections (JC1, JC2). Notice that in this ‘multiplicative’ scenario the difference
between the input drives of D1 and D2 MSNs increases proportionally with λCTX. Because the
cortical input to D1 MSNs is now enhanced by the larger synaptic strength (JC1 > JC2), the diagonal in Fig 2C is no longer in the regime where D2 MSNs dominate (ΔMSN < 0). Similarly to
scenario I, also here also we found only a very small regime of input drive for which either D1
or D2 MSNs can have higher firing rate as cortical input rate is changed (the oval area marked
in Fig 2D). Therefore, we argue that for a flexible switching between direct and indirect pathways the difference between the input rates/weights to the two MSN populations and their outputs rates should be small.
As in the ‘additive’ scenario, we performed a simplified analysis of the dynamics of Eqs 21
and 22 without leak term and a linear transfer function and found the fixed point rates λD1 and
λD2:
lD1 ¼
lFSI ð J2F J12 J1F J22 Þ þ lCTX ð JC1 J22 J12 JC2 Þ
ð J22 J11 J12 J21 Þ
ð10Þ
lD2 ¼
lFSI ð J2F J11 J1F J21 Þ þ lCTX ð JC1 J21 J11 JC2 Þ
ð J12 J21 J22 J11 Þ
ð11Þ
ΔMSN now reduces to
DMSN ¼
lFSI ð J2F ð J12 þ J11 Þ J1F ð J22 þ J21 ÞÞ þ lCTX ð JC1 ð J22 þ J21 Þ JC2 ð J12 þ J11 ÞÞ
ð J22 J11 J12 J21 Þ
ð12Þ
This expression can be split into the effective contributions of D1 (D1eff) and D2 (D2eff),
respectively:
lCTX JC1 ð J22 þ J21 Þ lFSI J1F ð J22 þ J21 Þ
ð J22 J11 J12 J21 Þ
ð13Þ
lCTX JC2 ð J12 þ J11 Þ þ lFSI J2F ð J12 þ J11 Þ
ð J22 J11 J12 J21 Þ
ð14Þ
D1eff ¼
D2eff ¼
Cortico-striatal projections are stronger than interneuronal projections to MSNs, therefore,
JC1 > J1F and JC2 > J2F, and hence, D1eff is positive and D2eff is negative. At low cortical rates, j
D1eff j is larger than j D2eff j due to the scaled cortical excitation for D1 (JC1 > JC2) and hence,
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Go/No-Go Decision Transition Threshold in the Striatum
the D1 type neurons dominate. However, for higher cortical driving rates, stronger feedforward
inhibition to the D1 MSNs ensures that D2 MSNs exceed the firing rates of D1 MSNs, reversing
the sign of ΔMSN (Fig 3D). The values are qualitatively similar to ΔMSN calculated numerically
from the full Eqs 21 and 22 with the leak term and non-linear transfer function (Fig 3F). In
general, as is evident from the Eq 12, the sign of ΔMSN is determined by both λCTX and λFSI.
However, in this input scenario the relation between ΔMSN and the firing rates λCTX and λFSI is
more complex than in the ‘additive’ input scenario. Similarly to the ‘additive’ input scenario,
for a fixed value of λCTX the sign of ΔMSN could be reversed by an increase in λFSI (Fig 3D).
However, for a constant value of λFSI, an increase in λCTX changes the sign of ΔMSN from negative to positive. This is because a large value of λCTX is needed to overcome the effect of preferential connections of the FSIs to D1 MSNs. Indeed, the higher the firing rate of FSIs, the
stronger cortical input would be needed to switch the sign of ΔMSN. Note that in Fig 3F and 3E
we showed the sign reversal of ΔMSN when both λFSI and λCTX increased in proportion (cf.
black lines in Fig 3D).
Although, qualitatively the two scenarios I and II yield the same result (compare Fig 2A and
2B with Fig 2C and 2D) their mechanisms are slightly different. The ‘multiplicative’ input scenario requires preferential inhibition from FSis, whereas the ‘additive’ input scenario relies on
asymmetrical projections between D1 and D2 MSNs (however, see also later in subsection “Effect of symmetrical FSI projections on the DTT”). In all subsequent analyses, we used the scenario II, keeping JC1 > JC2 and maintaining identical input rate λCTX to both D1 and D2 MSNs.
The reason is that scenario I is not suitable to compare the effect of input correlations, as this
requires the input rates for D1 and D2 MSNs to be equal.
Decision transition threshold in the striatum
To obtain further insight into the activity balance of D1 and D2 MSNs, we considered a state of
the striatum in which the cortico-striatal projections had been strengthened for a particular
‘Go’ task (JC1 > JC2, cf. the ‘multiplicative scenario’ for detail). This scenario is shown for the
firing rate model in Fig 1B and for the spiking neural network in Fig 4). In this state, we estimated the difference between D1 and D2 MSN output rates as a function of cortical input rate.
For a constant difference in synaptic strength to D1 and D2 MSNs (ΔJ = JC1 − JC2 = const), the
difference between the output firing rates of D1 and D2 MSNs (ΔMSN) changes from positive to
negative as the cortical input rate increases (Figs 1B and 4). That is, the response of the striatum
to a range of cortical input rates can be divided into two regimes: a regime where D1 firing rate
exceeds D2 firing rate and a regime where this situation is reversed. At the transition point
ltran
CTX , the D1 and D2 firing rates are equal. This behavior can be interpreted as a striatal bias towards the ‘Go’ pathway below a certain cortical input rate (lCTX < ltran
CTX ), switching to a bias
towards the ‘No-Go’ pathway for higher cortical input rates (lCTX > ltran
CTX ) (cf. Figs 4A and 4B
and 1B).
The cortical input rate (for a constant difference ΔCTX) at which ΔMSN changes its sign,
switching the striatum from a putative ‘Go’ to ‘No-Go’ mode, can be referred to as decision
transition threshold (DTT). That is, the striatum acts as a threshold device, indicating the
change in the cortical input rates by the sign of the difference between the firing rates of D1
and D2 MSNs—higher D1 activity reflecting lower cortical input rate and higher D2 activity reflecting higher cortical input rate. Therefore, we propose that the asymmetric striatal connectivity allows it to ‘sense’ small changes in the cortical activity reaching the striatum and to
modulate the balance between ‘Go’ and ‘No-Go’ states of the basal ganglia accordingly.
Here we considered the scenario in which the sign of ΔMSN changes from positive to negative, i.e. going from ‘Go’ to ‘NoGo’ bias. It is possible to tune the strength of FSI inputs such
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Go/No-Go Decision Transition Threshold in the Striatum
Fig 4. Decision transition threshold in the striatum. Mean firing rates of D1 and D2 MSNs populations for different cortical input rates with constant
difference between the drives to the two populations (scenario - II cf. Eq 9). The DTT in this case is 20 Hz. The rasters and PSTH for the spiking activity for
D1 and D2 MSNs are shown for two points, before and after crossing the DTT. (A) (top) Rasters for D1 and D2 populations at λCTX = 10.0Hz. (Middle) PSTHs
for D1 and D2. (Bottom) Difference between the firing rates of D1 and D2 (i.e ΔMSN). Notice that the firing rate of D1 MSNs is higher than that of D2 MSNs
(ΔMSN > 0). At the decision transition threshold, the bias switches from D1 to D2.(B) Same as (A) but at λCTX 25Hz, where ΔMSN < 0.
doi:10.1371/journal.pcbi.1004233.g004
that initially the striatum is biased towards the ‘NoGo’ pathway and a subsequent increase in
the cortical input changes the bias to the ‘Go’ pathway (cf. Fig 3A and 3D).
Previously, Lo and Wang [18] described the cortico-basal ganglia loop as a DTT for reaction
time tasks. However, in their model the striatum was considered a passive unit, the threshold
of which could be tuned by the strength of cortico-striatal synapses. In our model, for the first
time, we describe the striatum as an active participant in deciding the level of DTT. It should
however be noted, that our model explores the DTT under steady state conditions only. In this
regime, the striatal network settles into stable fixed points (cf. Materials and Methods) and,
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Go/No-Go Decision Transition Threshold in the Striatum
therefore, the DTT is a stable network state as well. A DTT might also emerge as a transient
state in the network dynamics in the presence of additional mechanisms like short term and/or
long term plasticity of cortico-striatal and/or striato-striatal synapses and intrinsic plasticity of
striatal neurons. Further exploration of such mechanisms is, however, beyond the scope of the
present work. The presence of such dynamic DTT suggests several possible control mechanisms, that may set the bias towards either of the two pathways along the basal ganglia downstream nuclei, depending on the animal behavioral state, motivation and acquired learning. In
the following sections, we describe three possible mechanisms that can modify the balance of
D1 and D2 activity by modulating the level of the DTT, thereby rendering it as a
dynamic variable.
In a standard feedforward model of the basal ganglia, the GPi activity can indicate the
higher activity of D1 (D2) MSNs by decreasing (increasing) its activity around the baseline.
However, recent data shows that the basal ganglia network is more complicated than the simple
feedforward model (cf. review by [19]). In future it would be important to show whether GPi
can change its activity according to the D1 and D2 MSNs firing rate differences, in a more updated model of the basal ganglia.
Robustness of DTT
In the above, the DTT was demonstrated assuming that the firing rate of the cortical input and
the difference between the input to the D1 and D2 MSNs (ΔCTX) remaines constant, but in
more realistic conditions the input rate and ΔCTX could both be random variables. Therefore,
we tested the viability of the concept of DTT when the mean firing rate of the cortical inputs to
the D1 and D2 MSNs was chosen from a low-pass filtered noise. In addition, the extra input to
the D1 MSNs was chosen from a low-pass filtered noise. To quantify the DTT we measured
ΔMSN in pre-DTT and post-DTT regions (Supplementary S1 Fig). It is obvious that when ΔCTX
is noisy the difference between the firing rate of the D1 and D2 MSNs around the DTT decreases, however ΔMSN in the pre-DTT and post-DTT remains large enough to be reliably measured (Supplementary S1 Fig). When the standard deviation is increased upto 50% of the mean
value of the ΔCTX, both pre-DTT and post-DTT areas decreased only by 20% (Supplementary
S1 Fig). Correlations in the spiking is another way of generating transient fluctuations in the
input to the D1 and D2 MSNs and the effect of correlation induced fluctuations is described
below (cf. subsection “Effect of cortical spiking activity correlations on the DTT”)
In addition to the input rate fluctuations, the connectivity between the D1 and D2 MSNs
could also be different from what we have previously used. Therefore, we tested the robustness
of the DTT for variation in the chosen values of striatal network connectivity (e.g. J11, J12 etc.).
There are up to eight network connectivity parameters (cortical input to the two MSNs populations, mutual and within MSNs connectivity and MSN and FSI connectivity) which could potentially affect the existence of the DTT. We systematically varied all the parameters (except
the FSI connectivity to the MSNs) by ±100% around their mean values (as used throughout
this manuscript). Visualisation of this high dimensional parameter space is difficult. Therefore,
here we checked if the DTT exists when only one of the parameters is varied systematically by
±100% of its mean value, while other parameters can take any value between ±100% of their
mean value. A DTT is said to exist if ΔMSN for increasing values of λctx and the given parameter
combination changes sign from positive to negative values exactly once. This was detected algorithmically as well as by visual inspection. Indeed, the DTT is a robust to such changes in the
network connectivity and for every parameter we tested here (cortical inputs to the MSNs and
within and mutual connectivity of MSNs), we can find the DTT (cf. Supplementary S2A–S2F
Fig). Only for very small values of the within connectivity of the D1 and D2 MSNs (J11 and J22),
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Go/No-Go Decision Transition Threshold in the Striatum
we could not find a solution for the DTT. In addition, we also checked the existence of the
DTT for various pairs of the network connectivity parameters (e.g. J21 vs. J11). While there are
some forbidden regions in which DTT does not exist, for a large variations in the striatal connectivity, the DTT can be observed (Supplementary S2 Fig). When the existence is visualised as
a scatter diagram as a function of a pair of the network connectivity parameters, the mean of
these clusters lie very close to the values we have used throughout this manuscript.
Effect of cortical spiking activity correlations on the DTT
Task related activity in the cortex is modulated both in firing rates and in spike correlations
[20]. Here we investigated how spike correlations in the cortical inputs to the striatum may affect the balance between D1 and D2 MSNs. Injection of correlated inputs to a population of
MSNs requires the separation of input correlations into two categories [12]: correlations arising
due to convergence of cortical projections onto individual neurons, i.e correlations within the
pool of pre-synaptic neurons seen by individual MSNs (in the following referred to as ‘within’correlations—W) and correlations arising due to divergence of cortical projections, i.e. correlations between the pools as seen by different MSNs, the so-called shared input (in the following
referred to as ‘between’ correlations—B) (cf. Materials and Methods and Fig 5A). Note that because both B and W correlations arise from convergent and divergent projections of the same
pre-synaptic population, between correlations are always smaller than or equal to within correlations(B W [21, 22],cf. Materials and Methods).
To understand the effect of input correlations on the balance of D1 and D2 MSN activity,
we independently varied both within and between input correlations while maintaining a constant input rate (MIP model in [23]). These simulations were only performed for the spiking
neural network model since modelling correlations in a mean field model is non-trivial, especially when post-synaptic neurons are recurrently connected. (cf. Materials and Methods, Fig
5A). As expected, within-correlations affected the individual firing rates in a non-monotonic
manner [23] for both D1 and D2 MSNs. For weak input correlations, as D1 MSNs form stronger synapses with cortical afferents, (JC1 > JC2; scenario II), D1 MSNs spike at higher rate than
D2 MSNs (Fig 5B). For high values of W the MSNs operated in a so called ‘spike-wasting’ regime, because the average count of coincident spikes (or the input variance) exceeded the spike
threshold [23] and therefore, W caused no difference in the output firing rates of D1 and D2
MSNs (Fig 5C).
That is, the difference in the firing rates of D1 and D2 MSNs (ΔMSN) changed non-monotonically as a function of W, reaching its maximal values for W = Wopt 6¼ 0 (Fig 6A and 6B).
The effect of W on ΔMSN was further modulated by the between-correlations (B). For W <
Wopt, increasing B resulted in a monotonic increase in ΔMSN. In this regime, B increased the
correlation within D1 and D2 MSNs and, thereby, increased the effective inhibition of one population by the other, and vice versa. Because D1 MSNs received input with a slightly higher
strength (JC1 > JC2; scenario II), B induced more synchrony among D1MSNs and, hence, they
were better able to inhibit the D2 MSNs (Fig 5E). However, this held only over a smaller parameter regime, because by design B cannot exceed W (cf. Materials and Methods). For W Wopt, ΔMSN decreased monotonically with increasing B. In this regime, an increase in B enhanced the correlations within D1 and D2 MSNs, hence amounting to a ‘wasting of recurrent
inhibition’ [12]. Increased correlations within D1 and D2 MSNs created strong but only short
lasting inhibition leaving out longer lasting time windows without any recurrent inhibition
(Fig 5D). In the absence of sufficiently long-lasting recurrent inhibition the difference ΔMSN
monotonically decreased (Fig 6A and 6B).
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Go/No-Go Decision Transition Threshold in the Striatum
Fig 5. Network activity of D1 and D2 MSNs for selected values of B and W. In all examples the striatum network was configured according to the
scenario-II and D1 MSNs received inputs with higher strength than D2 MSNS. Both MSN populations received cortical input at 7 Hz. (A) Scheme describing
the two types of input correlations: W refers to correlations among the pre-synaptic neurons of a single MSN. B0 refers to correlation among the pre-synaptic
neurons of two different MSNs.(B)B0 = 0.01, W = 0.01, raster(top) and population activity (bottom). For these inputs the activity of the D1 and D2 MSNS is
uncorrelated and, therefore, exerts less effective inhibition on the other population. Because D1 MSNs are configured in the scenario-II, D1 MSNs have a
higher average firing rate. (C) B0 = 0.01, W = 0.33. This value falls in the range of W < Wopt. Here D1 operates in the ‘spike-wasting’ regime and, hence,
cannot effectively inhibit D2, in spite of a higher drive from cortex. (D) B0 = 0.21, W = 0.33. High values of B0 and W leads to wasted inhibition due to periods of
large correlated activity, followed by long periods of quiescence. (E) B0 = 0.21, W = 0.01. For W < Wopt, B increases ΔMSN.
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Go/No-Go Decision Transition Threshold in the Striatum
Fig 6. Effect of B and W input correlations on the balance of D1 and D2 activity. (A) ΔMSN as a function B and W when the two MSNs population
received cortical input with a firing rate of 7 Hz. Here, we considered the scenario-II and set JC1 = 3.6 nS, JC2 = 3.0 nS. At this input rate and with uncorrelated
inputs (B = W = 0), D1 MSNS have higher firing rates. The black contour marks the region beyond which ΔMSN is close to zero. All the values are
concentrated below the diagonal, because of the constraint: B W. ΔMSN varies non-monotonically as a function of W for a constant value of B. Dotted arc
marks the Wopt for a given firing rate where ΔMSN is maximal. For W Wopt, increasing B decreases ΔMSN. For values W < Wopt, increasing B increases
ΔMSN. (B) same as in (A), for input rate of 23 Hz. Wopt has shifted to 0.3 as compared to 0.2 in panel A. (C) ΔMSN as a function of W and input firing rate for B0
= 0.01. The space spanned by the input firing rates and W can be divided into three distinct regimes. Low W and high firing rates, ΔMSN is negative(blue
colours) and the output of the striatum is biased towards D2 MSNs (‘No-Go’ pathway). In the regime where W Wopt, D1 MSNs have higher firing rate than
D2 MSNs and the striatum output is biased towards the ‘Go’ pathway. The third regime spans across very high values of W in which both D1 and D2 MSNs
operate in a spike wasting regime and, therefore, ΔMSN is very low. This regime we define as ‘Region of High-Conflict’ (RHC). Because higher firing rates for
D1 and D2 MSNs are observed in non-overlapping regions in the space spanned by input correlation and rates, we argue that the striatum may act as a
threshold detector and signal the state of cortical inputs by raising the relative activity of D1 or D2 MSNs over the other, respectively.
doi:10.1371/journal.pcbi.1004233.g006
Previously, we have shown that shared inputs (B > 0) reduce the signal-to-noise ratio and
the contrast enhancement in the striatum [12]. Our results here now show that the ‘between’correlations could play an even more important role in initiating the action-selection process
in the striatum. However, whether B correlations improve or impair action selection strongly
depends on the ‘within’-correlations. (Fig 6). With an increase in the cortical input rate, the
amount of inhibition experienced by a MSN in the striatal network also increases due to the increase in MSN firing rates. This requires more coincident excitatory inputs (i.e. higher W) to
overcome the increased inhibition and to drive the output neurons above firing threshold.
Thus, Wopt increases with an increase in input rates (compare Fig 6A and 6B). That is, an increase in cortical input rates broadens the dynamic range of the interaction between B and W
by increasing Wopt (Fig 6C, also discussed in [23]).
These results show that the striatum also acts as a threshold device for cortical input correlations, in the same way we have shown above for cortical input rates. To illustrate this, we plotted ΔMSN as a function of the cortical input rates and within correlations (W), for a fixed value
of B (Fig 6C, cf. Supplementary S4 Fig). Here, we can define three regions in the space of input
firing rates and input correlations for which ΔMSN is positive, negative and zero respectively
(Fig 6C), reflecting how the striatum will react to the first and second order statistics of the cortical inputs by activating the direct, the indirect or none of the two pathways. While the direct
and indirect pathways reflect the striatal preference for a ‘Go’ and ‘No-Go’ action, respectively,
ΔMSN 0 could reflect a state of high conflict where it is useful to halt the decision making process until more information is available.
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Go/No-Go Decision Transition Threshold in the Striatum
Effect of Dopamine on the DTT
MSNs express D1 and D2 types dopamine receptors forming the direct and indirect pathways
of the BG, respectively. Dopamine affects striatal function by modulating the intrinsic excitability of the MSNs and synaptic weights [24] and synaptic plasticity [25, 26] of the cortico-striatal
projections. Because we are interested in how striatum processes cortical inputs, here we studied the effect of dopamine-induced modulation of the strength of cortico-striatal projections.
Dopamine induces long term potentiation (LTP) in cortico-striatal synapses onto D1 MSNs
and long term depression (LTD) in cortico-striatal synapses onto D2 MSNs [24]. This steady
state effect of dopamine is often modelled by changing the weight of cortico-striatal synapses
for D1 (JC1) and for D2 (JC2) [27]. Therefore, in our model we simulated the effect of dopamine
depletion by decreasing JC1 and increasing JC2, whereas higher than normal dopamine was simulated by increasing JC1 and decreasing JC2.
Thus, in a low dopamine state (e.g. in PD patients), D2 MSNs received much stronger cortical input and, therefore, exerted more inhibition on to the D1 MSNs. This shifted the DTT towards lower cortical input rates (Fig 7A) reducing the regime under which the direct pathway
could dominate the indirect pathway. Consistent with experimental observations and previous
models [4, 5], our model suggests that dopamine depletion would (1) increase the firing rate of
D2 MSNs, thereby, (2) introducing a preference for ‘No-Go’ type actions.
By contrast, in a high dopamine state (e.g. as in PD patients with L-Dopa treatment) D1
MSNs received much stronger cortical input and were able to maintain higher output firing
Fig 7. Effect of Dopamine and GPe firing rates on the balance of D1 and D2 activity. (A) Mean firing rates of D1 MSNs and D2 MSNs populations plotted
for different levels of dopamine. Darker shades of blue (red) correspond to D1 (D2) MSN activity for higher levels of dopamine. For lower than normal levels of
dopamine, the DTT shifts to the left (from 19 Hz to 9 Hz). This decreases the regime with a bias towards D1 MSNs. For higher than normal levels of
dopamine, the DTT shifts to right (from 19 to 27Hz). This, in turns increases the regime with a bias towards D1 MSNs. (B) Effect of GPe firing rates on the
DTT. Solid lines refer to the normal state of the striatum. GPe inhibits the fast spiking interneurons, shifting the DTT to the right (from 9 to 14Hz) (dotted
lines). Therefore, the regime with a D2 bias (10 < λCTX < 14, D2 solid red line) now has a bias towards D1 MSNs (blue dashed line) (C) In dopamine depleted
conditions, inhibition of fast spiking interneurons via GPe is not able to switch the bias from D2 MSNs to D1 MSNs (compare dashed red and dashed blue
lines).
doi:10.1371/journal.pcbi.1004233.g007
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Go/No-Go Decision Transition Threshold in the Striatum
rate than D2 MSNs for a wider range of cortical inputs. That is, high dopamine shifted the
DTT towards higher cortical inputs (Fig 7A) and increased the regime under which the direct
pathway could dominate the indirect pathway. Consistent with experimental observations, our
model suggests that an increase in steady state dopamine levels would (1) increase the firing
rate of D1 MSNs [28], thereby (2) introducing a preference for ‘Go’ type actions or dyskinesia.
In summary, therefore, synaptic effects of dopamine can be seen as a change in the DTT of
the striatum.
Effect of GPe induced disinhibition of FSI activity on the DTT
Approximately 1% of the striatal neuronal population is composed of parvalbumin expressing
fast spiking interneurons (FSIs) [29]. Despite being only a small portion of the neuronal population in the striatum, they can strongly modulate MSNs activity due to their high degree of
convergent and divergent connections to the MSNs. Moreover, they form synaptic connections
near or at MSN’s somata as opposed to glutamergic synapses or local MSN axon collaterals,
which usually connect to spines or dendritic shafts [30]. Furthermore, recent data suggests that
FSIs preferentially inhibit direct pathway MSNs (D1) as compared to indirect pathway MSNs
(D2) [16].
This implies that the activity of FSIs could alter the balance between the direct and indirect
pathways and may even change the DTT. For instance, Fig 4A showed that for small ΔCTX, D1
MSNs fire at higher rate than D2 MSNs at low inputs, however an increase in FSI firing rate
would inhibit D1 MSNs more than D2 MSNs and, hence, reduce the DTT. By contrast, a reduction in the firing rate of FSIs will have an opposite effect. Thus, FSIs could play a vital role
in maintaining the default state of the striatal circuit as a ‘No-Go’ state, which could be changed
to a ‘Go’ state by exciting D1 MSNs more than D2 MSNs. Alternatively, a depletion of FSIs
(e.g. as seen in patients with Tourette syndrome), a reduction in their firing rate or connectivity
should reverse this scenario and enforce a ‘Go’ state as the default striatum state leading to impulsivity-related behavioral symptoms.
Besides cortical inputs, other factors such as back projections from the GPe could also modulate the effect of FSIs on MSNs by modulating their firing rates (Fig 7B). The effect of GPe on
FSIs was modelled in our network model as a constant amount of inhibition on FSI firing rates.
An increase in GPe activity (for instance due to an increase in STN activity) would effectively
release D1 and D2 MSNs from the feedforward inhibition. However, the effect will be further
amplified for D1 MSNs because these neurons receive more input from FSIs than D2 MSNs
do. This can shift the bias from a ‘No-Go’ to ‘Go’ state for high cortical input rate (as shown in
Fig 7B, dotted lines, for λCTX in the range (10–15 Hz)). The same effect of FSI activity can be
observed in the Fig 3D. The same effect is shown for mean field model in Supplementary S5
Fig. The temporal dynamics of such a modulation by GPe are shown in Supplementary S6 Fig
for a mean field model.
Thus, an increase in the activity of the FSIs would result in shifting the DTT to higher cortical rates. We argue that increased firing rates in STN might not only maintain a tight rein on
GPi to prevent a premature response in a high conflict decision, as suggested by [5], but also assist in resolving the conflict in the striatum via pallido-striatal back projections.
In the dopamine depleted condition when the DTT is very small, the effect of the GPe back
projection might not be able to modulate the DTT any further (Fig 7C). Similarly, tonically active interneuron (TANs) could exploit the FSI to D1 and D2 MSNs connectivities by modulating the strength of FSI synapses onto the respective MSNs [31], thereby, changing the
DTT indirectly.
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Go/No-Go Decision Transition Threshold in the Striatum
Effect of symmetrical FSI projections on the DTT
Our model is based on in vitro results from Taverna et al. [14] and Gittis et al. [16]. Taverna
et al. [14] showed that D2 MSNs inhibit D1 MSNs more strongly in terms of numbers of projections and synaptic strengths as compared to vice versa, whereas Gittis et al. [16] showed that
in the Lhx6-GFP transgenic mice, FSI has more projections to D1 than to D2, while the synaptic strengths of the projections to both MSN types are comparable. While it is unlikely that the
preferential connectivity of FSIs to D1 MSNs is specific to the mouse strain (A. Gittis personal
communication), another recent study by Planert et al. [15], while supporting the results of Taverna et al. [14], did not report any significant preference in the connectivity of FSIs to D1
MSNs. Thus, even though Planert et al. drew their conclusion based on only a small number of
neurons, the apparent inconsistency of the data in the two studies requires a reevaluation of
the connectivity of FSIs to MSNs. We, therefore, analysed the striatal dynamics for equal innervation of MSNs by FSIs.
Symmetrical FSI projections in the additive input scenario. In this case, the inhibition
from FSIs is considered same to be the same to both MSN subpopulations (J1F = J2F = JF).
Therefore, the expression for ΔMSN in Eq 6 modifies to:
DMSN ¼
ð J12 þ J11 J21 J22 Þð JF lFSI lCTX Þ þ DCTX ð J22 þ J21 Þ
ð J22 J11 J12 J21 Þ
ð15Þ
ð J12 þ J11 J21 J22 Þð JF lFSI lCTX Þ
ð J22 J11 J12 J21 Þ
ð16Þ
DCTX ð J22 þ J21 Þ
ð J22 J11 J12 J21 Þ
ð17Þ
Inpstr ¼
Inpadd ¼
That is, when J1F = J2F = JF the recurrent connectivity between the two MSN subpopulations
(D1 and D2) becomes a more important determinant of the sign of ΔMSN. Given the nature of
the MSNs inter-connectivity (J12 + J11 > J21 + J22) and the preference of FSI projections onto
MSNs (JF < 1), Inpstr remains negative and increases (decreases) with λFSI (λCTX). As a consequence, for a fixed value of λFSI, an increase in λCTX results in a sign reversal of ΔMSN, thereby,
imposing a DTT. On the other hand, for a fixed value of λCTX, ΔMSN increases with increasing
λFSI. However, note that if ΔMSN is positive, λFSI enhances the magnitude of ΔMSN, whereas
when ΔMSN is negative, λFSI attenuates the magnitude of ΔMSN (Fig 8A). Thus, in a scenario
where λFSI and λCTX are comodulated a DTT can be observed (cf. Fig 8A, solid black line and
Fig 8B).
Symmetrical FSI projections in the multiplicative input scenario. In the multiplicative
input scenario, when FSI projections to the D1 and D2 MSN subpopulations are equally strong
(J1F = J2F = JF), the Eqs 13 and 14 reduce to:
lCTX ð JC1 ð J22 þ J21 ÞÞ
ð J22 J11 J12 J21 Þ
ð18Þ
lCTX ð JC2 ð J12 þ J11 ÞÞ
ð J22 J11 J12 J21 Þ
ð19Þ
D1eff ¼
D2eff ¼
commeff ¼
PLOS Computational Biology | DOI:10.1371/journal.pcbi.1004233 April 24, 2015
JF lFSI ðð J12 þ J11 Þ ð J22 þ J21 ÞÞ
ð J22 J11 J12 J21 Þ
ð20Þ
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Fig 8. DTT in striatum with symmetrical FSI projections. (A) For the ‘additive’ input scenario, the striatum exhibits a DTT, in spite of the symmetrical
feedforward inhibition from FSIs, refer to Eqs 15–17. A DTT is encountered not only along the diagonal (i.e increasing cortical excitation as well as
feedforward inhibition from FSIs-dashed line), but also for clamped value of FSIs with increasing λCTX. (B) ΔMSN = Inpstr + Inpadd plotted for increasing values
of λCTX. (C) ΔMSN = λD1 − λD2 as calculated from Eqs 2, 3. (D) In a ‘multiplicative’ input scenario, however, ΔMSN always remains positive for symmetrical FSI
projections along the diagonal. A DTT can be imposed by clamping the FSI rates at a constant value, while increasing the cortical excitation. (E) ΔMSN = D1eff
+ D2eff + commeff plotted for increasing values of λCTX. (F) ΔMSN = λD1 − λD2 as calculated from Eqs 21, 22.
doi:10.1371/journal.pcbi.1004233.g008
where commeff is the common FSI feedforward inhibition to both types of MSNs. In this scenario, an increase in the λCTX for a constant value of λFSI enhances the activity of D2 MSNs
and, thereby, reverses the sign of ΔMSN from positive to negative, imposing a DTT.
By contrast, for a fixed value of λCTX, ΔMSN increases with increasing λFSI reversing the sign
of ΔMSN (Fig 8D) from negative to positive.
In a scenario where λFSI and λCTX are co-modulated, the sign reversal of ΔMSN opposes that
of the DTT (cf. Fig 8D, solid black line and Fig 8E). This is because commeff is always positive
due to higher effective inhibition to D1 (J12 + J11), and hence, the sum of D1eff, D2eff and commeff monotonically increases as shown in Fig 8D, 8E and 8F.
These results reveal an important role that FSIs can play in shaping the balance between D1
and D2 MSNs activities. In most cases, FSI activity is likely to be correlated with the cortical activity impinging onto the striatum. However, subtle changes in the correlation between the FSI
and cortical activity can dramatically change striatal function. Moreover, these results also provide a possible experimental scenario to test predictions of our model. For instance, selective
modulation and clamping of the activity of FSIs in a behavioural task could be used to determine whether the striatum operates in a multiplicative or additive mode, and whether FSIs
have same the effective connectivity to both types of MSNs.
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Go/No-Go Decision Transition Threshold in the Striatum
Discussion
Here we studied the dynamical properties of the network of D1 and D2 type MSNs in the striatum. Specifically, we focused on the firing rates of the D1 and D2 MSNs, because the relative
balance of firing in these neurons is crucial for setting the stage for action-selection in the basal
ganglia and reinforcement learning. Using both mean-field analysis and spiking network simulations of D1 and D2 MSNs we show that the experimentally observed asymmetry of recurrent
and mutual connections among the two types of neurons creates a ‘decision transition threshold’(DTT) in the striatum to choose between the direct (‘Go’) and indirect (‘No-Go’) pathways.
Based on this DTT, the striatum performs a threshold operation on the firing rates and spike
correlations reaching the striatum from the neocortex and uses the outcome to bias the actionselection.
The analysis of the striatal network dynamics revealed a narrow range of difference in cortical input rates (ΔCTX), for which it was easy to switch the balance of activity towards D1 or D2
MSNs. If ΔCTX is negative, the striatal dynamics are stuck in a state of constant bias towards D2
MSNs. By contrast, when ΔCTX is large and positive (more input to D1 MSNs), the striatal dynamics remain in a state of constant bias towards D1 MSNs. Thus, we predict that in order to
have a flexible state in which both D1 and D2 MSNs could have higher firing rates and implement a preference towards either ‘Go’ or ‘No-Go’ actions, the D1 MSNs should receive only
slightly higher cortical input than the D2 MSNs (cf. Fig 2 for details).
A more exhaustive parameter search revealed many possibilities of implementing the
thresholding operation. In an ‘additive’ input scenario, a DTT emerges for a constant firing
rate of FSI and increasing cortical rates and vice versa (Fig 3A) together with a third, biologically more realistic possibility of correlated firing rates of cortical inputs and FSIs.
A ‘multiplicative’ input scenario reveals a DTT for the biologically plausible case that the firing rates of the cortical inputs are correlated with the FSIs (Fig 3F and 3E). In addition, DTT
can also be observed for increasing FSI firing rates and a constant cortical excitation (Fig 3D).
This raises the issue of having an input-dependent DTT in the striatum, suggesting a multitude
of mechanisms which could control both the existence as well as the value of the striatal DTT.
When cortical input firing rates and spike correlations are increased, while maintaining the
difference in the firing rates of D1 and D2 MSNs, D1 (D2) MSNs show higher activity than D2
(D1) MSNs for low (high) input rates and spike correlations (Fig 6C). That is, in the space of
input properties (firing rates and spike correlations) two regions exist for which either D1
MSNs or D2 MSNs have relatively higher output firing rates. These two regions are separated
by the so called DTT. There is a third region, spanned by high input correlations ( 0.5) in the
input parameter space, in which both neuron populations have equal firing rates. The notion
of the DTT in the striatum naturally emerges as a consequence of the asymmetric connectivity
between and among D1 and D2 MSNs (Figs 1 and 6). However, this does not imply that the
DTT is fixed for the striatum. In fact, controlled modulation of the DTT is desired because it
could provide a neuronal substrate for implementing notions like learning history, behavioral
state, or motivation to perform a task. Furthermore, this view of striatal function can be extended to better understand neuronal mechanisms underlying various disorders of the basal
ganglia. We identified several components of the BG network structure and activity that could
potentially control and modulate the DTT.
Recent modelling work [32], shows that striatum maintains a balance of firing of D1 and D2
MSNs for different cortical input rates and this balance is maintained even when asymmetric
striatal connectivity is considered. We investigated this discrepancy between our observations
and those described by Damodaran et al. [32] and tuned the leaky integrate and fire neurons to
closely follow the input current and output firing rate (F-I) curves of the D1 and D2 MSNs by
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Go/No-Go Decision Transition Threshold in the Striatum
adjusting the passive properties of the neuron (i.e. membrane capacitance, resting membrane
potential and time constant) according to the values described by Gertler et al. [13](Supplementary S3A Fig). Using an asymmetric striatal network of these D1 and D2 MSNs, we show
that a DTT exist for Scenario I (Supplementary S3B Fig). That is, different F-I curves and excitability of the D1 and D2 MSNs is not sufficient to abolish the DTT and create a balance of D1
and D2 output rates. Damodaran et al. [32] used a more complicated neuron model with some
essential morphological features. To understand the reason for the discrepancy between our results and those described by Damodaran et al. [32], therefore, would require a more systematic
analysis of the striatal connectivity and morphology and integrative properties of the MSNs,
which is beyond the scope of the current manuscript.
Modulation of the DTT by cortical input correlations
Spike correlations in cortical input to the striatum influence the DTT in a complex fashion. To
better understand the effect of input correlations we separately studied the role of correlations
within and correlations between the pre-synaptic pools of the MSNs. Previously, we have
shown that correlations between the pre-synaptic pools of MSNs (B) tend to reduce the signalto-noise ratio of the striatal response to cortical inputs [12].
On the other hand correlations within the pre-synaptic pools of individual MSNs (W) affect
the signal-to-noise ratio of the striatal response in a non-monotonic manner. Here, we found
that W correlations also affect the difference between D1 and D2 MSNs activity (ΔMSN) in a
non-monotonic fashion (Fig 6). Interestingly, when input firing rates result in equal output firing rates of D1 and D2 MSNs, a modulation in W could break the symmetry and increase the
firing rate of either one of the two MSN subpopulations. The value of within-correlations
(Wopt) for which ΔMSN is maximal depends on the input firing rates and the value of betweencorrelations (B) (Fig 6).
The effect of between-correlations (or shared inputs) depends on the value of the withincorrelations. In the regime W Wopt, an increase in shared input (B) decreases the value of
ΔMSN. However, in the regime W Wopt, an increase in B increases the value of ΔMSN. Finally,
for high values of input correlation jΔMSNj is small and the two MSN populations have comparable firing rates, indicating the failure of the striatum to bias an action-selection process.
Such an operating regime could halt the decision making process in high-conflict decisionmaking tasks. Indeed, we predict that in high conflict choice tasks, there should be an increase
of correlation in the cortical activity reaching the striatum.
Modulation of the DTT by FSIs
The preferential connectivity of the FSIs to D1 MSNs gives the FSIs an ability to control both
the difference in the firing rate of the D1 and D2 MSNs and, hence, the DTT. For instance, an
increase in FSI activity would reduce D1 MSNs activity and decrease the DTT whereas a decrease in FSI activity would tip the bias in favour of D1, and hence the ‘Go’ pathway. Such a decrease in FSI activity could be due to insufficient excitation from the cortex, or to high input
from the pallido-striatal back backprojections. We suggest that selective modulation of the FSI
activity, which could shift the DTT in the striatum from ‘No-Go’ to ‘Go’, could be a mechanism
for decision making in a high-conflict situation. This functional concept may also help understand neuronal mechanisms underlying various behavioral phenomena. For instance, the arbitration of FSI via the STN-GPe pathway could explain longer reaction times, required to
respond to a high-conflict decision. Similarly, increased impulsivity in PD patients with DBS
could be explained by the fact that DBS increases GPe firing rates [33], which would decrease
the activity of FSIs by pallido-striatal backprojections, resulting in lowering the DTT.
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Go/No-Go Decision Transition Threshold in the Striatum
Being the primary source of feedforward inhibition, FSIs play an important role in both the
existence and the actual value of the DTT. For example, we find that a preferential inhibition of
D1 MSNs is required for a DTT to be present for ‘multiplicative’ input. By contrast, an ‘additive’ scenario reveals a DTT, irrespective of the asymmetry of FSI projections onto MSNs. We
showed that even in the ‘multiplicative’ input mode the striatum can exhibit a DTT, by clamping the FSIs rates, while increasing cortical input rates. We conclude that FSI activity plays a
prominent role in determining the DTT in the striatum, underlining once more the functional
importance of pallido-striatal backprojections to FSIs.
Modulation of the DTT by dopamine
At the synaptic level dopamine has been described to increase (decrease) the strength of cortico-striatal synapses onto D1 (D2) MSNs. This differential effect of dopamine provides a powerful mechanism to control the DTT and difference between the activities of the two neuronal
subpopulations activities and, hence, the DTT. For instance, a dopamine induced increase in
synaptic inputs to D1 MSNs shifts the DTT in the favour of D1 MSNs, and thereby increasing
the cortical input regime for a ‘Go’ bias. This, in fact, could be a putative mechanism by which
external administration of L-Dopa induces dyskinesia as a prominent side-effect. By contrast, a
reduction in dopamine level reduces the regime for a ‘Go’ bias, while increasing the regime for
a ‘No-Go’ bias. Thus, low dopamine results in higher firing rates in D2 MSNs, possibly manifesting in akinetic symptoms and inducing oscillations in the GPe-STN network [34].
Recent experimental data suggest that the activation of ‘Go’ and ‘No-Go’ pathways might
not be exclusive in vivo awake behaving animals. At least at the population level, both D1 and
D2 MSNs have been shown to be co-activated and neither of the two pathways were completely
shutdown [8]. Consistent with these experimental findings, both D1 and D2 MSNs in our
model are co-activated, however, the two populations differed in their firing rates. We note
that here our goal is not to reproduce the experimental results of Cui et al. 2013 [8] which involved behaving mice performing self-paced lever pressing to retrieve the reward. Apart from
the obvious simplicity of our network model, we think there might be an additional reason for
this. In our model, we considered the striatum as unit representing a single action. Indeed, Cui
et al. 2013 [8] suggest that a non-selective global deactivation of D2 MSNs in the striatum
would abolish suppression of most unwanted motor programs and, hence, lead to hyperkinesia. It can be argued that our model represents such a global activation and deactivation of D1
and D2 in the striatum. The concept of a DTT, if extended to include multiple action channels,
could be used to more closely address the results reported in [8].
Implications of asymmetry in the striatal circuit
Thus far, the striatum has been considered as a homogenous structure, with a weak and homogeneous inter-connectivity. Anatomical evidence, however, suggests that there is an inherent
asymmetry in the striatal circuit in terms of mutual connections between D1 and D2 MSNs, as
well as in feedforward projections from FSIs. In Fig 2, we showed that this asymmetry, for an
appropriate activity regime, has the ability to reinforce the cortical bias (bias towards D1 at low
cortical rates) or override it (bias towards D2 at high cortical rates). Taking cortical input correlations into consideration, these dynamics become even more complex, as is summarized in
Fig 9. Low dopamine levels decrease the regime with D1 bias, while increasing the regime with
D2 bias. Alternatively, an increase in dopamine levels increase the area for a D1 bias, while decreasing the area for a D2 bias. An increase in between-correlations in cortical input (B), however, pushes the contour towards the top left corner in Fig 9 (towards higher firing rates and
lower values of within-correlation (W)), thereby enlarging the area of conflict (RHC). We
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Go/No-Go Decision Transition Threshold in the Striatum
Fig 9. Modulation of the DTT in the striatum. (Left) Each contour encloses an area in the parameter space of input correlation (W) and input firing rates, in
which the firing rate of D1 MSNs exceeds that of D2 MSNs. The blue contour represents the healthy state of the striatum. Loss of dopamine results in a
decrease in the strength of excitatory inputs to the D1 MSNs and, therefore, the blue contour shrank to the red contour indicating that for a large range of firing
rates and correlations D2 MSNs have higher firing rate than D1 MSNs. The green contour on the other hand depicts a state in which there is a high level of
dopamine in the striatum (e.g. in PD patients on L-Dopa treatment) because excessive dopamine increases the overall excitability of D1 MSNs, thereby
expanding the region in which D1 MSNs have a higher firing rate. We refer to the region in which the firing rates of D1 and D2 MSNs are comparable as a
region of high-conflict. (Right) The blue contour is the same as in the left panel. The orange contour shows how the increase in shared correlation (B0 ) could
increase the region of high-conflict (RHC).
doi:10.1371/journal.pcbi.1004233.g009
therefore suggest, that the asymmetry in the striatal circuit is useful to cover different regimes
of D1 and D2 bias, accommodating characteristic features of cortical input, including rates,
input correlations and local dopamine levels.
Given the in vitro experimental conditions [14, 15], the measurement of the connectivities
within and between D1 and D2 MSNs could be biased. In more realistic in vivo conditions, the
connectivities of D1 and D2 MSNs may turn out to be different. In particular, if D1 and D2
MSNs would not differ in their connectivities and membrane properties, there is no reason to
consider the striatum as a two population network and the role of action selection would need
to be performed in the downstream nuclei. Such a single population striatum could initiate this
process by ‘winner-less competition’ [11] or by enhancing the contrast of the input [12]. If the
D1 and D2 MSNs would only differ in their membrane properties, then our analysis of asymmetric connectivity between the two neuron subpopulations would still be valid, because the effect of different neuron transfer functions could be reduced to a difference in effective
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Go/No-Go Decision Transition Threshold in the Striatum
connection strengths. In the extreme case, when D2 MSNs would receive more inhibition than
D1 MSNs, our analysis remains valid, but we would have to reverse the interpretation.
Implications for the understanding brain disorders involving the basal
ganglia
Many disorders in the basal ganglia are related to the imbalance of its functional pathways. PD
is thought to be a manifestation of dopamine depletion in the form of a domination of indirect
and hyperdirect pathways, while weakening the direct pathway. Dyskinesia, on the other hand,
is correlated with hyperactivity of the direct pathway and reduced activity of the indirect pathway [35]. The Tourette syndrome is correlated with decreased activity of FSIs [36]. This (dis)
balance of striatal activity can be systematically modeled using the concept of the DTT. In fact,
our model provides multiple neuronal mechanisms and a mechanistic understanding of several
brain disorders by linking their specific pathophysiology to the DTT. Specifically, we explored
three factors that may influence the DTT.
In summary our results show that the striatum, with its asymmetric connectivity among
and between D1 and D2 MSNs, can act as a threshold device, indicating the increase in cortical
input firing rates and correlations by increasing the relative firing rates of D1 and D2 MSNs.
The DTT (basically, a threshold on the difference between the two) is a dynamic variable
which may represent behavioral state, learning history and motivation level of the animal. Various mechanisms e.g. feedforward inhibition, dopamine, GPe backprojections, tonically active
neurons exist that can modulate the DTT and thereby, provide the striatum with a rich
computational repertoire.
We arrived at this functional description of the striatum as a DTT by considering the striatal
network dynamics emerging from including low-level properties such as synaptic weights and
connection probabilities. Moreover, we included several factors that can modulate the DTT
level by affecting chemical imbalances and changes in neuronal properties. Taken together, we
provide a high-level functional model of the striatum, which can be easily linked to low-level
properties. Typically, the striatum is included in large-scale functional models [5], acting according to winner-take-all dynamics; an idea no longer supported by experiments [10]. Instead, we argue that our functional description of the striatum provides a more biologically
realistic basis for large-scale functional models of basal ganglia function.
Model predictions and explanation of experimental data
High-conflict decision making. The cognitive concept of conflict is described as a quantity that should increase with the absolute amount of activation and the number of competing
representations [37]. In this condition, it is essential for the action selection system to stall the
decision for further deliberation. A stalled decision in our model implies that the activity in the
direct and indirect pathways are comparable i.e. ΔMSN 0 (in our model ΔMSN 0 for high
cortical firing rates and correlations (Fig 6A)). Especially an increase in B expands the region in
which ΔMSN 0 (Fig 9). Therefore, we predict that high-conflict tasks should be associated
with an increase in correlations in the activity of cortico-striatal projection neurons. Recent experimental data shows higher cortical firing rates in high-conflict situations [38]. According to
our model this is not sufficient for decision stalling, because it would lead to a higher firing rate
in D2 MSNs. Moreover, increased cortical rates should be accompanied with an increase
in correlations.
After stalling the decision, at some point a decision needs to be made in favor of either the
direct or the indirect pathway. A rather simple way to resolve the conflict would be to change
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Go/No-Go Decision Transition Threshold in the Striatum
the cortical firing rates and correlations (cf. Fig 9). The pallido-striatal backprojections may
also provide alternative mechanisms to resolve conflict and facilitate the decision making.
L-Dopa induced Dyskinesia. In our model, an increase in the steady state level of dopamine changes the DTT to higher cortical input rates, thereby increasing the region in which D1
MSNs can have their firing rate exceed those of D2 MSNs (Fig 7A). This observation could be
used to explain the dyskinesia often reported in patients with PD on L-Dopa therapy (L-Dopa
Induced Dyskinesia—LID). Undesired involuntary movements characteristic of LID can be attributed to the frequent undesired activation of D1 MSNs (direct pathway). This could be a direct consequence of a shift in DTT towards ‘Go’, which would result in a decrease in GPi
activity during LID. Indeed, injection of dopamine agonists in MPTP treated nonhuman primates induced dyskinesia and showed a marked decrease in GPi firing rates [35].
Increased reaction time in PD patients. According to our model, in a low dopamine
state, there is a very small parameter regime in which D1 MSNs can have higher firing rate
than D2 MSNs (Fig 7A). Therefore, we propose that in a low dopamine state even low-conflict
tasks might require the arbitration by pallido-striatal backprojections via FSIs (as shown in Fig
7B and S5B), thereby increasing the average reaction time for voluntary actions in Parkinson’s
disease patients.
Akinesia in low-dopamine state. In our model, a decrease in the steady state dopamine
level results in an increase in the range of cortical input for which D2 MSNs have a higher firing rate than D1 MSNs. This might give rise to induced akinesia observed in dopamine-depleted conditions. Such a bias towards ‘No-Go’ may have a two-fold effect on the basal ganglia
output. First, it would introduce an insufficient ‘Go’ bias in the GPi (firing rates do not decrease
sufficiently) and second, in extreme dopamine depleted conditions it might introduce a strong
inhibition on GPe, which can instigate beta band oscillations in the GPe-STN loop [34]. Furthermore, unlike in normal dopamine conditions, the STN-GPe loop is also insufficient to shift
the balance of activity towards D1 MSNs (Figs 7C and S5C).
Selective inhibition of FSIs causes dystonia in animal models. In several brain disorders,
the number of FSIs and/or their connectivity to MSNs are altered. For instance, FSI count is reduced in Tourette syndrome and dystonia [39, 40]. Because FSIs preferentially connect to D1
MSNs, according to our model the loss of FSIs or their connectivity to MSNs would increase
the activity in the direct pathway, compared to the indirect pathway (Fig 7B and 7C) and shift
the DTT to higher values. This might underlie the symptoms of Tourette’s syndrome and dystonia. This explanation is partially supported by recent experimental findings which show that
the symptoms of dystonia could be evoked in an animal model by a selective decrease of the firing rate of FSIs [36].
Re-wiring of FSIs towards D2 MSNs in PD animal models may be a compensatory
mechanism. In our model, based on the experimental data, FSIs preferentially innervate D1
MSNs [16] in order to maintain the default state of bias as ‘No-Go’ under equal cortical drive,
while allowing the cortical activity to switch the balance of D1 and D2 MSNs activity. Interestingly, in the mouse model of PD, three days after dopamine depletion, the FSIs doubled their
connectivity to D2 MSNs as compared to D1 MSNs [41]. Our model suggests that dopamine
depletion reduces the DTT such that the striatum mostly operates in a ‘No-Go’ state and,
hence, cortical activity no longer suffices to switch the balance of D1 and D2 MSNs activity. In
our model, an increase in the FSI to D2 MSNs connectivity will shift the DTT back in favour of
D1 (Fig 4B). Therefore, we argue that the experimentally observed rewiring of the FSIs to D2
MSNs may be a compensatory mechanism to maintain striatal function in a low
dopamine state.
Increased impulsivity in PD patients with deep brain stimulation. Impulsivity could be
considered as a too frequent activation of D1 MSNs or equivalently, a reduced DTT of the
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Go/No-Go Decision Transition Threshold in the Striatum
striatum. Deep brain stimulation (DBS) in the STN is known to increase GPe firing rates [33].
Such increased GPe firing rates would impose a sustained inhibition of FSIs, thereby shifting
the DTT to higher cortical input rates and expanding the regime with a bias towards the ‘Go’
pathway, which may manifest itself as impulsivity at the behavioral level. Notice,that a similar
trend was discussed in LID in Fig 7A. Therefore, we predict that striatal dynamics should show
similar characteristics during impulsivity and LID. Indeed, there is evidence for the potential
mechanistic overlap between behavioral ICDs (impulse control disorders) and motor (dyskinesia) dopaminergic ramifications (cf. review in [42]).
Materials and Methods
The schematic of the striatal circuit is shown in Fig 1. The parameters considered for the rate
model are included in Tables 1 and 2.
Mean field model
The mean population dynamics of the striatal circuit for scenario II were modelled using coupled non-linear differential equations:
_ ¼ 0:01l þ Sð J l þ J l þ J l þ J l Þ
lD1
D1
11 D1
12 D2
1F FSI
C1 CTX
ð21Þ
_ ¼ 0:01l þ Sð J l þ J l þ J l þ J l Þ
lD2
D2
21 D1
22 D2
2F FSI
C2 CTX
ð22Þ
The values of parameters used in Eqs 21 and 22 are listed in the Table 1.
The non-linear transfer function of the neuron S(z) has a sigmoidal function of the form:
z
ffi
SðzÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi
2
z þ1
ð23Þ
The parameters J11, J12, J21, J22 are calculated by considering the effective input received by a
neuron in the population. This can be quantified as:
Jxy ¼ Ny rxy IPSCxy Rx txy
ð24Þ
with x and y either 1 or 2. These parameters are listed in Table 2.
This measure calculates the relative inhibitory strength of projections between D1 and D2:
although the self connectivity of D1 is comparable to the number of projections it receives
from D2 (ρ11 = 0.26, ρ12 = 0.27), since D2 makes stronger inhibitory connections than D1, J12
is larger than J11.
Table 2. Parameters for Eq (24).
Parameter
Description
Value
N1,2
Number of neurons in the population
2000
Nfsi
Number of neurons in the population
80
ρ11,21,22,12
% Connectivity
0.26,0.07,0.36,0.27 [14]
ρd1fsi,d2fsi
% Connectivity
0.54,0.36 [16]
42pA,107pA,107pA,133pA [14]
IPSC11,21,22,12
% Connection strength
Rmsn
Input resistance
238MΩ [14]
τmsn,fsi
Synaptic time constant
12msec,8msec [14, 16]
doi:10.1371/journal.pcbi.1004233.t002
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Go/No-Go Decision Transition Threshold in the Striatum
Table 3. Parameters for model neurons in network simulations.
Parameter
MSN(D1,D2)
FSI
Number of neurons
4000
80
Vrest(mV)
-80 [45]
-80 [45]
Vexc(mV)
0
0
Vinh(mV)
-64 [48]
-76 [49]
Vth(mV)
-45 [50]
-54 [49]
τexc(ms)
0.3
0.3
τinh(ms)
2
2
C(pF)
200 [13]
500
Grest(nS)
12.5 [51]
25
doi:10.1371/journal.pcbi.1004233.t003
Stability of the striatal circuit. Linearizing around the fixed point with the Jacobian matrix:
2
3
!
!
1
2z12
1
2z12
pffiffiffiffiffiffiffiffiffiffiffiffiffi þ
ffiþ
J12
0:01 J11
6 pffiffiffiffiffiffiffiffiffiffiffiffi
7
6
7
z12 þ 1 ðz12 þ 1Þ1:5
z12 þ 1 ðz12 þ 1Þ1:5
6
7
6
7 ð25Þ
!
!
6
7
6
7
2
2
1
2z2
1
2z2
4
5
pffiffiffiffiffiffiffiffiffiffiffiffiffi þ
p
ffiffiffiffiffiffiffiffiffiffiffiffi
ffi
J
0:01
J
þ
21
22
z22 þ 1 ðz22 þ 1Þ1:5
z22 þ 1 ðz22 þ 1Þ1:5
where
z1 ¼ J11 lD1 þ J12 lD2 þ J1F lFSI þ JC1 lCTX
ð26Þ
z2 ¼ J21 lD1 þ J22 lD2 þ J2F lFSI þ JC2 lCTX
ð27Þ
allows to analyze the stability by calculating the eigenvalues of the fixed point. We found the eigenvalues to be real and negative for all values of the fixed points.
Network Simulations
The striatal network model was based on the spiking network model of the striatum as described in [12], except that the network connectivity was not considered to be homogeneous as
in [12]. The simulations were carried out in NEST [43] with networks of 4000 MSNs (2000 D1,
2000 D2) and 80 Fast Spiking Interneurons (FSI), consistent with the estimated ratio of MSN
and FSI neurons [44, 45]. The neuron model parameters for FSIs and MSNs are listed in
Table 3.
D1 MSNs, D2 MSNs and FSIs received independent excitatory Poisson inputs, mimicking
the background cortico-striatal inputs. The cortical input parameters are summarized in
Table 4. Connection probabilities for D1 MSNs, D2 MSNs and FSIs were taken from [14, 16],
also listed in Table 5.
The leaky-integrate-and-fire (LIF) neuron model was used to simulate the neurons in the
network with the subthreshold dynamics of the membrane potential Vx(t) described by:
x
¼ I syn ðtÞ
C x V_ x ðtÞ þ Gx ½V x ðtÞ Vrest
ð28Þ
I syn ðtÞ ¼ I D1 ðtÞ þ I D2 ðtÞ þ I FSI ðtÞ þ I Ctx ðtÞ
ð29Þ
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Go/No-Go Decision Transition Threshold in the Striatum
Table 4. Cortical input to striatal neurons.
Target
Background input rate (Hz)
Peak conductance(nS)
D1
2500
3.6
D2
2500
3.0
FSI
2500
5.0
doi:10.1371/journal.pcbi.1004233.t004
where x 2 D1, D2, FSI. In the above equations, Isyn(t) describes the total cortical excitatory, recurrent and feedforward inhibition to a neuron, Cx is the membrane capacitance, Gx is the leak
conductance, and Vrest is the resting membrane potential. When the membrane potential of the
neuron reached Vth, a spike was elicited and the membrane potential was reset to Vrest for a refractory duration (tref = 2 ms.)
An alpha function was used to model the excitatory synaptic input received by D1, D2 and
FSI:
8
t
x
t
< Jexc
e1texc for t 0
texc
x
gexc ðtÞ ¼
ð30Þ
:
0 for t < 0
where x 2 D1, D2, FSI. The rise times for the excitatory inputs τexc was set to be identical for all
neurons. In addition to excitatory input from the cortex, D1 and D2 MSNs received (self and
mutual) recurrent inhibition as well as feedforward inhibition from the FSIs. The corresponding inhibitory conductance changes were modelled as:
8
t
t
x
< Jinh
e1tinh for t 0
tinh
x
ð31Þ
ginh ðtÞ ¼
:
0 for t < 0
where x 2 D1, D2. The rise times τinh for all inhibitory synaptic conductance transients were
set to have identical values. Assuming the synaptic strengths as fixed, the total excitatory conductance Gxexc;i ðtÞ in a MSN i was given by
X X
Gxexc;i ¼
x
Ctx
gexc
ðt tmn
Þ
ð32Þ
x
m2Ki
n
where x 2 D1, D2. The sequence of spikes (n0 s) impinging on a particular-synapse m are added
Table 5. Inhibitory input to striatal neurons.
Source
Target
Probability
Peak conductance (nS)
Delay(ms)
D1
D1
0.26 [14]
0.5
2
D1
D2
0.07 [14]
1.0
2
D2
D2
0.36 [14]
1.0
2
D2
D1
0.27 [14]
1.2
2
FSI
D1
0.54 [16]
2.5 [50]
1
FSI
D2
0.36 [16]
2.5 [50]
1
doi:10.1371/journal.pcbi.1004233.t005
PLOS Computational Biology | DOI:10.1371/journal.pcbi.1004233 April 24, 2015
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Go/No-Go Decision Transition Threshold in the Striatum
by the inner sum, while the outer sum runs over all such excitatory synapses m in the set Kix
Ctx
projecting onto neuron i. The set tmn
represents the spike times of the excitatory neuron m.
The total inhibitory conductance Gxinh;i ðtÞ in a D1 or D2 MSN i was given by
X X
XX
FSI
FSI
D1
D1
ginh
ðt tmn
DFSI Þ þ
ginh
ðt tmn
DD1 Þ
Gxinh;i ¼
m2KiFSI
n
X X
D2
D2
ginh
ðt tmn
DD2 Þ
þ
m2KiD2
m2KiD1
n
ð33Þ
n
where x 2 D1, D2 and KiFSI , KiD1 , KiD2 refer to the pre-synaptic FSIs, D1 and D2 projecting onto
neuron i, respectively. The transmission delays of ΔFSI and ΔD1, ΔD2 were fixed to 1ms and
2ms, respectively. The total synaptic current to a MSN i was
x
x
Gxinh;i ðtÞ½Vix ðtÞ Vinh
Iix ðtÞ ¼ Gxexc;i ðtÞ½Vix ðtÞ Vexc
ð34Þ
x
x
, Vinh
denote the reversal potentials of excitatory and inhibitory synwhere x 2 D1, D2 and Vexc
aptic currents, respectively.
Similarly, for a FSI neuron i, the excitatory conductance was:
X X
GFSI
FSI
Ctx
exc;i ¼
gexc
ðt tmn
Þ
ð35Þ
FSI
m2Ki
n
and the total synaptic current for the FSI neuron i was
FSI
FSI
IiFSI ðtÞ ¼ GFSI
exc;i ðtÞ½Vi ðtÞ Vexc ð36Þ
The parameter values for both MSNs and FSIs in our network model are summarized in
Table 3.
Generation of B and W correlations
To separately control the correlations within and between the pre-synaptic pools of the striatal
neurons, we extended the multiple-interaction process (MIP) model of correlated ensemble of
Poisson type spike trains [12, 23]. The MIP model generates correlations by copying spikes
from a spike train (the mother spike train) with a fixed probability (the copy probability, which
determined the resulting correlation) to the individual spike trains. This process was implemented in the NEST simulator by introducing a synapse model that transmits spikes with a
fixed probability: the lossy synapse. By making many convergent connections using the ‘lossy
synapse’ we can mimic the random copying of spikes from the mother spike train to the children process. This way we controlled the correlations within (W) the pre-synaptic pool of
a neuron.
To introduce correlations between (B) the pre-synaptic pools of two neurons we created a
two-layered MIP (Fig 10). We started with a spike train M with a firing rate Rin. In the first
layer of MIP we generated N spike trains (C1, C2, , CN) with a fixed pair-wise correlation (B0 )
and firing rate (Rin B0 ). In the second layer, each of these spike trains acted as a mother spike
train to generate pairwise correlations in the pre-synaptic pools of individual striatal neurons.
Each spike train Ci was used to generate correlated spike trains (Ci0, Ci1, , Cim) that acted as
the pre-synaptic pool of each striatal neuron with a pairwise correlation W and firing rate Rin
B0 W (Fig 10). The effective correlation between the pre-synaptic pools of two neurons is then
defined as B = B0 W. Thus, in this model we can control the correlations between and within
the pre-synaptic pools by changing the copy probabilities in the first and second layer.
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Go/No-Go Decision Transition Threshold in the Striatum
Fig 10. Generation of within and between correlations. (A) The correlations are generated by copying spikes twice from the mother process (M). Let Rin
be the rate of the mother process. The spikes are copied into a set of children processes (C1 CN) with a copy probability B0 . This controls the shared input
to the postsynaptic population and leads to a rate Rin B0 . The spikes are copied for a second time with a copy probability W into a pool of input neurons (C11
C1N). Each neuron in the input pool spikes with a rate Rin B0 W. The correlation within the input pool can be increased by increasing W, whereas the shared
input can be increased by increasing the copy probability B0 . (B) B W. The means of the pairwise correlations measured from randomly chosen 2000 pairs
of MSN neurons (B) are plotted against the corresponding W values. The different colors represent different input rates in Hz. Notice that most of the points lie
below the diagonal.
doi:10.1371/journal.pcbi.1004233.g010
Relationship between B and W
B refers to the average correlation between the spiking activities of neurons in the pre-synaptic
pools of two striatal neurons. The spike trains in each pre-synaptic pool are themselves correlated with a correlation coefficient W. For such pooled random variables, Bedenbaugh and
Gerstein [22] derived the following relationship:
r1;2 ¼
B
1
þO
W þ n1 ð1 WÞ
n
where ρ1,2 is the correlation coefficient between the two pools of the pre-synaptic neurons, n is
the size of each pre-synaptic pool. Because 0 ρ1,2 1, it follows that B W.
Intuitively we can understand this relationship between B and W in the following way:
Imagine two pools of identical spike trains, but with individual spikes trains uncorrelated to
each other (i.e. B0 = 1 and W = 0). In this case, the average correlations between the two pools
will be small because each spike train has only copy in the other pool, while being uncorrelated
with all others. However, when W > 0, more such pairs with non-zero correlation will occur,
increasing the value of B. Because, B W, the difference between the firing rates of the two
PLOS Computational Biology | DOI:10.1371/journal.pcbi.1004233 April 24, 2015
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Go/No-Go Decision Transition Threshold in the Striatum
MSN neuronal population in Fig 6 is estimated for the lower triangular region. This relationship is explicitly shown also in Fig 10B, where the measured input correlation between the
pools (B) is shown to be less than W.
Model limitations
Our model considers the striatum as a single action unit with ‘Go’ and ‘No-go’ counterparts
and, hence, with a single DTT. But ideally, there are multiple channels of competing motor
programs which can be envisioned as mutually competing DTT’s. Thus, future work will include the extension of this model to incorporate multiple action channels. Secondly, the effect
of input correlations is yet not formulated into the analytical framework. This is a non-trivial
problem, due to the non-linear transfer of input correlations into output firing rates, specifically if the output neurons are recurrently interconnected. Thirdly, our model explores the striatal
dynamics at steady states, since the model showed stable fixed points. However, it is possible
that in the presence of additional phenomena, such as plasticity of cortico-striatal or striatostriatal projections or intrinsic plasticity of the striatal neurons etc., the model may show interesting transient dynamics. Recent work proposed the existence of a ‘decision threshold’ in a different context [50]. They considered the ‘decision threshold’ as a function of the RT (reaction
time) distribution of the rats in a decision task pertaining to the transient dynamics of the basal
ganglia. A future extension to this work can explore the effect of DTT due to the asymmetry in
the striatal connectivity on the distance to the ‘decision threshold’ in [50], i.e. the effect of DTT
on the RTs of a decision. And lastly, though our model abides by many experimental observations, it contradicts one experimental result, to the best of our knowledge. In [51], it was observed that FSIs increased their activity around a choice execution. In our model, a choice
execution will correspond to the activation of the D1 pathway, which requires reduced activity
in the FSIs, in contrast to the recordings in [51]. We think this contradiction is a consequence
of our model of the striatum considering only one channel in the ‘Go’ and ‘No-Go’ pathways.
Supporting Information
S1 Fig. Effect of temporal fluctuations on DTT in the striatum. (A) Cortical input rate applied to the striatal mean field model. The instantaneous input rate was choosen from a low
pass filtered noise (time constant = 5 msec) and standard deviation of ±0.5 around the mean.
The mean of the input rate was varied every 500 msec while the standard deviation remained
fixed. (B) To mimic a noisy ΔCTX we provided additional noisy inputs to the D1 and D2 MSNs
(Scenario I). D1 MSNs received additional input with a mean of 1 Hz and standard deviation σ
− Δ − ctx1. (C) Similarly, the D2 MSNs also received low pass filtered noisy input with 0 Hz
mean and standard deviation σ − Δ − ctx2. Both σ − Δ − ctx1 and σ − Δ − ctx2 were considered
free variables here.(D) The instantaneous firing rates of D1 and D2 MSNs in response to the
noisy cortical input and ΔCTX. A DTT can be observed in spite of noisy fluctuations. (E) To
quantify the effect of noisy Δctx, on the DTT we measured the areas around the cross-over of
the D1 and D2 MSNs firing rates. This is the area under the curve (yellow portion) for the time
interval [t − Δ,t ] (pre-DTT) and [t ,t + Δ] (post-DTT), where t is the time when DTT occurs. Because the gain of the D1 and D2 MSNs is different in pre-DTT (where λD1 > λD2) and
post-DTT regions (where λD1 < λD2) we separately measured the area in the pre-DTT and
post-DTT regimes. We normalised the pre-DTT and post-DTT areas with the areas measured
for zero noise case. Here, we show the normalised pre-DTT and post-DTT areas for different
values of σ − Δ − ctx1 and σ − Δ − ctx2. As expected the increase in the input noise progressively
decreases the pre-DTT area. Nevertheless, we can reliably measure DTT for large fluctuations
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Go/No-Go Decision Transition Threshold in the Striatum
compared to the mean.
(TIFF)
S2 Fig. Effect of striatal network connectivity parameters on the existence of the DTT.
(A-F) Striatal parameters, Δ-ctx1, Δ-ctx2, J11, J22, J21, J12 were varied by up to ±100% around
their means as specified in the model. It is difficult to visualise this six dimensional space. We
use two different visualisations. First, we show whether for ±100% variation in one of the parameters it is possible to find a DTT for any set of values of all other parameters (again within
±100% of their means). The “Solution” refers to when a DTT is observed for at least one parameter set, given a specific value of the parameter for which robustness is tested. The values of
parameters for which no combination of other parameters yielded a DTT were marked as “No
solution”. It can be observed that a DTT can be found for nearly all values of all parameters
(A-F). Only for very weak values of J11 and J22 we could not find a combination of other parameters that would yield a DTT. (G-J) Next, we show the existence DTT for pairs of striatum network parameters. Each dot shows the existence of a DTT. Not only do these solutions follow
the relational constraints as described in Taverna et al. (2010) and the model (e.g. J12 > J21),
but also the centroids (marker with red asterisk) of these clusters lie very close to the values
used in the model. This can be verified in the mean values shown in (A-F). This indicates that
these values are indeed a robust combination of striatal parameter values.
(TIFF)
S3 Fig. Presence of DTT in spiking neural network in which D1 and D2 MSNs have different F-I curves. (A) F-I curves of D1 and D2 MSNs. Passive neuron properties were tuned to
match the F-I curves of the D1 and D2 MSNs to match with the experimental measurement of
the F-I curves of these neurons (shown in Gertler et al. 2008). (B) Firing rate of the D1 and D2
MSNs as a function of cortical input rate. D1 MSNs received extra input according to the scenario I. D1 and D2 MSNs have different F-I curves shown in the panel A. These results show
that a DTT exists in the striatum even when D1 and D2 MSNs have different F-I curves. The
network connectivity is same as described in Table 5.
(TIFF)
S4 Fig. Effect of B0 correlations on striatum’s DTT. The blue and red regions mark the inputs
for which D2 and D1 MSNs have higher firing rates. White regions show the inputs for which
there is no difference between the firing rate of the two MSN populations. From these figures it
is clear that B0 results in the increasing the region of high conflict.
(TIFF)
S5 Fig. Effect of Dopamine and GPe firing rates on the balance of D1 and D2 activity
(mean field equations). (A) Fixed points for D1 and D2 MSNs plotted for different levels of
dopamine. Darker shades of blue (red) correspond to D1 (D2) MSN activity for higher levels of
dopamine. For lower than normal levels of dopamine, the DTT shifts to the left (from 11 Hz
to 6 Hz). This decreases the regime with the bias towards D1 MSNs. For higher than normal
levels of dopamine, the DTT shifts to right (from 11 to 19Hz). This in turn, increases the
regime with a bias towards D1 MSNs. (B) and (C) same as Fig 7 in the main text but calculated
for mean field equations.
(TIFF)
S6 Fig. Arbitration by GPe. (A) GPe activity is represented as an inhibition on FSI firing rates.
A regime with a D2 bias in the rate equations is shown(λD1 < λD2, ΔMSN < 0). At t = 400 ms, a
constant inhibition is given to FSIs. The FSI activity decreases (D) and the bias shifts in the favour of D1 (λD1 > λD2, ΔMSN > 0). (B,E) Shows the similar arbitration in spiking neural
PLOS Computational Biology | DOI:10.1371/journal.pcbi.1004233 April 24, 2015
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Go/No-Go Decision Transition Threshold in the Striatum
network activity. Dashed line represent the 95% confidence interval. (C,F) Inhibition on FSIs is
unable to resolve the bias in favour of D1 in dopamine depletion conditions.
(TIFF)
Acknowledgments
We thank Robert Schmidt, Alejandro Bujan, Ioannis Vlachos, Marko Filipovic, Jeanette Hellgren-Kotaleski and Stefan Rotter for their helpful comments on different versions of
this manuscript.
Author Contributions
Conceived and designed the experiments: JB AA AK. Performed the experiments: JB. Analyzed
the data: JB AK. Contributed reagents/materials/analysis tools: AK JB. Wrote the paper: AK JB
AA.
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