Hosoya, Baccus, and Meister (2005) "Dynamic predictive coding by the retina"

Hosoya, Baccus, and Meister (2005) "Dynamic predictive coding by the retina"
Vol 436|7 July 2005|doi:10.1038/nature03689
ARTICLES
Dynamic predictive coding by the retina
Toshihiko Hosoya1†, Stephen A. Baccus1† & Markus Meister1
Retinal ganglion cells convey the visual image from the eye to the brain. They generally encode local differences in space
and changes in time rather than the raw image intensity. This can be seen as a strategy of predictive coding, adapted
through evolution to the average image statistics of the natural environment. Yet animals encounter many environments
with visual statistics different from the average scene. Here we show that when this happens, the retina adjusts its
processing dynamically. The spatio-temporal receptive fields of retinal ganglion cells change after a few seconds in a
new environment. The changes are adaptive, in that the new receptive field improves predictive coding under the new
image statistics. We show that a network model with plastic synapses can account for the large variety of observed
adaptations.
Because the physical world is composed of discrete objects with
surfaces of fairly uniform reflectance, image points that are close in
space or in time tend to have very similar intensities1,2. This
redundancy permits efficient encoding of the visual image3. The
receptive field of many ganglion cell types shows spatial antagonism
between centre and surround, and a biphasic temporal antagonism4–6. For any given image point, neural circuits in the retina predict
the local intensity from values at nearby points in space and
preceding points in time, and subtract this predicted value from
the actual intensity3. Thus, ganglion cells signal not the raw visual
image, but the departures from the predictable structure, under the
assumption of spatial and temporal uniformity. This difference signal
obtained by predictive coding has a much smaller dynamic range
than the raw image, and is therefore more suited for transmission
through neural fibres with a limited firing rate7–9.
Although these receptive fields produce an efficient encoding of
the average visual scene, animals spend considerable time in environments that differ strongly from the average image statistics. For
example, sand or gravel surfaces contain finer spatial variation, such
that images are correlated only on short scales; forest or tall grass
introduce vertical structure, in which two image points are highly
correlated if separated vertically but not horizontally. Self-motion of
the observer generates optic flow on the retina, in which the intensity
at one point of the image is highly correlated with that at a different
point later in time. Under all these conditions, the rules for predictive
coding are markedly different from the average. An efficient encoder
of visual scenes would adapt its strategy accordingly10,11. If this
happens in the retina, one should find that ganglion-cell receptive
fields change dynamically depending on the correlation structure of
the visual environment, in a way that enhances predictive coding and
suppresses the dominant spatio-temporal structure in the stimulus.
To test this proposal, we recorded spike trains from ganglion cells
in the retinae of salamanders and rabbits. We manipulated the
statistics of the visual scene and tested whether adaptation to a
different environment altered the encoding of retinal signals. Furthermore, we asked whether these changes conformed to the notion
of dynamic predictive coding.
Adaptation to spatial image correlations
Because centre–surround antagonism has been explained as an
evolutionary adaptation to positive image correlations in space, we
first tested whether negative correlations would alter these receptive
fields. Environment A was designed as a flickering uniform field
(Fig. 1a, b) with perfect positive correlation between all image points,
whereas environment B was a flickering checkerboard with perfect
negative correlation between two sets of neighbouring tiles. After
adaptation to environment A or B, an uncorrelated stimulus P was
used to probe the ganglion cells’ spatio-temporal receptive field.
From the responses to P, we computed the sensitivity of each
ganglion cell to stimuli of type A or type B (see Methods). For
example, the salamander ganglion cell illustrated in Fig. 1c, d
experienced a large change in its receptive field. After adaptation to
environment B, the receptive field profile flattened, so the neuron was
almost equally sensitive to the two checkerboard regions (Fig. 1d). As
a result, the cell became less sensitive to stimuli drawn from the
checkerboard environment B by a factor of about 0.57, while
becoming more sensitive to stimuli from the uniform environment
A by a factor of about 1.4. The reverse changes occurred during
adaptation to environment A. Figure 1e plots these sensitivity
changes caused by adaptation for a large sample of ganglion cells.
For most cells, the sensitivity to the adapting stimulus decreased,
whereas that to the novel stimulus increased. In some cases the
sensitivity to both stimuli changed in the same direction, while still
enhancing the novel stimulus. Overall, most data points lie considerably above the diagonal, which means that the cell increased its
selectivity for the novel over the adapting stimulus.
The degree to which the novel stimulus is enhanced is summarized
by the adaptation index a (equation (2)); note that a ¼ 1 if a neuron
does not change its sensitivity or changes it equally to stimuli from
the two environments, but a . 1 if a neuron preferentially suppresses stimuli from the environment to which it is adapted. Across
the population of ganglion cells, about half had an adaptation index
significantly greater than 1 (Fig. 1f). These cells exhibited dynamic
predictive coding. A differential suppression of the adapting stimulus
by factors of a ¼ 2 or greater was not uncommon. Such adaptations
were observed with checkerboard tile sizes of both 200 mm and
400 mm (Fig. 1e), and earlier work suggests that they occur at least
over a range of 140–800 mm (ref. 12).
The gain change was not instantaneous after the switch to a new
environment, but occurred gradually (Fig. 1g; see also ref. 12). The
adaptation index adjusted with a time constant of several seconds,
and there may be even slower components beyond the 10-s range that
we measured. Because the immediate light response of a ganglion cell
is at least tenfold faster than this (Fig. 1d), one can regard this slow
1
Department of Molecular and Cellular Biology, Harvard University, Cambridge, Massachusetts 02138, USA. †Present addresses: RIKEN Brain Science Institute, 2-1 Hirosawa,
Wako-shi, Saitama 351-0198, Japan (T.H.); Department of Neurobiology, Stanford University, Stanford, California 94305, USA (S.A.B.).
© 2005 Nature Publishing Group
71
ARTICLES
NATURE|Vol 436|7 July 2005
adaptation as a gradual modulation of the rules by which the
ganglion cell combines information across space. In addition, it is
possible that fast changes in sensitivity13 occur immediately after the
switch and before our first probe measurement (see Supplementary
Information).
Adaptation to oriented stimuli
The two environments in Fig. 1 differ substantially in their spatial
frequency content. Retinal interneurons with small or large receptive
fields will be driven differently by these two stimuli, leading to the
potential that local neurons adapt their sensitivity at various sites in
the circuit12. Thus, we tested whether more subtle spatial correlations
are also effective in driving adaptation. In Fig. 2a the two environments consist of flickering bar gratings, oriented horizontally or
vertically. These two stimuli were perfectly matched for mean
intensity, contrast, and spatial and temporal power spectra; in fact,
all image statistics were identical except for the difference in orientation. As in Fig. 1b, we exposed the retina to one or the other
environment for several seconds, and then probed each ganglion
cell’s sensitivity to both horizontal and vertical gratings. Adaptation
to the oriented gratings produced changes in the receptive fields of
many ganglion cells. Figure 2c illustrates a sample cell whose
receptive field acquired a horizontally elongated shape after exposure
to the vertical bars. Across many cells, this adaptation systematically
had the effect expected from dynamic predictive coding, namely to
enhance sensitivity for the novel orientation relative to the adapting
orientation (Fig. 2d, e).
Again, one might suspect that individual cells within the retina are
driven differentially by the two stimuli and adapt their sensitivity
accordingly. For example, an interneuron might be located at an edge
between two bars of the grating such that it is strongly stimulated and
fatigued by one orientation but not by the other. We tested this by
randomly shifting the grating in each stimulus frame (see Supplementary Information and ref. 12). Under these conditions the
spatial correlations in the stimulus are maintained, but no neuron
consistently lies on a boundary. Still, many ganglion cells adapted
their sensitivity in the direction of predictive coding (Fig. 2e).
We performed the same experiments in rabbit retina. Again, about
half of the ganglion cells engaged in dynamic predictive coding
(Fig. 2f), with differential gain changes similar to those in the
salamander. Thus, dynamic pattern adaptation is a shared aspect of
retinal function between amphibians and mammals, animals that
differ greatly in ecology and physiology but share the challenge of
adjusting to a variable visual environment.
Figure 1 | Adaptation to spatial image correlations. a, Stimulus display
with square tiles of two intensities X and Y. b, Stimulus time course (see
Methods). An adapting environment (A or B) was presented for 13.5 s,
followed by a probe stimulus (P) for 1.5 s. c, Spatial receptive field of a
sample ganglion cell, measured by reverse correlation to a random flicker
stimulus38. Data in Figs 1, 2, 3 and 4 are from salamander retina, except Fig.
2f, which is from rabbit retina. d, Temporal response filter of the ganglion
cell to the two stimulus regions X and Y, after adaptation to environment A
(left) or B (right). Here and in all other figures, error bars show the s.e.m. of
values obtained from independent subsets of the data. e, Effects of
adaptation to a switch from A to B on the sensitivity S A for the novel
stimulus A (ordinate) and sensitivity S B for the adapting stimulus B
(abscissa). Each data point represents one ganglion cell from experiments
with tile size 400 mm (filled symbols) or 200 mm (open symbols). The dotted
line is the identity. The axes are logarithmic. f, The adaptation index a
(equation (2)) is the change in differential sensitivity resulting from
adaptation, and corresponds to the distance of data points above the line in
e. Histogram of the adaptation index for 35 cells. Of these, 43% had a . 1 at
P , 0.05. g, The adaptation index for the ganglion cell shown in d, plotted
against time in the uncorrelated probe environment, gradually relaxed to 1.
An exponential fit to loga has a time constant t of 2.7 s. Over seven cells, the
time constant was 3.2 ^ 0.8 s (mean ^ s.e.m.).
72
© 2005 Nature Publishing Group
ARTICLES
NATURE|Vol 436|7 July 2005
Adaptation to temporal and spatio-temporal structure
To test the retina’s strategy for predictive coding across time, we
constructed two environments matched in intensity and contrast but
with very different temporal correlations (Fig. 3a, see Supplementary
Information). The intensity at any given time could be predicted by
the intensity 60 ms earlier, but the predictive rule had opposite sign in
the two environments. In one case (positive correlation) a bright
frame was followed 60 ms later by a bright frame, and in the other
case (negative correlation) by a dark frame. Adaptation to one
environment or the other produced substantial changes in the
temporal response function of many ganglion cells. For example,
under positive correlations, the cell in Fig. 3b had a strongly biphasic
Figure 2 | Adaptation to oriented stimuli. a, The environment was a
flickering counterphase grating oriented horizontally (A) or vertically (B).
The sequence of stimuli was as in Fig. 1b. b, Spatial receptive field of a sample
ganglion cell. c, Sensitivity to the four regions coloured in b during the probe
interval P, after adaptation to environment A (left) or B (right). For each
region the peak amplitude of the response filter is plotted. d, Effects of
adaptation on the sensitivity to stimuli A and B, displayed as in Fig. 1e.
e, Histogram of the adaptation index a for 39 cells (grey); 41% of cells had
a . 1 at P , 0.05. In some experiments (line, 40 cells) the borders of the
bars were shifted randomly (see Supplementary Information). The means of
the two histograms are not significantly different (P ¼ 0.59). f, Histogram of
the adaptation index a for 34 ganglion cells from rabbit retina; 38% of cells
had a . 1 at P , 0.05.
response function, which suppressed its response to stimuli with
positive correlation. After adaptation to the other environment, the
response function became less biphasic. Many cells underwent
similar changes, with the effect being again a suppression of the
predictable stimulus features compared with novel features (Fig. 3c, d).
Finally, we tested the retina’s ability for predictions across space
and time. The stimulus was a flickering checkerboard (Fig. 1a) in
which one set of tiles was modulated with a time-shifted version of
the flicker in another set of tiles (Fig. 3e). Thus certain points on the
retina could serve to predict the intensity at other points, but the
order of the time shift, and thus the direction for prediction, was
inverted in the two environments. This is a somewhat abstract
stimulus, and the difference between the two environments is subtle
and almost imperceptible to the human observer. Nevertheless, many
retinal ganglion cells changed their response properties significantly
from one environment to the other (Fig. 3f, g), leading to a different
temporal summation of the intensity from the two stimulus regions.
When this occurred, it again had the effect of suppressing predictable
features over novel features (Fig. 3h), but the magnitude of the effect
was somewhat weaker than for the other stimulus correlations we
had tested.
A pattern detector hypothesis
How can the retina accomplish this diverse set of adaptations? A
popular hypothesis for phenomena of pattern adaptation postulates
that there are several parallel pathways within the circuit that
combine to make the output signal14,15, in this case at the retinal
ganglion cell (Fig. 4a). Within each pathway, the interneurons are
pattern detectors, selective for a particular stimulus feature; for
example, bars of a specific orientation. If that feature occurs frequently, those interneurons will be driven strongly, leading to their
fatigue and diminished contribution to the output signal. As a result,
the output becomes less sensitive to common stimulus features and
more sensitive to rare features, for example bars of the orthogonal
orientation.
In the retina, the most plausible candidates for parallel interneurons pooled by ganglion cells are the bipolar cells. To explain how
ganglion cells adapt to the orientation of a grating (Fig. 2), the
bipolar cells would need to be significantly orientation-selective and
there should be a range of bipolar cells with different selectivities
feeding each ganglion cell. Using intracellular recordings, we
measured the receptive fields of ten salamander bipolar cells directly
(Fig. 4b) and found that they were round or only slightly elongated
(Fig. 4c). Furthermore, the receptive-field centres were small compared with the bars of the gratings, and most of them fell entirely
within a single bar. Considering all possible positions and orientations of the receptive field, this panel of bipolar cells was selective
for one grating over the other by only a factor 1.06 ^ 0.07 (mean ^
s.d.). Moreover, bipolar cells adapt their gain only slightly (about
10%; refs 13, 16), even with a strong change in input amplitude.
Altogether, this suggests that adaptation in oriented bipolar cells
would contribute less than a 1% orientation selectivity to the
ganglion cells, much smaller than the observed gain changes by
factors of 1.5–2.5 (Fig. 2d, e).
Significant orientation selectivity has been reported in some other
retinal interneurons, notably amacrine cells17. However, these neurons are primarily inhibitory. In that case, the fatigue of an amacrine
cell would lead to less suppression of the adapting stimulus in the
ganglion cells, contrary to what we observed. Still, the role of patternselective interneurons in these adaptation phenomena deserves
further attention.
A network plasticity hypothesis
An alternative explanation invokes plasticity of synapses rather
than fatigue of interneurons. Consider a ganglion cell with a
centre–surround receptive field (Fig. 5a, left), in which the
surround inhibition is conveyed by amacrine cells18. Suppose that
© 2005 Nature Publishing Group
73
ARTICLES
NATURE|Vol 436|7 July 2005
the inhibitory synapse from an amacrine cell to a ganglion cell is
plastic in the following manner: when signals in the presynaptic and
postsynaptic neurons are correlated, the synapse becomes stronger; if
the two neurons are anticorrelated, the synapse becomes weaker.
Note that this is an anti-hebbian rule for plasticity, to be distinguished from the hebbian form, in which correlated activity leads to
increased excitation. If one applies a vertical grating stimulus (Fig. 5a,
middle), the ganglion cell will be strongly correlated with the
amacrine cells above and below, and anticorrelated with those to
the left and right. Thus, the synapses will change so as to strengthen
inhibition from above and below, and weaken inhibition from the
sides. This gives the ganglion cell a receptive field with distinct
horizontal orientation, and makes the neuron more sensitive to
horizontal gratings and less sensitive to vertical ones.
We formalized this idea as a simple feedforward network from a
layer of bipolar cells to ganglion cells (Fig. 5b). In this model, a
ganglion cell receives excitatory synapses of fixed strength from
bipolar cells, and these determine the default shape of the receptive
field in absence of stimulation. In addition, there are inhibitory
connections—mediated by amacrine cells—that are plastic as determined by the above correlation-based rule (equations (3) and (4); see
Methods). Each ganglion cell integrates its bipolar cell inputs linearly,
weighted by synaptic strength. Exposure of the network to different
stimulus environments induces change in the amacrine synapses,
which in turn alters the receptive field of the ganglion cell. In the
approximation of linear processing, one can solve the dynamics of
this network analytically (see Methods).
To illustrate the results, we consider a model ganglion cell connected to a 4 £ 4 patch of bipolar cells (Fig. 5c); only the centre 2 £ 2
bipolar cells provide direct excitatory connections, but all bipolar
cells make modifiable inhibitory connections through amacrine cells.
When the network is exposed to the kinds of stimulus used in the
experiments of Figs 1 and 2, the ganglion-cell receptive field changes
in a way that suppresses the correlated components of the stimulus.
For example, when driven by a spatially uniform stimulus, the
receptive field strengthens the antagonistic surround; when driven
by a horizontal grating, the receptive field develops a vertical
orientation that suppresses sensitivity to horizontal bars; when
driven by an uncorrelated stimulus, the receptive field becomes
attenuated uniformly, which is akin to contrast adaptation observed
previously12,13,19. More generally, one can show that such an adaptive
network performs a multidimensional scaling in the space of stimuli:
the sensitivity for any given axis in stimulus space is scaled down
according to the strength of stimulation along that axis (see Supplementary Information). Figure 5d illustrates how this stimulus
selectivity changes over time in the course of adaptation. The model
makes the intriguing prediction that adaptation to an increase in
stimulus strength occurs more rapidly than to a decrease, a feature
that is indeed consistently observed in animals from flies to
humans12,20,21.
Several features distinguish this adaptive network hypothesis
(Fig. 5a) from that of adaptive pattern detectors (Fig. 4a). First,
adaptation happens at each synapse, not in each neuron. Because
there are far more synapses than neurons, this allows a rich set of
adaptations. By contrast, the pattern detector model requires a
specific interneuron selective for each of the various types of
correlation to which the retina can adapt (Figs 1–3). Second,
amacrine cells come in a great variety of types, with differing
receptive field sizes, integration times, and latencies22,23. This
means that many different kinds of stimulus correlation across
space and time can be sensed and exploited for predictive coding.
Last, this hypothesis predicts that inhibition is essential for the
adaptation process, because it implements the subtraction of predictive signals. We tested this by repeating experiments on orientation-adaptation while blocking the inhibitory neurotransmitters
GABA (g-aminobutyric acid) and glycine23. These conditions profoundly influence retinal signalling: the firing rate of ganglion cells
increases and their stimulus selectivity changes as the antagonistic
surround weakens18,24. Still, one can ask whether these stimulus
selectivities are altered by adaptation, and the adaptation index a
(equation (2)) is independent of any absolute changes in sensitivity.
We found that, without inhibition, the ganglion cells indeed lost the
ability for dynamic adjustment of stimulus selectivity (Supplementary Fig. S3).
Figure 3 | Adaptation to temporal and spatio-temporal correlations. a, The
environment was a flickering uniform field whose intensity values 60 ms
apart (see markers) had a strong positive (A) or negative (B) correlation.
b, Filter waveform of a sample ganglion cell, after adaptation to
environment A (left) or B (right). c, Effects of adaptation on the sensitivity
to stimuli A and B, displayed as in Fig. 1e. d, Histogram of the adaptation
index a for 34 cells; 44% had a . 1 at P , 0.05. e, Environments with
different spatio-temporal correlations. Stimulus display as in Fig. 1a, but
with Y the same as X delayed by 60 ms (A) or vice versa (B). f, Response
kernel of a sample ganglion cell to X and Y, after adaptation to A (left) or B
(right). g, Effects of adaptation on the sensitivity to stimuli A and B,
displayed as in Fig. 1e. h, Histogram of the adaptation index a for 34 cells;
41% had a . 1 at P , 0.05.
74
Discussion
Although the notion of dynamic predictive coding captures the
essence of these effects, we also noted several departures from the
© 2005 Nature Publishing Group
ARTICLES
NATURE|Vol 436|7 July 2005
theoretically ideal encoder. First, only about half of the observed
ganglion cells adapted to each stimulus, and a breakdown by cell type
in the salamander retina revealed some systematic differences: in
particular, ‘fast OFF’ cells25 adapted to all the conditions tested,
whereas ‘slow OFF’ cells adapted only to spatial correlations in the
stimulus (Figs 1 and 2). Second, the absolute magnitude of adaptation was lower than for the ideal efficient encoder. In theory, a
perfectly predictable stimulus component could be completely suppressed, whereas the largest differential gain changes we observed
were a factor of 3. Last, some stimulus correlations induced greater
adaptation than others (compare Figs 3d with Fig. 3h). All these
limitations may provide clues to the circuit mechanisms that
underlie the effects. For example, under the adaptive network
hypothesis (Fig. 5) the range of conditions to which a given ganglion
cell type can adapt is limited by which classes of amacrine cells it
contacts in the inner plexiform layer. The degree of adaptation is
limited by how sensitive an amacrine cell synapse is to the correlation
signal, for example the parameter b in equation (4). Further
exploration of the range of adaptive behaviours will certainly be
instructive.
With regard to the mechanisms for these adaptive effects, we
considered two alternative explanations that place the changes either
at individual interneurons (Fig. 4) or at individual synapses (Fig. 5).
The former scheme invokes a pattern-selective interneuron for each
type of stimulus pattern to be sensed, and fatigue of those interneurons in the course of adaptation. This interpretation is widespread in the literature, to the point at which the observation of
pattern adaptation is accepted as evidence for the existence of pattern
detector neurons14,15. For example, in human vision, adaptation to an
oriented grating raises the contrast threshold for that same stimulus
by a factor of 2–4 (refs 26, 27). This is commonly thought to occur in
the visual cortex14,27–29, because it is there that one first encounters
Figure 4 | Pattern detector model for adaptation. a, Parallel patternselective pathways converge at the ganglion cell and adjust their gain
independently. Each pathway contains orientation-selective interneurons
(left) whose response becomes fatigued under prolonged exposure to the
preferred orientation (middle and right). b, Receptive field profiles of two
salamander bipolar cells (top, ON type; bottom, OFF type). Each profile was
fitted with a gaussian shape, and the line shows the contour of the gaussian at
1 s.d. from the centre. c, Receptive field shape of 10 salamander bipolar cells
(open symbols, ON type; closed symbols, OFF type), given by the long and
short diameters of the 1 s.d. ellipse from the gaussian fit.
overtly orientation-selective cells. However, given that many retinal
ganglion cells adapt to oriented gratings with a gain change of 1.5–2.5
(Fig. 2d, e), it is possible that half of the psychophysical after-effect
already arises in the retina. Furthermore, given the extensive knowledge of retinal circuits and the lack of orientation selectivity in retinal
bipolar cells (Fig. 4b), this is unlikely to occur through fatigue of
pattern-selective interneurons. Thus, one may need to reconsider
both the site and the mechanism of various perceptual adaptation
effects in human vision.
The alternative hypothesis of a modifiable network (Fig. 5) is
intriguing, because a single assumption for plasticity at amacrine cell
synapses can explain a host of seemingly different adaptations
Figure 5 | Network plasticity model for adaptation. a, Inhibitory amacrine
cells connect to a ganglion cell through modifiable synapses (top), providing
the receptive field’s antagonistic surround (bottom). Under patterned
stimulation (middle and right) each synapse changes strength depending on
the correlation between presynaptic and postsynaptic activity. This shapes
the receptive field in a way that reduces sensitivity to the predominant
stimulus pattern. b, Schematic circuit of the inner retina. Bipolar cells (B),
carrying input signals x j, connect to ganglion cells (G) with output signals y i,
through fixed excitatory synapses b ij and through modifiable inhibitory
synapses a ij of amacrine cells. c, Adaptive change in the receptive field of a
model ganglion cell connected to a 4 £ 4 array of bipolar cells. Top, sequence
of stimulus environments driving the adaptation: steady grey screen,
independently flickering pixels, flickering uniform field (Fig. 1b), flickering
checkerboard (Fig. 1b), flickering vertical bars (Fig. 2a), flickering horizontal
bars (Fig. 2a), steady grey screen. the Square outline marks the 4 £ 4 pixels in
the ganglion-cell receptive field; each pixel drives one bipolar cell. Bottom,
receptive field profile of the ganglion cell after adaptation to each
environment. Colours reflect the net synaptic connectivity to each pixel,
a ij þ b ij; red is positive, blue is negative. The fixed connections b ij are
limited to the central four bipolar cells. Pattern adaptation induces
contributions from the surrounding pixels in a manner that suppresses the
adapting stimulus. d, Time course of adaptation. Each environment is
applied for a period of five adaptation time constants t. Curves show the
sensitivity of the ganglion cell (see equation (12) in the Supplementary
Information) to four different types of stimulus: uniform (unif.),
checkerboard (chec.), vertical (vert.) and horizontal (hori.).
© 2005 Nature Publishing Group
75
ARTICLES
NATURE|Vol 436|7 July 2005
(Fig. 5c, d). The qualitative function of this network has a simple
interpretation: one can view each amacrine cell as ‘trying to predict’
the response of the ganglion cell from stimulus information in its
own spatio-temporal receptive field. Those neurons that are successful get ‘rewarded’ with a stronger inhibitory synapse. As a result, the
successful predictions are subtracted from the ganglion cell input,
and by slowly adjusting its synapses the network literally performs a
dynamical version of predictive coding. A precedent for such an
adaptation mechanism is found in the electrosensory system of
weakly electric fish, in which the prediction for sensory input is
performed using efferent motor signals30. Again, the predictions are
adjusted dynamically, in a recurrent network that has been shown to
use activity-dependent synapses with anti-hebbian plasticity. Plausibility notwithstanding, the central ingredient of this hypothesis is the
modifiable amacrine cell synapse and its dependence on presynaptic/
postsynaptic correlation. This kind of use-dependent plasticity has
been reported for inhibitory synapses in the cerebellum31 and elsewhere32, but it remains to be explored in the retina. Finally, the two
mechanisms considered here—changes in neuronal sensitivity versus
changes in connectivity—are not exclusive, and may each contribute
to adaptation.
Faced with the remarkable plasticity of retinal processing—light
adaptation33, contrast adaptation34 and pattern adaptation—the
question arises how the brain copes with the continually changing
rules of encoding in the sensory periphery. For the fly eye, it has been
suggested that the state of adaptation is communicated downstream
through certain slow statistics of visual spike trains21. However, there
is no necessity to inform the brain of adaptive changes in coding. The
goal of the visual system is not to construct internally a veridical
reproduction of the intensity pattern on the retina. Instead, the
system must reduce the onslaught of raw visual information and
extract the few bits of information that are relevant to behaviour.
This entails the discarding of signals that are less useful. The many
known visual illusions of brightness and contrast are evidence of such
information loss, as are the illusory after-effects of pattern adaptation. Thus, pattern adaptation is not merely a scheme for efficient
recoding but rather serves to strip from the visual stream predictable
and therefore less newsworthy signals.
segments were kept short (1.5 s) relative to the intervening adapting segments
(13.5 s). For the experiment of Fig. 1g, the probe lasted 10 s and the adaptation
50 s. Over several hours of experimenting, the ganglion cell firing rates remained
fairly stable, with a typical variation of 20%. Through the rapid interleaving of
different stimuli, the analysis was robust to any slow drifts in response properties.
The Supplementary Information gives details on the construction of the
stimuli.
Analysis. The visual responses of retinal ganglion cells can be approximated well
by a linear–nonlinear (LN) model36. This is a simple mathematical functional
that turns a visual stimulus into the neuron’s firing rate. The stimulus is passed
through a spatio-temporal linear filter, and the resulting variable is transformed
by a nonlinear function that can account for firing threshold and saturation. For
example (Supplementary Fig. S2), if the stimulus contains two spatial regions
with intensity time courses x(t) and y(t), the cell’s firing rate r(t) is
ð
ð
0
0
0
0
0
0
ð1Þ
rðtÞ ¼ NðgðtÞÞ ¼ Nð xðt ÞLX ðt 2 t Þdt þ yðt ÞLY ðt 2 t Þdt Þ
where L X(t) and L Y(t) are the time-dependent impulse responses of the filters
applied to stimulus variables x and y respectively, and N(g) is the nonlinearity.
The brief probe segments P were used to derive the best-fit LN model for each
of the adapting conditions A and B. From the measured response to gaussian
random flicker, we estimated both the filter and the nonlinearity by standard
reverse-correlation algorithms36. The waveforms of these filters for representative neurons are plotted in Figs 1d, 3b and 3f.
To assess the degree of predictive coding, we evaluated the sensitivities of the
neuron’s spatio-temporal filter to stimuli drawn from environments A and B,
denoted S A and S B, respectively (see Supplementary Information). The filter
computed under adapting condition A yielded sensitivities S A(A) and S B(A),
and the filter under adapting condition B yielded S A(B) and S B(B). In the course
of adaptation to B, the sensitivity S A changes by a factor S A(B)/S A(A), and S B
changes by a factor S B(B)/S B(A). These factors are plotted in Figs 1e, 2d, 3c and
3g. The ratio of these two factors is the adaptation index
a¼
SA ðBÞ=SA ðAÞ
SB ðBÞ=SB ðAÞ
ð2Þ
A1 P1 A2 P2 … A10 P10 B1 P11 B2 P12 … B10 P20 A11 P21 A12 P22 … A20 P30 …
which is plotted in Figs 1f, 1g, 2e, 2f, 3d and 3h. It measures the extent to which
adaptation selectively suppresses the adapting stimulus. The hypothesis of
dynamic predictive coding predicts that a . 1. For each cell, the experimental
uncertainty in a was determined using independent subsets of data, and the
P value for a . 1 was computed by means of a one-tailed t-test.
Pharmacology. After the addition of 10 mM strychnine and 100 mM picrotoxin
to the bathing solution, ganglion cell firing rates doubled on average. The light
responses did not saturate, as judged by the form of the nonlinearity in the LN
fits. After return to control solution, we did not achieve full reversal of the drug
effects within the available time; washout of these drugs from a whole-mount
preparation is exceedingly slow37.
Bipolar cell receptive fields. The spatio-temporal receptive field of each bipolar
cell was measured by reverse-correlating the membrane potential to a flickering
checkerboard stimulus35, and then approximated as the product of a spatial
profile and a temporal filter38 The spatial profile was fitted by a two-dimensional
gaussian bell, characterized by the long and short axis of the ellipse at 1 s.d. (Fig.
4b). To estimate how selective such a cell would be for oriented gratings, the
gaussian was convolved with two grating stimuli: one whose bars were aligned
with the long axis of the ellipse, yielding the maximal sensitivity, and the other
aligned with the short axis, yielding the lowest sensitivity. The ratio of the two
sensitivities was taken as the cell’s orientation selectivity, quoted in the text.
Anti-hebbian retina model. Consider a linear feedforward network as illustrated in Fig. 5b. The input layer represents bipolar cells, the output layer
ganglion cells. Each ganglion cell receives two kinds of synaptic input from
bipolar cells: excitatory synapses with fixed strength b ij, and inhibitory
synapses—through intermediate amacrine cells—with a variable strength a ij.
The fixed excitatory synapses set the ‘default’ receptive field of a ganglion cell; the
inhibitory synapses can weaken or strengthen depending on the stimulus history,
and modify the receptive field accordingly.
We suppose that the network operates linearly. For simplicity, we also ignore
the dynamics of the light response, and treat retinal processing as instantaneous.
If x j is the activity of bipolar cell j, and y i the activity of ganglion cell i, then
X
ðb þ aij Þxj
ð3Þ
yi ¼
j ij
where Ai, Bi and Pi represent stimulus segments with different pseudo-random
flicker sequences drawn from environments A, B and P. In a typical experiment,
500 different segments of P were collected for each adaptation state. The probe
daij =dt ¼ ð2aij 2 bkyi xj lÞ=t t; b . 0
METHODS
Recording. The dark-adapted retina of a larval tiger salamander or New Zealand
white rabbit was isolated under an infrared microscope into oxygenated Ringer’s
medium (salamander, 22 8C) or Ames’ solution (rabbit, 37 8C). A piece of retina,
2–4 mm (salamander) or about 5 mm (rabbit) on a side, was placed with the
ganglion cell side downwards onto an array of 61 electrodes to record action
potentials from ganglion cells, as described previously12,35. For intracellular
recordings from salamander bipolar cells13, sharp microelectrodes were filled
with 2 M potassium acetate and 1% Alexa 488, having a final impedance of 150–
250 MQ. After recording, cells were filled ionophoretically (21 to 22 nA pulses,
about 10–15 min), and imaged with a 40 £ water-immersion objective.
Stimulation. Visual stimuli were generated on a computer monitor and
projected through an objective lens onto a 3.25-mm diameter aperture of the
retina. The light was white, with a mean intensity of M < 4 mWm22 at the
retina, in the regime of photopic vision. All experiments employed random
flicker stimuli, whose single-point statistics were identical for every location on
the retina. Over time, each point experienced intensity values distributed as a
gaussian with mean M and standard deviation C ¼ 0.35 M. The intensity was
updated with another random value at periodic intervals of 15 or 30 ms. The
various stimulus environments differed in their two-point statistics, namely the
correlation between the intensity values at different points in space or time.
The goal was to probe retinal response properties after adaptation to two
different stimulus environments A and B. The probe environment P was always
neutral, in that it contained a superposition of stimuli of types A and B. We
interleaved segments of the two adapting stimuli and the probe stimulus in the
order
76
The network undergoes adaptation through slow modulation of the synapses a ij,
following the rule
© 2005 Nature Publishing Group
ð4Þ
ARTICLES
NATURE|Vol 436|7 July 2005
Here 2a ij/t is a decay term, to ensure that the network does not remember
stimulus history forever. The term 2bky ix jl/t is dependent on recent activity,
specifically the correlation between bipolar cell j and ganglion cell i. Adaptation
is driven by the statistics of the inputs x j to the network. Thus, a change in the
stimulus ensemble leads to a change in the ganglion cell’s synapses and thus its
receptive field. For the solution of these dynamic equations, and their application in Fig. 5c, d, see Supplementary Information.
Received 5 September 2004; accepted 27 April 2005.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
Dong, D. W. & Atick, J. J. Statistics of natural time-varying images. Network 6,
345–-358 (1995).
Field, D. J. Relations between the statistics of natural images and the response
properties of cortical cells. J. Opt. Soc. Am. A 4, 2379–-2394 (1987).
Srinivasan, M. V., Laughlin, S. B. & Dubs, A. Predictive coding: a fresh view of
inhibition in the retina. Proc. R. Soc. Lond. B 216, 427–-459 (1982).
Kuffler, S. W. Discharge patterns and functional organization of mammalian
retina. J. Neurophysiol. 16, 37–-68 (1953).
Barlow, H. B. Summation and inhibition in the frog’s retina. J. Physiol. (Lond.)
119, 69–-88 (1953).
Meister, M. & Berry, M. J. II The neural code of the retina. Neuron 22, 435–-450
(1999).
Barlow, H. B. in Sensory Communication (ed. Rosenblith, W. A.) 217–-234 (MIT
Press, Cambridge, Massachusetts, 1961).
van Hateren, J. H. Real and optimal neural images in early vision. Nature 360,
68–-70 (1992).
Atick, J. J. & Redlich, A. N. What does the retina know about natural scenes?
Neural Comput. 4, 196–-210 (1992).
Barlow, H. & Földiák, P. in The Computing Neuron (eds Durbin, R., Miall, C. &
Mitchison, G.) 54–-72 (Addison-Wesley, Wokingham, 1989).
Barlow, H. B. in Vision: Coding and Efficiency (ed. Blakemore, C.) 363–-375
(Cambridge Univ. Press, Cambridge, 1990).
Smirnakis, S. M., Berry, M. J., Warland, D. K., Bialek, W. & Meister, M.
Adaptation of retinal processing to image contrast and spatial scale. Nature
386, 69–-73 (1997).
Baccus, S. A. & Meister, M. Fast and slow contrast adaptation in retinal
circuitry. Neuron 36, 909–-919 (2002).
Graham, N. V. S. Visual Pattern Analyzers (Oxford Univ. Press, New York,
1989).
Mollon, J. D. in The Perceptual World (eds Von Fiendt, K. & Monstgaard, I. K.)
71–-97 (Academic, London, 1977).
Rieke, F. Temporal contrast adaptation in salamander bipolar cells. J. Neurosci.
21, 9445–-9454 (2001).
Bloomfield, S. A. Orientation-sensitive amacrine and ganglion cells in the rabbit
retina. J. Neurophysiol. 71, 1672–-1691 (1994).
Cook, P. B. & McReynolds, J. S. Lateral inhibition in the inner retina is important
for spatial tuning of ganglion cells. Nature Neurosci. 1, 714–-719 (1998).
Chander, D. & Chichilnisky, E. J. Adaptation to temporal contrast in primate
and salamander retina. J. Neurosci. 21, 9904–-9916 (2001).
Snippe, H. P., Poot, L. & van Hateren, J. H. Asymmetric dynamics of adaptation
after onset and offset of flicker. J. Vis. 4, 1–-12 (2004).
Fairhall, A. L., Lewen, G. D., Bialek, W. & van Steveninck, R. R. D. Efficiency and
ambiguity in an adaptive neural code. Nature 412, 787–-792 (2001).
Masland, R. H. The fundamental plan of the retina. Nature Neurosci. 4,
877–-886 (2001).
23. Yang, C. Y., Lukasiewicz, P., Maguire, G., Werblin, F. S. & Yazulla, S. Amacrine
cells in the tiger salamander retina: morphology, physiology, and
neurotransmitter identification. J. Comp. Neurol. 312, 19–-32 (1991).
24. Cook, P. B., Lukasiewicz, P. D. & McReynolds, J. S. Action potentials are
required for the lateral transmission of glycinergic transient inhibition in the
amphibian retina. J. Neurosci. 18, 2301–-2308 (1998).
25. Warland, D. K., Reinagel, P. & Meister, M. Decoding visual information from a
population of retinal ganglion cells. J. Neurophysiol. 78, 2336–-2350 (1997).
26. De Valois, K. K. Spatial frequency adaptation can enhance contrast sensitivity.
Vision Res. 17, 1057–-1065 (1977).
27. Blakemore, C. & Campbell, F. W. On the existence of neurones in the human
visual system selectively sensitive to the orientation and size of retinal images.
J. Physiol. (Lond.) 203, 237–-260 (1969).
28. Movshon, J. A. & Lennie, P. Pattern-selective adaptation in visual cortical
neurones. Nature 278, 850–-852 (1979).
29. Snowden, R. J. & Hammett, S. T. Subtractive and divisive adaptation in the
human visual system. Nature 355, 248–-250 (1992).
30. Bell, C. C. Memory-based expectations in electrosensory systems. Curr. Opin.
Neurobiol. 11, 481–-487 (2001).
31. Aizenman, C. D., Huang, E. J., Manis, P. B. & Linden, D. J. Use-dependent
changes in synaptic strength at the Purkinje cell to deep nuclear synapse. Prog.
Brain Res. 124, 257–-273 (2000).
32. Gaiarsa, J. L., Caillard, O. & Ben-Ari, Y. Long-term plasticity at GABAergic and
glycinergic synapses: mechanisms and functional significance. Trends Neurosci.
25, 564–-570 (2002).
33. Shapley, R. & Enroth-Cugell, C. in Progress in Retinal Research (eds Osborne, N.
& Chader, G.) Vol. 3 263–-346 (Pergamon, London, 1984).
34. Baccus, S. A. & Meister, M. Retina versus cortex; contrast adaptation in parallel
visual pathways. Neuron 42, 5–-7 (2004).
35. Meister, M., Pine, J. & Baylor, D. A. Multi-neuronal signals from the retina:
acquisition and analysis. J. Neurosci. Methods 51, 95–-106 (1994).
36. Chichilnisky, E. J. A simple white noise analysis of neuronal light responses.
Network 12, 199–-213 (2001).
37. Cook, P. B., Lukasiewicz, P. D. & McReynolds, J. S. GABAC receptors control
adaptive changes in a glycinergic inhibitory pathway in salamander retina.
J. Neurosci. 20, 806–-812 (2000).
38. Schnitzer, M. J. & Meister, M. Multineuronal firing patterns in the signal from
eye to brain. Neuron 37, 499–-511 (2003).
Supplementary Information is linked to the online version of the paper at
www.nature.com/nature.
Acknowledgements We thank members of the Meister laboratory,
H. Sompolinsky and D. Fisher for advice. This work was supported by grants
from the National Eye Institute (M.M. and S.A.B.) and the Human Frontier
Science Program (T.H.).
Author Contributions T.H. and M.M. planned the study, T.H. and S.A.B.
performed the experiments, and T.H. and M.M. completed the analysis and
wrote the manuscript.
Author Information Reprints and permissions information is available at
npg.nature.com/reprintsandpermissions. The authors declare no competing
financial interests. Correspondence and requests for materials should be
addressed to M.M. ([email protected]).
© 2005 Nature Publishing Group
77
Supplementary Methods
Stimulation
The following reports in detail the stimuli used in the various experiments. We
define the stimulus variable s( t ) as the intensity I (t ) normalized for mean and
contrast:
s( t ) = ( I ( t ) ! M ) C .
(5)
Also, we define the random variable n15 (t ) as a staircase waveform updated every 15
ms by independent draws from a normal distribution; similarly n30 ( t ) is updated every
30 ms.
Spatial correlation (Fig 1a): The field was divided into two sets of alternating square
tiles like a checkerboard (Fig S1a). One set was modulated with stimulus variable x ( t ) ,
the other with y (t ) . Environment A (positive correlation): y (t ) = x ( t ) = n30 (t ) .
Environment B (negative correlation): y (t ) = ! x (t ) = n30 ( t ) . P (probe): y (t ) and x ( t ) are
independent versions of n30 ( t ) . The tile size was chosen as 400 µm or 200 µm, similar to
the diameter of a typical receptive field center for salamander ganglion cells.
Spatial orientation (Figs 2a): The field was divided into square tiles belonging to 4
symmetrical sets (Fig S1b) modulated with stimulus variables x ( t ) , y (t ) ,u (t ), v (t ) . A
(horizontal bars): x ( t ) = y ( t ) = !u (t ) = ! v (t ) = n30 ( t ) . B (vertical bars):
x ( t ) = !y (t ) = u (t ) = ! v (t ) = n30 ( t ) . P (probe): x ( t ) , y (t ) ,u (t ), v (t ) all modulated
independently as n30 ( t ) . The tile size was chosen as 200 µm. For the "shifting border"
Hosoya, Baccus, & Meister: Supplementary Material
1
condition (Fig 2e), the tiling was shifted randomly on a fine square grid (40 or 67 µm) at
every stimulus update (30 ms).
Temporal correlation (Fig 3a): A uniform field was modulated with stimulus
variable x ( t ) . A (positive correlations across 60 ms):
2
x ( t ) = 0.97 ! x ( t " 60 ms ) + 1 " 0.97 ! n15 (t ) . B (negative correlations across 60 ms):
2
x ( t ) = !0.97 " x (t ! 60 ms) + 1! 0.97 " n15 (t ) . P (uncorrelated probe): x ( t ) = n15 (t ) .
Space-time delay (Fig 3e): The field was again divided like a checkerboard with 200
µm tiles (Fig S1a), and the two sets of tiles modulated with stimulus variables x ( t ) and
y (t ) . A (X advanced): y (t ) = x ( t ! 60 ms ) = n30 ( t ! 60 ms) . B (Y advanced):
x ( t ) = y ( t ! 60 ms ) = n30 ( t ! 60 ms) . P (probe): x ( t ) , y (t ) drawn independently as n30 ( t ) .
Analysis
To characterize each neuron’s light response, the brief probe segments P were used
to derive the best-fit LN model 37 for the firing rate. For example (Figs 1, S2), if the
stimulus contains two spatial regions modulated with x ( t ) and y (t ) , then the LN fit to
the firing rate is
(1)
r ( t ) = N ( g ( t )) = N
( # x( t! ) L
X
(t " t ! ) dt ! + # y ( t ! ) LY ( t " t ! ) dt ! ) .
The linear filter was obtained from the first-order kernel of the spike train with respect
to the stimulus variables.
L X (! ) =
(6)
LY (! ) =
1
T
1
T
T
# x (t " ! ) r ( t ) dt
0
T
# y (t " ! ) r ( t ) dt
0
Hosoya, Baccus, & Meister: Supplementary Material
2
where T is the duration of the spike train used for analysis. Then the nonlinearity N ( g)
was found by computing
g (t ) = # x (t ! ) LX (t " t ! ) dt ! + # y ( t ! ) LY ( t " t ! ) dt !
(7)
and plotting the measured firing rate r ( t ) against g (t ) .
In computing the filter, only the first 0.8 s of each P segment were used, to limit the
degree of adaptation to the probe stimulus itself. Moreover, the very beginning of each
P segment, amounting to the duration of the filter (typically 0.18 s), was ignored to
avoid contamination of the response from the preceding adapting stimulus. Note that
this also precluded the detection of any fast changes in the neuron’s sensitivity that may
occur instantaneously on switching from environments A or B to P. Such very rapid
changes are observed, for example, when the switch involves a simple change in
stimulus contrast 13.
Two models were computed for the adapting conditions A and B. For any given cell,
the shape of the nonlinearity was found to be essentially the same under both A and B,
and thus we used the same function N ( g) in fitting the model to both conditions. The
resulting LN model produced a good fit to the recorded spike trains, with an RMS
deviation between model and neuron 37 of typically 0.15 ± 0.01 spikes per 15 ms timebin (mean ± SEM, 15 cells, experiment of Fig 1).
To assess the degree of predictive coding, we evaluated how sensitive the neuron is
to stimuli drawn from environments A and B. Specifically, we measured the root-meansquare amplitude of the output g (t ) from the linear filter (Fig S2), if it were stimulated
with ensemble A or B. For example, consider the spatial correlation experiment (Fig 1,
Eqn (1)). In environment A, y (t ) = x ( t ) = n30 (t ) , and therefore
Hosoya, Baccus, & Meister: Supplementary Material
3
g (t ) = # x (t ! ) LX (t " t ! ) dt ! + # y ( t ! ) LY ( t " t ! ) dt !
(8)
= # n30 (t ! ) $% LX (t " t !) + LY ( t " t !) &' dt !
Because the values of n30 ( t ! ) are drawn independently from a normal distribution, the
variance of g (t ) , averaged over all stimuli in A, is to within some constant factor
(9)
g
2
2
A
! ' #$ LX ( t " ) + LY ( t " )%& dt " .
In environment B, y (t ) = ! x (t ) = n30 ( t ) , and therefore
(10)
g (t ) = ' n30 (t ! ) #$ LX (t " t !) " LY ( t " t !) %& dt !
and
(11)
g
2
2
B
! ( $% LX ( t " ) # LY ( t " )&' dt " .
The sensitivities SA and SB to stimuli of type A or B are then defined as
(12)
SA =
g
2
A
,
SB =
g
2
B
.
To test the effects of adaptation, we measured the filters after the retina was adapted
to A – yielding SA ( A ) and SB ( A ) – and then again after it was adapted to B – yielding
SA ( B ) and SB ( B ) . In the course of adaptation to B, the sensitivity SA changes by a factor
SA ( B ) SA ( A ) , and SB changes by a factor SB ( B ) SB ( A ) . The ratio of these two factors is
the adaptation index
(2)
!=
SA ( B ) SA ( A )
SB ( B ) SB ( A )
Hosoya, Baccus, & Meister: Supplementary Material
.
4
For the other adaptation experiments, the analysis proceeded in precisely the same
fashion. In each case, the probe stimulus spans a broad space that encompasses both
stimuli of type A and B. Thus the response filters derived from P allow a measurement
of the neuron's sensitivity to stimuli A and B, and ultimately the adaptation index.
Anti-Hebbian retina model
In the approximation discussed in the text, the retina’s instantaneous light response
to bipolar cell signals is given by Eqn (3), which becomes in matrix notation
y = (B + A ) ! x = R ! x
(13)
where
(14)
B = !" bij #$ , A = !" aij #$ , R = B + A = Response matrix .
The bipolar cell synapses bij are constant, but the amacrine cell synapses aij evolve as
given by Eqn (4), or in matrix notation,
(15)
d
1
T
A = "A " # $ y $ x
dt
!
(
) = !1 (" A " # $ (A + B) x $ x ) = " !1 (A + # ( A + B ) $ C) ,
T
where
(16)
C = x ! xT = stimulus covariance matrix .
Thus the response matrix R of the network changes according to
(17)
d
1
R = ! (( R ! B ) + # R $C) .
dt
"
After adaptation is complete, dR dt = 0 , and therefore
Hosoya, Baccus, & Meister: Supplementary Material
5
(18)
R (t = ! ) = B" (1 + # C)
$1
To interpret this response matrix, it is best to work in the eigenbasis of the covariance
matrix. C is symmetric real and therefore has n orthonormal eigenvectors, where n is
the number of bipolar cells. Let u j denote the jth eigenvector of C with eigenvalue c j . In
the basis of the u j , C is diagonal
(19)
! c1 0 0 $
C =# 0 O 0&
#
&
" 0 0 cn %
and therefore
(20)
$ 1
'
0
0
& 1+ # c
)
1
&
)
R ( !) = B " & 0
O
0 )
&
1 )
& 0
)
0
%
1 + # cn (
So in the final state after adaptation, the system behaves as though a multi-dimensional
scaling had been applied to the bipolar cell input: The component of the input vector
(
)
along the eigenvector u j of the covariance matrix gets suppressed by a factor 1 1 + ! c j .
In summary, the system learns to suppress highly correlated components of the
stimulus. It does so by subtracting from the ganglion cell input those signals that are
effective at predicting it.
The approach to the final adapted state follows a time course with multiple
exponentials. Define the deviation from the final state as
(21)
R ! ( t ) = R ( t ) " R (# ) .
Hosoya, Baccus, & Meister: Supplementary Material
6
Then Eqn (17) is solved by
(22)
n
& 1 + $ cj
R! ( t ) = R! (0 ) " , exp ( #
'
%
j =1
)
T
t + "u j "u j .
*
So the sensitivity of the system along the direction u j in stimulus space approaches the
(
)
final state exponentially with time constant ! 1 + " c j . Note the approach is faster the
higher the stimulus variance c j along that direction.
In the example of Figs 5d-e, the stimulus drives a 4x4 array of bipolar cells, which
are connected to a single ganglion cell as described above. The fixed synapses connect
only to the central 2x2 bipolar cells, with equal strength. This default receptive field R
is apparent when the stimulus is off ( 0 < t < 5! ). With the appearance of patterned
stimulation, the modifiable synapses gradually adjust according to the above rules to
suppress the correlated components. For example, stimulation with a flickering grating
results in a receptive field with preferred orientation orthogonal to that of the grating
( 20! < t < 30! ). For this illustration, the gain factor ! was set to 5.
Hosoya, Baccus, & Meister: Supplementary Material
7
Supplementary Figure Legends
Fig S1
Spatial layout of multi-variable flicker stimuli.
Fig S2
A Linear-Nonlinear cascade to model a neuron's firing rate in response to two stimulus
inputs. The inputs x ( t ) and y (t ) are each passed through a linear filter, with impulse
response LX (! ) and LY (! ) respectively. The results are summed and transformed by an
instantaneous nonlinear function N ( g) to yield the firing rate r ( t ) .
Fig S3
Inhibitory synapses are essential for pattern adaptation. Adaptation index measured
with horizontal and vertical gratings (as in Fig 2e), in normal Ringer's solution (left) and
after addition of 10 µM Strychnine and 100 µM Picrotoxin (right) to block inhibitory
transmission via glycine and GABA, respectively. In the drug condition, the average
adaptation index is not significantly different from 1 (p=0.19).
Hosoya, Baccus, & Meister: Supplementary Material
8
Fig S1
a
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
b
x
u
x
u
x
y
v
y
v
y
x
u
x
u
x
y
v
y
v
y
x
u
x
u
x
Fig S2
x (t )
L X (τ )
g (t )
y (t )
L Y (τ )
N (g)
r (t )
Fig S3
Control
PTX + STR
4
Cells
4
0
0
0.5
1
2
Adaptation Index, α
0.5
1
2
Adaptation Index, α
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement