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The Journal of Neuroscience, December 15, 2002, 22(24):10811-10818
Isolation of Relevant Visual Features from Random Stimuli for
Cortical Complex Cells
Jon
Touryan1,
Brian
Lau2, and
Yang
Dan1, 2
1 Group in Vision Science, University of California,
Berkeley, California 94720, and 2 Division of Neurobiology,
Department of Molecular and Cell Biology, University of California,
Berkeley, California 94720-3200
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ABSTRACT |
A crucial step in understanding the function of a neural circuit in
visual processing is to know what stimulus features are represented in
the spiking activity of the neurons. For neurons with complex,
nonlinear response properties, characterization of feature
representation requires measurement of their responses to a large
ensemble of visual stimuli and an analysis technique that allows
identification of relevant features in the stimuli. In the present
study, we recorded the responses of complex cells in the primary visual
cortex of the cat to spatiotemporal random-bar stimuli and applied
spike-triggered correlation analysis of the stimulus ensemble. For each
complex cell, we were able to isolate a small number of relevant
features from a large number of null features in the random-bar
stimuli. Using these features as visual stimuli, we found that each
relevant feature excited the neuron effectively in isolation and
contributed to the response additively when combined with other
features. In contrast, the null features evoked little or no response
in isolation and divisively suppressed the responses to relevant
features. Thus, for each cortical complex cell, visual inputs can be
decomposed into two distinct types of features (relevant and null), and
additive and divisive interactions between these features may
constitute the basic operations in visual cortical processing.
Key words:
complex cell; primary visual cortex; Wiener kernel; principal component analysis; spatiotemporal; nonlinear
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INTRODUCTION |
An important goal in studying the
receptive-field properties of visual neurons is to understand how they
respond to complex spatiotemporal inputs, including those encountered
in natural scenes. To analyze the responses to complex stimuli, a
useful approach is to decompose the stimuli into a set of basic
features (basis set) and to characterize how each feature contributes
to the neuronal response. Several methods have been used to define basis sets for the efficient representation of visual stimuli, including principal component analysis (PCA), independent component analysis (Bell and Sejnowski, 1997 ; van Hateren and Ruderman, 1998),
and/or analysis based on sparse coding (Olshausen and Field, 1996). For
studying the response properties of a given visual neuron, it is
desirable to construct a basis set so that the neuron responds to only
a small number of visual features in the set. The segregation between a
small number of "relevant" visual features and a large number of
"irrelevant" features can greatly facilitate experimental
characterization of the visual neuron.
For neurons with a linear stimulus-response relationship, relevant
visual features can be identified by estimating their linear receptive
fields using a spike-triggered average of the stimulus ensemble (also
called "reverse correlation") (de Boer and Kuyper, 1968). This
method has been widely used to measure the spatiotemporal receptive
fields of neurons in the early visual pathway (Jones and Palmer, 1987;
Reid et al., 1997); the resulting receptive fields can largely account
for the neuronal responses to complex spatiotemporal stimuli (Brodie et
al., 1978; Dan et al., 1996). However, in the visual cortex most of the
neurons are complex cells with nonlinear stimulus-response
relationships that cannot be characterized with the spike-triggered
average. In the present study, we have used spike-triggered correlation
analysis of the stimulus ensemble (de Ruyter van Steveninck and Bialek,
1988; Yamada and Lewis, 1999; Brenner et al., 2000) to construct the basis set for each complex cell. We found that visual features in such
a set are clearly segregated into two categories: a small number of
relevant features and a large number of null features. Using visual
stimuli consisting of either a single feature or a combination of
features, we directly measured the contribution of each type of feature
to the cortical responses.
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MATERIALS AND METHODS |
Physiological preparation. Adult cats (2-3 kg) were
initially anesthetized with isoflurane (3%, with
O2) followed by sodium pentothal (10 mg/kg, i.v.,
supplemented as needed). During recording, anesthesia was maintained
with sodium pentothal (3 mg · kg 1 · hr1,
i.v.), and paralysis was maintained with pancuronium bromide (0.1-0.2
mg · kg1 · hr1,
i.v.). The pupils were dilated with 1% atropine sulfate, nictitating membranes were retracted with 2.5% phenylephrine hydrochloride, and
the eyes were mechanically stabilized and optimally refracted. End-expiratory CO2 was maintained at 4%, the
core body temperature was kept at 38°C, and the
electrocardiogram and EEG were monitored continuously. All
experimental procedures were performed as approved by the Animal Care
and Use Committee at the University of California, Berkeley.
Recording. Extracellular recordings were made with tungsten
electrodes (A-M Systems, Carlsborg, WA). Unit isolation was based on
the cluster analysis of waveforms and the presence of a refractory period determined from the autocorrelograms. Cells were classified as
simple if their receptive fields had clear on and off subregions (Hubel
and Wiesel, 1962) and if the ratio of the first harmonic to the DC
component of the response to an optimally oriented drifting grating was
>1 (Skottun et al., 1991). All other cells were classified as complex.
Among the 61 complex cells recorded, one was excluded from analysis
because of its low firing rate in response to random-bar stimuli (<1
spike per second).
Visual stimulation. Visual stimuli were generated with a
personal computer and presented with a Barco monitor (size, 40 × 30 cm; refresh rate, 120 Hz; maximum luminance, 80 cd/m2). Luminance nonlinearities
were corrected using software written in our laboratory. The random-bar
stimuli were presented in a rectangular patch covering the receptive
field of each cell. This patch was divided into 16 bars aligned to the
optimal orientation of the cell; the length of the bars was equal to or
slightly longer than the receptive field. The contrast of each bar was
temporally modulated according to a pseudorandom binary m-sequence
(Sutter, 1987) (luminance, ±39 cd/m2 from
the mean of 40 cd/m2). The full m-sequence
was 32,767 frames long and was updated every other frame, for an
effective frame rate of 60 Hz. To measure the contrast-response
functions of individual features (see Fig. 5), we randomly interleaved
short movies (16 frames per movie) of relevant and null features, each
at a range of contrasts (positive and negative, see below for
definition of contrast), with no gap between movies. To measure the
interaction between two relevant features (see Fig. 6), we generated a
set of short movies containing all possible linear combinations of the
two features. The number of repetitions for each short movie varied
between 1 and 120, which was proportional to the probability of the
corresponding feature contrast in the random-bar stimuli (for example,
the probability of a high contrast for a given feature in the
random-bar stimuli is generally lower than the probability of a low
contrast for the same feature; thus, the movie of the feature at the
high contrast was repeated fewer times). Note that in each movie, which
contains either a single feature or a combination of features, the
luminance of each bar must be between  1 and 1 (corresponding to 0 and
80 cd/m2, respectively), which limits the
maximum contrast of each feature that can be presented (the definition
of contrast is described below).
Spike-triggered correlation analysis. In general, if certain
features in the visual stimuli affect the firing probability of the
cell, the spike-triggered stimulus ensemble should exhibit a different
probability distribution from the entire stimulus ensemble (see Fig.
1B; compare the distribution of the filled circles and the distribution of all of the circles).
Although a change in the probability distribution can be reflected in a change in the first-order (mean), second-order (variance), or higher-order moments, the correlation analysis aims to identify features with changed variance. Because PCA results in a set of components with their variance ranking from the highest to the lowest,
it is ideally suited for the identification of features with
outstanding variance. Practically, identification of relevant features
was achieved by finding eigenvalues of the spike-triggered correlation
matrix that were significantly different from the eigenvalues of the
control correlation matrix (computed by randomly sampling the entire
stimulus ensemble). For each cell, responses to three to four repeats
of the random-bar stimuli (~9 min) were used for spike-triggered
correlation analysis. Each pattern in the stimulus ensemble consisted
of luminance at 16 bar positions at 16 frames (assuming that neuronal
spiking probability depends only on the immediate stimulus history
within 16 frames, lasting for 268 msec), which was uniquely specified
by 256 parameters. The spike-triggered correlation matrix,
[Cm,n] (m,
n = 1, 2, ... , 256) was computed as follows:
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where Sm(i) and
Sn(i) are the
mth and nth parameters of the stimulus pattern
preceding the ith spike, respectively, and N is
the total number of spikes in the response. The resulting matrix is
closely related to the second-order Wiener kernel (Wiener, 1958;
Marmeralis and Marmeralis, 1978) of the neuron. Eigenvalues and
eigenvectors of this spike-triggered correlation matrix were then
computed. To compute each control correlation matrix, we generated a
random spike train with the same number of spikes as in the recorded
response but with random spike timing; the correlation matrix was
computed based on this simulated random spike train. Because subsequent
experiments required fast identification of the significant
eigenvectors, we computed only five control correlation matrices for
each cell during the experiments; the confidence interval for the
control eigenvalues was set at mean ± 5.2 SD (corresponding to
p < 104). Eigenvectors
with eigenvalues outside of the control confidence interval were
considered significant. Subsequent offline analyses with 100 control
matrices confirmed that the significant eigenvectors were identified
reliably using only five control matrices.
An important question is how the constraints of the above method affect
the outcome of the analysis. For example, the eigenvectors must be
orthogonal, which could affect the visual features identified. As shown
in Figure 4, the two significant eigenvectors for most complex cells
exhibited similar spatial frequencies; one might suspect that the
~90° spatial phase difference between them resulted from the
orthogonality between the two vectors. However, this is not the case.
First, both vectors are spatiotemporal patterns. The fact that the dot
product of them is 0 (summed over all temporal delays) does not
uniquely specify their spatial relationship at each temporal delay.
Second, two Gabor functions (which were used to fit the spatial
profiles of the vectors in Fig. 4) with a 90° phase difference are
generally not orthogonal to each other. Even if the two vectors are
orthogonal at each temporal delay (which is not imposed by the method),
their phase difference still may not be 90°. Thus, the spatial phase
relationship we have shown is not a trivial consequence of the method
but is a reflection of the response property of complex cells. Another
question is whether this method allows identification of visual
features that are not orthogonal to each other. Generally, even if the
features are not orthogonal, this method can still be used to identify linear combinations of the features. Subsequently, the relationship between the visual features and the neuronal response may be revealed by measuring the joint contrast-response function of the significant eigenvectors (see Fig. 6A). Finally, although PCA is
a linear method for decomposing each stimulus into the sum of multiple eigenvectors, it does not require additive interaction between different eigenvectors in the response of the neuron. Even if the cell
does not sum the responses to different features, this method can still
identify either the individual features or linear combinations of them.
The type of interaction between visual features can then be determined
through analysis of the joint contrast-response function. These points
can be demonstrated using simulated responses of model cells with the
feature selectivity described above (data not shown). Finally, it is
important to keep in mind that this method does not necessarily
identify all of the features that affect the responses of the neuron,
especially those that contribute weakly to the response.
Contrast-response function. For measuring the
contrast-response functions, the contrast of the kth
eigenvector in the stimulus, Vk, is
defined as the dot product between the stimulus vector and the
eigenvector, as follows:
where 1  S(x,t) 1 represents luminance at the tth temporal frame in the
xth bar position of the stimulus pattern. Because the
eigenvectors are normal:
the scaling factor 1/16 in the definition ensures that the
contrast of each stimulus pattern is bound between 1 and 1. In the
joint contrast-response function, the contrasts of both eigenvectors (Figs. 6A,B, 7A, contrast 1 and contrast
2) are defined the same as above.
Estimation of upper limit for correlation coefficient. To
estimate the upper limit for the correlation coefficient between the
predicted and measured contrast-response functions of relevant visual
features (see Fig. 6), we simulated the functions measured from a
finite number of repeats using a parametric bootstrap (Efron and
Tibshirani, 1993). Briefly, for each stimulus that was repeated L times in the experiment with recorded responses
r1,
r2, ... , rL, we simulated the response by
drawing random samples (r1', r2', ... , rL', from a Gaussian distribution with
the same mean and variance as the recorded responses
(r1,
r2, ... , rL) and computed the average of the
simulated responses, as follows:
Repeating this step for all of the contrast levels resulted in a
simulated contrast-response function with a noise level comparable
with that measured experimentally. We then computed the mean ± 95% confidence interval (obtained from 500 simulations) of the
correlation coefficient between contrast-response functions obtained
in different trials of the simulation. This was used as an estimate of
the upper limit for the correlation coefficient between the predicted
and measured contrast-response functions set by noise in the measured responses.
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RESULTS |
Segregation between two types of visual features
Single-unit recordings were made from complex cells in the striate
cortex of anesthetized adult cats. The stimuli consisted of 16 bars
along the preferred orientation of the cell, with each bar varying
randomly between light and dark at 60 Hz (Fig.
1A). To construct a
basis set for each neuron that isolates the relevant visual features,
we collected the spatiotemporal visual signals within a window of 268 msec (16 frames) before each spike and performed principal component
analysis of this spike-triggered stimulus ensemble (Fig.
1B, filled circle) (see Materials and Methods). Unlike the spike-triggered average, which is the mean of the
spike-triggered stimulus ensemble, the present method identifies a set
of visual features (represented by eigenvectors of the spike-triggered correlation matrix) that account for different amounts of variance (the
corresponding eigenvalues) in the ensemble. A visual feature with an
outstanding variance (significantly larger or smaller than the variance
of the control ensemble) (Fig. 1B, open
circle) is directly relevant to the spiking response of the
neuron.
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Figure 1.
Illustration of a spike-triggered correlation
analysis. A, Spatiotemporal random-bar stimuli
(top) and the spike train response
(bottom). Each bracket indicates a
spatiotemporal stimulus pattern preceding a spike (for actual analysis,
each pattern contained 16 rather than 3 frames). B,
Schematic representation of the spike-triggered stimulus ensemble
(filled circles) and the entire stimulus ensemble
(open and filled circles) in a
multidimensional parameter space. Each axis (black arrows)
represents luminance at a particular bar position and time frame, and
each point represents a stimulus pattern. Note that the
actual stimulus ensemble is represented in a 256-dimensional space (16 frames by 16 bars). a-c indicate stimulus patterns
shown in A. The gray arrow indicates an
eigenvector of the spike-triggered ensemble with its eigenvalue
(variance) greater than the eigenvalues of the entire
ensemble.
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Figure 2A shows the 30 largest eigenvalues of the spike-triggered correlation matrix for a
complex cell. Two eigenvalues (filled circles)
conspicuously stood out from the rest (open circles), suggesting that the corresponding visual features (eigenvectors) are
particularly relevant to the cell. The dashed lines indicate the confidence interval for eigenvalues of the control stimulus ensemble, sampled randomly from the random-bar stimuli (see Materials and Methods). The first two eigenvalues of the spike-triggered ensemble
were well above the control, indicating significance of the
corresponding eigenvectors. Figure 2B shows three
eigenvectors, two corresponding to the significant eigenvalues (first
and second) and one to a nonsignificant eigenvalue (nth).
Although the spatiotemporal structure of the nonsignificant eigenvector
appeared to be random, the significant eigenvectors had spatially
separate on and off subregions evolving smoothly over time. To further
confirm the distinction between these two types of eigenvectors, we
compared both their eigenvalues and the correlation in their structures (which is a measure of nonrandomness) for a population of complex cells
(n = 60). Figure 2C shows the distributions
of the significant and nonsignificant eigenvalues; Figure
2D shows the distributions of the correlation of the
eigenvectors (legend to Fig. 2). The two types of eigenvectors showed
little overlap in both properties, indicating an unambiguous
segregation between them.
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Figure 2.
Distinction between the significant
and nonsignificant eigenvectors. A, The 30 largest
eigenvalues of the spike-triggered correlation matrix of a complex
cell. Dashed line, Control confidence interval
(p = 104) obtained by
random sampling of the entire stimulus ensemble (100 repeats; see
Materials and Methods). Filled circles, Eigenvalues that
are significantly different from the control; open
circles, nonsignificant eigenvalues. B, Two
significant eigenvectors (first and second) and one nonsignificant
eigenvector (nth; n = 15 in this
case) whose eigenvalues are indicated by the large
circles in A. Arrow, 40 msec (the
delay at which the spatial profiles of the eigenvectors are shown in
Fig. 4A). Calibration: 1°, 100 msec.
C, Distributions of significant (sig.;
solid line) and nonsignificant (nonsig.;
dashed line) eigenvalues from 60 complex cells. Each
eigenvalue was normalized by the mean eigenvalue of the cell.
D, Distribution of temporal correlation in significant
and nonsignificant eigenvectors. The autocorrelation function of each
eigenvector was computed along the temporal axis; the eigenvector
correlation shown here was measured by the autocorrelation at the
delay of 1 frame (16.7 msec) normalized by the autocorrelation at 0 delay, as follows:
where V(x,t) represents the luminance
value of the eigenvector at pixel x and temporal delay
t (see Materials and Methods). This value provides a measure
of the nonrandomness of the eigenvector structure between 0 (completely
random) and 1 (temporally uniform).
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For most (47 of 60) of the complex cells studied, we found two
significant eigenvectors (Fig. 3),
corresponding to the two largest eigenvalues. These two eigenvectors
exhibited separate on and off spatial subregions (Fig.
2B), resembling the receptive fields of simple cells.
The relationship between the two vectors was revealed by fitting their
spatial profiles at the peak temporal delay (~40 msec preceding
spike) with Gabor functions (Fig.
4A). In all cases, the
Gabor fits for the two vectors exhibited similar spatial frequencies
but a difference of ~90° in phase (Fig. 4B), reminiscent of the relationship between different subunits in the
energy model for complex cells (Movshon et al., 1978; Pollen and
Ronner, 1981; Adelson and Bergen, 1985; Heeger, 1991). As explained in
detail in Materials and Methods, this phase relationship reflects the
response property of complex cells and is not a trivial consequence of
the orthogonality between eigenvectors, which is imposed by the method.
In a few cases (3 of 60), we found only one significant eigenvector for
each complex cell; these vectors also exhibited spatiotemporal profiles
resembling simple-cell receptive fields. In the remaining cases, more
than two eigenvalues reached significance. However, these additional
eigenvectors (corresponding to third, fourth, ... , largest
eigenvalues) tended to exhibit much less spatiotemporal structure than
the first two eigenvectors, and their eigenvalues were much smaller,
suggesting less functional importance.
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Figure 3.
Distribution of the number of significant
eigenvectors found for each cell, from a total of 60 complex
cells.
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Figure 4.
Relationship between the two most significant
eigenvectors. A, Spatial profiles (solid
lines) of the two significant eigenvectors shown in Figure
2B at 40 msec from spiking (arrows) and
their Gabor fits (dashed lines). The two Gabor fits had
a phase difference of 90.4°. Dotted line represents
mean luminance. B, Distribution of the spatial phase
difference between the two significant eigenvectors of each complex
cell.
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Responses of cortical neurons to individual visual features
The clear segregation between the significant and nonsignificant
eigenvalues suggests that the corresponding eigenvectors contribute
differently to the cortical responses. To test this idea directly, we
measured the responses of each complex cell to individual vectors in
both categories. Each vector (a 268 msec movie) was presented at a
range of positive and negative contrasts (see Materials and Methods for
the definition of contrast), and the peristimulus time histograms
(PSTHs) of the cell were measured (Fig.
5A). Note that only the last
bin (indicated by an arrow) of each PSTH reflects the
neuronal response to the complete spatiotemporal visual feature
represented by the eigenvector; its amplitude was used to measure the
contrast-response function. Figure 5B shows the
contrast-response functions of a complex cell for two significant eigenvectors and one nonsignificant eigenvector. For each significant eigenvector, the response increased with the absolute value of the
contrast at both positive and negative polarities, consistent with the
known polarity invariance of complex cells (Hubel and Wiesel, 1962).
The nonsignificant eigenvector, however, evoked no contrast-dependent
response. We fitted the left and right sides of each contrast-response
function separately with a power function, y(x) =  |x| , where
x and y represent the vector contrast and the
neuronal response, respectively, and and  are free parameters.
For the significant eigenvectors, the exponent was found to be
2.7 ± 0.1 (SEM; n = 34), similar to the exponent
of contrast-response functions measured with drifting gratings
(Albrecht and Geisler, 1991; Anzai et al., 1999). The ratio between the
response at maximal vector contrast and that at zero contrast was
96.6 ± 11.8. Thus, visual features represented by the significant
eigenvectors can each drive the cortical neuron effectively in a
contrast-dependent manner; they are referred to as relevant features.
For the nonsignificant eigenvectors, the ratio between the responses at
maximal and zero contrasts was 3.8 ± 0.9 (n = 24), much lower than that for the significant eigenvectors. Thus,
visual features represented by the nonsignificant eigenvectors evoked
little contrast-dependent response and were therefore termed null
features.
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Figure 5.
Responses of a complex cell (different from that
shown in Fig. 2) to individual features. A, Responses to
two significant eigenvectors (first and second) and one nonsignificant
eigenvector (nth, randomly chosen for each cell;
n = 8 in this case). The top panel of
each row shows the eigenvector presented at a range of positive
(right) and negative (left) contrasts.
Calibration: 1°, 100 msec. The PSTH in response to each stimulus is
plotted below, with the arrow indicating the last time
bin of the PSTH. B, Contrast-response function
(amplitude of the last bin in each PSTH) for each vector
(bars) and the fit of each side with a power function
(line). Error bars indicate SEM.
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Additive interaction between relevant visual features
Each spatiotemporal random-bar pattern can be decomposed into a
combination of relevant and null features in the basis set. To
understand cortical responses to arbitrary random-bar stimuli, it is
necessary to characterize not only the contrast-response functions for
individual features (Fig. 5) but also the interaction between features.
First, we measured the responses of each complex cell to combinations
of relevant features. For each neuron with two significant
eigenvectors, we constructed a set of visual stimuli, each of which was
a 268 msec movie consisting of a linear combination of the two
significant eigenvectors. Figure
6A shows the responses of a complex cell at various combinations of the two vectors, which is
referred to as the joint contrast-response function (see Materials and
Methods). The response increased with the absolute value of contrast of
either vector independently of their polarities, consistent with the
contrast-response functions measured with individual vectors (Fig.
5B). Note that each combination of the two vectors also
exhibited spatially separate on and off subregions (small outer
plots), with the spatial phase shifting with the relative weights
of the two vectors. The approximate circular symmetry of the joint
contrast-response function indicates that the response is insensitive
to the spatial phase of the stimuli, a well known property of cortical
complex cells (Hubel and Wiesel, 1962; Movshon et al., 1978).
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Figure 6.
Interaction between relevant visual
features. A, Joint contrast-response function of a
complex cell (different from Figs. 2 and 5), in which the amplitude of
the response (color-coded, with the scale shown on the
right in units of spikes per second) is plotted against
the contrasts of both relevant features. The outer plots
depict the stimulus patterns corresponding to selected points
(indicated by arrows) in the contrast-response
function. Calibration: 1°, 100 msec. The range of contrast
represented in the center box is 0.33 to 0.33; the
contrast of each feature varied at a step size of 0.033. Responses were
measured only at contrasts at which the luminance signals in the movie
do not exceed the range of the monitor. Gray indicates
contrast at which the response was not measured. B,
Prediction of the contrast-response function based on additive
interaction between the two vectors. The left and
bottom histograms represent the contrast-response
functions for the first and second eigenvectors, respectively, computed
from the joint contrast-response function in A in
the following manner: For eigenvector 1, the response at each contrast
(contrast 1) was computed by averaging the measured responses
across all values of contrast 2 (average across each row). Similarly,
the contrast-response function for vector 2 (bottom
histogram) was computed by averaging the responses in
A across all values of contrast 1. Prediction of the
response at each combination of contrast 1 and contrast 2 was then made
by summing the value in each of the histograms at the corresponding
contrast. Predictions were made only for contrasts at which the actual
responses were measured in A. C, Measured
response in A plotted against the additive prediction in
B at corresponding contrasts. Each circle
represents data at one pair of contrasts. D, Response
from one simulation (A) plotted against that from
another (B) (see Materials and Methods).
E, Correlation coefficients between simulated responses
plotted against the correlation coefficient between the predicted and
measured contrast-response functions for 13 complex cells.
Vertical bars, 95% confidence intervals of correlation
coefficients between simulated responses, with the mean indicated by
the point (the mean is not in the middle of the bar,
because the distribution of the correlation coefficient is
skewed).
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The approximate circular symmetry of the joint contrast-response
function also suggests additive interaction between the two significant
eigenvectors. To examine this idea quantitatively, we predicted the
response to each combination of the two vectors by summing the response
to each vector at the corresponding contrast. As shown in Figure
6B, this prediction reproduced well the overall profile of the measured contrast-response function. Figure
6C shows the predicted responses (Fig. 6B)
plotted against the measured responses (Fig. 6A) at
corresponding contrasts; the correlation coefficient between them was
found to be 0.84. To determine whether the difference between the
predicted and the measured responses was attributable to systematic
errors of the additive model or to the noise in the measured responses,
we estimated the upper limit of the correlation coefficient set by
noise in the responses. A contrast-response function measured in a
single experiment was simulated with a Monte Carlo method, taking into
consideration the variability of the measurement (see Materials and
Methods); the contrast-response functions simulated in different
trials were compared. As shown in Figure 6D, the
correlation between the responses simulated in different trials
(correlation coefficient, 0.87) was comparable with that between the
predicted and measured responses, indicating that the additive model is
consistent with the experimental results within the limit set by noise.
Figure 6E summarizes the correlation coefficients
between the measured and the predicted contrast-response functions for
the 13 complex cells analyzed. In 12 of the cells, the correlation
coefficient was not significantly different from that between simulated
responses (p > 0.05), indicating that the model
based on additive interaction provides an adequate description of the
cortical responses to combinations of relevant visual features.
Divisive effect of the null features
Certain visual stimuli that do not evoke spiking responses on
their own may nevertheless modulate cortical responses to other stimuli. Such nonlinear effects are well known for stimuli at nonpreferred orientations (Bonds, 1989) or nonclassical receptive fields (Allman et al., 1985; Walker et al., 2000) of cortical neurons.
Here we tested whether the null features, which evoked little response
when presented in isolation (Fig. 5), can modulate the cortical
responses evoked by relevant features. The interaction between the
relevant and null features was revealed by comparing the responses of
each complex cell to relevant features alone (Fig.
6A) and to the random-bar stimuli (Fig.
1A) that contain both the relevant and the null
features. Figure 7A shows the
joint contrast-response function of a complex cell for the two
relevant features, either in the absence (left) or presence
(right) of null features. Although the two
contrast-response functions exhibited similar shapes, the amplitude of
the response to the random-bar stimuli was much lower (Fig.
7B), indicating a suppressive effect of the null features.
Similar suppressive effects were observed for all of the cells
examined.
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Figure 7.
Suppressive effects of null features.
A, Contrast-response functions of a complex cell for
the two relevant features measured in the absence (left)
or presence (right) of null features, with a luminance
scale indicated in the middle (in spikes per second).
The range of contrast represented in the boxes is 0.33
to 0.33. Contrasts of both features varied at a step size of 0.033. The
black line in the right plot delineates
the range of contrasts shown in the left plot. Higher
contrasts for the relevant features were possible in the presence of
null features, because the superposition of certain null features
can reduce the extreme luminance values in the short movie to levels
within the monitor limit. B, Responses in the
left plot versus the responses in the right
plot in A at corresponding contrasts (in spikes
per second). Each circle represents data at one pair of
contrasts. The slope of the linear regression (dashed
line) is 0.24.
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The simplest models for this type of suppression are subtractive and
divisive, and we evaluated both models in describing the effects of the
null features. First, we fitted the contrast-response function for
each relevant feature, either in the presence or in the absence of null
features, with power functions (Fig. 5B). The average
scaling factor of the fit () was found to be 0.01 ± 0.002 (n = 52) in the presence of the null features and
0.03 ± 0.005 in the absence of them. The average exponents ()
were 2.65 ± 0.13 (n = 52) and 2.95 ± 0.13 in the presence and absence of the null features, respectively.
Although the null features reduced the scaling by a factor of ~3
(p < 0.0005; paired t test), they
did not change the exponent systematically (p > 0.10). This is consistent with the observation that the null features
changed the amplitude but not the shape of the contrast-response
functions (Fig. 7A), suggesting a divisive effect. To
compare directly the divisive and the subtractive models, we used both
models to predict the joint contrast-response function measured in the
presence of null features (Fig. 7A, right plot)
from the function in the absence of null features (Fig. 7A,
left plot). Each model contained a single free parameter (a
scaling factor for the divisive model and a subtractive constant for
the subtractive model) to ensure the fairness of the comparison. We
found that for all of the cells analyzed (n = 13), the
divisive model performed significantly better than the subtractive
model (p < 0.02), as measured by the correlation coefficient between the predicted and measured responses (Fig. 8). Finally, we also fitted the
predicted response based on the subtractive model with power functions.
We found that the mean exponent of the fit was 5.60 ± 1.07 (n = 52), significantly larger than that for the
measured responses (p < 0.005; paired t test). Together, these results support a divisive rather
than a subtractive model for the suppressive effect of the null
features.
View larger version (16K):
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Figure 8.
Comparison between the subtractive and divisive
models for null features. Correlation coefficients between the measured
responses and the prediction based on the divisive model were plotted
against the correlation coefficients between the measured responses and
the prediction based on the subtractive model. Each
point represents the result of one cell. Error bar, 95%
confidence interval, estimated using nonparametric bootstrap (see
Materials and Methods for the Monte Carlo method used in the
analysis).
|
|
| |
DISCUSSION |
In the present study, we have found that for each complex cell,
visual inputs can be decomposed into two types of visual features, each
having a distinct effect on the response of the cell. The two relevant
features found for most complex cells resemble the receptive fields of
simple cells, with a phase difference of ~90° in their spatial
profiles; their contrast-response functions exhibited contrast
polarity invariance and expansive nonlinearity reminiscent of a
squaring function. Thus, the additive interaction between these
relevant features corresponds closely to the energy model for complex
cells (Movshon et al., 1978; Pollen and Ronner, 1981; Adelson and
Bergen, 1985; Heeger, 1991). Although the energy model is well known,
it has not been tested quantitatively with complex spatiotemporal
stimuli in previous studies. The main difficulty in testing this model
with complex stimuli comes from the fact that the parameters describing
the underlying subunits of the energy model (simple-cell receptive
fields) could not be determined easily for each cell. In the present
study, relevant visual features were identified with spike-triggered
correlation analysis, which allowed us to measure the contribution of
each feature to the cortical response directly and to demonstrate the
additive interaction between them (Fig. 6). This result is also
consistent with the finding that neural networks trained to predict the
responses of complex cells to random-bar stimuli contained additive
subunits resembling simple cells (Lau et al., 2002).
Divisive interactions have also been used to model the responses of
both simple and complex cells (Heeger, 1992; Carandini et al., 1997);
they can account for the suppressive effects of visual stimuli at
nonpreferred orientations or nonclassical receptive fields of the
cortical neurons. Such divisive suppression may reduce the redundancy
in information carried by neighboring neurons and enhance the
efficiency of coding for natural scenes (Schwartz and Simoncelli,
2001). Here, identification of a small number of relevant features for
each cell allows us to specify the additive components in the visual
inputs and to predict their contributions to the neuronal response. The
number of null features that contribute to the suppression of cortical
responses may be considerably larger. A similar spike-triggered
analysis technique may be used to identify the null features that
contribute maximally to the divisive suppression of the responses
(Schwartz et al., 2001).
For sensory neurons with nonlinear stimulus-response relationships, it
is often difficult to know a priori what visual stimuli are
relevant for probing the response properties (Touryan and Dan, 2001).
In the present study, we first isolated relevant features from null
features for each cell using spike-triggered correlation analysis of
the responses to a large ensemble of random spatiotemporal stimuli.
This allowed us to construct new visual stimuli for each cell to
measure the contribution of each type of features to the cortical
response efficiently. Although this method has been used here to
analyze the responses to random-bar stimuli, it is also applicable to
studying cortical responses to more complex stimuli that vary in both
dimensions of space (although with an increased number of parameters
this analysis will require more data). Such a two-step approach may
also prove to be useful for understanding the response properties of
nonlinear neurons in other cortical areas and other sensory modalities.
| |
FOOTNOTES |
Received Aug. 7, 2002; revised Sept. 24, 2002; accepted Sept. 27, 2002.
This work was supported by National Eye Institute Grant R01 EY12561-01
and Office of Naval Research Grant N00014-00-1-0053. We thank William
Bialek, Timothy Kubow, and Gidon Felsen for helpful discussions.
Correspondence should be addressed to Dr. Yang Dan, Department of
Molecular and Cell Biology, University of California, Berkeley, CA
94720. E-mail: ydan{at}uclink4.berkeley.edu.
B. Lau's present address: Center for Neural Science, New York
University, New York, NY 10003.
| |
REFERENCES |
-
Adelson EH,
Bergen JR
(1985)
Spatiotemporal energy models for the perception of motion.
J Opt Soc Am A
2:284-299[ISI][Medline].
-
Albrecht DG,
Geisler WS
(1991)
Motion selectivity and the contrast-response function of simple cells in the visual cortex.
Vis Neurosci
7:531-546[ISI][Medline].
-
Allman J,
Miezin F,
McGuinness E
(1985)
Stimulus specific responses from beyond the classical receptive field: neurophysiological mechanisms for local-global comparisons in visual neurons.
Annu Rev Neurosci
8:407-430[ISI][Medline].
-
Anzai A,
Ohzawa I,
Freeman RD
(1999)
Neural mechanisms for processing binocular information. I. Simple cells.
J Neurophysiol
82:891-908[Abstract/Free Full Text].
-
Bell AJ,
Sejnowski TJ
(1997)
The "independent components" of natural scenes are edge filters.
Vision Res
37:3327-3338[ISI][Medline].
-
Bonds AB
(1989)
Role of inhibition in the specification of orientation selectivity of cells in the cat striate cortex.
Vis Neurosci
2:41-55[ISI][Medline].
-
Brenner N,
Bialek W,
de Ruyter van Steveninck R
(2000)
Adaptive rescaling maximizes information transmission.
Neuron
26:695-702[ISI][Medline].
-
Brodie SE,
Knight BW,
Ratliff F
(1978)
The response of the Limulus retina to moving stimuli: a prediction by Fourier synthesis.
J Gen Physiol
72:129-166[Abstract/Free Full Text].
-
Carandini M,
Heeger DJ,
Movshon JA
(1997)
Linearity and normalization in simple cells of the macaque primary visual cortex.
J Neurosci
17:8621-8644[Abstract/Free Full Text].
-
Dan Y,
Atick JJ,
Reid RC
(1996)
Efficient coding of natural scenes in the lateral geniculate nucleus: experimental test of a computational theory.
J Neurosci
16:3351-3362[Abstract/Free Full Text].
-
de Boer E,
Kuyper P
(1968)
Triggered correlation.
IEEE Trans Biomed Eng
15:169-179[Medline].
-
de Ruyter van Steveninck R,
Bialek W
(1988)
Real-time performance of a movement-sensitive neuron in the blowfly visual system: coding and information transfer in short spike sequences.
Proc R Soc Lond B Biol Sci
234:379-414.
-
Efron B,
Tibshirani RJ
(1993)
In: An introduction to the bootstrap. New York: Chapman and Hall.
-
Heeger DJ
(1991)
Nonlinear model of neural responses in cat visual cortex.
In: Computational models of visual processing (Landy MS,
Movshon JA,
eds), pp 119-133. Cambridge, MA: MIT.
-
Heeger DJ
(1992)
Normalization of cell responses in cat striate cortex.
Vis Neurosci
9:181-197[ISI][Medline].
-
Hubel DH,
Wiesel TN
(1962)
Receptive fields, binocular interaction and functional architecture in the cat's visual cortex.
J Physiol (Lond)
160:106-154[Free Full Text].
-
Jones JP,
Palmer LA
(1987)
The two-dimensional spatial structure of simple receptive fields in cat striate cortex.
J Neurophysiol
58:1187-1211[Abstract/Free Full Text].
-
Lau B,
Stanley GB,
Dan Y
(2002)
Computational subunits of visual cortical neurons revealed by artificial neural networks.
Proc Natl Acad Sci USA
99:8974-8979[Abstract/Free Full Text].
-
Marmeralis P,
Marmeralis V
(1978)
In: Analysis of physiological systems: the white noise approach. New York: Plenum.
-
Movshon JA,
Thompson ID,
Tolhurst DJ
(1978)
Receptive field organization of complex cells in the cat's striate cortex.
J Physiol (Lond)
283:79-99[Abstract/Free Full Text].
-
Olshausen BA,
Field DJ
(1996)
Emergence of simple-cell receptive field properties by learning a sparse code for natural images.
Nature
381:607-609[Medline].
-
Pollen DA,
Ronner SF
(1981)
Phase relationships between adjacent simple cells in the visual cortex.
Science
212:1409-1411[Abstract/Free Full Text].
-
Reid RC,
Victor JD,
Shapley RM
(1997)
The use of m-sequences in the analysis of visual neurons: linear receptive field properties.
Vis Neurosci
14:1015-1027[ISI][Medline].
-
Schwartz O,
Simoncelli EP
(2001)
Natural signal statistics and sensory gain control.
Nat Neurosci
4:819-825[ISI][Medline].
-
Schwartz O,
Chichilnisky EJ,
Simoncelli EP
(2001)
Characterizing neural gain control using spike-triggered covariance.
In: Advances in neural information processing systems (Dietterich TG,
Becker S,
Ghahramani Z,
eds). Cambridge, MA: MIT, in press.
-
Skottun BC,
De Valois RL,
Grosof DH,
Movshon JA,
Albrecht DG,
Bonds AB
(1991)
Classifying simple and complex cells on the basis of response modulation.
Vision Res
31:1079-1086[ISI][Medline].
-
Sutter EE
(1987)
A practical non-stochastic approach to nonlinear time-domain analysis.
In: Advanced methods of physiological systems modeling (Marmarelis VZ,
ed), pp 303-315. Los Angeles: University of Southern California.
-
Touryan J,
Dan Y
(2001)
Analysis of sensory coding with complex stimuli.
Curr Opin Neurobiol
11:443-448[Medline].
-
van Hateren JH,
Ruderman DL
(1998)
Independent component analysis of natural image sequences yields spatio-temporal filters similar to simple cells in primary visual cortex.
Proc R Soc Lond B Biol Sci
265:2315-2320[Medline].
-
Walker GA,
Ohzawa I,
Freeman RD
(2000)
Suppression outside the classical cortical receptive field.
Vis Neurosci
17:369-379[ISI][Medline].
-
Wiener N
(1958)
In: Nonlinear problems in random theory. New York: Wiley.
-
Yamada WM,
Lewis ER
(1999)
Predicting the temporal responses of non-phase-locking bullfrog auditory units to complex acoustic waveforms.
Hear Res
130:155-170[ISI][Medline].
Copyright © 2002 Society for Neuroscience 0270-6474/02/222410811-08$05.00/0
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ABSTRACT
INTRODUCTION
