1Laboratory of Biophysics, The Rockefeller University; and 2Department of Neurology and Neuroscience, Weill Medical College of Cornell University, New York, New York 10021
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ABSTRACT |
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Reich, Daniel S.,
Ferenc Mechler, and
Jonathan D. Victor.
Formal and Attribute-Specific Information in Primary Visual
Cortex.
J. Neurophysiol. 85: 305-318, 2001.
We estimate the rates at which neurons in the primary visual
cortex (V1) of anesthetized macaque monkeys transmit stimulus-related information in response to three types of visual stimulus. The stimulirandomly modulated checkerboard patterns, stationary
sinusoidal gratings, and drifting sinusoidal gratings
have very
different spatiotemporal structures. We obtain the overall rate of
information transmission, which we call formal information,
by a direct method. We find the highest information rates in
the responses of simple cells to drifting gratings (median: 10.3 bits/s, 0.92 bits/spike); responses to randomly modulated stimuli and
stationary gratings transmit information at significantly lower rates.
In general, simple cells transmit information at higher rates, and over
a larger range, than do complex cells. Thus in the responses of V1
neurons, stimuli that are rapidly modulated do not necessarily evoke
higher information rates, as might be the case with motion-sensitive neurons in area MT. By an extension of the direct method, we parse the
formal information into attribute-specific components, which provide estimates of the information transmitted about contrast and
spatiotemporal pattern. We find that contrast-specific information rates vary across neurons
about 0.3 to 2.1 bits/s or 0.05 to 0.22 bits/spike
but depend little on stimulus type. Spatiotemporal pattern-specific information rates, however, depend strongly on the
type of stimulus and neuron (simple or complex). The remaining information rate, typically between 10 and 32% of the formal
information rate for each neuron, cannot be unambiguously assigned to
either contrast or spatiotemporal pattern. This indicates that some
information concerning these two stimulus attributes is confounded in
the responses of single neurons in V1. A model that considers a simple cell to consist of a linear spatiotemporal filter followed by a static
rectifier predicts higher information rates than are found in real
neurons and completely fails to replicate the performance of real cells
in generating the confounded information.
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INTRODUCTION |
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Recent studies of the responses
of visual neurons to stimuli with rich temporal structure, such as
flickering checkerboard patterns and drifting gratings that abruptly
change direction, have pointed to overall information transmission
rates of between 5 and 100 bits/s (Berry and Meister
1998; Bura
as et al. 1998
; Reich et
al. 2000a
; Reinagel and Reid 2000
; Ruyter
van Steveninck et al. 1997
). The sensory systems analyzed in
these studies range from blowfly lobula plate to primate cortex. The
information calculations are based on the direct method
(Ruyter van Steveninck et al. 1997
; Strong et al.
1998
), which estimates the overall rate of information transmission in a set of responses to a single stimulus.
Earlier studies on neurons in primary visual cortex, based on slowly
fluctuating stimuli, report information rates an order of magnitude
lower (Gershon et al. 1998; Heller et al.
1995
; Mechler et al. 1998b
;
Richmond and Optican 1990
; Tolhurst 1989
;
Victor and Purpura 1996
). These studies use a variety of
methods other than the direct method to calculate the information rates
in responses to sets of stimuli that vary along some particular
parameter, such as contrast or spatial pattern. All of these methods
calculate information as a measure of the degree to which responses can be clustered into the appropriate stimulus classes.
Comparing these and similar results, Buraas and Albright
(1999)
argue that neurons, especially cortical neurons, more
effectively convey information about stimuli with rich temporal
structure than about stimuli with simpler structure. This argument is
incomplete, however, because the two sets of studies use qualitatively
different approaches to measuring transmitted information, both in
terms of the richness of the stimuli and in terms of the analysis
method. It is therefore impossible to draw conclusions about the types of stimulus that evoke the highest information rates from such a
comparison. Here we link the results of these two sorts of studies by
recording the responses of V1 neurons to a battery of stimuli of
different spatiotemporal structure and by analyzing the responses in a
uniform fashion (a variant of the direct method). Our major finding is
that the overall rate of information transmission
which we dub formal
information
does vary with stimulus type but that responses to rapidly
modulated stimuli do not necessarily convey the most information,
particularly in the case of simple cells.
Next we draw a distinction between formal and attribute-specific information rates. Formal information concerns all aspects of the response that depend on the stimulus. Attribute-specific information concerns only aspects of the response that allow the discrimination between stimuli that differ in some particular attribute, such as contrast, in the face of variation in other attributes, such as spatiotemporal pattern. The attribute-specific information is a measure of the degree to which responses to different stimuli cluster according to a particular stimulus attribute and is thus more comparable to the information measured in the second type of study mentioned above. By presenting each type of stimulus at multiple contrasts and appropriately modifying the direct method, we parse the formal information rate into attribute-specific components relating to contrast and spatiotemporal pattern. Here, spatiotemporal pattern refers to a broad category of stimulus attributes that includes temporal fluctuations as well as variations in spatial phase.
Overall, we find that information about contrast is transmitted at a
significantly slower rate than information about spatiotemporal pattern, although not for every type of neuron and stimulus. The rate
of contrast-specific information transmission depends little on
stimulus type. Contrast-specific information rates estimated by the
direct method are very similar to contrast-specific information rates
estimated by a method based on computing the distances between pairs of
spike trains (Victor and Purpura 1996).
We find that contrast- and spatiotemporal pattern-specific information
rates together account for less than the full formal information
ratetypically 68-90%. This indicates that a significant portion of
the information in V1 responses relates to a confounded representation of contrast and spatiotemporal pattern by which the
spatiotemporal pattern of the stimulus is encoded in a
contrast-dependent fashion and the contrast of the stimulus in a
spatiotemporal pattern-dependent fashion. An observer who is only aware
of the portion of a single neuron's response that contains the
confounded information cannot draw conclusions about contrast or
spatiotemporal pattern in isolation. The possibility still
exists
though it is not addressed in this paper
that the confounded
information may be parsed into individual components by considering the
simultaneous responses of additional neurons.
We ask whether a basic model of a V1 simple cell can account for our
results. This model consists of a linear spatiotemporal filter, which
we derive from the responses of a real neuron, followed by a static
rectifier and a Poisson spike generator. The responses of such a model
to the same stimuli presented to real neurons transmit formal
information at rates comparable to, but higher than, those of real
responses. However, unlike in the responses of real neurons, all of the
formal information in model responses can be parsed into
attribute-specific components: the model does not confound the encoding
of contrast and spatiotemporal pattern. We have shown that this
discrepancy is not due to differences in the underlying dynamics of
spike generation, which do not strongly determine information rates
(Reich et al. 2000a). Instead, the discrepancy occurs
because this basic model lacks certain features of real cortical
neurons, such as contrast gain control (Ohzawa et al.
1982
), contrast normalization (Albrecht and Geisler
1991
; Heeger 1992
), and pattern gain control
(Carandini et al. 1997a
), by which variations in
one stimulus attribute can affect the encoding of another.
Portions of this work have appeared in abstract form (Reich et
al. 2000b).
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METHODS |
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We present data from recordings of individual neurons in the
primary visual cortices of sufentanil-anesthetized macaque monkeys. Our
detailed experimental procedures have been described elsewhere (Reich et al. 2000a; Victor and Purpura
1998
). We use three types of stimulus: flickering checkerboards
modulated by m-sequences, drifting sinusoidal gratings, and transiently
presented, stationary sinusoidal gratings. All stimuli are presented on
a Tektronix 608 monitor with a mean luminance of approximately 150 cd/m2 and a frame rate of 270.329 Hz.
M-sequence stimuli
The principles and methodology of the m-sequence checkerboard
stimuli have been extensively described (Reich et al.
2000a; Reid et al. 1997
; Sutter
1992
). In the experiments reported here, we use two such
stimuli: a 12th-order m-sequence (4,095 stimulus frames) modulating 249 stimulus checks and a 9th-order m-sequence (511 frames) modulating 25 checks. In both cases, each frame lasts for 14.8 ms (four monitor
refreshes), so that the total 12th-order stimulus lasts 60.6 s and
the 9th-order sequence lasts 7.6 s. Individual checks typically
span 16 × 16 arc-min of visual angle and are arranged in a
square. The size and orientation of the array are sometimes adjusted
based on the neuron's spatial-frequency preferences, but only in cases
where the adjustment is expected to produce a dramatically larger response.
The 249-check (long) stimulus is surrounded by a black circular aperture, and the 25-check (short) stimulus is surrounded by a uniform field at the mean luminance. In both stimuli, every check is modulated by the same m-sequence, but the starting point in the sequence varies from check to check. The minimum offset between starting points is 237 ms (64 samples of the m-sequence). The use of m-sequences in this way ensures that there are essentially no pair-wise correlations in time within individual checks, or in space across checks, that are relevant to the neuron's response. The long stimulus is presented at a single contrast (1), and the short stimulus is presented at each of five geometrically spaced contrasts (0.0625, 0.125, 0.25, 0.5, and 1). Both standard and contrast-inverted (reversed dark and light checks) sequences, each repeated 12-16 times, are presented in the long stimulus. Standard and inverted sequences within a repeat are separated by a period of 23 s during which a uniform field at the mean luminance is presented; repeats are separated by 18 s. For the short stimulus, no inverted sequences are presented. Contrasts are presented in increasing order, separated by uniform-field presentations lasting 10 s, and the entire of set of contrasts is repeated 25-100 times with 10 s between repeats.
Drifting-grating stimuli
We use "optimal" sinusoidal gratings with spatial frequency,
temporal frequency, and orientation chosen to maximize either firing
rate (for complex cells) or response modulation at the driving
frequency (for simple cells) (Skottun et al. 1991). For simultaneously recorded neurons, we optimize the gratings for at least
one of the cells, usually the one with the most distinct extracellularly recorded waveform (since this is the neuron most easily
monitored during the experiment). The parameter choices based on the
response of this cell are likely to be similar to the parameters that
would have been chosen for the other simultaneously recorded neurons
(DeAngelis et al. 1999
), as we occasionally verified empirically. We present the gratings for 4 s at each of six
geometrically spaced contrasts (0, 0.0625, 0.125, 0.25, 0.5, and 1). We
repeat the entire set of contrasts five to eight times, with the order of contrast presentation randomized within blocks. For 4-Hz gratings, this yields 80-128 stimulus cycles at each contrast. Within each block, grating presentations at different contrasts are separated by
presentation of a uniform field of the same mean luminance for 8 s, and blocks are separated by presentation of the uniform field for
13 s.
Stationary-grating stimuli
We present stationary sinusoidal gratings at the same spatial
frequency and orientation as the drifting gratings. The spatial phase
of the stationary gratings is the one that maximizes the neuron's
firing rate in response to stationary gratings of unit contrast. For
each neuron, we present in increasing order either seven geometrically
spaced (0, 0.03125, 0.0625, 0.125, 0.25, 0.5, 1) or nine arithmetically
spaced (0, 0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875, 1) contrasts.
Gratings replace a uniform field of the same mean luminance for a
period of 237 ms, after which the uniform field reappears for a minimum
of 710 ms. The amount of time between grating presentations increases
as a function of the contrast of the preceding grating. For example,
the amount of off time following the 0.5 contrast presentation is
2.84 s and following the 0.875 contrast presentation is 4.26 s. This strategy is used to approximate a uniform state of contrast
adaptation (Sclar et al. 1989) prior to the presentation
of each different grating. The entire series of contrasts is usually
presented 100 times. We analyze only spikes that occur between 30 and
300 ms after stimulus onset.
Information rates
We use extensions of the direct method of calculating
information rates (Ruyter van Steveninck et al. 1997;
Strong et al. 1998
) to evaluate the responses to all
three types of stimulus. This method is based on a comparison of the
response variability across time to the response variability across
trials. The underlying principle behind the approach is that the
portion of variability that cannot be explained by intrinsic variations
in the response to a particular stimulus must represent
stimulus-related information.
Direct method
The straightforward application of the direct method, diagrammed
in Fig. 1A, evaluates what we
call the formal information rate. The spike train recorded during each
trial is divided into time bins, and the spike counts in each bin are
tabulated. These spike counts are considered letters in the neuron's
response alphabet. Several letters in a row constitute a word, and each
word has a probability, possibly stimulus-dependent, of being
"spoken" by the neuron. In this paper, we choose one-letter words
(single time bins) because our data sets are not typically large enough to obtain reliable multi-letter-word information estimates. Based on
others' results in different systems (Reinagel and Reid
2000; Strong et al. 1998
), as well as on
analysis of a limited number of V1 neurons from which large amounts of
data were collected, we estimate that information rates are likely to
differ by at most 25% in the two cases, but that the qualitative
results (formal vs. attribute-specific information and amount of
confounded information) are not likely to change greatly.
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We perform our calculations at a variety of bin (letter) sizes, ranging
from 0.9 to 59.2 ms. We choose the bin size that yields the highest
information rate. This choice is justified because the actual
information rate cannot decrease as the bin size decreases (Strong et al. 1998), even though our information-rate
estimates may not increase because of limitations set by the amount of
available data. It is important to emphasize that the quantities
estimated here are information rates for brief samples of the responses (single bins or letters), not the total information contained in an
extended response. In general, the conversion between information-rate estimates and information estimates over extended responses is subadditive, in part because the information encoded at different times
in the response may be redundant (DeWeese and Meister
1999
).
From the set of binned spike counts, we extract two quantities. The
first, called the total entropy
(HT), is a measure of the response
variability across timethat is, the uncertainty in spike count across
all bins. We calculate HT from the
distribution of spike counts in individual bins across all time and
trials (represented by the light gray rectangle in Fig. 1A).
We obtain a direct estimate of the probabilities
pn that n spikes are observed directly from the spike count statistics. We then apply Shannon's formula (Cover and Thomas 1991
) to obtain the entropy
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(1) |
An important caveat of the direct method is that it is only sensitive
to fluctuations within an analyzed response. If there is
little variation in the local spike-count distribution during the
course of a response, then the direct method yields a low information
rate. Such is the case for complex cells when the stimulus is a
drifting sinusoidal grating at fixed contrast. Since complex cells
respond to these stimuli primarily by elevating their discharge rates
(Skottun et al. 1991), the direct method only detects information if
the analyzed response includes both background and stimulus-driven
firing, because the appearance of the stimulus causes a change in
firing rate. In this chapter, the unit-contrast drifting-grating
responses, unlike the unit-contrast stationary-grating responses, do
not include any period of background firing. In the past, the direct
method has always been applied to the responses of neurons to rapidly
varying stimuli, which typically evoke a wide range of firing rates
that fluctuate over time (Bura
as et al. 1998
;
Reinagel and Reid 2000
; Ruyter van Steveninck et al.
1997
). However, there is no a priori reason to limit the
application of the direct method to rapidly varying stimuli, and we
will show that the method can yield useful results even when applied to
other sorts of responses.
Attribute-specific information
The formal information rate calculated by the direct method evaluates the overall rate of information transfer about all time-varying aspects of the stimulus. It does not evaluate the rates at which information about individual stimulus attributes is transmitted without the confounding influence of other attributes. We refer to these latter quantities as the attribute-specific information rates, and we now describe an extension of the direct method that allows us to estimate them. In our experiments, we concentrate on contrast and spatiotemporal pattern, but the idea can be applied to any situation in which two (or more) attributes are varied independently. As used here, spatiotemporal pattern is an omnibus term that refers to aspects of the stimulus that do not change as the contrast is varied. Our stimuli are completely defined by contrast and spatiotemporal pattern: indeed, each stimulus is defined by a spatiotemporal pattern and a contrast value by which it is multiplied.
As shown in Fig. 1B, attribute-specific information is
calculated by a procedure that elaborates on the one depicted in Fig. 1A. The total entropy
H,
is an
overall quantity that represents the uncertainty in spike count across
all bins in the entire data set
that is, across time, trials, and
contrasts
and is used in both formal and attribute-specific
information calculations. When restricted to a single contrast,
H
,
reduces
to HT, the total entropy from Fig.
1A. [We use the dot (
) notation to denote inclusion of
either all time or all contrasts in the information calculations.] To
obtain the overall noise entropy used in the calculation of formal
information rates, we average together all the estimates of the noise
entropy H
,
taken from spike counts measured on different trials at fixed times (
) and contrasts (
) (dark gray region). This is a direct
extension of the procedure used when only a single stimulus type is
presented, as in Fig. 1A, in which
H
,
is
equivalent to H
.
Attribute-specific information rates are obtained by averaging over one
or another stimulus attribute and over trials. Intuitively, by ignoring
the value of one stimulus attribute, we are considering stimulus
variations along the ignored attribute to be a source of "noise."
This potentially adds variability to the spike count and could reduce
the ability of stimulus-induced variations in spike count to transmit
information about the nonignored attribute. To obtain the
contrast-specific noise entropy
H,
, we consider spike counts recorded at contrast
, regardless of time bin
or trial number (medium gray), and then average those entropy estimates
across contrasts. This essentially represents the uncertainty in spike
count at each contrast, averaged across all contrasts. The
contrast-specific information is then the difference between the total
entropy and the averaged contrast-specific noise entropy: H
,
H
,
. To
obtain the spatiotemporal pattern-specific noise entropy
H
,
, we
average across time bins the entropies derived from spike counts at
time
, regardless of contrast (light gray). This corresponds to the
uncertainty in spike count at each time relative to the stimulus,
averaged across all times. The spatiotemporal pattern-specific
information is the difference between the total entropy and the
averaged spatiotemporal pattern-specific noise entropy:
H
,
H
,
. The sum of the two
pattern-specific information rates cannot exceed the formal information
except for measurement errors, and equality can only hold under
circumstances in which the two attributes are independently
represented. Proof of this statement and further background concerning
attribute-specific information can be found in the
APPENDIX.
Bias in the information estimates
Because we only have access to a limited amount of data, our
estimates of the total and noise entropies are both subject to a
downward bias. This is a generic property of information estimates from
limited data sets (Carlton 1969; Miller
1955
). Since the transmitted information is the difference
between these two entropies, the resulting information rate will be
either underestimated or overestimated depending on the relative
magnitude of the bias in the two entropy estimates. When the
probabilities of each word are directly estimated from the observed
probabilities, an asymptotic estimate of the bias is
(k
1)/2N ln (2), where k is
the number of distinct, observed words (here, spike counts per bin) and
N is the total number of observations (Panzeri and
Treves 1996
; Victor 2000). Because N
is large for the total entropy (number of bins times number of trials),
the correction is quite small (for m-sequence responses, about 0.01%).
On the other hand, in the calculation of the individual bin-specific
noise entropies, N can itself be small (as low as 12 for
m-sequences), making the correction much larger (sometimes on the order
of 10% or more).
In many cases, particularly with short bins, only one distinct spike
countzero
is observed. These bins contribute an entropy of 0 to the
averaged noise entropy, even with the asymptotic bias correction
(because k = 1). Because these bins are so common and because entropy is a logarithmic function of probability, as in Eq. 1, the noise entropy is potentially severely
underestimated. This can result in an overestimation of the transmitted
information. We address the problem of the zero-count bins by assuming
that the noise entropy varies slowly when the number of spikes is very low. This assumption allows us to group several consecutive bins together to generate a single estimate of the bin-specific noise entropy. Specifically, when we encounter a bin with no spikes in any
trial, we sequentially consider subsequent bins until we find one that
has at least one spike. For these m bins, we calculate the
noise entropy as described, applying the analytic bias correction with
N, the number of observations, equal to m times
the number of trials. We then assign this value of the noise entropy to
each of the bins that are grouped together in this way. The final value of the noise entropy is again the average of the individual
bin-specific entropies, where some of those entropies have been
calculated by grouping several bins together. This grouping occurs most
commonly in the calculation of the formal noise entropy for which the
number of observations is simply equal to the number of trials. The
effective number of trials in the calculation of the spatiotemporal
pattern-specific noise entropy is higher because time bins are grouped
together across contrasts. The effective number of trials in the
calculation of the contrast-specific noise entropy is vastly higher
because time bins and trials are grouped together at a single contrast so that we never encounter a bin with only one type of spike count.
We have verified that our assumptionthat the noise entropy varies
slowly when the number of spikes is very low
allows us to obtain
accurate information-rate estimates for synthetic spike trains that are
examples of modulated (inhomogeneous) Poisson processes with fewer than
half the number of trials that would otherwise be required (simulations
not shown). Indeed, for long stimuli (such as the
m-sequences), accurate information estimates can sometimes be
obtained with as few as four trials of the stimulus
we typically have
at least 16. For briefer stimuli (such as the gratings), more trials
are required. Because real neuronal responses are not examples of
modulated Poisson processes (Reich et al. 1998
), we
impose an additional criterion to eliminate data sets that do not
contain enough trials. In particular, we insist that the formal
information rate obtained from half the data (randomly chosen) be
within 10% of the corresponding information rate obtained from the
full data set. This requirement is quite strict and eliminates between
20-50% of the neurons recorded with each stimulus type. The neurons
that are retained tend to convey information at slightly higher rates.
Finally, to obtain an estimate for the scatter of individual
information estimates, we use the jackknife procedure (Efron 1998). Specifically, for each data set, we sequentially remove
of the trials and recalculate the information rates;
is chosen because we have access to only 16 trials for some
m-sequence data sets. From the resulting distribution of information
rates, we estimate the standard error of the information rate obtained
from the full data set. The jackknife estimate of the standard error is
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(2) |
Simple-cell model
To help explain our findings about formal and attribute-specific
information ratesin particular, the confounded information rate (see
RESULTS)
we model a V1 simple cell as a linear
spatiotemporal filter the output of which is subject to a static
nonlinear rectification and a Poisson spike generating mechanism
(Carandini et al. 1996
). Rather than assuming any
particular form for the linear filter, we use first-order kernels
derived from responses of real V1 neurons to unit-contrast m-sequence
stimuli (Reich et al. 2000a
; Reid et al.
1997
; Sutter 1992
); the calculation of this
kernel, as well as its normalization, is extensively discussed in the
references. The subset of 11 neurons modeled here includes five simple
and six complex cells. Complex cells, which often yield significant linear kernels (Reich et al. 2000a
), are modeled here in
exactly the same way as simple cells. However, because we only use the linear kernels in the model and because the model does not include any
full-wave rectification (Movshon et al. 1978a
), the
model neurons derived from complex cells respond like simple cells. Although we only model neurons with robust linear kernels or
receptive-field maps, we verified that these neurons have firing and
information rates not significantly different from the firing and
information rates of the entire population of simple cells, for unit
contrast responses (Kolmogorov-Smirnov test, P > 0.05).
To derive the parameters of the rectification (threshold and linear
gain), we first predict the linear response by convolving the
first-order kernel with the unit-contrast m-sequence stimulus. We then
find the constant offset and linear gain that, when applied to the
predicted linear response histogram, yield the best least-squares fit
to the histogram of the observed unit-contrast m-sequence response for
the neuron being modeled. We present the model neurons with the same
three stimuli that we present to real V1 neurons: m-sequences,
stationary sinusoidal gratings, and drifting sinusoidal gratings.
Kernels and stimuli are binned at 1.8-ms resolution (half of a single
display frame for the actual visual stimuli). Because the
spatiotemporal patterns in grating stimuliparticularly drifting
gratings
change faster than the spatiotemporal pattern in m-sequences
(even though such changes are not necessarily detected by the neuron),
we integrate the gratings over each m-sequence check and time bin
before presenting them to the model neuron.
The model yields a response histogram that serves as the modulation envelope of an inhomogeneous Poisson process, which is then used to determine the spike times in each trial. For the purpose of calculating information rates, the response histogram and the Poisson assumption are sufficient, because knowing the firing rate in each bin gives us the full spike-count probability distribution in that bin. Thus for model responses, we obtain an exact value for the information rate in one-letter words. In practice, these exact information rates are extremely close to the ones obtained by applying the method described in the preceding text for real data; indeed, this similarity is the primary justification for the applicability of our bias-correction methods.
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RESULTS |
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Formal information rates
Figure 2 shows the responses of a complex cell in monkey V1 to the three different stimuli used in this paper, each presented at unit contrast. Response histograms, obtained by averaging the number of spikes across all trials in consecutive 7.4 ms bins and then normalizing by the binwidth, are presented atop raster diagrams that show the spike times following stimulus onset in each trial. The top panel presents results from a flickering checkerboard stimulus, in which the time course of contrast modulation in each check is determined by a binary m-sequence (see METHODS). The stimulus lasts 60.6 s and is repeated 14 times; here, we only show spike times that occurred between 32 and 33 s after stimulus onset.
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A striking feature of these responses is that spike firing tends to be
clustered at particular times, which presumably follow transient
changes in the stimulus. However, since the stimulus changes every 14.8 ms (67.6 times during the 1 s display period), it is clear that not all
stimulus transitions are followed by a consistent change in firing
probabilitythat is, only some of the changes, such as the one that
causes firing shortly before 32.5 s, are effective in driving the
neuron. This event-like firing in response to stimuli of this sort has
been noted in other species and visual areas as well (Bair and
Koch 1996
; Berry et al. 1997
; Ruyter van
Steveninck et al. 1997
). As estimated by the direct method
(METHODS), the response to the m-sequence conveys 7.8 bits/s (0.75 bits/spike) of stimulus-related information, which is
within the range reported both for MT neurons (Bura
as et
al. 1998
) and for salamander and rabbit retinal ganglion cells
(Berry and Meister 1998
) in responses to similar stimuli.
Figure 2 also shows the responses of the same neuron to 100 presentations of a stationary sinusoidal grating of optimal orientation and spatial frequency. The grating appears at time 0 and disappears 237 ms later (times marked by solid vertical lines). The response histogram reveals three distinct firing levels. High firing rates begin abruptly 40-45 ms after the stimulus is presented and reappear at about 310 ms, after the grating is removed. In between, the firing rate decays to a lower level that is still higher than the baseline. The high firing rates that occur in the response transients resemble, in terms of peak rate and duration, the brief periods of high firing rate in the m-sequence response. However, the neuron spends a great deal more time firing spikes at the lower rate than at the higher rate. This results in a lower information rate of 4.3 bits/s (0.14 bits/spike), despite the fact that the mean firing rate is 72% higher than for the m-sequence response (29.8 vs. 17.3 spikes/s). Although both total and noise entropy are higher in the stationary-grating response, the noise entropy, which reflects the spike count variability across trials at particular times, is subject to a proportionately greater increase than the total entropy. Thus the variability in firing is larger for the stationary-grating response than for the m-sequence response, and the stimulus-induced modulation is relatively small. Note that information-rate calculations on stationary-grating responses only involve spikes that occur from 30 to 300 ms following stimulus onset (dashed lines), meaning that the off response of this neuron is effectively ignored.
The third stimulus is a sinusoidal grating, again presented at the
optimal orientation and spatial frequency, that drifts uniformly at 2.1 Hz. For this neuron, we recorded the responses to 40 cycles of the
drifting grating. The analysis treats individual cycles of the grating
as separate stimulus trials. As is the case for most complex cells, the
response to the drifting grating is only weakly modulated
(Skottun et al. 1991), and the most prominent feature is
an elevation of the mean firing rate (compare the average response
level in the bottom-right panel to the response level just
after stimulus onset in the bottom-left panel). There
are no periods of abruptly increased firing rate because, unlike the m-sequence and stationary-grating stimuli, the drifting-grating stimulus contains no temporal transients. The information rate in the
drifting-grating response is 4.4 bits/s (0.11 bits/spike). This is in
the top 15% of information rates measured in bits/s in our sample of
43 complex-cell responses to drifting gratings but near the median
information rate measured in bits/spike. The firing-rate elevation that
is such an important feature of the responses of complex cells to
drifting sinusoidal gratings does not contribute to the information
rate calculated by the direct method, which reflects only reproducible
modulations in the spike count probability during the course of the
response. Such slow modulation, superimposed on a relatively high
overall firing rate of 41.5 spikes/s, is evident in the response histogram.
Figure 3 and Table
1 summarize the results of similar
experiments performed on our entire population of V1 neurons, all at unit contrast. Panels A-F summarize across-neuron results,
and G shows within-neuron comparisons for the neurons that
convey significant information rates in response to at least two of the stimuli. The results reveal that differences between simple and complex
cells are most pronounced in the responses to drifting gratings, which
evoke the highest information rates in simple cells but the lowest
information rates in complex cells. For simple cells (Fig. 3,
A-C), the unexpected finding is that information rates are
higher in drifting-grating responses than in m-sequence responses (Fig.
3G, filled triangles are typically above the line of
equality). This is evidence against the hypothesis that high information rates in V1 neurons are more likely to be evoked by stimuli
that change rapidly in time than by stimuli that change slowly, as
might be the case in the motion-sensitive area MT (Buraas and Albright 1999
; Bura
as et al. 1998
).
Complex-cell responses to drifting gratings behave differently: the
filled triangles in Fig. 3G are typically below the line of
equality. This is because complex cells, unlike simple cells, transmit
very little information about the aspect of the drifting-grating
stimulus
its spatial phase
that varies during the course of the
experiment. We found (Table 1) that stimulus type has a significant
effect on firing and information rates for complex cells but only on
information rates in bits/spike for simple cells. However, we believe
that the lack of significance for simple cells is simply due to the smaller number of simple cells in the sample.
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Attribute-specific information rates
The direct method is usually applied only to neurons' responses
to rapidly modulated stimuli, such as the unit-contrast m-sequence responses in Figs. 2 and 3. Most stimulus sets used in neurophysiology experiments can be classified along two (or more) attributes: one (or
more) features, such as contrast and spatial phase, that are explicitly
varied from one stimulus presentation to the next; and the time course
of the stimulus itself. For responses to these sorts of stimulus,
straightforward implementation of the direct method measures only the
overall rate of information transmission, which we call the formal
information rate, and not information rates for individual stimulus
features, which we call the attribute-specific information rates.
Earlier methods, based on firing rates (Tolhurst 1989),
principal components (Richmond and Optican
1990
), stimulus reconstruction (Bialek et al.
1991
), and time structure of individual responses
(Panzeri and Schultz 2000
; Victor and Purpura
1996
), are expressly designed to measure attribute-specific information.
Here, we focus on contrast and spatiotemporal pattern. Formal information rates are parsed into components specific to contrast and spatiotemporal pattern. Responses to five contrasts (0.0625, 0.125, 0.25, 0.5, and 1) are analyzed. The m-sequence stimulus used in the contrast experiments is shorter, running for 7.6 s instead of 60.6 s and modulating 25 checks instead of 249.
Figure 4 shows the responses (raster diagrams and histograms) of a second complex cell, which has a maintained discharge (response to a uniform field at the mean luminance) of 2.4 spikes/s. For all three types of stimulus, the responses generally become more reproducible as contrast increases. Some features, such as the spikes that occur around 7,100 ms in the m-sequence response, or the transient firing-rate elevation at the beginning of the stationary-grating response, are particularly reproducible and give rise to peaks in the histograms.
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Figure 5A plots the neuron's
firing rate as a function of contrast for all three stimulus types.
Clearly, the curves are quite similar, to within the error bars (95%
confidence limits of the mean). As shown in Fig. 5, B and
C, the formal information rate is 3.2 bits/s (0.58 bits/spike) in the m-sequence response, 1.8 bits/s (0.28 bits/spike) in
the stationary-grating response, and 1.2 bits/s (0.06 bits/spike) in
the drifting-grating response. Responses to lower contrasts can
contribute extra information as well as extra variability so that
including them in the information calculations can both raise and lower
formal information rates. In this example, including those contrasts
decreases the information rate measured in bits/s (but not bits/spike)
in the m-sequence responses (), increases the
information rate measured in bits/spike (but not bits/s) in the
stationary-grating responses (
), and leaves virtually
unchanged the information rates in the drifting-grating responses
(
).
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To isolate the relative amount of information transmitted about different aspects of the stimulus, we modify the direct method by selectively changing our definition of noise entropy while leaving unchanged the definition of total entropy (see METHODS). For these stimuli, information is conveyed about contrast and spatiotemporal pattern. Spatiotemporal pattern refers to aspects of the stimuli that affect the response variation across time at a fixed contrast. The results are displayed in Fig. 5, B and C, together with error estimates derived from jackknife resampling (see METHODS).
The information rates due to contrast and spatiotemporal pattern alone do not sum to the full formal information rate, as they would if the two stimulus features were encoded independently (see APPENDIX). Instead, for this neuron, the sum of the two attribute-specific information rates fails to account for 19-46% of the formal information rate, depending on stimulus type. We call the information not accounted for by the attribute-specific information rates confounded. The presence of confounded information means that the dynamics of contrast- and spatiotemporal pattern-encoding are interdependent. The confounded information cannot be used to determine either contrast or spatiotemporal pattern based on the response of this neuron alone.
The results of the contrast and spatiotemporal pattern experiment over the population of neurons are shown in Fig. 6 and Table 2. Here, we have combined data from simple and complex cells for m-sequence and stationary-grating responses because we could not find significant differences among the distributions, probably due in part to the limited number of simple cells in our sample. Overall, as with unit-contrast responses, drifting gratings evoke the highest formal information rates in simple cells but the lowest formal information rates in complex cells. The same is true for spatiotemporal pattern-specific information rates.
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The strikingly high formal and pattern-specific information rates in
simple cell responses to drifting gratings are explained by the fact
that simple cells are exquisitely sensitive to spatial phase
(Hubel and Wiesel 1962; Movshon et al.
1978b; Victor and Purpura 1998
), which is the
only aspect of the stimulus that varies at fixed contrast. The
spatial-phase variation causes the firing rate to be deeply modulated,
which results in high spatiotemporal pattern-specific, and hence
formal, information rates. This is emphatically not the case for
complex cells, as discussed above. On the other hand, simple and
complex cells transmit contrast-specific information at the same rates
in response to drifting gratings. Indeed, the contrast-specific
information-rate distributions are relatively independent of stimulus
type, measured either in bits/s or bits/spike (P > 0.05, Kruskal-Wallis ANOVA).
For the neuron presented in Figs. 4 and 5, the attribute-specific
information rates do not account for all the formal information in the
neuron's response. This is also the case for the population results
(Fig. 6, right). The median confounded informationthe portion of information that cannot be resolved into contrast- or
spatiotemporal pattern-specific components
represents a substantial fraction of the formal information. Indeed, across all neurons and
stimulus types, the confounded information rate typically accounts for
10-32% (interquartile range) of the formal information rate.
Information transmission in a model V1 simple cell
The model described here considers the responses of a V1 simple
cell to derive from a linear spatiotemporal filter followed by a static
rectifier and a Poisson spike generator (see METHODS). The
shape and size of the filter, as well as the parameters of the
rectifier, are drawn from the responses of actual neurons to the long
m-sequence stimulus at unit contrast. Similar models have been used in
the past to describe the responses of visual neurons to various kinds
of stimuli (Carandini et al. 1996), even though it
is well known that V1 neurons, even simple cells, display many features
that are not captured in the model, including nonlinearities of
spatiotemporal summation (Movshon et al. 1978a
) and
contrast response (Albrecht and Hamilton 1982
).
Figure 7 shows the responses of a real simple cell (top) and its corresponding model neuron (bottom) to the three types of stimulus, each presented at unit contrast. Despite the rudimentary nature of the model, it successfully captures many of the features of the real data. In particular, the model replicates the location of the peaks in the m-sequence response and the existence of a transient period of elevated firing rate at the beginning of the stationary-grating response. Notable differences, especially in the grating responses, do exist. For example, the transient portion of the stationary-grating response is briefer in the real data than in the model, and the drifting-grating response is narrower in the model than in the real data. Moreover the distinct, brief period of very low firing rate that immediately follows the real neuron's transient response to the stationary grating is absent from the model's response. These differences are not surprising given that the model's parameters are fit to the linear part of the m-sequence response and not to the grating responses.
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The similarities and differences between real and model responses are also reflected in the rates of information transmission. The m-sequence response of the real neuron transmits information at a rate of 15.5 bits/s (1.6 bits/spike), compared with 19.2 bits/s (2.5 bits/spike) for the model. In response to the stationary and drifting-grating stimuli, the real neuron transmits 24.2 bits/s (0.43 bits/spike) and 49.4 bits/s (0.92 bits/spike) of information, respectively, whereas the model neuron transmits 20.8 bits/s (0.42 bits/spike) and 90.8 bits/s (1.2 bits/spike). It should be noted that the information in this real neuron's response to the drifting grating is at the top of the range of such information rates, even among simple cells (Fig. 3).
Figure 8 and Table 3 show that these results generalize to the population of 11 neurons that we modeled; the results should be compared with Fig. 6 (noting the sometimes-different vertical scales) and Table 2. Typically, the information rates of model responses are higher than the information rates of real responses. As is the case with the simple cell modeled in Fig. 7, and with real simple cells, the model neurons convey the most information about drifting gratings. The low spatiotemporal pattern-specific information rates in the modeled stationary-grating responses correspond to the low spatiotemporal pattern-specific information rates in the responses of real neurons to these stimuli. However, unlike in real neurons, the contrast-specific information rate conveyed by the model neurons does depend significantly on the type of stimulus.
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The most striking difference between real and model responses is in the confounded information rate, which is nearly zero in model responses. In model responses, contrast and spatiotemporal pattern can be independently determined from the response time course and depth of modulation. The prominence of the confounded information in real responses suggests that an interaction between the coding of contrast and spatiotemporal pattern constitutes one of the primary differences between real and model neurons.
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DISCUSSION |
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We used three types of stimulus to evaluate the ways in which the spatiotemporal features of a stimulus affect the rates at which V1 neurons transmit information. The first stimulus is based on pseudo-random m-sequences and appears as a rapidly modulated checkerboard pattern, in which the spatial pattern changes every 14.8 ms. The second stimulus is a sinusoidal grating at the cell's optimal orientation, spatial frequency, and spatial phase; it appears abruptly and is removed 237 ms later. The third stimulus is a sinusoidal grating that drifts at the cell's optimal temporal frequency and has the same orientation and spatial frequency as the stationary grating. We find that V1 simple cells typically transmit information at the highest rates in response to high-contrast, drifting-grating stimuli. The same stimuli evoke the lowest information rates in complex cell responses because those responses are not modulated in time.
Do cortical neurons transmit information at high or low rates?
Our results are surprising in light of the fact that comparisons
of information rates in a variety of neural systems suggest that
stimuli that change rapidly in time drive neurons to encode information
at rates often more than an order of magnitude higher than the
corresponding rates for slowly changing stimuli (Buraas and Albright 1999
). Based on the formal and attribute-specific information rates calculated for V1-neuron responses to different types
of stimulus, it is likely that the major cause of this discrepancy is
not simply the rate of variation of the stimulus, as some have argued
(Bura
as and Albright 1999
). Moreover, the cause does not lie in
the number of transient changes in a stimulus, because such transients
are absent from drifting gratings (which evoke the highest information
rates in simple cells) but present in both m-sequences (which evoke the
highest information rates in complex cells) and stationary gratings.
Instead, the results suggest that the magnitude of a measured information rate has a complicated dependence on the type of attribute-specific information that is being estimated and the sensitivity of the neuron under study to that stimulus attribute. For example, a stimulus (such as a drifting grating) that consists of rapid changes in spatial phase evokes high spatiotemporal pattern-specific information rates in simple cells and low spatiotemporal pattern-specific information rates in complex cells, but a stimulus (such as an m-sequence checkerboard) that consists of rapid changes in luminance evokes indistinguishable information rates in simple and complex cells.
Information rates and channel capacity
The information rates for complex cells can be compared with
estimates of the channel capacities of complex cells in the
supragranular layers of alert monkeys, which range from 6.7 to 8.5 bits/s (Wiener and Richmond 1999). The channel capacity
is a measure of the maximum information rate that a communications
channel can transmit (Cover and Thomas 1991
). In
response to the stimuli used here, which differ from the stimuli used
by Wiener and Richmond, complex cells transmit information at
approximately half of this estimated channel capacity. Since our
stimuli were not designed to evoke information rates that approach the
channel capacity, we consider this result to be rather impressive.
Relevance of the confounded information for visual processing
Our use of the direct method to calculate both formal and
attribute-specific information rates reveals a hitherto-overlooked aspect of the information conveyed by V1 neurons. We found that a
substantial fraction of the information (typically 10-32%) cannot be
attributed to either contrast or spatiotemporal pattern alone. This
portion of the information, which we call confounded, arises from an interdependence of contrast and spatiotemporal pattern in
generating neuronal responses. Confounded information is not present in
the model responses, where there is no such interdependence (see
APPENDIX). In other words, the amount of confounded
information quantifies the effects of changes in the spatiotemporal
profile of the stimulus on the contrast response and sensitivity
functions of V1 neurons (Albrecht 1995; Gawne et
al. 1996
; Maffei and Fiorentini 1973
;
Tolhurst and Movshon 1975
). Such changes may be mediated by variation in the "adaptive state" of a neuron under different spatiotemporal stimulus conditions, particularly between rapidly varying stimuli, like the m-sequence checkerboard, and gratings (Gaska et al. 1994
).
Another way in which contrast and spatiotemporal pattern can
potentially interact is through a refractory period that is intrinsic to a neuron's spike generating mechanism but that influences the responses to both stimuli. However, we do not believe that refractory periods contribute significantly to the confounded information reported
here. This is because as much or more confounded information is present
in data sets that have been resampled to effectively eliminate the
refractory period while preserving the overall rate modulation and
distribution of spike counts per trial (Reich et al.
2000a).
Whatever its basis, the finding that a substantial portion of the total information is confounded means that downstream neurons cannot use all of the information in the responses of their inputs to draw conclusions about either one of those stimulus attributes in isolation. But perhaps the task of the visual system is not simply to decompose stimuli into components relating to contrast and spatiotemporal pattern. For example, if we had determined attribute-specific information along the visual system's preferred axes, we might not have found any confounded information.
Alternatively, or perhaps in addition, it is possible that the messages
that correspond to the confounded information would separate into
contrast- and spatiotemporal pattern-specific components if the
concurrent responses of other neurons to the same stimulus were
considered (deCharms 1998). It is known, for example,
that spikes that are synchronous across two cat LGN neurons can convey additional information beyond what can be obtained from each neuron's individual response (Dan et al. 1998
), and similar
results have been obtained in various cortical systems (Maynard
et al. 1999
; Riehle et al. 1997
; Vaadia
et al. 1995
). However, it is important to point out that simply
averaging together the responses of redundant neurons, or even neurons
that have some degree of correlated variability but identical average
responses (Shadlen and Newsome 1998
), would not help to
disambiguate the confounded information. Thus, whether concurrent
decoding of responses of a cluster of neurons can reduce the amount of
confounded information is an issue that must be resolved
experimentally. It is relatively straightforward to do this by an
extension of the direct method to the responses of multiple neurons
recorded together.
What do we expect from a simple model?
We evaluated the degree to which a simple model of V1 simple cells
can replicate our experimental results. This model is quasi-linear and
therefore fails to account for many of the interesting nonlinearities displayed by V1 neurons, particularly complex cells (Movshon et al. 1978a) but also, to some extent, simple cells
(DeAngelis et al. 1993
; Mechler et al.
1998a
). In particular, the model does not account for
contrast-specific nonlinearities (Albrecht and Hamilton
1982
; Carandini et al. 1997b
; Dean
1981
). We find that this model significantly overestimates the
magnitude of formal and attribute-specific information rates, in
particular the contrast-specific information rates in
stationary-grating responses (compare Fig. 8 to Fig. 6). Most
significantly, the model responses contain no confounded information,
in stark contrast to the prominent confounded information found in real responses.
In earlier work (Reich et al. 2000a), we used a spike
train resampling technique to show that, for V1 neurons, the details of
spike generation do not have a large effect on the magnitude of formal
information rates. That result, together with additional resamplings
done in connection with the present study (not shown), indicates that
the discrepancies between real and model information rates (including
confounded information) are not likely to be due to the assumption of a
Poisson spike generating mechanism in the model. Moreover, these
discrepancies are also not likely to be due to cell-to-cell variation
in the shape of the linear filter or kernel, since such variation is
similar for real and model responses and in both cases has little
impact on information rates.
Instead the discrepancies between real and model responses almost
certainly relate to the fact that real responses to stimuli that differ
only in contrast are not simply related by a scaling factor but rather
depend strongly on factors such as spatiotemporal pattern and the level
of adaptation (Albrecht 1995; Bonds 1991
; Ohzawa et al. 1982
). It is possible that a single
mechanism
nonlinear suppression that is sometimes called
contrast normalization (Albrecht and Geisler
1991
; Heeger 1992
)
can account for all of these
discrepancies but only if the mechanism is sensitive to the
spatiotemporal parameters of the stimulus and can affect the dynamics
of the response. Indeed, such mechanisms are known to exist in the
retina (Shapley and Victor 1981
), lateral geniculate
nucleus (Sclar 1987
), and primary visual cortex
(Reid et al. 1992
). The sensitivity can be intrinsic to
the suppressive mechanism itself or, alternatively, might be derived
from a pooling of the responses of other V1 neurons with different
stimulus-response properties.
Summary
The major finding in this paper is that V1 neurons transmit formal information at high rates for a variety of stimulus types and that the amount of attribute-specific information is much lower. Contrast-specific information rates depend little on stimulus and cell type, whereas spatiotemporal pattern-specific information rates depend strongly on these factors. A substantial fraction of the formal information cannot be attributed to either contrast or spatiotemporal pattern if only the responses of single neurons are taken into account, and this confounded information is likely to be a result of dynamic interactions between stimulus attributes during response generation. Further work may determine the degree to which the confounded information can be sorted into stimulus-specific components on the basis of the simultaneous responses of groups of neurons.
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APPENDIX |
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In this appendix, we make rigorous the statement that there is no confounded information if and only if a neuron's response depends independently on the two stimulus attributes (in our experiments, contrast and spatiotemporal pattern). We assume that stimuli are defined by an independent choice of a stimulus s1 out of a set S1 (corresponding to the 1st attribute) and a stimulus s2 out of a set S2 (corresponding to the 2nd attribute). A corollary of this demonstration is that there is no confounded information if and only if, for each possible response r, there is no mutual information between the conditional distributions {S1|r} and {S2|r}. This is in turn equivalent to the statement that, for each response r, the conditional probability p(s1,s2|r) is a separable function of s1and s2: p(s1,s2|r) = p(s1|r)p(s2|r).
For notation, we follow the conventions of Cover and Thomas
(1991). The mutual information between two variables
X and Y is
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(A1) |
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(A2) |
![]() |
(A3) |
We define the confounded information C as
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(A4) |
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(A5) |
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(A6) |
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(A7) |
It is straightforward to show that Eq. A7 is equivalent to
our independence condition, given that the stimulus probabilities are
independent, i.e.,
p(s1,s2) = p(s1)p(s2).
By Bayes's rule
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(A8) |
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(A9) |
Examples
We now discuss eight simple examples to illustrate the concept of confounded information. In these examples, we consider systems with two independent inputs, each of which can take a value of 0 or 1 with equal probability, and a single output. There are therefore four different input configurations of the stimuli (s1,s2): (0,0), (0,1), (1,0), and (1,1). In the first six examples, the output is also binary, whereas in the last two examples, the output can take on more than two distinct values. The example systems are displayed in Table 4, together with the corresponding information values.
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In example 1, the response is independent of s1 and is completely determined by s2. In example 2, the response is independent of s2 and is completely determined by s1. Of the 2 bits of entropy in the stimuli, only 1 bit is conveyed as information, but it is transmitted perfectly. Moreover, the response to one stimulus is independent of the value of the other, so that there is no confounded information.
In example 3, the system responds with a 1 if the two inputs
are identical, and with a 0 if they are different. There is still 1 bit
of information transmitted, but in this case the response to one
stimulus depends completely on the value of the second, so that all the
information is confoundedthat is, there is no information transmitted
about either stimulus unless the value of the other stimulus is known.
In example 4, the system responds only if the values of both
stimuli are 1. Here, the response is symmetric in the two stimuli, so
that the response to each stimulus alone conveys the same amount of
information. However, because the response depends jointly on the
values of both stimuli, some of the information23% of the total
is confounded.
In the fifth and sixth examples, the system's response again has a complicated dependence on the two stimuli, so that the confounded information is nonzero. In example 5, the system responds at random if s2 = 0, and identically reflects s1 if s2 = 1. Confounded information arises because although no information is conveyed about s2, the response to s1 is more informative if s2 is known. In example 6, the system responds at random if the two stimuli are different and reflects their shared value if they are the same. The response thus conveys equal amounts of information about the two stimuli, but there is still some confounded information.
The first six examples illustrate that confounded information can arise even in very simple systems, so long as the conditional response probabilities are separable in the two stimuli, as in Eq. A9. The last two examples demonstrate that this requirement is not equivalent to a requirement that stimulus encoding be linear or additive. Example 7 is a system that simply sums the values of the two stimuli (and thus has three possible responses); the confounded information in this system is 33% of the total. On the other hand, example 8 is a system that generates distinct responses to each of the four stimuli, but its response is also additive in the sense that the response to any pair of stimuli is the sum of the responses to two pairs of stimuli that add up to the same input. For instance, the response to (1,1) is equal to the sum of the responses to (1,0) and (0,1). As with any system that maps each input to a distinct output, even a system that is not additive at all, the system in example 8 does not produce confounded information.
Thus it is not surprising that we observe substantial amounts of confounded information in real neuronal responses: real responses depend in a complicated way on both contrast and spatiotemporal pattern. In the simple examples considered here, in fact, only systems that ignore one or the other stimulus, as in examples 1 and 2, or systems that respond differently to each stimulus pair, as in example 8, can feature responses that convey information about both stimuli without confounding them.
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ACKNOWLEDGMENTS |
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We thank B. Knight and K. Purpura for much useful advice.
This work was supported by National Institutes of Health Grants GM-07739 and EY-07138 (D. S. Reich) and EY-9314 (J. D. Victor).
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FOOTNOTES |
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Address for reprint requests: D. Reich, The Rockefeller University, 1230 York Ave., Box 200, New York, NY 10021 (E-mail: reichd{at}rockefeller.edu).
Received 10 April 2000; accepted in final form 27 September 2000.
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REFERENCES |
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