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The Journal of Neuroscience, March 1, 2001, 21(5):1676-1697
Correlated Firing in Macaque Visual Area MT: Time Scales
and Relationship to Behavior
Wyeth
Bair1, 2,
Ehud
Zohary3, and
William T.
Newsome1, 4
1 Howard Hughes Medical Institute (HHMI),
2 Center for Neural Science, New York University, New York,
New York 10003, 3 Department of Neurobiology, Institute of
Life Science, Hebrew University, Jerusalem, 91904, Israel, and
4 Department of Neurobiology, Stanford University School of
Medicine, Stanford, California 94305
 |
ABSTRACT |
We studied the simultaneous activity of pairs of neurons recorded
with a single electrode in visual cortical area MT while monkeys
performed a direction discrimination task. Previously, we reported the
strength of interneuronal correlation of spike count on the time scale
of the behavioral epoch (2 sec) and noted its potential impact on
signal pooling (Zohary et al., 1994 ). We have now
examined correlation at longer and shorter time scales and found that
pair-wise cross-correlation was predominantly short term (10-100
msec). Narrow, central peaks in the spike train cross-correlograms were
largely responsible for correlated spike counts on the time scale of
the behavioral epoch. Longer-term (many seconds to minutes) changes in
the responsiveness of single neurons were observed in
auto-correlations; however, these slow changes in time were on average
uncorrelated between neurons. Knowledge of the limited time scale of
correlation allowed the derivation of a more efficient metric for spike
count correlation based on spike timing information, and it also
revealed a potential relative advantage of larger neuronal pools for
shorter integration times. Finally, correlation did not depend on the
presence of the visual stimulus or the behavioral choice of the animal.
It varied little with stimulus condition but was stronger between
neurons with similar direction tuning curves. Taken together, our
results strengthen the view that common input, common stimulus
selectivity, and common noise are tightly linked in functioning
cortical circuits.
Key words:
Area MT/V5; cross-correlation; neuronal pooling; visual
motion; extrastriate cortex; synchrony; stimulus-locked modulation; noise correlation; visual cortex
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INTRODUCTION |
A fundamental problem in sensory
neuroscience is to understand how psychophysical performance is related
to the signaling capacities of single sensory neurons. It is now widely
recognized that no satisfactory solution to this problem can be
achieved in the absence of detailed knowledge concerning correlated
firing within the pool of sensory neurons contributing to a particular psychophysical judgment (Johnson et al., 1973;
Johnson, 1980; van Kan et al., 1985;
Britten et al., 1992; Gawne and Richmond, 1993; Zohary et al., 1994; Geisler and
Albrecht, 1997; Parker and Newsome, 1998). For
example, combining signals across a pool of neurons can generate
superior psychophysical sensitivity if the noise carried by individual
members of the pool is averaged out. This benefit of pooling is only
achievable, however, to the extent that the noise carried by individual
neurons is independent (uncorrelated); noise that is common to the
entire pool cannot be averaged out. In general, the effect of
correlated noise depends on how signals are combined, and although
correlation may either aid or hinder noise removal (Johnson,
1980; Abbott and Dayan, 1999; Panzeri et
al., 1999), its impact on the amount of information conveyed by
a pool of neurons may be profound. Thus, empirical analysis of
correlated firing is central to a quantitative understanding of the
relationship between physiological responses and psychophysical judgments.
Extrastriate visual area MT is ideal for investigating pools of
sensory neurons that underlie psychophysical performance. MT contains a
preponderance of directionally selective neurons (Zeki,
1974; Maunsell and Van Essen, 1983;
Albright et al., 1984), the activity of which has been
linked compellingly to the psychophysical discrimination of direction
in stochastic motion stimuli (Newsome et al., 1989;
Britten et al., 1992; Salzman et al.,
1992; Murasugi et al., 1993; Salzman and
Newsome, 1994). In a previous study, therefore, we measured
correlated firing in MT and found that spike counts from adjacent
neurons were noisy and only weakly correlated but that even this small
amount of correlated noise placed substantial limits on the benefits of
signal averaging across a pool (Zohary et al., 1994).
Subsequently, Shadlen and colleagues (1996) incorporated
these insights into a computational model of the relationship between
the activity of MT neurons and psychophysical judgments of motion direction.
In the present study, our primary goals were to examine the time scale
at which correlation arises: in particular, to relate spike count
correlation to spike timing correlation and examine the dependence of
correlated firing on stimulus and behavioral parameters. Our most
intriguing finding is that trial-to-trial correlations in spike count,
measured over trials of 2 sec duration, are produced largely by the
same mechanisms that generate peaks in the spike train
cross-correlogram (CCG) on a time scale of a few tens of milliseconds.
For a given pair of MT neurons, a quantitative measurement based on the
CCG peak predicts with fair accuracy the level of correlation
calculated from spike counts over the full trial length. Furthermore,
the CCG-based measure is substantially more reliable than the measure
based on spike count correlation. The spike train CCG is typically used
as a qualitative indicator of functional connectivity among neurons. In
contrast, our results suggest that the spike train CCG can provide
quantitative measures of neuronal correlation that are of considerable
interest for models that seek to reconcile neuronal and psychophysical performance.
Some of these results have been published previously in abstract form
(Bair et al., 1996,
1999).
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MATERIALS AND METHODS |
Subjects, surgery, and daily routine. The experiments
were performed on three adult rhesus monkeys weighing between 7 and 9 kg (Macaca mulatta, two males and one female). Before the
experiments, each monkey was surgically implanted with a device for
stabilizing head position (Evarts, 1968), a scleral
search coil for measuring eye position (Judge et al.,
1980), and a recording cylinder that allowed microelectrode
access to cortex within the occipital lobe. All surgical procedures
were performed under aseptic conditions with halothane anesthesia.
After recovery from surgery, each animal engaged in daily training or
experimental sessions of 2-6 hr duration. Behavioral control was
accomplished by operant conditioning techniques using fluids as a
positive reward; fluid intake was therefore restricted during periods
of training or electrophysiological recording. The diet was
supplemented with moist monkey treats, fruits, and nuts. The animals
were maintained in accordance with guidelines set by the U.S.
Department of Health and Human Services (NIH) Guide for the Care and
Use of Laboratory Animals.
Visual stimuli. The visual stimuli used in this study were a
set of dynamic random dot patterns in which a unidirectional motion
signal was interspersed among random motion noise. The stimulus set has
been described extensively in previous publications (Britten et
al., 1992), and we simply summarize its essential features here.
Dynamic random dots were plotted sequentially on the face of a CRT
screen at a high rate (6.67 kHz). After 45 msec, a dot was either
displaced in a specified direction (coherent motion) or replaced by
another dot at a random location on the screen (noise). In one extreme
form of the display, all dots were positioned randomly so that the
display was pure noise. In this form, which we term 0% coherence, the
display contained many local motion events (caused by fortuitous
pairings of the dots in space and time) but on average no net motion in
any direction. At the other extreme (100% coherence), all dots were
displaced uniformly so that the display contained noise-free motion in
a specified direction. Our software permitted us to create any stimulus
intermediate between these two extremes by specifying the percentage of
dots that carried the "coherent" motion signal. The percentage of
dots engaged in coherent motion governed the strength of the motion signal without affecting the overall luminance, contrast, or average spatial and temporal structure of the stimulus. When a psychophysical subject was asked to discriminate the direction of motion in such displays, the difficulty of the discrimination was related directly to
the percentage of dots in coherent motion.
In early experiments (monkey E), the stimuli were generated by a PDP
11/73 computer and displayed on a large, electrostatic deflection
oscilloscope via a high speed DMA digital-to-analog converter. In later
experiments (monkeys R and K), the stimuli were created by means of an
IBM 386 equipped with a dedicated graphics board (SGT Pepper no. 9).
These stimuli were displayed on a raster scan CRT monitor with a 60 Hz
refresh rate. In all experiments, the display monitor was positioned 57 cm in front of the monkey.
A critical distinction must be made between two different methods of
presenting repeated stimuli for a particular condition (e.g., a 6.4%
coherence, upward stimulus). Our standard method used a new random
number sequence for each repeat, resulting in what we will refer to as
"ensemble stimuli," which differ in detail but have on average the
prescribed motion coherence. As a control for the effect of random
stimulus variation on neuronal responses, we recorded from four pairs
using repeated presentations of stimuli generated with exactly the same
sequence of random numbers. We will refer to the identical stimulus
repeats used by this method as "replicate stimuli."
Behavioral paradigms and selection of visual stimuli. We
used two behavioral paradigms in this study: a fixation task and a
discrimination task. In the fixation task, the monkey was required only
to maintain its eye position within an electronically defined window
around the fixation point for 2-4 sec. The monkey received a liquid
reward on successful completion of each trial. In most experiments, the
window permitted eye movements up to 1.5° away from the fixation
point, but in practice, the monkeys usually held their eye position
within 0.5° of the fixation point.
The monkeys performed the fixation task during the initial search for
well isolated pairs of neurons, during mapping of receptive fields, and
during quantitative measurement of the direction tuning properties of
the neurons. Receptive field boundaries were mapped qualitatively for
each neuron of the pair, and the stimulus aperture was positioned to
include both receptive fields. The receptive fields typically
overlapped substantially, so that the stimulus aperture only engaged a
small portion of the surround of either receptive field. The optimal
speed was estimated qualitatively for each neuron, and subsequent
experiments were conducted using a motion speed intermediate between
the two optima. To measure a direction tuning curve, a 100% coherence
dot pattern was presented in eight different directions of motion
equally spaced around the clock at 45° intervals. The different
directions were presented in a pseudorandom sequence until 10-20
repetitions were completed for each direction.
The two direction tuning curves were used to assign a
"preferred-null" axis of motion for use during the discrimination
task (below). Ostensibly, the preferred-null axis was the axis of
maximal directionality for the two neurons; motion in opposite
directions along the axis should yield a maximal difference in
responsiveness. In practice the axis chosen was usually a compromise
between the preferred directions of the two neurons measured
individually. Most pairs of neurons had similar preferred directions,
and the compromise therefore resulted in a near-optimal axis for both. On occasion we recorded from pairs of neurons with preferred directions that were nearly opposite each other. In this case again, the choice of
directional axis was easy because the signs of the two response were
simply reversed along the same axis. Occasionally, however, the
preferred directions of the two neurons were nearly orthogonal to each
other, or one of the neurons was not directional at all. In such cases,
we chose the preferred-null axis and the speed of the motion signal to
match the preferences of the more responsive, directional neuron. On
the whole, therefore, most neurons were studied during the
discrimination task with stimuli that matched their physiological
properties reasonably well. For a few neurons, the stimuli were
substantially suboptimal.
In the discrimination task, the monkey performed a two-alternative,
forced-choice discrimination of motion direction. This task has been
used extensively in our laboratory and is described in detail in
previous publications (Britten et al., 1992). On each
trial a random dot stimulus was presented for 2 sec within the aperture
covering both receptive fields. The direction of the coherent motion
signal was varied randomly from trial to trial between the preferred
direction of the neurons under study and the direction 180° opposite
(the "null" direction); the monkey's task was to discriminate
correctly the direction of motion. The strength of the motion signal
was varied among a range of coherence levels that spanned
psychophysical threshold. A minimum of 15 repetitions was obtained for
each stimulus condition (i.e., each combination of direction and
coherence), and all conditions were presented in pseudorandom order. We
will refer to the neuronal data from these experiments as "coherence
series data" to distinguish them from the direction tuning data.
Each trial began with the appearance of the fixation point. After the
monkey achieved fixation and held its gaze within the fixation window
for 300 msec, the visual stimulus was presented as described above. The
monkey was required to hold its gaze on the fixation point during
stimulus presentation so that the stimulus remained well positioned on
the receptive fields of the two neurons. At the end of the 2 sec
display interval, the random dot pattern and the fixation point
disappeared, and two small visual targets appeared, one corresponding
to each of the two possible directions of coherent motion. The monkey
made a saccadic eye movement to one of the two targets to indicate the
direction of motion perceived in the visual stimulus. Eye movements
were measured continuously with the scleral search coil technique,
permitting the computer to register correct and incorrect choices.
Correct choices were followed by a liquid reward; incorrect choices
were followed by a brief time-out period. On 0% coherence trials, the
monkey was rewarded randomly with a probability of 0.5 because there
was no "correct" answer on these trials. If the monkey broke
fixation prematurely during a trial, the trial was aborted, the data
were discarded, and a time-out period ensued.
Electrophysiological recording and spike sorting.
Electrophysiological recordings were made with tungsten microelectrodes inserted into the cortex through a transdural guide tube (electrode impedance = 0.5-2.0 M at 1 kHz) (Micro Probe, Potomac, MD).
The guide tube was held rigidly in a stable coordinate system by a plastic grid inside the recording cylinder (Crist et al.,
1988). We recorded through any particular guide tube for
several consecutive days.
The signal from the microelectrode was amplified and bandpass-filtered
(0.5-10 kHz), and action potentials from multiple single neurons were
discriminated using an on-line spike sorting system that was developed
originally in the laboratory of Dr. Moshe Abeles (Hebrew University,
Jerusalem) and was commercially available from Alpha Omega Engineering
(Nazareth, Israel). The filtered microelectrode signal was continuously
sampled at a rate of 14 kHz by a digital signal processing system
housed in an IBM 386 platform. (The apparent discrepancy between the 7 kHz cutoff frequency, implied by 14 kHz sampling, and the 10 kHz cutoff
of our bandpass filter was not a limiting factor, because in practice
the amplitude of the noise from 7 to 10 kHz was small relative to the
amplitude of all well isolated action potentials.) The computer
software provided a user interface to the spike sorting hardware and
included graphics displays of voltage waveforms, spike templates, and
distributions of matching errors (below). Spikes were discriminated
on-line using an eight-point template-matching algorithm
(Wörgötter et al., 1986). Each time the
voltage exceeded a threshold level, an eight-point voltage sample was
acquired and compared with the predefined templates that characterized
the waveform of each recorded neuron. If the root-mean-square error
(RMSE) of the match between the signal waveform and one of the
templates was below a criterion value, an action potential was
registered for that neuron. A template was defined by the software to
minimize the RMSE of the match to the template across a sample of 100 action potentials accepted by the experimenter as belonging to a
specific neuron.
The quality of unit isolation was determined by the separation of the
templates from each other and from the noise. Excellent separation of
the templates from each other was necessary to prevent "cross-talk"
between the two waveforms. Occasional misclassification of the two
action potentials could result in artifactual correlations that would
be deleterious to certain analyses. The only substantive insurance
against cross-talk were the rigor and attentiveness of the
experimenter both in selection of pairs for study and in maintaining
quality of isolation during the experiment. We attempted to be
exceedingly rigorous in selecting pairs for study, rejecting all
candidates except those with waveforms that were strikingly distinct
from each other. Similarly, we attempted to be unusually conservative
in on-line assessment of the quality of isolation. If either waveform
began to deteriorate, creating any doubt about isolation, we ceased
recording until the waveforms could be restored.
Separation of the two templates from the noise could be achieved more
objectively. For each template, the software compiled and displayed a
frequency histogram of RMSE values resulting from comparison of each
triggered waveform with that template. Excellent separation of the
template from the noise corresponded to a bimodal histogram of RMSE
values. A peak at low RMSE values corresponded to action potentials
from the neuron that defined the template; a larger peak at high RMSE
values reflected the substantial mismatch between noise waveforms and
the template. We insisted that both modes be visible and well separated
from each other in the RMSE histogram. The criterion RMSE value for
accepting an action potential as corresponding to a particular template
was set at the local minimum in the bimodal RMSE distribution. This
ensured a reasonable balance between minimizing noise contamination and
minimizing false negative matches to the template. We rejected
recordings for which the error distributions were judged by eye to
overlap in a manner that would produce more than ~5% false
positives, but we estimate that the contamination rate was typically
lower because the peaks in the bimodal error distribution often showed no sign of overlap after collecting hundreds of spikes. Admitting a
small percentage of spikes from other neurons to one or the other
template should have negligible effect on estimates of pair-wise interneuronal correlation because of the modest to weak correlations typical between cortical neurons.
Obviously, our technique of multi-unit recording with a single
electrode cannot detect simultaneous spikes because the two waveforms
superimpose, resulting in a poor match to either template. Because the
primary lobes of the action potential waveforms were generally  0.5
msec in duration (Mountcastle et al., 1969;
Funahashi and Inoue, 2000), this limitation only
resulted in an underestimation of spikes that were synchronous to
within 1 msec [for example, see Gawne and Richmond
(1993); Funahashi and Inoue (2000)]. For two
neurons with uncorrelated activity firing at rates <100 spikes per
second, the probability of spike synchrony at the millisecond time
scale is <0.12, which is reasonably uncommon.
However, the pairs of neurons studied here often have peaks in their
CCGs at time zero (see Fig. 5B), and therefore the
probability of simultaneous firing may be many times greater. This
problem can be compounded for cells that fire bursts during which
firing rates may reach 300-500 spikes per second. However,
multi-electrode cross-correlation studies in monkey and cat suggest
that CCG peaks in visual cortex are typically broader than 1-2 msec
(Ts'o et al., 1986; Krüger and Aiple,
1988; Ts'o and Gilbert, 1988; Cardoso de
Oliveira et al., 1997). The available evidence suggests that
the vast majority of CCGs do not have sudden discontinuities on the
time scale of 1 msec at the origin and that peaks of width 1 msec, when
they exist, are weak and could not account for a substantial fraction of the strength of interneuronal correlation commonly observed in
visual cortex. Therefore, we approximate the CCG value at time zero
using values at neighboring time lags, as described later.
Analysis of direction selectivity. To assess neuronal
direction selectivity, we determined which of two different models
could better match the direction tuning curves. The first model assumed that the neuron was not direction selective and that response variation
across direction was caused simply by sampling noise. It therefore
predicted that the level of activity was essentially invariant with
direction and was best estimated as the mean of the responses to all
directions. The second model assumed that the neuron was in fact
direction selective with a Gaussian distribution of responses centered
on the optimal direction of motion. This distribution had four free
parameters: the optimal direction of motion, the maximal response rate,
the bandwidth of the Gaussian function, and the baseline response (the
spontaneous firing rate). We performed maximum likelihood fits to the
two separate, nested models under the assumption of normal errors. The
likelihoods (L) obtained from these computations were transformed
by:
|
(1)
|
such that l is distributed as  2 with
three degrees of freedom (Hoel et al., 1971). If
l was below the criterion value (p = 0.05),
we concluded that the direction tuning function of the neuron was
better described by a Gaussian fit than by a constant response
independent of direction. We considered these neurons to be direction
selective, and the quantitative analyses in this paper used optimal
directions and bandwidths obtained from the Gaussian fit to the tuning
curve of each neuron. We will use the notation  PD to
refer to the difference (in degrees) between the preferred directions
of neurons within a pair.
Analysis of psychophysical data. Psychophysical data from
the discrimination experiments were compiled into psychometric
functions depicting the proportion of correct decisions as a function
of the strength of the motion signal (in % coherence). We used a maximum likelihood method (Watson, 1979) to fit these
data with sigmoidal functions of the form:
|
(2)
|
where p is the probability of a correct decision,
c is coherence, a is the coherence level that
supports threshold performance (82% correct), b is the
slope of the sigmoidal function, and d is the asymptotic
performance for strong motion signals (expressed as proportion of
correct decisions). The threshold parameter, a, and the
slope parameter, b, provide a succinct description of the
psychophysical data.
Equation 2, derived from the integral of a Weibull distribution
(Quick, 1974), provided acceptable fits to the bulk of
our psychophysical data. Thirty-four of 46 psychometric functions in
our data set were well fit [likelihood ratio test, p > 0.05; see the Appendix of Watson (1979)] when the
asymptotic performance, d, was constrained to be unity, and
the remaining functions were well fit by allowing d to vary.
The non-unity asymptote in the latter 12 experiments reflected the
monkey's occasional errors at the highest coherence levels.
Analysis of neural thresholds. We measured neural thresholds
to the stochastic motion stimuli in a manner that permitted direct comparison with psychophysical thresholds [see Britten et al. (1992) for a detailed description]. For each neuron, we first compiled for each motion coherence a frequency histogram of responses to preferred direction motion and a separate histogram of responses to
null direction motion. We considered a "response" to be the total
number of spikes generated by the neuron during the 2 sec stimulus. For
very strong (high coherence) motion signals, these "preferred" and
"null" response distributions were typically non-overlapping because most of our neurons were highly directional. At these coherence
levels, the direction of motion could be determined unambiguously on
any given trial simply by monitoring the response of the neuron. For
very weak motion signals, however, the preferred and null response
distributions overlapped almost completely, so judgments of motion
direction based on the responses of the neuron would be at chance.
Intermediate coherence levels resulted in partial overlap between the
two response distributions, leading to intermediate levels of performance.
Following these intuitions, we used a method based on signal detection
theory (Green and Swets, 1966; Britten et al.,
1992) to compute the performance expected of an ideal observer
who based judgments of motion direction on the measured neuronal
responses. For each neuron, this calculation was performed for each
coherence level (typically six non-zero levels, but as many as eight,
and the results were compiled into a neurometric function that plotted expected performance (in % correct decisions) as a function of coherence. A sigmoidal curve was fitted to the data using Equation 2,
and the threshold and slope parameters were extracted as described in
the preceding section. These parameters describe the sensitivity of a
single neuron to the motion signals in our displays in a manner that
can be compared directly with the psychophysical sensitivity measured
on the same trials. Equation 2 described our neurometric data well; the
fits were acceptable (likelihood ratio test, p > 0.05)
for all 83 of the neurons comprising the 46 pairs with valid
psychophysical data.
Assessment of correlated activity. We analyzed two main
types of correlation between the responses from each pair of neurons: signal correlation and noise correlation (Gawne and Richmond, 1993; Gawne et al., 1996; Lee et al.,
1998). Signal correlation, designated
rsignal, refers to the common modulation in a
set of paired mean responses associated with multiple stimulus
conditions. For our purposes, it is simply the correlation coefficient
computed for the mean spike rates from a pair of direction tuning
curves. Noise correlation, rnoise, refers to
common trial-to-trial fluctuations around the mean response for a
single stimulus condition, and its estimation and interpretation occupy
the bulk of this paper. The dichotomy implied by the names "signal"
and "noise" correlation is somewhat unfortunate because apparently
noisy variations in spike rate may carry information about neural
signals that we simply cannot access. However, we will adhere to these
terms for the sake of precedent. The traditional measure of noise
correlation is the interneuronal correlation coefficient (van
Kan et al., 1985; Bach and Krüger, 1986;
Gawne and Richmond, 1993; Zohary et al.,
1994; Gawne et al., 1996; Lee et al.,
1998), which measures correlation at a fixed time scale and
temporal relationship, i.e., the simultaneous trial. We will describe
two new methods for quantifying noise correlation, one at time scales
greater than and equal to the single trial that generalizes the
interneuronal correlation coefficient to non-simultaneous trials
(below) and another at the scale of milliseconds that is derived from
spike train correlograms (Appendix A). Table
1 provides a unified reference to all of
our notation regarding correlation.
The trial cross-covariance. The interneuronal correlation
coefficient is traditionally computed for the spike counts
N1 and N2 of neurons 1 and 2, respectively, according to:
![r<SUB><UP>SC</UP></SUB>=<FR><NU><UP>E</UP>[N<SUB>1</SUB>N<SUB>2</SUB>]−<UP>E</UP>N<SUB>1</SUB><UP>E</UP>N<SUB>2</SUB></NU><DE>&sfgr;<SUB>N<SUB>1</SUB></SUB>&sfgr;<SUB>N<SUB>2</SUB></SUB></DE></FR>,](/content/vol21/issue5/fulltext/1676/img003.gif)
|
(3)
|
where E is expected value and  is the SD computed across all
repetitions of a particular stimulus. However, an experiment yields
several sets of paired spike counts (one set for each stimulus condition), and rather than applying Equation 3 separately to each set,
the sets can be combined after performing a within-set normalization.
One simple normalization, the z-score, involves modifying
the spike count values within each set (i.e., for each stimulus
condition) by subtracting the mean and dividing by the SD for that set
of responses. The subtraction eliminates the mean stimulus-evoked
portion of the response, and the division scales the variance around
the mean so that random fluctuations at high firing rates (which are
known to be larger than those at low rates) are not unduly weighted.
Further empirical justification for this normalization comes from the
observation that rSC changes very little with
firing rate or stimulus condition, as shown in Results. The resulting
z-scores can be represented in the order in which they
occurred in the original experiment by the sequences
z1i and z2i,
1 i M, where M is the total
number of trials in the experiment. Because Ez1 = Ez2 = 0, and
z1 = z2 = 1, the equation for the correlation coefficient, Equation 3,
simplifies to:
![r<SUB><UP>SC</UP></SUB>=<UP>E</UP>[z<SUP><UP>i</UP></SUP><SUB><UP>1</UP></SUB>z<SUP><UP>i</UP></SUP><SUB><UP>2</UP></SUB>].](/content/vol21/issue5/fulltext/1676/img004.gif)
|
(4)
|
For a single set of paired responses, this equation is
equivalent to Equation 3 because neither subtraction nor division by a
positive constant (applied to the spike count data) changes the value
of the correlation coefficient. For multiple sets of responses, the
equation provides an aggregate correlation coefficient. Equation 4 can
be generalized from responses that occurred on the same trial to
responses that occurred on trials separated in time by a lag,  , in
units of experimental trials (~5 sec per unit; see below). This
generalization, which we will refer to as the trial cross-covariance
(TCC), is simply the cross-correlation of z1 and
z2:
![<UP>TCC</UP>(&phgr;)=<UP>E</UP>[z<SUP><UP>i</UP></SUP><SUB><UP>1</UP></SUB>z<SUP><UP>i+&phgr;</UP></SUP><SUB><UP>2</UP></SUB>].](/content/vol21/issue5/fulltext/1676/img005.gif)
|
(5)
|
The value at = 0 is equal to rSC
(Eq. 3) averaged across all stimulus conditions (with appropriate
weighting for the number of trials for each condition), and we use the
symbols TCC(0) and rSC interchangeably. For
 0, TCC() is the correlation coefficient, with values
from  1 to 1, for temporal offsets in arbitrary numbers of trials. For
a pair of uncorrelated neuronal responses, TCC() will approach zero
everywhere as the number of trials used in its estimate increases. The
trial auto-covariance, TAC(), is defined in a similar manner by
replacing z2 with z1 (or
vice versa) in Equation 5, and by definition it is equal to unity for
= 0.
We already know that TCCs will have positive values at = 0 for
neuronal pairs with rSC > 0. If
interneuronal correlation arises at a time scale shorter than the trial
duration, the positive value at = 0 will stand as a narrow,
isolated peak. However, if the correlation between neurons arises from
slow changes in their responsiveness, the positive value at = 0 will be part of a broader peak, i.e., the TCC will have positive
values for 0 as well. Similarly, the presence of a broad
peak around the origin in the TAC will indicate the presence of slow
variations in the excitation of individual neurons.
Our use of the TCC does not rest on whether the horizontal axis is
given in units of time or trials. We retained "trials" as the axis
unit to avoid the technical difficulty associated with the
cross-correlation of data sampled at somewhat irregular time intervals.
The irregularity in the mapping from trials to time was caused by the
monkey's failure to fixate immediately on 10-20% of trials during a
recording session. To estimate the time scale of slow correlation, we
will convert from trials to time using the average time between trial
starts, ~5 sec.
Previous studies by Eggermont and Smith
(1995, 1996)
have attempted to separate correlation at multiple time scales using a
method similar in concept to the TCC, but their time unit was 50 msec,
roughly two orders of magnitude faster than ours. Variations in firing
rate on the time scale of 10s to 100s of milliseconds (Nelson et
al., 1992; Eggermont and Smith, 1995;
Arieli et al., 1996) are considered by us to be short
term because they fall well within the duration of our behavioral epoch.
The spike train cross-correlogram. We measured correlation
at the time scale of milliseconds using spike train auto- and
cross-correlograms (ACGs and CCGs) (Perkel et al.,
1967a,b). Our CCG
is defined based on the trial-averaged cross-correlation,
Cjk( ) (defined in Appendix A, Eq. 14), of
binary spike sequences from neurons j and k
(typically, 1 and 2). In particular:
|
(6)
|
where  j and k are the mean firing
rates (in spikes per second) of neurons j and k.
For ACGs, j = k = 1 or 2. The function  () is
a triangle representing the extent of overlap of the spike trains as a
function of the discrete time lag , i.e.:
|
(7)
|
where T is the duration of the spike train segments
used to compute Cjk. Dividing
Cjk by () in Equation 6 changes the units of our CCG from raw coincidence count to coincidences per second and
corrects for the triangular shape of Cjk caused
by finite duration data.
In Equation 6, we chose to divide by the geometric mean spike rate
(GMSR),  , because under this normalization the area of our CCG peaks remained relatively constant as
firing rate varied [shown later; see also Krüger and
Aiple (1988)] and because it is symmetric with respect to the
two neurons. With this normalization, the CCG is the ratio of a
coincidence rate to a mean spike rate and ends up with units of
coincidences per spike. Once the shift-predictor (below) is subtracted,
this normalization is similar to that of many other studies
(Mastronarde, 1983a; Krüger and Aiple,
1988; Eggermont and Smith, 1996; Cardoso de Oliveira et al., 1997) and is conceptually similar to that proposed by Aertsen et al. (1989) for their "joint
peri-stimulus time histogram." A different normalization, dividing by
the product of the spike rates, has been favored less often
(Melssen and Epping, 1987, their Eq. 17; Das and
Gilbert, 1995), and for our data was less appropriate than
dividing by the GMSR.
Shift- (also known as shuffle-) predictors [defined in Perkel
et al. (1967b)] for CCGs and ACGs were computed using the same normalization as above but based on the average cross-correlation of
all M2 M pairings of
nonsimultaneous responses from neurons j and k
for a set of M trials. This "all-way" cross-correlation,
denoted C*jk(), can be computed
efficiently from the cross-correlation of the post-stimulus time
histograms (PSTHs), Sjk (defined in Appendix A,
Eq. 15), according to the following expression:
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(8)
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which approaches Sjk() as M
increases (Perkel et al., 1967b). Substituting
C*jk for Cjk in
Equation 6 gives the final shift-predictor. A shift-predictor computed
from responses to ensemble stimuli (i.e., those that resulted from
different sequences of random numbers; see above) will be referred to
as an ensemble shift-predictor. When computed for replicate stimuli
(i.e., repetitions of identical stimuli), it will be referred to
plainly as a shift-predictor.
CCGs, ACGs, and shift-predictors were computed from data in the
post-stimulus onset period 300-2000 msec to avoid processing the
initial transient response. This made shift-predictors flatter and
prevented changes in correlation strength that might be associated with
the stimulus onset transient from influencing the analysis. We computed
all quantitative results for the full trial as well and found only
negligible differences. CCGs and ACGs were computed individually for
each stimulus condition, shift-predictors were subtracted, and then
averages were taken across all valid stimulus conditions. We set
criteria for the minimum quantity of data required for neurons to be
accepted into the CCG and ACG analysis pool. These rules were applied
in order: (1) no trial was valid that had fewer than four spikes within
the analysis window, (2) no stimulus condition was valid that had fewer
than four valid trials or <64 spikes in total per neuron, and (3) no
pair of neurons (or neuron) was included that had fewer than four valid
stimulus conditions. These criteria eliminated 1 of 104 pairs from our direction tuning data set and 2 of 50 pairs from our coherence series
data set.
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RESULTS |
Our findings are organized as follows. The first section provides
a brief description of our data for a typical pair of neurons and shows
how all pairs are distributed according to the strength of their
interneuronal correlation and the similarity of their directional
tuning curves. The second major section is devoted to measuring the
time scale of interneuronal correlation, which involves (1) separating
long- and short-term correlation, (2) assessing the time scale of
short-term correlation using spike train CCGs, and (3) relating CCG
peaks to spike count correlation. A more efficient metric for spike
count correlation is derived here and in Appendix A. The next major
section of results reports the dependence of correlation, or synchrony,
on stimulus parameters and on the decision-making and behavioral state
of the animal. A brief section shows that neurons do not cluster with
respect to their sensitivity to the stimulus or their relationship to
behavior, and the final section describes control experiments for the
influence of stimulus variance on our estimates of correlation.
Basic measurements of response correlation
Our results are based on simultaneous recordings from 107 pairs of
MT neurons in three monkeys. We obtained directional tuning data for
104 pairs; we gathered discrimination data for a subset of 46 pairs.
All recordings admitted to our database conformed to two requirements:
both neurons were well isolated for at least 10 repetitions per
stimulus condition, and at least one of the neurons yielded reliable,
directionally selective responses to fully coherent random dot stimuli.
For analyses involving CCG and ACG computations, we further restricted
the database to pairs that satisfied criteria for a minimum number of
spikes (see Materials and Methods). For ease of reference and
consistency checking, the numbers of cells and pairs qualified for the
major analyses are summarized in Table
2.
Figure 1 depicts a complete set of
measurements for a representative pair of simultaneously recorded MT
neurons. A and B are direction tuning curves for
neurons 1 and 2, respectively. Both neurons were directionally
selective and exhibited similar preferred directions and tuning
bandwidths. C and D depict responses of the same
two neurons as a function of motion coherence for both the preferred
and null directions of motion. For these measurements, the preferred
direction was set to 90°, approximating the optimal directions of
both neurons. Off-line analysis of the data in A and
B revealed the preferred directions to be 58° and 82°
for neurons 1 and 2, respectively (see Materials and Methods). In C and D, the firing rates of both neurons
increased roughly linearly with motion coherence in the preferred
direction and decreased linearly with motion coherence in the null
direction, a typical pattern for MT neurons (Britten et al.,
1993).
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Figure 1.
Example of direction tuning curves and responses
versus coherence for a pair of neurons recorded simultaneously.
A and B show mean firing rate as a function of
stimulus direction for the two neurons (pair emu018).
Stimuli were 100% coherence moving dots. Error bars show ±1 SE.
Thin, flat lines show spontaneous firing rate.
C and D show mean firing rate as a function of
coherence for preferred direction (90°, thick lines) and
null direction (270°, thin lines) stimuli for the same
neurons as A and B, respectively. Note that the
minimum and maximum spike rates here do not reach those in A
and B because 100% coherence was not included in these
direction-discrimination experiments. Although some error bars are
occluded at low coherence for null direction stimuli, they were roughly
the same size as those for the preferred direction. A linear horizontal
axis is maintained to emphasize the roughly linear relation between
firing rate and coherence.
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Using the direction tuning data for each pair of neurons, we assessed
the strength of two distinct types of correlation, that of the mean
responses and that of the variations about the mean. The former,
commonly known as signal correlation, measures the similarity of tuning
curves for a pair of neurons and was computed here as the correlation
coefficient, rsignal, between the sets of data
points from the direction tuning curves. For the curves in Figure 1,
A and B, rsignal was 0.88, indicating
a high degree of match. The distribution of
rsignal for all pairs (Fig.
2A) was comparable to that of
a more conventional but less general metric, the difference between
preferred directions, PD, shown for comparison in
Figure 2B. The dominant modes in both distributions, i.e.,
high rsignal and low PD,
indicate that adjacent neurons in our study tended to have similar
direction tuning, consistent with the known columnar organization of MT
(Albright et al., 1984; DeAngelis and Newsome,
1999).
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Figure 2.
The joint distribution of signal and noise
correlation for pairs of MT neurons. A, The marginal
frequency distribution of rsignal for all pairs
represented in C. B, The frequency distribution of
PD (applicable to only those 69 pairs with both neurons
directional) was similar to that reported by Albright et al.
(1984) for pairs of MT neurons recorded successively at 50 µm
intervals. Their study showed a second, small mode in the distribution
for PD between 120 and 180°, corresponding to nearly
opposite preferred directions for the two neurons. This mode was not
readily apparent in our data; however, our sample size was smaller.
From 104 pairs, 163 of 196 individual neurons (84%) were directional
by the likelihood test described in Materials and Methods.
C, Noise correlation is plotted against signal correlation
for 103 pairs (direction tuning data). Squares indicate that
at least one cell in the pair did not meet the directionality criterion
(n = 34). Circles indicate directional pairs, and
filled circles (n = 57) indicate pairs with
similar preferred directions, i.e., PD < 90°.
For the 57 directional pairs with similar preferred directions,
rsignal was typically high (median 0.86), and
the mean rnoise value was 0.20 (SD 0.15), which
was significantly greater than zero (p < 10 6; t test). The two filled
circles in the top left quadrant represent pairs for
which PD was only slightly <90° (87 and 83°) and
for which peculiarities of the direction tuning curves caused
rsignal to be negative. For directional pairs
with PD  90°,
rnoise = 0.02 (SD 0.09; n = 12; not different from zero; p = 0.59). For pairs
with at least one nondirectional cell,
rnoise = 0.06 (SD 0.15; n = 34; not different from zero; p = 0.11). D,
Marginal frequency distribution of rnoise for
all pairs represented in C.
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The second type of correlation is assessed not from the mean responses
for all stimuli but from the trial-to-trial fluctuations (evidenced by
the error bars in Fig. 1) around the mean response for each stimulus
condition. This interneuronal correlation has therefore been dubbed
noise correlation (Gawne and Richmond, 1993; Gawne et al., 1996; Lee et al., 1998).
Noise correlation, or rnoise, is typically
estimated by computing the correlation coefficient, rSC, between the number of spikes generated by
one of the neurons and the number of spikes generated by the second,
simultaneously recorded neuron for a set of nominally identical
stimuli. However, we developed a lower-variance estimator for
rnoise (introduced and described in detail in
the next section of Results) and have plotted those estimates against
the values of rsignal for all pairs in Figure
2C (the marginal distribution of
rnoise is shown in D). The pairs
appear to fall into two general groups in C that are not
apparent from the marginal distributions alone. One group consists of
pairs with very similar direction tuning curves (i.e., high
rsignal values) and positive noise correlation.
A second group consists of pairs with low or negative signal
correlation and noise correlation near zero. Overall, the correlation
coefficient between rsignal and
rnoise is 0.61 (p < 106; n = 103; direction tuning data).
The correlation of rnoise with
rsignal is consistent with the notion that
shared common input endows nearby neurons with similar tuning
properties and makes them subject to similar noise sources. This
observation is not unique to our data set, but it allowed us to focus
our investigation of interneuronal correlation, when appropriate, on
the cluster of neurons associated with non-zero rnoise values. We will use
PD < 90° as a criterion for making this separation.
The time scale of interneuronal correlation
In this section, we determine the time scale at which
interneuronal correlation arises. We will quantify fluctuations in the neuronal response at time scales much slower and faster than the psychophysical trial and will show that the magnitude of
rnoise for our MT pairs can be accounted for by
the central peaks in their spike train CCGs on the order of 10s of
milliseconds wide.
Short-term and long-term correlation
Since the earliest attempts to estimate
rnoise in visual cortex, it has been recognized
that slow processes could play an important role in determining its
magnitude (van Kan et al., 1985; Bach and
Krüger, 1986). Changes in neuronal excitation caused by
motivational or attentional factors or fatigue could create a
correlation at a time scale of anywhere between seconds and many
minutes across a large population of neurons. On the other hand, common
synaptic input to multiple neurons that operates on a millisecond time
scale would also contribute to interneuronal correlation but across a
smaller population of neurons sharing similar tuning properties.
Because knowing the time scale of correlation may shed light on its
origin and on its effect on pooled signals, our first goal was to
determine to what extent long-term correlation was present and to
calculate the remaining short-term component of
rnoise once any long-term fluctuation of the
firing rates was factored out. Assessing the presence of slow
covariations in firing rate is also important because such covariation,
when combined with faster stimulus-locked modulation, can lead to
narrow CCG peaks that may be misinterpreted as evidence for fast
synchronization (Brody, 1998,
1999). To tackle the problem of estimating slow changes in
neuronal excitation for data collected in discrete epochs, i.e.,
trials, we developed a method called the TCC.
The TCC is a spike-count (as opposed to spike-train) -based
cross-covariance that operates on the deviations from the expected responses (instead of the actual responses) for the two neurons, given
the stimulus. Figure 3 outlines the TCC
computation for two pairs of neurons, one for which correlation was
predominantly long term, exceeding the duration of the 2 sec trial
(left column), and a second for which correlation was
predominantly short term (right column). The top
panels (A and D) show for the individual neurons
the z-score-normalized spike counts (see Materials and Methods) for trials in the order in which they occurred in the discrimination experiments. These traces estimate the levels of relative responsiveness of the neurons throughout the experiment. Beneath them, their auto-covariance functions, TAC(), are shown side-by-side (B and E; has units of
experimental trials, typically 5 sec per trial). Only the left or right
halves of the TACs are shown (the functions are symmetrical about the
origin), and the unity values at the origin are omitted. The gradual
rise to a positive value around the origin, which was typical for our
neurons, indicated that responses, or more precisely, response
deviations from the mean, on any particular trial were correlated to
those on earlier trials. For the two example pairs, the
cross-covariance functions, TCC(), for the data in the top
panels are shown at the bottom (C and
F). TCC(0) is the traditional interneuronal correlation
coefficient for spike counts, rSC (or more
generally rnoise), whereas TCC( 0)
is a generalization of rSC to responses occurring trials apart. In Figure 3C the broad central
rise in the TCC indicates that the positive correlation on simultaneous trials (TCC(0) = 0.1; indicated by the circled dot) is
related to a correlated drift in the activities of the neurons on a
time scale longer than one trial. The example in F shows an
entirely different outcome. Namely, the positive correlation
coefficient for simultaneous trials does not extend to neighboring
trials, despite the slow drifts in responsiveness of the two individual neurons evident from positive values near the origin in their TACs.
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Figure 3.
The trial cross-covariance (TCC) can
reveal the presence or absence of long-term correlation. A,
The z-score normalized spike counts for all 320 trials are
plotted in the order they occurred for two simultaneously recorded
neurons (emu080). Spikes were counted during the 2 sec
stimulus, but trials occurred on average 5 sec apart, so 100 trials
represent ~8.3 min. The dots show data points for
preferred direction, 100% coherence stimuli to demonstrate how trials
from one stimulus condition are interleaved among all others.
B, Trial auto-covariance (TAC) plots are shown
for the sequences from A, one on the left and one
on the right side of lag 0 (the TAC is necessarily
symmetrical about 0). Both have peaks of correlation around zero,
indicating that the responsiveness of these neurons was not independent
from trial to trial. The value at zero, 1 by definition, was omitted.
The smooth line was used to estimate the value that the plot approached
near the origin (arrows), referred to as
rAC or the long-term auto-correlation.
C, The TCC is the cross-correlation of the sequences in
A. TCC(0) (circled point) is the aggregate
rSC for the pair. For this pair, the value at
TCC(0), 0.1, is associated with a peak that extends over
lags of ±70 trials. The smooth line shows the TCC, with
center value replaced (see Results), convolved with a Gaussian (SD 4 trials) and was used to estimate the long-term cross-correlation,
rLT. D, Traces similar to those in
A, but for a different pair of neurons (emu090).
Data are shown for the first 400 of 1320 trials. E, TACs for
the traces in D. Both neurons show positive correlation
around zero. F, The isolated peak at TCC(0)
(circled point) for this pair indicates that correlation was
predominantly short term, i.e., not associated with drifts in the
responsiveness of the cells at time scales longer than the trial
duration.
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The TCC provides a framework for estimating long- and short-term
components of rnoise, which is represented at
TCC(0). Long-term correlation, rLT, is the value
of the TCC around, but not at, zero. We estimated
rLT by replacing the value at zero with the average of its neighbors (at lag ±1 trial), convolving with a Gaussian
of SD four trials, and reading off the new value at zero (very similar
results held for Gaussian SD two or eight trials). The traces from
which rLT was measured are shown as smooth
curves superimposed on the raw TCCs (which still have their
central values intact) in Figure 3, C and F. For
the neuronal pair in C, rLT was nearly the same
as the raw rnoise value (the circled
point is near the smooth line at lag 0), whereas in F,
rLT is close to zero and does not account for the
value of rnoise. We used the same method
(replacing the center and smoothing) to compute the long-term component
of the auto-covariance, rAC, from the two-sided,
symmetrical forms of the TACs (Fig. 3B, E, arrows mark values).
To estimate the short-term component of rnoise,
we removed the slow changes in responsiveness underlying
rLT by applying an ideal high-pass filter to the
z-scored spike counts. The filter's cutoff frequency, 0.1 trial1 (cutoff period 10 trials), was chosen to be
faster than the mean time scale of slow changes in excitability
observed in the TACs. The filtered data were subsequently renormalized
to z-scores and used to compute a TCC (denoted
TCChp) whose zero-lag value was our estimate of
short-term correlation, i.e., rST = TCChp(0). Figure 4 depicts
the TCC and TCChp (A and B,
respectively) for a pair of neurons that had substantial long- and
short-term correlation. The long-term correlation was no longer visible
in TCChp, but a narrow, central peak remained. A
simpler approach to computing rST is to subtract
rLT from rSC, i.e., from
TCC(0). However, this may yield less accurate results for many neurons
because it is not in general correct to assume that
rST and rLT are
additive.
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Figure 4.
Noise correlation can be divided into short-term
and long-term components. A, In the TCC
(points connected by line segments, for
pair emu008), the total noise correlation,
rSC, is the value at zero lag (circled
point). The long-term correlation, rLT, is
the value of the smooth line (computed according to text,
and Fig. 3C legend) at lag zero. B, For the same
data as in A, the TCChp was computed
after the z-scored spike counts were high-pass filtered.
This eliminated the long-term correlation visible in A. The
value at zero lag is taken to be the short-term correlation,
rST. C, Database averages of
correlation measures for coherence series data. Gray bar
shows long-term auto-correlation, rAC, averaged
across all individual neurons (n = 86). Long- and
short-term cross-correlation, rLT and
rST, are shown for directional pairs with
PD < 90 (black bars; n = 29) and other pairs (white bars; n = 19). Error bars show one SEM. D, The same measurements
shown in C are shown here for direction tuning data from a
larger set of pairs (n = 196 gray bars;
n = 57 black bars; n = 47 white
bars).
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Figure 4, C and D, shows database averages for
our estimates of long- and short-term correlation. Separate averages
are shown for pairs with PD < 90° (black
bars) and pairs with PD 90° or in which
one neuron was not directional (white bars). A distinction between coherence series data (C) and direction tuning data
(D) was maintained because we collected fewer total trials
(typically 80) and had more pairs for direction tuning experiments
(n = 104) than for discrimination experiments (at least
210 trials; n = 48). The database averages led to three
significant observations. First, the average long-term auto-covariance,
rAC, was positive (gray bars;
averaged across all individual cells), indicating that responses of
single cells were correlated on a time scale longer than the single
trial. For coherence series data, the mean long-term auto-covariance
was 0.14 (SD 0.12; n = 86), only 4 of 86 cells had
negative values, and the average TAC peak width at half-height was 48 trials (SD 51), corresponding to no less than 4 min. Second, however,
the long-term cross-correlation, rLT, was on
average no different from zero (t test; p = 0.39; coherence series data). For the coherence series data, the
distribution of rLT was roughly Gaussian with
mean 0.01 (SD 0.07; n = 48). Third,
rST accounted for roughly the entire magnitude
of rnoise for pairs in which both neurons were
directional and had PD < 90. For other pairs,
rST was not on average significantly different from zero, consistent with Figure 2C.
These results have the potentially counterintuitive implication that
two neurons have responses that are correlated with their own responses
on later trials and with each other's responses on simultaneous trials
but not with each other's responses on later trials. In other words,
long-term auto-correlation and short-term cross-correlation exist in
the absence of long-term cross-correlation. This situation could arise
if the sources of variance that caused the long-term auto-correlation
in the responses of the individual neurons were independent from each
other and from the source of variance that caused the short-term
cross-correlation. That long- and short-term correlation arise from
independent mechanisms would not be surprising, because they operate on
time scales separated by four orders of magnitude, i.e., several
minutes (shown above) versus 10s of milliseconds (shown in the next section).
In summary, slow drifts in the response strength of individual neurons
were present (rAC > 0) but on average
uncorrelated (rLT  0) between pairs of
neurons in MT, and therefore did not contribute significantly to the
magnitude of interneuronal correlation across our database. Thus,
rnoise was accounted for by the short-term component of correlation alone and must arise on a time scale no longer
than the behavioral trial.
Spike train auto- and cross-correlograms
The positive value of rnoise (~0.21)
associated with the cluster of points on the right side of Figure
2C did not result from long-term correlation, so we now test
for its relationship to faster sources of correlation, the presence of
which is revealed by spike train auto- and cross-correlograms.
Examining ACGs as well as CCGs is important because ACGs bear on the
interpretation of a CCG and because both are required for a
mathematical result that we will use below to derive a new metric for
rnoise. In this section, we establish,
consistent with a body of cross-correlation studies, that correlation
is largely limited in time to a small, central region of the ACGs and
CCG and show that for our MT pairs there is a strong empirical
relationship between that central region of the CCG and the traditional
measure of spike count correlation, rSC.
We computed the average spike train ACG for each individual neuron and
the average CCG for each pair of neurons as described in Materials and
Methods. Plots for one pair of neurons are shown on the left
in Figure 5, and database summaries
appear on the right. On the left, the ACGs
(A) and CCG (B) are plotted in excess of the
ensemble shift-predictor (see Materials and Methods) and are encased in
lines showing ±3 SD of the noise (estimated from the tails of the
plots for lags from 400 to 800 msec). The ACG for neuron 1 (Fig.
5A, top trace, shifted vertically for visibility) has
a dip near the origin, indicating that the likelihood of a spike
occurring within 5 msec of another is lower than expected if spikes
were fired independently of each other. This period of anti-correlation
in the ACG is followed by a period of positive correlation from 7 until
~80 msec after a spike. Periods of both correlation and
anti-correlation appeared in the ACG for neuron 2 as well (Fig.
5A, bottom trace). In addition, neuron 2 tended to
fire pairs, or bursts, of spikes; however, this is not evident in the
ACG plotted here because the positive values, at lags 2 and 3 msec, lay
above the upper vertical limit of the plot and are not shown. The
average CCG (Fig. 5B) for this pair of neurons had a
central, somewhat asymmetric peak that did not extend beyond 100 msec
from the origin.
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Figure 5.
The time scale of spike train auto- and
cross-correlation. A, ACG() is plotted for two
simultaneously recorded neurons (with ensemble shift-predictors
subtracted). The thick line is a smoothed (Gaussian
convolution, SD 2 msec) version of the raw ACG trace (thin
line). The three horizontal lines show zero and ±3 SD
of the noise (see Results). The top plot, ACG 1,
has been shifted upward for clarity. B, CCG() is plotted
for the pair with ACGs that are shown in A. Again, a
smoothed trace is superimposed, and horizontal lines
indicate zero and ±3 SD of the noise. C, The fraction of
neurons for which the ACGs were significantly (3 SDs) below (thin
line) and above (thick line) the ensemble
shift-predictor is plotted as a function of time lag (logarithmic
axes). D, The fraction of pairs for which the CCG was
significantly above (thick lines) and below (thin
lines) the ensemble shift-predictor is plotted versus time lag.
CCGs are two-sided, and results for negative time lags have been
reflected onto positive time lags (yielding two traces at each
thickness) for ease of comparison with the plot in C.
Significant correlation occurs much more frequently for time intervals
of several to tens of milliseconds than it does at or above 100 msec.
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Across our database, ACG shapes were diverse and varied in the presence
and size of (1) a narrow central peak associated with short bursts, (2)
a dip associated with a 1-3 msec absolute refractory period that was
sometimes extended by a longer relative refractory or integration
period (Abeles, 1982), and (3) a broader peak of positive correlation. The CCGs had mainly single, central peaks that
varied in size, shape, and symmetry. Peak shapes were consistent with
common synaptic input more so than with serial coupling (Moore et al., 1970). The shapes of our ACGs and CCGs were not
consistent with the oscillatory Gabor functions that Kreiter and
Singer (1996) used to describe CCGs in MT. In particular, we
did not observe rounded central peaks flanked by similar but damped
side-lobes.
We did not attempt a systematic classification of the subtleties of
correlogram shapes, which would have required more data than we were
able to collect for many of the pairs, but characterized only the
extent in time of the correlation. This was accomplished for both
correlation and anti-correlation by computing at each millisecond time
lag the fraction of cells that had correlation >3 SDs above the
ensemble shift-predictor and the fraction that had anti-correlation <3
SDs below the shift-predictor. Correlograms were smoothed with a
Gaussian of SD 2 msec before the test was applied to avoid counting
isolated points that exceeded the criterion (as observed frequently in
Fig. 5A,B). The results for the ACGs (Fig. 5C)
revealed that significant response correlation for individual neurons
was confined almost entirely to time lags <100 msec, was most
prevalent around 30 msec, and decreased at shorter times because of the
presence of anti-correlation associated with "non-burst" firing
patterns or inter-burst intervals [described by Bair et al.
(1994) for a comparable MT data set]. The extent of
correlation in the CCGs is summarized in Figure 5D and,
similar to that in the ACGs, was almost entirely confined to within 100 msec of the origin.
Two points deserve emphasis regarding these results. First, our
analysis does not preclude weaker, yet significant, correlation that
extends beyond 100 msec; it simply indicates that strong correlation,
i.e., that which caused 3 SD differences between the correlograms and
ensemble shift-predictors, was common at time scales on the order of
10s of milliseconds but was rare beyond 100 msec. Weaker, long-term
sources of correlation certainly exist in MT but are not likely to
contribute substantially to rnoise. Second, the
time scale of correlation in our ACGs and CCGs is intrinsic to the
visual system and does not result from temporal correlation in our
stimulus because the signal strength (amount of preferred motion) in
our dynamic dot stimulus was uncorrelated in time. In particular, the
number of signal dots in any epoch (or in one video frame) was
uncorrelated with that in any other epoch. The time scale of the
correlation observed in Figure 5, C and D,
matches both the integration times for visual neurons upstream from MT
(Hawken et al., 1996) and the temporal limits of motion
perception for dynamic dot stimuli (Morgan and Ward, 1980). When analysis was restricted to zero coherence stimuli (which were effectively white noise to beyond 1 kHz), we found the same
time scale of correlation across our database; therefore, the 45 msec
time between signal dots in our stimulus was not responsible for the
correlation observed here.
Having determined the typical time scale of correlation in our data, we
may now apply a simple test to assess whether
rnoise estimated in the traditional manner from
the spike count for the entire trial is related to the central peak in
the CCG. In Figure 6, the integral of
CCG() minus the ensemble shift-predictor (for = 32 to 32 msec) is plotted against rST for our database
(coherence series data). There is a clear relationship between these
two measures of correlation (overall, r = 0.76, p < 106, n = 48; for pairs with
PD < 90°, r = 0.71, p = 0.00001, n = 29, filled circles; for other pairs, r = 0.66, p = 0.002, n = 19, open circles). This may
seem striking because rST was derived from spike
counts for the entire trial without information regarding the temporal
structure of the spike trains, whereas the CCG area is based on the
interrelationship of spikes occurring within 32 msec of each other. The
significant positive correlation between the two metrics holds for
limits of integration down to ±2 msec (r = 0.48) but
does not grow much in the range from ±32 to ±128 msec (e.g.,
r = 0.80 at both ±64 and ±128 msec). The data
indicate that pairs of neurons with high spike count correlation also
tend to have a substantial peak around the origin in their CCGs. This relationship is not given a priori (van Kan et al.,
1985) and was not found in other studies of visual cortex
(Gawne and Richmond, 1993; Gawne et al.,
1996), although it was hinted at by Bach and Krüger (1986).
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Figure 6.
The area under the cross-correlogram peak is
correlated with interneuronal correlation,
rnoise. The area under CCG() from = 32 to 32 msec and in excess of the ensemble shift-predictor is
plotted against rST, an estimator of
rnoise. The correlation coefficient for all
points is 0.76 (p < 106; n = 48; coherence series data). A significant relationship also holds
for the subset of 29 pairs for which PD < 90°
( ) and for the remaining 19 pairs ( ; see Results for statistics).
The ordinate and abscissa have similar values here, but their units are
not the same. A metric that estimates rnoise by
appropriate normalization of the CCG peak area is demonstrated in
Figure 7.
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Assessing rnoise from
the cross-correlogram
We will now make a more rigorous connection between
rnoise and the area under the CCG by
defining a metric based on the CCG that estimates exactly the value of
rnoise under the condition that correlation has
a limited time scale. Our approach derives from the fact that the
equation for rSC (the well known Pearson's correlation coefficient) can be rewritten in a form that is based solely on the areas under the spike train CCG and ACGs as follows (from
Appendix A, Eq. 26):
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(9)
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where the areas are integrated across all lag times in the
correlograms. However, if correlation is limited to short time lags, as
suggested by results from the previous section, only those regions near
the origin will contribute to non-zero areas in Equation 9. The flanks
of the CCG and ACGs, which approach the shift-predictors, will
contribute on average nothing but noise. We therefore propose the use
of a metric, rCCG() (defined in Appendix A,
Eq. 27), which estimates rnoise by integrating only a limited central region (from to msec) of the CCG and ACGs. This metric eliminates the noise that would be contributed by the
flanks of the correlograms by simply not including the flanks in the
integration. In essence, it assumes that the correlograms beyond ±
are on average equal to the shift-predictors.
Before applying the rCCG() metric to our MT
data, we tested it on pairs of simulated spike trains that had a
central, Gaussian-shaped CCG peak (SD 4 msec) and an
rnoise value of exactly 0.2. For the simulated
data, all of the area in the CCG (and ACGs) was concentrated near the
center, and the expected value of the flanks (when the shift-predictor
was subtracted) was known to be zero. Figure
7A shows
rCCG() plotted for 10 sets of simulated spike
trains (details of the simulation are given in the Figure legend). As
increased, the average value of rCCG
increased until it reflected the true value, 0.2. A plateau occurred
when exceeded the time scale of the correlation, and further
increases in caused a loss of precision as noise from the ACG and
CCG flanks was integrated. When reached the full trial duration
(here 1700 msec), rCCG became equivalent to
rSC, according to Equation 28. This simulation shows vividly how noise from the tails of the ACGs and CCG corrupts rSC, and it demonstrates that a more accurate
estimate of rnoise can be obtained with
rCCG() when is shorter than the trial duration (but longer than the time scale of correlation).
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Figure 7.
Computing rCCG()
for simulated and neuronal data. A, rCCG() is
plotted against integration time for 10 blocks of simulated pairs
of spike trains in which the time scale and strength of correlation
were known. (For each trial, 2 simulated spike trains were generated by
selecting spike times independently and at random with probability 0.2 per spike from a Poisson spike train having mean rate 200 spikes per
second. The rate is arbitrary and does not affect the results. The
spike times in one of the resulting trains were jittered by adding a
Gaussian random variable with mean zero and SD 4 msec. The resulting
pairs of spike trains have rnoise = 0.2 and
a time scale of correlation matching the Gaussian jitter.) For the
simulated data, rCCG() approached the true
value of rnoise once exceeded the time scale
of correlation but became corrupted by noise as increased.
Arrows indicate where rCCG(32) and
rSC are read out. Here,
rCCG(32) = 0.200 SD 0.009, whereas
rSC = 0.197 SD 0.037 (n = 20). The ratio of SDs was 1:4. B, Results from an
analysis similar to A but for 11 coherence levels for
neuronal data (pair emu005). As in A, the curves
rose from zero as increased but became highly corrupted by noise as
exceeded 100 msec. For this pair, the width at one-half height of
the CCG (data not shown) was 9 msec. C, The mean value of
rCCG() across 29 pairs (averaged across all
coherence levels for each pair with PD < 90°) is
plotted as a function of in 1 msec steps (filled
circles mark octave steps). Vertical bar indicates ±1
SE at = 64 msec. The SD of
rCCG() is plotted at octave steps only
(open circles connected by straight lines). The
value of rCCG reached an asymptote of ~0.21
for around 30-100 msec, but the SD continued to increase with .
D, rCCG(32) (thick lines,
filled circles) and rSC (thin
lines, open circles) for 48 pairs of neurons (coherence
series data). Arrows indicate points corresponding to
example pairs from B (emu005) and from Figure 3C
(emu080). Points are sorted by increasing value
of rCCG. Vertical lines show ±1 SD
(computed across coherence levels) and were always larger for
rSC.
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We plotted rCCG() for our neuronal pairs and
found a similar pattern of results. Curves for one pair are shown in
Figure 7B for 11 coherence levels (from 100% preferred to
25.6% null direction motion, which satisfied the minimum data
requirements stated in Materials and Methods). The curves increased
together to r 0.16 as approached 30-40
msec but then diverged as grew larger. This was consistent with the
CCGs (data not shown), which had central peaks that fell to the level
of the shift-predictor at ~30-40 msec from the origin. The direction
of divergence of curves such as those in Figure 7B typically
did not depend on the stimulus condition (a systematic analysis is
given in the next section), so we averaged across conditions to get an
rCCG() curve for each pair, and we averaged
across pairs to get one database curve. The database curve for pairs
having PD < 90° (Fig. 7C, filled
circles) approached an asymptote of ~0.21 for values of above 32-64 msec. The value 0.21 was the same as that for short-term correlation for this database (Fig. 4C, right-hand
bar), and the timing of the approach to the asymptote was
consistent with the time scale of correlation observed in the ACGs and
CCGs (Fig. 5C,D). Figure 7C also shows the SD for
the rCCG estimate (open circles,
averaged across the same set of curves used to compute the mean). The
SD grew with increasing even after the mean of rCCG() had leveled off. This shows the
inefficiency of a long integration time such as that associated with
the rSC metric (i.e., the entire trial
duration). Finally, a direct comparison of rSC with rCCG(32) for individual pairs is provided
in Figure 7D. The SD was always smaller for
rCCG (thick lines) than for
rSC (thin lines). Two points are
labeled, one for the pair from B (emu005) and another
(emu080) from Figure 3C, that had a large
long-term component of rSC. For the latter,
rCCG(32) is much less than
rSC because rCCG(32)
discounts long-term correlation. It does so by integrating area over
only 1.9% (32 msec/1700 msec) of the CCG and therefore captures only
1.9% of the excess area that a source of long-term correlation spreads
evenly across a CCG.
Clearly, rCCG provided a more repeatable (less
noisy) estimate of interneuronal correlation (for < T) than did rSC, but we wanted to verify
that it also maintained the relationships that
rSC had with the measures for similarity of
neuronal tuning mentioned above, namely, rsignal
and PD. Compared with rSC,
rCCG() was more positively correlated with
rsignal (Pearson's r = 0.59, rather than 0.53, for both = 32 and 64 msec; n = 46) and was more negatively correlated with the logarithm of
PD (Pearson's r = 0.47, rather than
0.36, for both = 32 and 64 msec; n = 34,
where the logarithm was taken to correct the skew of the distribution in Fig. 2B).
In summary, it appears that rCCG() accurately
captures the amount of interneuronal correlation for our pairs. That it
does so for as small as 32 msec shows that most of the correlation observed at the time scale of the behavioral epoch can be accounted for
by CCG peaks at a time scale nearly two orders of magnitude shorter.
Therefore, mechanisms underlying narrow, central CCG peaks affect
response properties relevant to both temporal and rate coding.
Dependence of correlation on stimulus and behavior
Assessing the dependence of correlation on stimulus parameters is
necessary to justify averaging rnoise values and
CCGs across stimulus conditions as we have done. In addition, this
assessment is important with respect to both stimulus and behavioral
parameters because of the potential link between correlation, or
synchrony, and the perception of the animal as reflected by its
behavior. Here we examine how correlation changes with the firing rates of the neurons, the direction and coherence of stimulus motion, and the
presence of the stimulus, and we test whether synchronous activity
exerts extra influence on the monkey's decision and whether it varies
from passive fixation to active discrimination.
Correlation versus firing rate, direction, and
motion coherence
Because firing rate varied as our stimulus parameters changed, we
first established that our correlation metrics did not show a
substantial dependence on firing rate before testing for more interesting relationships between interneuronal correlation and other
variables. Figure 8, A and
C, shows scatter plots of the area under the CCG peak (from
32 to 32 msec) and rCCG(32) versus geometric
mean spike rate for each coherence level for the 29 directional pairs
with PD < 90°. Firing rate was not significantly correlated with CCG area and showed only a weak relationship with rCCG (see Figure legend for details). A
pair-by-pair analysis also revealed no overall trend, although several
individual pairs showed significant relationships (see Fig. 8 legend).
Similar results held for data from the direction tuning experiments,
for integration times ranging from several to hundreds of milliseconds, and when all pairs were included in the analysis.
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Figure 8.
CCG peak area and
rnoise plotted as a function of geometric mean
spike rate (GMSR) and motion coherence. A, The area of the
CCG (minus the ensemble shift-predictor) integrated from = 32 to 32 msec is plotted as a function of GMSR for pairs of
directional neurons with PD < 90°. Each point
represents data for one coherence level from 1 of 29 qualified pairs.
Pearson's correlation coefficient for this scatter was not
significantly different from zero (r = 0.02; p = 0.65; n = 329). We chose our CCG normalization (Eq. 6) to
realize this empirical observation. For individual pairs, 3 of 15 negative relationships and 6 of 14 positive relationships were
significant (p < 0.05). B, Mean CCG area (±32 msec)
averaged over the same 29 pairs as in A is plotted for
preferred (thick line) and null (thin line)
motion as a function of coherence (there are typically 24-29 pairs per
point because some pairs were not tested at all coherence levels).
Vertical bars show ±1 SEM. Values remained relatively
constant except for a 43% reduction at 100% coherence compared with
the average across all lower coherence stimuli (preferred and null
directions combined). C, A measure of
rnoise, here rCCG(32), is
plotted against GMSR for the same set of pairs. There is a small
overall positive correlation with mean spike rate (r = 0.14;
p = 0.01; n = 329). For individual pairs, 1 of 12 negative and 6 of 17 positive relationships were significant.
D, Like CCG area, the rnoise measure
varies little with motion direction and coherence except at 100%
coherence where it dropped by 32% relative to the average across all
lower coherence levels (preferred and null directions combined). As in
B, typically 24-29 pairs contributed data to each
point.
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The same two correlation metrics were largely constant when plotted
against stimulus direction and coherence, except at 100% coherence
where both measures were lower (B and D show CCG
area and rCCG, respectively, Fig. 8). The
numbers of individual pairs for which these metrics were significantly
correlated with coherence were almost identical to those for spike
rate. The drop in correlation strength at 100% coherence can be
related to the nature of MT responses to coherent and incoherent
motion. MT neurons typically show clear stimulus-locked modulation for
stimuli of <100% coherence, but at 100% coherence there is little or
no such modulation (Bair and Koch, 1996). How this
modulation impacts our measures of correlation is the subject of the
last section of Results. Whether the reduction in
rnoise at 100% coherence is also related to a
previous report that correlation is almost completely abolished during
high contrast motion in MT (Cardoso de Oliveira et al.,
1997) is discussed in the next section.
The consistency of rnoise in the face of large
changes in firing rate indicates that the underlying mechanism did not
act additively to alter neuronal firing rates, for if it did,
rnoise would be larger at lower firing rates. In
the absence of substantial overall relationships between our
correlation metrics and the stimulus direction and coherence or the
firing rate, we chose to average these metrics across all stimulus
conditions. The observed decrease at 100% coherence had little
influence on our statistics because <10% of our coherence series data
was collected at c = 100%.
Correlation during spontaneous and stimulus-driven activity
We tested the dependence of correlation on the presence of the
stimulus by computing rnoise for a 330 msec
epoch of spontaneous activity and for an equal duration epoch of
stimulus-driven activity. The spontaneous epoch began when the monkey
acquired fixation and ended 30 msec after stimulus onset, precluding
the arrival of stimulus-driven activity in MT (Raiguel et al.,
1999). The driven epoch began 30 msec after stimulus onset. We
limited analysis to pairs that had at least four stimulus conditions
each having at least 10 trials with at least one spike per trial per
cell during the 330 msec period. The value of
rCCG(32) for the spontaneous epoch was
significantly correlated with that for the driven epoch (r = 0.63; p = 0.00001; n = 40), and the average difference
between the values for spontaneous and driven activity, 0.018 (SD
0.14), was not significantly different from zero. Limiting the analysis to directional pairs with PD < 90° gave nearly
identical results. Similar results were found when (1)
rST, computed from the TCC, was substituted for
rCCG(32), or (2) the driven epoch was defined to
be the entire stimulus epoch, rather than just the first 330 msec. We
conclude that noise correlation during spontaneous activity is similar
to and predictive of the noise correlation during activity evoked by
our random dot stimuli.
This result stands in striking contrast to the report of Cardoso
de Oliveira et al. (1997) that interneuronal correlation in MT
is present during spontaneous activity but is practically abolished
during visual stimulation. To determine whether our correlation values
were more similar to their values for spontaneous or for driven
activity, we normalized our CCGs according to their methods (after
dividing by the geometric mean spike rate, we used a three-point boxcar
function to smooth the CCGs and then found the peak within ±100 msec
of zero) and computed peak height, position, and width statistics like
those presented in their Figure 5. All three measures from our data
were well matched to their results for spontaneous activity, indicating
that our results differ only during visual stimulation. If we assume
that high-contrast, coherently moving stimuli reduce correlation
strength between responses of nearby MT neurons, then it remains to be
determined why our strongest stimulus (100% coherence motion) caused a
decrease in correlation strength that was small compared with the
decrease caused by the square-wave grating of Cardoso de
Oliveira et al. (1997).
Does correlation change with behavior?
Investigators have hypothesized that synchronous firing among
cortical neurons underlies various coding or processing functions (for
review, see Singer and Gray, 1995; Roelfsema,
1998). Our data provide the opportunity to determine whether
synchrony among adjacent MT neurons is correlated with perceptual
choice or behavioral state.
The relation of synchrony to perceptual choice is best assessed at low
motion coherence where the monkey correctly identifies the direction of
motion on some trials but makes mistakes on others. The psychometric
function in Figure 9A
(thick line, filled circles) plots the monkey's
performance on trials for which the stimulus was optimized for the pair
of neurons the tuning curves of which are shown in Figure 1. We asked
whether synchrony was stronger on trials in which the animal chose the
direction preferred by the pair of neurons, as might be expected if
synchronously active neurons exert stronger effects on downstream
decision circuitry. To test this, we divided the trials for each
stimulus condition (i.e., for a particular coherence level and
direction) into two groups, one in which the animal chose the preferred
direction (for the pair) and one in which the animal chose the null
direction. Note that one group corresponds to correct decisions,
whereas the other corresponds to incorrect decisions (where the
correspondence depends on whether the direction of motion was null or
preferred for the stimulus condition) except at zero coherence where
there was no "correct" response.
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Figure 9.
Psychophysical and neuronal performance.
A, The monkey's performance (filled
circles) and the neuronal performance for neuron 1 (×'s) and neuron 2 (squares) are plotted as a
function of motion coherence (logarithmic axis) for the same pair of
neurons as in Figure 1. The lines show fits to Equation 2.
The thick line is for the monkey's psychophysical
responses. B, An example of average CCGs for preferred
(thick lines) and null (thin lines) direction
decisions. CCGs for coherence levels of ±6.4, ±3.2, and 0% were
averaged together; other coherence levels did not have a sufficient
number of choices in each direction. C, The area under the
CCG between 32 and 32 msec and in excess of the shift-predictor for
null decisions is plotted versus that for preferred decisions. Each
point (n = 137) shows data for a particular coherence
level and direction, so there are multiple points per neuronal pair
(n = 35). D, For the same data set as in C,
the comparison of CCG area is made for a narrower integration region,
from 2 to 2 msec. Results in B-D reflect a lack of
correlation between perceived direction of motion and the magnitude of
synchronous activity in the population of neurons that prefer the
perceived direction.
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We considered only stimulus conditions that had at least 10 trials with
preferred responses and 10 trials with null responses; therefore, 51.2 and 100% coherence conditions were rarely included because the monkey
rarely made 10 mistakes for such salient stimuli. This limited the
number of pairs for this analysis from 46 to 35. Figure 9B
shows CCGs for preferred and null decision trials for the same pair of
neurons illustrated in Figures 1 and 9A. The CCGs appear
virtually identical, which was typical for our data set. Figure 9,
C and D, depicts quantitative measurements of the
area under the CCG from 32 to +32 msec (C) and from 2 to
+2 msec (D) for preferred and null decision trials for 137 stimulus conditions from the 35 pairs of neurons. In both panels, the
points cluster around the unity diagonals, showing that synchronous firing did not differ between the two decision states (paired t test; p = 0.75 for C, p = 0.94 for D). This result also held for the
rCCG metric, for all integration times tested
(from ±2 to ±128 msec), and when only directional pairs were tested.
We also analyzed synchrony simply as a function of motion coherence,
regardless of perceptual choice. At low coherence, the dot patterns
appear to be a white noise stimulus and elicit no global motion
percept. As coherence increases, however, observers perceive the entire
stimulus to drift in the specified direction as though the disparate
motion signals provided by individual dot pairs are bound into a
perceptually coherent whole. Theories of perceptual binding that
postulate a unique role for synchronous neural activity might predict
that synchrony should be stronger for coherent (c = 100%) than for incoherent dot patterns (c = 0%).
However, we have already seen that the opposite is true (Fig. 8B,D).
Finally, we compared CCGs obtained during passive fixation (direction
tuning experiments) with those obtained during active discrimination to
determine whether the overall behavioral state of the animal was
correlated with neural synchrony. In the subset of experiments in which
both blocks of data were obtained, the area under the CCG did not
differ systematically between the two states (paired t test;
t = 0.06; p = 0.95; n = 46), and the
measurements were highly correlated between the two states
(r = 0.90; p < 106). In
short, we found no evidence that synchronous firing varied systematically as a function of perceptual decision or behavioral state.
Do sensitive or informative neurons cluster?
For each experiment in which we obtained psychophysical data, we
used analytic methods based on signal detection theory [see Materials
and Methods, or see Britten et al. (1992) for detailed methods] to compare the directional sensitivity of each neuron with
the monkey's psychophysical sensitivity. Figure 9A
illustrates the outcome of this analysis for the data depicted earlier
in Figure 1, C and D. The filled
circles represent the psychophysical performance of the monkey on
the direction discrimination task, which increased from nearly chance
at low coherence levels to perfection at the three highest levels.
Psychophysical threshold, defined as the motion coherence that
supported 82% correct performance, was 4.3% coherence. The
×'s and squares indicate the performance of the
two MT neurons measured on the same trials represented in the
psychometric curve. Neuron 2 was as sensitive to the directional signals as was the monkey psychophysically, yielding a neurometric threshold of 4.7% coherence. Close correspondence between neuronal and
psychophysical thresholds is common in MT (Britten et al., 1992). In contrast, neuron 1 was considerably less sensitive to motion signals in the displays, yielding a threshold of 13.8% coherence (sensitivity = 1/threshold). Across 72 directional
neurons studied in 41 discrimination experiments, the geometric mean
ratio of neuronal to psychophysical threshold was 1.72 (range,
0.27-11.5), a value higher than those previously observed in this
laboratory (Newsome et al., 1989; Britten et al.,
1992; Celebrini and Newsome, 1995). The
discrepancy arises because the inclusion criterion for direction
selectivity was less stern in the current study to maximize the number
of pairs. Interestingly, neither neuronal thresholds nor choice
probabilities [defined in Britten et al. (1996)] were
significantly correlated between adjacent MT neurons in our sample.
Thus we find no evidence for clustering of neurons that are
particularly sensitive to the stimulus or that have particularly close
relationships to behavior.
Controls for stimulus variance-replicate stimuli
Our estimates of interneuronal correlation have been based on
responses to ensembles of stochastic stimuli in which the random detail
of the dot patterns differed from repeat to repeat within a particular
stimulus category. In principle, such sets of nonidentical stimuli
could inflate rnoise estimates and increase CCG
peak sizes if the responses of the neurons were influenced by the
random variation across stimuli. For example, if 15 of 30 stimuli that were generated at 6.4% coherence had by chance slightly more motion in
the preferred direction than the other 15 stimuli, an ideal pair of
neurons with no common noise source but having identical direction
preferences would tend to fire on average more for the former than for
the latter 15 stimuli. This would yield an erroneous positive value of
rnoise, which should otherwise be zero. Below we
describe direct experimental controls as well as simulations that allow
us to estimate the magnitude of this effect in our data.
In our experiments, random variation from stimulus to stimulus was
necessary to prevent the monkeys from associating particular spatial
patterns with a reward. However, for four pairs of neurons we
interleaved experiments using replicate stimuli in which the dot
patterns for a particular stimulus condition were identical (see
Materials and Methods). Estimates of rnoise for
the four controls using both the rSC and
rCCG(32) metrics are presented in Figure
10. The values of
rSC (A) offered no evidence that
interneuronal correlation was greater for stochastic stimuli
(white bars) than for replicate stimuli (black
bars), but the lower-variance estimates provided by
rCCG(32) (B) painted a clearer
picture. For pairs emu034 and emu035, rCCG(32)
was higher for stochastic stimuli than for replicate stimuli
(p = 0.08 and p = 0.00001 respectively, t tests). For the other two control pairs, the difference
was negligible.
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Figure 10.
A comparison of interneuronal correlation
strength for stochastic versus replicate stimuli. A,
rSC is plotted for four pairs tested with both
ensemble (white bars) and replicate (black bars)
stimuli. No significant change was observed for the first three pairs.
For the fourth pair, the responses to replicate stimuli had a large
long-term component of correlation because of drifts in firing rate
during the experiment (rLT = 0.27, rST = 0.02). Error bars show SE. B,
rCCG(32) is shown for the same four pairs. In
the first three cases, rCCG(32) was less for
replicate stimuli. The difference was significant for emu035
(see Results). Lower values for replicate stimuli (black
bars) were consistent with the reduction in the CCG peak that
occurred when the true shift-predictor (Fig. 11C) was
subtracted. Cases emu034 and emu035 were based on
coherence series data; rt068 and rt072 were based
on direction tuning data (c = 100%). For
rt072, the low rnoise value was
consistent with PD > 90°.
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An examination of the PSTHs, CCGs, and shift-predictors for emu035 (the
pair that had a significant decline in rCCG for
replicate stimuli) reveals how stimulus-locked modulation can inflate
rnoise. For replicate stimuli, the
stimulus-locked modulation of firing rate is captured in the PSTHs
(Fig. 11A,B, thin
lines, neurons 1 and 2, respectively), but when the stimulus
varies from one repeat to the next (ensemble stimuli), the modulation
is washed out (A, B, thick lines). The difference in
the PSTHs carries over into the CCG shift-predictors because
shift-predictors are closely related to the cross-correlation of the
PSTHs [see Eq. 8 in Materials and Methods, Eq. 15 in Appendix A, and
Perkel et al. (1967b)]. The ensemble shift-predictor is
flat (Fig. 11C, thick line), whereas the
shift-predictor for replicate stimuli has a peak (line with dots). The peak indicates that the stimulus-locked
modulation in neuron 1 and 2 PSTHs (A, B, thin lines)
was correlated. The difference in area between the CCG (C,
thin line) and the two shift-predictors accounts for the
difference in rCCG plotted in Figure
10B for this pair (emu035). In summary, an
ensemble shift-predictor fails to capture correlated stimulus-locked
modulation, so subtracting it from the raw CCG yields an overestimate
of the correlation if correlated stimulus-locked modulation
existed in the first place. Thus, when there is little stimulus-locked
modulation (as was the case for rt068 and rt072
in Fig. 10B) or when modulation is present but largely
uncorrelated (e.g., emu034), using an ensemble shift-predictor is
acceptable. But for emu035, it caused an overestimate of the CCG peak
area and of rnoise.
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Figure 11.
Comparing PSTHs and shift-predictors for
responses to ensemble and replicate stimuli for pair emu035.
A, PSTHs (bin size 20 msec) for neuron 1 averaged across 60 trials of different 0% coherence stimuli (thick line,
ensemble stimuli) and averaged across 30 trials of one particular 0%
coherence stimulus (thin lines, replicate stimuli, broken
into 2 groups of 15 trials to demonstrate that the modulation is
reproducible). B, Similar to A, but for the
simultaneously recorded responses of neuron 2. C, The raw
CCG (without the shift-predictor subtracted; thin line) is
plotted for comparison against the ensemble shift-predictor
(thick line) and the actual shift-predictor
(points) computed from responses to replicate
stimuli. Plots show averages across 13 stimulus conditions ranging from
0 to 51.2% coherence, preferred and null directions. The
shift-predictor accounts for roughly half of the area of the CCG peak.
It is worth noting that the raw CCG for ensemble stimuli does not
differ on average from that for replicate stimuli because both result
from cross-correlation of simultaneous responses to stimuli with the
same underlying statistics; therefore, only one trace marked
CCG is shown here. Of course, any particular CCG from
repeats of one replicate stimulus will deviate from the average
ensemble CCG, but if raw CCGs from many different replicate stimulus
sets are averaged together, they will approach the raw ensemble CCG.
This is not true for shift-predictors, as seen here, because they are
based on responses from non-simultaneous trials.
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One method for estimating the inflation of
rnoise caused by stochastic stimuli across our
database is to compare results for 0 and 100% coherence stimuli. Such
a comparison is useful because there is little or no stimulus-locked
response modulation for c = 100% stimuli, whereas
modulation is strong at c = 0% (Bair and Koch,
1996). For the 20 pairs that we tested at both c = 0 and 100% and that consisted of two directional neurons with
PD < 90°, rCCG was on
average 0.20 (SD 0.13) at 0% coherence and 0.17 (SD 0.13) at 0%. This
15% decrease is consistent with our hypothesis but was not
statistically significant (paired t test; t = 1.45; p = 0.16). A similar, but unpaired, comparison can be made from the plot of rCCG in Figure
8D, which shows a 27% reduction from c = 0
to c = 100% (preferred direction only). Again, this change was not statistically significant (t test;
t = 1.48; p = 0.16). A broader unpaired comparison
between all data from discrimination experiments (the vast majority of
which was collected at low coherence) and all data from direction
tuning experiments (where c = 100%) for pairs with
PD < 90° showed only an 8% decline in
rCCG(32) for the direction tuning data set.
These results suggest that inflation of rnoise
caused by stochastic stimuli is modest across our database.
Finally, we used a simulation to estimate the inflation of
rnoise caused by the modulated drive resulting
from stochastic stimuli. The stimulus drive consisted of randomly
occurring bursts of stimulation that simulated those caused by the
random occurrence of coherent dots in our motion stimulus. Parameters
for the strength and frequency of occurrence of the random bursts
determined the amount of trial-to-trial variability and thus the value
of rnoise. The details of the simulation and a
solution for rnoise for all parameter values are
given in Appendix B. An example of the drive provided by a simulated
stimulus during a 1 sec epoch from one trial is shown in Figure
12A. The trace
represents the PSTH for both neurons, which are defined to be
identical. For a set of trials governed by the same statistics that
generated the trace in A (see legend for parameters), the
expected value of rnoise is 0.04. For a
simulation with stronger modulation (B), the expected value
of rnoise is higher, 0.24. Figure 12D
plots the value of rnoise for a wide range of
parameter combinations and shows (with white dots) the
parameters used to generate traces for the examples just described. A
comparison of the PSTH for a simulated pair of neurons (B)
with the measured PSTHs (C) for the pair of neurons from
Figure 11 reveals a critical difference: the neuronal PSTHs are not
identical. This was true although this pair of neurons was as closely
matched in preferred direction and bandwidth of direction tuning as any
in our database (PD = 9°;
rsignal = 0.97). Because nearby neurons
have responses that differ in fine detail (DeAngelis et al.,
1999), our simulation provides an upper bound on the strength
of correlation induced by stochastic stimuli. Furthermore, gauged by
responses to replicate stimuli here and in a previous study of MT
(Bair and Koch, 1996), the strength of modulation in
Figure 12A appears typical or above average, whereas that in
B represents an upper limit to what has been observed.
Therefore, our simulations suggest that stochastic stimuli are not
likely to inflate rnoise by more than ~0.04
units on average.
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Figure 12.
Modeling the interneuronal correlation caused by
stimulus variation for a pair of identical neurons. A,
Simulated time-varying instantaneous mean firing rate for p = 0.5, max = 100 spikes/sec,
min = 5 spikes/sec (smoothed with Gaussian SD 4 msec to achieve a realistic temporal resolution). The single
line represents the PSTHs for two identical neurons in an ideal
pair. See Appendix B for a description of the model. B,
Similar to A, but p = 0.1,
max = 400 spikes/sec. C, PSTHs for pair
emu035 for comparison to the simulations. The PSTHs for
neuron 1 (thin line) and neuron 2 (thick line)
represent a segment of the same data shown by the pairs of thin
lines in Figure 11, A and B, but are
smoothed like the simulation traces in A and B
here. D, For our model, rSC is
plotted as a function of all values of p and
max (see Eq. 39). Points A and
B mark parameters used to generate traces in A
and B. Black shading indicates low correlation,
white indicates high correlation. Values are given for the
contour lines.
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|
In summary, stochastic stimuli probably inflate our estimates of
rnoise but cannot be responsible for more than a
small fraction of the correlation that we measured. Experimental
controls, simulations, and comparisons of incoherent to coherent
stimuli suggest that this inflation is likely to range from negligible
to at most 20% of our average rnoise estimates.
Response variance caused by eye movements
One final potential source of error in our estimate of
rnoise is the movement of the monkey's eyes.
Small saccades executed during fixation could cause correlated signals
in neurons with similar direction preferences. The potential strength
of this effect has been estimated from the influence of eye movements on single-unit MT data (Bair and O'Keefe, 1998), and it
was concluded that fixational saccades are too brief and typically too
infrequent to create substantial correlation except when occurring on a
background of very low firing rate. We found no indication that
rnoise was higher at lower firing rates (Fig.
8C) and believe that eye movements did not substantially
affect estimates of correlation strength in this study.
| |
DISCUSSION |
Summary
We have investigated the time scale at which interneuronal
correlation arises for pairs of nearby cortical neurons and have explored the relationship between interneuronal correlation and behavioral and stimulus parameters in area MT.
We found that synchrony, revealed by CCG peaks, was closely linked to
correlated variability, rnoise, at the time
scale of the trial. In principle, these two phenomena need not be
related (van Kan et al., 1985), but several observations
showed that they were related for our MT pairs. First, the predominant
time scale of interneuronal correlation was on the order of 10-100
msec, consistent with numerous cross-correlation studies throughout the
visual system of both cat and monkey (Mastronarde,
1983b; Michalski et al., 1983; Ts'o et
al., 1986; Krüger and Aiple, 1988;
Nelson et al., 1992; Cardoso de Oliveira et al.,
1997) and in auditory cortex (Dickson and Gerstein,
1974; Abeles, 1982; Eggermont and Smith,
1996). Next, CCG peaks at this time scale (10-100 msec) were
strikingly predictive of rnoise for the
behavioral epoch. Although rnoise is
mathematically related to the total area under the CCG, such a result
need not apply to the central CCG region alone. For example, pairs
could have had central CCG peaks that were canceled by negative
side-lobes, or they could have had excess area distributed across the
entire CCG. Neither of these are consistent with our findings. Finally,
slow drifts in the gain of single neuronal responses occurred but were
not on average correlated between neurons and therefore had little impact on rnoise. This result was somewhat
surprising because it has been suggested that long-term
cross-correlation is common for neurons in primary visual cortex
(Bach and Krüger, 1986). Also, because nearby
cortical neurons share a large fraction of their inputs, it is unclear
how one cell can undergo gain changes that are independent from those
of its neighbors. However, if mechanical instability of the electrode
in the tissue was the source of the long-term gain changes, it is
conceivable that nearby neurons could be affected independently.
In the second part of this study, we found that synchronous activation
in pairs of neurons was not related to the monkey's decision on the
direction discrimination task and that synchrony was not stronger for
perceptually more salient or unified stimuli. Synchrony did not depend
on whether the monkey was actively discriminating or passively fixating
during stimulus presentation. Finally, the strength of synchrony was
similar with and without the stimulus, and it showed little systematic
variation with firing rate. We are unable to corroborate reports that
synchrony in MT changes with the unity of the stimulus (Kreiter
and Singer, 1996; Castelo-Branco et al., 2000)
or is nearly abolished during visual stimulation (Cardoso de
Oliveira et al., 1997). Experiments using more diverse stimulus
configurations will have to resolve these differences. Other studies
have suggested that synchrony could signal behavioral events in frontal
cortex (Vaadia et al., 1995), encode tone frequency in
auditory cortex (deCharms and Merzenich, 1996), indicate
attentional selection in somatosensory cortex (Steinmetz et al.,
2000), or be involved in arousal, attention, or learning in
sensorimotor cortex (Murthy and Fetz, 1996). In
contrast, our results portray synchrony and correlation as relatively
constant for a typical pair of MT neurons.
In the course of this analysis, we derived two metrics that are useful
for determining the strength and time scale of correlation. The TCC
provides a systematic way to extract short- and long-term components of
the traditional interneuronal correlation coefficient, rSC, for trial-based data, whereas
rCCG() offers an estimate of
rnoise with lower variance than
rSC when the time scale of correlation is
shorter than the period during which spikes are counted. We believe
that these techniques are potentially useful for comparing correlation
across a wide range of data.
Other studies of rsignal,
rnoise, and the CCG
Previous studies of visual cortex have examined
rnoise, rsignal, and
spike train CCGs (Gawne and Richmond, 1993; Gawne
et al., 1996). They reported r2
values, interpreted as percentage of explained variance, so we squared
our rnoise and rsignal
values (before averaging) for comparison. Their value of
rnoise2, ~5% for both inferotemporal
cortex (IT) and primary visual cortex (V1), was similar to our values:
4.5% for all pairs and 6.4% for directional pairs with
PD < 90°. They found
rsignal2 to be 19% in IT and V1 using
static, spatial (Walsh) patterns, but this increased to 40% in V1 for
conventional bar stimuli. The latter value was comparable to our mean,
48%, for MT. In spite of some similarity between our results,
including the fact that over half of their CCGs had significant peaks,
they did not comment on the relationship between
rnoise and the CCG and concluded that the
rsignal and the CCG were unrelated (they found
rsignal to be lower for pairs with CCG peaks in
IT, but the result failed a significance test). This outcome is
different from that depicted in our Figure 2C, which shows a
clear relationship between rsignal and
rnoise, where rnoise,
being rCCG, is a strong reflection of the CCG
peak. It seems likely that a relationship like this must exist between
rsignal and the CCG in both IT and V1 because
one consistent feature of CCGs from diverse regions of cortex is that peaks are more common between nearby neurons, particularly within distances associated with cortical columns (Fetz et al.,
1991). Cortical columns are clusters of neurons with similar
preferences, and such similarity is what
rsignal, in principle, measures. Maybe differences in the number of cells tested or in the method of estimating the strength of CCG peaks or rsignal
led to the differences between our results and those of Gawne and
collaborators (Gawne and Richmond, 1993; Gawne et
al., 1996). For example, the relationship between
two-dimensional Walsh patterns and the columnar structure in IT
(Fujita et al., 1992; Tanaka, 1996) may
be somehow fundamentally different than that between moving patterns
and direction columns in MT (Albright et al., 1984).
Consistent with our findings, Bach and Krüger
(1986) noted that excess area in the CCG (±30 msec) was
slightly larger for pairs of V1 neurons with strong common variability
(i.e., rnoise). Also, for both motor and
parietal cortex, Lee et al. (1998) found that
rsignal and rnoise were
higher for pairs with significant central CCG peaks. All of these
results are consistent with the simple notion that sources of common
input arrive onto nearby neurons through one or more synapses and
thereby create common noise, central peaks in CCGs, and similar tuning
curves in pairs of neurons (Shadlen and Newsome,
1998).
Stimulus variance
A major goal of the study from which the present paired MT data
arose (Zohary et al., 1994) was to estimate accurately
the strength of noise correlation for nearby MT neurons but to do so
when those neurons were generating signals that would underlie a
psychophysical judgment made by the monkey. The latter constraint led
to the use of stochastic stimuli to prevent the monkeys from associating particular stimulus patterns with a reward. In principle, however, stochastic stimuli can bias estimates of
rnoise upward, as demonstrated by our
simulations. We attempted to estimate this bias by comparing responses
for replicate and ensemble stimuli, by comparing c = 0% with c = 100% data, and by simulating the effect
of stochastic stimuli on neuronal responses. The results suggested that
the actual rnoise value for pairs with similar direction tuning was somewhat less than the measured value of 0.21, but
probably not by >20%.
Implications for pooling
Interneuronal correlation places limits on the effectiveness of
signal pooling (Johnson et al., 1973; for review, see
Parker and Newsome, 1998). Our previous studies showed
that the signal-to-noise ratio (SNR) for a pooled signal was sensitive
to even modest values of rnoise (Zohary
et al., 1994; Shadlen et al., 1996). We can now
use our estimates of the time scale of interneuronal correlation to
understand how rnoise and SNR change with the
length of the time window, T, in which signals are pooled.
We simulated pools of spike trains with correlation on the time scale
typical for MT (see Fig. 7A legend for methods) and computed
the SNR as in Zohary et al. (1994). The SNR for the
pooled signal is the expected value, µ , of the sum of
spikes from all neurons divided by the SD, , of that
sum, i.e.:
|
(10)
|
where µ and are the mean and SD for spike count from a
single neuron, and N is the number of neurons in the pool.
Our simulated data were Poisson, so 2 = µ and
doubling T would increase the SNR by a factor of  if rnoise remained constant, but because
correlation was spread over time (Fig.
13A, thick line),
rnoise was lower for shorter T (B,
thick line). Thus the SNR (Eq. 10) was enhanced for larger pools of neurons at shorter integration times, as shown in C
(thick curves are squeezed upward in the bottom right
corner; see legend for details).
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Figure 13.
Changes in rnoise and SNR
as a function of pooling time, T. A, CCGs of simulated spike
trains are plotted for two hypothetical neuronal populations, one with
a realistic time scale of pair-wise correlation (thick line,
approximates a Gaussian of SD 8 msec) and one with
instantaneous correlation (thin line, peak at zero is
truncated). Simulation method is described in the legend for Figure
7A. B, For pairs from the two hypothetical populations,
rnoise was computed as a function of
T, the period in which spikes would be counted to form a
population response. For instantaneous correlation,
rnoise was constant (here 0.2) for all
integration times (thin line). However, for broad
correlation (thick line), rnoise was
near zero for short T and increased to the veridical value
as T became large relative to the time scale of correlation.
Results for the simulated broad correlation were comparable to those
for our neuronal data (open circles;
rnoise averaged across 29 neuronal pairs that
were directional and had PD < 90°, coherence
series data). The negative value at 1 msec for the neuronal data
results from limitations in recording two nearly simultaneous action
potentials using one electrode. Error bars for the model show SD across
10 blocks of 200 trials (mean firing rate 40 spikes per second). Error
bars are smaller for smaller window sizes because, for example, there
are 1000 T = 1 msec windows for each T = 1000 msec window. C, Pooled signals (sums of spike
counts) from the hypothetical populations were compared in terms of
their SNR (Eq. 10) as a function of neuronal pool size for time windows
of various duration. For instantaneous correlation, the SNR curves
(thin lines) had the same shape for all T but
were scaled by when T doubled. For the more
realistic case of broad correlation, however, the SNR curves
(thick lines) increased more steeply with pool size for
short T because rnoise was less for
short T (as shown by the thick line in
B).
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Therefore, the time scale of correlation must be taken into account
when signals are pooled in short time windows. This may be of relevance
to the visual system, where it is likely that some processes underlying
visual discrimination operate with integration times from 10 to 100 msec (Oram and Perrett, 1992; Thorpe et al., 1996; Corthout et al., 1999). Here we have
focused on one particular pooling model that involves averaging across
redundant signals (Zohary et al., 1994; Shadlen
et al., 1996; Shadlen and Newsome, 1998). The
ultimate role of interneuronal correlation in computations underlying
perceptual decisions will depend on details of the actual mechanisms
that have yet to be worked out.
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FOOTNOTES |
Received Sept. 22, 2000; revised Dec. 5, 2000; accepted Dec. 12, 2000.
W.B. is supported by Howard Hughes Medical Institute (HHMI). Part of
this work was funded by the L. A. Hansen Fellowship to W.B. while in
the lab of Christof Koch at Caltech. W.T.N. is an investigator of HHMI.
We thank Michael N. Shadlen, Carlos Brody, J. Anthony Movshon, and
Christof Koch for suggestions and helpful discussion that has guided
the course of this work, and we owe additional thanks to M. N. Shadlen
and C. Brody for detailed comments on this manuscript.
Correspondence should be addressed to Wyeth Bair, Howard Hughes Medical
Institute, Center for Neural Science, New York University, 4 Washington
Place, Room 809, New York, NY 10003. E-mail:
wyeth{at}cns.nyu.edu.
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APPENDIX A: RELATING SPIKE COUNT CORRELATION TO SPIKE TRAIN
CORRELATION |
Here we derive an expression that relates the correlation
coefficient of spike count, rSC, to the area
under the CCG and the ACGs for a set of paired spike trains. A similar
relationship was noted earlier by Haim Sompolinsky (personal
communication of unpublished notes of 1992 entitled
"Statistics of spike counts and spike trains in a stationary
process," pp 1-6), and recently Brody (1999) has
noted the relationship between spike count covariance and the area of
the CCG, not involving the ACGs. On the basis of our derivation, we
propose a metric, rCCG(), which can provide a
lower variance estimate of rnoise when
interneuronal correlation is limited to a time scale shorter than the trial.
Spike trains from M trials for the two neurons are
represented as discrete binary signals of period T at the
millisecond resolution, i.e.:
|
(11)
|
where k = 1, 2 and 1 t T and 1 i M. The spike counts for the
ith trial are:
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(12)
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and the post-stimulus time histograms are:
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(13)
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The spike train auto-correlation and cross-correlation functions
are defined as:
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(14)
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where j = k for an auto-correlation and
j = 1, k = 2 for the cross-correlation,
C12(), between neurons 1 and 2. The
auto-correlation and cross-correlation of the PSTHs are:
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(15)
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For convenience in defining the correlation functions
above, we have allowed the time index (t + ) to take
values outside [1, T]; therefore, we define
xk(t) and
Pk(t) to be zero for t < 1 and t > T. The function
Sjk will be referred to as the shift-predictor for the purposes of this appendix because it approximates that portion
of the correlation that results from modulation in the PSTHs
(Perkel et al., 1967b).
The equation for the correlation coefficient of spike counts:
![r<SUB><UP>SC</UP></SUB>=<FR><NU><UP>E</UP>[N<SUB>1</SUB>N<SUB>2</SUB>]−<UP>E</UP>N<SUB>1</SUB><UP>E</UP>N<SUB>2</SUB></NU><DE>&sfgr;<SUB>1</SUB>&sfgr;<SUB>2</SUB></DE></FR>,](/content/vol21/issue5/fulltext/1676/img018.gif)
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(16)
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where E is expected value and k2 is the
variance of the spike count computed over trials, can be rewritten in
terms of the cross-correlation equations above. First, observe
that:
|
(17)
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(19)
|
![<UP>=</UP><LIM><OP>∑</OP><LL>&tgr;=<UP>−</UP>T</LL><UL>T</UL></LIM>[C<SUB><UP>kk</UP></SUB>(&tgr;)−S<SUB><UP>kk</UP></SUB>(&tgr;)].](/content/vol21/issue5/fulltext/1676/img024.gif)
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(20)
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A similar result holds for the numerator of Equation 16:
![<UP>E</UP>[N<SUB>1</SUB>N<SUB>2</SUB>]−<UP>E</UP>N<SUB>1</SUB><UP>E</UP>N<SUB>2</SUB>](/content/vol21/issue5/fulltext/1676/img025.gif)
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(21)
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(22)
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(23)
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![<UP>=</UP><LIM><OP>∑</OP><LL>&tgr;=<UP>−</UP>T</LL><UL>T</UL></LIM>[C<SUB>12</SUB>(&tgr;)−S<SUB>12</SUB>(&tgr;)].](/content/vol21/issue5/fulltext/1676/img030.gif)
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(24)
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The following generic expression:
![A<SUB><UP>jk</UP></SUB>(T)=<LIM><OP>∑</OP><LL>&tgr;=<UP>−</UP>T</LL><UL>T</UL></LIM>[C<SUB><UP>jk</UP></SUB>(&tgr;)−S<SUB><UP>jk</UP></SUB>(&tgr;)],](/content/vol21/issue5/fulltext/1676/img031.gif)
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(25)
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defines the area under the auto- and cross-correlation integrated
from T to T (after the shift-predictor is
subtracted). We can rewrite the expression for the correlation
coefficient in terms of these areas as follows:
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(26)
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We now define a metric:
|
(27)
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which will be used to estimate the inter-neuronal correlation
coefficient by integrating a limited central region of the CCG and
ACGs. This measure is equal to the traditional measure, rSC, when = T, i.e.:
|
(28)
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In Results, neuronal data and simulated data are used to
demonstrate that rCCG() can provide a lower
variance estimate of rnoise.
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APPENDIX B: COMPUTING rSC WHEN STIMULUS
STRENGTH VARIES |
Here we derive an expression for rSC, thus
rnoise, for a pair of simulated spike trains
that arise otherwise independently (i.e., with no common noise)
generated from a common stimulus that varies in strength from trial to trial.
Let fi(t) be the mean firing
rate on the ith trial as a function of
time (e.g., Fig. 12A), and let two spike trains be generated as independent realizations of an inhomogeneous Poisson processes according to fi(t). Assume
that fi(t) varies across trial
number, i, in such a way that the time-averaged firing rate,
i, for any trial has mean µ ,
variance , and probability density
g. To derive the correlation in spike count
induced by the trial-to-trial changes in
fi(t), we need only consider
the statistics of the mean rate, , and not the details of the
modulation of fi(t) during the
trial. In particular, to compute the correlation coefficient
rSC between the spike counts
N1 and N2 across trials, we must find the expected values and variances required by Equation 16.
The expected value of the product of the spike counts can be computed
as follows:
![<UP>E</UP>[N<SUB>1</SUB>N<SUB>2</SUB>]=<LIM><OP>∑</OP><LL>j=0</LL><UL>∞</UL></LIM> <LIM><OP>∑</OP><LL>k=0</LL><UL>∞</UL></LIM> jk <UP>Pr</UP>{N<SUB>1</SUB>=j, N<SUB>2</SUB>=k}](/content/vol21/issue5/fulltext/1676/img035.gif)
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(29)
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(30)
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(31)
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(32)
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(33)
|
where T is the duration of the trial. A derivation
similar to that above, but substituting N1 for
N2 or vice versa, leads to:
![<UP>E</UP>[N<SUB>1</SUB>N<SUB>1</SUB>]=<UP>E</UP>[N<SUB>2</SUB>N<SUB>2</SUB>]=T&mgr;<SUB>&lgr;</SUB>+T<SUP>2</SUP>(&mgr;<SUP>2</SUP><SUB>&lgr;</SUB>+&sfgr;<SUP>2</SUP><SUB>&lgr;</SUB>),](/content/vol21/issue5/fulltext/1676/img041.gif)
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(34)
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and a similar but even simpler derivation yields:
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(35)
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Using the identity VARx = Ex2 E2x
and substituting the results of Equations 33, 34, and 35 into the
equation for the correlation coefficient (Eq. 16), we arrive at:
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(36)
|
where µN = Tµ and
N2 = T22 are used to
express the results in terms of spike counts rather than mean rates. This equation states that our simulated spike trains have uncorrelated counts (rSC = 0) when there is no
trial-to-trial variation in the stimulus strength, i.e., when
N2 = 0.
To determine the values of µN and N2,
we must define the rate function,
fi(t). Many statistical
descriptions are possible, but we chose one that provided modulation
which was qualitatively similar to that observed in PSTHs analyzed in
our previous study (Bair and Koch, 1996) of responses to
replicate stimuli collected under stimulus conditions similar to those
of the present study. The rate function, defined as a discrete signal
at the resolution of 1 msec, was described by three parameters, a
spontaneous firing rate, min, a stimulated firing
rate max, and a probability, p, that
at each millisecond fi(t) = max (otherwise,
fi(t) = min). Because for any Bernoulli random variable,
X, E[X] = p and VAR[X] = pq (where
p is the probability of success and q = 1 p), it follows that the mean and variance of the trial spike count
generated by fi(t) for trials
of duration T seconds are:
![&mgr;<SUB><UP>N</UP></SUB>=T[p&lgr;<SUB><UP>max</UP></SUB>+(1−p)&lgr;<SUB><UP>min</UP></SUB>],](/content/vol21/issue5/fulltext/1676/img044.gif)
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(37)
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(38)
|
where min and max are given in
spikes per second and  = 0.001 sec. Substituting this into
Equation 36 yields:
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(39)
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This expression represents the strength of artifactual spike count
correlation induced by trial-to-trial stimulus variance for a model of
paired spike trains designed to be consistent with MT responses to our
dynamic dot stimulus. See Figure 12 and the final section of Results
for its application.
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[Abstract]
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D. B. Katz, S. A. Simon, and M. A. L. Nicolelis
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D. S. Reich, F. Mechler, and J. D. Victor
Independent and Redundant Information in Nearby Cortical Neurons
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[Abstract]
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C. Constantinidis, M. N. Franowicz, and P. S. Goldman-Rakic
Coding Specificity in Cortical Microcircuits: A Multiple-Electrode Analysis of Primate Prefrontal Cortex
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TOP
ABSTRACT
INTRODUCTION
