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The Journal of Neuroscience, May 15, 1998, 18(10):3870-3896
The Variable Discharge of Cortical Neurons: Implications for
Connectivity, Computation, and Information Coding
Michael N.
Shadlen1 and
William T.
Newsome2
1 Department of Physiology and Biophysics and Regional
Primate Research Center, University of Washington, Seattle, Washington
98195-7290, and 2 Howard Hughes Medical Institute and
Department of Neurobiology, Stanford University School of Medicine,
Stanford, California 94305
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ABSTRACT |
Cortical neurons exhibit tremendous variability in the number and
temporal distribution of spikes in their discharge patterns. Furthermore, this variability appears to be conserved over large regions of the cerebral cortex, suggesting that it is neither reduced
nor expanded from stage to stage within a processing pathway. To
investigate the principles underlying such statistical homogeneity, we
have analyzed a model of synaptic integration incorporating a highly
simplified integrate and fire mechanism with decay. We analyzed a
"high-input regime" in which neurons receive hundreds of excitatory
synaptic inputs during each interspike interval. To produce a graded
response in this regime, the neuron must balance excitation with
inhibition. We find that a simple integrate and fire mechanism with
balanced excitation and inhibition produces a highly variable
interspike interval, consistent with experimental data. Detailed
information about the temporal pattern of synaptic inputs cannot be
recovered from the pattern of output spikes, and we infer that cortical
neurons are unlikely to transmit information in the temporal pattern of
spike discharge. Rather, we suggest that quantities are represented as
rate codes in ensembles of 50-100 neurons. These column-like ensembles
tolerate large fractions of common synaptic input and yet covary only
weakly in their spike discharge. We find that an ensemble of 100 neurons provides a reliable estimate of rate in just one interspike
interval (10-50 msec). Finally, we derived an expression for the
variance of the neural spike count that leads to a stable propagation
of signal and noise in networks of neurons that is, conditions that do
not impose an accumulation or diminution of noise. The solution implies that single neurons perform simple algebra resembling averaging, and
that more sophisticated computations arise by virtue of the anatomical
convergence of novel combinations of inputs to the cortical column from
external sources.
Key words:
noise; rate code; temporal coding; correlation; interspike interval; spike count variance; response variability; visual
cortex; synaptic integration; neural model
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INTRODUCTION |
Since the earliest single-unit
recordings, it has been apparent that the irregularity of the neural
discharge might limit the sensitivity of the nervous system to sensory
stimuli (for review, see Rieke et al., 1997 ). In visual cortex, for
example, repeated presentations of an identical stimulus elicit a
variable number of action potentials (Schiller et al., 1976; Dean,
1981; Tolhurst et al., 1983; Vogels et al., 1989; Snowden et al., 1992; Britten et al., 1993), and the time between successive action potentials [interspike interval (ISI)] is highly irregular (Tomko and
Crapper, 1974; Softky and Koch, 1993). These observations have led to
numerous speculations on the nature of the neural code (Abeles, 1991;
Konig et al., 1996; Rieke et al., 1997). On the one hand, the irregular
timing of spikes could convey information, imparting broad information
bandwidth on the neural spike train, much like a Morse code.
Alternatively this irregularity may reflect noise, relegating the
signal carried by the neuron to a crude estimate of spike rate.
In principle we could ascertain which view is correct if we knew how
neurons integrate synaptic inputs to produce spike output. One
possibility is that specific patterns or coincidences of presynaptic events give rise to precisely timed postsynaptic spikes. Accordingly, the output spike train would reflect the precise timing of relevant presynaptic events (Abeles, 1982, 1991; Lestienne, 1996).
Alternatively, synaptic input might affect the probability of a
postsynaptic spike, whereas the precise timing is left to chance. Then
presynaptic inputs would determine the average rate of postsynaptic
discharge, but spike times, patterns, and intervals would not convey
information.
In this paper we propose that the irregular ISI arises as a consequence
of a specific problem that cortical neurons must solve: the problem of
dynamic range or gain control. Cortical neurons receive 3000-10,000
synaptic contacts, 85% of which are asymmetric and hence presumably
excitatory (Peters, 1987; Braitenberg and Schüz, 1991). More than
half of these contacts are thought to arise from neurons within a
100-200 µm radius of the target cell, reflecting the stereotypical
columnar organization of neocortex. Because neurons within a cortical
column typically share similar physiological properties, the conditions
that excite one neuron are likely to excite a considerable fraction of
its afferent input as well (Mountcastle, 1978; Peters and Sethares,
1991), creating a scenario in which saturation of the neuron's firing
rate could easily occur. This problem is exacerbated by the fact that
EPSPs from individual axons appear to exert substantial impact on the membrane potential (Mason et al., 1991; Otmakhov et al., 1993; Thomson
et al., 1993b; Matsumura et al., 1996). An individual EPSP depolarizes
the membrane by 3-10% of the necessary excursion from resting
potential to spike threshold, and this seems to hold for synaptic
contacts throughout the dendrite regardless of the distance between
synapse and soma (Hoffman et al., 1997), suggesting that a large
fraction of the synapses are capable of influencing somatic membrane
potential. Absent inhibition, a neuron ought to produce an action
potential whenever 10-40 input spikes arrive within 10-20 msec of
each other.
These findings begin to reveal the full extent of the cortical
neuron's problem. The neuron computes quantities from large numbers of
synaptic input, yet the excitatory drive from only 10-40 inputs,
discharging at an average rate of 100 spikes/sec, should cause the
postsynaptic neuron to discharge near 100 spikes/sec. If as few as 100 excitatory inputs are active (of the  3000 available), the
postsynaptic neuron should discharge at a rate of 200 spikes/sec. It
is a wonder, then, that the neuron can produce any graded spike output
at all. We need to understand how cortical neurons can operate in a
regime in which many (e.g., 100) excitatory inputs arrive for every
output spike. We will refer to this as a "high-input regime" to
distinguish it from situations common in subcortical structures in
which the activity of a few inputs determines the response of the
neuron. We emphasize that we refer only to the active inputs
of a neuron, which may be as few as 5-10% of its afferent synapses,
although our arguments apply to all larger fractions as well. The
actual fraction active is not known for any cortical neuron, but most
cortical physiologists realize that large numbers of neurons are
activated by simple stimuli (McIlwain, 1990) and would probably
estimate the fraction as considerably greater than 5-10%.
In this paper we analyze a simple model of synaptic integration that
permits presynaptic and postsynaptic neurons to respond over the same
dynamic range, solving the gain control problem. The model is a variant
of the random walk model proposed by Gerstein and Mandelbrot (1964) and
others (for review, see Tuckwell, 1988). Although constrained by neural
membrane biophysics, the model is not a biophysical implementation.
There are no synaptic or voltage-gated conductances, etc. Instead, we
have chosen to attack the problem of synaptic integration as a counting
problem, focusing on the consequences of counting input spikes to
produce output spikes. We show in Appendix , however, that a more
realistic, conductance-based model undergoes the same statistical
behavior.
The paper is divided into three main parts. The first concerns the
problem of synaptic integration in the high-input regime. Given a
plethora of synaptic input, how do neurons achieve an acceptable
dynamic range of response? It turns out that the solution to this
problem imposes a high degree of irregularity on the pattern of action
potentialsthe price of a reasonable dynamic range is noise. The rest
of the paper concerns implications of this noise on the reliability of
neural signals. Part 2 explores the consequences of shared connections
among neurons. Redundancy is a natural strategy for encoding
information in noisy neurons and is a well established principle of
cortical organization (Mountcastle, 1957). We examine its implications
for correlation, synchrony, and coding fidelity. In part 3 we consider
how neurons can receive variable inputs, compute with them, and produce
a response with variability that is, on average, neither greater nor
less than its inputs. We find a stable solution to the propagation of
noise in networks of neurons and in so doing gain insight into the
nature of neural computation itself. Together the exercise supports a
view of neuronal coding and computation that requires large numbers of
connections, much redundancy, and, consequentially, a great deal of
noise.
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BACKGROUND: THE OBSERVED VARIABILITY OF SINGLE NEURONS |
The variability of the neuronal response is characterized in two
ways: interval statistics and count statistics. Interval statistics
refer to the time between successive action potentials, known as the
ISI. For cortical neurons, the ISI is highly irregular. Because this
interval is simply the reciprocal of the discharge rate at any instant,
a neuron that modulates its discharge rate over time must exhibit
variability in its ISIs. Yet even a neuron that fires at a constant
rate over some epoch will exhibit considerable variability among its
ISIs. In fact the distribution of ISIs resembles the exponential
probability density of a random (Poisson) point process. To a first
approximation, the time to the next spike depends only on the expected
rate and is otherwise random.
Count statistics refer to the number of spikes produced in an epoch of
fixed duration. Under experimental conditions it is possible to
estimate the mean and variability of the spike count by repeating the
measurement many times. A typical example is the number of spikes
produced by a neuron in the primary visual cortex when a bar of light
is passed through its receptive field. For cortical neurons, repeated
presentations of the identical stimulus yield highly variable spike
counts. The variance of spike counts over repeated trials has been
measured in several visual cortical areas in monkey and cat. The
relationship between the count variance and the count mean is linear
when plotted on log-log graph, with slope just greater than unity. A
reasonable approximation is that the response variance is about 1.5 times the mean response (Dean, 1981; Tolhurst et al., 1983; Bradley et
al., 1987; Scobey and Gabor, 1989; Vogels et al., 1989; Snowden et al.,
1992; Britten et al., 1993; Softky and Koch, 1993; Geisler and
Albrecht, 1997).
What is particularly striking about both interval and counting
statistics is that they seem to be fairly homogeneous throughout the
cerebral cortex (Softky and Koch, 1993; Lee et al., 1998). Measurements
of ISI variability are difficult, because any measured variation is
only meaningful if the rate is a constant. Nevertheless, the sound of a
neural spike train played through a loudspeaker is remarkably similar
in most regions of the neocortex and contrasts remarkably with
subcortical spike trains, whose regularity often evokes tones. Such
gross homogeneity among cortical areas implies that the inputs to, and
the outputs from, a typical cortical neuron conform to common
statistical principles. To the electrophysiologist recording from
neurons in cortical columns, it is clear that nearby neurons respond
under similar conditions and that their response magnitudes are roughly
similar. Neurons encountered within the column are fairly
representative of the inputs of any one neuron and, in a rough sense,
the targets of any one neuron (Braitenberg and Schüz, 1991).
Again, we emphasize that it is only the neuron's active
inputs to which we refer.
Table 1 lists properties of the neural
response that apply more or less equivalently to a neuron as well as to
its inputs and its targets. These properties are to be interpreted as
rough rules of thumb, but they pose important constraints for the flow of impulses and information through networks of cortical neurons.
Figure 1 illustrates these properties for
a neuron recorded from the middle temporal visual area (MT or V5) of a
behaving monkey, a visual area that is specialized for processing
motion information (for review, see Albright, 1993). Figure 1 shows 210 repetitions of the spike train produced by this neuron when an identical sequence of random dots was displayed dynamically in the
receptive field of the neuron. The stimulus contained rapid fluctuations of luminance and random motion, which produced similarly rapid fluctuations in the neural discharge. The fluctuations in discharge appear stereotyped from trial to trial, as is evident from
the vertical stripe-like structure in the raster and from the
peristimulus time histogram (PSTH) below. The PSTH shows the average
response rate calculated in 2 msec epochs. The spike rate varied
between 15 and 220 impulses/sec (mean ± 2 ). A power spectral density analysis of this rate function reveals significant modulation at 50 Hz, suggesting that the neuron is capable of expressing a change
in its rate of discharge every 10 msec or less (Bair and Koch, 1996).
Thus the neuron is capable of computing quantities over an interval
comparable to the average ISI of an active neuron.
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Figure 1.
Response variability of a neuron recorded
from area MT of an alert monkey. A, Raster and peristimulus
time histogram (PSTH) depicting response for 210 presentations of an
identical random dot motion stimulus. The motion stimulus was shown for
2 sec. Raster points represent the occurrence of action potentials. The
PSTH plots the spike rate, averaged in 2 msec bins, as a function of
time from the onset of the visual stimulus. The response modulates
between 15 and 220 impulses/sec. Vertical lines delineate a
period in which spike rate was fairly constant. The gray
region shows 50 trials from this epoch, which were used to
construct B and C. B, Magnified view of the
shaded region of the raster in A. The spike rate, computed
in 5 msec bins, is fairly constant. Notice that the magnified raster
reveals substantial variability in the timing of individual spikes.
C, Frequency histogram depicting the spike intervals in
B. The solid line is the best fitting exponential
probability density function. D, Variance of the spike count
is plotted against the mean number of spikes obtained from randomly
chosen rectangular regions of the raster in A. Each
point represents the mean and variance of the spikes counted
from 50 to 200 adjacent trials in an epoch from 100 to 500 msec long.
The shaded region of A would be one such example.
The best fitting power law is shown by the solid curve. The
dashed line is the expected relationship for a Poisson point
process.
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At first glance, the pattern of spike arrival times appears fairly
consistent from trial to trial, but this turns out to be illusory. A
closer examination of any epoch reveals considerable variability in
both the time of spikes and their counts. Figure 1B
magnifies the spikes occurring between 360 and 460 msec after stimulus
onset for 50 consecutive trials, corresponding to the shaded region of
Figure 1A. We selected this subset of the raster, because
the discharge rate was fairly constant during this epoch and because it
represents one of the more consistent patterns of spiking in the
record. Nevertheless, the ISIs show marked variability. The mean is
7.35 msec, and the SD is 5.28. We will frequently refer to the ratio,
SD/mean, as the coefficient of variation of the ISI distribution
(CVISI). The value from these
intervals is 0.72. The ISI frequency histogram (Fig. 1C)
is fit reasonably well by an exponential distribution (solid
curve)the expected distribution of interarrival times for a
random (Poisson) point process. Although some of the variability in the
ISIs may be attributable to fluctuations in spike rate, the pattern of
spikes is clearly not the same from trial to trial.
This point is emphasized further by simply counting the spikes produced
during the epoch. If the pattern of spikes were at all reproducible, we
would expect consistency in the spike count. The mean for the
highlighted epoch was 12.8 spikes, and its variance was 8.22. We made
similar calculations of the mean count and its associated variance for
randomly selected epochs lasting from 100 to 500 msec, including from
50 to 200 consecutive trials. The mean and variance for 500 randomly
chosen epochs are shown by the scatter plot in Figure 1D.
The main diagonal in this graph, (Var = mean), is the expected
relationship for a Poisson process. Notice that the measured variance
typically exceeds the mean count. The example illustrated above
(highlighted region of Fig. 1A) is one of the rare
exceptions. The variance is commonly described by a power law function
of the mean count. The solid curve depicts the fit, Var = 0.4mean1.3, but the fit is only marginally better
than a simple proportional rule: Var  1.3mean. Both the timing
and count analyses suggest that the structured spike discharge apparent
in the raster could be explained as a random point process with varying
rate. The process is not exactly Poisson (e.g., the variance is too
large), a point to which we shall return in detail. However, the key
point is that the structure evident in the raster of Figure 1 is only a
manifestation of a time-varying spike rate. The visual stimulus causes
the neuron to modulate its spike rate consistently from trial to trial,
whereas the timing of individual spikestheir intervals and
patternsis effectively random, hence best regarded as noise.
The neuron in Figure 1 illustrates the four properties of statistical
homogeneity listed in Table 1: dynamic range, irregularity of the ISI,
spike count variance, and the time course of spike rate modulation. As
suggested above, it seems likely that these properties are
characteristic of the neuron's afferent inputs and its output targets
alike. Its dynamic range is typical of MT neurons, as well as of V1
neurons that project to MT (Movshon and Newsome, 1996) and neurons in
MST (Celebrini and Newsome, 1994), a major target of MT projections.
Indeed, neurons throughout the neocortex appear to be capable of
discharging over a dynamic range of ~0-200 impulses/sec. Second, the
ISIs from this neuron are characteristic of other neurons in its column
and elsewhere in the visual cortex (Softky and Koch, 1993). Where it
has been examined, the distribution of ISIs has a long
"exponential" tail that is suggestive of a Poisson process. Third,
the variance of the spike count of this neuron exceeds the mean by a
margin that is typical of neurons throughout the visual cortex.
Finally, the rapid modulation of the discharge rate occurs at a time
scale that is on the order of an ISI of any one of its inputs. Our goal is to understand the basis of these statistical properties in single
neurons and their conservation in networks of interconnected neurons.
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MATERIALS AND METHODS |
Physiology. Electrophysiological data (as in Fig. 1)
were obtained by standard extracellular recording of single neurons in the alert rhesus monkey (Macaca mulatta). A full description
of methods can be found in Britten et al. (1992). Experiments were in
compliance with the National Institutes of Health guidelines for care
and treatment of laboratory animals. The unit in Figure 1 was recorded
from the middle temporal visual area (MT or V5). These trials were
extracted from an experiment in which the monkey judged the net
direction of motion of a dynamic random dot kinematogram, which was
displayed for 2 sec in the receptive field while the monkey fixated a
small spot. In the particular trials shown in Figure 1, dots were
plotted at high speed at random locations on the screen, resulting in a
stochastic motion display with no net motion in any direction.
Importantly, however, the exact pattern of random dots was repeated for
each of the trials shown.
Model neuron. We performed computer simulations of neural
discharge using a simple counting model of synaptic integration. Both
excitatory and inhibitory inputs to the neuron are modeled as simple
time series. With a few key exceptions, they are constructed as a
sequence of spike arrival times with intervals that are drawn from an
exponential distribution. The model neuron counts these inputs; when
the count exceeds some threshold barrier, it emits an output spike and
resets the count to zero. Each excitatory synaptic input increments the
count by one unit step. The count decays to zero with a time constant,
 , representing the membrane time constant or integration time
constant. Each inhibitory input decrements the count by one unit. If
the count reaches a negative barrier, however, it can go no further.
Thus inhibition subtracts from any accumulated count, but it does not
hyperpolarize the neuron beyond this barrier. Except where noted, we
placed this reflecting barrier at the resting potential (zero) or one
step below it.
Figure 2, B, D, and
F, represents the count by the height of a particle.
Excitation drives the particle toward an absorbing barrier at spike
threshold, whereas inhibition drives the particle toward a reflecting
barrier (represented by the thick solid line) just below
zero. The particle represents the membrane voltage or the integrated
current arriving at the axon hillock. The height of the absorbing
barrier is inversely related to the size of an excitatory synaptic
potential. It is the number of synchronous excitatory inputs necessary
to depolarize the neuron from rest to spike threshold.
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Figure 2.
Three counting models for synaptic
integration in the high-input regime. The diagrams (B, D,
F) depict three strategies that would permit a neuron to count
many input spikes and yet produce a reasonable spike output. For each
of the strategies, model parameters were adjusted to produce an output
spike count that is the same, on average, as any one input. The
membrane state is represented by a particle that moves between a lower
barrier and spike threshold (top bar). The height of the
particle reflects the input count. Each EPSP drives the particle toward
spike threshold, but the height decays to the ground state with time
constant, (insets). When the particle reaches the top
barrier, an action potential occurs, and the process begins again with
the count reset to 0. A, Excitatory input to the model
neurons. The 300 input spike trains are depicted as rows of a raster.
Each input is modeled as a Poisson point process with a mean rate of 50 spikes/sec. The simulated epoch is 100 msec. C, E, G, Model
response. The particle height is interpreted as a membrane voltage that
is plotted as a function of time. These outputs were obtained using
input spikes in A and the model illustrated in the middle
column (B, D, F). B, C, Integrate-and-fire model
with negligible inhibition and 20 msec time constant. To achieve an
output of five spikes in the 100 msec interval, the spike threshold was
set to 150 steps above the resting/reset state. Notice the regular
interspike intervals in C. D, E, Coincidence detector. The
spike threshold is only 16 steps above rest/reset, but the time
constant must be 1 msec to achieve five spikes out. The coincidence
detector fires if and only if there is sufficient synchronous
excitation. F, G, Balanced excitation-inhibition. A second
set of inputs, like the ones shown in A, provide inhibitory
input. Each inhibitory event moves the particle toward the
lower barrier. The spike threshold is 15 steps above
rest/reset, and the time constant is 20 msec. The particle follows a
random walk, constrained by the lower barrier and the
absorption state at spike threshold. This model is most consistent with
known properties of cortical neurons. A more realistic implementation
is described in Appendix .
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The model makes a number of simplifying assumptions, which are known to
be incorrect. There are no active or passive electrical components in
the model. We have ignored electrochemical gradients or any other
factor that would influence the impact of a synaptic input on membrane
polarizationwith one exception. The barrier to hyperpolarization at
zero is a crude implementation of the reversal potential for the ionic
species that mediate inhibition. We have intentionally disregarded any
variation in synaptic efficacy. All excitatory synaptic events count
the same amount, and the same can be said of inhibitory inputs. Thus we
are considering only those synaptic events that influence the
postsynaptic neuron (no failures). We have ignored any variation in
synaptic amplitude that would affect spikes arriving from the same
inputbecause of adaptation, facilitation, potentiation, or
depressionand we have ignored any differences in synaptic strength
that would distinguish inputs. In this sense we have ignored the
geometry of the neuron. We will justify this simplification in
Discussion but state here that our strategy is conservative with
respect to our aims and the conclusions we draw. Finally, we did not
impose a refractory period or any variation that would occur on reset
after a spike (e.g., afterhyperpolarization). The model rarely produces
a spike within 1 msec of the one preceding, so we opted for simplicity. Appendix describes a more realistic model with several of the
biophysical properties omitted here.
We have used this model to study the statistics of the output spike
discharge. It is important to note that there is no noise intrinsic to
the neuron itself. Consistent with experimental data (Calvin and
Stevens, 1968; Mainen and Sejnowski, 1995; Nowak et al., 1997), all
variability is assumed to reflect the integration of synaptic inputs.
Because there are no stochastic components in the modeled postsynaptic
neuron, the variability of the spike output reflects the statistical
properties of the input spike patterns and the simple integration
process described above.
A key advantage to the model is its computational simplicity. It
enables large-scale simulations of synaptic integration under the
assumption of dense connectivity. Thus a unique feature of the present
exercise is to study the numerical properties of synaptic integration
in a high-input regime, in which one to several hundred excitatory inputs arrive at the dendrite for every action potential the
neuron produces.
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RESULTS |
1.1: Problem posed by high-input regime
Figure 2 illustrates three possible strategies for synaptic
integration in the high-input regime. Figure 2A depicts the
spike discharge from 300 excitatory input neurons over a 100 msec
epoch. Each input is modeled as a random (Poisson) spike train with an average discharge rate of 50 impulses/sec (five spikes in the 100 msec
epoch shown). The problem we wish to consider is how the postsynaptic
neuron can integrate this input and yet achieve a reasonable spike
rate. To be concrete, we seek conditions that allow the postsynaptic
neuron to discharge at 50 impulses/sec. There is nothing special about
the number 50, but we would like to conceive of a mechanism that
produces a graded response to input over a range of 0-200 spikes/sec.
One way to impose this constraint is to identify conditions that would
allow the neuron to respond at the average rate of any one of its
inputs (that is, output spike rate should approximate the number of
spikes per active input neuron per time).
A counting mechanism can achieve this goal through three types of
parameter manipulations: a high absorption barrier (spike threshold), a
short integration time (membrane time constant), or a balancing force
on the count (inhibition). Figure 2 shows how each of these
manipulations can lead to an output spike rate that is approximately
the same as the average input. The simplest way to get five spikes out
of the postsynaptic neuron is to impose a high spike threshold. Figure
2B depicts the output from a simple integrate-and-fire
mechanism when the threshold is set to 150 steps. Each synaptic input
increments the count toward the absorption barrier, but the count
decays with an integration time constant of 20 msec. The counts might
be interpreted as voltage steps of 50-100 µV, pushing the membrane
voltage from its resting potential ( 70 mV) to spike threshold (55
mV). This textbook integrate-and-fire neuron (Stein, 1965; Knight,
1972) responds at approximately the same rate as any one of its 300 excitatory inputs. There are problems, however, that render this
solution untenable. The mechanism grossly underestimates the impact of
individual excitatory synaptic inputs (Mason et al., 1991; Otmakhov et
al., 1993; Thomson et al., 1993a, b; Thomson and West, 1993), and it
produces a periodic output spike train. The regularity of the spike
output in Figure 2C contrasts markedly with the random ISIs
that constitute the inputs in Figure 2A. As suggested by
Softky and Koch (1993), these observations are clear enough indication
to jettison this mechanism.
If relatively few counts are required to reach the absorption barrier,
then the synaptic integration process must incorporate an elastic force
that pulls the count back toward the ground state. This can be
accomplished by shortening the integration time constant or by
incorporating a balancing inhibitory force that diminishes the count.
Figure 2D depicts a particle that steps toward the absorption barrier with each excitatory event. It takes only 16 steps
to reach spike threshold, but the count decays according to an
exponential with a short time constant ( = 1 msec). There is no
appreciable inhibitory input. The resulting output is shown in Figure
2E. The simulated spike train is quite irregular, reflecting occasional coincidences of spikes among the inputs. Because of the
short time constant, the coincidences are sensed with precision well
below the average interspike interval. Again, had we chosen a higher
threshold, we could have achieved a proper spike output with a longer
time constant, but only at the price of a regular ISI (even 3 msec is
too long). The mechanism illustrated in Figure 2, D and
E, detects coincidental synaptic input such that only the
synchronous excitatory events are represented in the output spike
train. Although the coincidence detector produces an irregular ISI, it
requires an unrealistically short membrane time constant (Mason et al.,
1991; Reyes and Fetz, 1993). This requirement can be relaxed somewhat
when spike rates are low and the inputs are sparse (Abeles, 1982), but
the mechanism is probably incompatible with the high-input regime
considered in this paper. This is disappointing because this model
would effectively time stamp presynaptic events that are sufficient to
produce a spike, providing the foundation for propagation of a precise
temporal code in the form of spike intervals (Abeles, 1991; Engel et
al., 1992; Abeles et al., 1993; Softky, 1994; Konig et al., 1996;
Meister, 1996).
The third strategy is to balance the excitation with inhibitory input.
This is illustrated in the bottom panels of Figure 2. For
each of the 300 excitatory inputs shown in Figure 2A, there is an equivalent amount of inhibitory drive (data not shown). Each
excitatory synaptic input drives the particle toward the absorption
barrier, as in Figure 2, B and C; each inhibitory
input moves the particle toward the ground state. The accumulated count decays with a time constant of 20 msec. The particle follows a random
walk to the absorption barrier situated 15 steps away. The lower
barrier just below the reset value crudely implements a synaptic
reversal potential for the inhibitory current. The membrane potential
is not permitted to fall below this value. In other words, inhibitory
synaptic input is only effective when the membrane is depolarized from
rest.
This model is an integrate-and-fire neuron with balanced excitation and
inhibition. It implies that the neuron varies its discharge rate as a
consequence of harmonious changes in its excitatory and inhibitory
drive. Conditions that lead to greater excitation also lead to greater
inhibition. This idea is reasonable because most of the inhibitory
input to neurons arises from smooth stellate cells within the same
cortical column (Somogyi et al., 1983a; DeFelipe and Jones, 1985;
Somogyi, 1989; Beaulieu et al., 1992). Thus excitatory and inhibitory
inputs are activated by the same stimuli; e.g., they share the same
preference for orientation (Ferster, 1986), or they are affected
similarly by somatosensory stimulation (Carvell and Simons, 1988;
McCasland and Hibbard, 1997). Contrast this idea with the standard
concept of a push-pull arrangement in which the neural response
reflects the degree of imbalance between excitation and
inhibition. In the high-input regime, more inhibition is needed to
balance the excitatory drive. The balance confers a proper firing rate
without diminishing the impact of single EPSPs or the membrane time
constant (but see Appendix ). The cost, however, is an irregular ISI.
Gerstein and Mandelbrot (1964) first proposed that such a process would give rise to an irregular ISI, and numerous investigators have implemented similar strategies, termed random walk or diffusion models
(Ricciardi and Sacerdote, 1979; Lansky and Lanska, 1987; for review see
Tuckwell, 1988). What is novel in our analysis is that the same idea
allows the neuron to respond over the same dynamic range as any one of
its many inputs. That is, it allows the neuron to operate in a
high-input regime. This simple idea has important implications for the
propagation of signal and noise through neural networks of the
neocortex.
1.2: Dynamic range
The counting model with balanced excitation and inhibition
achieves a proper dynamic range of response using reasonable
parameters. Figure 3A shows
the response of a model neuron as a function of the average response of
the inputs. We used 300 excitatory and 300 inhibitory inputs in these
simulations. The output response is nearly identical to the response of
any one input, on average. This neuron is performing a very simple
calculation, averaging, but it is doing so in a high-input regime.
Consider that there are ~300 excitatory synaptic inputs for every
spike, yet each excitatory input delivers  the
depolarization necessary to reach spike threshold. By balancing the
surfeit of excitation with a similar inhibitory drive, the neuron
effectively compresses a large number of presynaptic events into a more
manageable number of spikes. Sacrificed are details about the input
spike times; they are only reflected in the tiny bumps and wiggles that describe the membrane voltage during the interspike interval. This
capacity for compression permits the neuron to integrate inputs from
its dendrites and thus to perform calculations on large numbers of
inputs.
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Figure 3.
Conservation of response dynamic range. The spike
rate of the model neuron is plotted as a function of the average input
spike rate. A, Simulations with 300 excitatory inputs and
300 inhibitory inputs; parameters are the same as in Figure 2,
F and G (barrier height, 15 steps; = 20 msec). The balanced excitation-inhibition model produces a response
that is approximately the same as one of its many inputs. B,
Simulations with 600 excitatory and inhibitory inputs. Open
symbols and dashed curve show the response obtained
using the same model parameters as in A. Solid symbols
and curve show the response when the barrier height is
increased to 25 steps. These simulations suggest that a small
hyperpolarization could be applied to enforce a unity gain
input-output relationship when the number of active inputs is
large.
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The mechanism should also allow the neuron to adapt to a broad range of
activation in which more or fewer inputs are active. Figure
3B shows the results of simulations using twice the number of excitatory and inhibitory inputs. The dashed curve
depicts the model response using the identical parameters to those in Figure 3A. The output response is now a little larger than
the average input, and the relationship is approximately quadratic. The
departure from linearity is attributable to the membrane time constant.
At higher input rates, the count frequently accumulates toward spike
threshold before there is any time to decay. Although the range of
response is reasonable, it is not a sustainable solution. If every
neuron were to exhibit such amplification, the response would exceed
the observed dynamic range in very few synapses. Imagine a chain of
neurons, each squaring the response of its averaged input.
A small adjustment to the model repairs this. The solid
curve in this graph was obtained after changing the height of the threshold barrier from 15 to 25. The neuron can now accommodate a
doubling of the number of inputs. With what amounts to a few millivolts
of hyperpolarization, the neuron can achieve substantial control of its
gain. Such a mechanism has been shown to underlie the phenomenon of
contrast adaptation in visual cortical neurons (Carandini and Ferster,
1997). In addition, the curves in Figure 3B raise the
possibility that a neuron could compute the square of a quantity by a
small adjustment in its resting membrane potential or conductance. This
observation may be relevant to computational models that use
squaring-type nonlinearities (Adelson and Bergen, 1985; Heeger,
1992a).
Our central point is that a simple counting model can accommodate large
numbers of inputs with relatively modest adjustment of parameters. It
is essential, however, that a balance of inhibition holds. Because
excitatory synapses typically outnumber inhibitory inputs by about 6:1
for cortical neurons (Somogyi, 1989; Braitenberg and Schüz,
1991), it is possible that the control of excitation (e.g., presynaptic
release probability or synaptic depression) may play a role in
maintaining the balance (Markram and Tsodyks, 1996; Abbott et al.,
1997).
1.3: Irregularity of the interspike interval
As indicated in the preceding section, a consequence of the
balanced excitation-inhibition model is an irregular ISI. Figure 4A shows a representative
interval histogram for one simulation. The intervals were collated from
20 sec of simulated response at a nominal rate of 50 spikes/sec. The
solid curve is the best fitting exponential probability
density function.
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Figure 4.
Variability of the interspike interval.
A, Frequency histogram of ISIs from one simulation using 300 inputs at 50 spikes/sec. Notice the substantial variability. The SD
divided by the mean interval is known as the coefficient of variation
of the interspike interval (CVISI).
The value for this simulation is 0.9. The distribution is approximated
by an exponential probability density (solid curve), which
would predict CVISI = 1. B,
Coefficient of variation of the interspike interval
(CVISI) from 128 simulations using
300 and 600 inputs and a variety of spike rates. Each simulation
generated 20 sec of spike discharge using parameters that led to a
similar rate of discharge for input and output neurons (i.e., a common
dynamic range). The average CVISI was
0.87.
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The variability of the ISI is commonly measured by its coefficient of
variation (CVISI = SD/mean). The value for
the example in Figure 4A is 0.9, just less than the value
expected of a random process (for an exponential distribution,
CVISI = 1). The value is typical for these
simulations, appearing impervious to spike rate or the number of
inputs. Figure 4B shows the distribution of
CVISI obtained for 128 simulations
incorporating a variety of parameters including those used to produce
Figure 3 (solid symbols). The simulations encompassed a
broad range of spike rates, but all produced an irregular spike output.
The CVISI of 0.8-0.9 reflects a remarkable
degree of variation in the ISI. Because the model is effectively
integrating the response from a very large number of neurons, one might
expect such a process to effectively "clean up" the irregularity of
the inputs, as in Figure 2C (Softky and Koch, 1993). The
irregularity is a consequence of the balance between excitation and
inhibition, suggesting an analogy between the ISI and the distribution
of first passage times of random-walk (diffusion) processes.
In our simple counting model, the relationship between input and output
spikes is entirely deterministic. All inputs affect the neuron with the
same strength, and there is no chance for an input to fail. The only
source of irregularity in the model is the time of the input spikes
themselves. We simulated the input spike trains as random point
processes, and the counting mechanism nearly preserved the exponential
distribution of ISIs in its output. But it did not do so completely;
the CVISI was slightly <1. This raises a
possible concern. To what extent does the output spike irregularity
depend on our choice of inputs? Suppose the input spike trains are more
regular than Poisson spike trains; suppose they are only as irregular
as the spike trains produced by the model. Would the counting mechanism
reduce the irregularity further?
Figure 5 shows the results of a series of
simulations in which we varied the statistics of the input spike
trains. We used the same simulation parameters as in Figure
3A but constructed the input spike trains by drawing ISIs
from families of  distributions which lead to greater or less
irregular intervals than the Poisson case (Mood et al., 1963). By
varying the parameters of the distribution we maintained the same input
rate while affecting the degree of irregularity of the spike intervals.
Figure 5 plots the CVISI of our model
neuron as a function of the CVISI for the
inputs. The Poisson-like inputs would have a
CVISI of 1. Notice that for a wide range of
input CVISI, the output of the
counting model attains a CVISI that is
quite restricted and relatively large. The fit intersects the main
diagonal at CVISI = 0.8. This implies that
the mechanism would effectively randomize more structured input and
tend to regularize (slightly) a more irregular input. Most importantly,
the result indicates that the output irregularity is not merely a
reflection of input spike irregularity. The irregular ISI is a
consequence of the balanced excitation-inhibition model.
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Figure 5.
Irregularity of the spike discharge is not merely
a reflection of input spike irregularity. The graph compares the
irregularity of the ISI produced by the balanced excitation-inhibition
model with the irregularity of the intervals constituting the 300 excitatory and inhibitory input spike trains. The input spike trains
were constructed by drawing intervals randomly from a gamma
distribution. By varying the parameters of the gamma distribution, the
input CVISI was adjusted from relatively
regular to highly irregular (abscissa). Each
point represents the results of one simulation, using
different parameters for the input interval distribution. Notice that
the degree of input irregularity has only a weak effect on the
distribution of output interspike intervals. Points above
the main diagonal represent simulations in which the
counting model produced a more irregular discharge than the input spike
trains. Points below the main diagonal represent
simulations in which the output is less irregular than the input spike
trains. The dashed line is the least squares fit to the
data. This line intersects the main diagonal at
CVISI = 0.8. The best fitting line does not
extrapolate to the origin, because the inputs are not necessarily
synchronous.
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These ideas are consistent with Calvin and Steven's (1968) seminal
observations in motoneurons that the noise affecting spike timing is
attributable to synaptic input rather than stochastic properties of the
neuron itself (e.g., variable spike threshold) (Calvin and Stevens,
1968; Mainen and Sejnowski, 1995; Nowak et al., 1997). Nonetheless, if
the random walk to a barrier offers an adequate explanation of ISI
variability, then it is natural to view the irregular ISI as a
signature of noise and to reject the notion that it carries a rich
temporal code. The important insight is that the irregular ISI may be a
consequence of synaptic integration and yet may reflect little if any
information about the temporal structure of the synaptic inputs
themselves (Shadlen and Newsome, 1994; van Vreeswijk and Sompolinsky,
1996).
1.4: Variance of spike count
It is important to realize that the coefficient of variation that
we have calculated is an idealized quantity. It rests on the assumption
that the input rates are constant and that the input spike trains are
uncorrelated. Under these assumptions the number of input spikes
arriving in any epoch would be quite precise. For example, at an
average input rate of 100 spikes/sec, the number of spikes arising over
the ensemble of 300 inputs varies by only ~2% in any 100 msec
interval. Variability produced by the model is therefore telling us how
much noise the neuron would add to a simple computation (e.g., the
mean) when the solution ought to be the same in any epoch. This will
turn out to be a useful concept (see section 3 below), but it is not
anything that we can actually measure in a living brain. In reality,
inputs are not independent, and the number of spikes among the
population of inputs would be expected to be more variable. We will
attach numbers to these caveats in subsequent sections. For now, it is interesting to calculate one more idealized quantity.
If, over repeated epochs, the number of input spikes were indeed
identical (or nearly so), how would the spike count of the output
neuron vary over repeated measures? Using the same simulations as in
Figures 3 and 4, we divided each 20 sec simulation into 200 epochs of
100 msec. We computed the mean and variance of the spikes counted in
these epochs and calculated the ratio: variance/mean. Figure
6 shows the distribution of variance/mean
ratios for a variety of spike output rates and model parameters. The
ratios are concentrated between 0.7 and 0.8, just slightly less than the value expected for a random Poisson point process.
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Figure 6.
Frequency histogram of the spike count
variance-to-mean ratios obtained from the same simulations as in Figure
4B. For each of the simulations, the spikes were counted in
200 epochs of 100 msec duration. The variance in the number of spikes
produced by the model in each of these epochs is proportional to the
mean of the counts obtained for these epochs. Spike count variability
is therefore conveniently summarized by the variance-to-mean ratio. The
average ratio is 0.75 (arrow).
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There are two salient points. First, notice that the histogram of
variance/mean ratios appears similar to the histogram of CVISI from the same simulations (Fig.
4B). In fact, the ratios in Figure 6 are approximated by
squaring the values for CVISI in Figure
4B. This is a well known property of interval and count statistics for a class of stochastic processes known as renewals (Smith, 1959). We will elaborate this point in section 3. Second, the
variance/mean ratios fall short of the value measured in visual cortex
(i.e., 1-1.5). Clearly the variability observed in vivo reflects sources of noise beyond the mechanisms we have considered. In
contrast to our simulations, a real neuron does not receive an
identical number of input spikes in each epoch; the input is itself
variable. A key part of this variability arises from correlation among
the inputs. In the next section we turn attention to properties of
cortical neurons that lead to correlated discharge. We will return to
the issue of spike count variance in section 3.
2: Redundancy, correlation, and signal fidelity
The preceding considerations lead us to depict the neuronal spike
train as a nearly random realization of an underlying rate term
reflecting the average input spike rate (i.e., the number of input
spikes per input neuron per time), or some calculation thereon. Whether
we accept this argument on principle, there is little doubt that many
cortical neurons indeed transmit information via changes in their rate
of discharge. Yet, the irregular ISI precludes single neurons from
transmitting a reliable estimate of this very quantity. Because the
spike count from any one neuron is highly variable, several ISIs would
be required to estimate the mean firing rate accurately (Konig et al.,
1996). The irregular ISI therefore poses an important constraint on the
design of cortical architecture: to transmit rate information
rapidlysay, within a single ISIseveral copies of the
signal must be transmitted. In other words, the cortical design must
incorporate redundancy. In this section we will quantify the notion of
redundancy and explore its implications for the propagation of signal
and noise in the cortex.
2.1: Redundancy necessitates shared connections
By redundancy we refer to a group of neurons, each of which
encodes essentially the same signal. Ideally, each neuron would transmit an independent estimate of the signal through its rate of
discharge. If the variability of the spike trains were truly uncorrelated (independent), then an ensemble of neurons could convey,
in its average, a high-fidelity signal in a very short amount of time
(e.g., a fraction of an ISI; see below). Although this is a desirable
objective, the assumption of independence is unlikely to hold in real
neural systems. Redundancy implies that cortical neurons must share
connections and thus a certain amount of common variability.
The need for shared connections is illustrated in Figure
7A. The flow of information in
this figure is from the bottom layer of neurons to the top. The neurons
at the top of the diagram represent some quantity, . Many
neurons are required to represent accurately, because the discharge
from any one neuron is so variable. To compute its estimate of ,
each neuron in the upper layer requires an estimate of some other
quantity,  , supplied by the neurons in middle tier of the
diagram. To compute rapidly, however, each neuron at the
top of the diagram must receive many samples of . Note, however, that to compute , each of the neurons in the middle of the diagram needs an estimate of some other
quantity,  . What was said of the neurons at the top
applies to those in the middle panel as well. Thus each neuron must receive inputs from many neurons. The chain of
processing resembles a pyramid and is clearly untenable as a model for
how neurons deep in the CNS come to encode any particular quantity. We
cannot insist on geometrically large numbers of independent,
lower-order neurons to sustain the responses of a higher-order neuron
positioned a few synaptic links away. From this perspective, shared
connectivity is necessary to achieve redundancy, and hence rapid
processing, in a system of noisy neurons.
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Figure 7.
Redundancy necessitates shared connections. Three
ensembles of neurons represent the quantities , , and . Each
neuron that represents receives input from many neurons that
represent , and each neuron that represents receives input from
many neurons that represent . A, There are no shared
connections; each neuron receives a distinct set of inputs from its
neighbor. The shaded neurons receive no common input, and
the same can be said of any pair of neurons in the ensemble that
represents . The scheme would require an inordinately large number
of neurons. B, Neurons share a fraction of their inputs. The
shaded neurons receive some of the same inputs from the
ensemble that represents . Likewise, any pair of neurons in the ensemble receive some common input from the neurons that represent .
This architecture allows for redundancy without necessitating immense
numbers of neurons. Neither the number of neurons nor the number of
connections are drawn accurately. Simulations suggest that the pair of
shaded neurons might receive as much as 40% common input,
and each needs about 100 inputs to compute with the quantity .
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In Figure 7B, the same three tiers are illustrated, but the
neurons encoding receive some input in common. Each neuron projects to many neurons at the next stage. Viewed from the top, some
fraction of the inputs to any pair of neurons is shared. In principle, a shared input scheme, such as the one in Figure 7B, would
permit the cortex to represent quantities in a redundant fashion
without requiring astronomical numbers of neurons. There is, however, a
cost. If the neurons at the top of the diagram receive too
much input in common, the trial-to-trial variation in their responses will be similar; hence the ensemble response will be little more reliable than the response of any single neuron.
We therefore wish to explore the influence of shared inputs on the
responses of two cortical neurons such as the ones shaded at
the top of Figure 7B. How much correlated
variability results from differing amounts of shared input? How does
correlated variability among the input neurons themselves (as in the
middle tier of Fig. 7B) influence the estimate of
at the top tier? To what extent does shared connectivity
lead to synchronous action potentials among neurons at a given level?
How does synchrony among inputs influence the outputs of neurons in
higher tiers? We can use the counting model developed in the previous
section to explore these questions. Our goal is to clarify the
relationship among common input, synchronous spikes, and noise
covariance. A useful starting point is to consider the effect of shared
inputs on the correlation in spike discharge from pairs of neurons.
2.2: Shared connections lead to response correlation
We simulated the responses from a pair of neurons like the ones
shaded at the top of Figure 7B. Each
neuron received 100-600 excitatory inputs and the same number of
inhibitory inputs. A fraction of these inputs were identical for the
pair of neurons. We examined the consequences of varying the fraction
of shared inputs on the output spike trains. Except for this
manipulation, the model is the same one used to produce the results in
Figures 3 and 4. Thus each neuron responded approximately at the
average rate of its inputs. We now have a pair of spike trains to
analyze, and once again we are interested in interval and count
statistics. For a pair of neurons, interval statistics are commonly
summarized by the cross-correlation spike histogram (or
cross-correlogram); count statistics have their analogy in measures of
response covariance. We will proceed accordingly.
Figure 8 depicts a series of
cross-correlograms (CCGs) obtained for a variety of levels of common
input. We obtained these functions from 20 sec of simulated spike
trains using the same parameters as in Figure 3A. The
normalized cross-correlogram depicts the relative increase in the
probability of obtaining a spike from one neuron, given a spike from
the second neuron, at a time lag represented along the abscissa
(Melssen and Epping, 1987; Das and Gilbert, 1995b). The probabilities
are normalized to the expectation given the base firing rate for each
neuron. Two observations are notable. First, the narrow central peak in
the CCG reflects the amount of shared input to a neuron, as previously
suggested (Moore et al., 1970; Fetz et al., 1991; Nowak et al., 1995).
Second, no structure is visible in the correlograms until a rather
substantial fraction of the inputs are shared. This is despite several
simplifications in the model that should boost the effectiveness of
correlation. For example, introducing variation in synaptic amplitude
attenuates the correlation. Thus it is likely that the modest peak in
the correlation obtained with 40% shared excitatory and inhibitory inputs represents an exaggeration of the true state of affairs.
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Figure 8.
Cross-correlation response histograms from a pair
of simulated neurons. The correlograms represent the relative change in
response from one neuron when a spike has occurred in the other neuron
at a time lag indicated along the abscissa. The spike train
for each neuron was simulated using the random walk counting model with
300 excitatory and 300 inhibitory inputs. Plots A-F differ
in the amount of common input that is shared by the simulated
pair. A small central peak in the correlogram is apparent when the pair
of neurons share 20-50% of their inputs.
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Rather than viewing the entire CCG for each combination of shared
excitation and inhibition, we have integrated the area above the
baseline and used it to derive a simpler scalar value:
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(1)
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where A11 and A22
represent the area under the normalized autocorrelograms for neurons 1 and 2, respectively, and A12 is the area under
the normalized cross-correlogram (the autocorrelogram is the
cross-correlogram of one neural spike train with itself). The value of
rc reflects the strength of the correlation
between the two neurons on a scale from 1 to 1. This value is
equivalent to the correlation coefficient that would be computed from
pairs of spike counts obtained from the two neurons across many
stimulus repetitions (W. Bair, personal
communication).a
It provides a much simpler measure of correlation than the entire CCG
function.
Using rc we can summarize the effect of shared
excitation and inhibition in a single graph. Figure
9 is a plot of rc
as a function of the fraction of shared excitatory and shared
inhibitory inputs. The points represent correlation coefficients from
simulations using 100, 300, and 600 excitatory and inhibitory inputs
and a variety of spike rates. The threshold barrier was adjusted to confer a reasonable dynamic range of response (input spike rate divided
by output spike rate was 0.75-1.5). Over the range of simulated
values, the correlation coefficient is approximated by the plane:
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(2)
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where  E and I are the fraction of
shared excitatory and inhibitory inputs, respectively. The graph shows
that both the fractions of shared excitatory and shared inhibitory
connections affect the correlation coefficient. Shared excitation has a
greater impact, because it can lead directly to a spike from both
neurons.
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Figure 9.
Effect of common input on response covariance. The
correlation coefficient is plotted as a function of the fraction of
shared excitatory and shared inhibitory input to a pair of model
neurons. Each point was obtained from 20 sec of simulated
spike discharge using a variety of model parameters (input spike rate,
number of inputs, and barrier height). In each simulation, the output
spike rate was approximately the same as the average of any one input
(within a factor of ±0.25). The best fitting plane through the origin
is shown. A substantial degree of shared input is required to achieve
even modest correlation.
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Over the range of counting model parameters tested, we find this planar
approximation to be fairly robust (the fraction of variance of
rc accounted for by Eq. 2 is 42%). We can
improve the fit with a more complicated model (e.g., spike rate has a modest effect), but such detail is unimportant for the exercise at
hand. Of course, Equation 2 must fail as the fraction of shared input
approaches 1; the two neurons will follow identical random walks to
spike threshold, and the correlation coefficient must therefore
approach 1.
The most striking observation from Figure 9 is that only modest
correlation is obtained when nearly half of the inputs are identical.
The counting model is impressively resilient to common input,
especially from inhibitory neurons. Electrophysiological recordings in
visual cortex indicate that adjacent neurons covary weakly from trial
to trial on repeated presentations of the same visual stimulus, with
measured correlation coefficients typically ranging from 0.1 to 0.2 (van Kan et al., 1985; Gawne and Richmond, 1993; Zohary et al., 1994).
The counting model suggests that such modest correlation might entail
rather substantial common input, ~30% shared connections, by
Equation 2. This is larger than the amount of common input that might
be expected from anatomical considerations. The probability that a pair
of nearby neurons receive an excitatory synapse from the same axon is
believed to be ~0.09 (Braitenberg and Schüz, 1991; Hellwig et
al., 1994). Comparable estimates are not known for the axons from
inhibitory neurons, although the probability is likely to be
considerably larger (Thomson et al., 1996), because there are fewer
inhibitory neurons to begin with. Still, it is unlikely that pairs of
neurons share 50% of their inhibitory input; yet this is the value for I needed to attain a correlation of 0.15 (when
E = 0.09, Eq. 2). We suspect that this discrepancy
arises in part because the covariation measured electrophysiologically
exists not only because of common input to a pair of neurons at the
anatomical level, but also because the signals actually transmitted by
the input neurons are contaminated by common noise arising at earlier
levels of the system. As an extreme example, small eye movements could introduce shared variability among all neurons performing similar visual computations (Gur et al., 1997).
2.3: Response correlation limits fidelity
Why should we care about such modest correlation? The reason is
that even weak correlation severely limits the ability of the neural
population to represent a particular quantity reliably (Johnson, 1980;
Britten et al., 1992; Seung and Sompolinsy, 1993; Abbott, 1994; Zohary
et al., 1994; Shadlen et al., 1996). Importantly for our present
purposes, developing an intuition for this principle will help us
understand a major component of the variability in the discharge of a
cortical neuron.
Consider one of the shaded neurons shown in the top tier of
Figure 7B. Its rate of discharge is supposed to represent
the result of some computation involving the quantity . For present purposes we need not worry about exactly what the neuron is computing with this value. What is important is that in any epoch, all that the
shaded neuron knows about is the number of spikes it receives from
neurons in the middle tier of Figure 7B. Clearly,
the variability of the shaded neuron's spike output depends to some
extent on the variability of the number of input spikes, no matter
what the neuron is calculating. If the shaded neuron receives input from hundreds of neurons, each contributing an independent
estimate of , then the number of input spikes per neuron per unit
time would vary minimally. For example, suppose that some visual
stimulus contains a feature represented by the quantity = 40 spikes/sec. Each of the neurons representing this quantity would be
expected to produce four spikes in a 100 msec epoch, but because the
spike train of any neuron is highly variable, each produces from zero to eight spikes. This range reflects the 95% confidence interval for a
Poisson process with an expected count of four. We might say that the
number of spikes from any one neuron is associated with an uncertainty
of 50% (because the SD is two spikes; we use the term uncertainty
here, rather than coefficient of variation, to avoid confusion with
CVISI). In contrast, the
average number of spikes from 100 independent
neurons should almost always fall between 3.6 and 4.4 spikes per input
(i.e., ± 2 SE of the mean). The shaded neuron would receive a fairly
reliable estimate of , which it would incorporate into its
calculation of . In this example, the uncertainty associated with
the average input spike count is 5%, that is, a 10-fold
reduction because of averaging from 100 neurons. With more neurons, the
uncertainty can be further reduced, as illustrated by the gray
line (r = 0) in Figure
10.
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Figure 10.
Weak correlation limits the fidelity of a neural
code. The plot shows the variability in the number of spikes that
arrive in an average ISI from a pool of input neurons modeled as
Poisson point processes. Pool size is varied along the
abscissa. In one ISI, the expected number of input spikes
equals the number of neurons. Uncertainty is the SD of the input spike
count divided by the mean. For one input neuron, the uncertainty is
100%. The diagonal gray line shows the expected
relationship for independent Poisson inputs; uncertainty is reduced by
the square root of the number of neurons. If the input neurons are
weakly correlated, then uncertainty approaches an asymptote of
 (see Appendix ). For an average
correlation of 0.2, the uncertainty from a pool of 100 neurons
(arrow) is approximately the same as for five independent
neurons or, equivalently, the count from one neuron in an epoch of five
average ISIs.
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Unfortunately, the neurons representing , or any other quantity, do
not respond independently of each other. Some covariation in response
is inevitable, because any pair of neurons receive a fraction of their
inputs in common, a necessity illustrated by Figure 7. The preceding
section suggests that the amount of shared input necessary to elicit a
small covariation in spike discharge may be quite substantial, but even
a small departure from independence turns out to be important. It is
easy to see why; any noise that is transmitted via common inputs cannot
be averaged away. This is true even when the number of inputs is very
large. For example, Zohary et al. (1994) showed that the signal-to-noise ratio of the averaged response cannot exceed
  1/2 where is
the average correlation coefficient among pairs of neurons.
We would like to know how correlation among input neurons affects the
variability of neural responses at the next level of processing. We can
start by asking how variable are the quantities that a neuron inherits
to incorporate in its own computation. From the perspective of one of
the neurons in the top tier of Figure 7B, what is
the variability in the number of spikes that it receives from neurons
in the middle layer? In other words, how unreliable is the estimate of
?
The answer is shown in Figure 10. We have calculated the uncertainty in
the number of spikes arriving from an ensemble of neurons in the middle
layer. Each curve in Figure 10 shows uncertainty as a
function of the number of neurons in the input ensemble, where
uncertainty is expressed as the percentage variation (SD/mean) in the
number of spikes that a neuron in the top layer would receive from the
middle layer in an epoch lasting one typical ISI. This characterization
of variability is appealing, because it bears directly on neural
computation at a natural time scale.
If there is just one neuron, then the mean number of spikes arriving in
an average ISI is one, of course, and so is the SD, assuming a Poisson
spike train. Hence the uncertainty is 100%. If there are 100 inputs
from the middle layer, then the expected number of spikes is 100: one
spike per neuron. If each spike train is an independent Poisson process
(Fig. 10, gray line), then the SD is 10 spikes (0.1 spikes
per neuron), for a percentage uncertainty of 10%. If the spike trains
are weakly correlated, however, then the percentage uncertainty is
given by:
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(3)
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where m is the number of neurons, and
is the average correlation coefficient among all
pairs of input neurons (see Appendix ). Each of the curves
in Figure 10 was calculated using a different value for
. The solid curves indicate the
approximate level of correlation that is believed to be present among
pairs of cortical neurons and that is consistent with our simulations using a large fraction of common input ( = 0.1-0.3).
Even at the lower end of this range, there is a substantial amount of variability that cannot be averaged away. For an average correlation coefficient of 0.2, the percentage uncertainty for 100 neurons is 45%;
only a twofold improvement (approximately) over a single neuron!
Three important points follow from this analysis. First, modest amounts
of correlated noise will indeed lead to substantial uncertainty in the
quantity , received by the top tier neurons in Figure
7B that compute , even if 100 neurons provide the ensemble input. This variability in the input quantity will influence the variance of the responses of the top tier neurons, an issue to
which we shall return in section 3. Second, the modest reduction in
uncertainty achieved by pooling hardly seems worth the trouble until
one recalls that what is gained by this strategy is the capacity to
transmit information quickly. For example, using 100 neurons with an
average correlation of 0.19, the brain transmits information in one ISI
with the same fidelity as would be achieved from one neuron for five
times this duration. This fact is shown by the dotted lines
in Fig. 10. If we interpret the abscissa for the r = 0
curve as m ISIs from one neuron (instead of one ISI from
m input neurons), we can appreciate that the uncertainty reduction achieved in five ISIs is approximately the same as the uncertainty achieved by about 100 weakly correlated neurons
( = 0.2; Fig. 10, arrow). Third, the
fidelity of signal transmission approaches an asymptote at 50-100
input neurons; there is little to be gained by adding more inputs. This
observation holds for any realistic amount of correlation, suggesting
that 50-100 neurons might constitute a minimal signaling unit in
cortex. Here lies an important design principle for neocortical
circuits. Returning to Figure 7, we can appreciate that the more
neurons that are used to transmit a signal, the more common inputs the
brain is likely to use. The strategy pays off until an asymptotic limit in speed and accuracy is approached: ~50-100 neurons.
A most surprising finding of sensory neurophysiology in recent years is
that single neurons in visual cortex can encode near-threshold stimuli
with a fidelity that approximates the psychophysical fidelity of the
entire organism (Parker and Hawken, 1985; Hawken and Parker, 1990;
Britten et al., 1992; Celebrini and Newsome, 1994; Shadlen et al.,
1996). This finding is understandable, however, in light of Equation 3,
which implies that psychophysical sensitivity can exceed neural
sensitivity by little more than a factor of 2, given a modest amount of
correlation in the pool of sensory neurons.
2.4: Synchrony among input neurons
If pairs of neurons carrying similar signals are indeed
correlated, it is natural to inquire whether such correlation
influences the spiking interval statistics considered earlier. How do
synchronous spikes such as those reflected in the cross-correlograms of
Figure 8 influence the postsynaptic neuron? What is the effect on ISI variability of weak correlation and synchronization in the input spike
trains themselves (recall that the inputs in the simulations of Fig. 8
were independent)?
We tested this by simulating the response of two neurons using inputs
with pairwise correlation that resembles Figure 8E. We generated a
large pool of spike trains using our counting model with 300 excitatory
and 300 inhibitory inputs. Each spike train was generated by drawing
300 inputs from a common pool of 750 independent Poisson spike trains
representing excitation and another 300 inputs from a common pool of
750 Poisson spike trains, which represented the inhibitory input. The
strategy ensures that, on average, any pair of spike trains was
produced using 40% common excitatory input and 40% common inhibitory
input. Thus any pair of spike trains has an expected correlation of
0.25-0.3 and a correlogram like the one in Figure 8E. We simulated
several thousand responses in this fashion and used these as the input
spike trains for a second round of simulations. The correlated spike
trains now served as inputs to a pair of neurons using the identical model. Again, 40% of the inputs to the pair were identical. By using
the responses from the first round of simulations as input to the
second, we introduced numerous synchronized spikes to the input
ensemble.
The result is summarized in Figure 11.
The cross-correlogram among the output neurons (Fig. 11A)
resembles the correlogram obtained from the inputs (Fig.
11B). We failed to detect an increase in synchrony. In fact
the correlation coefficient among the pair of outputs was 0.29, compared with 0.30 for the inputs. The synchronized spikes among the
input ensemble did not lead to more synchronized spikes in the two
output neurons. Nor did input correlation boost the spike rate or cause
any detectable change in the pattern of spiking. As in our earlier
simulations, the output response was approximately the same as any one
of the 600 inputs. Moreover, the spike trains were highly irregular,
the distribution of ISIs approximating an exponential probability
density function (Fig. 11, inset;
CVISI = 0.94). We detected no structure in
the output response or in the unit autocorrelation functions.
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Figure 11.
Homogeneity of synchrony among input and output
ensembles of neurons. A, Normalized cross-correlogram from a
pair of neurons receiving 300 excitatory and inhibitory inputs, the
typical pairwise cross-correlogram of which is shown in B.
The pair share 40% common excitatory and inhibitory input. The CCG was
computed from 80 1 sec epochs. The simulation produced a correlation
coefficient of 0.29. B, The average correlogram for pairs of
neurons serving as input to the pair of neurons, whose CCG is shown in
A. The correlogram was obtained from 80 1 sec epochs using
randomly selected pairs of input neurons. The mean correlation
coefficient, , was 0.3. Vertical scale reflects
percent change in the odds of a spike, relation to background.
C, Spike interval histogram for the output neurons.
Synchrony among input neurons does not lead to detectable structure in
the output spike trains (CVISI = 0.94).
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The finding contradicts the common assumption that synchronous spikes
must exert an exaggerated influence on networks of neurons (Abeles,
1991; Singer, 1994; Aertsen et al., 1996; Lumer et al., 1997). This
idea only holds practically when input is relatively sparse so that a
few presynaptic inputs are likely to yield a postsynaptic spike
(Abeles, 1982; Kenyon et al., 1990; Murthy and Fetz, 1994). The key
insight here is that the cortical neuron is operating in a high-input
regime in which the majority of inputs are effectively synchronous.
Given the number of input spikes that arrive within a membrane time
constant, there is little that distinguishes the synchronous spikes
that arise through common input. If as few as 5% of the ~3000 inputs
to a neuron are active at an average rate of 50 spikes/sec, the neuron
receives an average of 75 input spikes every 10 msec. The random walk
mechanism effectively buffers the neuron from the detailed rhythms of
the input spike trains just as it allows the neuron to discharge in a
graded fashion over a limited dynamic range.
3: Noise propagation and neural computation
As indicated previously, a remarkable property of the neocortex is
that neurons display similar statistical variation in their spike
discharge at many levels of processing. For example, throughout the
primary and extrastriate visual cortex, neurons exhibit comparable irregularity of their ISIs and spike count variability. When an identical visual stimulus is presented for several repetitions, the
variance of the neural spike count has been found to exceed the mean
spike count by a factor of ~1-1.5 wherever it has been measured (see
Background). The apparent consistency implies that neurons receive
noisy synaptic input, but they neither compound this noise nor average
it away. Some balancing equilibrium is at play.
Recall that our simulations led to a spike count variance that was
considerably less than the mean count (Fig. 6), in striking contrast to
real cortical neurons. Because the variance of the spike count affects
signal reliability, it is important to gain some understanding of this
fundamental property of the response. In this section we will develop a
framework for understanding the relationship between the mean response
and its variance under experimental conditions involving repetitions of
nominally identical stimuli. Why does the variance exceed the mean
spike count, and how is the ratio of variance to mean count preserved
across levels of processing? The elements of the variance turn out to
be the very quantities we have enumerated in the preceding sections: irregular ISIs and weak correlation.
3.1: Background and terminology
At first glance it may seem odd that investigators have measured
the variance of the spike count; after all it is the
SD that bears on the fidelity of the neural discharge.
However, a linear relationship between the mean count and its variance
would be expected for a family of stochastic point processes known as
renewal processes. A stochastic point process is a random sequence of stereotyped events (e.g., spikes) that are only distinguishable on the
basis of their time of occurrence. In a renewal process the intervals
from one event to the next (e.g., ISIs) are independent of one another
and drawn from a common distribution (i.e., they are independent and
identically distributed; Karlin and Taylor, 1975). The Poisson process
(e.g., radioactive decay) is a well known example. Recall that the
intervals of a Poisson process are described by the exponential
probability density function:
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(4)
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where  is the average event rate, and 1 is
both the mean interval and SD (i.e., CV = 1).
The number of events observed in an epoch of duration, T, is
also a random number. The count, which we shall denote,
N(T), obeys a Poisson distribution; the mean count
is T, as is the variance. We will use angled brackets to indicate the mean of many repetitions and summarize the Poisson case by
writing, Var[N(T)] =  N(T) = T.
For any renewal process (not just the Poisson process), the number of
events counted in an epoch, N(T), is a random number
with variance that scales linearly with the mean count. The constant of
proportionality is the squared coefficient of variation
(CV) of the interval distribution:
Var[N(T)] = CV2N(T) (Smith,
1959).
Spike trains from cortical neurons bear certain similarities to Poisson
processes, and it is presumably for this reason that investigators have
sought a lawful relationship between the mean spike count and its
variance. However, as we have noted, the spike counts from cortical
neurons exhibit even greater variance than the Poisson case:
Var[N(T)] > N(T). In general, real
spike trains are not renewal processes. ISIs often fail independence
(e.g., during bursts), and, unless the spike rate is constant, ISIs
cannot be described by a common distribution (Teich et al., 1997). The latter concern is particularly important when we consider the behavior
of the neuron over many repetitions, for as we will see in a moment,
the spike rate is emphatically not the same from epoch to epoch. In
contrast to real spike trains, the random walk model described in
section 1 produced a sequence of independent ISIs, described by a
common distribution, whether viewed in one or several epochs; it
describes a renewal process. We would like to relate this process to
the variance of the spike count that would be observed over many
stimulus repetitions.
Table 2 lists the main mathematical
symbols used in our argument as well as some guidance to their
interpretation. The scenario we will develop can be summarized as
follows. In an epoch of duration, T, the neuron receives
ni(T) synaptic inputs from
m input neurons (i = 1m). The
postsynaptic neuron computes some quantity from these inputs, which it
attempts to represent as a spike rate, , by emitting a sequence of
spikes modeled as a renewal. From here on we will no longer simulate
spike trains using the random walk model and instead identify a desired
spike ratethe result of some neural computationand adopt
1 as the expected ISI of a renewal process. The
strategy will allow us to write equations for the spike count
variance.
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Table 2.
Mathematical symbols used in the description of a counting
process and their interpretation in a model of neural computation
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3.2: Two components of response variance
The observed spike count variance can be divided into two
components. The first is related to the irregular ISIs that would arise
from a renewal process when the rate is known. We will refer to this
source of variance as conditional noise, because it assumes precise knowledge of the desired output spike rate. It is the variance
in the spike count that we would anticipate if a neuron were to compute
the same quantity on each stimulus repetition. It is the kind of
variability that we might associate with a Poisson process (or some
other renewal) when the rate is known. The second source of noise
reflects the fact that for any one spike train, we do not really know
the expected rate with certainty. For repeated presentations of the
same stimulus, the quantity actually computed by the neuron varies from
trial to trial. As we discovered in section 2, this is because the
inputs are weakly correlated. This source of variability might be
called extrinsic noise or, more accurately, the variability
of the computed expectation.
In general, when a random quantity, y, depends on a second
quantity, x, which is also variable, the variance of
y can be expressed as the sum:
![<UP>Var</UP>[y]=⟨<UP>Var</UP>[y‖x]⟩+<UP>Var</UP>[⟨y‖x⟩],](/content/vol18/issue10/fulltext/3870/img009.gif)
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(5)
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where ··· denotes the average over all values of
x. The expression y|x denotes the
random value, y, conditional on knowledge that x
has taken on some particular value. In words, Equation 5 says that the
observed variance of y is the sum of its mean conditional variance plus the variance of its conditional
mean. The equation applies intuitively to random numbers drawn
from a distribution with mean and variance that change on every pick. The conditional variance describes the variance that would apply on any
one draw, whereas the variance of conditional mean describes the
distribution of expectations across the picks.
In a similar fashion, we can conceive of the spike count as a random
number drawn from a distribution with expectation that changes for each
repetition of the stimulus or trial. The expectation reflects some
computation on the input spike trains. Thus on any one trial, the
neuron computes a spike rate, , and emits a random spike train with
an expected count of T. Representing the spike count by
N(T), we can rewrite Equation 5 as follows:
![<UP>Var</UP>[N(T)]=⟨<UP>Var</UP>[N(T)‖&lgr;]⟩+<UP>Var</UP>[⟨N(T)‖&lgr;⟩]](/content/vol18/issue10/fulltext/3870/img010.gif)
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(6)
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The idea is that the variance of the spike count can be divided
into a portion that would be present even if the neuron were to compute
the same value for on every trial and another portion that reflects
the fact that it does not compute the same on every
trial.b
It is helpful to relate this concept briefly to the counting model used
in previous sections. In section 1, we designed the model to respond at
a spike rate equal to the average of its inputs. For m
inputs, the computation is:
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(7)
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where ni(T) is the input spike
count from the ith input neuron during
the epoch T. Thus, reflects the average spike rate among
the active inputs, and the computation satisfies our desire for
homogeneity of input and output spike rates. Recall from section 1 that
the model produced a variable spike count even though the average input
rate was the same on each repetition. Because the inputs were modeled
as independent, there was little variation in the mean rate of the
inputs from trial to trial. Hence was known precisely, or nearly
so, implying that the variance of the conditional mean was negligible.
In short, all of the spike count variance is attributed to the renewal
process. When correlation was introduced among the inputs, however
(section 2), additional variance arose because of the uncertainty in
. Equation 6 tells us that these two sources of variance add.
Figure 12 is intended to convey an
intuition for the two sources of variance and their importance for
networks of interconnected neurons. The three neurons at the
top of Figure 12 are idealized as Poisson generators, and
all compute the same quantity, , which is the mean input spike rate
(Eq. 7). The neurons at the bottom provide the input spikes;
they are also Poisson generators and represent the same expected rate.
(By Poisson generator, we mean that the neuron computes a rate value
and generates a spike train with intervals that are independent and
exponentially distributed.) Suppose that the input spike rate is 100 spikes/sec and T = 100 msec. Then the expected spike
count from the input neurons is ni(T) = 10 spikes, and
the variance is 10 spikes2, as shown by distribution
in the bottom graph. Each of the neurons shown at the
top of Figure 12 computes the average spike rate among the
inputs: 100 spikes/sec, or 10 spikes in the 100 msec epoch. The
top graphs depict the distribution of spike counts emitted by these "output" neurons.
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Figure 12.
Propagation of noise among ensembles of neurons.
Each of the neurons depicted in this figure emits a spike train
idealized as a Poisson point process. The three neurons at the
top compute the average spike rate among their inputs and
emit the answer as a Poisson spike train. During a 100 msec epoch, the
input neurons (bottom) discharge at 100 spikes/sec. Each
input neuron is therefore expected to generate 10 spikes, but in any
one epoch the count may vary. The bottom graph shows the
Poisson distribution of spike counts from one input neuron. The
middle row of graphs shows the probability
density of the quantity that the output neuron has computed: the number
of spikes per input neuron. The neuron obtains an estimate of the input
spike count by calculating with one (A) or more neurons
(B, C). The value is represented as an expectation,
T, which can be thought of as a desired rate times
the epoch duration. In any one epoch, the neuron emits a Poisson spike
train at a rate, , resulting in N(T) spikes. The
distribution of N(T) from many 100 msec repetitions is shown
at the top. A, If the output neuron receives
input from just one input neuron, the variance of the input count,
var[T], is 10. The output neuron emits an
average of 10 spikes, but the variance is 20, reflecting the sum of
input and (its own) Poisson variance. B, If there are many
independent inputs, then the variance of the mean input count is
negligible (delta function; middle plot). The output neuron
emits an average of 10 spikes, and the variance is 10, the amount of
variance expected for a Poisson spike train. C, If there are
many weakly correlated input neurons, then the variance of the mean
input count is approximately 10 times the average correlation
coefficient among the input neurons. If = 0.2,
then the variance is 2. The output neuron emits an average of 10 spikes, but the total variance is 12. Notice that in A and
C, the variance of the output spike count exceeds the
variance of the inputs.
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If there were just one input neuron, as in Figure 12A, then
two sources of variance, extrinsic and conditional in the terms established above, would combine to produce an output that is more
variable than the single neuron input. Extrinsic noise reflects the
fact that the input is not constant on each trial but instead takes the
form of a random pick from the probability distribution shown in the
middle row. This is the distribution of T
estimated by applying Equation 7 to just one neuron. Thus, we imagine
that the output neuron receives a random pick of T from
this distribution and generates a Poisson spike train with this
expected count. The total variance is the sum of the extrinsic (input)
variance and the conditional variance arising from the Poisson renewal process. In this case the mean is 10 spikes, and the variance is 20 spikes2. In just one synapse the variance-to-mean
ratio has doubled. Such a scheme is obviously untenable for extended
chains of neural processing.
If the neuron were to compute the average spike rate from many
independent neurons, then the extrinsic noise would approach zero. This
is represented by the delta function in Figure 12B (middle row). On every repetition, the computation produces
100 spikes/sec (10 spikes/100 msec per neuron) by pooling across the ensemble of input neurons. The output neuron necessarily incorporates conditional noise and therefore generates a Poisson spike train with an
expected count of 10 and a variance of 10. The output response is only
as variable as a Poisson process with rate 100 spikes/sec. This
situation resembles the simulations performed in section 1.4. The
variance of the mean input count was negligible because we simulated
each of the inputs independently. Here we have simply bypassed the
random walk model and instead postulated that the neuron computes = 100 spikes/sec and emits a Poisson spike train accordingly.
A more realistic situation, involving correlated noise among the
inputs, is shown in Figure 12C. Again, we suppose that the neuron computes the mean of many input spike counts, but that the
response of any one input neuron is correlated weakly with the response
of the others. Averaging no longer leads to a precise value but retains
some variability. For the computation in Equation 7, the variance would
be estimated by multiplying the average correlation coefficient,
, by the variance of a single input:
![<UP>Var</UP>[⟨&lgr;T⟩]≈<A><AC>r</AC><AC>&cjs1171;</AC></A> <UP>Var</UP>[n(T)].](/content/vol18/issue10/fulltext/3870/img013.gif)
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(8)
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This approximation is implied by Equation 3 for large numbers of
inputs (see Appendix ). If = 0.2, then the variance of the computed quantity would be 2 spikes2. The
value transmitted to the neuron at the top of Figure 12 is 10 spikes per neuron per epoch, with a variance of 2. Once again, the
top neuron represents this quantity by emitting a Poisson spike train.
The total variance in this case is 12 spikes2, 1.2 times the mean.
The idea that spike count variance arises as a sum of conditional and
input variances is appealing for its simplicity. It provides some
intuition about how the spike count variance would exceed its mean, and
it suggests that the ratio of spike count variance to the mean count
ought to be a constant. However, we have yet to achieve an adequate
explanation for the apparent homogeneity of this relationship among
input and output neurons. The example in Figure 12B would be
regarded as successful were it not for the unrealistic assumption of
independent inputs, hence noise-free computation. The example in Figure
12C comes close, but the variance of the input neurons at
the bottom does not match the variance of the spike count
produced by the neuron at the top (10 and 12, respectively).
It would be untenable for neurons to boost the variance by 20% at each
stage of synaptic integration.
3.3: A stable solution for spike count variance
We therefore seek a stable ratio of variance to mean spike count,
one that encompasses input neurons and output neurons alike and one
that more accurately reflects the variability of real neurons. For the
moment, we will continue to make our simplifying assumption that the
neuron computes the average firing frequency of its active inputs (Eq. 7). This case allows us to elaborate the scheme depicted in Figure 12
and serves as a reference point for further discussion.
In section 1.2 we established that a random walk model for synaptic
integration allows the neuron to respond at a rate that is
approximately equal to the average rate of its inputs, but that the
output spike train is highly irregular. Recall that the coefficient of
variation of the ISIs (CVISI) was
0.8-0.9 for a variety of model parameters. When we computed the spike
counts from many epochs, we found that the variance-to-mean ratio of the spike counts was 0.75 on average (Fig. 6). This value is sensible because the random walk model produced spike trains that conformed to a
renewal process: spike intervals were independent, and the expected
spike rate was the same for all repetitions. As noted earlier, the
predicted relationship between the spike interval distribution and the
variance in the number of spikes counted in an epoch, T, is
given by Var[N(T)] = CVISI2N(T), which
would predict a variance-to-mean ratio of 0.64-0.81. This relationship
presumes a fixed spike rate, . If is variable, however, this
same relationship then applies only to the average conditional
variance:
![⟨<UP>Var</UP>[N(T)‖&lgr;]⟩≈⟨C<SUP><UP>2</UP></SUP><SUB><UP>V</UP><SUB><UP>ISI</UP></SUB></SUB> N(T)‖&lgr;⟩=C<SUP>2</SUP><SUB><UP>V<SUB>ISI</SUB></UP></SUB>⟨N(T)⟩.](/content/vol18/issue10/fulltext/3870/img014.gif)
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(9)
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Equation 9 works because the CVISI is
the same regardless of spike
rate.c The left side of Equation 9 is the average of the variances computed at each fixed spike rate. We
referred to this as the mean conditional variance in Equation 5. It is
the amount of variance that we would observe if only the neuron were
instructed to produce the same spike rate in each epoch.
In reality, of course, correlation among the inputs guarantees that the
neuron receives somewhat different instructions on each trial. This
additional variance is represented by the second term in Equation 6:
the variance of the conditional mean. The amount of this additional
variation depends on what the neuron is calculating. If the neuron
computes the number of spikes per active input, then we can use the
expression in Equation 8. The total variance is therefore:
![<UP>Var</UP>[N(T)]=C<SUP>2</SUP><SUB><UP>V</UP><SUB><UP>ISI</UP></SUB></SUB>⟨N(T)⟩+<A><AC>r</AC><AC>&cjs1171;</AC></A> <UP>Var</UP>[n(T)].](/content/vol18/issue10/fulltext/3870/img015.gif)
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(10)
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This equation is obtained by combining Equations 6-9. It says
that the variance of the spike count reflects (1) the irregularity of
the spike train, and (2) an inability to average out common noise.
We seek conditions in which the input and output neurons exhibit the
same degree of variability. This would describe a steady state in which
noise neither compounds nor attenuates in a neural processing chain.
For the mean spike rate calculation (Eq. 7), the expected counts are
the same for input and output neurons. In the steady state, the
expected variance of the input and output neurons must be equal as
well: var[N(T)] = var[n(T)].
This equality allows us to rearrange Equation 10 to yield a
steady-state solution:
![<UP>Var</UP>[N(T)]=<FR><NU>C<SUP>2</SUP><SUB><UP>V</UP><SUB><UP>ISI</UP></SUB></SUB></NU><DE>1−<A><AC>r</AC><AC>&cjs1171;</AC></A></DE></FR>⟨N(T)⟩.](/content/vol18/issue10/fulltext/3870/img016.gif)
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(11)
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This expression tells us the relationship between the variance and
the mean spike count for any neuron in any tier of a network that is
designed to compute the mean response of its inputs.
In Figure 12, we assumed that the neuron behaves as a Poisson
generator. Choosing CVISI = 1 and
= 0.2, Equation 11 indicates that the variance of
the spike count should be 1.25 times its mean value. This is the
homogeneous solution for populations of neurons that behave as Poisson
generators whose spike rate is the mean of the input spike rates. If
the input neurons in Figure 12C had a spike count variance
of 12.5 (instead of 10), then the variance of the output, shown at the
top, would have been 12.5 as well. The neuron at the
top of Figure 12C would inherit a variance of 2.5 (because Var[T] Var[n(T)] = (0.2)(12.5) = 2.5), which adds to the conditional variance of the
Poisson generator (10 spikes2). The simulations in
section 1 lead us to suggest that a better estimate of the
CVISI might be 0.8 (Figs. 4 and 5). The
stable solution for the variance is then 0.8 times the mean (by Eq. 11), which is less than the value of 1-1.5, measured
experimentally.
It is possible that this discrepancy between theory and experiment
arises from additional sources of signal variability in vivo, such as fluctuations in eye position in awake animals (Gur et al., 1997) or fluctuations in cortical excitability (Arieli et al.,
1996). More likely, in our opinion, is that our theoretical analysis
underestimates the real variance, because we have assumed a very simple
computationaveraging. In the next section we will explore this
proposition.
3.4: Neural calculations other than the mean
Throughout this paper we have assumed that a cortical neuron
computes something like the average rate of its inputs. This was a
convenient choice to satisfy the homogeneity constraints listed in
Table 1. In general, however, a postsynaptic neuron is likely to
compute something more complicated than the average of its inputs, and
the variance of this quantity will typically exceed the amount deduced
for averaging (Eq. 8). Even to compute the average number of spikes per
active input, the neuron needs an estimate of the number of active
inputs (m in Eq. 7). We have tacitly assumed knowledge of
this value in Equation 7 and in the adjustment of barrier height in the
random walk model. The neuron needs an estimate of this quantity,
however, and to the extent that this estimate carries
uncertainty, it will augment the variance associated with
computation Var[T]. This is just one of many possible sources of uncertainty that would boost the variance of the
neural discharge. We have referred to this additional variance as
pooling noise in previous work (Shadlen et al., 1996).
How much additional variance can be anticipated if the neural
computation combines several quantities, each carrying its own uncertainty? The following contrivance offers surprising insight. Suppose that in addition to computing the average rate among some set
of inputs, the neuron were to add and subtract two other quantities. Let the expected spike rate reflect the addition of two values and the
subtraction of a third, as in a + b c.
Analogous to Equation 7, we can represent the computation as:
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(12)
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where ma, mb, and
mc are the number of inputs that convey the
quantities a, b, and c. We are not proposing that
this is how a neuron might add and subtract quantities; we use this
contrived example only to illustrate a computation with more variance
than a simple average. The important aspect of Equation 15 is that it combines distinct entities after the averaging steps. Each
term therefore introduces variance into the computation of
T. Thus, Equation 15 might be interpreted as a special
case of a class of computations that would include multiplication and
logical comparisons between inputs arriving at distinct parts of the
dendrite. This particular calculation allows us to extend the
intuitions gained from the preceding section. If we take the average
spike rates of all the inputs to be the same, then the expected output
rate is the same as any one of the inputs, and we can apply the same strategy as before to find a stable solution for the spike count variance.
Assume that 100 or more inputs represent each of the three quantities,
a, b, and c, and that the neurons that comprise
the pools are weakly correlated among themselves.d
The extra quantities boost the uncertainty of the computation. Using
the same notation as before, the variance of the conditional expectation reflects the sum of the variances from each of the three
terms:
![<UP>Var</UP>[⟨&lgr;T⟩]≈3<A><AC>r</AC><AC>&cjs1171;</AC></A> <UP>Var</UP>[n(T)],](/content/vol18/issue10/fulltext/3870/img018.gif)
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(13)
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triple the amount in Equation 8. The rest of the argument is
identical to the last section. The observed spike count 18 variance reflects this uncertainty plus the variability that arises because of
the stochastic nature of the ISI (i.e., the fact that the neuron realizes its intended rate as a renewal process):
![<UP>Var</UP>[N(T)]=C<SUP>2</SUP><SUB><UP>V</UP><SUB><UP>ISI</UP></SUB></SUB>⟨N(T)⟩+3<A><AC>r</AC><AC>&cjs1171;</AC></A> <UP>Var</UP>[n(T)].](/content/vol18/issue10/fulltext/3870/img019.gif)
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(14)
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Because the expected spike count is the same for input and output,
we can equate the variances to find a stable solution:
![<UP>Var</UP>[N(T)]=<FR><NU>C<SUP>2</SUP><SUB><UP>V</UP><SUB><UP>ISI</UP></SUB></SUB></NU><DE>1−3<A><AC>r</AC><AC>&cjs1171;</AC></A></DE></FR>⟨N(T)⟩.](/content/vol18/issue10/fulltext/3870/img020.gif)
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(15)
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Plugging in 0.8 for the CVISI and 0.2 for , we obtain a stable variance-to-mean ratio of
1.6. If input neurons were less variable than this, then a neuron that
computes the quantity in Eq. 15 would be more variable than its inputs.
If the inputs had a variance exceeding 1.6 times the mean count, then
the output response would be less variable than the inputs.
This point is illustrated by the numerical simulation in Figure
13A. The variance of the
spike count is shown for a neuron that computes the sum and difference
described by Equation 15. We used 100 neurons in each of the three
pools representing the quantities a, b, and c,
and we assumed an average correlation of 0.2 among all neuronal pairs
within the pools. We adjusted the variance of the input neuron spike
counts, using a variance-to-mean ratio of from 0.6 to 1.8, as shown on
the abscissa. The variance of the output neuron response reflects the
uncertainty in the computation of T and the additional
variation attributable to the irregularity of the ISI, which we modeled
as a renewal with CVISI = 0.8. Notice that
the input and output neurons share the same variability when the
variance-to-mean ratio is 1.6, as predicted by Equation 15. This is the
stable value for the variance-to-mean ratio in a network of neurons
that compute quantities that necessitate combining sources of variance,
in the manner of the sum and difference in Equation 12. In Figure
13B we show the stable solution for a range of correlation
strengths. For between 0.1 and 0.2, the variance-to-mean ratio approximates the value measured experimentally (Table 1).
View larger version (13K):
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Figure 13.
Stable solution for the variance-to-mean spike
count in networks of neurons when the computation involves three
independent quantities. A, Numerical approximation to the
stable solution. We simulated the response by estimating three values
using 100 weakly correlated neurons per value ( = 0.2). The three means were combined as a sum and difference
(a + b c) to generate the expected spike count
from the output neuron, the spike train of which was modeled as a
renewal process with CVISI = 0.8. The
variance-to-mean ratio for the input neurons is shown along the
abscissa. The output variance-to-mean ratio is plotted along
the ordinate. The least squares fit line crosses the main
diagonal at a ratio of 1.6. This variance-to-mean ratio would be common
to input and output neurons that compute similarly complex quantities.
B, Effect of correlation on the stable variance-to-mean
ratio. Each point represents a numerical approximation like
the one obtained in A. The procedure was repeated for a
range of average correlation coefficients. The solid line is
the theoretical result (Eq. 18). Simulations led to larger estimates of
variance, because the number of neurons is finite.
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It is sobering to realize that such a simple calculation leads to such
large variance in the neural response. This observation is particularly
ominous for models in which separate computations are performed on
major dendritic branches, with the resulting quantities being summed or
multiplied at the soma (Koch et al., 1983; Shepherd and Brayton, 1987;
Segev, 1992; Mel, 1993). Our results suggest that in terms of response
variance, little headroom is available for such algebra to occur within
the postsynaptic neuron. How, then, do neurons compute complex
quantities such as orientation, disparity, and motion while keeping
response variance stable? In thinking about this question, we find it
useful to distinguish between computations that are performed
within the postsynaptic neuron and computations that are
imposed from outside the postsynaptic neuron by virtue of
its afferent anatomical connections. Our results suggest that the
primary computation performed within most neurons is a relatively
simple weighted average of its many inputs (a slight elaboration of Eq. 7 yields a weighted average). On the other hand, sophisticated
quantities can emerge from a weighted average, provided that the inputs
and their weights are arranged properly. This computational principle
lies at the heart of most neural network models; complex quantities are
computed by means of a network of intricately connected "units,"
each of which calculates a weighted average of its inputs (Pouget et
al., 1998). This principle is also congenial with recent
"normalization" models in which neurons in different visual areas
compute in a similar manner (linear summation followed by response
normalization), whereas the diverse response properties that
characterize different cortical areas emerge from different patterns of
input (Heeger et al., 1996; Simoncelli and Heeger, 1998).
From this point of view, the burden of neural computation falls on the
pattern of connections to each postsynaptic neuron or, by extension, on
the pattern of connections to the cortical column. The present analysis
predicts, in other words, that cortically derived properties such as
orientation and direction tuning will be apparent in the organization
of synaptic inputs to layer 4 of a given cortical column (Ferster et
al., 1996). Outside of layer 4, the majority of neurons receive
ascending input predominantly from other neurons in the same column and
will thus inherit substantially similar response properties. The
intrinsic circuitry of the column may amplify and modulate the
quantities generated by the novel anatomical convergence onto layer 4 (Douglas et al., 1995; Somers et al., 1995).
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DISCUSSION |
We have identified several principles of statistical homogeneity
that pertain to networks of neurons operating in a high-input regime.
The central idea is that from the point of view of statistics, input
and output neurons should be indistinguishable. They respond over the
same dynamic range; they exhibit the same irregularity in their spike
output patterns; and they manifest the same uncertainty in their spike
counts over epochs of a few ISIs. These constraints are satisfied by a
simple integrate-and-fire model with decay, assuming balanced
excitation and inhibition. If cortical neurons operate in this fashion,
then the irregular spike patterns they exhibit ought to be interpreted
as noise. This implies that the intervals themselves do not convey
meaningful information beyond their expression of spike rate. More
specifically, precise arrangements of spikes and constellations of
intervals among ensembles of neurons are unlikely to play a role in
neural
signaling.e The
idea that neurons encode information through modulation of their spike
rate is hardly novel, but we have shown that it takes an ensemble of
neurons to do so reliably and, moreover, that the ensemble need not
exceed 100 neurons. Because the rate can change quickly, our argument
is not inconsistent with the broad concept of temporal coding. Indeed,
Figure 1 illustrates responses of a neuron with a rate that varies
rapidly with changes in the visual stimulus.
Our arguments presuppose that neurons in neocortex receive a surfeit of
excitatory synaptic input. This is likely to be so because the
conditions that cause a neuron to discharge also cause some
considerable fraction of its inputs to discharge as well. This
proposition is difficult to quantify precisely, but it is consistent
with optical imaging studies, which suggest that simple visual patterns
elicit activity in very large populations of neurons (Grinvald et al.,
1994; Das and Gilbert, 1995a), and it conforms to the informal
observations of a great many physiologists. Where response properties
respect a columnar organization, it can be surmised that neurons within
a cylindrical radius of 50-100 µm respond under similar conditions.
Such a cylinder accounts for approximately half of the 3000-10,000
excitatory synapses to a neuron (Braitenberg and Schüz, 1991;
Douglas et al., 1995). Moreover, many of the inputs from outside this
cylinder come via horizontal connections from columns with overlapping
response properties as well (Ts'o et al., 1986; Gilbert and Wiesel,
1989; Weliky et al., 1995; Bosking et al., 1997). These rules are best
documented for the primate visual cortex, but they are probably valid
for other regions of neocortex as well (Braitenberg and Schüz,
1991; Amir et al., 1993; Lund et al., 1993). Thus stimulus conditions that induce one neuron to respond also affect a large number of the
neuron's inputs.
As long as the neuron operates in such a high-input regime, excitation
needs to be balanced with inhibition to maintain a proper dynamic range
of responses. Several considerations suggest that the net excitatory
and inhibitory inputs may be approximately balanced. At a membrane
potential below spike threshold, for example, outward currents follow
weaker electrochemical gradients, but they are longer-lasting than
inward currents, and their synaptic contacts are positioned closer to
the soma (Beaulieu et al., 1992; Kisvarday et al., 1993). In addition,
inhibitory inputs tend to make multiple synapses on a given neuron,
ensuring more secure transmission (e.g., fewer failures) (Somogyi et
al., 1983b; Thomson et al., 1996). To the extent that excitation of the
distal dendrite relies on active (voltage-dependent) dendritic
conductances to reach the soma (Amitai et al., 1993; Schwindt and
Crill, 1995; Stuart and Sakmann, 1995; Johnston et al., 1996; Schwindt
and Crill, 1997a), inhibitory input may have a disproportionately large
impact by affecting the gain of this amplification mechanism (Bernander
et al., 1994; Hoffman et al., 1997; Schwindt and Crill, 1997b).
Finally, powerful evidence favoring such a balance emerges from
intracellular recordings performed by Ferster (1986) in simple and
complex cells of the visual cortex in the cat (also see Douglas et al.,
1991; Nelson et al., 1995; Borg-Graham et al., 1996). Contrary to
previous expectations, IPSPs, like EPSPs, were elicited predominantly
by a bar of the preferred orientation of the cell. This
finding is entirely sensible from the point of view of the counting
model; the massive excitatory input characteristic of the high-input
regime must be balanced by inhibitory input of similar
stimulus selectivity.
In the high-input regime, in which excitation and inhibition are in
approximate balance, synaptic integration may be viewed as a counting
process resembling a random walk. We emphasize that our conclusions
apply only to neurons operating in the high-input regime. In such a
setting the neural response reflects the activity of a large ensemble
of inputs, and the arrival of any one presynaptic spike has negligible
effect on the time of a postsynaptic spike. In some brain structures,
of course, the high-input regime does not hold, and a small number of
inputs reliably produces an action potential. This appears to be the
case for the visual relay neurons in thalamus (Hubel and Wiesel, 1961;
Hoffmann et al., 1972; Kaplan and Shapley, 1984) and is paramount in
brainstem auditory structures that preserve precise temporal
information in their spike trains (Oertel, 1983). Such "privileged"
connections may also exist at the initial input stage of the neocortex.
For example, individual afferents from the thalamus seem to exert a
strong influence on visual cortical neurons (Tanaka, 1983; Reid and
Alonso, 1995; Stratford et al., 1996). We suspect, however, that this
is true for only a small minority of neocortical connections, and we
have ignored them in this paper by adopting the generic perspective that the statistical properties of input and output neurons are grossly
indistinguishable.
Simplifications in the counting model
The counting model we have analyzed incorporates several flawed
assumptions, most for the sake of simplicity. For example, the model
assumes that all PSPs are of identical magnitude. Obviously, this is an
overly simple view; different synapses can vary in strength as a
function of their location on the dendritic tree, their history of
activation on long time scales as in long-term potentiation and
depression, and short time scales as in adaptation. Inputs arriving via
stronger synapses might be expected to produce a postsynaptic spike
with greater probability, but the importance of this stratagem deserves
further scrutiny in the context of the high-input regime under
consideration here. Although it is true that synchronous input from
just a few strong synapses would produce a spike, such events occur
against a setting in which the neuron is inundated with synaptic input.
Thus it is likely that such spikes, secure and well timed as they may
be, are responsible for only a minority of the output spikes of a
neuron.
Interestingly, recent advances in synaptic physiology raise the
possibility that variation in synaptic efficacy plays a less pivotal
role in shaping the response of the neuron than previously thought.
Active conductances along the dendrite can increase the effect at the
soma of distal synaptic input, thus ensuring more equal access to the
spike-generating region of the neuron for all synaptic currents
(Cauller and Connors, 1994; Johnston et al., 1996; Cook and Johnston,
1997). Importantly, increased access to the soma via active dendritic
conductances reinforces the central problem motivating our
preoccupation with the high-input regimen: all 3000-10,000
synaptic inputs to a neuron can influence the discharge of the cell,
regardless of their distance from the soma.
In our model we have ignored conductance changes and active membrane
properties. Obviously, synaptic integration is not really a counting
process but an electrical one involving dynamic conductance changes. A
proper model would incorporate the variety of ion channels, their
locations on the dendrite, and the propagation of current through the
dendrite to the soma. These models are terribly complicated and are
therefore difficult to incorporate into network simulations. One
salient example of this type of model, moreover, does not predict an
irregular ISI (Softky and Koch, 1993). Douglas et al. (1995) have
proposed a conduction-based model that makes explicit account of the
network of connections of the neuron, but their model does not deal
with the time of synaptic inputs and therefore offers limited insight
into the statistics of the neural spike train. Recently, Troyer and
Miller (1996) analyzed a realistic model of the layer 5 pyramidal cell
using a balance of excitation and inhibition. Their model is especially
elegant because the parameters (e.g., conductances) were derived from
intracellular recording in the striate cortex of the cat (McCormick et
al., 1985). This model produces an irregular ISI and therefore supports our contention that the simplified counting process captures essential characteristics of more realistic neural models. In Appendix we
implement a conductance-based model similar to the one used by Troyer
and Miller (1996) to determine how it performs in the high-input regime
with which we are concerned. The model replicates the main results
obtained in this paper with the simpler counting model, although it
seems to require a substantially larger synaptic conductance
(approximately a factor of eight) than has been measured experimentally
(Borg-Graham et al., 1996). In fact, it is very difficult to measure
synaptic conductance changes in vivo under conditions
leading to the kind of response shown in Figure 1. We suspect that
existing physiological studies underestimate the actual conductance,
but we also believe that single-compartment models such as ours tend to
overestimate conductance (see Appendix ). Plainly, substantially more
work is needed on both the experimental and theoretical fronts to
resolve this discrepancy.
Fortunately, the key insight of the random walk model does not depend
critically on biophysical details. It is that the path followed by the
membrane voltage from one spike to the next is complex, reflecting a
large number of synaptic events. The detailed history of input activity
cannot be revealed in the time between output spikes, because there are
many possible paths leading to each ISI. Whether it is balanced
inhibition or some other mechanism that allows hundreds of synaptic
inputs to result in just one spike, the detailed timing relationships
of inputs at the dendrites cannot be reconstructed from the output
spike discharge. A similar argument has been put forth by van Vreeswijk
and Sompolinsky (1996) using an entirely different mathematical
framework.
Principles of statistical homogeneity
We have emphasized the notion that the statistical properties of
the neural response are similar for input and output neurons alike.
These properties, summarized in Table 1, should be interpreted as loose
approximations. Some neurons, of course, respond more strongly than
others, and spiking patterns can vary to some extent (McCormick et al.,
1985; Gray and McCormick, 1996). Nevertheless, it seems likely that the
stimulus conditions that drive a given neuron produce similar changes
in an appreciable minority of its inputs and immediate targets.
Moreover, the neuron's inputs and its targets probably respond over a
similar range of spike rates. This general point of view is central to
our analysis, but it is little more than a caricature of a somewhat
heterogeneous population of neurons. The problem of excess excitation
and the consequences of weak correlation that we have discussed do not
depend on an exact sense of homogeneity but on this relaxed sense of a
common dynamic range.
In addition to a common dynamic range, cortical neurons exhibit similar
response statistics. Interspike intervals and spike counts obey similar
properties for cortical inputs and targets (Softky and Koch, 1993). The
balanced excitation-inhibition model presented in section 1 of Results
mimics this property of cortical neurons, consistently leading to a
CVISI near 0.8. Interestingly, the model
suggests that this degree of variability in the output spike train
would result nearly independently of the degree of variability of the
input spike trains, as indicated in Figure 5. Thus a network of neurons
conforming to our model of synaptic integration will find a steady
state near the cited value. This finding might help resolve the puzzle
of how relatively regular spike trains from thalamus (Funke and
Worgotter, 1995) give rise to such irregular spike trains in
cortex.
In section 3 of Results we tried to identify conditions that would
allow the variance of the spike count to maintain a fixed relationship
to its mean through several stages of processing. This is a critically
important property of cortical circuitry, because a quantity computed
by neurons at one location in the cortex should not be overwhelmed by
an accumulation of noise across subsequent synaptic connections leading
to a motor response. Across a wide range of cortical areas, the
variance of the neural response appears to remain approximately
constant at ~1-1.5 times the mean count.
We quantified two sources of response variance in cortical neurons: the
near-Poisson nature of the synaptic integration process (the random
walk model) and response covariance resulting from common input.
Together these factors account for a surprisingly large amount of the
experimentally observed variance, yielding a variance-to-mean ratio of
~0.8. This result is quite unexpected, because it leaves little
headroom for additional variance that would accompany computations more
complex than taking a mean (as in our simulations). For example, a
relatively simple operation that incorporates just two additional terms
leads to a stable variance-to-mean ratio of 1.6, fully accounting for
experimentally observed variability. Disconcertingly, Gur et al. (1997)
have recently raised the possibility that experimentally measured
variance is in fact exaggerated (but see Bair and O'Keefe, 1998),
implying that neural calculations may be simpler even than the
three-term operation considered above.
As we have shown, response covariance resulting from common input
sacrifices fidelity in the sense of signal/noise, but it must be
remembered that this redundancy yields a great benefit: speedy
transmission of signals from point to point in the cortex. Input and
output neurons can only calculate over the same time scale if there is
no accumulation of waiting time for the requisite information to arrive
at subsequent stages (Knight, 1972). We have assumed that a natural
time scale for computation is on the order of 10 msec, which is about
half of a membrane time constant, or an average ISI of a neuron with a
rate of discharge of 100 spikes/sec. Computation on this time scale is
only possible if the neuron receives many samples of the quantities
that it needs for its calculations. This convergence, or fan-in, of
many inputs imposes a requirement of shared connections and,
consequently, limits the fidelity of signals embedded in noise, even
after averaging.
Finally, our simulations revealed new insights into the effects of
spike synchrony in the high-input regime, insights that may ultimately
form a fifth homogeneity principle if confirmed by more extensive
measurement and simulations. Succinctly, the fraction of synchronous
spikes appears to be conserved from an input ensemble to an output
ensemble when a common signal is passed from stage to stage in a
processing pathway (Fig. 11). Common input leads to a modest percentage
of synchronous spikes in an ensemble of neurons, but the synchronous
spikes do not appear to propagate consistently or to establish a
consistent pattern of activity among neurons at the next level of
processing. We failed to observe any tendency for synchronous spikes to
dominate the population response as signals propagate from one stage to
the next. Again, this observation follows naturally from the exigencies
of the high-input regime. Given the large numbers of input spikes that arrive in a given membrane time constant, each arriving spike is
effectively synchronous with some population of inputs, although the
composition of the "synchronous" population changes randomly from
epoch to epoch. Thus synchrony is ubiquitous in the high-input regime;
"special" spikes defined by consistent synchrony among a specific
subpopulation of input neurons will have no more effect postsynaptically than the random collections of synchronous inputs that
arise stochastically in the high-input regime. Synchronous spikes do
not represent anything extraordinary.
Concluding remarks
Together, these principles inform a view of information coding by
cortical neurons that draws diverse physiological and anatomical data
into a coherent framework. In the high-input regime, neurons attain
their dynamic range of response by balancing excitation with
inhibition. As a consequence, they spike with great irregularity and
lack the capacity to transmit information in patterns of spikes. The
spike output for a single neuron therefore resembles a stochastic point
process, and the rate term, (t), is the currency of
information transfer from neuron to neuron. Because the spikes from any
one neuron arrive with such irregularity, however, (t)
must be represented in a population of neurons to gain signal/noise
and, more importantly, speed. Neurons in such a signaling population
must share common input, resulting in a certain amount of common noise
that ultimately limits the fidelity of signal transmission. Within this
limit, rate changes can be signaled in approximately one ISI using
pools of ~100 neurons, but little or no improvement is gained with
larger pools.
Finally, quantitative comparison of known noise sources and
experimentally observed response variance suggests that a single neuron
is unable to compute quantities much more complicated than a weighted
average of the spikes arriving at its dendritic field. This implies
that interesting neural response properties do not arise from
sophisticated computation within single neurons but rather
reflect the anatomical convergence of novel combinations of inputs to
the ensemble (the column) from external sources. This insight is
consistent with the basic computational principles incorporated in
neural network models. From this point of view, the basic function of
the column is to amplify these convergent inputs while controlling the
gain of single neurons so that in any interspike interval sufficient
spikes are available to represent appropriate quantities for subsequent
computation. Several investigators have emphasized the importance of
cortical amplification and gain control (Douglas and Martin, 1991;
Heeger, 1992b; Douglas et al., 1995; Stratford et al., 1996).
Ideas about synaptic integration and ideas about neural computation
tend to use different languagesthe first involving conductances, membrane properties and spikes and the second having reference primarily to formal mathematical operations. Our analysis attempts to
form a framework for unifying the view of the neuron as an integrator
of synaptic input with the problem of what and how a neuron
computes.
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FOOTNOTES |
Received Sept. 15, 1997; revised Feb. 25, 1998; accepted March 3, 1998.
This research was supported by National Institutes of Health Grants
EY05603, RR00166, EY11378 and the McKnight Foundation. W.T.N. is an
Investigator of the Howard Hughes Medical Institute.
We are grateful to Richard Olshen, Wyeth Bair, and Haim Sompolinsky for
advice on mathematics, Marjorie Domenowske for help with illustrations,
and Crista Barberini, Bruce Cumming, Greg DeAngelis, Eb Fetz, Greg
Horwitz, Kevan Martin, Mark Mazurek, Jamie Nichols, and Fred Rieke for
helpful suggestions. We are also grateful to two anonymous reviewers
whose thoughtful remarks improved this paper considerably.
Correspondence should be addressed to Dr. Michael N. Shadlen,
Department of Physiology, University of Washington Medical School, Box
357290, Seattle, WA 98195-7290.
dWeak correlation between the pools
representing a, b, and c would not affect the
calculations substantially, because the resulting covariance terms
would offset each other due to the sum and difference.
a
We were advised of this relationship by
H. Sompolinsky and W. Bair. The area of the correlogram is:
where  () = T || is a triangular
weighting function with a peak that lies at the center of the trial
epoch of duration, T msec, and
is the normalized cross- or autocorrelation function computed
from bins of binary values,
xj|k(i), denoting the presence or
absence of a spike in the ith millisecond
from neuron j or k. Mathematical details and a
proof of Equation 1 will appear in a paper by E. Zohary, W. Bair, and W. T. Newsome (unpublished data).
b
Knowledge of does not imply exact
knowledge of the input spike trains. Presumably there are many patterns
of inputs that give rise to the same result, . Because our model for
synaptic integration is deterministic, identical inputs would produce
identical outputs. If x were an exhaustive description of
the input spike trains, that is, the time of every spike among all
excitatory and inhibitory inputs, then var[N(T)|x] = 0. To make sense of Equation 5, we need to make it clear that what we
know about the inputs, as reflected in the conditional probabilities,
is a scalar value that is computed from them, i.e., .
c
For many stochastic processes,
including our random walk model, a change in rate is equivalent to
scaling time. Hence the CV is constant. For
these cases, Equation 9 also holds for time varying spike rates,
(t), as long as the rate function can be repeated for
each epoch contributing to the conditional variance. Thus, for a
nonstationary Poisson process, the variance of the counts equals the
mean of the counts.
e
One way to appreciate this is to try to
reconstruct the spike intervals among a subset of input neurons from
the pattern of output spikes. This is obviously impossible with just
one output neuron, but it is equally hopeless with an arbitrarily large
number of output neurons. The requisite information is encoded in the path of the membrane voltage between spikes, but such information is
jettisoned from the spike code. If, in principle, there is no way to
reconstruct such intervals, then they cannot encode information, except
insofar as they reflect changes in spike rate.
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APPENDIX 1 |
Here we describe a simple conductance-based model with properties
similar to the random walk counting model described in the text.
Instead of counting to a threshold, we have modeled synaptic inputs as
brief changes in Na+ (excitatory inputs) or
Cl (inhibitory inputs) conductance, and solved for
the membrane voltage. The membrane voltage, V(t), is
described by the differential equation:
|
(A1.1)
|
where Cm is the membrane capacitance, and
I represents the transmembrane current attributable to
excitatory synaptic input (IEx), inhibitory
input (IIn), a spike-triggered potassium current (IK), and a leak current
(Ileak). The model contains just one compartment
(i.e., no variation in V as a function of location).
The synaptic currents were computed by multiplying a time-varying
conductance by the driving force:
|
(A1.2)
|
The neuron receives 300 excitatory inputs and 300 inhibitory
inputs. Each excitatory synaptic conductance was modeled by a simple
exponential decay:
|
(A1.3)
|
with gE_peak = 6 nS, and Ex = 2 msec. The peak conductance was set so that 15 synchronous inputs
would elicit a spike from rest (70 mV), as illustrated in Fig.
14B. Spike threshold was 55 mV, which implies that the average EPSP was just >1 mV when the
membrane was at rest. The inhibitory conductances were modeled similarly, but they were larger and longer-lasting
(gI_peak = 11.25 nS; In = 4 msec). These parameters were chosen to balance the excitation,
thereby yielding an output spike rate equal to the spike rates of the
inputs (see below). An IPSP had no impact on the resting membrane,
because ECl = Vrest = 70 mV. In addition to the synaptic conductance, we modeled a leak
conductance, Ileak = gleak(V Vrest),
which was not time-varying. The leak conductance and membrane
capacitance were chosen to achieve a 40 M input resistance and 20 msec time constant at rest (gleak = 25 nS).
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Figure 14.
A single-compartment model using brief synaptic
conductance changes instead of voltage steps. The model behaves like
the simpler random walk counting model. The resting potential is 70
mV, m = 20 msec, and spike threshold is 55 mV.
A, Response to 0.75 nA current step. The trace lasts 250 msec. B, Fifteen synchronous excitatory synaptic inputs
depolarize the neuron sufficiently to trigger a spike. C,
Response to simulated current steps of varying size. The
f-I plot depicts the steady-state firing rate as a function
of injected current. D, Response to 300 excitatory and 300 inhibitory inputs. Each input neuron spikes at an average of 80 spikes/sec. The size of the inhibitory conductance was adjusted to
produce an average spike rate similar to the input rate. Notice that
the membrane voltage hovers near spike threshold. E, ISI
distribution from the simulated response, which includes the trace in
C. Solid line is an exponential fit to the
distribution. F, Average synaptic conductance preceding a
spike. The spike-triggered average excitatory conductance is shown by
the solid curve (left ordinate scale); the
inhibitory conductance is shown by the dashed curve
(right ordinate scale). Although the effective membrane time
constant is ~0.5 msec, the synaptic activity affects the neuron for
5-10 msec preceding a spike.
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Spikes were modeled as brief voltage transients, which we simply added
to the membrane voltage:  Vm = 55e2t mV (i.e., a decay time constant
of 0.5 msec). This model does not incorporate a voltage reset, but each
spike triggered a K conductance (peak conductance, 0.15 µS;
EK = 90 mV; K = 40 msec), leading to the spike frequency adaptation shown in Figure
14A. The K current, leak conductance and membrane
capacitance produce the f-I curve shown (Fig.
14C), which plots the steady-state firing rate as a function
of current step amplitude. The model is similar to one described by
Troyer and Miller (1996) except for the large number of synaptic inputs
and absence of any reset voltage.
We stimulated the model with 300 excitatory inputs, each modeled as
independent Poisson spike trains with an average rate of 80 spikes/sec.
The inhibitory inputs were of the same number and rate. We adjusted the
peak inhibitory conductance so that the model produced an output of
78.9 spikes/sec. A half-second sample of the model neuron response is
shown in Figure 14D. As with our counting model, the cost of
a reasonable firing rate is an irregular ISI. This is apparent in the
membrane voltage sample and in the distribution of ISIs (Fig.
14E) obtained from a 10 sec simulation. The
CVISI was 0.77 for this simulation, which is near the upper bound for a Poisson process with a refractory dead
zone (Troyer and Miller, 1996).
To achieve a firing rate that approximates an average input, the
peak inhibitory conductance was set to 11.25 nS (In = 4 msec). This led to a very large conductance, as shown in Figure 14F. With 300 excitatory and inhibitory inputs, the model
incorporated a total synaptic conductance of >600 nS, which is ~25
times the conductance at rest. Interestingly, this massive conductance
change does not produce a large net current; only 2-4 nA is sufficient to produce the simulated spike trains in Figure 14D. In
contrast to the large conductances incorporated in our model,
Borg-Graham et al. (1996) measured a 200-300% change in conductance
using whole-cell patch recordings of cortical neurons in
vivo during periods of modest discharge (we know of no other
comparable measurements of synaptically activated neurons under the
high-input conditions that interest us). We suspect that the
discrepancy results both from an underestimate of the actual
conductance by Borg-Graham et al. (1996) and to an overestimate in our
model. Borg-Graham et al. (1996) estimated the conductance by measuring
current-voltage relationships during voltage clamp of the cell soma.
To the extent that the synapses are electrotonically distant from the
soma, however, these measurements will inevitably underestimate
synaptic conductance changes. On the other hand, our single-compartment model probably overestimates the total conductance. In a more realistic
model, which includes the dendrites, it is likely that less inhibitory
conductance would be required to balance a given amount of excitatory
input [e.g., if excitatory synapses are further from the soma or if
inhibitory synapses were to interfere with dendritic amplification
(Cook and Johnston, 1997; Hoffman et al., 1997; Schwindt and Crill,
1997b)]. We therefore suspect that the actual state of affairs is
intermediate between that estimated by Borg-Graham et al. (1996) and
that incorporated in our model. Measurements of the dendritic
conductance during vigorous sensory stimulation will be necessary to
resolve this discrepancy fully.
There is one aspect of the simulation that we regard as deceptive. The
membrane voltage appears to hover near spike threshold and to cross
occasionally at random intervals, resembling the coincidence detector
depicted in Figure 2, D and E. In light of the
large conductance, it might be argued that the membrane time constant
is effectively shortened from 20 to <1 msec (Koch et al., 1995), and
that therefore the model neuron only spikes when it senses rare
coincidences of excitatory input within a time frame of ~1 msec.
Although the effective integration time is indeed short, spikes are not
the result of an occasional volley of excitation. As shown in Figure
14F, the events preceding a spike include a pause in the
inhibition as well as a rise in excitation. In this simulation the
neuron was affected by synaptic input throughout much of the ISI (mean
ISI was 14 msec). Moreover, the increased excitation immediately
preceding a spike hardly constitutes a rare coincidence. The change in
excitatory conductance (Fig. 14F, solid line) was just
20 nS, on average, consistent with just three or four extra synaptic
inputs over the background. In the simulation depicted here, such
excesses occur in 1 of 4 msec. In other words, the putative
coincidences are ubiquitous. In general, spikes are not the consequence
of particular patterns of excitatory input, because the sequence of
inhibitory inputs also affects the time of the postsynaptic spike.
Inhibition does not merely shorten the integration time for excitation.
Thus, unlike the coincidence detector portrayed in Figure 2,
D and E, the present model would require a fine
orchestration of excitatory and inhibitory synaptic input to produce
reliably timed spikes. This insight is likely to hold for other
biophysical implementations of balanced excitation-inhibition, as long
as the balancing inhibition comes from spiking neurons.
The conductance-based model described here embodies the same
constraints as our simpler counting model, and it achieves a highly
irregular ISI. It is just one example of a biophysical implementation
of the random walk to a barrier idea that solves the problem faced by
neurons that receive a surfeit of excitatory input. Although the
conduction-based model may be more biophysically plausible than the
counting model pursued in the main text, it remains highly simplified
and sacrifices much of the conceptual simplicity inherent in counting
to a barrier. It would be useful to determine whether a more realistic
model using active dendritic conductances (e.g., Cook and Johnston,
1997) would behave similarly.
| |
APPENDIX 2 |
Here we derive the standard error of the mean spike count from
m weakly correlated neurons over a duration lasting one
average ISI. Throughout the paper we refer to a computation of the
average input spike rate, in which the expected output spike rate is
the same, on average, as any one of its active inputs. Equation 7 expresses the expected spike rate as a function of the m
input spike counts transmitted in an epoch, T. The expected
spike count is:
|
(A2.1)
|
Each of the m inputs is a random quantity with variance
denoted var[ni(T)]. If the
m input counts are independent, then the variance of the
mean is just the average variance of one input, divided by
m:
![<UP>Var</UP>[⟨&lgr;T⟩]=<FR><NU>1</NU><DE>m</DE></FR>⟨<UP>Var</UP>[n<SUB><UP>i</UP></SUB>(T)]⟩<SUB><UP>i</UP></SUB>=<FR><NU>1</NU><DE>m<SUP>2</SUP></DE></FR> <LIM><OP>∑</OP><LL>i<UP>=</UP>1</LL><UL>m</UL></LIM> <UP>Var</UP>[n<SUB><UP>i</UP></SUB>(T)],](/content/vol18/issue10/fulltext/3870/img026.gif)
|
(A2.2)
|
where ···i denotes the average across all
values of i. If the m inputs are weakly
correlated, then the variance of the mean is derived from the sum of
the covariance among the m inputs:
![<UP>Var</UP>[⟨&lgr;T⟩]=<FR><NU>1</NU><DE>m<SUP>2</SUP></DE></FR> <LIM><OP>∑</OP><LL>i<UP>=</UP>1</LL><UL><UP>m</UP></UL></LIM> <LIM><OP>∑</OP><LL>j<UP>=</UP>1</LL><UL><UP>m</UP></UL></LIM> <UP>Cov</UP>[n<SUB><UP>i</UP></SUB>, n<SUB><UP>j</UP></SUB>]](/content/vol18/issue10/fulltext/3870/img027.gif)
|
(A2.3)
|
where rij is the correlation coefficient
between the ith and
jth input. This is just the sum of the
elements forming the m-by-m covariance matrix. If
all m inputs are identically distributed, then we can
eliminate the subscripts on the terms for variance and substitute the
mean correlation coefficient, , for
rij:
![<UP>Var</UP>[⟨&lgr;T⟩]=<FR><NU>1</NU><DE>m<SUP>2</SUP></DE></FR>(m <UP>Var</UP>[n(T)]+m(m−1)<A><AC>r</AC><AC>&cjs1171;</AC></A> <UP>Var</UP>[n(T)])](/content/vol18/issue10/fulltext/3870/img029.gif)
|
(A2.4)
|
If the m inputs do not share the same mean and
variance, as is often the case, then a useful approximation is to
substitute the average variance:
![<UP>Var</UP>[⟨&lgr;T⟩]≈<A><AC>r</AC><AC>&cjs1171;</AC></A>⟨<UP>Var</UP>[n<SUB><UP>i</UP></SUB>(T)]⟩<SUB><UP>i</UP></SUB>, <UP>for large </UP>m.](/content/vol18/issue10/fulltext/3870/img031.gif)
|
(A2.5)
|
In practice, m need only be 100 for the
approximations in Equations A2.4 and A2.5 to hold.
For the special case in which
ni(T) is the count of events
in a Poisson point process, and the duration,
TISI = 1, is the expected ISI,
we have:
![<UP>Var</UP>[n<SUB><UP>i</UP></SUB>(T<SUB><UP>ISI</UP></SUB>)]=⟨n<SUB><UP>i</UP></SUB>(T<SUB><UP>ISI</UP></SUB>)⟩=1.](/content/vol18/issue10/fulltext/3870/img032.gif)
|
(A2.6)
|
Substituting into Equation A2.4 yields:
![<UP>Var</UP>[⟨&lgr;T<SUB><UP>ISI</UP></SUB>⟩]=<FR><NU>1+m<A><AC>r</AC><AC>&cjs1171;</AC></A>−<A><AC>r</AC><AC>&cjs1171;</AC></A></NU><DE>m</DE></FR>,](/content/vol18/issue10/fulltext/3870/img033.gif)
|
(A2.7)
|
from which follows the expression in Equation 3 and the curves in
Figure 10.
| |
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Top
Abstract
Introduction
![=⟨<UP>Var</UP>[N(T)‖&lgr;]⟩+<UP>Var</UP>[⟨&lgr;T⟩].](/content/vol18/issue10/fulltext/3870/img011.gif)
![A<SUB><UP>jk</UP></SUB>=<LIM><OP>∑</OP><LL>&tgr;<UP>=−</UP>50</LL><UL>50</UL></LIM> &THgr;(&tgr;)[C<SUB><UP>jk</UP></SUB>(&tgr;)−1],](/content/vol18/issue10/fulltext/3870/img034.gif)
![=<FR><NU>1</NU><DE>m<SUP>2</SUP></DE></FR><FENCE><LIM><OP>∑</OP><LL>i<UP>=</UP>1</LL><UL><UP>m</UP></UL></LIM> <UP>Var</UP>[n<SUB><UP>i</UP></SUB>(T)]+<LIM><OP>∑</OP><LL>i<UP>=</UP>1</LL><UL><UP>m</UP></UL></LIM> <LIM><OP>∑</OP><LL>j<UP>≠i</UP></LL><UL><UP>m</UP></UL></LIM> r<SUB><UP>ij</UP></SUB><RAD><RCD><UP>Var</UP>[n<SUB><UP>i</UP></SUB>(T)]<UP>Var</UP>[n<SUB><UP>j</UP></SUB>(T)]</RCD></RAD></FENCE>,](/content/vol18/issue10/fulltext/3870/img028.gif)
![≈<A><AC>r</AC><AC>&cjs1171;</AC></A> <UP>Var</UP>[n(T)], <UP>for large </UP>m.](/content/vol18/issue10/fulltext/3870/img030.gif)
