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The Journal of Neuroscience, August 15, 2000, 20(16):6193-6209
Impact of Correlated Synaptic Input on Output Firing Rate and
Variability in Simple Neuronal Models
Emilio
Salinas1 and
Terrence J.
Sejnowski2
1 Computational Neurobiology Laboratory,
Howard Hughes Medical Institute, The Salk Institute for Biological
Studies, La Jolla, California 92037, and
2 Department of Biology, University of California
at San Diego, La Jolla, California 92093
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ABSTRACT |
Cortical neurons are typically driven by thousands of synaptic
inputs. The arrival of a spike from one input may or may not be
correlated with the arrival of other spikes from different inputs. How
does this interdependence alter the probability that the postsynaptic
neuron will fire? We constructed a simple random walk model in which
the membrane potential of a target neuron fluctuates stochastically,
driven by excitatory and inhibitory spikes arriving at random times. An
analytic expression was derived for the mean output firing rate as a
function of the firing rates and pairwise correlations of the inputs.
This stochastic model made three quantitative predictions. (1)
Correlations between pairs of excitatory or inhibitory inputs increase
the fluctuations in synaptic drive, whereas correlations between
excitatory-inhibitory pairs decrease them. (2) When excitation and
inhibition are fully balanced (the mean net synaptic drive is zero),
firing is caused by the fluctuations only. (3) In the balanced case,
firing is irregular. These theoretical predictions were in excellent
agreement with simulations of an integrate-and-fire neuron that
included multiple conductances and received hundreds of synaptic
inputs. The results show that, in the balanced regime, weak
correlations caused by signals shared among inputs may have a
multiplicative effect on the input-output rate curve of a postsynaptic
neuron, i.e. they may regulate its gain; in the unbalanced regime,
correlations may increase firing probability mainly around threshold,
when output rate is low; and in all cases correlations are expected to
increase the variability of the output spike train.
Key words:
random-walk; integrate-and-fire; computer simulation; spike synchrony; oscillations; cross-correlation; balanced inhibition; cerebral cortex
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INTRODUCTION |
The output of a typical cortical
neuron depends on the activity of a large number of synaptic
inputs several thousands of them, as estimated by anatomical
techniques (White, 1989 ; Braitenberg and
Shüz, 1997). What kind of response should be expected
from a postsynaptic neuron driven by so many inputs? Answering this question in detail requires a deep understanding of dendritic integration, synaptic function, and spike generation mechanisms; but,
given the large numbers commonly involved, as a first approximation it
is natural to cast the problem in statistical terms. The strategy then
is to compute the output responses of a model neuron (or their
statistics), given a set of driving inputs with known statistical properties. These inputs may be either independent or temporally correlated. In the latter case, spikes from different input neurons arrive close together in time more often or less often than expected by chance.
In general, the situation with independent inputs is easier to analyze,
and for many applications it is probably a good approximation. However,
there are at least three reasons why the effects of correlations on
single cells should be fully characterized. First, correlations in
spike counts have indeed been observed (Gawne and Richmond, 1993; Zohary et al., 1994; Salinas et
al., 2000) and, based on the convergent connectivity of the
cortex (White, 1989; Braitenberg and
Schüz, 1997), they must be ubiquitous (Shadlen and
Newsome, 1998; Bair et al., 1999). Second, such
correlations may alter the coding capacity of a neuronal population
(Gawne and Richmond, 1993; Zohary et al.,
1994; Abbott and Dayan, 1999). Third,
synchrony and oscillations, two forms of correlated activity that have
been intensely studied, may also be important for information encoding (DeCharms and Merzenich, 1995; Riehle et al.,
1997; Dan et al., 1998) or for other aspects of
cortical function (Engel et al., 1992; Singer and
Gray, 1995). This paper, however, does not focus on the
possible higher-level functional roles of coordinated spike firing;
instead, it addresses a more elementary problem: how does a typical
cortical neuron react to synaptic inputs that are correlated, compared
to synaptic inputs that are uncorrelated?
This problem has been investigated in the past (Bernander et
al., 1994; Murthy and Fetz, 1994; Shadlen
and Newsome, 1998), but the model neurons used earlier have
often been examined with limited sets of parameters, and sometimes in
regimes outside the normal operating range of cortical neurons; for
instance, some studies have ignored the effects of inhibition. This
study attempts to provide a broad framework within which the impact of
input correlations on a single postsynaptic neuron can be better
understood. Using a simple theoretical model, the mean firing rate of a
postsynaptic neuron is solved as a function of the firing rates and
pairwise correlations of its excitatory and inhibitory inputs. This
model also provides qualitative insight on how correlations affect
output variability. The analytic expressions are then compared to
computer simulations of a conductance-based model neuron with more
realistic dynamics. We find that correlations affect both the firing
rate and variability of the output and that the strength and details of
these effects depend strongly on the balance between excitation and inhibition.
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MATERIALS AND METHODS |
A theoretical model with random walk dynamics.
Consider a simple stochastic model neuron in which an incoming
excitatory spike increases the membrane potential by an amount
 E, and each incoming inhibitory spike
decreases the membrane potential by an amount I. These voltage steps are fixed, and are
independent of the input statistics. In the absence of synaptic input,
the voltage of the model neuron, termed V, decreases by a
fixed amount d in each time step, but there is a fixed
minimum Vrest below which the voltage cannot be
driven, even if inhibition is strong. The d term makes the
voltage decrease linearly with time toward
Vrest. Because of leak currents, membrane
potentials of real neurons actually relax exponentially to their rest
values, but approximating this with a linear term may be reasonable
when V remains relatively far from rest. In addition,
whenever the voltage exceeds a threshold V ,
an action potential is fired, and the voltage is instantaneously reset
to the value Vreset. Given specific values for
these six parameters, the output of the model neuron will be entirely
determined by the statistics of the inputs. The advantage of these
simple dynamics is that, if the input statistics are known and certain simplifying assumptions are made, then the output firing rate may be
computed analytically, revealing the explicit dependence on the input
statistics. This is shown in the following sections.
The analysis follows in the tradition of classic results from the
theory of stochastic processes (Ricciardi, 1977;
Tuckwell, 1989; Risken, 1996). Many of
the previous studies that applied these random walk methods to the
problem of synaptic integration were aimed at understanding, in terms
of a simple mechanistic explanation, how spike firing in a neuron is
triggered by the stochastic fluctuations of its membrane potential
(Tuckwell, 1988; Smith, 1992). In other
studies the goal was to develop models that could account in detail for
the measured firing statistics of real neurons (Gerstein and
Mandelbrot, 1964; Shinomoto et al., 1999). As
shown below, this framework is also of heuristic value to the problem
of input correlations and their impact on firing probability
(Feng and Brown, 2000).
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RESULTS |
Changes in voltage modeled as random walk steps
According to the above description, at each time step
t the voltage jumps by an amount:
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(1)
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where nE and nI are
the total numbers of incoming excitatory and inhibitory spikes that
arrived in that interval t. By defining the net number of
excitatory spikes as:
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(2)
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the change in voltage can be written as:
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(3)
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The net number of excitatory spikes will vary randomly from one
time step to the next. The chance of n having a given
specific value at any particular time step is characterized by the
probability distribution P(n), such that µ =  n and  2 = (n  µ)2 correspond, respectively, to the mean and
variance of n. Throughout the paper, angle brackets are used
to indicate an average over time steps. A positive value of µ indicates a mean excess of excitatory drive versus inhibitory drive in
each t, whereas represents the
fluctuations in the drive. Because changes in voltage are proportional
to n, V will be linearly related to the net number of
excitatory spikes that have accumulated since the last output spike was
emitted:
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(4)
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Thus, N changes by n in each time step, it
has a lower limit of Nrest, it needs to reach a
critical value N for the postsynaptic neuron
to fire again, and is reset to Nreset after each
postsynaptic spike. N is obtained when
V = V in the above equation, and the
same is true for the other values specified by their subscripts. For
convenience we will set Nrest = 0; this
choice does not alter the results in any significant way, because what
counts is the difference between N and
N.
Given that in each time step N changes by a random amount,
N (and therefore V) is equivalent to the
net displacement of a one-dimensional random walk process with drift in
which there is a reflecting barrier at one end and an absorbing barrier
at the other. What we want to know is the average number of steps  that it takes for N to go from reset to threshold. This is
the same as asking how much time it typically takes for V to
go from Vreset to V.
The total amount of time will be:
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(5)
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This is the mean interspike interval of the output neuron. For a
random walk, this time is known as the mean time to capture (Berg, 1993). This, or its reciprocal, the mean firing
rate rout, can be computed making some
assumptions about the probability distribution of n. The
derivation is left for the Appendix, but the main intuition is this: on
average, in each time step the net change in N is µ. If
is small, should be approximately N/µ. Now suppose instead that
µ = 0 so there is no drift. In this case N
just fluctuates around its initial value. After steps, however, the
typical displacement (positive or negative, in the root mean square
sense) relative to the starting point is 
(Feynman et al., 1963). Hence, now it should take
on the order of (N/)2 steps
for N to reach a point N units
away. In general, then, it would seem that either µ or may drive
the neuron to fire. A more detailed analysis confirms this idea and
leads to the following expressions (see Appendix). When µ  0,
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(6)
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When µ > 0 there is a net excitatory drive and,
in general, both µ and tend to increase the firing rate, although
this is not true for all combinations of these two parameters. This solution is not exact, but it should be quite good as long as remains smaller than N (see Appendix). On the
other hand, when µ  0,
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(7)
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where c is a constant. In this case the negative drive
acts to effectively decrease by an amount proportional to µ. This happens up to the point where + cµ = 0,
beyond which the output firing rate is set to zero (otherwise,
+ cµ would correspond to a negative effective
SD). This approximation is partly based on simulation results
shown below, where it is discussed further.
Equations 6 and 7 are useful for three reasons. First, they are valid
for small and large (small or large relative to the distance from
rest to threshold), second, they combine µ and seamlessly, in the
sense that cases with and without drift also fall under the same
formulation, and third, the approximations are best when the underlying
distribution P(n) is Gaussian but they are quite good even
when the distribution is very different. Other theoretical models are
usually restricted in one or more of these ways (Gerstein and
Mandelbrot, 1964; Tuckwell, 1988; Smith,
1992). The rest of the paper examines the behavior of these expressions: first, as functions of µ and , second, as functions of the mean firing rate and variability of the input spike trains, which determine µ and , and finally, in comparison to simulations of a more realistic, conductance-based model.
Robustness of the random walk approximations
A crucial assumption underlying the above results was that the
full probability distribution of n could be represented by its mean and SD. How good is this approximation? We explored this through computer simulations in which, at each time step, n
was drawn from a specified distribution, using a random number
generator (Press et al., 1992). Each simulation
cycle started with N = Nreset. Then, in
each step, the update rule N  N + n was applied
until N reached the threshold value, in which case the total
number of steps elapsed was saved, and a new cycle was started. This was repeated 5000 times, after which the average number of steps was obtained. For the results shown in Figure
1c-h,
N = 40, Nreset = 20,
and varies along the x axes. The different curves in
Figure 1c-h correspond to different values of µ. The insets depict the type of distribution function for n used
in the corresponding panels. The dots indicate the simulation results, and the continuous lines in Figure 1c are the analytic
approximations given by Equations 6 and 7; these are the same
regardless of the distribution. The analytic results are most accurate
when n is distributed in a Gaussian fashion, but the random
walk approximation is qualitatively accurate when the distribution of
n is uniform (Fig. 1d), and even when it is
sharply skewed (Fig. 1e). The approximations are good even
when is almost as large as N.
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Figure 1.
Computer simulations of the stochastic neuron
model. The two traces on the top illustrate how the accumulated number
of net excitatory spikes, N, varies over time. In each time
step, N changes to N + n, where n
is drawn from a distribution with mean µ and SD . When
N reaches the threshold (dotted line), a spike is
emitted (vertical bars), and N is lowered to its
reset value. In this figure N = 40 and
Nreset = 20. a, Drift dominates
over the fluctuations, so the neuron fires regularly; n was
drawn from a Gaussian distribution with µ = 0.71, = 2. b, The neuron is driven exclusively by the
fluctuations, so it fires irregularly; n was drawn from a
Gaussian distribution with µ = 0, = 8.
Notice N cannot fall below the reflecting barrier at 0. c-e, Output firing rate
(rout) as a function of . Red
dots correspond to µ = 1.5, blue dots
to µ = 0, and green dots to µ = 3. Insets indicate the distribution of n
in each case; vertical lines mark the mean values. Gaussian,
uniform, and exponential distributions were tested. The
continuous lines in c are the analytic results
from Equations 6 and 7. f-h, Coefficient of variation of
the output interspike intervals as a function of . The three panels
correspond to the three distributions for n shown in the
above insets. Colors indicate same parameter values as in panels
above.
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Through these simulations we also investigated what happens when
N relaxes exponentially to its rest value. In this case the simulations proceeded exactly as described above, except that the
update rule for N was N hN + n, where
h is a constant <1 (h = 1 is the original
case without exponential decay). This is equivalent to having a leak
term proportional to V in Equation 1 instead of the
constant d. We found that the shapes of the resulting curves
were very similar to those obtained using the linear decay term that
contributes to µ (Eq. 2), except that they corresponded to more
negative values of µ. For instance, the results of a simulation with
h = 0.95 and µ = 0 were almost
identical to the results obtained with h = 1 and
µ = 1. Therefore, the exact shape of the
distribution of n and the precise way in which V
relaxes to rest do not affect the results qualitatively.
Two output modes: mean excitatory drive versus fluctuations
The dynamics of the output neuron may be understood intuitively in
the two limits mentioned before, when the drift is positive and much
larger than the fluctuations, and when the drift is zero (Troyer and
Miller, 1997). If the net drive is positive and is close to 0 (Gerstein and Mandelbrot, 1964; Tuckwell,
1988; Usher et al., 1994; Koch,
1999), Equation 6 is reduced to
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(8)
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In this case rout depends linearly on the
average drive, which brings V closer to threshold.
Fluctuations produce some jitter in the path from rest to threshold
(Tuckwell, 1988; Koch, 1999), but the
interspike intervals of the model neuron should be rather regular.
Figure 1a shows that this is indeed what happens. Here an
individual sequence of N values from one of the simulations is shown; for this we set µ = 0.71 and = 2. The trajectories from reset to threshold are similar because
they are dominated by the constant drift, producing fairly regular
interspike intervals.
Previous stochastic models arrived at the above expression regarding µ as the sole contributor to the mean firing rate (Gerstein and Mandelbrot, 1964; Tuckwell, 1988;
Usher et al., 1994). In these models the fluctuations
were considered so small relative to the distance from reset to
threshold, that, in the absence of drift, it took an infinite amount of
time for V to reach threshold. In the present model,
however, fluctuations are not infinitesimal (Feynman et al.,
1963) so, when µ = 0,
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(9)
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In this case the output firing rate increases monotonically
with up to the limit 1/t. The t of the
model has a functional interpretation: it represents the refractory
period, because only one spike is allowed per t. In this
mode the neuron fires because there are fluctuations in the numbers of
excitatory and inhibitory input spikes that arrive per t,
even though on average excitatory and inhibitory contributions balance
each other out (Smith, 1992; Shadlen and Newsome,
1995; Bell et al., 1995). If the fluctuations are large, the average drive may even be negative, and this will not
prevent the neuron from firing. As mentioned above, when µ is
negative, the output firing rate can be accurately approximated by
Equation 7, which was used in Figure 1c (continuous line
over green dots). We found that c = 1.7 fitted the
simulation results fairly well. In Figures 1c-e the curves
for negative µ are very much like shifted versions of the curves with
µ = 0, which is precisely why the approximation works.
When the postsynaptic neuron is driven by fluctuations, the interspike
interval distribution of the evoked spike trains is expected to be
wide, because it follows an entirely stochastic process. As shown in
Figure 1b, individual trajectories of N are widely differentthey are also independent, and this produces highly
variable interspike intervals. The two dynamical modes described by
Equations 8 and 9 are thus distinct.
Figure 1f-h quantifies the variability of the interspike
intervals produced by the simulations. The y axes indicate
the coefficient of variation of the interspike interval distribution,
or CVISI. This is equal to the SD of the
interspike intervals divided by their mean and is shown as a function
of using the same results used in Figure 1c-e. The
plots confirm the intuitive picture discussed in the previous
paragraphs: when is large in relation to µ, the coefficient of
variation is close to 1, as expected from a Poisson process. On the
other hand, as approaches 0, µ becomes relatively large, and the
variability in the interspike intervals decreases sharply (Fig.
1f-h, red dots). This drop in variability has been viewed
as support for a large in real cortical neurons, that is, as
evidence of a balance between excitation and inhibition (Shadlen
and Newsome, 1994; Troyer and Miller, 1997).
Impact of input correlations
Now we quantify how the relative magnitudes of the fluctuations
and the mean of the total synaptic drive may change according to the
synaptic input statistics.
Assume that the model neuron receives ME and
MI excitatory and inhibitory inputs,
respectively. We denote the number of spikes fired by excitatory input
j in a time step t as
nEj; analogously,
nIk corresponds to the number of spikes
fired by inhibitory neuron k. Recalling that
nE and nI are the total
numbers of excitatory and inhibitory spikes, Equation 2 can be written
as:
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(10)
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We are interested in the mean and the variance of n,
which are µ and 2. To calculate them, we assume that
all excitatory inputs fire at the same mean rate
rE, such that the average number of spikes per
time step fired by any excitatory neuron is:
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(11)
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Similarly, all inhibitory neurons fire at a mean rate
rI but, furthermore, we will assume that
inhibitory and excitatory rates are proportional, such that:
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(12)
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where  is the constant of proportionality. With these
definitions, the mean value of n is simply:
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(13)
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The fraction inside the square brackets reflects the balance
between excitation and inhibition,
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(14)
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An analogous quantity is defined below (Eq. 29) for more realistic
neurons. They differ because, in the random walk model, the effect of
each input spike is characterized by a single, instantaneous voltage
step. When  RW = 1 the neuron is fully
balanced, and the mean drift in voltage attributable to synaptic inputs
is zero. Notice, however, that µ in Equation 13 includes another
negative term caused by leakage that is independent of the balance.
To compute the variance of n, we need to specify the
variance of the individual inputs as well as their pairwise
correlations. The variances in the spike counts of single excitatory
and inhibitory inputs are represented by sE2
and sI2, such that:
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(15)
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The j and k subscripts were dropped from the
right-hand sides of these expressions because all excitatory or
inhibitory neurons were assumed to be statistically identical. The
coordinated fluctuations in the spike counts of pairs of neurons are
quantified by linear (or Pearson's) correlation coefficients
(Press et al., 1992). The correlation coefficient
between random variables x and y is:
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(16)
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So, using the above definitions for the variances
sE2 and sI2, the
pairwise correlation coefficients for the inputs are:
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(17)
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Again, all excitatory-excitatory, inhibitory-inhibitory, and
excitatory-inhibitory pairs are assumed to be equivalent. Combining Equations 10 and 15 and 17, it is straightforward to compute the variance of n, which is:
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(18)
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This expression already shows the dependence of on the
correlation structure of the inputs. However, the link can be made clearer. Assume further that the time step t is small,
such that each input fires either one or zero spikes in each time step. In that case, the number of spikes per time step fired by neuron j, nEj, has a binomial probability
distribution with mean rEt and variance rEt(1 rEt). Thus, the
relationship between and the input statistics in the case of the
binomial approximation is:
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(19)
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To better appreciate the interplay between correlation terms, for
the moment we will consider a simplified version of this expression.
First, assume that rEt is small relative to
1, in which case the variance is approximately equal to the mean, both for excitatory and inhibitory neurons. Second, take
ME = MI = M, = 1,
and I = E. These
simplifications allow a better comparison of the different terms
contributing to the variance of n without altering the
conclusions in a qualitative way. The result is:
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(20)
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This simple equation reveals the great impact that the statistical
structure of a set of inputs may have on their target neuron. Two
important points must be highlighted. First, the correlation terms are
all multiplied by M2, where M
is the number of inputs to the model neuron. Therefore, if the
postsynaptic neuron is integrating the activity of hundreds or
thousands of other active input neurons, even small correlations in
their fluctuations will produce large variations in the net driving
input from one time step to the other. We already showed that, if the
postsynaptic neuron is working in the regime in which the net
excitatory input is close to zero, then a large will lead to a high
output firing rate, as indicated by Equation 9. In this situation input
correlations determine the gain of the neuron, and their effect can be
extremely powerful.
The second key element of Equation 20 is that correlations between
inhibitory inputs have the same effect as correlations between excitatory inputs, whereas correlations between excitatory and inhibitory inputs have the opposite effect. Synchronous inhibition is
an effective way to increase variance, but an inhibitory spike that
comes close in time to an excitatory one counteracts it, reducing
variance. Thus, the three individual correlation terms could have
relatively large values but still cancel out to produce practically no
effect. This is what happened in simulation studies by Shadlen
and Newsome (1998). They did not detect any changes in output
rate when inputs were highly correlated because their choice of
parameters was such that the three terms cancelled out exactly. Of
course, in this situation any change in the balance between positive
and negative correlation terms will produce a large change in
2.
At some point of the input-output rate curve, even an unbalanced
neuron with much <1 will be affected by correlations, as described
by Equations 9 and 18. The negative term caused by leakage in Equation 13 is independent of input rate and of . Therefore, whatever the
balance of the neuron, there will be a positive value of
rE for which µ = 0 and
2 > 0. Around such value, the membrane
voltage will have zero drift, but the neuron will be able to fire,
driven exclusively by input fluctuations. Thus, there will always be a
range of values of rE such that the target
neuron fires according to the zero-drift classic random walk dynamics.
In this range, correlations are expected to have the effects just described.
Input-output rate relationships predicted by the theory
Here and in the rest of the paper we explore the relative effects
of the three correlation terms. For the sake of simplicity, we
illustrate three cases: (1)  EE > 0, II = EI = 0, (2)
II > 0, EE = EI = 0, and (3) EE = II = EI > 0. However, the reader should keep in mind that it is the final weighted sum of the
three terms that determines 2, and that the
first two cases are also representative of the situation in which all
the terms are greater than zero but the final sum is also greater than
zero. For instance, suppose that Equation 20 applies, that
EE and II are
positive and equal, and that EI is also
positive but smaller than the other terms. In this case what counts is
EE + II 2EI, so this situation would be indistinguishable
from cases 1 or 2 above. Notice also that, in general, case 3, in which
all correlations are identical, does not automatically lead to an exact
cancellation, because the three terms have different coefficients in
Equation 19. As with the balance between excitation and inhibition, it
is hard to assess what the real biological situation is; the selected
cases are meant to illustrate a range of possibilities.
Figure 2 illustrates the results derived
in the previous section for two cases with different relative
contributions of µ and to the output rate. In this figure the
expressions for µ and derived above using the binomial
approximation (Equations 13 and 19) were used to compute the firing
rate of the output neuron, as given by Equations 6 and 7. A total of
1000 active inputs were considered, 20% of which were inhibitory. The
percentage of inhibitory neurons alters the input-output rate curve
that results with uncorrelated inputs, whereas the total number of
neurons modifies the weight of the correlation terms. Inhibitory
neurons fired at 1.7 times the rate of excitatory ones. The voltage
decay was set to d = 0.3 mV; this corresponds to a
decrease in voltage of 0.3 mV/msec, because t = 1
msec. This value is comparable to the 0.49 mV decrease that occurs in 1 msec when the voltage starts 10 mV above rest and relaxes exponentially
with a 20 msec time constant. The difference between resting potential
and threshold was 20 mV, with the reset voltage falling halfway in
between. Finally, the remaining parameters were chosen in two ways, to
obtain results for balanced and unbalanced neurons, but in all cases
the size of the individual excitatory depolarization
E was chosen to produce an output firing rate of ~75 spikes/sec at an input rate rE of 100 spikes/sec (Shadlen and Newsome, 1995, 1998).
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Figure 2.
Analytic results from the random walk model.
Output firing rate rout is plotted as a function
of input rate rE for different parameter values
and correlations. To obtain these curves, first, µ and were
computed from Equations 13 and 19, then Equations 6 and 7 were used. In
all plots, the continuous line corresponds to all
correlation coefficients equal to zero (uncorrelated inputs),
filled circles indicate positive correlations between
excitatory pairs only, open circles indicate positive
correlations between inhibitory pairs only, and dots
indicate identical, positive correlations between all pairs.
a, Input-output rate curves for a balanced postsynaptic
neuron for fixed values of the correlation coefficients. In this case
E = 0.5 mV and
I/E = 2.35, which
gives RW = 1. For the continuous
line EE = 0, II = 0, and EI = 0. For the filled
circles EE = 0.0033, II = 0, and EI = 0. For the open circles II = 0.0033, EE = 0, and
EI = 0. For the small dots
all three coefficients were equal to 0.0033. b,
Input-output rate curve for an unbalanced postsynaptic neuron with
E = 0.023 mV and
I/E = 0.8, giving
RW = 0.34. For these curves all nonzero
correlation coefficients were equal to 0.8. Other parameters were, for
all plots, as follows: ME = 800, MI = 200, = 1.7, d = 0.3 mV,
t = 1 msec, V Vrest = 20 mV, Vreset Vrest = 10 mV, and c = 1.7.
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Figure 2a shows the output firing rate as a function of
input rate rE for a balanced neuron, for which
RW = 1 (Eq. 13). Here the neuron is
driven exclusively by the fluctuations of its inputs, and a fixed
amount of net correlation has a multiplicative effect on the firing
rate curve, as expected from Equation 19. Figure 2b shows
similar curves for a case in which RW = 0.34; for these curves the ratio
I/E was modified to
obtain an unbalanced neuron that on average received more excitation
than inhibition. The input-output rate curve obtained with independent
inputs recovers the threshold-linear function typically used in
modeling work (Hertz et al., 1991; Abbott,
1994; Koch, 1999). In this case correlations no
longer have a multiplicative effect on the rout
versus rE curve, and the fractional change in
output rate caused by a given amount of correlation is much smaller
than for a balanced neuron. However, a net excess of excitatory
correlations still increases the output rate significantly, especially
around threshold (Kenyon et al., 1990; Bernander
et al., 1991). This is the point around which the neuron is
driven almost exclusively by fluctuations (Bell et al.,
1995).
According to this simple stochastic model, the synaptic input that
drives a postsynaptic neuron may be thought of as having two
components, a mean component, which depends on the net balance between
excitation and inhibition, plus another component that represents the
fluctuations around the mean, and both components may drive the
recipient neuron to fire. The fluctuations depend strongly on the
correlations between input spike trains, so it is through their effect
on the fluctuations that input correlations may greatly enhance the
resulting output firing rate. Such fluctuations may be the main driving
force around threshold. The next section explores the validity of these
conclusions using more realistic model neurons and computer simulations.
Simulations of a conductance-based integrate-and-fire neuron
Results in this section are based on simulations of an
integrate-and-fire neuron model receiving 160 excitatory and 40 inhibitory inputs with Poisson statistics at given mean rates. The
amplitudes of the synaptic conductances were varied so that balanced
and unbalanced situations could be studied and compared to the
predictions from the stochastic model and to Figure 2.
In the random walk model discussed above, input correlations were
synonymous with synchrony, because they referred exclusively to the
chances of two input spikes arriving in the same time slice t. We will show that the results apply to correlations in
a wider sense, that is, to situations in which the probability of
firing of one input is not independent of the probabilities of the rest of the inputs. We will consider two ways of generating correlated activity, through the equivalent of shared connections and through oscillations in the instantaneous firing rate of the inputs.
Description of the model and parameters
The conductance-based integrate-and-fire model we use is similar
to the one described by Troyer and Miller (1997)
(Knight, 1972; Tuckwell, 1988;
Shadlen and Newsome, 1998; Koch, 1999). The main difference is that we included a mechanism that reproduces the
spike rate adaptation typical of most excitatory cortical neurons
(McCormick et al., 1985). Subthreshold currents are
included, but currents that generate spikes are not. The membrane
voltage V(t) changes in time according to the differential
equation:
|
(21)
|
where the first term on the right corresponds to a leak current,
and EL is the resting potential. Here we have
written the membrane capacitance Cm as
 m/Rm, where
m is the membrane time constant and
Rm is the input resistance of the neuron,
which is equal to the inverse of the leak conductance
gL. The I terms stand for specific
types of current flowing through the membrane:
IAPP corresponds to externally applied
(injected) current, and the rest consist of a time-varying conductance
g times a driving force, such that:
|
(22)
|
ISRA represents a spike-triggered potassium
current that produces adaptation in firing frequency, which is
characteristic of most excitatory neurons in the cortex
(McCormick et al., 1985). IAMPA
and IGABA are the currents produced by fast
excitatory and fast inhibitory synapses, respectively. A single
isopotential compartment is considered (no spatial variations in
V). The above equations determine the subthreshold
behavior of the neuron; whenever V exceeds the threshold
V, an output action potential is produced and
the neuron enters a refractory period. In practice, when V
increases beyond threshold, a spike reaching 0 mV is pasted onto the
voltage trace and V is clamped to the value
Vreset for a time
refrac, after which it continues to evolve
according to Equation 21.
The conductance changes underlying spike rate adaptation are
implemented as follows. Whenever V exceeds
V and a postsynaptic spike is elicited, the
potassium conductance increases instantaneously by an amount
gSRA. The flow of potassium ions tends
to hyperpolarize the cell and slows down the firing. The change
in conductance decays exponentially toward zero with a time
constant SRA,
|
(23)
|
Here t0 corresponds to the time at which
the output spike was produced, and gSRA is
zero for all t < t0. Each subsequent output spike adds an identical conductance change at the corresponding point in time, so the total potassium conductance at any time can be
written as the sum of all changes:
|
(24)
|
where tj is the time of output spike
j, and the index runs over all output spikes.
The intrinsic model parameters, those independent of synaptic input,
were tuned to approximate the neurophysiological measurements of
McCormick et al. (1985, their Fig. 1C,D; see
also Troyer and Miller, 1997). The following values are
used: EL = 74 mV,
EK = 80 mV, V = 54 mV, Vreset = 60 mV,
m = 20 msec,
refrac = 1.72 msec,
SRA = 100 msec, and
 SRA = 0.14 gL. These
numbers are constant throughout the paper. Figure
3 illustrates the behavior of the model
in terms of its responses to IAPP, the injected
current. In these simulations the input resistance
Rm was set to 40 M (that is,
gL = 25 nS), but notice that in the rest of
the paper, where IAPP = 0, this parameter
is eliminated by expressing all conductances as fractions of
gL. Figure 3a shows the firing evoked by a stepwise change in IAPP and illustrates the
increase in the interspike intervals that results from the spike rate
adaptation current. Instantaneous firing frequency is plotted in Figure
3b as a function of applied current. For the curve with
circles, frequency was computed as the inverse of the first interspike interval, with 1 sec of zero current between simulated injections; thus the graph corresponds to the unadapted state. For the curve with
triangles, frequency was computed as the inverse of the last interspike
interval obtained after 1 sec of current injection, at which point
firing frequency had adapted fully, reaching a steady state. Again,
between current pulses there was a 1 sec intermission. Figure
3c plots the lengths of consecutive interspike intervals
evoked by different amounts of injected current. The curves rise,
reflecting the gradual lengthening of the intervals between output
action potentials. ISRA reduces the steady state firing rate to approximately half of the initial, unadapted rate.
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Figure 3.
Responses of the conductance-based
integrate-and-fire model to current injection. In these simulations,
Equation 21 was used with Rm = 40 M
(gL = 25 nS). The applied current
IAPP was varied, but no synaptic inputs were
included, so IAMPA = IGABA = 0. Other model parameters as specified after Equation 21.
a, The bottom trace shows the time course of a
step change in injected current from 0 to 1 nA. The top
trace shows the membrane potential of the model. A spike train is
elicited by the depolarizing current. The interspike intervals lengthen
because of the spike rate adaptation current,
ISRA. When the current pulse turns off, the
voltage falls below rest (74 mV) to a minimum of 75.7 mV,
later recovering. b, Instantaneous firing frequency as
a function of injected current. For the curve with
circles (unadapted), instantaneous
frequency is equal to the inverse of the first interspike interval
elicited by a current pulse; for the curve with
triangles (adapted), instantaneous
frequency is equal to the inverse of the last interspike interval
evoked after 1 sec of current injection. Membrane potential was allowed
to relax to rest before all step pulses. c, Lengths of
consecutive interspike intervals evoked by step current pulses. These
curves reveal the timecourse of adaptation. Each one corresponds to a
different current intensity, as is indicated to the right, in
nanoamperes. Compare to Figure 1 of McCormick et al. (1985).
|
|
Expressions similar to Equation 24 are used to model the conductance
changes caused by excitatory synaptic inputs. When an excitatory spike
arrives, gAMPA increases by
gAMPA. This increase is fast, so a single
exponential describing the subsequent decay is sufficient in this case
too,
|
(25)
|
Now t0 corresponds to the time at which the
excitatory input arrived, and the transient increase in
gAMPA falls off with a time constant
AMPA. Subsequent input spikes add identical
conductance changes, so that:
|
(26)
|
Now tj is the time of input spike
j, and the index runs over all excitatory input spikes.
Inhibitory spikes increase the GABA conductance. The rise in
gGABA after an inhibitory spike is somewhat
slow, so the timecourse of gGABA is better
described by the difference of two exponentials,
|
(27)
|
Here the D factor is a normalization term that
guarantees that the maximum of gGABA is equal
to GABA. The two time constants GABA(1) and
GABA(2) determine the characteristic rise
and fall times, as well as D. In this case,
t0 corresponds to the time at which the
inhibitory input spike arrived, and the total GABA conductance is the
sum of the effects of all inhibitory spikes,
|
(28)
|
Additional simulations were performed to explore whether a second,
slower inhibitory conductance would affect the results. In these runs
the additional conductance followed the same dynamics just described
but had a decay time constant of 150 msec. This slow component did not
alter the results in any significant way (data not shown) and is not
discussed further.
This model neuron does not include any intrinsic sources of noise. In
fact, the synapses themselves do not contribute any noise either,
because all excitatory or inhibitory spikes cause the same conductance
change (the effect of synaptic and intrinsic variability is explored in
a separate section below). This allows us to study the impact of input
variability in isolation from other noise sources. The model neuron is
driven by ME excitatory and
MI inhibitory inputs, and each input provides an
individual spike train. The mean spike rates are the same for all
excitatory and inhibitory inputs; these are rE
and rI, respectively. In all simulations, we
assume that these rates are proportional, so that rI = rE, with being the
constant of proportionality. These rates are constant, except when an
explicit time dependence is indicated.
The balance of the neuron, , refers to the ratio between the mean
amount of inhibition and excitation that it receives. We measure it as
Troyer and Miller (1997) did. This quantity depends on
the relative numbers of excitatory and inhibitory inputs, the relative
magnitudes of their firing rates, and the relative impacts of
excitatory and inhibitory spikes on the postsynaptic voltage. To
compute the latter, one should take into account the total changes in
conductance integrated over time and the driving forces, so that:
|
(29)
|
where
|
(30)
|
When = 1, there is no mean drift in voltage
caused by synaptic inputs; we refer to this as the balanced condition.
When is different from 1 the neuron is unbalanced. Notice, however, that in the literature a balanced neuron is often one that receives some amount of inhibition ( > 0), as opposed to an
unbalanced one which receives only excitation ( = 0). It seems more appropriate to use the term balanced when
excitation and inhibition are truly equilibrated, so in this paper we
apply it when = 1.
The results shown below are based on simulations that included
ME = 160 excitatory and
MI = 40 inhibitory inputs. In addition, separate simulations confirmed that the results still hold when the
numbers of inputs are increased (data not shown). In these runs the
ratio MI/ME was kept
constant, and maximal conductance changes were modified accordingly, so
that the balance and gain of the neuron remained approximately the same
as with the standard numbers of neurons. For the rest of the
parameters, the following values are used: = 1.7, EAMPA = 0 mV, ECl = 61 mV, AMPA = 5 msec,
GABA(1) = 5.6 msec, and
GABA(2) = 0.285 msec. These numbers are
the same in all simulations. For the amplitudes of the conductance
changes, two sets of values are considered. In the balanced condition
we use AMPA = 0.0806 gL, and GABA = 1.1143 gL, which gives = 1.
With these parameters, a single excitatory spike yields a maximum
depolarization of 0.7 mV at threshold, and a single inhibitory spike
yields a maximum hyperpolarization of 1.4 mV at threshold. In the
unbalanced condition we use AMPA = 0.0222 gL, and GABA = 0.1382 gL, which gives = 0.45. In this case a single excitatory spike yields a maximum depolarization of 0.2 mV at threshold, and a single inhibitory spike
yields a maximum hyperpolarization of 0.2 mV at threshold. Having
fixed the ratio GI/GE, the
maximal conductances were scaled so that, for all conditions tested,
the output firing rate rout was close to 75 spikes/sec when the input firing rate rE was set to 100 spikes/sec. This was to allow the neuron to fire at a rate similar to that of any of its inputs (Shadlen and Newsome 1995, 1998). Notice that, by expressing the maximal unitary
conductances in units of gL, this parameter can
be factored out of Equation 21, as long as IAPP = 0, which is the case for the rest of the paper. Equation 21 was
integrated numerically using a fixed time step of 0.05 msec.
Calculation of cross-correlation histograms
In the simulations, a number of excitatory and inhibitory inputs
drive a conductance-based integrate-and-fire neuron. We used two
methods to generate temporal dependencies between inputs, common drive
and temporal comodulations in rate (Brody, 1999). In
both cases we computed cross-correlation histograms (Perkel et
al., 1967; Fetz et al., 1991; Nelson et
al., 1992; Brody, 1999) to visualize the
resulting dependencies between pairs of input spike trains. For this,
we recorded the firing times of two of the inputs, a and
b, for a period of time of length
Tmax. Simulation parameters did not change
during this time, so the firing statistics of a and
b were the same throughout. We use the notation
Sa(t) =  j  (t tja) to indicate the spike train of input
a, where is the Dirac delta function, and
tja is the time at which input a
fired spike j. The cross-correlation histogram
evaluated at time lag can be written as:
|
(31)
|
The result from the integrals is the number of spikes in the data
record that were fired by input b between and + milliseconds after input
a fired; a negative corresponds to spikes from
b fired before a spike from a. The integral over t' simply corresponds to the binning associated with the
histogram, which has a bin width . The
cross-correlation is normalized by  ab,
which is the value of the integrals expected for two independent
Poisson processes with the same, constant mean rates as the inputs.
This factor is:
|
(32)
|
where ra and rb are
the mean firing rates of the two inputs. With this normalization, two
inputs that fire independently of each other should produce
Cab() = 1 at all lags. In practice, because of the discrete character of action potentials,
cross-correlograms are quite noisy. To obtain smooth
cross-correlograms, we used Tmax between 20 and
30 sec and = 1 msec, and we averaged over several pairs of inputs (80-120) whose spikes were collected
simultaneously from the same simulation.
Generating correlations via common drive
To generate correlated spike trains to be used as inputs to the
integrate-and-fire neuron, we used a procedure similar to the one used
by Shadlen and Newsome (1998) but somewhat simpler, because it takes advantage of the results in Figure 1. For each of the
input lines that impinge on the model postsynaptic neuron, there is a
random walk variable that generates the spikes of that line. Variable
Ni corresponds to input i and behaves
exactly as the model in Figure 1c (blue points), having no
net drift and producing a spike whenever Ni
exceeds a threshold. All random walk variables have the same
parameters. At each time step, a set of Mpool
independent Gaussian random samples with zero mean and unit variance
are drawn, and each Ni is updated by adding exactly Min of those samples, weighted by an
overall gain factor gin. The sum of Gaussian
samples acts as the net input to the random walk units. This is
implemented through a matrix multiplication:
|
(33)
|
where sj is Gaussian sample j
and the entries of matrix w can only be 0 or 1, with exactly
Min nonzero terms in each row. If the column
indices of the nonzero terms in each row of w are chosen
randomly, on average, pairs of input units will share a fraction:
|
(34)
|
of their Gaussian samples (Shadlen and Newsome,
1998). That is, on average, an input unit will share
 Min of its Min samples with any other unit. Any measure of correlation or synchrony between output spike trains generated in this way will be a function of ,
where = 0 corresponds to independent trains and
= 1 corresponds to identical trains. The gain
factor gin can be set so that spikes are
produced at any desired mean rate. This is done by using Equation 9 and
noting that the summation term in Equation 33 has a Gaussian distribution with variance equal to
ginMpool. Cross-correlograms between
spike trains generated in this way were similar to those reported by
Shadlen and Newsome (1998), and varying was
equivalent to varying the fraction of shared connections in their
network model.
With this method we generated the 160 excitatory and 40 inhibitory
inputs that arrived at the integrate-and-fire model neuron. In the
simulations, E refers to the fraction of
samples that each excitatory input draws from the common pool,
according to Equation 34 and, similarly, I
refers the fraction of samples that each inhibitory neuron draws from
the pool. A value of I = 0 means that
inhibitory spike trains are independent of each other and of all
excitatory spike trains and, analogously, E = 0 means that excitatory spike trains are all independent.
Simultaneous nonzero values for E and
I correspond to excitatory and inhibitory neurons sharing the same common pool of Gaussian samples, thus being
correlated between themselves and across each other's type as well.
Notice that these correlated spike trains have a
CVISI of approximately one independently of the
fraction of shared samples. This is because these neurons are always
random walkers drawing their steps from Gaussian distributions with
zero mean, as illustrated in Figure 1.
Impact of correlations generated by common drive
We tested how the output firing rate changed as a function of the
fractions of common drive E and
I. Figure 4
shows examples of the output spike trains produced by the balanced
integrate-and-fire model. In each panel, the plots below the spike
rasters are examples of cross-correlation histograms (Perkel et
al., 1967; Brody, 1999) between pairs of input
spike trains. The height of the cross-correlation at time lag indicates the probability of recording a spike from one neuron between
and +
milliseconds after (or before, for negative ) a spike from another
neuron. These histograms have been normalized so that 1 represents the
probability expected based on the mean firing rates when the two
neurons are independent and follow Poisson statistics. When pairs of
input neurons share some of the Gaussian samples, they become
correlatedthe chances of recording a spike from one neuron are higher
around the times when the other neuron has fired. The peaked
cross-correlograms shown in the figure were obtained with
= 0.1, which is roughly comparable with the
measured probability that two nearby pyramidal neurons are connected
(Braitenberg and Schüz, 1997). Larger peaks, reflecting stronger synchronization, are common in correlograms constructed from experimental data (Fetz et al., 1991;
Nelson et al., 1992).
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Figure 4.
Sample output spike trains from a balanced
( = 1) model neuron for various possible correlation
patterns generated by common drive. In each panel, the spike raster
shows a 10-sec-long spike train from the output neuron. The full train
is shown subdivided into 10 segments of 1 sec duration; each row of the
raster corresponds to one segment. The three plots below each raster
are cross-correlation histograms between inputs. The ones on the left
correspond to excitatory-excitatory (EE) pairs, the ones in
the middle correspond to inhibitory-inhibitory (II) pairs,
and the ones on the right correspond to excitatory-inhibitory
(EI) pairs. Input statistics were constant for each one of
the panels. The numbers to the right of the rasters indicate
the mean output firing rate (top) and the coefficient of
variation of the interspike interval distribution,
CVISI (bottom). In all cases, the
output neuron was driven by 160 excitatory and 40 inhibitory inputs
firing at rates rE = 40 and
rI = rE spikes/sec.
a, All inputs were uncorrelated; E = 0, I = 0. b, Only excitatory
inputs were correlated; E = 0.1, I = 0. c, Only inhibitory inputs
were correlated; E = 0, I = 0.1. d, All inputs were equally correlated;
E = 0.1, I = 0.1.
e, All inputs were correlated, but excitatory-excitatory
pairs were most correlated; E = 0.2, I = 0.1. Calibration: 200 msec. Other
parameters as indicated for the balanced condition.
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In Figure 4, three histograms are shown below each spike raster. They
show the mean correlation between excitatory-excitatory (EE),
inhibitory-inhibitory (II), and excitatory-inhibitory (EI) input
pairs that drove the output neuron under each condition. These
statistical dependencies did not change during the periods in which the
shown output spikes were generated. Figure 4a shows a spike
train evoked by uncorrelated inputs whose cross-correlations are,
consequently, flat. In Figure 4b, only excitatory neurons are correlated, which is evidenced by the peak in the corresponding EE
cross-correlation. The small E of 0.1 that
was used makes the output rate increase by ~60%. As shown in Figure
4c, correlations between inhibitory neurons also increase
the rate, but the effect is less strong. Correlations in excitatory
inputs make a bigger difference because of the parameters chosen. As
can be seen from Equations 19 and 20, excitatory or inhibitory inputs
may have the largest weight in determining 2, depending
on these choices. With the parameters used in Figure 4, when all
neurons are equally correlated the increase in rate practically
disappears, as illustrated in Figure 4d. However, Figure
4e shows that a large increase in output rate is again seen
when all neurons are correlated but excitatory-excitatory pairs are
most strongly correlated. Thus, as expected from Equation 19 or 20, it
is the balance between the three correlation terms that determines the
final contribution of the fluctuations to the output firing rate.
The examples in Figure 4 also show that input correlations can increase
the variability of the output neuron. This is most obvious in Figure 4,
b and e, which has CVISI
values of 1.5, much larger than the 1.1 obtained with uncorrelated
inputs. In these cases correlations tend to produce more bursts of
spikes. With correlations present only between inhibitory neurons (Fig.
4c) or when all neurons are equally correlated (Fig.
4d), the CVISI still increases
slightly, to 1.3, although in the latter case the output rate does not
change. Thus, correlations in the input generated through common drive
may lead to an increase in output firing rate but, in general, also
produce more irregular firing.
Figure 5 expands these results by varying
systematically either the input rates or the correlations. Figure
5a shows the output firing rate as a function of input rate
for uncorrelated inputs, excitatory-excitatory correlations only,
inhibitory-inhibitory correlations only, and all possible pairs
equally correlated. Figure 5a should be compared with Figure
2a, which depicts the analogous curves expected from the
random walk model. To compare the two sets of results, parameter
E was scaled so that the analytic curves with
uncorrelated inputs would have approximately the same gain as in the
simulation, and the value of 0.004 used in Figure 2a for the
correlation coefficients was chosen to match the effects seen in Figure
5a. The curves obtained with the conductance-based model are
in excellent agreement with the analytical predictions from the random
walk equations, in spite of the numerous details that distinguish the
two models. Figure 5c shows how the output rate changes as a
function of correlation strength for a fixed input rate,
rE = 40 spikes/sec. As expected, higher
correlations have stronger effects. For a balanced neuron, even a weak
correlation between its excitatory inputs may have a large impact on
the output; a fraction of shared samples of ~0.15
(E = 0.15) is enough to double the
output firing rate.
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Figure 5.
Effect of input correlations generated by common
drive on the firing rate and variability of the same balanced
( = 1) model neuron used in Figure 4. For each data
point, the output spike train was recorded for 30-90 sec of simulation
time, and the mean rate and coefficient of variation were computed from
this segment. a, Mean output firing rate
rout as a function of input rate
rE, for four conditions. The continuous
line indicates uncorrelated inputs (E = 0, I = 0), filled circles indicate
correlations between excitatory inputs only
(E = 0.1, I = 0),
open circles indicate correlations among inhibitory inputs
only (E = 0, I = 0.1), and dots indicate all pairs equally correlated
(E = 0.1, I = 0.1).
b, CVISI of the output spike trains
as a function of input rate, computed from the same simulations as in
a; symbols have identical meaning. The
dashed line marks a CVISI of 1, expected from a Poisson process. c, Mean output firing rate
rout as a function of correlation strength, for
a fixed input rate rE = 40 spikes/sec.
Filled circles correspond to correlations between excitatory
neurons only (E varies along the x
axis and I = 0), open circles
correspond to correlations between inhibitory neurons only
(E = 0 and I
varies along the x axis), and dots correspond to
all pairs equally correlated (E and
I vary identically along the x
axis). d, CVISI of the output spike
trains as a function of correlation strength, computed from the same
simulations as in c; symbols have identical
meaning. Other parameters as indicated for the balanced
condition.
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Figure 5, b and d, shows that correlations also
increase the variability of the output neuron, as measured by the
CVISI. For the balanced condition, the increase
is larger at low output rates. Notice that, even in the absence of
correlations, CVISI increases with increasing
input rate, as shown by the continuous line in Figure 5b.
With high input rates or with modest correlations, output variability
may easily increase past a CVISI of 1. This constitutes a major difference between the random walk description and
the conductance-based model. When the changes in membrane potential are
described by a random walk, the neuron is memoryless: what happens in
one t has no influence on what happens in the next and,
similarly, one interspike interval has no relation to the next. This
leads to values of CVISI 1, if there is some
net drift. Conductance changes, however, are not instantaneous so, for
instance, several synchronous excitatory spikes may produce a burst of
output spikes instead of the single spike expected from the random walk
model. Thus, conductances allow much greater variability in the output
spike train than instantaneous "adding" of spikes.
It is well known that refractoriness also affects spike train
variability (Koch, 1999). The simulations in Figures 4
and 5 were obtained with synaptic time constants of ~5 msec and a
refractory period of 1.72 msec. We also ran simulations in which all
synaptic time constants were divided by a factor of 2.5, and the
maximal conductances were increased to produce the same gain and
balance. This made the refractory period much larger relative to the
timescale of unitary changes in postsynaptic conductance. The only
difference observed in these simulations was that all
CVISI values were smaller than those of Figures
4 and 5 by ~0.2; otherwise, correlations had the same effects on rate
and caused the same relative increases in
CVISI.
The simulation results presented so far were obtained with a balanced
neuron. However, it is not certain whether values of as large as 1 are within the physiologically plausible range (Berman et al.,
1991; Douglas and Martin, 1998; Stevens
and Zador, 1998). Therefore, it was important to investigate
whether the results were also valid with an unbalanced neuron with low
. The corresponding simulation results are shown in Figures 6 and 7,
in the same format as Figures 4 and 5 but for an integrate-and-fire model with = 0.45. In viewing Figure 6 it is
important to recall that the plots below the rasters are
cross-correlation histograms between pairs of input spike trains. These
histograms reveal the statistical dependencies of excitatory and
inhibitory inputs, which were constant in each one of the panels.
The spike rasters in Figure 6 show that
the same amount of correlation now produces, overall, a small
enhancement in rate that is not multiplicative. Nevertheless, a
fraction E = 0.1 still raises the output
rate by ~10 spikes/sec when the input rate is <80 spikes/sec or so.
On the other hand, correlations still cause a large increase in
CVISI. Note, in particular, that the increase is
seen even when the three correlation terms are identical. This is
because the relative values of their coefficients in Equation 19 do not
produce a full cancellation. Here, however, the neuron is much less
variable to begin with (that is, with uncorrelated inputs), and the
CVISI is almost flat as a function of input
rate. Thus, as shown in Figure
7b, in the presence of weak
correlations CVISI stays around or <1 at all
input rates considered. As in the balanced condition, additional
simulations were also run using faster synaptic timescales, but this
made practically no difference on the results; refractoriness played a
minor role in this case.
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Figure 6.
Sample output spike trains from an unbalanced
( = 0.45) model neuron for various possible
correlation patterns generated by common drive. Same format and
correlation values as in Figure 4, except that
rE = 60 spikes/sec. Other parameters as
indicated for an unbalanced neuron.
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Figure 7.
Effect of input correlations generated by common
drive on the firing rate and variability of the same unbalanced
( = 0.45) model neuron used in Figure 6. Same format
and input parameters as in Figure 5, except that, in c and
d, rE = 60 spikes/sec.
Parameters of the postsynaptic neuron as indicated for the unbalanced
condition.
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Figure 7a shows the impact of input correlations on the
input-output firing rate curve of an unbalanced neuron. Note, in
particular, that correlations effectively decrease the threshold
(Kenyon et al., 1990; Bernander et al.,
1991; Bell et al., 1995). These curves should be
compared with those in Figure 2b, which correspond to the
random walk model in the presence of a large drift, or net excitatory
drive. The match between the two sets of curves is not perfect, but the
shape of the rate curve and the changes caused by the presence of input
correlations are well described by the theoretical expressions. Thus,
although parameter values used in Figure 2, a and
b, were adjusted to optimize overall agreement between the
two models, the predictions of the stochastic model, particularly the
differences between balanced and unbalanced regimes, are remarkably
accurate, considering its simplicity.
Impact of correlations generated by oscillations
The firing probabilities of two neurons may fluctuate in time
around some fixed average. If such temporal fluctuations tend to occur
together, the two neurons will be correlated; the probability of one of
them firing will be higher when the other one also fires, because this
will reflect an upward increase in the underlying firing rates. We
investigated whether inputs that become correlated in this way produce
effects similar to the ones observed through common drive. For this, we
modeled input firing rates as periodic functions of time, such
that:
|
(35)
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(36)
|
where f is the frequency of the oscillations, and the
mean firing rates are AE for excitatory and
AE for inhibitory neurons, respectively. The
probability that a particular excitatory neuron fires a spike at time
t is then rE(t)t, and similarly
for inhibitory inputs. Parameters  E and
I control the modulation amplitude around the
mean. When the rates vary according to the above equations, any measure
of correlation is a function of these two numbers. For example,
E = 0 implies that excitatory neurons
are uncorrelated. In the simulations these parameters were always
between 0 and 1.
Figure 8 shows sample spike trains from
the balanced integrate-and-fire neuron when the input rates vary
periodically at a frequency f = 40 Hz. As before, the
plots below the voltage traces are cross-correlation histograms, which
show the statistical dependencies between inputs in each condition. The
different panels correspond to different values of
E and I. For
comparison, Figure 8a shows a response to uncorrelated
inputs; this case corresponds to all cross-correlation histograms being
flat. When only rE fluctuates in time, as shown
in Figure 8b, excitatory inputs become correlated, their
cross-correlation histogram reflects the periodicity of the underlying
rates, and the output neuron fires more strongly. Based on the results
of previous sections, this is precisely what is expected when net
excitatory correlations are present. In addition, the output neuron
fires in bursts, reflecting the underlying input oscillations. Other
possible correlation patterns also modify the output firing rate as
predicted: co-fluctuations between inhibitory neurons also enhance the
output rate, and when all possible pairs are equally correlated the
enhancement practically disappears. Figure 8e shows a
variant of this latter case in which a phase difference exists between
excitatory and inhibitory rates (for a similar situation in real neural
circuits, see Skaggs et al., 1996; Tsodyks et
al., 1997). For this plot, E and
I were both equal to 0.6, but we used a
cosine instead of a sine in Equation 36. This introduced a time delay
of 6.25 msec between the peaks of rE(t) and
rI(t), which is reflected in the
cross-correlogram between excitatory and inhibitory neurons (Fig.
8e, EI). In this situation the output rate
is greatly enhanced; it is almost twice that obtained with uncorrelated
inputs. Here all input pairs are correlated, but at slightly different
times. This produces the greatest excitation at a time when inhibition
is not at its peak. This result demonstrates that the timing of
correlations also plays a crucial role in determining the output firing
rate.
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Figure 8.
Sample output spike trains from a balanced
( = 1) model neuron driven by inputs whose rates
oscillate in time. In each panel, the 500 msec voltage trace shows the
response of the model neuron. The three plots below each raster are
cross-correlation histograms between inputs. The ones on the
left correspond to excitatory-excitatory
(EE) pairs, the ones in the middle correspond to
inhibitory-inhibitory (II) pairs, and the ones on the
right correspond to excitatory-inhibitory (EI)
pairs. Input statistics were constant for each one of the panels. The
numbers to the right of the rasters indicate the
mean output firing rate from the traces shown. In all cases, the output
neuron was driven by 160 excitatory and 40 inhibitory inputs firing at
rates given by Equations 35 and 36 with AE = 40 spikes/sec and f = 40 Hz. a, All
input rates were constant; E = 0, I = 0. b, Only excitatory inputs
were oscillating; E = 0.6, I = 0. c, Only inhibitory inputs
were oscillating; E = 0, I = 0.6. d, All inputs were
oscillating with the same frequency and phase;
E = 0.6, I = 0.6.
e, All inputs were oscillating at the same frequency, but a
cosine instead of a sine was used in Equation 36 for
rI(t). A phase difference between excitatory and
inhibitory rates is apparent in the EI cross-correlation. For this plot
E = 0.6 and I = 0.6. Calibration: 100 msec. Other parameters as indicated for
the balanced condition.
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Figure 9 expands these results by varying
the input rates and correlations throughout a range. This figure has
the same format as Figure 5, except that E
and I vary along the x axes in
Figure 9, c and d, because they determine
the correlation strengths in this case. The symbols also have the same
meanings as in Figure 5, except that nonzero values of
E and I now
correspond to nonzero values of E and
I. The curves in Figure 9a
obtained with the conductance-based model are again in good agreement
with the analytical results from the random walk model. The major
difference is the greater effect that inhibitory oscillations have on
output rate. This occurs because, with oscillatory rates, the variance
in the number of spikes produced by each input per t
depends more strongly on the firing rate than what was assumed before
(the difference is the step between Equations 18 and 19: with
oscillatory rates, different expressions for
sE2 and sI2
should be used, and this results in different coefficients for the terms). Inhibitory correlations are stronger in this case because
inhibitory inputs fire faster than excitatory ones.
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Figure 9.
Effects of correlations generated by oscillating
input rates on the firing rate and variability of the same balanced
( = 1) model neuron used in Figure 8. For each data
point, the output spike train was recorded for 30-90 sec of simulation
time, and the mean rate and coefficient of variation were computed from
this segment. a, Mean output firing rate
rout as a function of mean input rate
AE, for four conditions. The continuous
line indicates uncorrelated inputs (E = 0, I = 0), filled circles indicate
oscillating excitatory inputs (E = 0.6, I = 0), open circles indicate
oscillating inhibitory inputs (E = 0, I = 0.6), and dots indicate all
input rates oscillating identically (E = 0.6, I = 0.6). b,
CVISI of the output spike trains as a function
of mean input rate, computed from the same simulations as in
a; symbols have identical meaning. The
dashed line marks a CVISI of 1. c, Mean output firing rate rout
as a function of correlation strength for a fixed mean input rate
AE = 40 spikes/sec. Filled
circles correspond to oscillating excitatory inputs
(E varies along the x axis and
I = 0), open circles
correspond to oscillating inhibitory inputs
(E = 0 and I
varies along the x axis), and dots correspond to
all inputs oscillating with the same frequency and phase
(E and I vary
identically along the x axis). d,
CVISI of the output spike trains as a function
of correlation strength, computed from the same simulations as in
c; symbols have identical meaning. Other
parameters as indicated for the balanced condition.
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As shown in Figure 9c, larger correlations still produce
larger increases in rate. Overall, the effects on output rate of correlations induced by temporal co-fluctuations in firing probability are similar to those produced by common drive to the inputs. On the
other hand, as shown in Figure 9, b and d, the
variability of the output spike trains tends to decrease when input
rates oscillate, as indicated by the CVISI. This
is not surprising because, in this case, input firing is more regular,
and the output spikes tend to follow the periodic increases in excitation.
Figures 10 and
11 show the corresponding results for
the unbalanced postsynaptic neuron with = 0.45. As
was shown above, the unbalanced neuron is less sensitive to
correlations than the balanced neuron. However, as seen in Figures 10
and 11a, the temporal modulation of excitatory input rates
using E = 0.6 still raises the output rate by ~10 spikes/sec when the mean input rate is <80 spikes/sec approximately, and the effect still increases monotonically as correlations become stronger, as indicated in Figure 11c.
Oscillations, however, seem to be less effective in driving the target
neuron when it is below threshold, as can be observed by comparing
Figures 11a and 7a (insets). It should also be
borne in mind that higher rates will be evoked whenever there is a
phase difference between excitatory and inhibitory inputs, as
illustrated in Figure 8e.
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Figure 10.
Sample output spike trains from an unbalanced
( = 0.45) model neuron driven by inputs whose rates
oscillate in time. Same format and correlation values as in Figure 8,
except that AE = 60 spikes/sec. Other
parameters as indicated for the unbalanced neuron.
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Figure 11.
Effects of correlations generated by oscillating
input rates on the firing rate and variability of the same unbalanced
( = 0.45) model neuron used in Figure 10. Same
format and input parameters as in Figure 9, except that, in
c and d, rE = 60
spikes/sec. Parameters of the postsynaptic neuron as indicated for the
unbalanced condition.
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In summary, as far as output firing rate goes, the presence of input
correlations has similar effects regardless of the mechanism by which
those correlations are generated, and such effects are well described
by the theoretical model developed above. A comparison between Figures
2a, 5a, and 9a shows that this is true
for a balanced model neuron, and a comparison between Figures
2b, 7a, and 11a shows that this is
also true for an unbalanced neuron. In contrast to the rate, the
variability of the interspike intervals is sensitive to the dynamics
that give rise to the correlations. Correlations arising from common
drive tend to increase the CVISI of the output, whether the postsynaptic neuron is balanced (Figures 5b, d)
or not (Figures 7b, d), whereas correlations arising from
periodic, temporal co-fluctuations in rate have a weaker tendency to
decrease the CVISI when the postsynaptic neuron
is balanced (Figures 9b, d), and their effect on an
unbalanced neuron may be either upward or downward (Figure
11b).
Impact of other factors contributing to input variance
A general implication of the analytic results presented here is
that a balanced neuron responds to variance in synaptic input: variance
is its driving force. In the stochastic model, variability in the
statistics of input spikes provided all the variance, but synapses
themselves also behave stochastically, and so do the channels and
receptors on the postsynaptic membrane (Calvin and Stevens,
1968). Including synaptic variability corresponds to considering a E that is not constant, but
rather comes from a distribution. If the theoretical analysis is
correct, unreliable, stochastic synapses that on average produce a
change in voltage of a given size should give rise to larger firing
rates than perfectly reliable synapses that always produce a
depolarization of the same mean size. In this case, failures in
synaptic transmission may actually boost the output firing rate,
because large but infrequent depolarizations have a better chance of
causing a spike than small and frequent ones.
To study this, we allowed synaptic failures in the conductance-based
model. In this case, whenever an input spike arrives, the synaptic
conductance can do two things, either increase by an amount
g(t)/PT (as in Eqs. 25 and 27), or remain the
same, as if no spike had arrived. The first option corresponds to
successful transmission and occurs with probability
PT; the second option corresponds to a failure
and occurs with probability 1 PT. With this scheme, the mean conductance change averaged over many input spikes is always the same (and equal to g(t)) regardless
of the probability of transmission PT. The case
PT = 1 corresponds to zero failures, as was
considered in all previous results.
Figure 12 shows how the output rate and
CVISI change when synapses are allowed to fail.
For this figure, the dynamics described in the previous paragraph were
applied to AMPA and GABA conductances, with separate
PT values for each. Based on experimental
reports (Murthy et al., 1997), we chose
PT = 0.15. Figure 12, a and
b, corresponds to a balanced postsynaptic neuron, and Figure
12, c and d, corresponds to an unbalanced one,
with the same sets of parameters used before. For these curves, all
inputs were uncorrelated. The curves are in good agreement with the
theory: failures increase input variance and, when the neuron is
balanced, this produces an increase in gain. In contrast to spike train
correlations, inhibitory failures cause larger effects than excitatory
ones. This can be understood by calculating the variance of
V across time steps. Starting from Equations 2 and 3, we
proceed as in the calculation of Equation 18, but assume that the
unitary voltage changes E and
I have binomial distributions with means
E and
I. In the absence of correlations, the
variance of V is then:
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(37)
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where PTE and PTI
are the transmission probabilities for excitatory and inhibitory
synapses, respectively. When these are equal to 1, only the first two
terms in the expression survive, and the original variance equal to
E2 2 is
recovered (this is obtained from Eqs. 3 and 18 with zero correlations and constant E). When the probabilities are
<1, contributions from excitatory and inhibitory failures increase the
variance. In the balanced condition MI
I2 is much larger than
ME E2 (Fig. 2,
legend) so, for similar transmission probabilities, the contribution
from inhibitory failures is bigger than the contribution from
excitatory ones, in agreement with Figure 12a. The effect of
failures can also be thought of as an increase in
2. To see this, first, obtain an effective
2 by factorizing the quantity
E2 from the above
equation; the result is:
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(38)
|
Second, compare this to Equation 18, assuming zero correlations.
For fixed values of E and
I, failures behave as if they changed
2. This explains why in the results of Figure
12a it appears as if failures increased the variance of the
spike counts.
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Figure 12.
Effects of synaptic failures on the firing rate
and variability of a target model neuron. In all cases inputs were
uncorrelated. For every incoming spike, a failure in transmission could
occur with probability 1 PT, in which
case no conductance change was elicited. Transmission occurred with
probability PT, and in this case a conductance
change took place as in previous simulations (Eqs. 25, 27), except that
maximal conductances were multiplied by a factor of
1/PT. Continuous lines indicate
PT = 1 for all inputs; filled
circles indicate PT = 0.15 for
excitatory and PT = 1 for inhibitory
synapses; open circles indicate PT = 1 for excitatory and PT = 0.15 for
inhibitory synapses; and dots indicate
PT = 0.15 for both excitatory and
inhibitory synapses. For each data point, the output of the
postsynaptic neuron was recorded for 30-90 sec of simulation time, and
the mean rate and coefficient of variation were computed from this
segment. a, Mean output firing rate
rout as a function of mean input rate
rE for a balanced neuron with = 1. b, CVISI of the same target
neuron used in a, as a function of mean input rate. c,
d, as in a and b but for an unbalanced
neuron with = 0.45. Inset amplifies the
region around threshold. Other parameters as indicated for balanced and
unbalanced conditions.
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In contrast to correlations, synaptic failures always add more
variance. This is because excitatory and inhibitory synapses release
their neurotransmitters independently of each other, so there is no
negative term equivalent to EI. The small
dots in Figure 12a show that a larger increase in mean
output rate is obtained when both excitatory and inhibitory synaptic
failures are included, as expected from the two equations above. As
shown in Figure 12c, for the unbalanced neuron the increase
in rate is much smaller, but still appreciable, especially near
threshold. Lowering of firing threshold attributable to increased
variance in membrane potential fluctuations has also been observed in
more detailed neuronal models (Bernander et al., 1991;
Destexhe and Hô, 1999; Hô et al.,
2000).
According to this result, an excitatory presynaptic neuron may be more
effective, on average, when it makes a synaptic contact of strength
2S that fails half the time, than when it makes a contact of
strengths S that never fails. Thus, synaptic failures and
synchronous spikes give rise to similar effects on the postsynaptic membrane. This may seem counterintuitive, but the key is that both
increase the variance in synaptic drive, and variance drives the neuron
when the mean drive is small in comparison.
The intrinsic variability of the postsynaptic membrane also contributes
to the variance of the membrane potential. This means that, in a
balanced neuron driven exclusively by fluctuations, intrinsic membrane
noise should produce an upward shift in the output versus input firing
rate curve. We confirmed this through simulations in which, on each
integration time step, independent Gaussian noise with zero mean was
added directly to the voltage of the postsynaptic neuron. The output
rate of the target neuron indeed increased with added Gaussian noise as
expected for balanced and unbalanced conditions and, interestingly,
Gaussian noise could also decrease the CVISI in
both cases (data not shown; Tiesinga and José,
1999). To observe these effects it was necessary to inject
relatively large amounts of noise (we used a SD of 0.17 mV).
Experimental findings (Calvin and Stevens, 1968;
Mainen and Sejnowski, 1995; Holt et al.,
1996) and theoretical arguments (Softky and Koch,
1993) indicate that statistical fluctuations caused by
spike-generating mechanisms should be much smaller than those produced
by variations in spike arrival times. In agreement with these
observations, we found that intrinsic membrane noise had much less
impact on firing rate and interspike interval variability than input
correlations. The reason for this, from Equations 19 and 20, is that
the correlation terms are multiplied by the square of the corresponding
numbers of inputs, which are at least in the hundreds. Therefore, even
weak correlations may contribute much more than other noise sources to
fluctuations in membrane potential.
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DISCUSSION |
The impact of input correlations on the mean firing rate of a
postsynaptic neuron was calculated using a simple stochastic model. The
analytical results (Fig. 2) were confirmed through simulations of a
conductance-based, integrate-and-fire neuron driven by hundreds of
synaptic inputs. In this framework, the process of synaptic integration
can be cast as follows. Both the drift, or mean net drive µaverage
excitation minus inhibition minus decay caused by leakageand the
fluctuations around this mean may cause a postsynaptic neuron to fire
(Troyer and Miller, 1997). Correlations between excitatory neurons or
between inhibitory neurons increase the variance 2 of
the fluctuations, whereas correlations between excitatory-inhibitory pairs decrease it. Correlations have their largest impact upon fully
balanced neurons (µ 0), because these are driven exclusively by fluctuations. However, even in unbalanced neurons weak correlations may have a significant impact around firing threshold, because at that
point it is usually the case that µ  0. By weak we mean shared input with = 0.1, which corresponds roughly
to the probability that two nearby excitatory cortical neurons are
connected (Braitenberg and Schüz, 1997;
Shadlen and Newsome, 1998).
The theoretical model did not provide an expression for the variability
of the interspike interval distribution of the postsynaptic neuron, but
it did bound its coefficient of variation CVISI
between 0 and 1, depending on which contributed more to the firing of the neuron, µ or . When = 0 and µ > 0, a neuron fires regularly, like a clock, and when
> 0 and µ 0, it fires
irregularly, like a Geiger counter. These facts are useful in reviewing
previous studies regarding neuronal variability.
The spike trains of neurons recorded in awake animals are extremely
variable (Burns and Webb, 1976; Dean,
1981; Softky and Koch, 1993; Holt et al.,
1996; Stevens and Zador, 1998; Shinomoto et al., 1999), but spike generation mechanisms themselves seem to be highly reliable (Calvin and Stevens, 1968;
Mainen and Sejnowski, 1995; Holt et al.,
1996). Therefore, although intrinsic properties may still be
important (Bell et al., 1995; Troyer and Miller, 1997), the variability of a neuron in an intact microcircuit
should come mostly from the variability of its inputs. Softky
and Koch (1992, 1993) pointed out that, although the
CVISI of typical cortical neurons is close to 1, this number should be much lower for an integrator that adds up many
small contributions to fire, especially at high output rates. They
found that, in the absence of inhibition, high
CVISI values could be obtained either with an
unrealistically small (i.e. submillisecond) membrane time constant, or
using active dendrites that worked as isolated coincidence detectors
(Abeles, 1982, 1991; Softky, 1993;
Shadlen and Newsome, 1994). In our integrate-and-fire model, a smaller time constant m also
produced higher CVISI values, but otherwise it
did not alter the effects of correlations (time constants down to 1 msec were tested; results not shown). Shadlen and Newsome
(1994) later showed that, as in earlier stochastic models
(Gerstein and Mandelbrot, 1964; Tuckwell,
1988; Smith, 1992), including incoming
inhibitory spikes produces high CVISI values.
Their model was nearly balanced. Troyer and Miller
(1997) extended these results by carefully tuning an
integrate-and-fire neuron to produce a relatively high
CVISI while using biologically plausible amounts
of inhibition (Bell et al., 1995). Their simulation results fell in the lower half of the range 0.5 to 1 reported by
Softky and Koch (1993). More recently, Stevens
and Zador (1998) (see also Destexhe and Paré,
2000) have also suggested, based on experiments in a slice
preparation, that input synchrony is required to produce the high
variability observed in vivo. This is consistent with
the suggestion that stochastic eye movements, which provide a common,
correlating signal, are responsible for a large fraction of the
variability observed in primary visual neurons (Gur et al.,
1997). It also agrees with recent theoretical results
(Feng and Brown, 2000), and with simulation studies in which network interactions produce synchronized recurrent input, which
also leads to high variability (Usher et al., 1994;
Tsodyks and Sejnowski, 1995; Van Vreeswijk and
Sompolinsky, 1996).
In the unbalanced condition, we found that weak correlations generated
by common drive to the inputs could raise the
CVISI from ~0.6 to 1 (Stevens and
Zador, 1998). This could result from correlations between
excitatory neurons only or between all inputs (Fig. 7b;
Feng and Brown, 2000). We also observed that
correlations could raise the CVISI of the
conductance-based model well beyond 1 (Figure 5b, d), even
in the unbalanced condition (Fig. 7d). This is in marked
contrast to our random walk model and to other models based on
memoryless processes where, it should be noted, the refractory period
(or dead time) may play an important role in limiting the
CVISI (Smith, 1992; see also
Tuckwell, 1988; Shadlen and Newsome,
1998; Koch, 1999; Shinomoto et al.,
1999).
Earlier studies also explored the relationship between input
correlation and output rate. Bernander et al., (1994)
and Murthy and Fetz (1994) found that synchrony could
increase the firing rate of the target cell but only up to a point,
after which the rate tended to decrease. This happened because, on one
hand, leakage tends to erase the effect of previous inputs, so these
should come close together in time to avoid wasting depolarization. But on the other hand, synaptic inputs that arrive during the refractory period have little or no effect, and to avoid this situation inputs should be somewhat spread out in time. The tradeoff between these two
conditions, as well as the degree of correlation between inputs, gives
rise to an optimal time window for synaptic integration. However, this
scenario is valid when excitation is overwhelmingly larger than
inhibition and when the output neuron is firing intensely, such that
refractory effects become important. In this unbalanced regime the
postsynaptic cell operates at a low gainin these studies, at least
100 (net) excitatory input spikes were required to produce a single
output spikeand firing is much more regular than observed experimentally. Because neurons fire irregularly and appear to operate
at high gain (Softky and Koch, 1992, 1993;
Shadlen and Newsome, 1994, 1998; Troyer and
Miller, 1997), the effects in low gain models may be small
within the normal dynamic range of real neurons. In our simulations,
the effects of correlations did not vary appreciably with the
refractory period.
Softky and Koch (1993) (see also Ritz and
Sejnowski, 1997) also noticed that synchronized inputs produced
higher output firing rates than independent inputs. They observed this
for positive correlations between excitatory pairs. Other studies
(Sejnowski, 1976; Kenyon et al., 1990;
Bernander et al., 1991; Bell et al., 1995) are also consistent with our finding that an unbalanced neuron should be considerably more sensitive to synchronous excitatory spikes than to uncorrelated ones. In contrast, Shadlen and
Newsome (1998) studied the effect of input synchrony and found
that even fractions of shared connections among inputs as large as 0.4 did not elicit any change in the statistics of a target neuron. Their model was fully balanced, so the output rate must have been most sensitive to the presence of correlations. However, they chose parameters for which 2 is given by Equation 20 and,
because they used identical correlations among all possible pairs, the
three correlation terms cancelled out exactly.
This brings us to the two experimental questions that are crucial to
evaluate the impact of correlations. First, what is the actual balance
between excitation and inhibition? In our models, inhibition countered
excitation through changes in conductance and firing rate, but other
factors like synaptic location are also important. Whatever the
biophysical implementation, balance is crucial because, according to
our models, it determines whether input correlations affect the gain of
a neuron either throughout its full dynamic range or mostly at low
firing rates. Second, what are the relative magnitudes of the three
correlation terms in typical cortical circuits? At the moment,
experimental data seem insufficient to determine this. Based on
anatomical considerations (White, 1989;
Braitenberg and Schüz, 1997) and
neurophysiological measurements (Fetz et al., 1991;
Nelson et al., 1992; Zohary et al., 1994;
Salinas et al., 2000), it seems likely that all terms are different from zero, at least for local microcircuits; but what
really needs to be known is the final weighted sum. This final sum
might not be constant, neither in time nor across cortical areas. Here
we illustrated various possible combinations of the three terms and
found that even a small net deviation from zero may have a large impact
on output gain and variability. Given such high sensitivity, perhaps
the correlation structure between neurons is dynamically modulated.
Recent experiments suggest that synchrony may covary with attention
(Steinmetz et al., 2000). Furthermore, because the
timing of correlations may also enhance their effects (Fig.
8e), correlations could also interact with normal plasticity
mechanisms that depend on spike timing (Linden, 1999;
Sejnowski, 1999; Paulsen and Sejnowski,
2000). Therefore, dynamic changes in correlations could have
profound functional implications (Singer and Gray, 1995;
Shadlen and Newsome, 1998).
It is interesting that our model neurons may encode the statistical
properties of their inputs in different ways, depending on their
balance. In a balanced neuron, output firing rate and variability are
both sensitive to input rate and to correlations, and the input-output
rate curve is nonlinear (Fig. 5a,b). On the other hand, in
an unbalanced neuron output rate is only modestly sensitive to input
correlations, the input-output rate curve is practically linear,
except for the threshold, and variability is sensitive to correlations
but much less so to input rate (Fig. 7a,b). The next step is
to explore the statistics of spike trains in feedback models and to
relate these to the functional properties of cortical circuits
(Singer and Gray, 1995; Tsodyks and Sejnowski, 1995; Van Vreeswijk and Sompolinsky, 1996;
Shadlen and Newsome, 1998).
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FOOTNOTES |
Received Feb. 28, 2000; revised May 16, 2000; accepted June 7, 2000.
This work was supported by the Howard Hughes Medical Institute. We
thank Larry Abbott and Paul Tiesinga for helpful comments.
Correspondence should be addressed to Emilio Salinas, Computational
Neurobiology Laboratory, The Salk Institute for Biological Studies,
10010 North Torrey Pines Road, La Jolla, CA 92037. E-mail: emilio{at}salk.edu.
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APPENDIX |
Here we derive Equations 6 and 7, which give the firing rate of
the stochastic neuron as a function of µ and , the mean and SD,
respectively, of the net number of excitatory inputs that arrive at the
neuron in one t. We make use of Equations 1-5.
The methods used to develop the stochastic model are standard in the
mathematics and physics literature. The present model is closely
related to the Ornstein-Uhlenbeck process, for which a closed-form
solution is not known, although asymptotic expansions have been found
(Ricciardi, 1977; Smith, 1992;
Shinomoto et al., 1999). The major difference is that
the Ornstein-Uhlenbeck process includes a term proportional to
V in Equation 1, a true leak, which drives the membrane
potential toward rest. The constant decay assumed here and represented
by the term d is thus a key simplification. As general
references on stochastic processes and on the computation of first
passage times the reader may consult, for example, the books by
Ricciardi (1977), Tuckwell (1989), or Risken (1996); see also Berg (1993) for
applications to other problems in biology.
First consider a situation analogous to a purely random walk without
drift, for which µ = 0. Assume that the distribution function P is symmetric, so that P(n) = P(n) for any n. In this case, the mean number of
steps needed to reach a threshold, as a function of N, obeys
the relationship:
![&ngr;(N)=1+ <LIM><OP>∫</OP><LL>0</LL><UL>∞</UL></LIM> [&ngr;(N+n)+&ngr;(N−n)]P(n)dn.](/content/vol20/issue16/fulltext/6193/img047.gif)
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(39)
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This equation is a generalization of the classic random walk in
which the magnitude of the step is constant and positive and negative
steps are equally probable (Berg, 1993) (see also Feynman et al., 1963). In that case the integral
corresponds to a sum of only two terms, with P = 1/2.
This expression follows from the fact that, in a single time step,
N can go from its present value to either N + n or N n with equal probabilities, hence the
requirement that P be symmetric. From this expression, a
differential equation for can be derived assuming that n
is small relative to N (Berg, 1993). The
alternative approach that follows leads to a similar solution but
highlights its key properties and limitations more clearly.
Recalling that the integral over all probabilities must be equal to 1, the above expression can be rearranged as:
![1+ <LIM><OP>∫</OP><LL>0</LL><UL>∞</UL></LIM> [&ngr;(N+n)+&ngr;(N−n)−2&ngr;(N)]P(n)dn=0.](/content/vol20/issue16/fulltext/6193/img048.gif)
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(40)
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Suppose the solution is a polynomial:
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(41)
|
where the ci are constant coefficients. For
the function (N) to satisfy Equation 40, the term in
brackets must be a function of n only; it cannot depend on
N. This condition is satisfied when:
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(42)
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and all other terms have cj = 0. This
can be verified by direct substitution. If higher coefficients are
nonzero, terms containing N survive, invalidating the
solution. By substituting the above quadratic function into Equation 40
we find:
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(43)
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The other two constants can be found by imposing boundary
conditions. The reflecting barrier can be taken into account by setting
the derivative of with respect to N equal to zero when evaluated at N = 0. This makes
c1 = 0, so:
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(44)
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The remaining constant can be set by specifying what happens when
N reaches threshold. Typically, the condition used for this
is (N) = 0 (Berg,
1993). This, however, is not the best choice for our
application, because should never be less than 1; it must always
take at least one time step for N to exceed threshold.
Therefore, should be zero when evaluated at N +  , where is the typical change in N during a
single time step. This is simply so, imposing the condition
(N + ) = 0 in the previous
equation, we finally obtain:
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(45)
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The average number of steps between consecutive output spikes
results when N = Nreset. The in the
numerator comes from the last boundary condition. It guarantees that
will be >1, but does not alter the scaling of the random walk,
namely, if N, N, and are multiplied by
the same factor, the expected number of steps to reach threshold does
not change.
This solution satisfies Equation 40 exactly. The caveat here is that
Equation 39 itself is an approximation, because at the boundaries it is
not true that increases and decreases in N are equally
probable. Nevertheless, the solution does not require that individual
steps be infinitesimally small, as in the limit where the diffusion
equation holds (Gerstein and Mandelbrot, 1964; Tuckwell, 1988; Smith, 1992). Instead,
Equation 45 should break down only when is comparable to
N and boundary effects become significant.
The case where there is a constant drift and µ is different from zero
can be approximated by substituting Nreset + µ for N in the right-hand side of Equation 45.
This can be seen as follows. After steps, the drift must have
contributed an amount µ to the current value of
N, but this is equivalent to starting from the initial value
N = Nreset + µ with zero drift.
Equivalently, one may think that it is the lower limit and the
threshold that move at a constant speed equal to µ so that after
steps Nrest and N
have both changed by an amount µ. Either way, the result is a
quadratic equation for ,
|
(46)
|
Here we have made the substitution N = Nreset and, as before, have assumed that
Nrest = 0. This equation gives the expected number of time steps elapsed between the firing of two consecutive action potentials as a function of µ and . To express the same result in terms of the mean firing rate of the output neuron, rout, use Equation 5 and note that
rout = 1/T; then:
|
(47)
|
which is Equation 6.
Including µ as we have done in the last two equations is valid when
µ 0, or when µ is negative but small in absolute
value with respect to . Otherwise, N hits the lower
boundary frequently, and Nreset + µ
falls below zero. Through simulations, we found that the case in which µ is negative can be approximated accurately by reducing by an
amount proportional to µ and eliminating the µ terms in the above
equations; the result is:
|
(48)
|
where c is a constant. This is Equation 7.
| |
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TOP
ABSTRACT
INTRODUCTION
