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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 |
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ABSTRACT
INTRODUCTION
