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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


    ABSTRACT
TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

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


    INTRODUCTION
TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

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.


    MATERIALS AND METHODS
TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

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 Delta E, and each incoming inhibitory spike decreases the membrane potential by an amount Delta 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 Vtheta , 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).


    RESULTS
TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

Changes in voltage modeled as random walk steps

According to the above description, at each time step Delta t the voltage jumps by an amount:
&Dgr;V=n<SUB>E</SUB>&Dgr;<SUB>E</SUB>−n<SUB>I</SUB>&Dgr;<SUB>I</SUB>−d, (1)
where nE and nI are the total numbers of incoming excitatory and inhibitory spikes that arrived in that interval Delta t. By defining the net number of excitatory spikes as:
n≡<FR><NU>&Dgr;V</NU><DE>&Dgr;<SUB>E</SUB></DE></FR>=n<SUB>E</SUB>−<FR><NU>&Dgr;<SUB>I</SUB></NU><DE>&Dgr;<SUB>E</SUB></DE></FR>n<SUB>I</SUB>−<FR><NU>d</NU><DE>&Dgr;<SUB>E</SUB></DE></FR>, (2)
the change in voltage can be written as:
&Dgr;V=n&Dgr;<SUB>E</SUB>. (3)
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 sigma 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 Delta t, whereas sigma 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:
N=N<SUB>rest</SUB>+<FR><NU>V−V<SUB>rest</SUB></NU><DE>&Dgr;<SUB>E</SUB></DE></FR>. (4)
Thus, N changes by n in each time step, it has a lower limit of Nrest, it needs to reach a critical value Ntheta for the postsynaptic neuron to fire again, and is reset to Nreset after each postsynaptic spike. Ntheta is obtained when V = Vtheta 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 Ntheta .

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 nu  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 Vtheta . The total amount of time will be:
T=&ngr;&Dgr;t. (5)
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 sigma  is small, nu  should be approximately Ntheta /µ. Now suppose instead that µ = 0 so there is no drift. In this case N just fluctuates around its initial value. After nu  steps, however, the typical displacement (positive or negative, in the root mean square sense) relative to the starting point is sigma <RAD><RCD><IT>&ngr;</IT></RCD></RAD> (Feynman et al., 1963). Hence, now it should take on the order of (Ntheta /sigma )2 steps for N to reach a point Ntheta units away. In general, then, it would seem that either µ or sigma  may drive the neuron to fire. A more detailed analysis confirms this idea and leads to the following expressions (see Appendix). When µ >=  0,
r<SUP>2</SUP><SUB>out</SUB>&Dgr;t<SUP>2</SUP>((N<SUB>&thgr;</SUB>+&sfgr;)<SUP>2</SUP>−N<SUP>2</SUP><SUB>reset</SUB>)−r<SUB>out</SUB>&Dgr;t(2&mgr;N<SUB>reset</SUB>+&sfgr;<SUP>2</SUP>)−&mgr;<SUP>2</SUP>=0. (6)
When µ > 0 there is a net excitatory drive and, in general, both µ and sigma  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 sigma  remains smaller than Ntheta (see Appendix). On the other hand, when µ <=  0,
r<SUB>out</SUB>=<FR><NU>(&sfgr;+c&mgr;)<SUP>2</SUP></NU><DE>&Dgr;t<FENCE>(N<SUB>&thgr;</SUB>+&sfgr;+c&mgr;)<SUP>2</SUP>−N<SUB>reset</SUB></FENCE></DE></FR>, (7)
where c is a constant. In this case the negative drive acts to effectively decrease sigma  by an amount proportional to µ. This happens up to the point where sigma  + cµ = 0, beyond which the output firing rate is set to zero (otherwise, sigma  + 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 sigma  (small or large relative to the distance from rest to threshold), second, they combine µ and sigma  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 sigma , second, as functions of the mean firing rate and variability of the input spike trains, which determine µ and sigma , 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 right-arrow 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 nu  was obtained. For the results shown in Figure 1c-h, Ntheta  = 40, Nreset = 20, and sigma  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 sigma  is almost as large as Ntheta .



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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 sigma . When N reaches the threshold (dotted line), a spike is emitted (vertical bars), and N is lowered to its reset value. In this figure Ntheta  = 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, sigma  = 2. b, The neuron is driven exclusively by the fluctuations, so it fires irregularly; n was drawn from a Gaussian distribution with µ = 0, sigma  = 8. Notice N cannot fall below the reflecting barrier at 0. c-e, Output firing rate (rout) as a function of sigma . 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 sigma . The three panels correspond to the three distributions for n shown in the above insets. Colors indicate same parameter values as in panels above.

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 right-arrow 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 sigma  is close to 0 (Gerstein and Mandelbrot, 1964; Tuckwell, 1988; Usher et al., 1994; Koch, 1999), Equation 6 is reduced to
r<SUB>out</SUB>≈<FR><NU>&mgr;</NU><DE>&Dgr;t(N<SUB>&thgr;</SUB>−N<SUB>reset</SUB>)</DE></FR>. (8)
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 sigma  = 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,
r<SUB>out</SUB>=<FR><NU>&sfgr;<SUP>2</SUP></NU><DE>&Dgr;t<FENCE>(N<SUB>&thgr;</SUB>+&sfgr;)<SUP>2</SUP>−N<SUP>2</SUP><SUB>reset</SUB></FENCE></DE></FR>. (9)
In this case the output firing rate increases monotonically with sigma  up to the limit 1/Delta t. The Delta t of the model has a functional interpretation: it represents the refractory period, because only one spike is allowed per Delta t. In this mode the neuron fires because there are fluctuations in the numbers of excitatory and inhibitory input spikes that arrive per Delta 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 different---they 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 sigma  using the same results used in Figure 1c-e. The plots confirm the intuitive picture discussed in the previous paragraphs: when sigma  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 sigma  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 sigma  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 Delta 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:
n= <LIM><OP>∑</OP><LL>j</LL><UL>M<SUB>E</SUB></UL></LIM> n<SUP>j</SUP><SUB>E</SUB>−<FR><NU>&Dgr;<SUB>I</SUB></NU><DE>&Dgr;<SUB>E</SUB></DE></FR> <LIM><OP>∑</OP><LL>k</LL><UL>M<SUB>I</SUB></UL></LIM> n<SUP>k</SUP><SUB>I</SUB>−<FR><NU>d</NU><DE>&Dgr;<SUB>E</SUB></DE></FR>. (10)
We are interested in the mean and the variance of n, which are µ and sigma 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:
⟨n<SUP>j</SUP><SUB>E</SUB>⟩=r<SUB>E</SUB>&Dgr;t. (11)
Similarly, all inhibitory neurons fire at a mean rate rI but, furthermore, we will assume that inhibitory and excitatory rates are proportional, such that:
⟨n<SUP>k</SUP><SUB>I</SUB>⟩=r<SUB>I</SUB>&Dgr;t=&agr;r<SUB>E</SUB>&Dgr;t, (12)
where alpha  is the constant of proportionality. With these definitions, the mean value of n is simply:
&mgr;=⟨n⟩=r<SUB>E</SUB>&Dgr;t M<SUB>E</SUB><FENCE>1−&agr;<FR><NU>M<SUB>I</SUB></NU><DE>M<SUB>E</SUB></DE></FR> <FR><NU>&Dgr;<SUB>I</SUB></NU><DE>&Dgr;<SUB>E</SUB></DE></FR> </FENCE>−<FR><NU>d</NU><DE>&Dgr;<SUB>E</SUB></DE></FR>. (13)
The fraction inside the square brackets reflects the balance between excitation and inhibition,
&bgr;<SUB>RW</SUB>≡&agr;<FR><NU>M<SUB>I</SUB></NU><DE>M<SUB>E</SUB></DE></FR> <FR><NU>&Dgr;<SUB>I</SUB></NU><DE>&Dgr;<SUB>E</SUB></DE></FR>. (14)
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 beta 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:
<AR><R><C>⟨(n<SUP>j</SUP><SUB>E</SUB>−⟨n<SUP>j</SUP><SUB>E</SUB>⟩)<SUP>2</SUP>⟩</C><C>=</C><C>s<SUP>2</SUP><SUB>E</SUB></C></R><R><C>⟨(n<SUP>k</SUP><SUB>I</SUB>−⟨n<SUP>k</SUP><SUB>I</SUB>⟩)<SUP>2</SUP>⟩</C><C>=</C><C>s<SUP>2</SUP><SUB>I</SUB>.</C></R></AR> (15)
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:
&rgr;<SUB>xy</SUB>=<FR><NU>⟨(x−⟨x⟩)(y−⟨y⟩)⟩</NU><DE><RAD><RCD>⟨(x−⟨x⟩)<SUP>2</SUP>⟩</RCD></RAD> <RAD><RCD>⟨(y−⟨y⟩)<SUP>2</SUP>⟩</RCD></RAD></DE></FR>. (16)
So, using the above definitions for the variances sE2 and sI2, the pairwise correlation coefficients for the inputs are:
<AR><R><C><FR><NU>⟨(n<SUP>j</SUP><SUB>E</SUB>−⟨n<SUP>j</SUP><SUB>E</SUB>⟩) (n<SUP>k</SUP><SUB>E</SUB>−⟨n<SUP>k</SUP><SUB>E</SUB>⟩)⟩</NU><DE>s<SUP>2</SUP><SUB>E</SUB></DE></FR></C><C>=</C><C>&rgr;<SUB>EE</SUB></C></R><R><C><FR><NU>⟨(n<SUP>j</SUP><SUB>I</SUB>−⟨n<SUP>j</SUP><SUB>I</SUB>⟩) (n<SUP>k</SUP><SUB>I</SUB>−⟨n<SUP>k</SUP><SUB>I</SUB>⟩)⟩</NU><DE>s<SUP>2</SUP><SUB>I</SUB></DE></FR></C><C>=</C><C>&rgr;<SUB>II</SUB></C></R><R><C><FR><NU>⟨(n<SUP>j</SUP><SUB>E</SUB>−⟨n<SUP>j</SUP><SUB>E</SUB>⟩) (n<SUP>k</SUP><SUB>I</SUB>−⟨n<SUP>k</SUP><SUB>I</SUB>⟩)⟩</NU><DE>s<SUB>E</SUB>s<SUB>I</SUB></DE></FR></C><C>=</C><C>&rgr;<SUB>EI</SUB>.</C></R></AR> (17)
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:
&sfgr;<SUP>2</SUP>=s<SUP>2</SUP><SUB>E</SUB>M<SUB>E</SUB>(1+M<SUB>E</SUB>&rgr;<SUB>EE</SUB>)+s<SUP>2</SUP><SUB>I</SUB>M<SUB>I</SUB><FR><NU>&Dgr;<SUP>2</SUP><SUB>I</SUB></NU><DE>&Dgr;<SUP>2</SUP><SUB>E</SUB></DE></FR>(1+M<SUB>I</SUB>&rgr;<SUB>II</SUB>) (18)

−2s<SUB>E</SUB>s<SUB>I</SUB>M<SUB>E</SUB>M<SUB>I</SUB><FR><NU>&Dgr;<SUB>I</SUB></NU><DE>&Dgr;<SUB>E</SUB></DE></FR>&rgr;<SUB>EI</SUB>.
This expression already shows the dependence of sigma  on the correlation structure of the inputs. However, the link can be made clearer. Assume further that the time step Delta 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 rEDelta t and variance rEDelta t(1 - rEDelta t). Thus, the relationship between sigma  and the input statistics in the case of the binomial approximation is:
&sfgr;<SUP>2</SUP>=r<SUB>E</SUB>&Dgr;tM<SUB>E</SUB><FENCE>(1−r<SUB>E</SUB>&Dgr;t)(1+M<SUB>E</SUB>&rgr;<SUB>EE</SUB>)</FENCE> (19)

+&agr;<FR><NU>M<SUB>I</SUB></NU><DE>M<SUB>E</SUB></DE></FR> <FR><NU>&Dgr;<SUP>2</SUP><SUB>I</SUB></NU><DE>&Dgr;<SUP>2</SUP><SUB>E</SUB></DE></FR> (1−&agr;r<SUB>E</SUB>&Dgr;t)(1+M<SUB>I</SUB>&rgr;<SUB>II</SUB>)

<FENCE>−2M<SUB>I</SUB><FR><NU>&Dgr;<SUB>I</SUB></NU><DE>&Dgr;<SUB>E</SUB></DE></FR> <RAD><RCD>&agr;(1−r<SUB>E</SUB>&Dgr;t)(1−&agr;r<SUB>E</SUB>&Dgr;t)</RCD></RAD>&rgr;<SUB>EI</SUB></FENCE>.
To better appreciate the interplay between correlation terms, for the moment we will consider a simplified version of this expression. First, assume that rEDelta t 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, alpha  = 1, and Delta I = Delta 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:
&sfgr;<SUP>2</SUP>=r<SUB>E</SUB>&Dgr;tM(2+M(&rgr;<SUB>EE</SUB>+&rgr;<SUB>II</SUB>−2&rgr;<SUB>EI</SUB>)). (20)
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 sigma  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 sigma 2.

At some point of the input-output rate curve, even an unbalanced neuron with beta  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 beta . Therefore, whatever the balance of the neuron, there will be a positive value of rE for which µ = 0 and sigma 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) rho EE > 0, rho II = rho EI = 0, (2) rho II > 0, rho EE = rho EI = 0, and (3) rho EE = rho II = rho EI > 0. However, the reader should keep in mind that it is the final weighted sum of the three terms that determines sigma 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 rho EE and rho II are positive and equal, and that rho EI is also positive but smaller than the other terms. In this case what counts is rho EE + rho II - 2rho EI, 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 sigma  to the output rate. In this figure the expressions for µ and sigma  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 Delta 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 Delta 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 sigma  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 Delta E = 0.5 mV and Delta I/Delta E = 2.35, which gives beta RW = 1. For the continuous line rho EE = 0, rho II = 0, and rho EI = 0. For the filled circles rho EE = 0.0033, rho II = 0, and rho EI = 0. For the open circles rho II = 0.0033, rho EE = 0, and rho 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 Delta E = 0.023 mV and Delta I/Delta E = 0.8, giving beta 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, alpha  = 1.7, d = 0.3 mV, Delta t = 1 msec, Vtheta  - Vrest = 20 mV, Vreset - Vrest = 10 mV, and c = 1.7.

Figure 2a shows the output firing rate as a function of input rate rE for a balanced neuron, for which beta 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 beta RW = 0.34; for these curves the ratio Delta I/Delta 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 Delta 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:
&tgr;<SUB>m</SUB> g<SUB>L</SUB><FR><NU>dV</NU><DE>dt</DE></FR>=−g<SUB>L</SUB>(V−E<SUB>L</SUB>)−I<SUB>SRA</SUB>−I<SUB>AMPA</SUB>−I<SUB>GABA</SUB>+I<SUB>APP</SUB>, (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 tau m/Rm, where tau 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:
<AR><R><C>I<SUB>SRA</SUB></C><C>=</C><C>g<SUB>SRA</SUB>(V−E<SUB>K</SUB>)</C></R><R><C>I<SUB>AMPA</SUB></C><C>=</C><C>g<SUB>AMPA</SUB>(V−E<SUB>AMPA</SUB>)</C></R><R><C>I<SUB>GABA</SUB></C><C>=</C><C>g<SUB>GABA</SUB>(V−E<SUB>Cl</SUB>).</C></R></AR> (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 Vtheta , 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 tau 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 Vtheta and a postsynaptic spike is elicited, the potassium conductance increases instantaneously by an amount Delta 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 tau SRA,
&Dgr;g<SUB>SRA</SUB>(t−t<SUB>0</SUB>)=<A><AC>g</AC><AC>&cjs1171;</AC></A><SUB>SRA</SUB><UP>exp</UP><FENCE>−<FR><NU>t−t<SUB>0</SUB></NU><DE>&tgr;<SUB>SRA</SUB></DE></FR></FENCE>, t>t<SUB>0</SUB>. (23)
Here t0 corresponds to the time at which the output spike was produced, and Delta 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:
g<SUB>SRA</SUB>(t)=<LIM><OP>∑</OP><LL>j</LL></LIM> &Dgr;g<SUB>SRA</SUB>(t−t<SUB>j</SUB>), (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, Vtheta  = -54 mV, Vreset = -60 mV, tau m = 20 msec, tau refrac = 1.72 msec, tau SRA = 100 msec, and <A><AC>g</AC><AC>&cjs1171;</AC></A>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 MOmega (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 MOmega (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 Delta gAMPA. This increase is fast, so a single exponential describing the subsequent decay is sufficient in this case too,
&Dgr;g<SUB>AMPA</SUB>(t−t<SUB>0</SUB>)=<A><AC>g</AC><AC>&cjs1171;</AC></A><SUB>AMPA</SUB><UP>exp</UP><FENCE>−<FR><NU>t−t<SUB>0</SUB></NU><DE>&tgr;<SUB>AMPA</SUB></DE></FR></FENCE>, t>t<SUB>0</SUB>. (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 tau AMPA. Subsequent input spikes add identical conductance changes, so that:
g<SUB>AMPA</SUB>(t)=<LIM><OP>∑</OP><LL>j</LL></LIM> &Dgr;g<SUB>AMPA</SUB>(t−t<SUB>j</SUB>). (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 Delta gGABA is better described by the difference of two exponentials,
&Dgr;g<SUB>GABA</SUB>(t−t<SUB>0</SUB>)= (27)

<FR><NU><A><AC>g</AC><AC>&cjs1171;</AC></A><SUB>GABA</SUB></NU><DE>D</DE></FR> <FENCE><UP>exp</UP><FENCE>− <FR><NU>t−t<SUB>0</SUB></NU><DE>&tgr;<SUP>(1)</SUP><SUB>GABA</SUB></DE></FR></FENCE>−<UP>exp</UP><FENCE>− <FR><NU>t−t<SUB>0</SUB></NU><DE>&tgr;<SUP>(2)</SUP><SUB>GABA</SUB></DE></FR></FENCE></FENCE>, t>t<SUB>0</SUB>.
Here the D factor is a normalization term that guarantees that the maximum of Delta gGABA is equal to <A><AC>g</AC><AC>&cjs1171;</AC></A>GABA. The two time constants tau GABA(1) and tau 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,
g<SUB>GABA</SUB>(t)=<LIM><OP>∑</OP><LL>j</LL></LIM> &Dgr;g<SUB>GABA</SUB>(t−t<SUB>j</SUB>). (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 = alpha rE, with alpha  being the constant of proportionality. These rates are constant, except when an explicit time dependence is indicated.

The balance of the neuron, beta , 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:
&bgr;≡&agr;<FR><NU>M<SUB>I</SUB></NU><DE>M<SUB>E</SUB></DE></FR> <FR><NU>G<SUB>I</SUB></NU><DE>G<SUB>E</SUB></DE></FR>, (29)
where
<AR><R><C>G<SUB>E</SUB></C><C>=</C><C>‖V<SUB>&thgr;</SUB>−E<SUB>AMPA</SUB>‖ <LIM><OP>∫</OP><LL>0</LL><UL>∞</UL></LIM> dt&Dgr;g<SUB>AMPA</SUB>(t)</C></R><R><C>G<SUB>I</SUB></C><C>=</C><C>‖V<SUB>&thgr;</SUB>−E<SUB>Cl</SUB>‖ <LIM><OP>∫</OP><LL>0</LL><UL>∞</UL></LIM> dt&Dgr;g<SUB>GABA</SUB>(t).</C></R></AR> (30)
When beta  = 1, there is no mean drift in voltage caused by synaptic inputs; we refer to this as the balanced condition. When beta  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 (beta  > 0), as opposed to an unbalanced one which receives only excitation (beta  = 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 beta  = 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: alpha  = 1.7, EAMPA = 0 mV, ECl = -61 mV, tau AMPA = 5 msec, tau GABA(1) = 5.6 msec, and tau 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 <A><AC>g</AC><AC>&cjs1171;</AC></A>AMPA = 0.0806 gL, and <A><AC>g</AC><AC>&cjs1171;</AC></A>GABA = 1.1143 gL, which gives beta  = 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 <A><AC>g</AC><AC>&cjs1171;</AC></A>AMPA = 0.0222 gL, and <A><AC>g</AC><AC>&cjs1171;</AC></A>GABA = 0.1382 gL, which gives beta  = 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