In this section, we derive, starting from a small set of assumptions, an integral equation for the population activity. The essential idea of the mathematical formulation is that we work as much as possible on the macroscopic level without reference to a specific model of neuronal dynamics. We will see that the interval distribution PI(t|) that has already been introduced in Chapter 5 plays a central role in the formulation of the population equation. Both the activity variable A and the interval distribution PI(t|) are `macroscopic' spike-based quantities that could, in principle, be measured by extracellular electrodes. If we have access to the interval distribution PI(t|) for arbitrary input I(t), then this knowledge is enough to formulate the population equations. In particular, there is no need to know anything about the internal state of the neuron, e.g., the current values of the membrane potential of the neurons.
Integral formulations of the population dynamics have been developed by (Gerstner, 1995,2000b; Wilson and Cowan, 1972; Gerstner and van Hemmen, 1992; Knight, 1972a). The integral equation (6.44) that we have derived in Section 6.2 via integration of the density equations turns out to be a specific instance of the general framework presented in this section.
We consider a homogeneous and fully connected network of spiking neurons in the limit of N. We aim for a dynamic equation that describes the evolution of the population activity A(t) over time.
We have seen in Eq. (6.8) that, given the population activity A(t') and the external input Iext(t') in the past, we can calculate the current input potential hPSP(t|) of a neuron that has fired its last spike at , but we have no means yet to transform the potential hPSP back into a population activity. What we need is therefore another equation which allows us to determine the present activity A(t) given the past. The equation for the activity dynamics will be derived from three observations:
Because of observation (ii) we know that the input-dependent interval distribution PI(t | ) contains all relevant information. We recall that PI(t | ) gives the probability density that a neuron fires at time t given its last spike at and an input I(t') for t' < t. Integration of the probability density over time PI(s | ) ds gives the probability that a neuron which has fired at fires its next spike at some arbitrary time between and t. Just as in Chapter 5, we can define a survival probability,
We now return to the homogeneous population of neurons in the limit of N and use observation (iii). We consider the network state at time t and label all neurons by their last firing time . The proportion of neurons at time t which have fired their last spike between t0 and t1 < t (and have not fired since) is expected to be
For an interpretation of the integral on the right-hand side of Eq. (6.72), we recall that A() is the fraction of neurons that have fired in the interval [, + ]. Of these a fraction SI(t|) are expected to survive from to t without firing. Thus among the neurons that we observe at time t the proportion of neurons that have fired their last spike between t0 and t1 is expected to be SI(t | ) A() d ; cf. Fig. 6.7.
Finally, we make use of observation (i). All neurons have fired at some point in the past6.2. Thus, if we extend the lower bound t0 of the integral on the right-hand side of Eq. (6.72) to - and the upper bound to t, the left-hand side becomes equal to one,
Since Eq. (6.73) is rather abstract, we try to put it into a form that is easier to grasp intuitively. To do so, we take the derivative of Eq. (6.73) with respect to t. We find
|0 = SI(t| t) A(t) + A() d .||(6.74)|
Eq. (6.75) is easy to understand. The kernel PI(t | ) is the probability density that the next spike of a neuron which is under the influence of an input I occurs at time t given that its last spike was at . The number of neurons which have fired at is proportional to A() and the integral runs over all times in the past. The interval distribution PI(t|) depends upon the total input (both external input and synaptic input from other neurons in the population) and hence upon the postsynaptic potential (6.8). Eqs. (6.8) and (6.75) together with an appropriate noise model yield a closed system of equations for the population dynamics. Eq. (6.75) is exact in the limit of N. Corrections for finite N, have been discussed by Meyer and van Vreeswijk (2001) and Spiridon and Gerstner (1999).
An important remark concerns the proper normalization of the activity. Since Eq. (6.75) is defined as the derivative of Eq. (6.73), the integration constant on the left-hand side of (6.73) is lost. This is most easily seen for constant activity A(t) A0. In this case the variable A0 can be eliminated on both sides of Eq. (6.75) so that Eq. (6.75) yields the trivial statement that the interval distribution is normalized to unity. Equation (6.75) is therefore invariant under a rescaling of the activity A0 c A0. with any constant c. To get the correct normalization of the activity A0 we have to go back to Eq. (6.73).
We conclude this section with a final remark on the form of Eq. (6.75). Even though (6.75) looks linear, it is in fact a highly non-linear equation because the kernel PI(t | ) depends non-linearly on hPSP, and hPSP in turn depends on the activity via Eq. (6.8).
Let us consider a population of Poisson neurons with an absolute refractory period . A neuron that is not refractory, fires stochastically with a rate f[h(t)] where h(t) is the total input potential, viz., the sum of the postsynaptic potentials caused by presynaptic spike arrival. After firing, a neuron is inactive during the time . The population activity of a homogeneous group of Poisson neurons with absolute refractoriness is (Wilson and Cowan, 1972)
The Wilson-Cowan integral equation (6.76) has a simple interpretation. Neurons stimulated by a total postsynaptic potential h(t) fire with an instantaneous rate f[h(t)]. If there were no refractoriness, we would expect a population activity A(t) = f[h(t)]. Not all neurons may, however, fire since some of the neurons are in the absolute refractory period. The fraction of neurons that participate in firing is 1 - A(t') dt' which explains the factor in curly brackets.
For constant input potential, h(t) = h0, the population activity has a stationary solution,
The function f in Eq. (6.76) was motivated by an instantaneous `escape rate' due to a noisy threshold in a homogeneous population. In this interpretation, Eq. (6.76) is the exact equation for the population activity of neurons with absolute refractoriness. In their original paper, Wilson and Cowan motivated the function f by a distribution of threshold values in an inhomogeneous population. In this case, the population equation (6.76) is an approximation, since correlations are neglected (Wilson and Cowan, 1972).
We apply the population equation (6.75) to SRM0 neurons with escape noise; cf. Chapter 5. The escape rate f (u - ) is a function of the distance between the membrane potential and the threshold. For the sake of notational convenience, we set = 0. The neuron model is specified by a refractory function as follows. During an absolute refractory period 0 < s, we set (s) to - . For s, we set (s) = 0. Thus the neuron exhibits absolute refractoriness only; cf. Eq. (5.57). The total membrane potential is u(t) = (t - ) + h(t) with
Given it seems natural to split the integral in the activity equation (6.75) into two parts
|PI(t | ) = f[h(t) + (t - )] exp - f[h(t') + (t' - )]dt' .||(6.80)|
Let us evaluate the two terms on the right-hand side of Eq. (6.79). Since spiking is impossible during the absolute refractory time, i.e., f[- ] = 0, the second term in Eq. (6.79) vanishes. In the first term we can move a factor f[h(t) + (t - )] = f[h(t)] in front of the integral since vanishes for t - > . The exponential factor is the survivor function of neurons with escape noise as defined in Eq. (5.7); cf. Chapter 5. Therefore Eq. (6.79) reduces to
Integral equations are often difficult to handle. Wilson and Cowan aimed therefore at a transformation of their equation into a differential equation (Wilson and Cowan, 1972). To do so, they had to assume that the population activity changes slowly during the time and adopted a procedure of time coarse-graining. Here we present a slightly modified version of their argument (Pinto et al., 1996; Gerstner, 1995).
We start with the observation that the total postsynaptic potential,
As a second step, we transform Eq. (6.85) into a differential equation. If the response kernels are exponentials, i.e., (s) = (s) = exp(- s/), the derivative of Eq. (6.85) is
Equation (6.87) is a differential equation for the membrane potential. Alternatively, the integral equation (6.76) can also be approximated by a differential equation for the activity A. We start from the observation that in a stationary state the activity A can be written as a function of the input current, i.e, A0 = g(I0) where I0 = Iext + J0 A0 is the sum of the external driving current and the postsynaptic current caused by lateral connections within the population. What happens if the input current changes? The population activity of neurons with a large amount of escape noise will not react to rapid changes in the input instantaneously, but follow slowly with a certain delay similar to the characteristics of a low-pass filter. An equation that qualitatively reproduces the low-pass behavior is
The Wilson-Cowan integral equation that has been discussed above is valid for neurons with absolute refractoriness only. We now generalize some of the arguments to a Spike Response Model with relative refractoriness. We suppose that refractoriness is over after a time so that (s) = 0 for s. For 0 < s < , the refractory kernel may have any arbitrary shape. Furthermore we assume that for t > the kernels (t - , s) and (t - , s) do not depend on t - . For 0 < t - < we allow for an arbitrary time-dependence. Thus, the postsynaptic potential is
We start from the population equation (6.75) and split the integral into two parts
The integrals on the right-hand side of (6.91) have finite support which makes Eq. (6.91) more convenient for numerical implementation than the standard formulation (6.75). For a rapid implementation scheme, it is convenient to introduce discretized refractory densities as discussed in Section 6.2; cf. Eq. (6.57).
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