We consider a homogeneous population of independent neurons. All neurons receive the same time-dependent input current which varies about the mean . For constant input the population would fire at an activity which we can derive from the neuronal gain function. We require that the variations of the input
are small enough for the population activity to stay close to the value
In that case, we may expand the right-hand side of the population equation into a Taylor series about to linear order in . In this section, we want to show that for spiking neuron models (either integrate-and-fire or SRM neurons) the linearized population equation can be written in the form
where is the interval distribution for constant input , is a real-valued function that plays the role of an integral kernel, and
is the input potential generated by the time-dependent part of the input current. The first term of the right-hand side of Eq. (14.43) takes into account that previous perturbations with have an after-effect one inter-spike interval later. The second term describes the immediate response to a change in the input potential. If we want to understand the response of the population to an input current , we need to know the characteristics of the kernel . The main task of this section is therefore the calculation of to be performed in Section 14.3.1.
The linearization of the integral equation (183) is analogous to the linearization of the membrane potential density equations (78) that was presented in Chapter 13. In order to arrive at the standard formula for the linear response,
Hats denote transformed quantities, i.e., is the Fourier transform of the response kernel; is the Fourier transform of the interval distribution ; and is the transform of the kernel . Note that for we have and since and are constant.
The frequency dependent gain describes the linear response of the population activity to a periodic input current . The linear response filter in the time-domain is found by inverse Fourier transform
is the mean rate for constant drive . The filter plays an important role for the analysis of the stability of the stationary state in recurrent networks (Section 14.2).
In order to derive the linearized response of the population activity to a change in the input we start from the conservation law,
For constant input , the population activity has a constant value . We consider a small perturbation of the stationary state, , that is caused by a small change in the input current, . The time-dependent input generates a total postsynaptic potential, where is the postsynaptic potential for constant input and
is the change of the postsynaptic potential generated by . Note that we keep the notation general and include a dependence upon the last firing time . For leaky integrate-and-fire neurons, we set whereas for SRM neurons we set . We expand Eq. (14.49) to linear order in and and find
We have used the notation for the survivor function of the asynchronous firing state. To take the derivative of the first term in Eq. (14.51) we use and . This yields
We note that the first term on the right-hand side of Eq. (14.52) has the same form as the population integral equation (14.5), except that is the interval distribution in the stationary state of asynchronous firing.
To make some progress in the treatment of the second term on the right-hand side of Eq. (14.52), we now restrict the choice of neuron model and focus on either SRM or integrate-and-fire neurons.
(i) For SRM neurons, we may drop the dependence of the potential and set where is the input potential caused by the time-dependent current ; compare Eqs. (14.44) and (14.50). This allows us to pull the variable in front of the integral over and write Eq. (14.52) in the form
with a kernel
(ii) For leaky integrate-and-fire neurons we set , because of the reinitialization of the membrane potential after the reset (183). After some rearrangements of the terms, Eq. (14.52) becomes identical to Eq. (14.53) with a kernel
Let us discuss Eq. (14.53). The first term on the right-hand side of Eq. (14.53) is of the same form as the dynamic equation (14.5) and describes how perturbations in the past influence the present activity . The second term gives an additional contribution which is proportional to the derivative of a filtered version of the potential .
We see from Fig. 14.10 that the width of the kernel depends on the noise level. For low noise, it is significantly sharper than for high noise.
Absolute refractoriness is defined by a refractory kernel for and zero otherwise. We take an arbitrary escape rate . The only condition on is that the escape rate goes rapidly to zero for voltages far below threshold: .
This yields and hence
The survivor function is unity for and decays as for . Integration of Eq. (14.58) yields
As we have seen in Section 14.1, absolute refractoriness leads to the Wilson-Cowan integral equation (14.10). Thus defined in (14.60) is the kernel relating to Eq. (14.10). It could have been derived directly from the linearization of the Wilson-Cowan integral equation (see Exercises). We note that it is a low-pass filter with cut-off frequency , which depends on the input potential .
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