Example: Wilson-Cowan differential equation

Sometimes one finds in the literature rate models of the form

which have an additional time constant $\tau_{A}$. Therefore the transient response to a step in the input would be even slower than that of the rate model in Eq. (15.13).

In the derivation of Wilson and Cowan (553) the time constant $\tau_{A}$ arises from time-averaging over the ’raw’ activity variable with a sliding window of duration $\tau_{A}$. Thus, even if the ’raw’ activity has sharp transients, the time-averaged variable $\bar{A}(t)$ in Eq. (15.15) is smooth. In the theory of Wilson and Cowan, the differential equation for the population activity takes the form (552; 553)

so as to account for absolute refractoriness of duration $\Delta^{\rm abs}$; cf. the integral equation (14.10) in Chapter 14. $F$ has a sigmoidal shape. The input potential $h(t)$ comprises input from the same as well as from other populations.

Since there is no reason to introduce the additional low-pass filter with time constant $\tau_{A}$, we advise against the use of the model defined in Eqs. (15.15) or (15.16). However, we may set

where $I(t)$ is the input current (as opposed to the input potential). The current can be attributed to input from other populations or from recurrent coupling within the population. The low-pass filter in Eq. (15.17) replaces the low-pass filter in Eq. (15.14) so that the two rate models are equivalent after an appropriate rescaling of the input, even for complex networks (345); see also Exercises.

Example: Population with self-coupling

If we have a single population, we can drop the indices in Eq. (15.19) so as to arrive at

where $J$ is the strength of the feedback. Thus, a single population with self-coupling is described by a single differential equation for the input potential $h$. The population activity is simply $A(t)=F(h(t))$.

Stationary states are found as discussed in Chapter 12. If there are three fixed points, the middle one is unstable while the other two (at low and high activity, respectively) are stable under the dynamics (15.20). The fixed points calculated from Eq. (15.20) are correct. However, stability under the dynamics (15.20) does not guarantee stability of the original network of spiking neurons. Indeed, as we have seen in Chapters 13 and 14, the fixed points of high and low activity may loose stability with respect to oscillations, even in a single homogeneous population. The rate dynamics of Eq. (15.20) cannot correctly account for these oscillations. In fact, for instantaneous synaptic current pulses $\alpha(s)=q\delta(s)$, where $\delta$ denotes the Dirac $\delta$-function, Eq. (15.20) reduces to a one-dimensional differential equation which can never give rise to oscillatory solutions.

Example: Effective rate model for Exponential Integrate-and-Fire neurons

We denote the stationary gain function of exponential integrate-and-fire neurons by $g_{\sigma}(I)$ and the linear filter arising from linearization around a stationary activity $A_{0}$ by $G_{I}(s)$. For exponential integrate-and-fire neurons $G_{I}(s)$ has the high-frequency behavior of a low-pass filter $\propto A_{0}{R\over\Delta_{T}\tau_{m}}{1\over{\omega}}$; see Tab. 15.2.

We recall that the Fourier-transform of an exponential filter also yields a high-frequency behavior $\propto 1/\omega$ with the inverse filter time constant as cut-off frequency. This suggests that, for the exponential integrate-and-fire model, we can approximate the time course of $G_{I}(s)$ as an exponential filter with an effective time constant $\tau_{\rm eff}\propto\tau_{m}/A_{0}$. This leads back to Eq. (15.22), but with an exponential filter $G(s)$. Hence, we are now nearly back to Eq. (15.13), except that the time constant of the exponential is different.

We now switch from the frequency current curve $g_{\sigma}(I)$ to the equivalent description of the gain function $F(h)=g_{\sigma}(h/R)$ where the argument of $F$ has units of a potential. It is convenient to implement the exponential filter in the form of a differential equation for the the input potential (373)

with an effective time constant

where $A_{0}(t)$ is the activity averaged over one or a few previous time steps and $F^{\prime}={\text{d}}F/{\text{d}}h$ is the derivative of the gain function at an appropriately chosen reference point $h_{0}$, so that $F(h_{0})\approx\langle A_{0}(t)\rangle$ is the long-term average of the activity.

In each time step, we update the input using Eq. (15.23) and calculate the activity as $A(t)=F(h(t))$ with $F(h)=g_{\sigma}(h/R)$. Such a rate model gives an excellent approximation of the activity in a population of exponential integrate-and-fire neurons (Fig. 15.10). Note that fast transients are well described, because the effective time constant is shortened as soon as the population activity increases.

Example: Effective adaptation filter

Suppose that $a_{\infty}(A)=c\,A$ is linear in the population activity with a constant $c>0$ and $\tau_{a}(A)=\tau_{a}$ is independent of $A$. Then Eq. (15.26) can be integrated and yields $a(t)=\int_{0}^{\infty}\gamma(s)\,A(t-s){\text{d}}s$ with a filter $\gamma(s)=(c/\tau_{a})\,\exp(-s/\tau_{a})$. We insert the result in Eq. (15.25) and find

Eq. (15.27) nicely describes the adaptation process but it misses, like other rate models, the sharp initial transient, and potential transient oscillation, caused by the synchronization of the population at the moment of the step (358)