Integral equations for population activity

## Review of Several Concepts

### Population activity $A(t)$

1. From equations

$$\begin{equation}\partial_t p(u,t)= \cdots + A(t)\delta(u-u_r)\end{equation}$$

We integrate over a range of potential

$$\begin{equation} \partial_t \int_{u_1}^{u_2} p(u,t) = \cdots + A(t), \end{equation}$$

provided that $u_r$ is within $[u_1,u_2]$.

So it’s some kind of flux. It works as a source term of faction of neurons at $u=u_r$, which is identical to fraction of neurons that spiked per unit time.

2. In fact, we have

$$\begin{equation} A(t) = J(\theta,t) \end{equation}$$

### Spike interval distribution $P_I(t\vert \hat t)$

$P_I(t\vert \hat t)$ is the probability density of firing, i.e.,

$$\begin{equation} \int_{t_1}^{t_2} P_I(t\vert \hat t) dt \end{equation}$$

calculates the probability of finding spikes during time interval $[t_1,t_2]$.

Meanwhile we have this survival probability

$$\begin{equation} S_I(t\vert \hat t) = 1 - \int_{\hat t}^t P_I(s\vert \hat t) ds \end{equation}$$

or identically

$$\begin{equation} P_I(t\vert \hat t) = \partial_t S_I(t\vert \hat t) \end{equation}$$

## Motivation

What we need for a complete description of network activities is to calculate $A(t+\Delta t)$ given $A(t)$ as well as the external input at $t$.

Using whatever we have up to this point, the procedure should be

1. Apply $A(t)$ and $I^{\mathrm{ext}}(t)$ to equation (6.8). Within a small time interval $\Delta t$, we obtain the PSP potential, i.e.,

$$h_{\mathrm{PSP}}(t + \Delta t\vert \hat t) = J_0 \int_0^\infty \epsilon(t - \hat t, s) A(t - s) ds + \int_0^\infty \kappa(t-\hat t,s) I^{\mathrm{ext}}(t-s) ds$$

My thought is, calculating next step is impossible, since we have an integral to infinity? I do not really get it.

2. The ultimate reason is that we have insufficient equations compared to the unknown quantities. Unknown: $A(t)$, $h_{\mathrm{PSP}}$, but we have only one equation.

Question: What about other equation? Eqn (6.21), one equation, two variables. Same fate.

So we need another equation. What we get is

$$A(t) = \int_{-\infty}^t P_I(t\vert \hat t) A(\hat t) d\hat t.$$

Oh wait, new function $P_I(t\vert \hat t)$ has been introduced to the function. How is that going to help? $P_I$ actually can be determined by $h$.

Eq. 6.78, 6.79, 6.80

### With Absolute refractoriness

Wilson-Cowan integral form

$$\begin{equation} A(t) = f[h(t)]\left( 1 - \int_{t-\Delta^{\mathrm{abs}}}^t A(t’) dt’ \right), \end{equation}$$

where

• $f[h(t)]$ rate of firing for a neuron that is not refractory,
• given the togal input potential $h(t)$.

Constant input potential $h(t)=h_0$,

$$\begin{equation} A_0 = \frac{f(h_0)}{1-\Delta^{\mathrm{abs}} f(h_0)} \equiv g(h_0). \end{equation}$$

To solve the population activity for homogeneous, isotropic and stationary network, all we need is the property of single neuron.

### Time Coarse-Graining

Remove the integral.

Planted: by ;