# 19.Population Activity

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## Review of Several Concepts

### Population activity $A(t)$

1. From equations

We integrate over a range of potential

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

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

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

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

Meanwhile we have this survival probability

or identically

## 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.,

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

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

where

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

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

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.