# 18.SRM with Escape Noise

Live chat with others.

## SRM with Escape Noise

1. Define a parameter $r=t-\hat t$.
2. Define density for $r$, i.e., fraction of neurons with parameter $[r_0,r_0+\Delta r]$ is given by $\int_{r_0}^{r_0+\Delta r} q(r’,t)dr’$.
3. Continuity equation: $\partial_t q(r,t) = -\partial_r J_{\mathrm{refr}}(r,t)$.
4. $J_{\mathrm{refr}}=q(r,t)\partial_t r=q(r,t)$ is the continuous flux.
5. Hazard function $\rho(t\vert t-r) =f(\eta(r)+h_{\mathrm{PSP}}(t\vert t-r))$ tells us about the firing rate of a neuron.
6. Loss per unit time $J_{\mathrm{loss}}=- \rho(t\vert t-r)q(r,t)$.
7. At time $t$, total number of neurons that fire, which is also called population activity $A(t)=\int_0^\infty (-J_{\mathrm{loss}})dr$.

The change in the fraction of neurons with parameter $r$ depends on

1. continuous flow passing by $r$,
2. the loss flux derivative,
3. the population activity,

so that we obtain

Population activity is the quantity we would love to obtain. By rewriting the previous equation

where