# 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

comments powered by Disqus