18.SRM with Escape Noise

SRM with Escape Noise

$$ \begin{equation} u(t) = \eta(t-\hat t) + h_{\mathrm{PSP}}(t\lvert \hat t). \end{equation} $$

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

$$ \begin{equation} \partial_t q(r,t) = -\partial_r q(r,t) - \rho(r\vert t-r) q(r,t) + A(t) delta(r) \end{equation} $$

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

$$ \begin{equation} A(t)= \int_{-\infty}^t P_I(t\vert \hat t) A(\hat t)d\hat t, \end{equation} $$

where

$$ \begin{equation} P_I(t\vert \hat t)= \rho(t\vert \hat t) \exp \left( - \int_{\hat t}^t \rho(t’\vert \hat t) dt’ \right) \end{equation} $$