18.SRM with Escape Noise

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: .
  4. $J_{\mathrm{refr}}=q(r,t)\partial_t r=q(r,t)$ is the continuous flux.
  5. Hazard function tells us about the firing rate of a neuron.
  6. Loss per unit time .
  7. At time $t$, total number of neurons that fire, which is also called population activity .

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

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