09.Escape Noise

Escape Noise

Escape noise

Two ways to introduce noise in formal spiking neuron models:

  • noisy threshold(escape model or hazard model)
  • noisy integration(stochastic spike arrival model or diffusion model)

In the escape model, the neuron may fire when $u<\vartheta$ the neuron may stay quiescent when $u>\vartheta$

Escape rate and hazard function

In the escape model, spikes can occur at any time with a probability density,


Since $u$ is a function of time,$\rho$ is also time dependent,


Required condition of function $f$, when $u\rightarrow-\infty$, $f\rightarrow0$


$$f(u-\vartheta)=\begin{cases} 0 & for &u<\vartheta\
\Delta^{-1}& for &u\ge\vartheta \end{cases}$$


$$f(u-\vartheta)=\beta[u-\vartheta]_{+}=\big{\begin{array}{lcc} 0 &for&u<\vartheta\
\beta(u-\vartheta)&for&u\ge\vartheta \end{array}$$



Interval distribution and mean fire rate

the expect value of interval distribution = $\frac{1}{mean\space fire\space rate}$ = mean period

use $\rho$ we can get interval distribution,

$$P_{I}(t|\hat{t})=\rho\space \exp[-\int_{\hat{t}}^{t}\rho dt]$$


use $SRM_{0}$,



use non-leaky integrate-and-fire,


use leaky integrate-and-fire,


use $SRM_0$ with periodic input,we get periodic response,

$$h(t)=h_0+h_1cos(\Omega t+\varphi_1)$$

$$\eta(s)=\begin{cases} -\infty & for & s<\Delta^{abs}\
-\eta_0 \exp{\big(-\frac{s-\Delta^{abs}}{\tau}\big)} & for & s>\Delta^{abs} \end{cases}$$

Planted: by ;

Current Ref:

  • snm/12.escape_noise.md