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,

$$\rho =f(u-\vartheta )$$

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

$${\rho }_I(t|\hat{t})=f[u(t|\hat{t})-\vartheta ]$$

Required condition of function $f$, when $u\to -\mathrm{\infty }$, $f\to 0$

Example

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

$$f(u-\vartheta )=\frac{1}{{\tau }_0}$$

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

$$f(u-\vartheta )=\frac{1}{2\mathrm{\Delta }}[1+erf(\frac{u-\vartheta }{\sqrt{2}\sigma })]$$

$$erf(x)=\frac{2}{\sqrt{\pi }}{\int }_0^x\exp (-y^2)dy$$

Interval distribution and mean fire rate

the expect value of interval distribution = $\frac{1}{meanfirerate}$ = mean period

use $\rho$ we can get interval distribution,

$$P_I(t|\hat{t})=\rho \exp [-{\int }_{\hat{t}}^t\rho dt]$$

$$\rho =f[u(t|\hat{t})-\vartheta ]$$

use $SR{M}_0$,

$$u(t|\hat{t})=\eta (t-\hat{t})+h(t)$$

$$h(t)={\int }_0^{\mathrm{\infty }}\kappa (s)I(t-s)ds$$

use non-leaky integrate-and-fire,

$$u(t|\hat{t})=u_r+\frac{1}{C}{\int }_{\hat{t}}^{t}I(t^{\prime })d{t}^{\prime }$$

use leaky integrate-and-fire,

$$u(t|\hat{t})=R{I}_0[1-e^{(-t-\hat{t})/{\tau }_m}]$$

use $SR{M}_0$ with periodic input,we get periodic response,

$$h(t)=h_0+h_{1}cos(\mathrm{\Omega }t+{\phi }_1)$$

$$\eta (s)=\{\begin{array}{lllll}-\mathrm{\infty }& for& s<{\mathrm{\Delta }}^{abs}-{\eta }_0\exp (-\frac{s-{\mathrm{\Delta }}^{abs}}{\tau })& for& s>{\mathrm{\Delta }}^{abs}\end{array}$$