Diffusive Noise and The subthreshold Regime

Diffusive noise

Interval distribution

Because the diffusive noise generated by stochastic spike arrival, we cannot predict the exact value of the neuronal membrane potential, but only the probability that the membrane potential is in a certain interval$[u_0,u_1]$

$$Prob{u_0<u(t)<u_1|u(\hat{t})-u_r}=\int_{u_0}^{u_1}p(u,t)du$$

thus ,we can get the survivor function,

$$S_I(t|\hat{t})=\int_{-\infty}^{\vartheta}p(u,t)du$$

the interval distribution,

$$P_I(t|\hat{t})=-\frac{d}{dt}\int_{-\infty}^{\vartheta}p(u,t)du$$

The sub-threshold regime

sub and super threshold stimulation

Input $I(t)$ is sub-threshold, if it generates a membrane potential that stays below the firing threshold

Input $I(t)$ is sup-threshold, if it generates a membrane potential higher than the threshold

In the super-threshold regime,

noise broadens the interspike interval distribution.

In the sub-threshold regime,

spikes are generated by the fluctuations of the membrane potential, rather than by its mean.

coefficient of variation C

$$C^2_V=\frac{\langle\Delta s^2\rangle}{\langle s\rangle^2}$$

$C_v>1$,broader than that generate by Poisson distribution

$C_v=1$,noise generate by Poisson distribution

$C_v<1$,more regular

$C_v=0$,regular firing with a fixed period $\langle s \rangle$

Poisson neuron with absolute refractoriness,

$$C_V=1-\frac{\Delta^{abs}}{\langle s \rangle}$$

refractoriness make the spikes more regularly than a Poisson neuron.