14.Noise in Refractory Kernel and Diffusive Noise

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Noise Using Stochastic Parameter

For a short review of refractory kernel, please refer to page 114 of the textbook (4.2.3 Simplified model SRM0), i.e.,

For noise models, we define refractory kernel

where

which in turn is plugged back into the refractory kernel,

We require that $r$ to be a parameter with mean $\langle r \rangle = 0$.

We discussed threshold in 5.3.1 where we said spikes occur with probability density

in which $\theta$ is the threshold. For noise model the threshold is a noisy function.

“Noise reset” model

Spike occur at $t$ when the potential reaches threshold $u(t\vert \hat t,r)=\theta$, thus the interval of spike is given by

Interval distribution:

SRM0 model:

The stochastic parameter $r$ work as a shift of the spikes on time axis.

Diffusive Noise

Integrate-and-fire model:

• Membrane time constant $\tau_m$;
• Input resistance $R$;
• Input current $I$.

Introducing noise: add noise to the RHS,

where $\xi(t)$ is a stochastic term thus the equation becomes a stochastic differential equation.

Figure 5.12 is a very nice plot showing the effect of $\xi$ on threshold.

For a Gaussian white noise

• $\sigma$ amplitude of noise;
• $\tau_m$ membrane time constant.

Stochastic Spike Arrival

In a network, a integrate-and-fire neuron will take in

• input $I^{ext}(t)$,
• input spikes at $t^{(f)}_j$, where $j$ means the spike from neuron $j$,
• stochastic spikes (from the background of the brain that we are not really interested in for now) $t_k^{(f)}$,

so that

which is called Stein’s model.

The stochastic spike arrivals are Poissonian.

Example: Membrane Potential Fluctuations

• Poisson process with rate $\nu$
• Input spike train $S(t) = \sum_{k=1}^N \sum_{t_k^{(f)}} \delta(t-t_k^{(f)}),$

which has an average

and autocorrelation

$\nu_0^2$ is from the constant hazard $\rho_0(t-\hat t) = \nu$ and Poisson has autocorrelation $C_{ii}(s) = \nu \delta(s) + \nu^2$.

• Neglect both threshold and reset, which basically means weak input so that neuron doesn’t reach firing threshold. $u(t) = w_0 \int_0^\infty \epsilon_0(s) S(t-s) ds$

Also neglect the term $-u/\tau_m$?

• Average over time we have $u_0 \equiv \langle u(t) \rangle = w_0 \nu_0 \int_0^\infty \epsilon_0(s)ds.$
• Variance of potential %

Figure 5.14: We have equation 5.83 $\langle \delta u^2 \rangle = 0.5 \tau_m \sum_k w_k^2 \nu_k$, larger $w_k$ will give us larger variance of potential so that the spikes are more probable.

Diffusion Limit

Stein model

After each firing, probability density of membrane potential can be calculated.

Between $\Delta t$, the probability of firing is $\sum_k\nu_k \Delta t$. As a result, the probability of quite is

During this time the membrane potential will decay

Incoming spike at synapse $k$: