Slow Noise in parameters and diffusive noise (Part 1)

## 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.,

$$u_i(t) = \sum_f \eta_0 (t-t_i^{(f)}) + \sum_j w_{ij} \sum_f \epsilon_0 (t-t_j^{(f)}) + \int_0^\infty \kappa_0 (s) I_i ^{ext} (t-s) ds.$$

For noise models, we define refractory kernel

$$\eta(s) = \eta_0 e^{-s/\tau},$$

where

$$\eta_0 \equiv \eta_0(r) = \tilde \eta_0 e^{r/\tau},$$

which in turn is plugged back into the refractory kernel,

$$\eta(s) = \tilde \eta_0 e^{- (s-r)/\tau}.$$

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

$$\rho = f(u-\theta),$$

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

“Noise reset” model

$$\theta = u(t \vert \hat t, r) = \eta_0(r) e^{-(t-\hat t)/\tau} + \int_0^\infty \kappa (t-\hat t,s) I(t-s) ds.$$

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

$$T(\hat t,r) = \mathrm{min}\left[ t -\hat t \vert u(t\vert \hat t,r)=\theta \right].$$

Interval distribution:

$$P_I(t\lvert \hat t) = \int dr \delta ( t-\hat t - T(\hat t, r) ) \mathscr{G}_0(r).$$

SRM0 model:

\begin{align} u(t\lvert \hat t,r) &= \eta_r(t-\hat t) + h(t)\ &= \tilde \eta_0 e^{-(t-\hat t- r)/\tau} + h(t). \end{align}

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

## Diffusive Noise

Integrate-and-fire model:

$$\tau_m \frac{d}{d t} u = - u + R I(t).$$

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

Introducing noise: add noise to the RHS,

$$\tau_m \frac{d}{d t} u = - u + R I(t) + \xi (t),$$

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

$$\langle \xi(t) \xi(t’) \rangle = \sigma^2 \tau_m \delta(t-t’).$$

• $\sigma$ 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

$$\frac{d}{dt} u = - \frac{u}{\tau_m} + \frac{1}{C}I^{ext}(t) + \sum_j \sum_{t_j^{(f)} > \hat t} w_j \delta(t- t_j^{(f)}) + \sum_k \sum_{t_k^{(f)} < \hat t} w_k \delta(t-t_k^{(f)}),$$

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

$$\langle S(t) \rangle = \nu_0,$$

and autocorrelation

$$\langle S(t) S(t’)\rangle - \nu_0^2 = N\nu_0 \delta(t-t’).$$

$\nu_0^2$ 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 \begin{align} \langle (u-u_0)^2 \rangle &= w_0^2 \left\langle \left( \int_0^\infty \epsilon_0(s) S(t-s) - w_0 \nu_0 \int_0^\infty \epsilon_0(s)ds \right)^2 \right\rangle \ & = \left\langle w_0^2 \int_0^\infty \int_0^\infty \epsilon_0(s) \epsilon_0(s’) S(t-s) S(t-s’) ds’ ds - 2 u_0 w_0 \nu_0 \int_0^\infty \epsilon_0(s) S(t-s)ds + u_0^2 \right\rangle \ & = \left\langle w_0^2 \int_0^\infty \int_0^\infty \epsilon_0(s) \epsilon_0(s’) S(t-s) S(t-s’) ds’ ds \right\rangle - 2 u_0 w_0 \left\langle \nu_0 \int_0^\infty \epsilon_0(s) S(t-s)ds \right\rangle + u_0^2 \ & = \left\langle w_0^2 \int_0^\infty \int_0^\infty \epsilon_0(s) \epsilon_0(s’) S(t-s) S(t-s’) ds’ ds \right\rangle - 2 u_0^2 + u_0^2 \ & = w_0^2 \nu_0 \int_0^\infty \epsilon_0(s)^2ds \end{align}

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

$$\frac{d}{dt} u = - \frac{u}{\tau_m} + \frac{1}{C}I^{ext}(t) + \sum_j \sum_{t_j^{(f)} > \hat t} w_j \delta(t- t_j^{(f)}) + \sum_k \sum_{t_k^{(f)} < \hat t} w_k \delta(t-t_k^{(f)}),$$

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

$$1 - \sum_k\nu_k \Delta t.$$

During this time the membrane potential will decay

$$u(t+\Delta t) = u(t) e^{-\Delta t/\tau_m}.$$

Incoming spike at synapse $k$:

$$u(t+\Delta t) = u(t) e^{-\Delta t/\tau_m} + w_k.$$

Planted: by ;