Leaky Integrate-and-Fire Dynamics

The purpose here is to calculate the interspike interval distribution for a population of leaky integrate-and-fire (IF) neurons driven by statistically-steady Poisson input. We assume that each input event elevates the membrane potential of a neuron by $\epsilon$ millivolts. If each cell's rest potential coincides with its leakage reversal potential, as is the case for retinal ganglion cells, then the dynamical equation for each neuron can be written as

$$\frac{dv}{dt}=-\gamma v+\epsilon \sum _{k}\delta \left(t-t_{k}\right)\, .$$ (1)

Here, the voltage has been non-dimensionalized, the threshold potential scaled to unity, and the reset scaled to zero. Thus the reset condition

$$v\left(t^{-}\right)=1\: \: \Rightarrow \: \: v\left(t^{+}\right)=0\, .$$ (2)

Time is still dimensional, and the leakage rate $\gamma$ has units of inverse milliseconds. The set of input event times $\left\{ t_{k}\right\}$ are Poisson distributed with a mean rate that hereafter will be denoted by $\sigma$. The waiting time between input spikes, call it $\Delta t$, is therefore governed by an exponential distribution: $p(\Delta t)=\sigma e^{-\sigma \Delta t}$.

The probability density function, $\rho (v,t),$ for a population of leaky IF neurons evolves according to the conservation law

$$\frac{\partial \rho }{\partial t} = -\frac{\partial J}{\partial v}$$ (3)

$$= \frac{\partial }{\partial v}\left(\gamma v\rho \right)+\sigma \left[\rho (v-\epsilon ,t)-\rho (v,t)\right]+\delta (v)J(1,t)\, .$$ (4)

The mean per-neuron firing rate, $r(t)$, is given by

$$r(t)=J(1,t)=\sigma \int _{1-\epsilon }^{1}\rho (V,t)\, dV\, ,$$ (5)

which is obtained from the probability flux

$$J(v,t)=-\gamma v\rho +\sigma \int _{v-\epsilon }^{v}\rho (V,t)\, dV$$ (6)

and the boundary condition $\rho (1,t)=0$ (which prevents backward drift through the threshold at $v=1$).

To obtain interspike interval statistics, one may initialize the probability density as a delta function at the reset, $\rho (v,0)=\delta (v)$, and track the firing rate as a function of time as $\rho (v,t)$ evolves under equation (4) without the reset flux. The probability flux, evaluated at threshold, is then identical to the interspike interval distribution, call it $p(\tau )$. Our goal is to derive an analytical expression for $p(\tau )$.