Fokker-Planck for LIF Populations

Imagine thousands of neurons as people slowly climbing toward a cliff edge. The climb is the membrane voltage rising; the cliff edge is the firing threshold. Each person also gets random shoves — that is the noisy synaptic input — so they do not all move together. The instant someone reaches the edge they "fire" and are carried straight back to a reset spot, then start climbing again. Instead of tracking every individual, we just ask how the crowd is spread across the hill. That spread is a probability density $p(V)$ over the voltage $V$.

Two effects move the crowd. A steady pull, the drift, comes from the leak (which tugs $V$ back toward rest $E_L$) together with the input current $I$, setting a mean voltage $\mu = E_L + RI$. The random shoves, the diffusion, come from noise of size $\sigma_V$ that smears the crowd out. If the mean $\mu$ sits well below threshold, almost nobody reaches the edge on drift alone and only lucky noise kicks cause firing. Push $I$ up so $\mu$ nears $V_{th}$ and the whole crowd spills over regularly. The rate at which people cross the edge is the population firing rate $\nu$.

Made precise, the density obeys $\partial_t p = -\partial_V\!\left[\frac{\mu - V}{\tau_m}\,p\right] + \frac{\sigma_V^2}{2}\,\partial_V^2 p$, where the first term is drift and the second is diffusion. The threshold is an absorbing boundary, $p(V_{th})=0$, and the escaping probability current is reinjected at $V_{reset}$. In steady state that current is constant and equals $\nu$; carrying out the integral gives the Siegert formula in Section 3. In this sim the Input I slider raises the mean drive $\mu$, while the parameter sliders scale the leak and the noise that decide how tightly the density clusters and how often it spills over the edge.

Set Input I to its maximum and watch the drive carry the population over threshold so the firing rate saturates. Then drop Input I low: now only the noise pushes neurons across the edge, giving sparse, irregular firing — the fluctuation-driven regime that real cortex often lives in. Finally raise Input I gradually from low to high and notice the firing-rate curve climbs smoothly rather than switching on sharply: noise has "softened" the threshold.

LIF SDE: dV = [(-V+μ)/τ_m]dt + σ_V dW. Itô formula → Fokker-Planck: ∂p/∂t = -∂(f p)/∂V + (σ²/2)∂²p/∂V². Boundary: p(V_th)=0 (absorption), probability current J at V_reset includes re-injection.

∂p/∂t=0 → J = ν = const (probability current = firing rate). Solve ODE: p(V) = (2ν/σ²) e^{-(V-μ)²/σ²} ∫_V^{V_th} e^{+(V'-μ)²/σ²} dV'. Normalise ∫p dV = 1 → Siegert formula for ν.

The time-dependent FP solution gives PSTH: ν(t) = J(V_th, t). Population responds like a linear filter + static nonlinearity (LNP model). Transient responses have fast (diffusion across threshold) and slow (drift) components. This enables exact PSTH prediction from single-neuron parameters.