Theory of input spike auto- and cross-correlations and their effect on the response of spiking neurons

Spike correlations between neurons are ubiquitous in the cortex, but their role is at present not understood. Here we describe the firing response of a leaky integrate-and-fire neuron (LIF) when it receives a temporarily correlated input generated by presynaptic correlated neuronal populations. Input correlations are characterized in terms of the firing rates, Fano factors, correlation coefficients and correlation timescale of the neurons driving the target neuron. We show that the sum of the presynaptic spike trains cannot be well described by a Poisson process. Solutions of the output firing rate are found in the limit of short and long correlation time scales.

💡 Research Summary

The paper addresses a fundamental question in cortical neuroscience: how temporally correlated spike inputs from presynaptic populations shape the firing response of a single leaky integrate‑and‑fire (LIF) neuron. Rather than treating the summed presynaptic activity as a homogeneous Poisson process, the authors model each presynaptic neuron by four statistical descriptors – its mean firing rate (ν), Fano factor (F), pairwise correlation coefficient (ρ), and correlation timescale (τc). By aggregating N such correlated spike trains, they derive explicit expressions for the mean input current μ and its variance σ², showing that the variance contains an additional term proportional to ρ·F·τc that cannot be captured by a simple Poisson approximation.

The membrane dynamics of the LIF neuron obey τm dV/dt = –V + I(t), where I(t) is the composite input current. Translating this stochastic differential equation into a Fokker‑Planck framework, the authors obtain a probability‑flux equation whose coefficients depend on the statistics of I(t). They then consider two asymptotic regimes for the correlation timescale relative to the membrane time constant τm.

Short‑correlation regime (τc ≪ τm). In this limit the input behaves like white noise with an effective diffusion coefficient D_eff = D0 + ΔD, where ΔD = w² ν F ρ τc (w is the synaptic weight). Solving the stationary Fokker‑Planck equation with absorbing (threshold Vth) and reflecting (reset Vreset) boundaries yields an analytical firing‑rate formula that involves an integral of the error function. The key prediction is that the output rate νout grows supra‑linearly with the product ρ·F·τc: modest increases in pairwise correlation or spike count variability can dramatically boost firing.

Long‑correlation regime (τc ≫ τm). Here the input varies slowly compared with the membrane, so the dominant effect is a shift in the mean drive μ_eff = μ + w ν ρ τc, while the diffusion term remains essentially D0. The firing‑rate solution reduces to a logarithmic expression νout ≈