Auto and crosscorrelograms for the spike response of LIF neurons with slow synapses

An analytical description of the response properties of simple but realistic neuron models in the presence of noise is still lacking. We determine completely up to the second order the firing statistics of a single and a pair of leaky integrate-and-fire neurons (LIFs) receiving some common slowly filtered white noise. In particular, the auto- and cross-correlation functions of the output spike trains of pairs of cells are obtained from an improvement of the adiabatic approximation introduced in \cite{Mor+04}. These two functions define the firing variability and firing synchronization between neurons, and are of much importance for understanding neuron communication.

💡 Research Summary

The paper addresses a long‑standing gap in theoretical neuroscience: a precise analytical description of the second‑order firing statistics of leaky integrate‑and‑fire (LIF) neurons when the input noise is filtered by slow synaptic dynamics. While many studies have treated LIF neurons driven by white (δ‑correlated) noise, real synaptic currents decay over tens of milliseconds, introducing a low‑pass filter that makes the noise “slow” compared to the membrane time constant. This slow component profoundly influences both the variability of a single neuron’s spike train (auto‑correlation) and the synchrony between two neurons receiving shared input (cross‑correlation).

Model and assumptions
Each neuron obeys
τ_m dV/dt = –(V–V_rest) + I(t),
with a spike emitted when V reaches a threshold θ, after which V is reset to V_r. The input current I(t) consists of three parts: a constant drive μ, a private Gaussian white noise term σ ξ_i(t), and a common Gaussian white noise term σ_c ξ_c(t). Both noise terms are passed through an exponential low‑pass filter with time constant τ_s, yielding Ornstein‑Uhlenbeck processes. The key small parameter is ε = τ_m/τ_s ≪ 1, expressing the separation of the fast membrane dynamics and the slow synaptic filtering.

Improved adiabatic approximation
The classic adiabatic approximation (Moreno et al., 2004) treats the slow noise as quasi‑static, allowing the membrane potential to equilibrate instantaneously for a given noise realization. This yields an exact expression for the mean firing rate but only a crude estimate of second‑order statistics. The authors refine this approach in two ways: (1) they solve the full two‑dimensional Fokker‑Planck equation for the joint distribution of membrane voltage and filtered noise, imposing the exact absorbing boundary at θ and the reset condition at V_r; (2) they perform a systematic perturbation expansion in ε, retaining both the zeroth‑order (static) term and the first‑order correction that captures the slow drift of the noise during an inter‑spike interval.

Analytical results
The analysis leads to closed‑form expressions for the auto‑correlation C_aa(Δt) and cross‑correlation C_ab(Δt) of the output spike trains:

C_aa(Δt) = r δ(Δt) + r²