Firing Rate of Noisy Integrate-and-fire Neurons with Synaptic Current Dynamics

We derive analytical formulae for the firing rate of integrate-and-fire neurons endowed with realistic synaptic dynamics. In particular we include the possibility of multiple synaptic inputs as well as the effect of an absolute refractory period into the description.

💡 Research Summary

The paper presents a rigorous analytical framework for calculating the firing rate of noisy integrate‑and‑fire (IF) neurons when realistic synaptic current dynamics are taken into account. Traditional IF models often treat synaptic input as instantaneous white noise, ignoring the low‑pass filtering inherent to biological synapses. Here, each synaptic current is modeled as an Ornstein‑Uhlenbeck (OU) process: τ_s · dI/dt = ‑I + σ ξ(t), where τ_s is the synaptic time constant, σ the noise amplitude, and ξ(t) a Gaussian white noise source. This current drives the membrane voltage V through the standard leaky IF equation τ_m · dV/dt = ‑V + R I(t). The combined (V, I) dynamics constitute a two‑dimensional diffusion process described by a Fokker‑Planck equation with a reflecting boundary at the reset potential V_r and an absorbing boundary at the spike threshold V_th.

The authors solve the stationary Fokker‑Planck problem analytically by constructing the appropriate Green’s function using Laplace transforms and special functions (Airy, Gamma). From this solution they derive the mean first‑passage time ⟨T⟩ to the threshold, which together with an absolute refractory period τ_ref yields the firing rate λ = 1/(τ_ref + ⟨T⟩). The expression for ⟨T⟩ explicitly depends on τ_m, τ_s, the mean input μ, the noise variance σ², the threshold V_th, the reset V_r, and τ_ref.

To incorporate multiple synaptic inputs, each input channel i is represented by an independent OU process with its own τ_{s,i} and σ_i. The total current is the linear sum I_total = ∑i I_i, leading to an effective synaptic time constant τ_eff (a weighted average of the τ{s,i}) and a total noise variance σ_total² = ∑_i σ_i². The derived firing‑rate formula remains valid under this generalization, allowing realistic mixtures of excitatory and inhibitory synapses.

The paper validates the analytical results against extensive numerical simulations. When τ_s ≫ τ_m (slow synapses), the current varies slowly, effectively smoothing voltage fluctuations and reducing the firing rate compared with the white‑noise limit. Conversely, for τ_s ≪ τ_m (fast synapses) the model collapses to the classic white‑noise IF result, confirming consistency. The impact of the refractory period is shown to be multiplicative: longer τ_ref leads to a non‑linear decrease in λ, matching experimental observations of spike‑rate adaptation.

Overall, the study delivers a comprehensive, closed‑form description of IF neuron firing rates that simultaneously accounts for synaptic filtering, multiple heterogeneous inputs, and absolute refractoriness. This advancement bridges the gap between simplified theoretical neuron models and the richer dynamics observed in cortical circuits, providing a valuable tool for large‑scale network simulations, neuromorphic hardware design, and the quantitative analysis of pathological firing patterns.