Finite-state Markovian surrogates for long-time neuronal state distributions and firing rates

Abstract

Spiking neuronal networks connect cellular and synaptic mechanisms to collective activity, but estimating their long-time statistics often requires costly spike-by-spike simulation. We construct finite-state Markovian surrogates for neuronal populations with one- to three-dimensional intrinsic dynamics. A finite phase-space partition represents each population by its occupancy across single-neuron states, and the firing rate is defined uniformly as probability flux through model-specific spike transitions. We compare two closures: Type I enforces a stationary mean synaptic drive, whereas Type II retains its temporal evolution. In the leaky linear integrate-and-fire (LIF) benchmarks, the Type I estimator is accurate in temporally homogeneous cases but misses rate changes driven by synaptic timescales. In the synaptic-timescale sweep, the Type II estimator achieves small relative errors in most cases. Exponential integrate-and-fire (EIF) results exhibit an analogous qualitative pattern. FitzHugh--Nagumo (FHN) and reduced Hodgkin--Huxley (HH) examples extend the construction from one-dimensional threshold--reset dynamics to continuous trajectories in nonlinear two- and three-dimensional phase spaces. Together, these examples show how the finite-state construction generates model-dependent estimates of long-time state distributions and firing rates without resolving every neuron and spike.