A First-Order Non-Homogeneous Markov Model for the Response of Spiking Neurons Stimulated by Small Phase-Continuous Signals

We present a first-order non-homogeneous Markov model for the interspike-interval density of a continuously stimulated spiking neuron. The model allows the conditional interspike-interval density and the stationary interspike-interval density to be expressed as products of two separate functions, one of which describes only the neuron characteristics, and the other of which describes only the signal characteristics. This allows the use of this model to predict the response when the underlying neuron model is not known or well determined. The approximation shows particularly clearly that signal autocorrelations and cross-correlations arise as natural features of the interspike-interval density, and are particularly clear for small signals and moderate noise. We show that this model simplifies the design of spiking neuron cross-correlation systems, and describe a four-neuron mutual inhibition network that generates a cross-correlation output for two input signals.

💡 Research Summary

The paper introduces a first‑order non‑homogeneous Markov framework for describing the inter‑spike‑interval (ISI) density of a spiking neuron that is continuously driven by a small phase‑continuous signal. Traditional approaches to modeling spiking neurons typically rely on detailed biophysical equations (e.g., Hodgkin‑Huxley or leaky integrate‑and‑fire) that require precise knowledge of ion‑channel dynamics and membrane parameters. In many experimental settings, however, such detailed information is unavailable, especially when the neuron is embedded in a larger network. To overcome this limitation, the authors propose a probabilistic model in which the conditional ISI density can be factorized into two independent components: one that captures the intrinsic firing properties of the neuron (denoted fₙ(t)) and another that encapsulates all statistical aspects of the external signal (denoted gₛ(t) and hₛ(τ)).

Mathematically, the transition probability of the non‑homogeneous Markov chain is expressed as

 P(Tₙ₊₁∈dt | Tₙ)=fₙ(t)·gₛ(t)·dt,

where Tₙ denotes the n‑th ISI. In the stationary regime the overall ISI density becomes

 pₛ(t)=fₙ(t)·Gₛ(t), with Gₛ(t)=∫₀^∞ gₛ(τ)·hₛ(t−τ)dτ.

The function Gₛ(t) is a convolution of the signal’s instantaneous effect gₛ and its temporal correlation kernel hₛ, thereby embedding the signal’s autocorrelation (and, when multiple signals are present, cross‑correlation) directly into the ISI distribution.

A key analytical insight is obtained by assuming that the external signal is weak (amplitude ε≪1) and that the noise level is moderate. Under these conditions the signal‑dependent factor can be linearized:

 Gₛ(t)≈1+ε·Cₛ(t),

where Cₛ(t) is the autocorrelation function of the stimulus. Consequently, the first‑order correction to the ISI density is proportional to the stimulus’ second‑order statistics. This result explains why, for small signals, the ISI histogram carries a clear imprint of the stimulus’ correlation structure, while the neuron’s own dynamics remain encoded in fₙ(t).

The authors validate the theory using numerical simulations of a leaky integrate‑and‑fire neuron with additive Gaussian white noise. They inject sinusoidal phase‑continuous signals of varying amplitude (0.01–0.2 of the firing threshold) and compare the simulated ISI histograms with the analytical predictions. The Kullback‑Leibler divergence between model and simulation remains below 0.02 for signal amplitudes up to 0.1, confirming the high fidelity of the approximation. When two signals with a fixed phase offset are presented to two separate neurons, the model correctly predicts the emergence of a cross‑correlation term in the joint ISI density.

Beyond theoretical analysis, the paper demonstrates a practical application: a four‑neuron mutual‑inhibition network that extracts the cross‑correlation of two input signals. Two “input” neurons receive the individual stimuli; two “output” neurons are coupled via reciprocal inhibition. The output neurons fire preferentially when the two inputs are temporally aligned, producing a spike‑rate that mirrors the analytical cross‑correlation function. This architecture illustrates how spiking neurons can implement correlation‑based signal processing without explicit digital computation, suggesting a route toward low‑power neuromorphic hardware for tasks such as sensor fusion or auditory localization.

In summary, the work makes three major contributions: (1) it provides a compact, analytically tractable Markov model that separates neuronal intrinsic dynamics from stimulus statistics; (2) it clarifies the mechanistic origin of stimulus autocorrelation and cross‑correlation in the ISI distribution, especially under weak‑signal, moderate‑noise conditions; and (3) it leverages this insight to design a simple spiking network that performs correlation detection. The approach opens avenues for modeling neural responses when the underlying biophysical details are unknown and for constructing biologically inspired correlation processors in neuromorphic systems. Future work should explore extensions to non‑stationary signals, stronger nonlinear stimulus effects, and validation with in‑vivo spike train recordings.