CONSISTENT TRANSITION MAPS FOR THE FITZHUGH-NAGUMO MODEL WITH NOISE VIA SHORT-TIME GAUSSIAN KERNELS

Noise from stochastic ion-channel gating shapes spike timing and variability, yet global phase-space analysis of noisy neuron models remains challenging because transition densities are rarely available in closed form. We develop a computational framework to construct a discrete Markov operator for a FitzHugh-Nagumo model with noise; as a representative case, we add additive noise to the refractoriness variable. Using a Kunitomo–Takahashi-type small-noise asymptotic expansion around the deterministic trajectory, we approximate the short-time transition density by a Gaussian whose mean follows the ODE solution and whose covariance is obtained from an associated matrix differential equation, avoiding expensive covariance time-integrals. Accuracy of the Gaussian kernel is assessed in whitened coordinates by comparing Monte Carlo transition samples with the standard normal reference using mean/covariance errors and density distances computed from binned KDE (Kullback-Leibler divergence and Hellinger distance). For noise intensity 0.1, time steps tau=0.01, 0.02, and 0.1 satisfy the acceptance criterion KL <= 5e-3. We then discretize the kernel on a grid to build a transition matrix for a single time step τ and estimate an effective reachable region in which column sums are approximately one, enabling stable composition of transitions. Consistency of the constructed transition map is further checked by (i) a time-step consistency test: the transition over 2τ computed directly matches the result of applying the τ-step transition twice, measured by total variation distance, and (ii) comparison of leading eigenvalues using residual and eigenvalue-gap metrics. Together, these checks support a practical, quantitatively assessed approximation of short-time stochastic neuron transitions.