FitzHugh–Nagumo Model in Neutral Delay Differential Equation Representation

Abstract

We present a neutral delay differential equation (NDDE) modification of the FitzHugh–Nagumo (FHN) model. One linear and three nonlinear variants are proposed to capture the interaction between delayed effects in the membrane potential and the stimulating current, including a delayed rate (neutral) term. We derive fixed points, compute eigenvalues, and obtain Hopf conditions to map regions of stability, instability, and oscillation. Parameter studies show how the neutral delay reorganizes dynamics and modulates the robustness of nerve firing. Numerical simulations quantify frequency, ISI, amplitude, and spike counts and reproduce canonical phenotypes: regular spiking adapts to large stimulus delays, intrinsic bursting yields rapid spikes, chattering shows high-frequency bursts with moderate delays, fast-spiking exhibits high rates, and low-threshold neurons adapt to high frequencies with small delays. Overall, the NDDE representation provides a compact, physically motivated framework in which time delay governs stability and synchronization, offering insight for diagnostic modeling and potential therapies for disorders with irregular firing.