Analog of the Hutchinson equation in biophysical neurodynamics: from the Morris–Lecar model to a delay differential equation

Abstract

Starting with the classical biophysical Morris–Lecar model of neuronal excitability, we introduce a functional analog of the Hutchinson equation initially obtained for population dynamics with delayed negative feedback. It is shown that the resulting equation with a fixed time delay qualitatively reproduces the dynamics of the original model upon direct current stimulation, preserving both the initial type of neuronal excitability and biophysically realistic spike shape within a wide range of the delay values. If the delay becomes very small (2 ms or less), the simplified delay-based model exhibits a distinct transition from the 1st to the 2nd excitability type.