Analytical solution of linearized equations of the Morris-Lecar neuron model at large constant stimulation

TL;DR

This paper derives an explicit analytical solution for the Morris-Lecar neuron model's behavior under large constant stimulation, revealing detailed damping dynamics of neuronal spikes.

Contribution

It provides an explicit analytical reduction of the Morris-Lecar equations near the asymptote, linking model parameters to damping behavior.

Findings

Explicit formulas for damping coefficients derived

Analytical and numerical solutions compared successfully

Characterization of spike amplitude damping stages

Abstract

The classical biophysical Morris-Lecar model of neuronal excitability predicts that upon stimulation of the neuron with a sufficiently large constant depolarizing current there exists a finite interval of the current values where periodic spike generation occurs. Above the upper boundary of this interval, there is four-stage damping of the spike amplitude: 1) minor primary damping, which reflects a typical transient to stationary dynamic state, 2) plateau of nearly undamped periodic oscillations, 3) strong damping, and 4) reaching a constant asymptotic value of the neuron potential. We have shown that in the vicinity of the asymptote the Morris-Lecar equations can be reduced to the standard equation for exponentially damped harmonic oscillations. Importantly, all coefficients of this equation can be explicitly expressed through parameters of the original Morris-Lecar model, enabling direct comparison of the numerical and analytical solutions for the neuron potential dynamics at later stages of the spike amplitude damping.