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

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 a four-stage damping of the spike amplitude: 1) minor primary damping, which reflects a typical transient to stationary state, 2) plateau of nearly undamped periodic oscillations, 3) strong damping, and 4) reaching a constant stationary asymptotic value Vst of the neuron potential. We have linearized the Morris-Lecar model equations at the vicinity of Vst and have shown that the linearized equations can be reduced to a 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 (i.e. without any fitting) of the numerical and analytical solutions for the neuron potential dynamics at later stages of the spike amplitude damping. This allows to explore quantitatively the applicability boundary of linear stability analysis that implies exponential damping.