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Monday, July 20 • 8:00pm - 9:00pm
P156: Analytical solution of linearized equations of the Morris-Lecar neuron model at large constant stimulation

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Tatiana Zemskova (zemskova.ts@phystech.edu)
, Alexander Paraskevov

The Morris-Lecar model (MLM) [1] is a classical biophysical model of spike generation by the neuron, which takes into account dynamics of voltage- dependent ion channels and realistically describes the spike waveform. MLM predict that upon stimulation of the neuron with sufficiently large constant depolarizing current _I stim_, there exists a finite interval of _I stim_ values where periodic spike generation occurs [2-4]. Numerical simulations show that the cessation of periodic generation of spikes above the upper boundary of this interval occurs through damping of the spike amplitude, arising with a delay inversely proportional to _I stim_ value. In particular, the damped dynamics can be divided into four successive stages: 1) minor primary damping, which reflects a typical transient to stationary state, 2) plateau of nearly undamped periodic oscillations, which determines the aforementioned delay, 3) strong damping, and 4) reaching a constant asymptotic value. As the last two stages resemble the well-known exponentially-damped harmonic oscillations, we tackled to find an analytical description for these stages [5].

First, we have linearized the MLM equations at the vicinity of the stationary asymptotic value of the neuronal potential. The resulting equations have been then reduced to an inhomogeneous Volterra integral equation of the 2nd kind. In turn, the latter has been transformed into an ordinary differential equation of the second order with a time-dependent coefficient at the first- order derivative. As this time dependence was just an exponential decay, we considered its asymptotic value and analytically solved the final equation. In order to verify the analytical solution found, we have compared it with the numerical solution obtained using the standard MATLAB tools for systems of ordinary differential equations.

We have accurately shown that the linearized system of equations of the MLM can be reduced to a standard equation of damped harmonic oscillations for the neuron potential. Since all coefficients of this equation are explicitly expressed through the parameters of the original MLM, one can directly (i.e. without any fitting) compare the numerical and analytical solutions for dynamics of the neuron potential at the stages of strong damping and reaching a constant asymptotic value. The results allow a quantitative study of the applicability boundary of linear stability analysis that implies exponential damping.


1. Morris C, Lecar. H. Voltage oscillations in the barnacle giant muscle fiber. Biophys. J., 1981, 35(1): 193-213. https://doi.org/10.1016/S0006-3495(81)84782-0

2. Tateno T, Harsch A, Robinson H. Threshold firing frequency-current relationships of neurons in rat somatosensory cortex: type 1 and type 2 dynamics. J. Neurophysiol., 2004, 92(4): 2283-2294. https://doi.org/10.1152/jn.00109.2004

3. Tsumoto K. et al. Bifurcations in Morris-Lecar neuron model. Neurocomputing, 2006, 69(4-6): 293-316. https://doi.org/10.1016/j.neucom.2005.03.006

4. Nguyen L.H, Hong K.S, Park S. Bifurcation control of the Morris-Lecar neuron model via a dynamic state-feedback control. Biol. Cybern., 2012, 106(10): 587-594. https://doi.org/10.1007/s00422-012-0508-4

5. Paraskevov A.V, Zemskova T.S. Analytical solution of linearized equations of the Morris-Lecar neuron model at large constant stimulation, bioRxiv, 2019. https://doi.org/10.1101/869875

Google meet for poster session: meet.google.com/rcy-qakc-oei

avatar for Tatiana Zemskova

Tatiana Zemskova

Ecole Polytechnique

Monday July 20, 2020 8:00pm - 9:00pm CEST
Slot 16