Nonlinear Observer Design and Synchronization Analysis for Classical Models of Neural Oscillators

TL;DR

This paper develops nonlinear observers and analyzes synchronization for classical neural oscillator models using contraction theory, providing methods for estimating unmeasured variables and synchronizing neural systems.

Contribution

It introduces observer designs and synchronization analysis for four classical neural oscillator models using nonlinear contraction theory.

Findings

Observer rate convergence for Fitzhugh-Nagumo model

Partial state observer design for Morris-Lecar model

Synchronization analysis for Fitzhugh-Nagumo and Hodgkin-Huxley models

Abstract

This work explores four nonlinear classical models of neural oscillators, the Hodgkin-Huxley model, the Fitzhugh-Nagumo model, the Morris-Lecar model, and the Hindmarsh-Rose model. Nonlinear contraction theory is used to develop observers and perform synchronization analysis on these systems. Neural oscillation and signaling models are based on the biological function of the neuron, with behavior mediated through the channeling of ions across the cell membrane. The variable assumed to be measured is the membrane potential, which may be obtained empirically through the use of a neuronal force-clamp system, or may be transmitted through the axon to other neurons. All other variables are estimated by using partial state or full state observers. Basic observer rate convergence analysis is performed for the Fitzhugh Nagumo system, partial state observer design is performed for the Morris-Lecar system, and basic synchronization analysis is performed for both the Fitzhugh-Nagumo and the Hodgkin-Huxley systems.