Nonlinear Dynamical Analysis and Modeling of Excitable Systems

Goyal, Srashti (2018) Nonlinear Dynamical Analysis and Modeling of Excitable Systems. Masters thesis, Indian Institute of Science Education and Research Kolkata.

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Abstract

In chapter 1, we address a few questions on Izheikevich model of a neuron, which has a state jump kind of discontinuity. We present a numerical continuation algorithm and derive the expression of state jump saltation matrix used for calculating the Jacobian and hence the stability of a periodic orbit. This algorithm can be used to find periodic orbits in nonsmooth systems with any combination of maps and flows. Overall, we analyze the effect of non-smoothness on bifurcations and transients of the Izheikevich model. In chapter 2, we report coherent activity patterns in a two-dimensional monolayer of cardiac cells obtained from chicken embryos. On increasing the external potassium concentration, monolayer shows the transition from spontaneous oscillations to bursting and finally to quiescence. We show that the bursting activity it exhibits has a parabolic variation in frequency during the bursting phase. Our analysis of the experimental data confirms that the mechanism of that parabolic bursting is of the circle/ circle type as per Izheikevichs definition. We reproduce the experimental behavior by our simulations using the extended Morris Leccar model and investigate the effect of coupling on overall dynamics. In simulations, we take into consideration both homogeneous and heterogeneous cell population and show the emergence of spirals, waves, and pacemaker switching at optimal coupling strength.

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