Development of a bifurcation identification interface applied to the analysis of neuronal excitability

Abstract

This master thesis concerns the implementation of a novel, computationally efficient bifurcation numerical analysis interface in the Julia compiled programming language. The interface involves the use of the well-known bisection or Newton-Raphson methods in order to locate bifurcations in the neuron models, as well as the use of numerical approximation methods of Jacobian matrices through forward numerical differentiation of the system's equations. The interface that is built aims at the identification of the bifurcations in neuron models in order to determine their excitability type. A recent paper-motivated canonical model is chosen as an example to which the interface can be applied as a proof of concept. This numerical analysis of the example model outputs results that highlight the importance of dynamical analysis of neuron models, i.e. analysis over a range of time-scale parameters, versus the more common static analysis of models through the visual inspection of their phase plane representation. Normal form identification based on visual inspection only is at considerable risk that the original system is identified to may not be the correct one. The results obtained through the use of this interface on a two-dimensional therefore motivate the need for extensive numerical analysis of original high-dimensional neuron models for various values of time-scale separation in order to reliably identify the bifurcation normal form that they can be reduced to.