The 24th January of 2024 Jordi Penalva will present his PhD thesis: Neuronal piecewise linear models reproducing bursting dynamics.
Title: Neuronal piecewise linear models reproducing bursting dynamics.
Author: Jordi Penalva
Directors: Dr. Antonio E. Teruel, Dra. Catalina Vich and Dr. Mathieu Desroches
Abstract: In this thesis, we propose a piecewise linear version of the original planar Morris-Lecar model which we study qualitatively and for which we characterize several bifurcations related to different types of bursting dynamics obtained by adding one or two slow variables. In doing so, we will make the first theoretical study of the slow-passage phenomenon in the context of piecewise linear slow-fast systems.
We divide the contents of the thesis in two parts. In the first part, we introduce our piecewise linear version of the Morris-Lecar model (PWL-ML). This model reproduce some dynamic behaviors observed in the original Morris-Lecar model and we identify and study adequate dynamical regimes where all parameters but three can be fixed. We study and categorize the bifurcation structure inherent to the PWL-ML model, revealing the presence of diverse bifurcation phenomena, primarily encompassing Hopf-like bifurcations, SNIC bifurcations, and homoclinic bifurcations, with the potential for other yet unexplored forms.
In the second part, we study slow-passage phenomena through different bifurcations of the PWL-ML model. In this endeavor, we adopt a slow-fast system perspective, with the PWL-ML model constituting the fast subsystem and the main bifurcation parameter studied in part I serving as the slow variable. Slow-passage phenomena are the results of the interplay between fast and slow components, emerging as a consequence of the gradual drift close to a bifurcation point of the fast subsystem. In our case, we study the slow-passage through a Hopf-like bifurcation, through a homoclinic connection and finally through a SNIC bifurcation. The synthesis of these findings opens the door to explore the intricate bursting dynamics appearing in the PWL-ML model, which includes: elliptic bursting, for the Hopf-like bifurcation; square-wave bursting, for the homoclinic connection; and parabolic bursting, for the SNIC bifurcation.
