Bifurcation of Orbits and Synchrony in Inferior Olive Neurons

Authors

Keum W. Lee, Kwandong University, GangwonFollow
Sahjendra N. Singh, University of Nevada, Las VegasFollow

Document Type

Article

Publication Date

9-2012

Publication Title

Journal of Mathematical Biology

Volume

65

Issue

3

First page number:

465

Last page number:

491

Abstract

Inferior olive neurons (IONs) have rich dynamics and can exhibit stable, unstable, periodic, and even chaotic trajectories. This paper presents an analysis of bifurcation of periodic orbits of an ION when its two key parameters (a, μ) are varied in a two-dimensional plane. The parameter a describes the shape of the parabolic nonlinearity in the model and μ is the extracellular stimulus. The four-dimensional ION model considered here is a cascade connection of two subsystems (S_a and S_b). The parameter plane (a − μ) is delineated into several subregions. The ION has distinct orbit structure and stability property in each subregion. It is shown that the subsystem S_a or S_b undergoes supercritical Poincare–Andronov–Hopf (PAH) bifurcation at a critical value μ_c(a) of the extracellular stimulus and periodic orbits of the neuron are born. Based on the center manifold theory, the existence of periodic orbits in the asymptotically stable S_a, when the subsystem S_b undergoes PAH bifurcation, is established. In such a case, both subsystems exhibit periodic orbits. Interestingly when S_b is under PAH bifurcation and S_a is unstable, the trajectory of Sa exhibits periodic bursting, interrupted by periods of quiescence. The bifurcation analysis is followed by the design of (i) a linear first-order filter and (ii) a nonlinear control system for the synchronization of IONs. The first controller uses a single output of each ION, but the nonlinear control system uses two state variables for feedback. The open-loop and closed-loop responses are presented which show bifurcation of orbits and synchronization of oscillating neurons.

Keywords

Bifurcation of orbits; Inferior olive neurons; Linear and nonlinear control; Synchronization

Disciplines

Controls and Control Theory | Electrical and Computer Engineering | Nuclear Engineering | Signal Processing

Repository Citation

Lee, K. W., Singh, S. N. (2012). Bifurcation of Orbits and Synchrony in Inferior Olive Neurons. Journal of Mathematical Biology, 65(3), 465-491.