Bifurcation of Orbits and Synchrony in Inferior Olive Neurons

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Journal of Mathematical Biology





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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 (Sa and Sb). 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 Sa or Sb undergoes supercritical Poincare–Andronov–Hopf (PAH) bifurcation at a critical value μc(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 Sa, when the subsystem Sb undergoes PAH bifurcation, is established. In such a case, both subsystems exhibit periodic orbits. Interestingly when Sb is under PAH bifurcation and Sa 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.


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


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

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