Oleksandr Burylko

An Interactive Channel Model of the Basal Ganglia: Bifurcation Analysis Under Healthy and Parkinsonian Conditions

Oscillations in the basal ganglia are an active area of research and have been shown to relate to the hypokinetic motor symptoms of Parkinson’s disease. We study oscillations in a multi-channel mean field model, where each channel consists of an interconnected pair of subthalamic nucleus and globus pallidus sub-populations.To study how the channels interact, we perform two-dimensional bifurcation analysis of a model of an individual channel, which reveals the critical boundaries in parameter space that separate different dynamical modes; these modes include steady-state, oscillatory, and bi-stable behaviour. Without self-excitation in the subthalamic nucleus a single channel cannot generate oscillations, yet there is little experimental evidence for such self-excitation. Our results show that the interactive channel model with coupling via pallidal sub-populations demonstrates robust oscillatory behaviour without subthalamic self-excitation, provided the coupling is sufficiently strong. We study the model under healthy and Parkinsonian conditions and demonstrate that it exhibits oscillations for a much wider range of parameters in the Parkinsonian case. In the discussion, we show how our results compare with experimental findings and discuss their possible physiological interpretation. For example, experiments have found that increased lateral coupling in the rat basal ganglia is correlated with oscillations under Parkinsonian conditions.

Desynchronization transitions in nonlinearly coupled phase oscillators

Abstract We consider the nonlinear extension of the Kuramoto model of globally coupled phase oscillators where the phase shift in the coupling function depends on the order parameter. A bifurcation analysis of the transition from fully synchronous state to partial synchrony is performed. We demonstrate that for small ensembles it is typically mediated by stable cluster states, that disappear with creation of heteroclinic cycles, while for a larger number of oscillators a direct transition from full synchrony to a periodic or a quasiperiodic regime occurs.

Identical Phase Oscillator Networks: Bifurcations, Symmetry and Reversibility for Generalized Coupling

For a system of coupled identical phase oscillators with full permutation symmetry, any broken symmetries in dynamical behaviour must come from spontaneous symmetry breaking, i.e. from the nonlinear dynamics of the system. The dynamics of phase differences for such a system depends only on the coupling (phase interaction) function

Winner-take-all in a phase oscillator system with adaptation.

g(\varphi)

Mechanism of desynchronization in the finite-dimensional Kuramoto model

and the number of oscillators

Multistability in the Kuramoto model with synaptic plasticity

N

Weak chimeras in minimal networks of coupled phase oscillators

. This paper briefly reviews some results for such systems in the case of general coupling

Bifurcation to heteroclinic cycles and sensitivity in three and four coupled phase oscillators

g

Extreme Sensitivity to Detuning for Globally Coupled Phase Oscillators

before exploring two cases in detail: (a) general two harmonic form:

Competition for synchronization in a phase oscillator system

g(\varphi)=q\sin(\varphi-\alpha)+r\sin(2\varphi-\beta)