Winnerless Competition in Neural Dynamics; Cluster Synchronisation of Coupled Oscillators
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Thesis or dissertation
University of Exeter
Systems of globally coupled phase oscillators can have robust attractors that are heteroclinic networks. Such a heteroclinic network is generated, where the phases cluster into three groups, within a specific regime of parameters when the phase oscillators are globally coupled using the function $g(\varphi) = -\sin(\varphi + \alpha) + r \sin(2\varphi + \beta)$. The resulting network switches between 30 partially synchronised states for a system of $N=5$ oscillators. Considering the states that are visited and the time spent at those states a spatio-temporal code can be generated for a given navigation around the network. We explore this phenomenon further by investigating the effect that noise has on the system, how this system can be used to generate a spatio-temporal code derived from specific inputs and how observation of a spatio-temporal code can be used to determine the inputs that were presented to the system to generate a given coding. We show that it is possible to find chaotic attractors for certain parameters and that it is possible to detail a genetic algorithm that can find the parameters required to generate a specific spatio-temporal code, even in the presence of noise. In closing we briefly explore the dynamics where $N>5$ and discuss this work in relation to winnerless competition.
Peter Ashwin, Gabor Orosz, John Wordsworth and Stuart Townley. Dynamics on networks of cluster states for globally coupled phase oscillators. SIAM Journal on Applied Dynamical Systems 6(4):728-758, 2007.
John Wordsworth and Peter Ashwin. Spatiotemporal coding of inputs for a system of globally coupled phase oscillators. Physical Review E, 78(6):066203, 2008.
PhD in Mathematics