A Fast Eulerian Approach for Computation of Global Isochrons in High Dimensions

We present a novel Eulerian numerical method to compute global isochrons of a stable periodic orbit in high dimensions. Our approach is to formulate the asymptotic phase as a solution to a first-order boundary value problem and solve the resulting Hamilton--Jacobi equation with the parallel fast sweeping method. All isochrons are then given as isocontours of the phase. We apply this method to the Hodgkin--Huxley equations and a model of a dopaminergic neuron which exhibits mixed mode oscillations. Our results show that this Eulerian scheme is an efficient, accurate method for computing the asymptotic phase of a periodic dynamical system. Furthermore, by computing the phase on a Cartesian grid, it is simple to compute the gradient of phase, and thus compute an “almost phaseless” target set for the purposes of desynchronization of a system of oscillators.

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