SEARCHING FOR SPECIFIC PERIODIC AND CHAOTIC OSCILLATIONS IN A PERIODICALLY-EXCITED HODGKIN–HUXLEY MODEL

Abstract

This work has a twofold aim: to present a numerical analysis of the Hodgkin–Huxley model in a nonsmooth excitation scenario — which is both challenging and theoretically relevant — and to use the established framework as a basis for testing a method to search for specific oscillating patterns in dynamical systems. The analysis is founded on classical qualitative methods — bifurcation diagrams, phase space and spectral analysis — and on the calculation of the system Lyapunov spectrum. This calculation is carried out by means of an algorithm particularly suited to deal with nonsmooth excitation and the complexity of the state equations. The obtained Lyapunov exponents are then used to build a robust cost function (invariant with respect to the initial conditions or specific trajectories in a given basin of attraction) for seeking predefined dynamical patterns that are optimized using the particle swarm optimization algorithm. This bioinspired method possesses two desirable features: it has a significant global search potential and does not demand cost function manipulation. The proposed approach, which was tested here in different representative scenarios for the Hodgkin–Huxley model, has a promising application potential in general dynamical contexts and can also be a valuable tool in the planning of drug administration and electrical stimulation of neuronal and cardiac cells.