From quasiharmonic oscillations to neural spikes and bursts: a variety of hyperbolic chaotic regimes based on Smale – Williams attractor

In the present paper we consider a family of coupled self-oscillatory systems presented by pairs of coupled van der Pol generators and FitzHugh–Nagumo neural models, with the parameters being periodically modulated in anti-phase, so that the subsystems undergo alternate excitation with a successive transmission of the phase of oscillations from one subsystem to another. It is shown that, due to the choice of the parameter modulation and coupling methods, one can observe a whole spectrum of robust chaotic dynamical regimes, taking the form ranging from quasiharmonic ones (with a chaotically floating phase) to the well-defined neural oscillations, which represent a sequence of amplitude bursts, in which the phase dynamics of oscillatory spikes is described by a chaotic mapping of Bernoulli type. It is also shown that 4D maps arising in a stroboscopic Poincaré section of the model flow systems universally possess a hyperbolic strange attractor of the Smale–Williams type. The results are confirmed by analysis of phase portraits and time series, by numerical calculation of Lyapunov exponents and their parameter dependencies, as well as by direct computation of the distributions of angles between stable and unstable tangent subspaces of chaotic trajectories.