A study of complex dynamical regimes in a system of coupled non-linear odes medeling the small neuronal circuit in dependence on the coupling parameters

To single out possible effects related to the modulation of characteristic times, we consider a small circuit of coupled identical elements and vary only the parameters, which control the inter-connections between oscillatory elements. Each element is described by the FitzHugh-Nagumo (FHN) model that well reproduces the dynamical behavior of typical neuron. This model consists of two ordinary differential equations (ODE). A coupling between elements is introduced by the variable, which depends on a threshold function and defines two states of this variable: zero (non-active) or one (active). We demonstrate that a variation of the coupling parameters results in complex regimes, which differ in the frequency and amplitude as well as in the shape of oscillations.

To understand how transitions between these regimes occur, we used an approach based on the application of the continuous wavelet transform (CWT) [E.B. Postnikov, et al., Int. J. Bifurcations and Chaos, 2012] instead of the classical bifurcation analysis of periodical orbits. One of the principal advantages of this method is a possibility to distinguish between continual transitions (they correspond to adiabatic invariants of ODEs) and bifurcational jumps from one attractor to another, when the control parameter reaches a critical value. In addition, the CWT-based method allows the revealing of details of dynamic transitions between different solutions with a multi-frequency composition (e.g. bursting), that is out of power of the conventional methods related to the analysis of periodical orbits.