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Michael Zaks

ON A POSSIBLE MECHANISM OF BIRHYTMICITY IN SIMPLE MODELS OF NEURONAL ENSEMBLES

 

Polyrhytmicity belongs to important attributes of large neuronal assemblies: recorded extracellular oscillations of human neurons demonstrate alternating epochs of fast and slow oscillations (i.e. gamma- and theta- rhythms). To take account of this phenomenon, most of the existing models include interaction of different units whose intrinsic timescales strongly differ. Here, we discuss a possible mechanism which ensures birhytmicity in simple models of neuronal ensembles in which all elements share the intrinsic timescale. Although the considered dynamical systems are non-generic, they involve typical properties of many existing models in neuroscience. We consider networks built of oscillatory units with the same eigenfrequency; coupling terms in the governing equations are proportional to velocities of the elements. No restrictions are put either on the symmetry of the coupling or on its  pattern (mean field, next neighbors, pairwise or triple interactions etc.).

In the parameter space of the ensemble, destabilization of the equilibrium occurs by means of the Hopf bifurcation. On the large part of the stability boundary, the spectrum of the linearized flow contains not one (as usually) but two pairs of purely imaginary eigenvalues. Of the two resulting frequencies, one is typically much lower than the individual frequency of an element, whereas the other one is distinctly higher. Accordingly, in the nonlinear regime the respective ensembles are potentially capable of performing both slow and fast modes of oscillations. We illustrate this general phenomenon by numerical data obtained from ensembles of oscillators with different coupling patterns and coupling functions.