Abstract
Recently, neuronal avalanches have been observed to display oscillations, a phenomenon regarded as the co-existence of a scale-free behaviour (the avalanches close to criticality) and scale-dependent dynamics (the oscillations). Ordinary continuous-time branching processes with constant extinction and branching rates are commonly used as models of neuronal activity, yet they lack any such time-dependence. In the present work, we extend a basic branching process by allowing the extinction rate to oscillate in time as a new model to describe cortical dynamics. By means of a perturbative field theory, we derive relevant observables in closed form. We support our findings by quantitative comparison to numerics and qualitative comparison to available experimental results.
Highlights
Neuronal avalanches have been observed to display oscillations, a phenomenon regarded as the co-existence of a scale-free behaviour and scale-dependent dynamics
In order to study the signaling in larger regions of neurons, multielectrode arrays, comprising about 60 electrodes spread across ≈ 4 mm[2 ], are used to capture the collective occurrence of spikes as local field potentials (LFPs)
We focus on modelling neuronal avalanches as branching processes
Summary
We analyse observables that are somewhat more involved to derive in the present framework. They are rooted in the periodic extinction, as an analytical calculation shows, see “Appendix: Avalanche shape to first order” for details These oscillations remain visible in the shape V(t, T) at all T and all t, but become vivid whenever the termination time T is commensurate with the period of the oscillations, 2π/ν. As we have used our field-theoretic scheme only to first order, the slight mismatch with simulation results at larger A, such as those shown, is not surprising At such large amplitudes, the resulting shape resembles that of recent experimental results, where γ-oscillation modulated the average shape of neuronal avalanches[38]. Varying small amplitudes do not seem to alter this shape and appear to converge to the universal parabola shape when criticality is approached r → 0
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