A dynamical study of pulse-coupled oscillators in the brain
© Luke et al; licensee BioMed Central Ltd. 2012
Published: 16 July 2012
In 1967, Winfree  proposed a novel mathematical approach to describe phenomena of collective synchrony in nature (i.e. flashing of fireflies, clapping in a theatre, alpha rhythms, etc.) using a large coupled network of phase oscillators with a diversity of natural frequencies. By analyzing this large heterogeneous network from a “mean field” approach, the spontaneous synchrony can be understood as a critical phase transition similar to most statistical mechanical systems.
In this work, we employ this approach to model the phase transitions and bifurcation structures of a large network of pulse-coupled theta neurons  by appropriate choice of Winfree's "response" and "influence" functions, the latter of which is parameterized by a "sharpness" parameter n . As this parameter increases, the influence function approximates the behavior of a pulse-coupled synapse. Assuming a Lorentzian distribution of natural frequencies of width Δ and mean value ω 0 , taking the thermodynamic limit, and employing the Ott-Antonsen reduction method , the collective dynamics of the pulse-coupled network can be analytically reduced to a single low-dimensional dynamical equation for the mean field parameter z(t).
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