Abstract
An ensemble of pulse-coupled phase oscillators is thoroughly analyzed in the presence of a mean-field coupling and a dispersion of their natural frequencies. In spite of the analogies with the Kuramoto setup, a much richer scenario is observed. The “synchronized” phase, which emerges upon increasing the coupling strength, is characterized by highly irregular fluctuations: A time-series analysis reveals that the dynamics of the order parameter is indeed high dimensional. The complex dynamics appears to be the result of the nonperturbative action of a suitably shaped phase-response curve. Such a mechanism differs from the often-invoked balance between excitation and inhibition and might provide an alternative basis to account for the self-sustained brain activity in the resting state. The potential interest of this dynamical regime is further strengthened by its (microscopic) linear stability, which makes it quite suited for computational tasks. The overall study has been performed by combining analytical and numerical studies, starting from the linear stability analysis of the asynchronous regime, to include the Fourier analysis of the Kuramoto order parameter, the computation of various types of Lyapunov exponents, and a microscopic study of the interspike intervals.
- Received 27 August 2015
DOI:https://doi.org/10.1103/PhysRevX.6.011015
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Published by the American Physical Society
Physics Subject Headings (PhySH)
Erratum
Erratum: Self-Sustained Irregular Activity in an Ensemble of Neural Oscillators [Phys. Rev. X 6, 011015 (2016)]
Ekkehard Ullner and Antonio Politi
Phys. Rev. X 7, 029901 (2017)
Popular Summary
The mammalian brain is characterized by dynamical phenomena occurring over many different scales, as revealed by the temporal structure of electroencephalogram signals. These phenomena are present even when the brain is in its resting state. Despite many efforts to clarify the origin of the irregular collective behavior of neurons, a convincing explanation has not yet been found. Quite often, a balance between inhibitory and excitatory activity is invoked as a necessary requisite. Here, we analytically and numerically study an ensemble of pulse-coupled oscillators and show that this condition is not strictly necessary.
Our setup is similar to the Kuramoto model typically invoked to describe the emergence of synchronization among globally coupled phase oscillators. However, the different coupling mechanism that we assume, which is more appropriate to describe a neural system, leads to a much richer collective motion with many degrees of freedom simultaneously active; our model is accordingly more prone to respond to external stimuli. We simulate three different system sizes (4000–64000 units) and find that the onset of such complex macroscopic dynamics is a manifestation of the rather subtle link that may exist between microscopic and macroscopic descriptions in systems out of thermodynamic equilibrium. Finally, the potential interest of this dynamical regime is further reinforced by its microscopic linear stability; stability is indeed a necessary requisite for a system that is expected to reliably perform computations.
We expect that our findings will inform future studies of ensembles of coupled components such as neurons.
