Analytical bounds on the critical coupling strength in a population of heterogeneous biological oscillators

Synchronization of nonlinear oscillators is a ubiquitous phenomenon in the biological sciences; however, existing analytical techniques are ill-equipped to handle the large amount of heterogeneity present in realistic populations of biological oscillators. Using phase reduction, we derive upper and lower bounds on the critical coupling strength required to achieve frequency synchronization in a population with both arbitrarily distributed natural frequencies and phase response properties. Numerical simulations reveal that these bounds are reasonably tight in a network of oscillatory neurons as might be relevant to diseases characterized by pathological neural synchronization such as epilepsy or Parkinson's disease. Furthermore, we show how the upper bounds can be altered by including the influence of a periodic external perturbation.

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