Stable Periodic Solutions to Discrete and Continuum Arrays of Weakly Coupled Nonlinear Oscillators

The existence and stability of periodic solutions to spatially distributed arrays of neural oscillators is analyzed. Conditions are found that guarantee that phase-locked patterns are orbitally stable. These conditions allow the solutions to be extended as some parameter varies. For continuum arrays with some differentiability conditions, it is shown that locking is lost when certain phase gradients become unbounded. Numerical methods are used to show that the results apply to realistic synaptically coupled oscillating neural networks. In particular, it is shown that synchronized solutions cannot generally be expected even if the neurons are identical.