Interaction of neuronal populations with delay: effect of frequency mismatch and feedback gain.

The effect of frequency mismatch, signal transduction delay and inter-population feedback gain on the interaction of neuronal populations, mediated by long-range excitation, is investigated using physiologically realistic system parameters. Self-consistent solutions for the frequency, amplitude, and relative phase of the component signals are derived for limit cycle oscillations. These solutions predict important qualitative features including discontinuous changes in frequency of oscillation and phase reversal between symmetric and antisymmetric limit cycles. A singularity in the solutions is used to predict parameter regions in which limit cycles do not exist. If limit cycles exist at zero delay, it is shown that limit cycles and quasi-periodic attractors alternate as a function of delay. The implications of these results for estimating physiologically meaningful delays from observed phase shifts in EEG time series are discussed. Spectral peaks for the quasi-periodic attractor occur at m nu 1 +/- n nu 2, where the difference nu 1 - nu 2 is approximately equal to the intrinsic population frequency mismatch delta nu. The cross-correlation function is amplitude modulated with a frequency equal to delta nu/2, indicating that the two populations slip in and out of phase with a mean correlation duration equal to 1/delta nu. These findings underpin the dynamical basis of delay induced "desynchronization" of oscillations reported in computer simulations. Bifurcation diagrams indicate that quasi-periodic attractors exist for a wide range of parameters in the presence of delay in long-range excitation and non-zero frequency mismatch. If the frequency mismatch is sufficiently large and the feedback gain is sufficiently small, quasi-periodic attractors exist for all delays. Delays of a few milliseconds, much smaller than the system time scale, can destabilize limit cycle oscillations. The role of synaptic change in inducing bifurcations of limit cycles to quasi-periodic attractors and vice versa is discussed. The implication of these findings for the generation of chaos in distributed neural systems is discussed.