Synchronization of Neural Oscillators: Data Analysis and Control

Two applications of the synchronization theory in neuroscience will be considered. First, the synchronization approach to multivariate data analysis will be discussed, with an emphasis on the analysis of brain activity. In particular; the following three tasks will be considered: quantification of a weak interaction of different oscillatory sources, analysis of directionality of interaction, and estimation of delay in coupling. The synchronization approach will be compared with the standard coherence technique. Next, the control of collective synchrony in a population of neuronal oscillators will be discussed from the viewpoint of a possible application to suppression of the Parkinsonian activity by means of deep brain stimulation. The method is based on the time-delayed feedback via the local field potential (mean field). The method is illustrated by numerical simulation of the dynamics of globally or randomly coupled neuronal oscillators. It will be shown that efficient suppression is possible for large domains of control parameters (delay and amplification in the feedback loop). It is important that, as soon as the undesired synchrony is suppressed, the intervention into the neuronal population becomes very small (it is then determined by the noise level in the desynchronized population). This means that the suggested control scheme is non-invasive. A theory based on the consideration of the synchronization transition as a Hopf bifurcation will be presented and compared with the numeric case. Several practically important problems will be discussed: the influence of the measurement noise, imperfect measurement, finite-size effects, etc.