Stochastic synchronization of neuronal populations with intrinsic and extrinsic noise

Synchronous oscillations are prevalent in many areas of the brain including sensory cortices, thalamus and hippocampus [1]. Recordings of population activity based on the electroencephalogram (EEG) or the local field potential (LFP) often exhibit strong peaks in the power spectrum at certain characteristic frequencies. For example, in the visual system of mammals, cortical oscillations in the γ frequency band (20-70 Hz) are generated with a spatially distributed phase that is modulated by the nature of a visual stimulus. Stimulus-induced phase synchronization of different populations of neurons has been proposed as a potential solution to the binding problem, that is, how various components of a visual image are combined into a single coherently perceived object [2,3]. An alternative suggestion is that such oscillations provide a mechanism for attentionally gating the flow of neural information [4,5]. Neuronal oscillations may be generated by intrinsic properties of single cells or may arise through excitatory and inhibitory synaptic interactions within a local population of cells. Irrespective of the identity of the basic oscillating unit, synchronization can occur via mutual interactions between the oscillators or via entrainment to a common periodic stimulus in the absence of coupling.From a dynamical systems perspective, self-sustained oscillations in biological, physical and chemical systems are often described in terms of limit cycle oscillators where the timing along each limit cycle is specified in terms of a single phase variable. The phase-reduction method can then be used to analyze synchronization of an ensemble of oscillators by approximating the high-dimensional limit cycle dynamics as a closed system of equations for the corresponding phase variables [6,7]. Although the phase-reduction method has traditionally been applied to deterministic limit cycle oscillators, there is growing interest in extending the method to take into account the effects of no