Oscillations are a prevalent phenomenon in biological neural networks and manifest themselves experimentally in electroencephalograms (EEG), recordings of local field potentials (LFP), and multi-unit recordings. Oscillations of the spike activity are particularly interesting from a functional point of view. Synchronization of different populations of neurons has, for example, been proposed as a potential solution to the binding problem (Eckhorn et al., 1988; Singer, 1994; Gray et al., 1989; Wang et al., 1990; König and Schillen, 1991; Eckhorn and Brosch, 1993). Oscillations play an important role in the coding of sensory information. In the olfactory system an ongoing oscillation of the population activity provides a temporal frame of reference for neurons coding information about the odorant (Laurent, 1996; Desmaison et al., 1999). Similarly, place cells in the hippocampus exhibit phase-dependent firing activity relative to a background oscillation (O'Keefe, 1993). Finally, rhythmic spike patterns in the inferior olive may be involved in various timing tasks and motor coordination (Welsh et al., 1995; Llinás, 1991; Kistler et al., 2000).
In this chapter we do not discuss all the interesting computational applications, but restrict ourselves to the analysis of mechanisms underlying oscillatory activity and synchronization. We start in Section 8.1 with a stability analysis of the state of asynchronous firing encountered in Chapter 6.4. In recurrent networks of spiking neurons we find that a state of asynchronous activity is unstable in the absence of noise. As a consequence, neurons tend to form clusters of cells that fire synchronously. In Section 8.2 we investigate network modes where neurons fire ``in lockstep'' and derive a stability criterion for synchronized activity. Finally, in Section 8.3 we explore the possibility that sparse recurrent networks exhibit an oscillatory population activity while generating irregular spike trains. Sparse networks producing irregular spike trains form a promising starting point for an understanding of the neuronal activity observed in various parts of the brain.
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