Inhibition in oscillatory networks of neurons can have apparently paradoxical e ects, sometimes creating dispersion of phases, sometimes fostering synchrony in the network. We analyze a pair of biophysically modeled neurons and show h o w the rates of onset and decay of inhibition interact with the time scales of the intrinsic oscillators to determine when stable synchrony is possible. We show that there are two di erent regimes in parameter space in which di erent combinations of the time constants and other parameters regulate whether the synchronous state is stable. We also discuss the construction and stability of non-synchronous solutions, and the implications of the analysis for larger networks. The analysis uses geometric techniques of singular perturbation theory that allow one to combine estimates from slow ows and fast jumps.

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In this study we have constructed a mathematical model of a recently proposed functional model known to be responsible for inducing waking, NREMS and REMS. Simulation studies using this model reproduced sleep-wake patterns as reported in normal animals. The model helps to explain neural mechanism(s) that underlie the transitions between wake, NREMS and REMS as well as how both the homeostatic sleep-drive and the circadian rhythm shape the duration of each of these episodes. In particular, this mathematical model demonstrates and confirms that an underlying mechanism for REMS generation is pre-synaptic inhibition from substantia nigra onto the REM-off terminals that project on REM-on neurons, as has been recently proposed. The importance of orexinergic neurons in stabilizing the wake-sleep cycle is demonstrated by showing how even small changes in inputs to or from those neurons can have a large impact on the ensuing dynamics. The results from this model allow us to make predictions of the neural mechanisms of regulation and patho-physiology of REMS.