A new approach, based on a Markov transition matrix, is proposed to explain frequent sleep and wake transitions during sleep. The matrix is determined by analyzing hypnograms of 113 obstructive sleep apnea patients. Our approach shows that the statistics of sleep can be constructed via a single Markov process and that durations of all states have modified exponential distributions, in contrast to recent reports of a scale-free form for the wake stage and an exponential form for the sleep stage. Hypnograms of the same subjects, but treated with Continuous Positive Airway Pressure, are analyzed and compared quantitatively with the pretreatment ones, suggesting potential clinical applications.

Responding to various stimuli, some neurons either remain resting or can fire several distinct patterns of action potentials, such as spiking, bursting, subthreshold oscillations, and chaotic firing. In particular, Wilson's conductance-based neocortical neuron model, derived from the Hodgkin-Huxley model, is explored to understand underlying mechanisms of the firing patterns. Phase diagrams describing boundaries between the domains of different firing patterns are obtained via extensive numerical computations. The boundaries are further studied by standard instability analyses, which demonstrates that the chaotic neural firing could develop via period-doubling and/or period- adding cascades. Sequences of the firing patterns often observed in many neural experiments are also discussed in the phase diagram framework developed. Our results lay the groundwork for wider use of the model, especially for incorporating it into neural field modeling of the brain.