In the first part of the dissertation, we discuss noise-induced phenomena in neuronal dynamics. Many neuronal systems exhibit slow random alternations and sudden switches in activity states. Models with noisy relaxation dynamics (oscillatory, excitable or bistable) account for these temporal, slow wave, patterns and the uctuations within states. The noise-induced transitions in a relaxation dynamics are analogous to escape by a particle in a slowly changing double-well potential. In this formalism, we obtain semi-analytically the first and second order statistical properties: the distributions of the slow process at the transitions and the temporal correlations of successive switching events.

We find that the temporal correlations can be used to distinguish among biophysical mechanisms for the slow negative feedback, such as divisive or subtractive. We develop our results in the context of models for cellular pacemaker neurons; they also apply to mean-field models for spontaneously active networks with slow wave dynamics.

We extend the analysis of noise-induced phenomena to the fast-slow system with a higher dimensional fast subsystem. We apply an appropriate theory and numerical algorithms that provide the asymptote of noise-induced transition rate in a general system. We further discuss the noise-induced phenomena in a particular limit of slow time constant and noise amplitude.

In the second part, we construct a model of perceptual bi-stability with a network in a balanced state of excitatory and inhibitory activity. The model implementing neuronal competition by mutually inhibitory populations generates slow alternations in dominance between the two populations via depression in network connectivity. Due to the linear response of a balanced network to the external input, the oscillations' frequency depends on the external input monotonically, as observed in the experiments.