Deterministic and Stochastic Dynamics of Multi-Variable Neuron Models: Resonance, Filtered Fluctuations and Sodium-Current Inactivation

Neurons are the basic elements of the networks that constitute the computational units of the brain. They dynamically transform input information into sequences of electrical pulses. To conceive the complex function of the brain, it is crucial to understand this transformation and identify simple neuron models which accurately reproduce the known features of biological neurons. This thesis addresses three different features of neurons. We start by exploring the effect of subthreshold resonance on the response of a periodically forced neuron using a simple threshold model. The response is studied in terms of an implicit one-dimensional time map that corresponds to the Poincar´e map of the forced system. Qualitatively distinct responses are found, including mode locking and chaos. We analytically find the stability regions of mode-locking solutions, and identify the transition to chaos through period-adding bifurcations. We show that the response becomes chaotic when the forcing frequency is close to the resonant frequency. Then we will consider an experimentally verified model with realistic spikegenerating mechanism and study the effect of filtered synaptic fluctuations on the firing-rate response of the neuron. Using a population density method as well as an efficient numerical method, we find the steady-state firing rate in two limits of fast and slow synaptic inputs and present the linear response theory for the firing rate of the model in response to both time-dependent mean inputs and time-dependent noise intensity. Finally, a novel model is introduced that incorporates threshold variability of neurons. We determine the modulation of the input-output properties of the model due to oscillatory inputs and in the presence of filtered synaptic fluctuations.