[résumé trop long] / The important relationship between structure and function has always been a fundamental question in neuroscience research. In particular in the case of movement, brain controls large groups of muscles and combines it with sensory informations from the environment to execute purposeful motor behavior. Mapping dynamics encoded in a high dimensional neural space onto low-dimensional behavioral space has always been a difficult challenge as far as theory is concerned. Here, we develope a framework to study spike/burst dynamics having low dimensional phase description, which can readily be extended under certain biological constraints on the coupling to low dimensional functional descriptions. In general, phase models are amongst the simplest of neuron models reproducing spike-burst behavior, excitability and bifurcations towards periodic firing. However, the coupling among neurons has only been considered using generic arguments valid close to the bifurcation point, and the distinction between electric and synaptic coupling remains an open question. In this thesis we aim to address this question and derive a mathematical formulation for the various forms of biologically realistic coupling. We begin by constructing a mathematical model based on a planar simplification of the Morris-Lecar model. Using geometric arguments we then derive a phase description of a network of neurons with biologically realistic electric coupling and subsequently with chemical coupling under the fast synapse approximation. We then demonstrate that electric and synaptic coupling are expressed differently on the level of the network’s phase description, exhibiting qualitatively different dynamics. Our numerical investigations confirm these findings and show excellent correspondence between the dynamics of the full network and the network’s phase description. Following the success of the phase description of the spiking neural network, we extend this approach in order to propose a generating mechanism for parabolic bursting captured by only a single phase variable. This is the first model in the literature which captures bursting dynamics in one dimension. In order to study the emergent behavior we extend this to a network of bursters with global coupling and analytically reduce a high dimensional system to only two dimensions. Further, we investigate the bifurcation properties numerically as well as analytically. One of the key conclusion is that the stability states remain invariant to the increasing number of spikes per burst. Finally we investigate a spikeburst neuron network coupled via mean field type of fast synapses developed in this thesis and systematically carry out a detailed bifurcation analysis of the model, for a tractable special case. Numerical simulations investigate this mean field model beyond special case and clearly reveals qualitative correspondence with the full network model. Moreover, these network displays rich collective dynamics as a function of two parameters, mainly the synaptic coupling strength and the width of the distribution in applied stimulus. Besides incoherence, frequency locking, and oscillator death (a total cessation of firing caused by excessively strong coupling), there exist multistable solutions in the full and the phase network of neurons.

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