[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.

Un nombre de sujets concernant la microrobotique numérique ont été abordés dans le cadre de cette the` se. Une nouvelle génération du microrobot numérique ”DiMiBot” a e´ te´ proposé ce qui rend le DiMiBot plus précis, plus contrôlable et plus petit. La nouvelle structure est formée de deux modules multistables seulement, ce qui ajoute des fonctionnalités´ s importantes comme l’augmentation du nombre de positions avec une taille plus réduite et la capacité´ de réaliser des trajectoires complexes dans l’espace de travail. Le principe du nouveau module multistable combine les avantages des microactionneurs pas à pas en termes du principe et du concept numérique en termes de la répétabilité et la robustesse en boucle ouverte. Un mécanisme de positionnement précis, capable de compenser les incertitudes de fabrication a e´ te´ développé et utilise´ pour assurer un positionnement précis. En parallèle, des modèles analytiques ont e´ te´ développés pour les principaux composants dans le DiMiBot: poutres flambées préformées et actionneurs e´ électrothermiques en U. Des méthodes de conception ont été développées par la suite qui permettent de choisir les dimensions optimales garantissant les performances requises en respectant les spécifications et limites de design. Des prototypes de modules multistables, fabrique´ s dans la salle Blanche MIMENTO, ont montré´ un bon Fonctionnement dans les expériences. / A number of topics concerning digital microrobotics were addressed in this thesis. A new generation of the digital microrobot ”DiMiBot” was proposed with several advantages making the DiMiBot more accurate, more controllable and smaller. The new structure consists of only two multistable modules which adds some important features such as increasing the number of positions with smaller size and the ability to realize complex trajectories in the workspace. The principle of the new multistable module combines the advantages of the stepping microactuators in terms of the principle and of the digital concept in terms of the repeatability and robustness without feedback. The accuracy is ensured with an accurate positioning mechanism that compensate the fabrication tolerances. In parallel, analytical models was developed for the main components in the DiMiBot: preshaped curved beams and U-shaped electrothermal actuators. Subsequently, design methods were developed that allow choosing the optimal dimensions that ensure the desired outputs and respecting the design specifications and limitations. Multistable module prototypes, fabricated in the clean room MIMENTO, showed a proper functioning in the experiments.