This study uses a conductance-based computer simulation to test the feasibility of a mechanism underlying a newly-described dynamic form of neuromodulation, called spike-timing dependent neuromodulation (STDN). In the mollusc, Tritonia diomedea, it was recently found that a serotonergic neuron (called DSI) alters the synaptic strength of another neuron (VSI-B) in a temporally biphasic-bidirectional manner, with an initial potentiation followed by prolonged synaptic depression (Sakurai and Katz 2003). Physiological evidence suggested that the depression phase is due to serotonin enhancing the A-current in VSI-B, thereby causing spike-narrowing or a decrease in spike amplitude, and thus a decrease in transmitter release. We sought to test the feasibility of this mechanism by developing a conductance-based model of VSI-B using a Hodgkin-Huxley style simulation with a minimal number of ion conductances: A-current, delayed rectifier potassium, fast sodium, and leak channels. From our model, we conducted simulations in order to study how the spike shape of the VSI-B action potential changes as the A-current conductance is enhanced, from which we are able to predict the amount of depression in the post-synaptic cell. Our model indicates that the depression due to the narrowing of the spike with A-current enhancement is sufficient to account for the empirically observed depression during STDN, although it suggests a greater effect of serotonin at the terminals than is observed in the soma. Additionally, the model suggested that the slow inactivation kinetics of the A-current cannot explain the dynamics of the depression phase of STDN. These modeling results suggest that serotonergic modulation of the A-current plays a role in STDN but does not account for its dynamics.

Complex networks of cells called neurons in the brain enable human learning and memory. The topology and electrophysiological function of these networks are affected by nano and microscale geometries of neurons. Understanding of these structure-function relationships in neurons is an important component of neuroscience in which simulation plays a fundamental role. This thesis addresses four specific geometric problems raised by modeling and simulation of intricate neuronal structure and behavior at the nanoscale. The first two problems deal with 3D surface reconstruction: neurons are geometrically complex structures that are tightly intertwined in the brain, presenting great challenges in reconstruction. We present the first algorithm that reconstructs surface meshes from polygonal contours that provably guarantees watertight, manifold, and intersection-free forests of densely packed structures. Many algorithms exist that produce surfaces from cross-sectional contours, but all either use heuristics in fitting the surface or they fail when presented with tortuous objects in close proximity. Our algorithm reconstructs surfaces that are not only internally correct, but are also free of intersections with other reconstructed objects in the same region. We also present a novel surface remeshing algorithm suitable for models of neuronal dual space. The last two problems treated by this thesis deal with producing derivative models from surface meshes. A range of neuronal simulation methodologies exist and we offer a framework to derive appropriate models for each from surface meshes. We present two specific algorithms that yield analysis-ready 1D cable models in one case, and proposed "aligned cell" models in the other. In the creation of aligned cells we also present a novel adaptive distance transform. Finally, we present a software package called VolRoverN in which we have implemented many of our algorithms and which we expect will serve as a repository of important tools for the neuronal modeling community. Our algorithms are designed to meet the immediate needs of the neuroscience community, but as we show in this thesis, they are general and suitable for a variety of applications. / text

In the first part of this thesis, motivated by the development of deep brain stimulation for Parkinson's disease, we consider the problem of reducing the synchrony of a neuronal population via a closed-loop electrical stimulation. This, under the constraints that only the mean membrane voltage of the ensemble is measured and that only one stimulation signal is available (mean-field feedback). The neuronal population is modeled as a network of interconnected Landau-Stuart oscillators controlled by a linear single-input single-output feedback device. Based on the associated phase dynamics, we analyze existence and robustness of phase-locked solutions, modeling the pathological state, and derive necessary conditions for an effective desynchronization via mean-field feedback. Sufficient conditions are then derived for two control objectives: neuronal inhibition and desynchronization. Our analysis suggests that, depending on the strength of feedback gain, a proportional mean-field feedback can either block the collective oscillation (neuronal inhibition) or desynchronize the ensemble.In the second part, we explore two possible ways to analyze related problems on more biologically sound models. In the first, the neuronal population is modeled as the interconnection of nonlinear input-output operators and neuronal synchronization is analyzed within a recently developed input-output approach. In the second, excitability and synchronizability properties of neurons are analyzed via the underlying bifurcations. Based on the theory of normal forms, a novel reduced model is derived to capture the behavior of a large class of neurons remaining unexplained in other existing reduced models.

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Dans la première partie de cette thèse, motivée par le développement de la stimulation cérébrale profonde comme traitement des symptômes moteurs de la maladie de Parkinson, nous considérons le problème de réduire la synchronie d'une population neuronale par l'intermédiaire d'une stimulation électrique en boucle fermée. Ceci, sous les contraintes que seule la tension de membrane moyenne de l'ensemble est mesurée et qu'un seul signal de stimulation est disponible (retour du champ moyen). La population neuronale est modélisée comme un réseau d'oscillateurs de Landau-Stuart contrôlé par un dispositif de rétroaction mono-entrée mono-sortie. En nous basant sur la dynamique de phase associée au système, nous analysons l'existence et la robustesse des solutions à verrouillage de phase, modélisant l'état pathologique, et nous dérivons des conditions nécessaires à une désynchronisation efficace par retour du champ moyen. Des conditions suffisantes sont ensuite dérivées pour deux objectifs de contrôle: l'inhibition et la désynchronisation neuronale. Notre analyse suggère que, en fonction de l'intensité du gain de rétroaction, le retour du champ moyen peut soit bloquer l'oscillation collective (inhibition neuronale) soit désynchroniser l'ensemble.Dans la deuxième partie, nous explorons deux voies possibles pour l'analyse des problèmes similaires dans des modèles biologiquement plus plausibles. Dans la première, la population est modélisée comme une interconnexion d'opérateurs entrée-sortie non-linéaires et la synchronisation neuronale est analysée en s'appuyant sur une approche entré-sortie récemment développée. Dans la seconde, les propriétés d'excitabilité et de synchronisabilité des neurones sont analysées via les bifurcations sous-jacentes. En nous basant sur la théorie des formes normales, un nouveau modèle réduit est dérivé pour capturer les comportements d'une grande classe de neurones qui restent inexpliqués dans les modèles réduits existants. / In the first part of this thesis, motivated by the development of deep brain stimulation for Parkinson's disease, we consider the problem of reducing the synchrony of a neuronal population via a closed-loop electrical stimulation. This, under the constraints that only the mean membrane voltage of the ensemble is measured and that only one stimulation signal is available (mean-field feedback). The neuronal population is modeled as a network of interconnected Landau-Stuart oscillators controlled by a linear single-input single-output feedback device. Based on the associated phase dynamics, we analyze existence and robustness of phase-locked solutions, modeling the pathological state, and derive necessary conditions for an effective desynchronization via mean-field feedback. Sufficient conditions are then derived for two control objectives: neuronal inhibition and desynchronization. Our analysis suggests that, depending on the strength of feedback gain, a proportional mean-field feedback can either block the collective oscillation (neuronal inhibition) or desynchronize the ensemble.In the second part, we explore two possible ways to analyze related problems on more biologically sound models. In the first, the neuronal population is modeled as the interconnection of nonlinear input-output operators and neuronal synchronization is analyzed within a recently developed input-output approach. In the second, excitability and synchronizability properties of neurons are analyzed via the underlying bifurcations. Based on the theory of normal forms, a novel reduced model is derived to capture the behavior of a large class of neurons remaining unexplained in other existing reduced models.