The topic of this dissertation is the study of the emergence of higher-order correlations in recurrentlyconnected populations of brain cells.Neurons have been experimentally shown to form vast networks in the brain. In these networks, eachbrain cell communicates with tens of thousands of its neighbors by sending out and receiving electricalsignals, known as action potentials or spikes. The effect of a single action potential can propagate throughthe network and cause additional spikes to be generated. Thus, the connectivity of the neuronal networkgreatly influences the network's spiking dynamics. However, while the methods of action potentialgeneration are very well studied, many dynamical features of neuronal networks are still only vaguelyunderstood.The reasons for this mostly have to do with the difficulties of keeping track of the collective, non-linearbehavior of hundreds of millions of brain cells. Even when one focuses on small groups of neurons, all butthe most trivial questions about coordinated activity remain unanswered, due to the combinatorialexplosion that arises in all questions of this sort. In theoretical neuroscience one often needs to resort tomathematical models that try to explain the most important dynamical phenomena while abstractingaway many of the morphological features of real neurons.On the other hand, advances in experimental methods are making simultaneous recording of largeneuronal populations possible. Datasets consisting of collective spike trains of thousands of neurons arebecoming available. With these new developments comes the possibility of finally understanding the wayin which connectivity gives rise to the many interesting dynamical aspects of spiking networks.The main research question, addressed in this thesis, is how connectivity between neurons influences thedegree of synchrony between their respective spike trains. Using a linear model of spiking neurondynamics, we show that there is a mathematical relationship between the network's connectivity and theso-called higher-order cumulants, which quantify beyond-chance-level coordinated activity of groups ofneurons. Our equations describe the specific connectivity patterns that give rise to higher-ordercorrelations. In addition, we explore the special case of correlations of third-order and find that, in large,regular networks, it is the presence of a single subtree that is responsible for third-order synchrony.In summary, the results presented in this dissertation advance our understanding of how higher-ordercorrelations between spike trains of neurons are affected by certain patterns in synaptic connectivity.Our hope is that a better understanding of such complicated neuronal dynamics can lead to a consistenttheory of the network's functional properties. / <p>QC 20161003</p>

Thesis--University of Wisconsin--Madison. / Typescript. Vita. Description based on print version record. Includes bibliographical references (leaves 128-129).

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It is well known that very few models of interacting systems particularly those in dimension higher than two, can be solved exactly. The mean-field treatment is the first step in approximate calculations for such models. Although mean-field approximation leads to sufficiently accurate results of the thermodynamic properties of these systems away from critical points, most often it fails miserably close to the critical points. In this thesis, we propose to study direct methods (not based on any mean-field type approximations) for proving the exponential decay of the two point-correlation functions and the analyticity of the pressure (free energy per unit volume) for models of Kac type. The methods are based on the Helffer-Sjöstrand formula for the covariance in terms of Witten's Laplacians.

Thesis (Ph. D. in Physics)--Vanderbilt University, Aug. 2008. / Title from title screen. Includes bibliographical references.

Thesis (Ph. D.)--University of Texas at Austin, 2006. / Vita. Includes bibliographical references.