On Linear Difference Equations for Which the Global Periodicity Implies the Existence of an Equilibrium

It is proved that any first-order globally periodic linear inhomogeneous autonomous difference equation defined by a linear operator with closed range in a Banach space has an equilibrium. This result is extended for higher order linear inhomogeneous system in a real or complex Euclidean space. The work was highly motivated by the early works of Smith (1934, 1941) and the papers of Kister (1961) and Bas (2011)

Metastability in a stochastic neural network modeled as a velocity jump Markov process

One of the major challenges in neuroscience is to determine how noise that is present at the molecular and cellular levels affects dynamics and information processing at the macroscopic level of synaptically coupled neuronal populations. Often noise is incorprated into deterministic network models using extrinsic noise sources. An alternative approach is to assume that noise arises intrinsically as a collective population effect, which has led to a master equation formulation of stochastic neural networks. In this paper we extend the master equation formulation by introducing a stochastic model of neural population dynamics in the form of a velocity jump Markov process. The latter has the advantage of keeping track of synaptic processing as well as spiking activity, and reduces to the neural master equation in a particular limit. The population synaptic variables evolve according to piecewise deterministic dynamics, which depends on population spiking activity. The latter is characterised by a set of discrete stochastic variables evolving according to a jump Markov process, with transition rates that depend on the synaptic variables. We consider the particular problem of rare transitions between metastable states of a network operating in a bistable regime in the deterministic limit. Assuming that the synaptic dynamics is much slower than the transitions between discrete spiking states, we use a WKB approximation and singular perturbation theory to determine the mean first passage time to cross the separatrix between the two metastable states. Such an analysis can also be applied to other velocity jump Markov processes, including stochastic voltage-gated ion channels and stochastic gene networks

A Dynamical System Approach to modeling Mental Exploration

The hippocampal-entorhinal complex plays an essential role within the brain in spatial navigation, mapping a spatial path onto a sequence of cells that reaction potentials. During rest or sleep, these sequences are replayed in either reverse or forward temporal order; in some cases, novel sequences occur that may represent paths not yet taken, but connecting contiguous spatial locations. These sequences potentially play a role in the planning of future paths. In particular, mental exploration is needed to discover short-cuts or plan alternative routes. Hopeld proposed a two-dimensional planar attractor network as a substrate for the mental exploration. He extended the concept of a line attractor used for the ocular-motor apparatus, to a planar attractor that can memorize any spatial path and then recall this path in memory. Such a planar attractor contains an infinite number of fixed points for the dynamics, each fixed point corresponding to a spatial location. For symmetric connections in the network, the dynamics generally admits a Lyapunov energy function L. Movement through different fixed points is possible because of the continuous attractor structure. In this model, a key role is played by the evolution of a localized activation of the network, a "bump", that moves across this neural sheet that topographically represents space. For this to occur, the history of paths already taken is imprinted on the synaptic couplings between the neurons. Yet attractor dynamics would seem to preclude the bump from moving; hence, a mechanism that destabilizes the bump is required. The mechanism to destabilize such an activity bump and move it to other locations of the network involves an adaptation current that provides a form of delayed inhibition. Both a spin-glass and a graded-response approach are applied to investigating the dynamics of mental exploration mathematically. Simplifying the neural network proposed by Hopfield to a spin glass, I study the problem of recalling temporal sequences and explore an alternative proposal, that relies on storing the correlation of network activity across time, adding a sequence transition term to the classical instantaneous correlation term during the learning of the synaptic "adaptation current" is interpreted as a local field that can destabilize the equilibrium causing the bump to move. We can also combine the adaptation and transition term to show how the dynamics of exploration is affected. To obtain goal-directed searching, I introduce a weak external field associated with a rewarded location. We show how the bump trajectory then follows a suitable path to get to the target. For networks of graded-response neurons with weak external stimulation, amplitude equations known from pattern formation studies in bio-chemico- physical systems are developed. This allows me to predict the modes of network activity that can be selected by an external stimulus and how these modes evolve. Using perturbation theory and coarse graining, the dynamical equations for the evolution of the system are reduced from many sets of nonlinear integro-dierential equations for each neuron to a single macroscopic equation. This equation, in particular close to the transition to pattern formation, takes the form of the Landau Ginzburg equation. The parameters for the connections between the neurons are shown to be related to the parameters of the Landau-Ginzburg equation that governs the bump of activity. The role of adaptation within this approximation is studied, which leads to the discovery that the macroscopic dynamical equation for the system has the same structure of the coupled equations used to describe the propagation of the electrical activity within one single neuron as given by the Fitzhugh-Nagumo equations

Asymptotic Homogenized SP2 Approximations to the Neutron Transport Equation.

Many current-generation reactor analysis codes use the diffusion approximation to efficiently calculate neutron fluxes. As a result, there is considerable interest in methods that provide a more accurate diffusion solution without significantly increasing computational costs. In this work, an asymptotic analysis, previously used to derive a homogenized diffusion equation for lattice-geometry systems, is generalized to derive a one-dimensional, one-group homogenized SP2 equation as a more accurate alternative to the standard homogenized diffusion equation. This analysis results in new diffusion coefficients and an improved formula for flux reconstruction. The asymptotic SP2 formulation is compared to standard SP2, asymptotic diffusion, and standard diffusion for several test problems. Both the eigenvalue and reconstructed fluxes are examined. In general, the asymptotic equations are more accurate than the standard equations, and SP2 is more accurate than diffusion theory, especially for optically small systems. The calculation of more accurate multigroup cross sections is considered. Standard multigroup cross sections are designed to preserve both the (multigroup) infinite medium neutron spectrum and eigenvalue; this property still holds if the multigroup cross sections are modified by a multiplicative scaling factor. In this thesis, a formula for the scaling factor is derived that makes the modified multigroup cross sections satisfy the asymptotic diffusion or SP2 limit of the neutron transport equation. Numerical simulations demonstrate that the scaled multigroup cross sections yield more accurate results than unscaled cross sections for multigroup eigenvalue problems in finite media. Finally, the asymptotic analysis is then extended to a hypothesized multigroup, spatially homogenized SP2 equation. The hypothesized equation uses standard homogenized cross section definitions, but leaves the diffusion coefficients undefined. The asymptotic analysis of the multigroup SP2 equation results in a monoenergetic SP2 equation, similar to the one obtained for the continuous energy transport equation. By requiring that the hypothesized multigroup SP2 equation have the same asymptotic limit as the continuous energy transport equation, we establish a condition that the additional multigroup diffusion coefficient, D2g, must satisfy. Two logical definitions for D2g are chosen, but numerical results indicate that they are inconsistent in their accuracy, and are frequently outperformed by standard multigroup diffusion and SP2.PhDNuclear Engineering and Radiological SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/116754/1/tgsaller_1.pd

The Dynamics of Shape

This thesis consists of two parts, connected by one central theme: the dynamics of the "shape of space". The first part of the thesis concerns the construction of a theory of gravity dynamically equivalent to general relativity (GR) in 3+1 form (ADM). What is special about this theory is that it does not possess foliation invariance, as does ADM. It replaces that "symmetry" by another: local conformal invariance. In so doing it more accurately reflects a theory of the "shape of space", giving us reason to call it \emph{shape dynamics} (SD). In the first part we will try to present some of the highlights of results so far, and indicate what we can and cannot do with shape dynamics. Because this is a young, rapidly moving field, we have necessarily left out some interesting new results which are not yet in print and were developed alongside the writing of the thesis. The second part of the thesis will develop a gauge theory for "shape of space"--theories. To be more precise, if one admits that the physically relevant observables are given by shape, our descriptions of Nature carry a lot of redundancy, namely absolute local size and absolute spatial position. This redundancy is related to the action of the infinite-dimensional conformal and diffeomorphism groups on the geometry of space. We will show that the action of these groups can be put into a language of infinite-dimensional gauge theory, taking place in the configuration space of 3+1 gravity. In this context gauge connections acquire new and interesting meanings, and can be used as "relational tools".Comment: 137 pages, 7 figures. PhD thesi