Komparativ analyse av heterogene og homogene nevrale feltmodeller

The present thesis is devoted to the comparative analysis of heterogeneous and homogeneous neural field models. The motivation for this work stems from the fact that there is considerable interest in processes in neural tissue, which can underlie both natural and pathological neurobiological phenomena (e.g., orientation tuning in primary visual cortex, short term working memory, control of head direction, motion perception, visual hallucinations and EEG rhythms). The main aim of this thesis is to investigate the outcome of the analysis of a heterogeneous neural field model and its homogeneous counterpart. Another goal is to get more realistic dynamical models for the brain function which takes into account microscopic effects. Mathematically, this approach is formulated in terms of (a system of) nonlinear integro-differential equations. These models describe nonlinear interactions between neuron populations. They are used as a starting point to study traveling wave fronts, localized stationary solutions (bumps) and pattern formation. The first part of the thesis consists of the introduction. Here we first give a short review of the neurophysical background. Secondly, we introduce the key mathematical objects of the present thesis, namely a neural field model of the Amari type and a 2-population homogenized neural field model. We also review the basic ideas of homogenization theory and the two-scale convergence method. Then we summarize the results and give ideas for future works. The second part of the thesis consists of three papers. Paper I deals with the existence and linear stability of stationary periodic bump solutions to a neural field model of the Amari type. In Paper II and III we focus on 2-population homogenized neural field models where the cortical microstructure is taken into account in the connectivity strength. We study the existence and stability of localized stationary single bump solutions (Paper II). In Paper III we investigate pattern forming processes in the same neural field model. The key methods in the present study are a pinning function technique for the existence of bumps, spectral methods and properties, block diagonalization and the Fourier decomposition method in the stability assessment and numerical simulations. We believe that the present thesis contributes to the understanding of the brain functions, both in normal and pathological cases.I denne avhandlingen utføres en komparativ analyse av heterogene og homogene nevrale nettverksmodeller. Motivasjonen for dette arbeidet er interessen for prosesser i hjernebarken, som kan være grunnlag for både naturlige og patologiske nevrobiologiske fenomener (for eksempel i orienteringsinnstilling i den primære visuelle hjernebarken, korttidsminne, kontroll av hoderetning, bevegelsesoppfattelse, visuelle hallusinasjoner og EEG-rytmer). Hovedformålet med denne avhandlingen er å analysere en heterogen nevral nettverksmodell og dens homogene motstykke. Et annet mål er å få mer realistiske dynamiske modeller for hjernefunksjonen, som tar hensyn til mikroskopiske effekter. Disse modellene er gitt som (et system av) ikke-lineære integro-differensiallikninger. Disse modellene beskriver ikke-lineære interaksjoner mellom nevronpopulasjoner. De brukes som utgangspunkt for å studere bølgeforplantning, lokaliserte stasjonære løsninger (bumps) og mønsterdannelse. I introduksjonen presenterer vi en oversikt over den nevrofysiologiske bakgrunnen. Videre introduserer vi de matematiske modellene som er sentrale i denne avhandlingen, det vil si en nevral nettverksmodell av Amari- typen og en homogenisert 2-populasjon nevral nettverksmodell. Vi gjennomgår også grunnbegrepene i homogeniseringsteori og toskala konvergensmetoden. Deretter oppsummerer vi resultatene og fremlegger ideer for videre arbeid. Den andre delen av denne avhandlingen består av tre artikler. Artikkel I omhandler eksistensen og den lineære stabiliteten til stasjonære periodiske bump-løsninger i en nettverksmodell av Amari-typen. I artikkel II og III fokuserer vi på en homogenisert 2-populasjons nevral nettverksmodell, hvor mikrostrukturen i hjernebarken tas med i beregningen av konnektivitetsstyrken. Vi undersøker eksistensen og stabiliteten til lokaliserte stasjonære bump-løsninger (artikkel II). I artikkel III studerer vi mønsterdannende prosesser i den samme nevrale nettverksmodellen. De sentrale metodene i denne studien er en pinning-funksjonsteknikk for eksistens av bumps. Stabilitetsanalysen er gjennomført ved hjelp av spektral metoder , blokk diagonalisering og Fouriertransformasjon og numeriske simuleringer. Vi mener at denne avhandlingen bidrar til forståelsen av hjernens funksjoner, både under normale og patologiske omstendigheter

Pinning of interfaces in random media

For a model for the propagation of a curvature sensitive interface in a time independent random medium, as well as for a linearized version which is commonly referred to as Quenched Edwards-Wilkinson equation, we prove existence of a stationary positive supersolution at non-vanishing applied load. This leads to the emergence of a hysteresis that does not vanish for slow loading, even though the local evolution law is viscous (in particular, the velocity of the interface in the model is linear in the driving force).Comment: 15 Page

Dynamics of the Desai-Zwanzig model in multiwell and random energy landscapes

We analyze a variant of the Desai-Zwanzig model [J. Stat. Phys. {\bf 19}1-24 (1978)]. In particular, we study stationary states of the mean field limit for a system of weakly interacting diffusions moving in a multi-well potential energy landscape, coupled via a Curie-Weiss type (quadratic) interaction potential. The location and depth of the local minima of the potential are either deterministic or random. We characterize the structure and nature of bifurcations and phase transitions for this system, by means of extensive numerical simulations and of analytical calculations for an explicitly solvable model. Our numerical experiments are based on Monte Carlo simulations, the numerical solution of the time-dependent nonlinear Fokker-Planck (McKean-Vlasov equation), the minimization of the free energy functional and a continuation algorithm for the stationary solutions

Guided transition waves in multistable mechanical metamaterials

Transition fronts, moving through solids and fluids in the form of propagating domain or phase boundaries, have recently been mimicked at the structural level in bistable architectures. What has been limited to simple one-dimensional (1D) examples is here cast into a blueprint for higher dimensions, demonstrated through 2D experiments and described by a continuum mechanical model that draws inspiration from phase transition theory in crystalline solids. Unlike materials, the presented structural analogs admit precise control of the transition wave’s direction, shape, and velocity through spatially tailoring the underlying periodic network architecture (locally varying the shape or stiffness of the fundamental building blocks, and exploiting interactions of transition fronts with lattice defects such as point defects and free surfaces). The outcome is a predictable and programmable strongly nonlinear metamaterial motion with potential for, for example, propulsion in soft robotics, morphing surfaces, reconfigurable devices, mechanical logic, and controlled energy absorption

Dynamics of Patterns

This workshop focused on the dynamics of nonlinear waves and spatio-temporal patterns, which arise in functional and partial differential equations. Among the outstanding problems in this area are the dynamical selection of patterns, gaining a theoretical understanding of transient dynamics, the nonlinear stability of patterns in unbounded domains, and the development of efficient numerical techniques to capture specific dynamical effects