Hopf Bifurcation in an Oscillatory-Excitatory Reaction-Diffusion Model with Spatial heterogeneity

We focus on the qualitative analysis of a reaction-diffusion with spatial heterogeneity. The system is a generalization of the well known FitzHugh-Nagumo system in which the excitability parameter is space dependent. This heterogeneity allows to exhibit concomitant stationary and oscillatory phenomena. We prove the existence of an Hopf bifurcation and determine an equation of the center-manifold in which the solution asymptotically evolves. Numerical simulations illustrate the phenomenon

Weakly coupled two slow- two fast systems, folded node and mixed mode oscillationsM

We study Mixed Mode Oscillations (MMOs) in systems of two weakly coupled slow/fast oscillators. We focus on the existence and properties of a folded singularity called FSN II that allows the emergence of MMOs in the presence of a suitable global return mechanism. As FSN II corresponds to a transcritical bifurcation for a desingularized reduced system, we prove that, under certain non-degeneracy conditions, such a transcritical bifurcation exists. We then apply this result to the case of two coupled systems of FitzHugh- Nagumo type. This leads to a non trivial condition on the coupling that enables the existence of MMOs

Beyond the brain: towards a mathematical modeling of emotions

Emotions are a central key for understanding human beings and of fundamental importance regarding their impact in human and animal behaviors. They have been for a long time a subject of study for various scholars including in particular philosophers and mystics. In modern science, the emotional phenomenon has attracted for a few decades an increasing number of studies, notably in the fields of Psychology, Psychiatry, Neuroscience and Biochemistry. However, since our perception of emotions is not, so far, directly detectable nor recordable by our measure instruments, Physics and Mathematics have not been so far used academically to provide a precise description of the phenomenon of feeling an emotion. Relying upon the works of O. Elahi and on the hypothesis that the human soul and its psyche may manifest in ourselves (in both conscious and unconscious manner) in an analog way as electromagnetic waves, we propose here a few mathematical descriptions consistent with the human personal experience, of the feeling and cognition of emotions. As far as we know, such a mathematical description has never been provided before. It allows a quantitative (intensity) and qualitative (nature of feelings/frequency) of the emotional phenomenon which provides a novel scientific approach of the nature of the mind, complementary to the on going research of physiological manifestation of emotions. We anticipate such an approach and the associated mathematical modeling to become an important tool to describe emotions and their subsequent behavior. In complement of the modeling of oscillations and brain dynamics, it provides a fruitful direction of research with potentially broad and deep impacts in both applied mathematics, physics, cognitive and behavioral sciences

An infrared origin of leptonic mixing and its test at DeepCore

Fermion mixing is generally believed to be a low-energy manifestation of an underlying theory whose energy scale is much larger than the electroweak scale. In this paper we investigate the possibility that the parameters describing lepton mixing actually arise from the low-energy behavior of the neutrino interacting fields. In particular, we conjecture that the measured value of the mixing angles for a given process depends on the number of unobservable flavor states at the energy of the process. We provide a covariant implementation of such conjecture, draw its consequences in a two neutrino family approximation and compare these findings with current experimental data. Finally we show that this infrared origin of mixing will be manifest at the Ice Cube DeepCore array, which measures atmospheric oscillations at energies much larger than the tau lepton mass; it will hence be experimentally tested in a short time scale.Comment: 14 pages, 1 figure; version to appear in Int.J.Mod.Phys.

Generalized Wasserstein distance and its application to transport equations with source

In this article, we generalize the Wasserstein distance to measures with different masses. We study the properties of such distance. In particular, we show that it metrizes weak convergence for tight sequences. We use this generalized Wasserstein distance to study a transport equation with source, in which both the vector field and the source depend on the measure itself. We prove existence and uniqueness of the solution to the Cauchy problem when the vector field and the source are Lipschitzian with respect to the generalized Wasserstein distance

A strong form of the Quantitative Isoperimetric inequality

We give a refinement of the quantitative isoperimetric inequality. We prove that the isoperimetric gap controls not only the Fraenkel asymmetry but also the oscillation of the boundary

Large Deviations in Stochastic Heat-Conduction Processes Provide a Gradient-Flow Structure for Heat Conduction

We consider three one-dimensional continuous-time Markov processes on a lattice, each of which models the conduction of heat: the family of Brownian Energy Processes with parameter $m$, a Generalized Brownian Energy Process, and the Kipnis-Marchioro-Presutti process. The hydrodynamic limit of each of these three processes is a parabolic equation, the linear heat equation in the case of the BEP$(m)$ and the KMP, and a nonlinear heat equation for the GBEP($a$). We prove the hydrodynamic limit rigorously for the BEP$(m)$, and give a formal derivation for the GBEP($a$). We then formally derive the pathwise large-deviation rate functional for the empirical measure of the three processes. These rate functionals imply gradient-flow structures for the limiting linear and nonlinear heat equations. We contrast these gradient-flow structures with those for processes describing the diffusion of mass, most importantly the class of Wasserstein gradient-flow systems. The linear and nonlinear heat-equation gradient-flow structures are each driven by entropy terms of the form $-\log \rho$; they involve dissipation or mobility terms of order $\rho^2$ for the linear heat equation, and a nonlinear function of $\rho$ for the nonlinear heat equation.Comment: 29 page

Existence and approximation of probability measure solutions to models of collective behaviors

In this paper we consider first order differential models of collective behaviors of groups of agents based on the mass conservation equation. Models are formulated taking the spatial distribution of the agents as the main unknown, expressed in terms of a probability measure evolving in time. We develop an existence and approximation theory of the solutions to such models and we show that some recently proposed models of crowd and swarm dynamics fit our theoretic paradigm.Comment: 31 pages, 1 figur