The FitzHugh-Nagumo equations have been used as a caricature of the Hodgkin-Huxley equations of neuron firing and to capture, qualitatively, the general properties of an excitable membrane. In this paper, we utilize a modified version of the FitzHugh-Nagumo equations to model the spatial propagation of neuron firing; we assume that this propagation is (at least, partially) caused by the cross-d...

In this paper we derive a lattice model with infinite distributed delay to describe the growth of a single-species population in a 2D patchy environment with infinite number of patches connected locally by diffusion and global interaction. We consider the existence of traveling wave solutions when the birth rate is large enough that each patch can sustain a positive equilibrium. When the birth ...

This paper builds on the past study of single-spike waves in one-dimensional integrate-and-fire networks to provide a framework for the study of waves with arbitrary (finite or countably infinite) collections of spike times. Based on this framework, we prove an existence theorem for single-spike traveling waves, and we combine analysis and numerics to study two-spike traveling waves, periodic t...

The Quota Traveling Salesman Problem is a generalization of the well known Traveling Salesman Problem. The goal of the traveling salesman is, in this case, to reach a given quota of the sales, minimizing the amount of time. In this paper we address the on-line version of the problem, where requests are given over time. We present algorithms for various metric spaces, and analyze their performan...

This paper focuses on regular traveling waves, or waves that propagate smoothly, in a one-dimensional network of theta neurons. We show that when coupling strength is sufficiently large, there exist two traveling wave solutions for this network. Moreover, the cells in the network spike more than one time after joining the wave; that is, some form of synaptic depression or other adaptive mechani...

We consider traveling front and pulse solutions to a system of integro-differential equations used to describe the activity of synaptically coupled neuronal networks in a single spatial dimension. Our first goal is to establish a series of direct links between the abstract nature of the equations and their interpretation in terms of experimental findings in the cortex and other brain regions. T...

Abstract. In this paper, we deal with the existence, asymptotic behavior and uniqueness of travelingwaves for nonlocal diffusion systems with delay and global response. We first obtain the existence of traveling wave front by using upperlower solutions method and Schauder’s fixed point theorem for c > c∗ and using a limiting argument for c = c∗. Secondly, we find a priori asymptotic behavior of...

This paper is devoted to the mathematical analysis of a reaction and diffusion model for Lyme disease. In the case of a bounded spatial habitat, we obtain the global stability of either disease-free or endemic steady state in terms of the basic reproduction number R₀. In the case of an unbounded spatial habitat, we establish the existence of the spreading speed of the disease and its coincidenc...

Abstract. We study flow-induced enhancement of the speed of pulsating traveling fronts for reaction-diffusion equations, and quenching of reaction by fluid flows. We prove, for periodic flows in two dimensions and any combustion-type reaction, that the front speed is proportional to the square root of the (homogenized) effective diffusivity of the flow. We show that this result does not hold in...

The wheels of planetary rovers typically have grousers (i.e., lugs, cleats) on their surface to provide grip for climbing in loose soil. However, from the viewpoint of designing wheels, there exist few theoretical methods or models for determining the grouser interval for a wheel. The goal of this study is to provide a fundamental guideline for determining the grouser interval on a wheel. To th...