William C. Troy

Symmetry properties in systems of semilinear elliptic equations

Abstract We investigate symmetry properties of solutions of systems of semilinear elliptic equations. The two main tools we use consist of the maximum principle and the device of moving parallel planes to a critical position and then showing that the solution is symmetric about the limiting plane. An application to the Belousov-Zhabotinskii chemical reaction is given.

EXACT HOMOCLINIC AND HETEROCLINIC SOLUTIONS OF THE GRAY-SCOTT MODEL FOR AUTOCATALYSIS*

In this paper we obtain explicit nontrivial stationary patterns in the one-dimensional Gray--Scott model for cubic autocatalysis. Involved in the reaction are two chemicals, A and B, whose diffusion coefficients are denoted by DA and DB, respectively. The chemical A is fed into the system at a rate kf, reacts with the catalyst B at a rate k1, and the catalyst decays at a rate k2 .If these parameters obey the relation (*) kf /DA = k2 /DB, then, for appropriate values of the rate constants, we present explicit expressions for two families of pulses and one kink. We also show that if (*) is only satisfied approximately, these families of pulses are preserved, and there exists a smooth branch of kinks leading from the explicit one obtained when (*) is satisfied. We determine the local behavior of this branch near the explicit kink.

Two-bump solutions of Amari-type models of neuronal pattern formation

Abstract We study a partial integro-differential equation defined on a spatially extended domain that arises in the modeling of pattern formation in neuronal networks. For a one-dimensional domain we develop criteria for the existence and stability of equal-width “2-bump” solutions under the assumption that the firing rate function is the Heaviside function. We apply these criteria to an example for which the connectivity is of lateral inhibition type (i.e. the coupling function has one positive zero) and find that families of 2-bump solutions exist, but none of the solutions are stable. Extensive numerical searches suggest that this is true for all coupling functions of this form. However, for a large class of coupling functions which have three positive zeros, we find the coexistence of both stable and unstable 2-bump solutions. We also extend our investigation to two spatial dimensions and give numerical evidence for the coexistence of 1-bump and 2-bump solutions. Our results imply that lateral inhibition type coupling is not sufficient to produce stable patterns that are more complex than single isolated patches of high activity.

Global branches of multi-bump periodic solutions of the Swift-Hohenberg equation

Abstract: In this paper we study the existence of single- and multi-bump periodic solutions of a class of fourth order ordinary differential equations arising in problems of pattern formation. Measuring the tendency to form patterns by a parameter q∈ℝ, we view the problem as a nonlinear eigenvalue problem. With the use of analytical as well as numerical methods, branches of periodic solutions are investigated, both locally and globally.

Solutions of third-order differential equations relevant to draining and coating flows

The behavior of solutions of two differential equations is investigated. The first is a model for a viscous fluid draining over a wet surface. The second equation is derived from the first as a result of an inner expansion as a parameter

Radial solutions of Δu + f(u) = 0 with prescribed numbers of zeros

\delta

Sustained Spatial Patterns of Activity in Neuronal Populations without Recurrent Excitation

tends to zero. The existence of the appropriate solution is proved for both equations.

The existence of bounded solutions of a semilinear heat equation

u’(0) = 0, u(r) --f 0 as r-+03. (1.2) Here n > 1 is a real parameter, and the function f satisfies the following hypotheses: (f 1) f: R + R is locally Lipschitz continuous, (f2) uf(u) 0 and j?’ < 0 such that F(u) < 0 on (0, PI, f(u) ’ 0 on UC ~0) F(u) <0 on (B’. 01, f(u) < 0 on !-a, PI, where F(u) = f;r f(s) ds, (f4) f(u)=k(u) jujP-’ u+g(u), where

The existence of traveling wave front solutions of a model of the Belousov-Zhabotinskii chemical reaction☆

Spatial patterns of neuronal activity arise in a variety of experimental studies. Previous theoretical work has demonstrated that a synaptic architecture featuring recurrent excitation and long-range inhibition can support sustained, spatially patterned solutions in integrodifferential equation models for activity in neuronal populations. However, this architecture is absent in some areas of the brain where persistent activity patterns are observed. Here we show that sustained, spatially localized activity patterns, or bumps, can exist and be linearly stable in neuronal population models without recurrent excitation. These models support at most one bump for each background input level, in contrast to the pairs of bumps found with recurrent excitation. We explore the shape of this bump as well as the mechanisms by which this bump is born and destroyed as background input level changes. Further, we introduce spatial inhomogeneity in coupling and show that this induces bump pinning: for a given starting pos...

Bifurcation phenomena in FitzHugh's nerve conduction equations

We investigate the existence of bounded solutions of