We consider infinite systems of ODE's on the two-dimensional integer lattice, given by a bistable scalar ODE at each point, with a nearest neighbor coupling between lattice points. For a class of ideal nonlinearities, we obtain traveling wave solutions in each direction e i`, and we explore the relation between the wave speed c, the angle `, and the detuning parameter a of the nonlinearity. Of particular interest is the phenomenon of "propagation failure, " and we study how the critical value a = a

We obtain existence of traveling wave solutions for a class of spatially discrete systems, namely lattice differential equations. Uniqueness of the wave speed c, and uniqueness of the solution with c 6= 0, are also shown. More generally, the global structure of the set of all traveling wave solutions is shown to be a smooth manifold where c 6= 0. Convergence results for solutions are obtained at the singular perturbation limit c ! 0. 1 Introduction We are interested in lattice differential equations, namely infinite systems of ordinary differential equations indexed by points on a spatial lattice, such as the D-dimensional integer lattice Z D . Our focus in this paper is the global structure of the set of traveling wave solutions for such systems. This entails results on existence and uniqueness, and on continuous (or smooth) dependence of traveling waves and their speeds on parameters, as well as some delicate convergence results in the singular perturbation case c ! 0 of the wav...

A VLSI architecture is proposed for the realization of real-time two-dimensional (2-D) image filtering in an addressevent -representation (AER) vision system. The architecture is capable of implementing any convolutional kernel F (x; y) as long as it is decomposable into x-axis and y-axis components, i.e., F (x; y)=H(x)V (y), for some rotated coordinate system fx; yg and if this product can be approximated safely by a signed minimum operation. The proposed architecture is intended to be used in a complete vision system, known as the boundary contour system and feature contour system (BCS-FCS) vision model, proposed by Grossberg and collaborators. The present paper proposes the architecture, provides a circuit implementation using MOS transistors operated in weak inversion, and shows behavioral simulation results at the system level operation and some electrical simulations. Index Terms---Analog integrated circuits, communication systems, convolution circuits, Gabor filters, image anal...

We consider an array of scalar nonlinear dynamical systems u = \Gammaf (u), arranged on the sites of a spatial lattice, for example on the integer lattice ZZ 2 in the plane IR 2 . We impose a coupling between nearest neighbors, and also between next-nearest neighbors, in the form of discrete Laplacians with +- and \Theta-shaped stencils. These couplings can be of any strength, and of either sign (positive or negative), and the resulting infinite systems of ODE's need not be near a PDE continuum limit. We study stable equilibria for such systems, from the point of view of pattern formation and spatial chaos, where these terms mean that the spatial entropy of the set of stable equilibria is zero, respectively, positive. In particular, for an idealized class of nonlinearities f corresponding to a "double obstacle" at u = \Sigma1 with f(u) = flu in between, it is natural to consider "mosaic solutions," namely equilibria which assume only the values u i;j 2 f\Gamma1; 0; 1g at each (i; ...

In this paper, we study the existence and stability of traveling waves in lattice dynamical systems, in particular, in lattice ordinary differential equations (lattice ODE's) and in coupled map lattices (CML's). Instead of employing the moving coordinate approach as for partial differential equations, we construct a local coordinate system around a traveling wave solution of a lattice ODE, analogous to the local coordinate system around a periodic solution of an ODE. In this coordinate system the lattice ODE becomes a nonautonomous periodic differential equation, and the traveling wave corresponds to a periodic solution of this equation. We prove the asymptotic stability with asymptotic phase shift of the traveling wave solution under appropriate spectral conditions. We also show the existence of traveling waves in CML's which arise as time-discretizations of lattice ODE's. Finally, we show that these results apply to the discrete Nagumo equation. 1 Introduction This paper is concer...

We survey a class of continuous-time lattice dynamical systems, with an idealized nonlinear. We introduce a class of equilibria called mosaic solutions, which are composed of the elements 1, \Gamma1, and 0, placed at each lattice point. A stability criterion for such solutions is given. The spatial entropy h of the set of all such stable solutions is defined, and we study how this quantity varies with parameters. Systems are qualitatively distinguished according to whether h = 0 (termed pattern formatio), or h ? 0 (termed spatial chaos). Numerical techniques for calculating h are described. 1. Mosaic Solutions As described in the companion paper [4], we study the phenomenon of pattern formation and spatial chaos in lattice dynamical systems. In order for us to see these phenomena globally, we consider a special class of equilibrium solutions, called mosaic solutions, introduced in [5], and studied there and in [6]. We work here with the system (1:1) u i;j = \Gammafi + \Delta + ...

In order to take advantage of the massive parallelism offered by artificial neural networks, hardware implementations are essential. However, most standard neural network models are not very suitable for implementation in hardware and adaptations are needed. In this section an overview is given of the various issues that are encountered when mapping an ideal neural network model onto a compact and reliable neural network hardware implementation, like quantization, handling non-uniformities and non-ideal responses, and restraining computational complexity. Furthermore, a broad range of hardware-friendly learning rules is presented, which allow for simpler and more reliable hardware implementations. The relevance of these neural network adaptations to hardware is illustrated by their application in existing hardware implementations.

We consider traveling wave solutions to a class of differential-difference equations. Our interest is in understanding propagation failure, directional dependence due to the discrete Laplacian, and the relationship between traveling wave solutions of the spatially continuous and spatially discrete limits of this equation. The differential-difference equations that we study include damped and undamped nonlinear wave and reaction-diffusion equations as well as their spatially discrete counterparts. Both analytical and numerical results are given.

. In this paper recent work on the dynamics of lattice differential equations is surveyed. In particular, results on propagation failure and lattice induced anisotropy for traveling wave or plane wave solutions in higher space dimensions spatially discrete bistable reaction-diffusion systems are considered. In addition, analysis of and spatial chaos in the equilibrium states of spatially discrete reaction-diffusion systems are discussed. Key words. lattice differential equations, traveling wave solutions, propogation failure, lattice anisotropy, equilibrium solutions, stability, spatial entropy Abbreviated title. Lattice Differential Equations 1 School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332 USA. The work of this author was supported in part by ARO Contract DAAH04-93-G-0199 and by NSF Grant DMS-9005420. 2 Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912 USA. The work of this author was supported in part by NSF Grant D...

In this paper we present the agent architecture development environment ADE, intended for the design, implementation, and testing of distributed robotic agent architectures. ADE is unique among robotic architecture development environments in that it is based on a universal agent architecture framework called APOC, which allows it to implement architectures in any design methodology, and in that it uses an underlying multi-agent system to allow for the the distribution of architectural components over multiple host computers. After a short exposition of the theory behind ADE, we present the multi-agent system setup and give an example of using ADE in a multi-robot setting. A general discussion then highlights some of the novel features of ADE and illustrates how ADE can be used for designing, implementing, testing, and running agent architectures.