Education

Abstract

What do the kick of a soccer ball, the swallowing of a dose of medicine, and the firing of a neuron all have in common? They can all be modeled as an impulsive forcing—essentially, a change in a system variable on a timescale that is rapid compared to the evolution of the system. Mathematically, this leads to the idea of the Dirac $\delta$-function, often found as a forcing term in a differential equation and commonly introduced in mechanics, systems engineering, and differential equations courses. Our first paper this issue, “On Spiking Models for Synaptic Activity and Impulsive Differential Equations,” examines an ambiguity that arises in these models. If the amplitude of the $\delta$-function forcing depends upon the variable whose value is jumping (i.e., how hard you can kick the soccer ball depends on how fast it is moving), there are several plausible interpretations which yield different solutions. The authors demonstrate that an idea from the theory of distributions, namely, representing the $\delta$-function as a limit of a sequence of functions, resolves this conundrum. They then apply these ideas to a model of the transmission of an electrochemical signal between two neurons. This paper nicely draws together a variety of subjects, yet is accessible to students who have taken courses in differential equations and a modest amount of analysis. It would be a natural starting point for a supplemental lecture or project in a course on dynamical systems or asymptotic analysis. It also demonstrates the growing role of mathematical models in the neurosciences. Our second paper, “Unstable Solutions of Nonautonomous Linear Differential Equations,” poses a paradox. Consider the linear first-order system $x'=Ax$ where A has constant negative eigenvalues. Can the solutions for $x(t)$ grow without bound? The surprising answer is yes, if the coefficient matrix A is time-dependent. The authors provide a geometrical interpretation of this result by first showing that in some sectors of the plane the solution's amplitude grows even if A has negative eigenvalues. In the time-dependent problem these growing sectors can align with the solution, creating a net amplification. An animation of this phenomena is available on the web, which clearly shows how this counterintuitive result can be achieved. The examples provided here are a natural bridge from the study of autonomous (i.e., constant coefficient) linear systems of ODEs to their nonautonomous cousins and could easily make an appearance toward the end of most differential equations courses.