The Asymptotic Shapes of Periodic Solutions of a Singular Delay Differential System

Let f(•, λ): → be given so that f(0, λ)=0 and f(x, λ)=(1+λ) x+ax2+bx3+o(x3) as x→0. We characterize those small values of >0 and λ for which there are periodic solutions of periods approximately with k of the following system arising from a network of neurons[formula] The periodic solutions are synchronized if k is even and phase-locked if k is odd. We show that, as →0, these periodic solutions approach square waves if a=0 and b<0, and pulses if a=0 and b>0 or if a≠0.