Influence of Large Mean Delay on Distributed Delay Differential Equations Dynamics: Application to a Neural Mass Model

Abstract: Delay differential equations (DDEs) with large delays play a pivotal role in understanding stability and bifurcations in systems ranging from neural networks to laser dynamics. While prior work has extensively studied DDEs with discrete delays, the impact of distributed delays has been less explored. This paper investigates the spectrum of linear DDEs with a uniformly distributed delay kernel, with mean delay $\tau_m$ and a half-width of $\rho$. When $\tau_m\to\infty$, we carry out asymptotic analysis and show that the spectrum splits into (i) a strong critical spectrum referring to a finite set of isolated, pure imaginary eigenvalues that are unaffected by delay, (ii) an asymptotic strong spectrum consisting of a finite set of eigenvalues with limits that are determined by non-delayed terms in the model and (iii) a pseudo-continuous spectrum consisting of infinitely many eigenvalues that limit on the imaginary axis, with real parts that scale linearly with the delay. Although this behavior is similar to the fixed delay case, the distributed delay introduces additional spectral features, including an infinite countable number of horizontal asymptotes in the pseudo-continuous spectrum at frequencies $\omega= k\pi/\rho$, where $k\in \mathbb{Z}\setminus{0}$. We validate our theoretical result through several examples and compare our findings with fixed-delay results from the literature. Finally, we apply the results to study the stability and bifurcations of a Wilson-Cowan model with a delayed self-coupling, large mean delay, and homeostatic plasticity.