Equivalence of stationary dynamical solutions in a directed chain and a Delay Differential Equation of neuroscientific relevance

While synchronized states, and the dynamical pathways through which they emerge, are often regarded as the paradigm to understand the dynamics of information spreading on undirected networks of nonlinear dynamical systems, when we consider directed network architectures, dynamical stationary states can arise. To study this phenomenon we consider the simplest directed network, a single cycle, and excitable FitzHugh–Nagumo (FHN) neurons. We show numerically that a stationary dynamical state emerges in the form of a self-sustained travelling wave, through a saddle-point bifurcation of limit cycles that does not destabilize the global fixed point of the system. We then formulate an effective model for the dynamical steady state of the cycle in terms of a single-neuron Delay Differential Equation (DDE) featuring an explicitly delayed feedback, demonstrating numerically the possibility of mapping stationary solutions between the two models. The DDE based model is shown to reproduce the entire bifurcation, which also in this case does not destabilize the global fixed point, even though global properties differ in general between the systems. The discrete nature of the cycle graph is revealed as the origin of these coordinated states by the parametric analysis of solutions, and the DDE effective model is shown to preserve this feature accurately. Finally, the scaling of the intersite propagation times hints to a solitary wave nature of the dynamical steady state in the limit of large chain size.