Abstract
We investigate the dynamics of unidirectional semi-infinite chains of type-I oscillators that are periodically forced at their root node, as an archetype of wave generation in neural networks. In previous studies, numerical simulations based on uniform forcing have revealed that trajectories approach a traveling wave in the far-downstream, large time limit. While this phenomenon seems typical, it is hardly anticipated because the system does not exhibit any of the crucial properties employed in available proofs of existence of traveling waves in lattice dynamical systems. Here, we give a full mathematical proof of generation under uniform forcing in a simple piecewise affine setting for which the dynamics can be solved explicitly. In particular, our analysis proves existence, global stability, and robustness with respect to perturbations of the forcing, of families of waves with arbitrary period/wave number in some range, for every value of the parameters in the system.
Highlights
Signal propagation in the form of waves is a ubiquitous feature of the functioning of neural networks
For simple feed-forward chains of type-I oscillators, our analysis proved that periodic wave trains can be generated from arbitrary initial condition, even when the root node is forced using an unrelated signal
These stable waves exist for an open interval of wave number and period
Summary
Signal propagation in the form of waves is a ubiquitous feature of the functioning of neural networks. In brief terms, traveling waves are typically observed in the far-downstream, large time limit; see Fig. 1 Speaking, this means that, letting {θs(f )(t)}s∈N denote the solution to (1) with forcing f (and say, typical initial condition), there exist a periodic function f0 and a time shift α ∈ R+ such that we have lim s→+∞. We prove the existence of TW with arbitrary period in some interval, and their global stability with respect to initial perturbations in the phase space TN, when the forcing at s = 0 is chosen to be a TW shape and for an open set of periodic signals with identical period This open set is shown to contain uniform forcing f (t) = t/τ provided that the coupling intensity is sufficiently small. Solutions of Eq (2) are in general denoted by {θs(f )(t)}s∈N but the notation {θs(f )(θ1, . . . , θs, t)}s∈N is employed when the dependence on initial condition needs to be explicitly mentioned
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