Learning to Oscillate: Hebbian Plasticity Imprints Limit Cycles

in Neural Networks

Oscillatory activity is a hallmark of brain dynamics, yet how neural circuits learn to generate rhythmic patterns through biologically plausible mechanisms remains an open question. In this work, we investigate high-dimensional, non-linear dynamical systems—prototypical models of recurrent neural networks—exposed to incoherent periodic inputs and Hebbian-like synaptic plasticity. Our analysis reveals a striking phenomenon: depending on the interplay between the strength of the periodic drive and synaptic plasticity, the system’s phase diagram exhibits a regime where, once both inputs are removed, the network spontaneously settles into a limit cycle . This demonstrates that periodic drives can "imprint" lasting rhythmic patterns into the network’s connectivity, effectively teaching it to oscillate autonomously. Using dynamical mean-field theory (DMFT) and finite-size numerical simulations, we characterize this memorization phase , where the system retains a "memory" of the rhythm. We identify distinct dynamical attractors—chaotic, fixed-point, and limit cycle phases—and show that the limit cycle’s robustness depends on the metabolic time window between stimulation and plasticity halting. This work bridges the gap between driven and autonomous dynamics, offering a theoretically tractable yet biologically inspired mechanism for rhythm learning.