Stability and Long Term Behavior of a Hebbian Network of Kuramoto Oscillators

We investigate the limit sets of a network of coupled Kuramoto oscillators with a coupling matrix determined by a Hebb rule. These limit sets are the output of the network if used for the recognition of a defective binary pattern out of several given patterns, with the output pattern encoded in the oscillators’ phases. We show that if all pairs of given patterns have maximum Hamming distance, there exists a degenerate attractive limit set that contains the steady states corresponding to each of the given patterns. As a result, switching between output patterns occurs for arbitrarily small modifications of the dynamics (for example, due to frequency inaccuracies). Even if the maximum Hamming distance constraint is dropped, numerical results suggest that the structural instability of the vector field persists. We conclude that the unique interchangeability of output patterns in Hebbian networks of Kuramoto oscillators, while sacrificing robustness, makes these networks more flexible than similar neural networks with separated, attractive output states.