A continuous-time RNN trained on a modulo-3 arithmetic task can learn a low-dimensional limit-cycle phase code whose input-induced phase shifts implement the computation (one solution among several)

- Murray trains continuous-time RNNs (100 recurrent units, hyperparameter grid over tanh/ReLU, tau and noise, 90 networks) on a modulo-3 arithmetic task (three integer pulses; output +1 if the sum is congruent to 0 mod 3) with no oscillation-promoting bias, and in a subset of networks finds a phase-code solution: PCA trajectories evolve along a limit cycle, each integer pulse shifts the phase by a nearly constant amount (phase response curves), and trial endpoints cluster into three regions matching the residue classes. Other networks learn qualitatively different solutions (hexagon and lattice arrangements of fixed-point attractors, a star-shaped fast oscillation, a line of attractors), so the limit cycle is one outcome rather than universal. The phase-code example is a single network from a toy task; the abstract-level and v1 description of the arXiv paper differ (v1, titled "Recurrent networks recognize patterns with low-dimensional oscillations", used a SET-card-inspired task with a tanh network, tau = 10 ms, and a handcrafted oscillator model; the current version does not state which hyperparameters gave the phase-code network). [murray-2023-phase-codes-emerge-in-recurrent-neural-networks-optimized-for-modular-arithmetic]