Network topology determines neuronal phase-of-firing codes

Brain rhythms organize neural activity in time. When a shared oscillation is broadcast across a population of neurons, the phase at which the neurons fire with respect to the oscillation can itself carry information, a strategy the brain appears to use for navigation, memory, and sensory perception. In such a phase-of-firing code, information is conveyed not by how rapidly a neuron fires but by when it fires relative to the phase of the ongoing rhythm. Models that implement such phase-of-firing codes consider the neurons as independent. How such codes arise in cortical and hippocampal circuits, where neurons are densely interconnected rather than independent, remains unclear. Here we show that recurrent connectivity does not merely permit phase-of-firing codes but actively determines which patterns the network can hold onto. Using networks of model neurons wired in a manner determined by the constraints of a Sudoku puzzle, we find that the network supports an enormous repertoire of distinct rhythmic firing patterns, each corresponding to a valid Sudoku solution and selectable by brief targeted inputs against a common background oscillation. Some patterns persist after the input is removed while others dissolve. We show that a factor deciding this selectivity is network symmetry. Patterns that respect the symmetries of the wiring are stable, while those that break them are not, and small perturbations can switch the network between stable patterns. Adjusting synaptic weights to remove asymmetry stabilizes patterns that were previously unstable, giving the network a tunable memory. More broadly, our results suggest that a common oscillatory drive, targeted biases to the neurons, and recurrent network structure, act together as complementary control variables. The oscillation sets up a temporal scaffold, the biases select spatiotemporal patterns, and the connectivity determines the stability of the chosen patterns. Tuning these complementary variables can predictably and flexibly reshape the repertoire of activity a circuit expresses.