On the arithmetic of phase locking: Coupled neurons as a lattice on R2

Using methods from the geometry of numbers, we derive an explicit, global solution for the phase-locking behavior of a simple integrate-and-fire model of coupled neurons. The solution gives the ratios of phase locking (rotation numbers) attained as functions of the parameters of natural frequency and bidirectional coupling. The ordering of the ratios is related to Farey-type series and to simple continued fractions. A transition between two ratios, say $\text{a}\text{b}$ to $\text{c}\text{d}$, is possible if, and only if, ad?bc=±1. Empirically, similar ordering is evident in published data from various neuron analogues. We compare and contrast the present results with those from models based on Caianiello's equation and on more general mappings on the torus.