| The phase oscillator serves as a paradigm of many rhythmic phenomena in nature. Among these are the electrical impulses arising in neurons. We model a predominantly feedforward network of conductance-based neurons by interpreting each member of the network as a phase oscillator, where signals arising from neighbors in the network serve as temporary adjustments to the oscillator's frequency. We use the theory of autonomous flows on the N-dimensional torus to provide a logical framework for the study of this system through analysis and computer simulations. We find that a network of this type exhibits a high degree of self-organization independent of its initial state. We show that while a feedforward network of two such oscillators cannot phase lock in (1:1) or (1:K) fashion, it exhibits a statistical synchronization: the two members have similar phases for a large proportion of the system running time. We also show that predominantly feedforward networks of this type with N ≥ 3 oscillators exhibit a steady state for which the phases of oscillators 3 through N are entirely determined by the phases of the first two oscillators, a phenomenon we term generalized phase locking. Lastly, we show that in long feedforward networks, an oscillator far down the chain is approximately phase locked to its immediate predecessor; this relative phase shift between neighbors is increasingly restricted and approaches an asymptotic value (in the infinite network-length limit). We compute this value and interpret its consequences for the collective signal output of large feedforward chains, conjecturing that the phenomenon persists for infinite feedforward chains as well as finite chains with some feedback. |
