### Self-organization Phenomena in Networks of Pulse-Coupled Phase
Oscillators

Stan Mintchev

Abstract.
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
\textit{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 $N \geq 3$ 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 \textit{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.