Abstract

The paper concerns a chain of identical phase oscillators, each coupled to only its nearest neighbour on one side. The governing equations are

where
${\theta }_0\left(t\right)$

is some prescribed function of time. Each
${\theta }_i\in [0,2\pi )$

;
$\omega$

and
$ϵ$

are constants,
$Z$

is taken to be
$Z\left(\theta \right)=1-cos\theta$

and
$g$

is a particular “pulse” function. (The results do not depend on the exact form of these, nor the values of parameters.) The model can be regarded as describing a feedforward network of theta neurons. The authors are interested in waves that travel with an approximately uniform profile and speed. They prove (under certain hypotheses) for a doubly-infinite chain (i.e. 
$i=-\infty ,\cdots ,\infty$

) that a family of such waves, each with constant speed and profile, does exist. (The family is parametrised by the wave’s temporal period.) They also prove that such a wave is stable to a large class of specified perturbations. The authors then give the results of careful numerical experiments which suggest that the hypotheses needed above are true.