Independent of my study, Somers and Kopell [53] investigated relaxation oscillators with two time scales: slow and fast. They analyzed two oscillators with mutual nonlinear coupling (so-called chemical synapses), and proved that the two oscillators reach rapid synchronization within some basin of attraction. They further simulated a ring of such oscillators and demonstrated fast synchronization within the ring. Terman and Wang (for a brief report see Wang and Terman [68]) later analyzed general networks of relaxation oscillators. They proved that with local excitatory coupling (nearest-neighbor being the simplest case) of nonlinear nature, a network of relaxation oscillators reach synchronization in exponential rates. Also, they studied the role of dynamic weight normalization, and found that, though it improves the quality of synchronization, such normalization is not required for achieving emergent synchronization [59]. A subsequent study using the Wilson-Cowan oscillators, similar to (2), was conducted by [9]. They proved that a chain of such oscillators can reach synchronization, and provided a technique for achieving rapid synchronization.
None of the above studies have taken into consideration time delays
in coupling. This is partially justified since coupling is local,
and potential conduction time between coupled oscillators should be small.
However, it has remained to be a question whether some delays would
disrupt synchronous behavior in networks of relaxation oscillators.
The conduction speed of neural fibers can be as slow as 1 m/s
in unmyelinated axons of the brain
[31], and it
is not entirely clear how fast lateral connections in the visual
cortex conduct neural
impulses
.
A preliminary study by Campbell
and Wang
[68] addresses this issue. They found that within a
considerable range of time delays, networks of locally coupled
relaxation oscillators still reach fast synchrony. But the
precision of synchrony is somewhat reduced (they called this
loose synchrony). It is interesting to see that loose synchrony
coincides well with the experimental observations of synchronous
oscillations
[35].
In sum, these theoretical analyses have answered positively and definitively the question of whether local coupling can give rise to global synchrony. One important point that emerges from these studies is that intrinsic properties of single oscillators and their coupling can dramatically influence global properties of synchronization. In particular, sinusoidal (harmonic) oscillators have very different properties than relaxation oscillators. This point has been also noticed by Somers and Kopell ( [53] for a later discussion see [54]). It may be useful to view harmonic oscillators and relaxation oscillators as two extremes of a spectrum. The former have very smooth trajectories, while the latter have rapid changes in their trajectories. Compared to phase models, the activity traces of relaxation oscillators are much more similar to the Hodgkin-Huxley equations of action potential generation resulting from interacting voltage-dependent ion channels [29]. The oscillators defined in (2) are somewhere between the two extremes. These oscillators, as shown in this chapter, can reach synchrony with local coupling. But intrinsic rates of synchronization are not as fast as those of relaxation oscillators, although some techniques can speed up synchronization substantially [9].