This chapter describes a variety of different ways in which oscillators can interact to create a phase-locked network,where the network behavior may be synchronous (zero-phase difference) or not (e.g. an antiphase relationship). The mechanisms are grouped according to mathematical structure, and in each subsection of the chapter we describe the basic mathematical mechanism as well as variations in more complicated settings.

After the introduction, the first subsection deals with oscillators that can be reduced (without loss of generality) to phase oscillations, with interactions through the differences of the phases. In one a pplication of the method, we show the surprising result that weak electrical coupling between neurons can, in some robust parameter ranges, lead to antiphase behavior of the circuit. (This was shown numerically by Sherman and Rinzel, but never explained in print.) Another application is to work of Ermentrout and Crook on synaptic coupling of two cells through their dendrites. Here the method shows how the length of the dendrites affects whether the cells will synchronize.

The next subsection deals with the interaction of spiking neurons. We discuss the "integrated kernel method" of analysis, which gives a formalism for how the history of spikes in one cell affects the ability of the other cell to fire. Periodic trajectories are computed as self- consistent solutions to the formalism. From the formalism, one can get the phase relationships of the cells (synchrony or not) and the period; with some extra work, one can do a stability analysis. There are several variations on this theme that we discuss, some applications and one other using another set of ideas. We discuss why time scales of inhibition and excitation can effect whether the cells synchronize, and in particular why inhibition with adequately slow onset and decay is synchronizing. A new application of these ideas is to work of Chow and Kopell on spiking cells interacting by electrical coupling, which is mathematically like discrete diffusion, rather than via pulses. Here one sees the surprising result that increasing the size of the spikes in the cells is bad for synchronization, while overshoot in the recovery, which acts like inhibition, helps the synchronization. Finally, we discuss a model to explain the results of Traub and collaborators that oscillating networks of excitatory and inhibitory neurons, interacting with a finite conduction time, can synchronize if the inhibitory neurons display doublets instead of single spikes.

The final set of mechanisms concern bursting neurons, which are described by singularly perturbed relaxation oscillators. Here the mathematical methods are combinations of slow flows and maps. In one application, we show that the synchronization properties of relaxation oscillators are different from oscillators interacting via phase differences. In another, we continue the theme how synaptic and intrinsic times scales combine to determine whether cells synchronize. We also discuss rebound-induced synchronization.

In the final section, we expect to comment on the implications of the work on two oscillators for networks of larger size. Here we will likely present lots of references, but not go deeply into the new mechanisms associated with larger size.

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G. B. Ermentrout and N. Kopell
Jan 11 1998
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