Dynamical Systems in Neuroscience:

Chapter 10Synchronization (seewww.izhikevich.com)This chapter, found on the author’s webpage www.izhikevich.com, considers networksof tonically spiking neurons. As any other kind of physical, chemical, or biologicaloscillators, such neurons could synchronize and exhibit collective behavior that isnot intrinsic to any individual neuron. For example, partial synchrony in cortical networksis believed to generate various brain oscillations, such as the alpha and gammaEEG rhythms. Increased synchrony may result in pathological types of activity, suchas epilepsy. Coordinated synchrony is needed for locomotion and swim pattern generationin fish. There is an ongoing debate on the role of synchrony in neural computation,see e.g., the special issue of Neuron (September 1999) devoted to the binding problem.Depending on the circumstances, synchrony could be good or bad, and it is importantto know what factors contribute to synchrony and how to control it. This is thesubject of the present chapter – the most advanced chapter of the book. It provides anice application of the theory developed earlier and hopefully gives some insight intowhy the previous chapters might be worth mastering. Unfortunately, it is too long tobe included into the book, so reviewers recommended to put it on the web.Our goal is to understand how the behavior of two coupled neurons depends on theirintrinsic dynamics. First, we introduce the method of description of an oscillation byits phase. Then, we describe various methods of reduction of coupled oscillators tophase models. The reduction method and the exact form of the phase model dependson the type of coupling, i.e., whether it is pulsed, weak, or slow, and on the type ofin-phase anti-phase out-of-phaseFigure 10.1: Different types of synchronization.403