Dynamical Systems in Neuroscience:

Chapter 10Synchronization (seewww.izhikevich.com)In this chapter we consider networks of tonically spiking neurons. As any other kind ofphysical, chemical, or biological oscillators, such neurons could synchronize and exhibitcollective behavior that is not intrinsic to any individual neuron. For example, partialsynchrony in cortical networks is believed to generate various brain oscillations, such asthe alpha and gamma EEG rhythms. Increased synchrony may result in pathologicaltypes of activity, such as epilepsy. Coordinated synchrony is needed for locomotion andswim pattern generation in fish. There is an ongoing debate on the role of synchronyin neural computation, see e.g., the special issue of Neuron (September 1999) devotedto 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.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 tosimple phase models. The reduction method and the exact form of the phase modeldepends on the type of coupling, i.e., whether it is pulsed, weak, or slow, and on thetype of bifurcations of the limit cycle attractor generating tonic spiking. Finally, weshow how to use phase models to understand the collective dynamics of many coupledoscillators.10.1 Pulsed CouplingIn this section we consider oscillators of the formẋ = f(x) + Aδ(t − t s ) , x ∈ R m , (10.1)457