Dynamical Systems in Neuroscience:

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402 Bursting28. [Ph.D.] Develop an asymptotic theory of singularly perturbed systems of theformẋ = f(x, u) (fast subsystem),˙u = µg(x, u) (slow modulation),that can deal with transitions between equilibria and limit cycle attractors of thefast subsystem.
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
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Eugene M. IzhikevichThe Neuroscienc
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6.3.6 Saddle-node homoclinic orbit
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9.4.2 Integrators vs. Resonators .
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xPrefaceversely, cells having quite
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2 Introductionapical dendritessomar
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6 Introduction1.1.3 Why are neurons
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8 Introductionconcepts of dynamical
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20 Introductionwith coupled neurons
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22 IntroductionFigure 1.17: John Ri
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24 Introduction
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26 Electrophysiology of NeuronsInsi
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28 Electrophysiology of Neuronsouts
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30 Electrophysiology of NeuronsRest
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32 Electrophysiology of NeuronsFigu
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52 Electrophysiology of Neuronsvolt
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56 One-Dimensional SystemsActivatio
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226 Excitabilityspike?spikerestrest
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Page 418 and 419: 408 Solutions to Exercises, Chap. 3
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Page 454 and 455: 444 ReferencesDickson C.T., Magistr
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452 ReferencesRosenblum M.G. and Pi
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454 ReferencesTuckwell H.C. (1988)
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456 References9
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458 Synchronization (see www.izhike
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