Dynamical Systems in Neuroscience:

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404 Synchronization (see www.izhikevich.com)bifurcations of the limit cycle attractor generating tonic spiking. Finally, we show howto use phase models to understand the collective dynamics of many coupled oscillators.
Synchronization (see www.izhikevich.com) 405Review of Important Concepts• Oscillations are described by their phase variables ϑ rotating on a circleS 1 . We define ϑ as the time since the last spike.• The phase response curve, PRC (ϑ), describes the magnitude of the phaseshift of an oscillator caused by a strong pulsed input arriving at phase ϑ.• PRC depends on the bifurcations of spiking limit cycle, and it definessynchronization properties of an oscillator.• Two oscillators are synchronized in-phase, anti-phase, or out-of-phase,when their phase difference, ϑ 2 − ϑ 1 , equals 0, half-period, or some othervalue, respectively; see Fig. 10.1.• Synchronized states of pulse-coupled oscillators are fixed points of thecorresponding Poincare phase map.• Weakly coupled oscillatorscan be reduced to phase modelsẋ i = f(x i ) + ε ∑ g ij (x j )˙ϑ i = 1 + ε Q(ϑ i ) ∑ g ij (x j (ϑ j )) ,where Q(ϑ) is the infinitesimal PRC defined by Malkin’s equation.• Weak coupling induces a slow phase deviation of the natural oscillation,ϑ i (t) = t + ϕ i , described by the averaged model(˙ϕ i = ε ω i + ∑ )H ij (ϕ j − ϕ i ) ,where the ω i denote the frequency deviations, andH ij (ϕ j − ϕ i ) = 1 T∫ Tdescribe the interactions between the phases.0Q(t) g ij (x j (t + ϕ j − ϕ i )) dt• Synchronization of two coupled oscillators correspond to equilibria of theone-dimensional system˙χ = ε(ω + G(χ)) , χ = ϕ 2 − ϕ 1 ,where G(χ) = H 21 (−χ) − H 12 (χ) describes how the phase difference χcompensates for the frequency mismatch ω = ω 2 − ω 1 .
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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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36 Electrophysiology of Neurons10.8
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38 Electrophysiology of Neurons2.3
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52 Electrophysiology of Neuronsvolt
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56 One-Dimensional SystemsActivatio
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80 One-Dimensional Systems20 mV100
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134 Conductance-Based Models• If
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226 Excitabilityspike?spikerestrest
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Page 452 and 453: 442 Referencesterneurons mediated b
Page 454 and 455: 444 ReferencesDickson C.T., Magistr
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454 ReferencesTuckwell H.C. (1988)
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456 References9
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