Dynamical Systems in Neuroscience:

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9.4.2 Integrators vs. Resonators . . . . . . . . . . . . . . . . . . . . . 3849.4.3 Bistability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3869.4.4 Bursts as a unit of neuronal information . . . . . . . . . . . . . 3879.4.5 Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . 389Summary and Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . 392Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39510 Synchronization (see www.izhikevich.com) 403Solutions to Exercises 407References 44110 Synchronization (see www.izhikevich.com) 45710.1 Pulsed Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45710.1.1 Phase of oscillation . . . . . . . . . . . . . . . . . . . . . . . . . 45810.1.2 Isochrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45910.1.3 PRC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46010.1.4 Type 0 and 1 phase response . . . . . . . . . . . . . . . . . . . . 46310.1.5 Poincare phase map . . . . . . . . . . . . . . . . . . . . . . . . 46610.1.6 Fixed points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46710.1.7 Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . 46810.1.8 Phase locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46910.1.9 Arnold tongues . . . . . . . . . . . . . . . . . . . . . . . . . . . 47010.2 Weak Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47110.2.1 Winfree’s approach . . . . . . . . . . . . . . . . . . . . . . . . . 47210.2.2 Kuramoto’s approach . . . . . . . . . . . . . . . . . . . . . . . . 47410.2.3 Malkin’s approach . . . . . . . . . . . . . . . . . . . . . . . . . 47510.2.4 Measuring PRCs experimentally . . . . . . . . . . . . . . . . . . 47610.2.5 Phase model for coupled oscillators . . . . . . . . . . . . . . . . 47910.3 Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48110.3.1 Two oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . 48310.3.2 Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48510.3.3 Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48710.3.4 Mean-field approximations . . . . . . . . . . . . . . . . . . . . . 48810.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48910.4.1 Phase oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . 48910.4.2 SNIC oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . 49010.4.3 Homoclinic oscillators . . . . . . . . . . . . . . . . . . . . . . . 49610.4.4 Relaxation oscillators and FTM . . . . . . . . . . . . . . . . . . 49810.4.5 Bursting oscillators . . . . . . . . . . . . . . . . . . . . . . . . . 500Summary and Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . 501Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511
PrefaceHistorically, much of theoretical neuroscience research concerned neuronal circuits andsynaptic organization. The neurons were divided into excitatory and inhibitory types,but their electrophysiological properties were largely neglected or taken to be identicalto those of Hodgkin-Huxley’s squid axon. The present awareness of the importance ofthe electrophysiology of individual neurons is best summarized by David McCormick inthe fifth edition of Gordon Shepherd’s book “The Synaptic Organization of the Brain”:“Information processing depends not only on the anatomical substrates of synapticcircuits but also on the electrophysiological properties of neurons... . Even iftwo neurons in different regions of the nervous system possess identical morphologicalfeatures, they may respond to the same synaptic input in very differentmanners because of each cell’s intrinsic properties.”David A. McCormick (2004)Much of present neuroscience research concerns voltage- and second-messengergatedcurrents in individual cells with the goal to understand the cell’s intrinsic neurocomputationalproperties. It is widely accepted that knowing the currents suffices todetermine what the cell is doing and why. This, however, contradicts a half-century oldobservation that cells having similar currents can still exhibit quite different dynamics.Indeed, studying isolated axons having presumably similar electrophysiology (all arefrom crustacean Carcinus maenas), Hodgkin (1948) injected a dc-current of varyingamplitude, and discovered that some preparations could exhibit repetitive spiking witharbitrarily low frequencies, while the others discharged in a narrow frequency band.This observation was largely ignored by the neuroscience community until the seminalpaper by Rinzel and Ermentrout (1989), who showed that the difference in behavior isdue to different bifurcation mechanisms of excitability.Let us treat the amplitude of the injected current in Hodgkin’s experiments as abifurcation parameter: When the amplitude is small, the cell is quiescent; when theamplitude is large, the cell fires repetitive spikes. When we change the amplitude of theinjected current, the cell undergoes a transition from quiescence to repetitive spiking.From the dynamical systems point of view the transition corresponds to a bifurcationfrom equilibrium to a limit cycle attractor. The type of bifurcation determines the mostfundamental computational properties of neurons, such as the class of excitability, theexistence or non-existence of threshold, all-or-none spikes, subthreshold oscillations,the ability to generate post-inhibitory rebound spikes, bistability of resting and spikingstates, whether the neuron is an integrator or resonator, etc.This book is devoted to a systematic study of the relationship between electrophysiology,bifurcations, and computational properties of neurons. The reader willlearn why cells having nearly identical currents may undergo distinct bifurcations, andhence they will have fundamentally different neuro-computational properties. (Conix
Page 1: Eugene M. IzhikevichThe Neuroscienc
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402 Bursting28. [Ph.D.] Develop an
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408 Solutions to Exercises, Chap. 3
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442 Referencesterneurons mediated b
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446 ReferencesGuckenheimer J. (1975
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448 Referencestional Journal of Bif
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450 ReferencesMarkram H, Toledo-Rod
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