Dynamical Systems in Neuroscience:

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438 Solutions to Exercises, Chap. 94slow variable u220-2averagedfull-4-4 -2 0 2 4slow variable u 1Figure 10.40: Ex. 18.The interspike period, T , is defined by v(T ) = +∞, given by the formulaT (u) = √ 1 ( πu 2 + atan √ 1 ). uThe result follows from the integraland the relationships1T (u)∫ T (u)0d i δ(t − T (u)) dt = d 1T (u)f(u) = 1T (u)and atan 1 √ u= arcot √ u .Periodic solutions of the averaged system (focus case) and the full system are depicted inFig. 10.40. The deviation is due to the finite size of the parameters µ 1 and µ 2 in Fig. 9.35.19. There are only two co-dimension-1 bifurcations of an equilibrium that result in transitions toanother equilibrium: saddle-node off limit cycle and subcritical Andronov-Hopf bifurcation.Hence, there are four point-point hysteresis loops, depicted in Fig. 10.41. More details areprovided in Izhikevich (2000).20. This figures are modified from (Izhikevich 2000), where one can find two models exhibitingthis phenomenon. The key feature is that the slow subsystem is not too slow, and the rateof attraction to the upper equilibrium is relatively weak. The spikes are actually dampedoscillations that are generated by the fast subsystem while it converges to the equilibrium.Periodic bursting is generated via the point-point hysteresis loop.21. There are only two co-dimension-1 bifurcations of a small limit cycle attractor (subthresholdoscillation) on a plane that result in sharp transitions to a large-amplitude limit cycle attractor(spiking): Fold limit cycle bifurcation and saddle-homoclinic orbit bifurcation; see Fig. 10.42.These two bifurcations paired with any of the four bifurcations of the large-amplitude limitcycle attractor result in 8 planar co-dimension-1 cycle-cycle bursters; see Fig. 10.43. Moredetails are provided by Izhikevich (2000).
Solutions to Exercises, Chap. 9 439FoldBifurcationFoldBifurcationFoldBifurcationSaddle HomoclinicOrbit BifurcationSubcriticalAndronov-HopfBifurcation"Fold/Fold"Hysteresis Loop"Fold/SubHopf"Hysteresis LoopSubcriticalAndrono-HopfBifurcationSaddle HomoclinicOrbit BifurcationFoldBifurcationSubcriticalAndrono-HopfBifurcationSaddleHomoclinicOrbitBifurcationsSubcriticalAndronov-HopfBifurcation"SubHopf/Fold"Hysteresis Loop"SubHopf/SubHopf"Hysteresis LoopFigure 10.41: Classification of point-point co-dimension-1 hysteresis loops.Fold Limit Cycle BifurcationSaddle Homoclinic Orbit BifurcationFigure 10.42: Co-dimension-1 bifurcations of a stable limit cycle in planar systems that result insharp loss of stability and transition to a large-amplitude (spiking) limit cycle attractor, not shownin the figure. Fold limit cycle: Stable limit cycle is approached by an unstable one, they coalesce, andthen disappear. Saddle homoclinic orbit: A limit cycle grows into a saddle. The unstable manifold ofthe saddle makes a loop and returns via the stable manifold (separatrix).
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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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4 Introduction(a)spikes(b)spikes cu
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6 Introduction1.1.3 Why are neurons
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8 Introductionconcepts of dynamical
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10 Introductionspikingmoderesting m
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12 Introduction(a)spikinglimitcycle
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14 Introductionco-existence of rest
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16 Introduction1.2.4 Neuro-computat
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18 IntroductionspikeFigure 1.16: Ph
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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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34 Electrophysiology of Neuronsm (V
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36 Electrophysiology of Neurons10.8
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38 Electrophysiology of Neurons2.3
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42 Electrophysiology of NeuronsDepo
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44 Electrophysiology of NeuronsV(x,
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46 Electrophysiology of Neurons1m (
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48 Electrophysiology of NeuronsPara
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52 Electrophysiology of Neuronsvolt
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54 Electrophysiology of Neurons
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56 One-Dimensional SystemsActivatio
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58 One-Dimensional Systemsinwardcur
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60 One-Dimensional Systems40bistabi
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62 One-Dimensional Systemsat every
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64 One-Dimensional Systems?F(V)>0+
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66 One-Dimensional SystemsUnstableE
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70 One-Dimensional Systems+ - - + +
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72 One-Dimensional Systemsλ(V-Veq)
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74 One-Dimensional Systemsbifurcati
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76 One-Dimensional Systems100806040
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78 One-Dimensional Systemsbifurcati
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80 One-Dimensional Systems20 mV100
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82 One-Dimensional Systemsmembrane
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84 One-Dimensional Systemssteady-st
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86 One-Dimensional Systemsspike rig
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88 One-Dimensional SystemsVF (V)1VV
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90 One-Dimensional Systems10.80.60.
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92 One-Dimensional Systems
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94 Two-Dimensional SystemsI Na,pneu
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96 Two-Dimensional Systemsdefines a
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120 Two-Dimensional Systemseigenval
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126 Two-Dimensional SystemsReview o
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128 Two-Dimensional SystemsabcdFigu
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130 Two-Dimensional SystemsFigure 4
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134 Conductance-Based Models• If
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162 Conductance-Based Modelscurrent
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164 Conductance-Based Modelsmodels
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166 Conductance-Based Models
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168 Bifurcationssaddle-node bifurca
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170 Bifurcationssaddle-node bifurca
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174 BifurcationsBoth types of the b
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178 Bifurcationsrstable limit cycle
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¡ ¡¡ ¡¡ ¡ ¡¡ ¡ ¡182 Bifur
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186 Bifurcationsmembrane potential,
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188 BifurcationsBifurcation of a li
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190 Bifurcationsstableunstablelimit
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192 Bifurcationsamplitude (max-min)
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196 Bifurcations0.80.60.4stable lim
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198 Bifurcationsfrequency (Hz)40030
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200 BifurcationsSaddle-Focus Homocl
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202 Bifurcationsc 1c 2xFigure 6.34:
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204 BifurcationseigenvaluesHopffold
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206 Bifurcationsfast nullclineslow
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216 Bifurcationsinvariant circlesad
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218 Bifurcationsbifurcationssaddle-
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220 BifurcationsExercises1. (Transc
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222 Bifurcationsv 2v 11a-11-1Figure
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224 Bifurcations19. [M.S.] A leaky
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226 Excitabilityspike?spikerestrest
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228 ExcitabilityAlternatively, the
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230 Excitability50 ms 20 mV 1 ms100
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234 Excitability0.20.150.10.05I 1I
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278 Excitability7. Show that the re
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280 Simple Modelsspikemembrane pote
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282 Simple Models1thresholdyz reset
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366 Burstingbifurcation of spiking
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spiking380 Burstingfoldbifurcations
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386 Bursting9.4.3 BistabilitySuppos
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488 Synchronization (see www.izhike
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512 Solutions to Exercises, Chap. 1
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