Dynamical Systems in Neuroscience:

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192 Bifurcationsamplitude (max-min), mV8070605040302010fold limit cyclebifurcationstable limit cyclesunstable limit cyclessubcriticalAndronov-Hopfbifurcation040 41 42 43 44 45 46 47 48 49 50injected dc-current, IFigure 6.24: Bifurcation diagram of the I Na,p +I K -model. Parameters as in Fig. 6.16.appear or disappear, but it does not explain how stable periodic spiking behaviorappears. Indeed, let us start with I = 42 in Fig. 6.23 and slowly increase the parameter.The state of the I Na,p +I K -model is at the stable equilibrium. When I passes thebifurcation value, a large-amplitude stable limit cycle corresponding to periodic spikingappears, yet the model is still quiescent, because it is still at the stable equilibrium.Thus, the limit cycle is just a geometrical object in the phase space that corresponds tospiking behavior. However, to actually exhibit spiking, the state of the system must besomehow pushed into the attraction domain of the cycle, say by external stimulation.This issue is related to the computational properties of neurons, and it is discussed indetail in the next chapter.In Fig. 6.24 we depict the bifurcation diagram of the I Na,p +I K -model. For eachvalue of I, we simulate the model forward (t → ∞) to find the stable limit cycle andbackward (t → −∞) to find the unstable limit cycle. Then we plot their amplitudes(maximal voltage minus minimal voltage along the limit cycle) on the (I, V )-plane.One can clearly see that there is a fold limit cycle bifurcation (left) and a subcriticalAndronov-Hopf bifurcation (right). The left part of the bifurcation diagram looksexactly like the one for saddle-node bifurcation, which explains why the fold limitcycle bifurcation is often referred to as fold or saddle-node of periodics.The similarity of the fold limit cycle bifurcation and the saddle-node bifurcationis not a coincidence. Stability of limit cycles can be studied using Floquet theory,Poincaré cross-section maps (Kuznetsov 1995), or by brute force, e.g., by reducingthe model to an appropriate polar coordinate system. When a limit cycle attractorundergoes fold limit cycle bifurcation, its radius undergoes saddle-node bifurcation(this is a hint to Ex. 10).
Bifurcations 193homoclinic orbitstablelimit cyclesaddle saddle saddlea. supercritical saddle homoclinic orbit bifurcationhomoclinic orbitunstablelimit cyclesaddle saddle saddleb. subcritical saddle homoclinic orbit bifurcationFigure 6.25: Saddle homoclinic orbit bifurcation.
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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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¡ ¡ ¡¡ ¡ ¡¡ ¡ ¡¡ ¡ ¡¡
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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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98 Two-Dimensional Systems(f(x(t),y
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100 Two-Dimensional Systems0.70acti
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102 Two-Dimensional Systems0.70.6K
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104 Two-Dimensional Systemsy1.510.5
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106 Two-Dimensional SystemsFor exam
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108 Two-Dimensional SystemsIn gener
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110 Two-Dimensional Systemsv 2 v 2v
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112 Two-Dimensional SystemsV-nullcl
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114 Two-Dimensional SystemsV-nullcl
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116 Two-Dimensional Systemsheterocl
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118 Two-Dimensional Systems0.60.5n-
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120 Two-Dimensional Systemseigenval
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122 Two-Dimensional SystemsK + acti
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124 Two-Dimensional Systemsexcitati
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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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132 Two-Dimensional Systems
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134 Conductance-Based Models• If
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136 Conductance-Based Modelsinward(
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138 Conductance-Based Models1I=01I=
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140 Conductance-Based Modelsleak cu
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Page 164 and 165: 154 Conductance-Based Modelsvoltage
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Page 168 and 169: 158 Conductance-Based Models5.2.2 E
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Page 172 and 173: 162 Conductance-Based Modelscurrent
Page 174 and 175: 164 Conductance-Based Modelsmodels
Page 176 and 177: 166 Conductance-Based Models
Page 178 and 179: 168 Bifurcationssaddle-node bifurca
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Page 182 and 183: 172 Bifurcations0.5v 2V-nullclineK
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Page 188 and 189: 178 Bifurcationsrstable limit cycle
Page 190 and 191: 180 BifurcationsnVstable limit cycl
Page 192 and 193: ¡ ¡¡ ¡¡ ¡ ¡¡ ¡ ¡182 Bifur
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Page 196 and 197: 186 Bifurcationsmembrane potential,
Page 198 and 199: 188 BifurcationsBifurcation of a li
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Page 208 and 209: 198 Bifurcationsfrequency (Hz)40030
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Page 214 and 215: 204 BifurcationseigenvaluesHopffold
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Page 234 and 235: 224 Bifurcations19. [M.S.] A leaky
Page 236 and 237: 226 Excitabilityspike?spikerestrest
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Page 242 and 243: 232 ExcitabilityClass 3 excitable n
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Page 246 and 247: 236 Excitabilitysaddle-node bifurca
Page 248 and 249: 238 Excitability(a) resting spiking
Page 250 and 251: 240 Excitabilityproperties integrat
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242 ExcitabilityThe existence of fa
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244 Excitability1coincidencedetecti
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246 Excitabilityspikeexcitatory pul
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248 Excitability(g)9 ms(f)(e)(d)(c)
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250 Excitabilitysquid axonmodel0 mV
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252 Excitability(a) integrator(b) r
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254 Excitability(a) integrator(b) r
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256 Excitabilitymembrane potential,
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258 Excitability0.1saddle-node bifu
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260 Excitability100 ms10 mVoscillat
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262 ExcitabilityBogdanov-Takens bif
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264 Excitabilitya pulse of current.
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266 Excitabilitymembrane potential
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268 Excitability(a)150100I K(M)(b)3
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270 Excitability50 mV300 msFigure 7
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272 Excitability20 mVADP100 msAHP-6
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274 ExcitabilityK + activation gate
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276 ExcitabilityBibliographical Not
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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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284 Simple Models1v reset =|b| 1/2b
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286 Simple Modelsbe an integrator o
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288 Simple Modelsintegrate-and-fire
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290 Simple Models(A) tonic spiking(
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292 Simple Modelsgeneral algorithm
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294 Simple Modelscha, cha — real
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296 Simple Modelslayer 5 neuronsimp
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298 Simple Modelsb=-240200saddle-no
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300 Simple Models(a)simple modellay
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302 Simple Models(a)burstingspiking
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304 Simple Models(a)dendriticspike2
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306 Simple Models(a)74 5 6 3210(b)C
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308 Simple Modelschattering neuron
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310 Simple ModelsLTS neuron (in vit
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312 Simple ModelsFS neuron (in vitr
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314 Simple Models(1) fast oscillati
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316 Simple Modelsthrough an appropr
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318 Simple Modelsthey are able to g
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320 Simple Modelsv r = −80 mV, an
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322 Simple Modelsshow in Fig. 7.36.
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324 Simple ModelsBibliographical No
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326 Simple ModelsExercises1. (Integ
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328 Simple Models17. [M.S.] Analyze
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330 Simple Modelsa35 mV350 msc-NAC
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332 Simple Modelsrat RTN neuronsimp
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334 Simple ModelsFigure 8.34: Class
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336 Simple Modelsspiny neuronlatenc
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338 Simple Models(a)stellate cellof
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340 Simple Modelsrat's mitral cell
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342 Bursting(a) cortical chattering
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344 Burstingmembranepotential (mV)-
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346 Burstingvoltage-gatedCa2+-gated
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348 Burstingslow dynamicsneuronvolt
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350 Burstingslow inactivation of in
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352 Bursting9.2.1 Fast-slow burster
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354 Burstingn-nullclinen slow =-0.0
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356 Bursting0maxmembrane potential,
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358 Bursting0membrane potential, V
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360 BurstingI=0I=4.5425 ms 25 mVI=5
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362 Burstingmembrane potential, V (
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364 Bursting9.3 ClassificationIn Fi
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366 Burstingbifurcation of spiking
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368 BurstingFigure 9.26: Putative
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370 Bursting108spikingslow variable
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372 Bursting(a)membrane potential,
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374 Burstingdepending on the type o
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376 BurstingsubcriticalAndronov-Hop
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378 Bursting(a)membrane potential,
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spiking380 Burstingfoldbifurcations
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382 BurstingspikingsupercriticalAnd
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384 Burstingaction potentials cut2
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386 Bursting9.4.3 BistabilitySuppos
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388 BurstingFigure 9.49: The instan
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esting390 Burstingspikesynchronizat
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392 BurstingReview of Important Con
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394 BurstingspikingrestingFigure 9.
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396 Bursting0-10membrane potential,
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398 Bursting0membrane potential, V
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400 BurstingFigure 9.61: A cycle-cy
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402 Bursting28. [Ph.D.] Develop an
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404 Synchronization (see www.izhike
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408 Solutions to Exercises, Chap. 3
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442 Referencesterneurons mediated b
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444 ReferencesDickson C.T., Magistr
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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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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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