Dynamical Systems in Neuroscience:

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200 BifurcationsSaddle-Focus Homoclinic Orbit BifurcationSubcritical Flip BifurcationSubcritical Neimark-Sacker BifurcationBlue-Sky CatastropheFold Limit Cycle on Homoclinic Torus BifurcationFigure 6.33: Some codimension-1 bifurcations of limit cycles in three-dimensional phasespace (modified from Izhikevich 2000).
Bifurcations 201the real eigenvalue (as in Fig. 6.33) and terminates along the subspace spanned by theeigenvectors corresponding to the complex-conjugate pair. The reverse direction is alsopossible. Depending on the direction and the relative magnitude of the eigenvalues,this bifurcation can result in the (dis)appearance or a stable (supercritical) or unstable(subcritical) twisted large-period orbit.The subcritical flip bifurcation in Fig. 6.33 occurs when a stable periodic orbitis surrounded by an unstable orbit of twice the period. The unstable periodic orbitshrinks to the stable one and makes it lose stability. This bifurcation is similar to thepitchfork bifurcation studied below, except it has codimension 1 (pitchfork bifurcationhas infinite codimension unless one considers dynamical systems with symmetry). Asupercritical flip bifurcation is similar, except that an unstable cycle is surrounded bya stable double-period cycle.The subcritical Neimark-Sacker bifurcation in Fig. 6.33 occurs when a stable periodicorbit is surrounded by an unstable invariant torus. The latter shrinks and makesthe periodic orbit lose its stability. In some sense, which we will not elaborate on here,this bifurcation is similar to the supercritical Andronov-Hopf bifurcation of an equilibrium.The supercritical Neimark-Sacker bifurcation occurs when an unstable orbit issurrounded by a stable invariant torus.The blue-sky catastrophe in Fig. 6.33 occurs when a small amplitude stable limitcycle disappears and a large-amplitude large-period stable orbit appears out of nowhere(from the blue sky): The orbit has an infinite period at the bifurcation, yet it isnot homoclinic to any equilibrium. A careful analysis shows that the large orbit ishomoclinic to the small limit cycle at the moment the cycle disappears. In somesense, which we elaborate on later, this bifurcation is similar to the saddle-node oninvariant circle bifurcation (see Ex. 20). In particular, both bifurcations share thesame asymptotics.The fold limit cycle on homoclinic torus bifurcation in Fig. 6.33 is similar to theblue-sky catastrophe except that the disappearance of the small periodic orbit resultsin a large-amplitude torus (quasi-periodic) attractor.6.3.2 Cusp and pitchforkRecall that an equilibrium x b of a one-dimensional system ẋ = f(x, b) is at a saddlenodebifurcation when f x = 0 (first derivative of f) but f xx ≠ 0 (second derivativeof f) at the equilibrium. The latter is called the non-degeneracy condition, and itguarantees that the system dynamics is equivalent to that of ẋ = c(b) + x 2 .If f x = 0 and f xx = 0, but f xxx ≠ 0, then the equilibrium is at the codimension-2cusp bifurcation, and the behavior of the system near the equilibrium can be describedby the topological normal formẋ = c 1 (b) + c 2 (b)x + ax 3 ,wherec 1 (b) = f(x b , b) , c 2 (b) = f x (x b , b) , a = f xxx /6 ≠ 0 ,
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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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142 Conductance-Based Modelshyperpo
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144 Conductance-Based Models0I=0ina
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146 Conductance-Based Models0I=9.75
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148 Conductance-Based Models1I=651I
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Page 178 and 179: 168 Bifurcationssaddle-node bifurca
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Page 196 and 197: 186 Bifurcationsmembrane potential,
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Page 236 and 237: 226 Excitabilityspike?spikerestrest
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Page 242 and 243: 232 ExcitabilityClass 3 excitable n
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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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306 Simple Models(a)74 5 6 3210(b)C
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308 Simple Modelschattering neuron
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318 Simple Modelsthey are able to g
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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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