Dynamical Systems in Neuroscience:

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206 Bifurcationsfast nullclineslow nullclineBogdanov-Takenspoint(a)(b)Figure 6.39: Intersection of nullclines of a two-dimensional system resulting inBogdanov-Takens bifurcation.I 0 =0 I 1 I 2bighomoclinic orbitI 3 I 4 I 5smallhomoclinic orbitsubcriticalAndronov-HopfbifurcationFigure 6.40: Transformations of phase portraits of a neuronal model near subcriticalBogdanov-Takens bifurcation point as the magnitude of the injected current I increases(here I k+1 > I k ). Shaded regions are the attraction domains of the equilibrium correspondingto the resting state.
Bifurcations 207(a) (b) (c) (d)g=0f=0(e) (f) (g) (h)Figure 6.41: Canard (French duck) limit cycles in a relaxation oscillator (hand drawing).Interestingly, the global vector field structure of neuronal models with nullclinesas in Fig. 6.39a results in the birth of a spiking limit cycle attractor via a big saddlehomoclinic orbit bifurcation, so the neuronal model undergoes a cascade of bifurcationsdepicted in Fig. 6.40 as the amplitude of the injected current I increases. The localphase portraits corresponding to I 0 , I 1 , and I 2 are topologically equivalent to the phaseportrait “1” in Fig. 6.38, right. (The equivalence is only local near the left knee; thereis no global equivalence because of the extra equilibrium in Fig. 6.40 and because ofthe big homoclinic or periodic orbit.) As I increases, a stable large-amplitude spikinglimit cycle appears via a big supercritical homoclinic orbit bifurcation at some I 1 . Itcoexists with the stable resting state for all I 1 int I 2 , thesaddle quantity, i.e., the sum of its eigenvalues, changes from negative to positive(it is zero at the Bogdanov-Takens bifurcation), so another saddle homoclinic orbitbifurcation (at some I 3 ) occurs, which is subcritical, giving birth to an unstable limitcycle. The phase portrait at I 3 is locally topologically equivalent to the one markedSHO in Fig. 6.38. Similarly, the phase portrait at I 4 is locally equivalent to the onemarked “2” in Fig. 6.38. The unstable cycle shrinks to the equilibrium and makes it losestability via a subcritical Andronov-Hopf bifurcation at some I 5 , which corresponds tocase AH in Fig. 6.38. Further increase of I converts the unstable focus into an unstablenode, which approaches the saddle and disappears via the saddle-node bifurcation SN 1in Fig. 6.38 (not shown in Fig. 6.40).6.3.4 Relaxation oscillators and CanardsLet us consider a relaxation oscillatorẋ = f(x, y, b) (fast variable)ẏ = µg(x, y, b) (slow variable)
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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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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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144 Conductance-Based Models0I=0ina
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148 Conductance-Based Models1I=651I
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150 Conductance-Based Modelshave co
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Page 236 and 237: 226 Excitabilityspike?spikerestrest
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256 Excitabilitymembrane potential,
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270 Excitability50 mV300 msFigure 7
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272 Excitability20 mVADP100 msAHP-6
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280 Simple Modelsspikemembrane pote
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352 Bursting9.2.1 Fast-slow burster
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366 Burstingbifurcation of spiking
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372 Bursting(a)membrane potential,
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spiking380 Burstingfoldbifurcations
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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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