Dynamical Systems in Neuroscience:

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180 BifurcationsnVstable limit cyclesI=40stableunstableII=30max/min of oscillations ofmembrane potential, mV0-10-20-30-40-50-60-70-80supercriticalAndronov-Hopfbifurcationstablestable cyclesunstable equiliblia0 20 40 60 80injected dc-current, II=20I=25I=15I=10I=0I=-100.70.60.50.4supercritical Andronov-Hopfbifurcation at I=14.66Ceigenvalues c + iωV-nullcline0.30.20.1n-nullclineimaginary part, ω00-80 -60 -40 -20 0 20membrane voltage, V (mV)stable unstable0real part, cFigure 6.11: Supercritical Andronov-Hopf bifurcation in the I Na,p +I K -model with lowthresholdK + current: As a bifurcation parameter I increases, an equilibrium losesstability and gives birth to a stable limit cycle with growing amplitude. Parameters asin Fig. 4.1b.
Bifurcations 1813210eigenvalues = c(I) + ω(I)ilinearization1510514.66 14.66-1000 10 20 30 0 10 20 30 0 10 20 30injected dc-current, I injected dc-current, I injected dc-current, Inumericalω(I)c(I)membrane voltage, (mV)amplitudenumericaltheoreticalfrequency, 2π/period (rads/ms)2.521.510.5frequencytheoreticalnumericalFigure 6.12: Supercritical Andronov-Hopf bifurcation in the I Na,p +I K -model with lowthresholdK + current (see Fig. 6.11). Dots — numerical simulation of the full model,continuous curves — analytical results using the topological normal form (6.8, 6.9).we conclude that r = 0 is an equilibrium for any value of c(b). Since(c(b)r + ar 3 ) r = c(b) at r = 0,the equilibrium is stable for c(b) < 0 and unstable for c(b) > 0, as we illustrate inFig. 6.13. Indeed, the rest state in the I Na,p +I K -model is stable when I < 14.66 andunstable when I > 14.66.When c(b) > 0, the normal form has a family of stable periodic solutions withamplituder = √ c(b)/|a| and (frequency) = ω(b) + d c(b)/|a| .Hence, the I Na,p +I K -model has a family of periodic attractors withandr = √ 0.03{I − 14.66}/0.0026(frequency) = (2.14 + 0.04{I − 14.66}) − 0.0029 · 0.03{I − 14.66}/0.0026 ,depicted in Fig. 6.12. We see that the topological normal form describes not onlyqualitatively but also quantitatively the full I Na,p +I K -model near the Andronov-Hopfbifurcation.6.1.4 Subcritical Andronov-HopfNeuronal models with monotonic steady-state I-V relations can also exhibit subcriticalAndronov-Hopf bifurcations, as we illustrate in Fig. 6.16 using the I Na,p +I K -modelhaving low-threshold K + current and a steep activation curve for Na + current. The
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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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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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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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112 Two-Dimensional SystemsV-nullcl
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114 Two-Dimensional SystemsV-nullcl
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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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230 Excitability50 ms 20 mV 1 ms100
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272 Excitability20 mVADP100 msAHP-6
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280 Simple Modelsspikemembrane pote
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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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