Dynamical Systems in Neuroscience:

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4.3.1 Bistability and attraction domains . . . . . . . . . . . . . . . . 1134.3.2 Stable/unstable manifolds . . . . . . . . . . . . . . . . . . . . . 1144.3.3 Homoclinic/heteroclinic trajectories . . . . . . . . . . . . . . . . 1164.3.4 Saddle-node bifurcation . . . . . . . . . . . . . . . . . . . . . . 1164.3.5 Andronov-Hopf bifurcation . . . . . . . . . . . . . . . . . . . . . 122Summary and Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . 124Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1275 Conductance-Based Models and Their Reductions 1335.1 Minimal Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1335.1.1 Amplifying and resonant gating variables . . . . . . . . . . . . . 1355.1.2 I Na,p +I K -model . . . . . . . . . . . . . . . . . . . . . . . . . . . 1385.1.3 I Na,t -model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395.1.4 I Na,p +I h -model . . . . . . . . . . . . . . . . . . . . . . . . . . . 1425.1.5 I h +I Kir -model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1445.1.6 I K +I Kir -model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1455.1.7 I A -model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1485.1.8 Ca 2+ -gated minimal models . . . . . . . . . . . . . . . . . . . . 1525.2 Reduction of multi-dimensional models . . . . . . . . . . . . . . . . . . 1555.2.1 Hodgkin-Huxley model . . . . . . . . . . . . . . . . . . . . . . . 1555.2.2 Equivalent potentials . . . . . . . . . . . . . . . . . . . . . . . . 1585.2.3 Nullclines and I-V recordings . . . . . . . . . . . . . . . . . . . 1585.2.4 Reduction to simple model . . . . . . . . . . . . . . . . . . . . . 161Summary and Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . 163Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1646 Bifurcations 1676.1 Equilibrium (Rest State) . . . . . . . . . . . . . . . . . . . . . . . . . . 1676.1.1 Saddle-node (fold) . . . . . . . . . . . . . . . . . . . . . . . . . 1706.1.2 Saddle-node on invariant circle . . . . . . . . . . . . . . . . . . 1736.1.3 Supercritical Andronov-Hopf . . . . . . . . . . . . . . . . . . . . 1776.1.4 Subcritical Andronov-Hopf . . . . . . . . . . . . . . . . . . . . . 1816.2 Limit Cycle (Spiking State) . . . . . . . . . . . . . . . . . . . . . . . . 1866.2.1 Saddle-node on invariant circle . . . . . . . . . . . . . . . . . . 1886.2.2 Supercritical Andronov-Hopf . . . . . . . . . . . . . . . . . . . . 1896.2.3 Fold limit cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . 1906.2.4 Homoclinic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1946.3 Other Interesting Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 1996.3.1 Three-dimensional phase space . . . . . . . . . . . . . . . . . . 1996.3.2 Cusp and pitchfork . . . . . . . . . . . . . . . . . . . . . . . . . 2016.3.3 Bogdanov-Takens . . . . . . . . . . . . . . . . . . . . . . . . . . 2026.3.4 Relaxation oscillators and Canards . . . . . . . . . . . . . . . . 2076.3.5 Bautin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
Page 1: Eugene M. IzhikevichThe Neuroscienc
Page 7 and 8: 8.2.1 Regular spiking (RS) neurons
Page 9 and 10: PrefaceHistorically, much of theore
Page 11 and 12: Chapter 1IntroductionThis chapter h
Page 13 and 14: Introduction 3Figure 1.2: What make
Page 15 and 16: Introduction 510 ms10 mV-45 mV0 pA-
Page 17 and 18: Introduction 7Figure 1.8: Neurons a
Page 19 and 20: Introduction 9K + activation gate,
Page 21 and 22: Introduction 11may look like the on
Page 23 and 24: Introduction 13trajectory describin
Page 25 and 26: Introduction 15neurons, is disconti
Page 27 and 28: Introduction 17saddle-node bifurcat
Page 29 and 30: Introduction 19(IB), and chattering
Page 31 and 32: Introduction 21Review of Important
Page 33 and 34: Introduction 23Figure 1.19: Richard
Page 35 and 36: Chapter 2Electrophysiology of Neuro
Page 37 and 38: Electrophysiology of Neurons 27Insi
Page 39 and 40: Electrophysiology of Neurons 29If t
Page 41 and 42: Electrophysiology of Neurons 31V cV
Page 43 and 44: Electrophysiology of Neurons 33Extr
Page 45 and 46: Electrophysiology of Neurons 35-30
Page 47 and 48: Electrophysiology of Neurons 37pre-
Page 49 and 50: Electrophysiology of Neurons 391h (
Page 51 and 52: Electrophysiology of Neurons 41E Na
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Electrophysiology of Neurons 43puls
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Electrophysiology of Neurons 45ear)
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Electrophysiology of Neurons 47Para
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Electrophysiology of Neurons 49Para
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Electrophysiology of Neurons 51Bibl
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Electrophysiology of Neurons 53+50
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Chapter 3One-Dimensional SystemsIn
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One-Dimensional Systems 57currentsi
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One-Dimensional Systems 59current (
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One-Dimensional Systems 61membrane
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One-Dimensional Systems 63F(V)stabl
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¡ ¡ ¡ ¡¡ ¡ ¡ ¡¡ ¡ ¡ ¡¡
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¡ ¡ ¡ ¡¡ ¡ ¡ ¡¡ ¡ ¡ ¡¡
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One-Dimensional Systems 6915 mV-59
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One-Dimensional Systems 71F (V)1? F
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¡ ¡ ¡ ¡¡ ¡ ¡ ¡¡ ¡ ¡ ¡¡
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One-Dimensional Systems 75result in
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One-Dimensional Systems 77F(V)I=18I
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One-Dimensional Systems 79F(V)10 mV
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One-Dimensional Systems 81200I-V re
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One-Dimensional Systems 8300Bistabi
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One-Dimensional Systems 85000I 0, n
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One-Dimensional Systems 87follow ap
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One-Dimensional Systems 8910.80.60.
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One-Dimensional Systems 9115. Prove
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Chapter 4Two-Dimensional SystemsIn
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Two-Dimensional Systems 9510886644y
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Two-Dimensional Systems 970.7K + ac
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Two-Dimensional Systems 99108642y0-
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Two-Dimensional Systems 101stableun
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Two-Dimensional Systems 103derivati
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Two-Dimensional Systems 105Figure 4
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Two-Dimensional Systems 107is calle
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Two-Dimensional Systems 109τ0eigen
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Two-Dimensional Systems 1110.60.5n-
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Two-Dimensional Systems 113It is ea
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Two-Dimensional Systems 1150.60.5n-
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Two-Dimensional Systems 117rest sta
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Two-Dimensional Systems 119membrane
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Two-Dimensional Systems 121-20-50-3
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Two-Dimensional Systems 1230-100-10
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Two-Dimensional Systems 125membrane
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Two-Dimensional Systems 127Bibliogr
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Chapter 5Conductance-Based Models a
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Conductance-Based Models 135Hodgkin
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Conductance-Based Models 137resonan
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Conductance-Based Models 139This mo
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V-nullclineConductance-Based Models
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Conductance-Based Models 143off) by
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Conductance-Based Models 145leak cu
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Conductance-Based Models 147brings
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Conductance-Based Models 149de-inac
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Conductance-Based Models 151K + act
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Conductance-Based Models 153Voltage
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Conductance-Based Models 15510050V(
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Conductance-Based Models 157150orig
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Conductance-Based Models 159current
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Conductance-Based Models 1615.2.4 R
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Conductance-Based Models 163v peak
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Conductance-Based Models 165500I sl
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Chapter 6BifurcationsNeuronal model
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Bifurcations 169fastBifurcation of
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Bifurcations 171I(V, b) = 0 (equili
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Bifurcations 173(a) saddle-node bif
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Bifurcations 175fast K+ current0.60
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Bifurcations 177matches numerically
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Bifurcations 179r0ϕFigure 6.10: Po
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Bifurcations 1813210eigenvalues = c
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Bifurcations 1831K+ activation vari
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Bifurcations 185unstable limit cycl
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Bifurcations 187saddle-node on inva
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Bifurcations 189bifurcation paramet
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Bifurcations 191I=43limitcyclesstab
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Bifurcations 193homoclinic orbitsta
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Bifurcations 195stable manifoldouti
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Bifurcations 1970.8-100.70.040.60.0
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Bifurcations 199heteroclinic orbitF
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Bifurcations 201the real eigenvalue
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Bifurcations 203xc 2 (b)xbbxxbxxbbb
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Bifurcations 2051 2b1 2SN23 AH 41BT
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Bifurcations 207(a) (b) (c) (d)g=0f
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Bifurcations 209(stable) branches.
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Bifurcations 211homoclinic orbithom
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Bifurcations 213homoclinic orbit bi
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Bifurcations 215nodesaddlesaddle-no
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Bifurcations 217Figure 6.48: Richar
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Bifurcations 219Figure 6.50: The fo
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Bifurcations 221600K + activation,
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Bifurcations 223• Andronov-Hopf b
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Chapter 7Neuronal ExcitabilityNeuro
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Excitability 227excitable bistable
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Excitability 229Layer 5 pyramidal c
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Excitability 231layer 5 pyramidal c
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Excitability 2330.250.20.15w0.10.05
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Excitability 235(a)(b)frequencypuls
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Excitability 23720 mV50 msaverage f
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Excitability 239co-existence of res
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Excitability 241K + activation, n10
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Excitability 243membrane potential
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Excitability 2455ms 10ms 15msnon-re
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Excitability 247AResonant for BReso
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Excitability 2491ms10 mVthreshold ?
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Excitability 251K + activationK + a
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Excitability 25310 ms10 mV-45 mV0 p
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Excitability 2557.2.8 Inhibition-in
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Excitability 25720 mV50 ms-75 mV0 p
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Excitability 2592) Neuronal respons
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Excitability 261integrator(saddle-n
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Excitability 263membrane potential,
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Excitability 265modelI K(M)I Na,p +
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Excitability 267membrane potential
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Excitability 269(a)current, I300200
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Excitability 271Figure 7.48: Reboun
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Excitability 273(a)(b)(c)spikes cut
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Excitability 275membrane potential,
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Excitability 277Figure 7.55: Ex. 6:
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Chapter 8Simple ModelsThe advantage
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Simple Models 281• Well-defined t
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Simple Models 283which we considere
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Simple Models 2851recovery variable
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Simple Models 287membrane potential
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Simple Models 289clever numerical m
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Simple Models 291Figure 8.9:neuronf
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Simple Models 293regular spiking (R
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Simple Models 2958.2.1 Regular spik
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Simple Models 2975 mV-30 mVS-30 mV1
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Simple Models 29920 mV100 ms49 pAre
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Simple Models 301Fig. 8.17a, show a
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Simple Models 303layer 5 neuronsimp
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Simple Models 305layer 5 neuronsimp
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Simple Models 307purpose is to coll
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Simple Models 309100080010 msrecove
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Simple Models 311membrane potential
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Simple Models 313LS neuron (in vitr
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Simple Models 315cells, but mostly
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Simple Models 317figure are v r =
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Simple Models 319The classification
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Simple Models 3211989, Alonso and K
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Simple Models 323the down-state and
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Simple Models 325tional papers can
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Simple Models 3278. (Another theta
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Simple Models 329Figure 8.29: An al
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Simple Models 331cat TC neuronsimpl
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Simple Models 333cat thalamic inter
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Simple Models 335A B C D ENB HTB LT
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Simple Models 337800resetI=600 pA40
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Simple Models 339rat's mitral cell
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Chapter 9BurstingA neuron can fire
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Bursting 343Figure 9.2: Is bursting
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Bursting 345interburst periodquiesc
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Bursting 347possibly resulting in a
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Bursting 349voltage-gatedCa2+-gated
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Bursting 351SpikingBistabilityu(t)R
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Bursting 35300−10−10membrane po
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Bursting 355to the stable equilibri
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Bursting 357membranepotential, V (m
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Bursting 359Spikingu(t)activation o
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Bursting 361eters. For example, in
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Bursting 363abdcburstinga 0 cc 0 b
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Bursting 365bifurcations of limit c
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spikingBursting 367foldbifurcationx
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Bursting 369homoclinic orbit bifurc
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Bursting 371Figure 9.31: Putative
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Bursting 373fast variable, v1086420
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Bursting 37510stable node4fast vari
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Bursting 37710 mV-39 mVFigure 9.38:
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Bursting 379stable equilibriumBauti
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Bursting 3819.3.4 fold/fold cycleWh
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spikingrestingBursting 383saddle-no
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Bursting 387spiking(a)Vfastnfoldthr
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Bursting 389spike synchronizationbu
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Bursting 391in Chap. 7, and lead to
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Bursting 393Saddle-Node Saddle Supe
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Bursting 3951K + activation gate, n
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Bursting 3970.5x 1Figure 9.58: Hopf
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Bursting 399˙u = µ(u − u 3 −
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Chapter 10Synchronization (seewww.i
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Synchronization (see www.izhikevich
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Solutions to ExercisesSolutions to
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Solutions to Exercises, Chap. 3 409
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ReferencesAcebron J. A., Bonilla L.
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References 443Canavier C. C. , Clar
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References 445FitzHugh R. (1960) Th
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References 447Hoppensteadt F.C. (20
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References 449Kuramoto, Y., (1975)
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References 451Pernarowski M. (1994)
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References 453Smolen P., Terman D.,
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References 455Oxford University Pre
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Chapter 10Synchronization (seewww.i
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