Dynamical Systems in Neuroscience:

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228 ExcitabilityAlternatively, the equilibrium may remain stable and co-exist with the newly bornlimit cycle, as it happens during saddle homoclinic orbit or fold limit cycle bifurcationsin Fig. 7.2. The dynamical system is no longer excitable, but bistable, though manyscientists still treat bistable systems as excitable. An appropriate synaptic input canswitch the behavior from resting to spiking and back. Notice that we considered onlybifurcations of a limit cycle so far.• Bifurcation of the equilibrium. Suppose the system is bistable, as in Fig. 7.2.Since the equilibrium is near the cycle, a small modification of the vector field inthe shaded neighborhood can make it disappear via saddle-node bifurcation, orlose stability via subcritical Andronov-Hopf bifurcation.In any case, excitable dynamical systems can bifurcate into oscillatory systems eitherdirectly or indirectly through bistable systems. All these cases are summarized inFig. 7.2.7.1.2 Hodgkin’s classificationAs we mentioned in the introduction chapter, the first one to study bifurcation mechanismsof excitability (years before mathematicians discovered such bifurcations) wasHodgkin (1948), who injected steps of currents of various amplitudes into excitablemembranes and looked at the resulting spiking behavior. We illustrate his experimentsin Fig. 7.3 using recordings of rat neocortical and brainstem neurons. When the currentstrength is small, the neurons are quiescent. When the current is strong, the neuronsfire trains of action potentials. Depending on the average frequency of such firing,Hodgkin identified two major classes of excitability:• Class 1 neural excitability. Action potentials can be generated with arbitrarilylow frequency, depending on the strength of the applied current.• Class 2 neural excitability. Action potentials are generated in a certainfrequency band that is relatively insensitive to changes in the strength of theapplied current.Class 1 neurons, sometimes called type I neurons, fire with a frequency that may varysmoothly over a broad range of about 2 to 100 Hz or even higher. The important observationhere is that the frequency can be changed tenfold. In contrast, the frequencyband of Class 2 neurons is quite limited, e.g., 150−200 Hz, but it can vary from neuronto neuron. The exact numbers are not important to us here. The qualitative distinctionbetween the classes noticed by Hodgkin is that the frequency-current relation (theF-I curve in Fig. 7.3, bottom) starts from zero and continuously increases for Class 1neurons, but is discontinuous for Class 2 neurons.Obviously, the two classes of excitability have different neuro-computational properties.Class 1 excitable neurons can smoothly encode the strength of an input, e.g.,
Excitability 229Layer 5 pyramidal cellBrainstem mesV cellI=320I=1000I=80I=600I=60I=40I=20100 ms 40 mVI=500I=400I=20020 ms 20 mVI (pA)I (pA)0 pA0 pAClass 1 excitabilityClass 2 excitabilityasymptotic firing frequency, Hz40302010F-I curve00 100 200 300injected dc-current, I (pA)asymptotic firing frequency, Hz25020015010050F-I curve00 500 1000 1500injected dc-current, I (pA)Figure 7.3: Top: Typical responses of membrane potentials of two neurons to steps ofdc-current of various magnitudes I. Bottom: Corresponding frequency-current (F-I)relations are qualitatively different. Shown are recordings of layer 5 pyramidal neuronsfrom rat primary visual cortex (left) and mesV neuron from rat brainstem (right). Theasymptotic frequency is 1000/T ∞ , where T ∞ is taken to be the interval between thelast two spikes in a long spike train.
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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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150 Conductance-Based Modelshave co
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152 Conductance-Based Models1 sec10
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154 Conductance-Based Modelsvoltage
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156 Conductance-Based Models1h=0.89
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158 Conductance-Based Models5.2.2 E
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160 Conductance-Based Models1recove
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162 Conductance-Based Modelscurrent
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164 Conductance-Based Modelsmodels
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166 Conductance-Based Models
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168 Bifurcationssaddle-node bifurca
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170 Bifurcationssaddle-node bifurca
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172 Bifurcations0.5v 2V-nullclineK
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174 BifurcationsBoth types of the b
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176 BifurcationsV-nullcline0.5K + g
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Page 196 and 197: 186 Bifurcationsmembrane potential,
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Page 214 and 215: 204 BifurcationseigenvaluesHopffold
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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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Page 246 and 247: 236 Excitabilitysaddle-node bifurca
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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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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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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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