Dynamical Systems in Neuroscience:

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460 Synchronization (see www.izhikevich.com)it is not periodic. However, it approaches the cycle as t → +∞. Hence, there is somepoint x 0 on the limit cycle, not necessarily the closest to y 0 , such thaty(t) → x(t) as t → +∞ . (10.2)Now take the phase of the non-periodic solution y(t) to be the phase of its periodicproxy x(t).Alternatively, we can consider a point on the limit cycle x 0 and find all the otherpoints y 0 that satisfy (10.2). The set of all such points is called the stable manifoldof x 0 . Since any solution starting on the stable manifold has an asymptotic behaviorindistinguishable from that of x(t), its phase is the same as that of x(t). For thisreason, the manifold represents solutions having equal phases, and it is often referredto as being the isochron of x 0 (iso: equal and chronos: time), a notion going back toBernoulli and Leibniz.Every point on the plane in Fig. 10.2, except the unstable equilibrium, gives rise toa trajectory that approaches the limit cycle. Therefore, every point has some phase.Let ϑ(x) denote the phase of the point x. Then, isochrons are level contours of thefunction ϑ(x), since the function is constant on each isochron.The entire plane is foliated by isochrons. We depict only 40 representative onesin Fig. 10.2. In this chapter we consider neighborhoods of exponentially stable limitcycles, where the foliation is continuous and invariant (Guckenheimer 1975):• Continuity: The function ϑ(x) is continuous so that nearby points have nearbyphases.• Invariance: If ϑ(x(0)) = ϑ(y(0)), then ϑ(x(t)) = ϑ(y(t)) for all t. Isochrons aremapped to isochrons by the flow of the vector-field.Fig. 10.3 shows the geometry of isochrons of various oscillators. The Andronov-Hopfoscillator in the figure is often called a radial isochron clock for the obvious reason.It is simple enough to be solved explicitly, see Ex. 1. In general, finding isochrons isa daunting mathematical task. In Ex. 3 we present a MATLAB program that findsisochrons numerically.10.1.3 PRCConsider a periodically spiking neuron (10.1) receiving a single brief pulse of currentthat increases the membrane potential by A = 1 mV, as in Fig. 10.4, left. Such aperturbation may not elicit an immediate spike, but it can change the timing, i.e., thephase, of the following spikes. For example, the perturbed trajectory (solid line inFig. 10.4, left) fires earlier than the free-running unperturbed trajectory (dashed line).That is, right after the perturbation, the phase, ϑ new , is greater that the old phase,ϑ. The magnitude of the phase shift of the spike train depends on the exact timing ofthe stimulus relative to the phase of oscillation ϑ. Stimulating the neuron at different
Synchronization (see www.izhikevich.com) 4611.5Andronov-Hopf oscillator2van der Pol oscillator10.51.510.5Im z0y0-0.5-1-0.5-1-1.5-1.5-1.5 -1 -0.5 0 0.5 1 1.5Re z-2-1.5 -1 -0.5 0 0.5 1 1.5xI Na +I K -model (Class 1) I Na +I K -model (Class 2)0.60.50.4n 0.3n0.80.70.60.50.40.20.30.10.200.10-80 -60 -40 -20 0 20V-80 -60 -40 -20 0VFigure 10.3: Isochrons of various oscillators. Andronov-Hopf oscillator: ż = (1 + i)z −z|z| 2 , z ∈ C. van der Pol oscillator: ẋ = x − x 3 − y, ẏ = x. The I Na + I K -model withparameters as in Fig. 4.1a and I = 10 (Class 1) and I = 35 (Class 2). Only isochronscorresponding to phases nT/20, n = 1, . . . , 20, are shown.membrane potential, V (mV)phase of stimulation,PRC= new20 0 T/2 T0new-20-40-60-80I(t)0 10 20 30 40 50 60 70 80time (ms)phase resetting, newPRCT/4T/800 T/2phase of stimulation,TFigure 10.4: Phase response of the I Na + I K -model with parameters as in Fig. 4.1a andI = 4.7. The dashed voltage trace is the free-running trajectory.
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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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178 Bifurcationsrstable limit cycle
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180 BifurcationsnVstable limit cycl
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¡ ¡¡ ¡¡ ¡ ¡¡ ¡ ¡182 Bifur
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¡ ¡ ¡¡ ¡ ¡¡ ¡ ¡¡ ¡ ¡184
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186 Bifurcationsmembrane potential,
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188 BifurcationsBifurcation of a li
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190 Bifurcationsstableunstablelimit
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192 Bifurcationsamplitude (max-min)
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194 Bifurcationshomoclinic orbithom
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196 Bifurcations0.80.60.4stable lim
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198 Bifurcationsfrequency (Hz)40030
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200 BifurcationsSaddle-Focus Homocl
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202 Bifurcationsc 1c 2xFigure 6.34:
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204 BifurcationseigenvaluesHopffold
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206 Bifurcationsfast nullclineslow
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208 Bifurcationswith fast and slow
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210 BifurcationsSupercritical Andro
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212 BifurcationsK + conductance tim
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214 BifurcationsIn contrast, if the
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216 Bifurcationsinvariant circlesad
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218 Bifurcationsbifurcationssaddle-
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220 BifurcationsExercises1. (Transc
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222 Bifurcationsv 2v 11a-11-1Figure
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224 Bifurcations19. [M.S.] A leaky
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226 Excitabilityspike?spikerestrest
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228 ExcitabilityAlternatively, the
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230 Excitability50 ms 20 mV 1 ms100
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232 ExcitabilityClass 3 excitable n
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234 Excitability0.20.150.10.05I 1I
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236 Excitabilitysaddle-node bifurca
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238 Excitability(a) resting spiking
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240 Excitabilityproperties integrat
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242 ExcitabilityThe existence of fa
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244 Excitability1coincidencedetecti
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246 Excitabilityspikeexcitatory pul
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248 Excitability(g)9 ms(f)(e)(d)(c)
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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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300 Simple Models(a)simple modellay
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302 Simple Models(a)burstingspiking
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304 Simple Models(a)dendriticspike2
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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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310 Simple ModelsLTS neuron (in vit
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312 Simple ModelsFS neuron (in vitr
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314 Simple Models(1) fast oscillati
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316 Simple Modelsthrough an appropr
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318 Simple Modelsthey are able to g
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320 Simple Modelsv r = −80 mV, an
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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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Page 452 and 453: 442 Referencesterneurons mediated b
Page 454 and 455: 444 ReferencesDickson C.T., Magistr
Page 456 and 457: 446 ReferencesGuckenheimer J. (1975
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Page 464 and 465: 454 ReferencesTuckwell H.C. (1988)
Page 466 and 467: 456 References9
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