Dynamical Systems in Neuroscience:

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298 Simple Modelsb=-240200saddle-node oninvariant circle bifurcationI=40 I=52 I=60u-nullclinev-nullcline-20spikerecovery variable, u-40b=0.54030saddle-node oninvariant circle bifurcationI=70 I=75 I=8020100spike-10-20-60 -55 -50 -45 -40-60 -55 -50 -45 -40 -60 -55 -50 -45 -40membrane potential, v (mV)Figure 8.14: Saddle-node on invariant circle bifurcations in the RS neuron model asthe magnitude of the injected current I increases.10090big saddle homoclinicorbit bifurcationI=120 I=124.5 I=1258070spikespikingbig homocliniclimit cycle60recovery variable, u504010090saddle homoclinicorbit bifurcationorbitsubcritical Andronov-HopfbifurcationI=126.5 I=127 I=127.58070605040-60 -55 -50 -45 -40-60 -55 -50 -45 -40 -60 -55 -50 -45 -40membrane potential, v (mV)Figure 8.15: The sequence of bifurcations of the RS model neuron (8.5, 8.6) in resonatorregime. Parameters as in Fig. 8.12 and b = 5; see also Fig. 6.40.
Simple Models 29920 mV100 ms49 pArecovery variablespiking limitcycleBABA-60 mVmembrane potentialFigure 8.16: Stuttering behavior of an RS neuron (data were provided by Dr. Klaus M.Stiefel, P28-36 adult mouse, coronal slices, 300µm, layer II/III pyramid, visual cortex).attraction domain of the focus (shaded region in the figure) is bounded by the stablemanifold of the saddle. As I increases, the stable manifold makes a loop and becomesa big homoclinic orbit giving birth to a spiking limit cycle attractor. When I = 125,stable resting and spiking states co-exist, which plays an important role in explainingthe paradoxical stuttering behavior of some neocortical neurons discussed later. AsI increases, the saddle quantity, i.e., the sum of eigenvalues of the saddle, becomespositive. When the stable manifold makes another, smaller loop, it gives birth toan unstable limit cycle, which then shrinks to the resting equilibrium and results insubcritical Andnronov-Hopf bifurcation.What is the excitability class of the RS model neuron in Fig. 8.15? If a slow ramp ofcurrent is injected, the resting state of the neuron becomes a stable focus and then losesstability via subcritical Andronov-Hopf bifurcation. Hence the neuron is a resonatorexhibiting Class 2 excitability. Now suppose steps of dc-current of amplitude I = 125pA or less are injected. The trajectory starts at the initial point (v, u) = (−60, 0),which is the resting state when I = 0, and then approaches the spiking limit cycle.Because the limit cycle was born via a homoclinic bifurcation to the saddle, it has alarge period and hence the neuron is Class 1 excitable. Thus, depending on the natureof stimulation, i.e., ramps vs. pulses, we can observe small or large spiking frequencies,at least in principle.In practice, it is quite difficult to catch homoclinic orbits to saddles because they aresensitive to noise. Injection of a constant current just below the neuron’s rheobase inFig. 8.15 would result in random transitions between the resting state and a periodicspiking state. Indeed, the two attractors co-exist and are near each other, so weakmembrane noise can push the trajectory in and out of the shaded region resulting ina stuttering spiking, illustrated in Fig. 8.16, mingled with subthreshold oscillations.Such a behavior is also exhibited by FS interneurons studied later in this section.
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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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106 Two-Dimensional SystemsFor exam
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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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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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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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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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378 Bursting(a)membrane potential,
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spiking380 Burstingfoldbifurcations
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382 BurstingspikingsupercriticalAnd
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392 BurstingReview of Important Con
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396 Bursting0-10membrane potential,
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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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