Dynamical Systems in Neuroscience:

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44 Electrophysiology of NeuronsV(x,t )1V(x,t )2V(x,t )3V(x,t )4100Traveling pulseMembrane voltage (mV)x0AxonFigure 2.18: Traveling pulse solution of the Hodgkin-Huxley cable equation at foursuccessive moments.(a) (b) (c) (d)neuron 1neuron 2dendriteVdsomaVsFigure 2.19: Dendritic tree of a neuron (a) is replaced by a network of compartments(b), each modeled by a Hodgkin-Huxley-type model. Two-compartment neuronalmodel (c) may be equivalent to two neurons coupled via gap junctions (electricalsynapse).
Electrophysiology of Neurons 45ear), semi-infinite, and satisfy Rall’s branching law (Rall 1959). Much of the insightcan be obtained via simulations, which typically substitute the continuous dendriticstructure in Fig. 2.19a by a network of discrete compartments in Fig. 2.19b. Dynamicsof each compartment is simulated by a Hodgkin-Huxley-type model, and the compartmentsare coupled via conductances. For example, if V s and V d denote the membranepotential at the soma and in the dendritic tree, as in Fig. 2.19c, thenC s ˙Vs = −I s (V s , t) + g s (V d − V s ) , and C d ˙Vd = −I d (V d , t) + g d (V s − V d ) ,where each I(V, t) represents the sum of all voltage-, Ca 2+ -, and time-dependent currentsin the compartment, and g s and g d are the coupling conductances that dependon the relative sizes of dendritic and somatic compartments. One can obtain manyspiking and bursting patterns by changing the conductances and keeping all the otherparameters fixed (Pinsky and Rinzel 1994, Mainen and Sejnowski 1996).Once we understand how to couple two compartments, we can do it for hundreds orthousands of compartments. GENESIS and NEURON simulation environments couldbe useful here, especially since they contain databases of dendritic trees reconstructedfrom real neurons.Interestingly, the somatic-dendritic pair in Fig. 2.19c is equivalent to a pair ofneurons in Fig. 2.19d coupled via gap-junctions. These are electrical contacts thatallow ions and small molecules to pass freely between the cells. Gap junctions areoften called electrical synapses, because they allow potentials to be conducted directlyfrom one neuron to another.Computational study of multi-compartment dendritic processing is outside of thescope of this book. We consider multi-compartment models of cortical pyramidal neuronsin Chap. 8 and gap-junction coupled neurons in Chap. 10.2.3.5 Summary of voltage-gated currentsThroughout this book we model kinetics of various voltage-sensitive currents using theHodgkin-Huxley gate modelI = ḡ m a h b (V − E)whereI (µA/cm 2 ) currentV (mV) membrane voltageE (mV) reverse potentialḡ (mS/cm 2 ) maximal conductancemprobability of activation gate to be openhprobability of inactivation gate to be openathe number of activation gates per channelbthe number of inactivation gates per channelThe gating variables m and n satisfy linear first order differential equations (2.9) and(2.10), respectively. We approximate the steady-state activation curve m ∞ (V ) by the
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Eugene M. IzhikevichThe Neuroscienc
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Page 16 and 17: 6 Introduction1.1.3 Why are neurons
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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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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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308 Simple Modelschattering neuron
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316 Simple Modelsthrough an appropr
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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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