Dynamical Systems in Neuroscience:

More documents

Recommendations

Info

6.3.6 Saddle-node homoclinic orbit . . . . . . . . . . . . . . . . . . . 2106.3.7 Hard and soft loss of stability . . . . . . . . . . . . . . . . . . . 213Summary and Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . 214Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2207 Neuronal Excitability 2257.1 Excitability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2257.1.1 Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2267.1.2 Hodgkin’s classification . . . . . . . . . . . . . . . . . . . . . . . 2287.1.3 Classes 1 and 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2317.1.4 Class 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2327.1.5 Ramps, steps, and shocks . . . . . . . . . . . . . . . . . . . . . 2347.1.6 Bistability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2367.1.7 Class 1 and 2 spiking . . . . . . . . . . . . . . . . . . . . . . . . 2387.2 Integrators vs. Resonators . . . . . . . . . . . . . . . . . . . . . . . . . 2397.2.1 Fast subthreshold oscillations . . . . . . . . . . . . . . . . . . . 2407.2.2 Frequency preference and resonance . . . . . . . . . . . . . . . . 2427.2.3 Frequency preference in vivo . . . . . . . . . . . . . . . . . . . . 2477.2.4 Thresholds and action potentials . . . . . . . . . . . . . . . . . 2497.2.5 Threshold manifolds . . . . . . . . . . . . . . . . . . . . . . . . 2507.2.6 Rheobase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2527.2.7 Post-inhibitory spike . . . . . . . . . . . . . . . . . . . . . . . . 2537.2.8 Inhibition-induced spiking . . . . . . . . . . . . . . . . . . . . . 2557.2.9 Spike latency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2577.2.10 Flipping from an integrator to a resonator . . . . . . . . . . . . 2597.2.11 Transition between integrators and resonators . . . . . . . . . . 2617.3 Slow Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2647.3.1 Spike-frequency modulation . . . . . . . . . . . . . . . . . . . . 2657.3.2 I-V relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2677.3.3 Slow Subthreshold oscillation . . . . . . . . . . . . . . . . . . . 2707.3.4 Rebound response and voltage sag . . . . . . . . . . . . . . . . 2707.3.5 AHP and ADP . . . . . . . . . . . . . . . . . . . . . . . . . . . 272Summary and Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . 274Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2768 Simple Models 2798.1 Simplest Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2798.1.1 Integrate-and-fire . . . . . . . . . . . . . . . . . . . . . . . . . . 2808.1.2 Resonate-and-fire . . . . . . . . . . . . . . . . . . . . . . . . . . 2818.1.3 Quadratic integrate-and-fire . . . . . . . . . . . . . . . . . . . . 2828.1.4 Simple model of choice . . . . . . . . . . . . . . . . . . . . . . . 2838.1.5 Canonical models . . . . . . . . . . . . . . . . . . . . . . . . . . 2898.2 Cortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
8.2.1 Regular spiking (RS) neurons . . . . . . . . . . . . . . . . . . . 2958.2.2 Intrinsically bursting (IB) neurons . . . . . . . . . . . . . . . . . 3018.2.3 Multi-compartment dendritic tree . . . . . . . . . . . . . . . . . 3058.2.4 Chattering (CH) neurons . . . . . . . . . . . . . . . . . . . . . . 3078.2.5 Low-threshold spiking (LTS) interneurons . . . . . . . . . . . . 3088.2.6 Fast spiking (FS) interneurons . . . . . . . . . . . . . . . . . . . 3118.2.7 Late spiking (LS) interneurons . . . . . . . . . . . . . . . . . . . 3138.2.8 Diversity of inhibitory interneurons . . . . . . . . . . . . . . . . 3148.3 Thalamus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3158.3.1 Thalamo-cortical (TC) relay neurons . . . . . . . . . . . . . . . 3168.3.2 Reticular thalamic nucleus (RTN) neurons . . . . . . . . . . . . 3178.3.3 Thalamic interneurons . . . . . . . . . . . . . . . . . . . . . . . 3178.4 Other interesting cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 3188.4.1 Hippocampal CA1 pyramidal neurons . . . . . . . . . . . . . . . 3188.4.2 Spiny projection neurons of neostriatum and basal ganglia . . . 3198.4.3 Mesencephalic V neurons of brainstem . . . . . . . . . . . . . . 3208.4.4 Stellate cells of entorhinal cortex . . . . . . . . . . . . . . . . . 3208.4.5 Mitral neurons of olfactory bulb . . . . . . . . . . . . . . . . . . 321Summary and Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . 323Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3269 Bursting 3419.1 Electrophysiology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3419.1.1 Example: The I Na,p +I K +I K(M) -model . . . . . . . . . . . . . . . 3439.1.2 Fast-Slow Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 3459.1.3 Minimal models . . . . . . . . . . . . . . . . . . . . . . . . . . . 3479.1.4 Central pattern generators and half-center oscillators . . . . . . 3519.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3519.2.1 Fast-slow bursters . . . . . . . . . . . . . . . . . . . . . . . . . . 3529.2.2 Phase portraits . . . . . . . . . . . . . . . . . . . . . . . . . . . 3529.2.3 Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3559.2.4 Equivalent voltage . . . . . . . . . . . . . . . . . . . . . . . . . 3579.2.5 Hysteresis loops and slow waves . . . . . . . . . . . . . . . . . . 3589.2.6 Bifurcations “resting ↔ bursting ↔ spiking” . . . . . . . . . . . 3609.3 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3649.3.1 fold/homoclinic . . . . . . . . . . . . . . . . . . . . . . . . . . . 3659.3.2 circle/circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3719.3.3 subHopf/fold cycle . . . . . . . . . . . . . . . . . . . . . . . . . 3749.3.4 fold/fold cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3819.3.5 fold/Hopf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3819.3.6 fold/circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3819.4 Neuro-Computational Properties . . . . . . . . . . . . . . . . . . . . . 3839.4.1 How to distinguish? . . . . . . . . . . . . . . . . . . . . . . . . . 383
Page 1: Eugene M. IzhikevichThe Neuroscienc
Page 8 and 9: 9.4.2 Integrators vs. Resonators .
Page 10 and 11: xPrefaceversely, cells having quite
Page 12 and 13: 2 Introductionapical dendritessomar
Page 14 and 15: 4 Introduction(a)spikes(b)spikes cu
Page 16 and 17: 6 Introduction1.1.3 Why are neurons
Page 18 and 19: 8 Introductionconcepts of dynamical
Page 20 and 21: 10 Introductionspikingmoderesting m
Page 22 and 23: 12 Introduction(a)spikinglimitcycle
Page 24 and 25: 14 Introductionco-existence of rest
Page 26 and 27: 16 Introduction1.2.4 Neuro-computat
Page 28 and 29: 18 IntroductionspikeFigure 1.16: Ph
Page 30 and 31: 20 Introductionwith coupled neurons
Page 32 and 33: 22 IntroductionFigure 1.17: John Ri
Page 34 and 35: 24 Introduction
Page 36 and 37: 26 Electrophysiology of NeuronsInsi
Page 38 and 39: 28 Electrophysiology of Neuronsouts
Page 40 and 41: 30 Electrophysiology of NeuronsRest
Page 42 and 43: 32 Electrophysiology of NeuronsFigu
Page 44 and 45: 34 Electrophysiology of Neuronsm (V
Page 46 and 47: 36 Electrophysiology of Neurons10.8
Page 48 and 49: 38 Electrophysiology of Neurons2.3
Page 50 and 51: 40 Electrophysiology of NeuronsFigu
Page 52 and 53: 42 Electrophysiology of NeuronsDepo
Page 54 and 55: 44 Electrophysiology of NeuronsV(x,
Page 56 and 57:
46 Electrophysiology of Neurons1m (
Page 58 and 59:
48 Electrophysiology of NeuronsPara
Page 60 and 61:
50 Electrophysiology of NeuronsFigu
Page 62 and 63:
52 Electrophysiology of Neuronsvolt
Page 64 and 65:
54 Electrophysiology of Neurons
Page 66 and 67:
56 One-Dimensional SystemsActivatio
Page 68 and 69:
58 One-Dimensional Systemsinwardcur
Page 70 and 71:
60 One-Dimensional Systems40bistabi
Page 72 and 73:
62 One-Dimensional Systemsat every
Page 74 and 75:
64 One-Dimensional Systems?F(V)>0+
Page 76 and 77:
66 One-Dimensional SystemsUnstableE
Page 78 and 79:
¡ ¡ ¡¡ ¡ ¡¡ ¡ ¡¡ ¡ ¡¡
Page 80 and 81:
70 One-Dimensional Systems+ - - + +
Page 82 and 83:
72 One-Dimensional Systemsλ(V-Veq)
Page 84 and 85:
74 One-Dimensional Systemsbifurcati
Page 86 and 87:
76 One-Dimensional Systems100806040
Page 88 and 89:
78 One-Dimensional Systemsbifurcati
Page 90 and 91:
80 One-Dimensional Systems20 mV100
Page 92 and 93:
82 One-Dimensional Systemsmembrane
Page 94 and 95:
84 One-Dimensional Systemssteady-st
Page 96 and 97:
86 One-Dimensional Systemsspike rig
Page 98 and 99:
88 One-Dimensional SystemsVF (V)1VV
Page 100 and 101:
90 One-Dimensional Systems10.80.60.
Page 102 and 103:
92 One-Dimensional Systems
Page 104 and 105:
94 Two-Dimensional SystemsI Na,pneu
Page 106 and 107:
96 Two-Dimensional Systemsdefines a
Page 108 and 109:
98 Two-Dimensional Systems(f(x(t),y
Page 110 and 111:
100 Two-Dimensional Systems0.70acti
Page 112 and 113:
102 Two-Dimensional Systems0.70.6K
Page 114 and 115:
104 Two-Dimensional Systemsy1.510.5
Page 116 and 117:
106 Two-Dimensional SystemsFor exam
Page 118 and 119:
108 Two-Dimensional SystemsIn gener
Page 120 and 121:
110 Two-Dimensional Systemsv 2 v 2v
Page 122 and 123:
112 Two-Dimensional SystemsV-nullcl
Page 124 and 125:
114 Two-Dimensional SystemsV-nullcl
Page 126 and 127:
116 Two-Dimensional Systemsheterocl
Page 128 and 129:
118 Two-Dimensional Systems0.60.5n-
Page 130 and 131:
120 Two-Dimensional Systemseigenval
Page 132 and 133:
122 Two-Dimensional SystemsK + acti
Page 134 and 135:
124 Two-Dimensional Systemsexcitati
Page 136 and 137:
126 Two-Dimensional SystemsReview o
Page 138 and 139:
128 Two-Dimensional SystemsabcdFigu
Page 140 and 141:
130 Two-Dimensional SystemsFigure 4
Page 142 and 143:
132 Two-Dimensional Systems
Page 144 and 145:
134 Conductance-Based Models• If
Page 146 and 147:
136 Conductance-Based Modelsinward(
Page 148 and 149:
138 Conductance-Based Models1I=01I=
Page 150 and 151:
140 Conductance-Based Modelsleak cu
Page 152 and 153:
142 Conductance-Based Modelshyperpo
Page 154 and 155:
144 Conductance-Based Models0I=0ina
Page 156 and 157:
146 Conductance-Based Models0I=9.75
Page 158 and 159:
148 Conductance-Based Models1I=651I
Page 160 and 161:
150 Conductance-Based Modelshave co
Page 162 and 163:
152 Conductance-Based Models1 sec10
Page 164 and 165:
154 Conductance-Based Modelsvoltage
Page 166 and 167:
156 Conductance-Based Models1h=0.89
Page 168 and 169:
158 Conductance-Based Models5.2.2 E
Page 170 and 171:
160 Conductance-Based Models1recove
Page 172 and 173:
162 Conductance-Based Modelscurrent
Page 174 and 175:
164 Conductance-Based Modelsmodels
Page 176 and 177:
166 Conductance-Based Models
Page 178 and 179:
168 Bifurcationssaddle-node bifurca
Page 180 and 181:
170 Bifurcationssaddle-node bifurca
Page 182 and 183:
172 Bifurcations0.5v 2V-nullclineK
Page 184 and 185:
174 BifurcationsBoth types of the b
Page 186 and 187:
176 BifurcationsV-nullcline0.5K + g
Page 188 and 189:
178 Bifurcationsrstable limit cycle
Page 190 and 191:
180 BifurcationsnVstable limit cycl
Page 192 and 193:
¡ ¡¡ ¡¡ ¡ ¡¡ ¡ ¡182 Bifur
Page 194 and 195:
¡ ¡ ¡¡ ¡ ¡¡ ¡ ¡¡ ¡ ¡184
Page 196 and 197:
186 Bifurcationsmembrane potential,
Page 198 and 199:
188 BifurcationsBifurcation of a li
Page 200 and 201:
190 Bifurcationsstableunstablelimit
Page 202 and 203:
192 Bifurcationsamplitude (max-min)
Page 204 and 205:
194 Bifurcationshomoclinic orbithom
Page 206 and 207:
196 Bifurcations0.80.60.4stable lim
Page 208 and 209:
198 Bifurcationsfrequency (Hz)40030
Page 210 and 211:
200 BifurcationsSaddle-Focus Homocl
Page 212 and 213:
202 Bifurcationsc 1c 2xFigure 6.34:
Page 214 and 215:
204 BifurcationseigenvaluesHopffold
Page 216 and 217:
206 Bifurcationsfast nullclineslow
Page 218 and 219:
208 Bifurcationswith fast and slow
Page 220 and 221:
210 BifurcationsSupercritical Andro
Page 222 and 223:
212 BifurcationsK + conductance tim
Page 224 and 225:
214 BifurcationsIn contrast, if the
Page 226 and 227:
216 Bifurcationsinvariant circlesad
Page 228 and 229:
218 Bifurcationsbifurcationssaddle-
Page 230 and 231:
220 BifurcationsExercises1. (Transc
Page 232 and 233:
222 Bifurcationsv 2v 11a-11-1Figure
Page 234 and 235:
224 Bifurcations19. [M.S.] A leaky
Page 236 and 237:
226 Excitabilityspike?spikerestrest
Page 238 and 239:
228 ExcitabilityAlternatively, the
Page 240 and 241:
230 Excitability50 ms 20 mV 1 ms100
Page 242 and 243:
232 ExcitabilityClass 3 excitable n
Page 244 and 245:
234 Excitability0.20.150.10.05I 1I
Page 246 and 247:
236 Excitabilitysaddle-node bifurca
Page 248 and 249:
238 Excitability(a) resting spiking
Page 250 and 251:
240 Excitabilityproperties integrat
Page 252 and 253:
242 ExcitabilityThe existence of fa
Page 254 and 255:
244 Excitability1coincidencedetecti
Page 256 and 257:
246 Excitabilityspikeexcitatory pul
Page 258 and 259:
248 Excitability(g)9 ms(f)(e)(d)(c)
Page 260 and 261:
250 Excitabilitysquid axonmodel0 mV
Page 262 and 263:
252 Excitability(a) integrator(b) r
Page 264 and 265:
254 Excitability(a) integrator(b) r
Page 266 and 267:
256 Excitabilitymembrane potential,
Page 268 and 269:
258 Excitability0.1saddle-node bifu
Page 270 and 271:
260 Excitability100 ms10 mVoscillat
Page 272 and 273:
262 ExcitabilityBogdanov-Takens bif
Page 274 and 275:
264 Excitabilitya pulse of current.
Page 276 and 277:
266 Excitabilitymembrane potential
Page 278 and 279:
268 Excitability(a)150100I K(M)(b)3
Page 280 and 281:
270 Excitability50 mV300 msFigure 7
Page 282 and 283:
272 Excitability20 mVADP100 msAHP-6
Page 284 and 285:
274 ExcitabilityK + activation gate
Page 286 and 287:
276 ExcitabilityBibliographical Not
Page 288 and 289:
278 Excitability7. Show that the re
Page 290 and 291:
280 Simple Modelsspikemembrane pote
Page 292 and 293:
282 Simple Models1thresholdyz reset
Page 294 and 295:
284 Simple Models1v reset =|b| 1/2b
Page 296 and 297:
286 Simple Modelsbe an integrator o
Page 298 and 299:
288 Simple Modelsintegrate-and-fire
Page 300 and 301:
290 Simple Models(A) tonic spiking(
Page 302 and 303:
292 Simple Modelsgeneral algorithm
Page 304 and 305:
294 Simple Modelscha, cha — real
Page 306 and 307:
296 Simple Modelslayer 5 neuronsimp
Page 308 and 309:
298 Simple Modelsb=-240200saddle-no
Page 310 and 311:
300 Simple Models(a)simple modellay
Page 312 and 313:
302 Simple Models(a)burstingspiking
Page 314 and 315:
304 Simple Models(a)dendriticspike2
Page 316 and 317:
306 Simple Models(a)74 5 6 3210(b)C
Page 318 and 319:
308 Simple Modelschattering neuron
Page 320 and 321:
310 Simple ModelsLTS neuron (in vit
Page 322 and 323:
312 Simple ModelsFS neuron (in vitr
Page 324 and 325:
314 Simple Models(1) fast oscillati
Page 326 and 327:
316 Simple Modelsthrough an appropr
Page 328 and 329:
318 Simple Modelsthey are able to g
Page 330 and 331:
320 Simple Modelsv r = −80 mV, an
Page 332 and 333:
322 Simple Modelsshow in Fig. 7.36.
Page 334 and 335:
324 Simple ModelsBibliographical No
Page 336 and 337:
326 Simple ModelsExercises1. (Integ
Page 338 and 339:
328 Simple Models17. [M.S.] Analyze
Page 340 and 341:
330 Simple Modelsa35 mV350 msc-NAC
Page 342 and 343:
332 Simple Modelsrat RTN neuronsimp
Page 344 and 345:
334 Simple ModelsFigure 8.34: Class
Page 346 and 347:
336 Simple Modelsspiny neuronlatenc
Page 348 and 349:
338 Simple Models(a)stellate cellof
Page 350 and 351:
340 Simple Modelsrat's mitral cell
Page 352 and 353:
342 Bursting(a) cortical chattering
Page 354 and 355:
344 Burstingmembranepotential (mV)-
Page 356 and 357:
346 Burstingvoltage-gatedCa2+-gated
Page 358 and 359:
348 Burstingslow dynamicsneuronvolt
Page 360 and 361:
350 Burstingslow inactivation of in
Page 362 and 363:
352 Bursting9.2.1 Fast-slow burster
Page 364 and 365:
354 Burstingn-nullclinen slow =-0.0
Page 366 and 367:
356 Bursting0maxmembrane potential,
Page 368 and 369:
358 Bursting0membrane potential, V
Page 370 and 371:
360 BurstingI=0I=4.5425 ms 25 mVI=5
Page 372 and 373:
362 Burstingmembrane potential, V (
Page 374 and 375:
364 Bursting9.3 ClassificationIn Fi
Page 376 and 377:
366 Burstingbifurcation of spiking
Page 378 and 379:
368 BurstingFigure 9.26: Putative
Page 380 and 381:
370 Bursting108spikingslow variable
Page 382 and 383:
372 Bursting(a)membrane potential,
Page 384 and 385:
374 Burstingdepending on the type o
Page 386 and 387:
376 BurstingsubcriticalAndronov-Hop
Page 388 and 389:
378 Bursting(a)membrane potential,
Page 390 and 391:
spiking380 Burstingfoldbifurcations
Page 392 and 393:
382 BurstingspikingsupercriticalAnd
Page 394 and 395:
384 Burstingaction potentials cut2
Page 396 and 397:
386 Bursting9.4.3 BistabilitySuppos
Page 398 and 399:
388 BurstingFigure 9.49: The instan
Page 400 and 401:
esting390 Burstingspikesynchronizat
Page 402 and 403:
392 BurstingReview of Important Con
Page 404 and 405:
394 BurstingspikingrestingFigure 9.
Page 406 and 407:
396 Bursting0-10membrane potential,
Page 408 and 409:
398 Bursting0membrane potential, V
Page 410 and 411:
400 BurstingFigure 9.61: A cycle-cy
Page 412 and 413:
402 Bursting28. [Ph.D.] Develop an
Page 414 and 415:
404 Synchronization (see www.izhike
Page 416 and 417:
Page 418 and 419:
408 Solutions to Exercises, Chap. 3
Page 420 and 421:
Page 422 and 423:
Page 424 and 425:
Page 426 and 427:
Page 428 and 429:
Page 430 and 431:
Page 432 and 433:
Page 434 and 435:
Page 436 and 437:
Page 438 and 439:
Page 440 and 441:
Page 442 and 443:
Page 444 and 445:
Page 446 and 447:
Page 448 and 449:
Page 450 and 451:
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
Page 458 and 459:
448 Referencestional Journal of Bif
Page 460 and 461:
450 ReferencesMarkram H, Toledo-Rod
Page 462 and 463:
452 ReferencesRosenblum M.G. and Pi
Page 464 and 465:
454 ReferencesTuckwell H.C. (1988)
Page 466 and 467:
456 References9
Page 468 and 469:
Page 470 and 471:
Page 472 and 473:
Page 474 and 475:
Page 476 and 477:
Page 478 and 479:
Page 480 and 481:
Page 482 and 483:
Page 484 and 485:
Page 486 and 487:
Page 488 and 489:
Page 490 and 491:
Page 492 and 493:
Page 494 and 495:
Page 496 and 497:
Page 498 and 499:
Page 500 and 501:
Page 502 and 503:
Page 504 and 505:
Page 506 and 507:
Page 508 and 509:
Page 510 and 511:
Page 512 and 513:
Page 514 and 515:
Page 516 and 517:
Page 518 and 519:
Page 520 and 521:
Page 522 and 523:
Page 524 and 525:
Page 526 and 527:
Page 528:
show all

Dynamical Systems in Neuroscience:

Create successful ePaper yourself

Delete template?

Save as template?