Table of Contents
Foreword v
List of Contributors xi
Acronyms xv
Introduction xxiii
1 Phase transitions in single neurons and neural populations: Critical slowing, anesthesia, and sleep cycles D.A. Steyn-Ross M.L. Steyn-Ross M.T. Wilson J.W. Sleigh 1
1.1 Introduction 1
1.2 Phase transitions in single neurons 2
1.2.1 H.R. Wilson spiking neuron model 3
1.2.2 Type-I and type-II subthreshold fluctuations 5
1.2.3 Theoretical fluctuation statistics for approach to criticality 7
1.3 The anesthesia state 11
1.3.1 Effect of anesthetics on bioluminescence 11
1.3.2 Effect of propofol anesthetic on EEG 13
1.4 SWS-REM sleep transition 15
1.4.1 Modeling the SWS-REM sleep transition 17
1.5 The hypnic jerk and the wake-sleep transition 20
1.6 Discussion 23
References 24
2 Generalized state-space models for modeling nonstationary EEG time-series A. Galka K.K.F. Wong T. Ozaki 27
2.1 Introduction 27
2.2 Innovation approach to time-series modeling 28
2.3 Maximum-likelihood estimation of parameters 28
2.4 State-space modeling 30
2.4.1 State-space representation of ARMA models 30
2.4.2 Modal representation of state-space models 32
2.4.3 The dynamics of AR(1) and ARMA(2,1) processes 33
2.4.4 State-space models with component structure 35
2.5 State-space GARCH modeling 36
2.5.1 State prediction error estimate 36
2.5.2 State-space GARCH dynamical equation 37
2.5.3 Interface to Kalman filtering 38
2.5.4 Some remarks on practical model fitting 38
2.6 Application examples 40
2.6.1 Transition to anesthesia 41
2.6.2 Sleep stage transition 43
2.6.3 Temporal-lobe epilepsy 45
2.7 Discussion and summary 48
References 51
3 Spatiotemporal instabilities in neural fields and the effects of additive noise Axel Hutt 53
3.1 Introduction 53
3.1.1 The basic model 54
3.1.2 Model properties and the extended model 57
3.2 Linear stability in the deterministic system 58
3.2.1 Specific model 60
3.2.2 Stationary (Turing) instability 61
3.2.3 Oscillatory instability 63
3.3 External noise 66
3.3.1 Stochastic stability 68
3.3.2 Noise-induced critical fluctuations 70
3.4 Nonlinear analysis of the Turing instability 71
3.4.1 Deterministic analysis 71
3.4.2 Stochastic analysis at order Ο(ε3/2) 74
3.4.3 Stochastic analysis at order (ε5/2) 76
3.5 Conclusion 77
References 78
4 Spontaneous brain dynamics emerges at the edge of instability V.K. Jirsa A. Ghosh 81
4.1 Introduction 81
4.2 Concept of instability, noise, and dynamic repertoire 82
4.3 Exploration of the brain's instabilities during rest 86
4.4 Dynamical invariants of the human resting-state EEG 89
4.4.1 Time-series analysis 90
4.4.2 Spatiotemporal analysis 93
4.5 Final remarks 94
References 97
5 Limited spreading: How hierarchical networks prevent the transition to the epileptic state M. Kaiser J. Simonotto 99
5.1 Introduction 99
5.1.1 Self-organized criticality and avalanches 100
5.1.2 Epilepsy as large-scale critical synchronized event 101
5.1.3 Hierarchical cluster organization of neural systems 101
5.2 Phase transition to the epileptic state 103
5.2.1 Information flow model for brain/hippocampus 103
5.2.2 Change during epileptogenesis 104
5.3 Spreading in hierarchical cluster networks 105
5.3.1 Model of hierarchical cluster networks 105
5.3.2 Model of activity spreading 107
5.3.3 Spreading simulation outcomes 107
5.4 Discussion 111
5.5 Outlook 112
References 114
6 Bifurcations and state changes in the human alpha rhythm: Theory and experiment D.T.J. Liley I. Bojak M.P. Dafilis L. van Veen F. Frascoli B.L. Foster 117
6.1 Introduction 117
6.2 An overview of alpha activity 118
6.2.1 Basic phenomenology of alpha activity 119
6.2.2 Genesis of alpha activity 120
6.2.3 Modeling alpha activity 121
6.3 Mean-field models of brain activity 122
6.3.1 Outline of the extended Liley model 124
6.3.2 Linearization and numerical solutions 128
6.3.3 Obtaining physiologically plausible dynamics 129
6.3.4 Characteristics of the model dynamics 130
6.4 Determination of state transitions in experimental EEG 136
6.4.1 Surrogate data generation and nonlinear statistics 137
6.4.2 Nonlinear time-series analysis of real EEG 137
6.5 Discussion 138
6.5.1 Metastability and brain dynamics 140
References 141
7 Inducing transitions in mesoscopic brain dynamics Hans Liljenström 147
7.1 Introduction 147
7.1.1 Mesoscopic brain dynamics 148
7.1.2 Computational methods 149
7.2 Internally-induced phase transitions 150
7.2.1 Noise-induced transitions 150
7.2.2 Neuromodulatory-induced phase transitions 155
7.2.3 Attention-induced transitions 156
7.3 Externally-induced phase transitions 162
7.3.1 Electrical stimulation 162
7.3.2 Anesthetic-induced phase transitions 167
7.4 Discussion 170
References 173
8 Phase transitions in physiologically-based multiscale mean-field brain models P.A. Robinson C.J. Rennie A.J.K. Phillips J.W. Kim J.A. Roberts 179
8.1 Introduction 179
8.2 Mean-field theory 181
8.2.1 Mean-field modeling 181
8.2.2 Measurements 184
8.3 Corticothalamic mean-field modeling and phase transitions 184
8.3.1 Corticothalamic connectivities 184
8.3.2 Corticothalamic parameters 185
8.3.3 Specific equations 187
8.3.4 Steady states 187
8.3.5 Transfer functions and linear waves 189
8.3.6 Spectra 189
8.3.7 Stability zone, instabilities, seizures, and phase transitions 191
8.4 Mean-field modeling of the brainstem and hypothalamus, and sleep transitions 194
8.4.1 Ascending Arousal System model 194
8.5 Summary and discussion 198
References 198
9 A continuum model for the dynamics of the phase transition from slow-wave sleep to REM sleep J.W. Sleigh M.T. Wilson L.J. Voss D.A. Steyn-Ross M.L. Steyn-Ross X. Li 203
9.1 Introduction 203
9.2 Methods 204
9.2.1 Continuum model of cortical activity 204
9.2.2 Modeling the transition to REM sleep 207
9.2.3 Modeling the slow oscillation of SWS 208
9.2.4 Experimental Methods 209
9.3 Results 210
9.4 Discussion 212
9.5 Appendix 215
9.5.1 Mean-field cortical equations 215
9.5.2 Comparison of model mean-soma potential and experimentally-measured local-field potential 217
9.5.3 Spectrogram and coscalogram analysis 217
References 219
10 What can a mean-field model tell us about the dynamics of the cortex? M.T. Wilson M.L. Steyn-Ross D.A. Steyn-Ross J.W. Sleigh I.P. Gillies D.J. Hailstone 223
10.1 Introduction 223
10.2 A mean-field model of the cortex 224
10.3 Stationary states 226
10.4 Hopf bifurcations 227
10.4.1 Stability analysis 227
10.4.2 Stability of the stationary states 228
10.5 Dynamic simulations 229
10.5.1 Breathing modes 230
10.5.2 Response to localized perturbations 233
10.5.3 K-complex revisited 237
10.5.4 Spiral waves 240
10.6 Conclusions 241
References 241
11 Phase transitions, cortical gamma, and the selection and read-out of information stored in synapses J.J. Wright 243
11.1 Introduction 243
11.2 Basis of simulations 244
11.3 Results 245
11.3.1 Nonspecific flux, transcortical flux, and control of gamma activity 245
11.3.2 Transition to autonomous gamma 246
11.3.3 Power spectra 248
11.3.4 Selective resonance near the threshold for gamma oscillation 248
11.3.5 Synchronous oscillation and traveling waves 251
11.4 Comparisons to experimental results, and an overview of cortical dynamics 252
11.4.1 Comparability to classic experimental data 253
11.4.2 Intracortical regulation of gamma synchrony 253
11.4.3 Synchrony, traveling waves, and phase cones 254
11.4.4 Phase transitions and null spikes 255
11.5 Implications for cortical information processing 257
11.6 Appendix 260
11.6.1 Model equations 260
11.6.2 Hilbert transform and null spikes 264
References 265
12 Cortical patterns and gamma genesis are modulated by reversal potentials and gap-junction diffusion M.L. Steyn-Ross D.A. Steyn-Ross M.T. Wilson J.W. Sleigh 271
12.1 Introduction 271
12.1.1 Continuum modeling of the cortex 272
12.1.2 Reversal potentials 272
12.1.3 Gap-junction diffusion 273
12.2 Theory 274
12.2.1 Input from chemical synapses 274
12.2.2 Input from electrical synapses 280
12.3 Results 282
12.3.1 Stability predictions 282
12.3.2 Slow-soma stability 284
12.3.3 Fast-soma stability 284
12.3.4 Grid simulations 287
12.3.5 Slow-soma simulations 288
12.3.6 Fast-soma simulations 290
12.3.7 Response to inhibitory diffusion and subcortical excitation 290
12.4 Discussion 294
Appendix 297
References 298
Index 301
