ISBN: 3-540-65697-9
TITLE: Phase Resetting in Medicine and Biology
AUTHOR: Tass, Peter A.
TOC:

1. Introduction 1
1.1 Goal 1
1.2 Physiological Motivation 3
1.2.1 Resetting Cerebral Rhythms 3
1.2.2 Deep Brain Stimulation 5
1.3 Stochastic Approach 6
1.4 Synergetics 9
2. Resetting an Ensemble of Oscillators 11
2.1 Introductory Remarks 11
2.2 Deterministic Models 11
2.2.1 Macroscopic Level 12
2.2.2 Cluster of Oscillators 19
2.3 Stochastic Model 21
2.4 FokkerPlanck Equation 22
2.4.1 Stationary Solution 23
2.4.2 Fourier Transformation24
2.5 Spontaneous Behavior 25
2.6 Black Holes Without Noise 26
2.7 Ensemble Dynamics During Stimulation 28
2.7.1 Black Holes in the Presence of Noise 28
2.7.2 Stimulation Induced Frequency Shift 35
2.7.3 Stimulation Mechanism with Higher Harmonics 39
2.8 Firing Patterns 46
2.9 Summary and Discussion 51
3. Synchronization Patterns 57
3.1 Introductory Remarks 57
3.2 Pattern Recognition 57
3.3 Clustering 59
3.4 Populations of Neurons 60
3.4.1 Model Neuron 61
3.4.2 Neuronal Interactions 62
3.5 Populations of Phase Oscillators 63
3.6 Slaving Principle and Center Manifold 66
3.6.1 Center Manifold Theorem 68
3.6.2 Strategy. 71
3.7 n-Cluster States 71
3.7.1 Configuration of Cluster States 71
3.7.2 One Cluster 73
3.7.3 Two Clusters 77
3.7.4 Three Clusters 79
3.7.5 Four Clusters 80
3.8 Complexity of Synchronized States 81
3.8.1 Hierarchy of Frequency Levels 81
3.8.2 Phase and Frequency Shifts 84
3.8.3 Cluster Variables 87
3.8.4 Frozen States 89
3.8.5 Transient Behavior 91
3.8.6 Coupling Mechanism 92
3.9 Neural Coding 95
3.9.1 Information Compression 95
3.9.2 Coding by Clusters 95
3.10 Summary 98
4. Stochastic Model 101
4.1 Introductory Remarks 101
4.2 Derivation of the Model Equation 102
4.3 Fourier Transformation 107
4.4 Summary and Discussion 108
5. Clustering in the Presence of Noise 109
5.1 Introductory Remarks 109
5.2 Modelling Emerging Synchronization 109
5.3 Comparison with the Ensemble's Dynamics 110
5.4 Noisy Cluster States 111
5.4.1 Linear Problem 112
5.4.2 First-Mode Instability. 113
5.4.3 Second-Mode Instability 119
5.4.4 Third-Mode Instability 120
5.4.5 Fourth-Mode Instability 121
5.4.6 Two-Mode Instability 122
5.5 Scaling of Noisy Cluster States 127
5.6 The Experimentalist's Inverse Problem 129
5.6.1 Travelling Waves 130
5.6.2 Firing Patterns 134
5.7 Neural Coding Revisited 137
5.8 Summary and Discussion 139
6. Single Pulse Stimulation 147
6.1 Introductory Remarks 147
6.2 How Stimulation Affects Order Parameters 149
6.2.1 Cluster Variables and Order Parameters 150
6.2.2 Uniform and Partial Desynchronization 151
6.2.3 Stimulating a One-Cluster State 152
6.2.4 Stimulating a Two-Cluster State 161
6.3 Transient Mode Excitation and Early Response 165
6.3.1 Excitation of Higher Order 167
6.3.2 Excitation of Lower Order 171
6.4 Couplings Determine Reaction to Stimulation 174
6.4.1 Rapid Recovery 175
6.4.2 Harmonic Early Response 176
6.5 Vulnerability and Recovery 178
6.5.1 Phase Errors Versus Duration Errors 179
6.5.2 Protective Effect of Couplings 183
6.5.3 Partial Desynchronization and Transient Phenomena 185
6.6 Black Hole and Recovery 187
6.7 Subcritical Long Pulses 189
6.7.1 Spiraling Towards the Desynchronized State 191
6.7.2 Excitation of Higher Order 193
6.7.3 Excitation of Lower Order 197
6.8 Summary and Discussion 197
7. Periodic Stimulation 205
7.1 Introductory Remarks 205
7.2 Smooth Periodic Stimulation 206
7.2.1 1:1 Phase Locking 207
7.2.2 1:2 Phase Locking 209
7.2.3 Changes of the Synchronization Pattern 212
7.3 Pulsatile Periodic Stimulation 214
7.4 Annihilation of Rhythms 216
7.5 Summary and Discussion 217
8. Data Analysis 221
8.1 Introductory Remarks 221
8.2 Phases and Amplitudes 222
8.2.1 Marker Events 223
8.2.2 Reconstruction of the Modes' Dynamics 224
8.2.3 Slaving Principle and Transients 226
8.3 Tracking Down the Black Holes 227
8.4 MEG and EEG Analysis 228
8.4.1 Triggered Averaging 229
8.4.2 Phase Dependent Triggered Averaging 229
8.4.3 Stimulus Locked n : m Transients 231
8.4.4 Stimulus Locked n : m Transients with Delay 238
8.4.5 Multiple Stimulus Locked n : m Transients 240
8.4.6 Detection of Multiple Stimulus Locked n : m Transients 242
8.4.7 Multiple Stimulus Locked n : m Transients with Delay 244
8.4.8 n : m Phase Synchronization 246
8.4.9 Self-Synchronization Versus Transients 250
8.4.10 The Flow of Synchronized Cerebral Activity 251
8.4.11 Inverse Problems 252
8.5 Summary and Discussion 253
9. Modelling Perspectives 259
9.1 Neural Oscillators 259
9.1.1 Time-Delayed Interactions 260
9.1.2 Anatomy of Interacting Clusters 262
9.2 Limit Cycle Oscillators 262
9.3 Chaotic Oscillators 263
9.4 Macroscopic Versus Microscopic 264
10. Neurological Perspectives 267
10.1 Therapeutic Stimulation Techniques 267
10.2 Parkinsonian Resting Tremor 268
10.2.1 Disease Mechanism 268
10.2.2 Stereotactic Treatment 272
10.2.3 Resetting the Tremor Rhythm 274
11. Epilogue 277
11.1 Natural Sensory Stimulation 277
11.2 Experimental Electrical and Magnetic Stimulation 281
11.3 Therapeutic Stimulation 282
Appendices 284
A. Numerical Analysis of the Partial Differential Equations 287
B. Phase and Frequency Shifts Occurring in Chap. 3 289
B.1 Two Clusters 289
B.2 Three Clusters 290
B.3 Four Clusters 290
C. Single-Mode Instability 293
C.1 First-Mode Instability 293
C.2 Second-Mode Instability 293
C.3 Third-Mode Instability 294
C.4 Fourth-Mode Instability 294
D. Two-Mode Instability 295
D.1 Center Manifold 295
D.2 Order Parameter Equation 295
D.3 Linear Problem (Type I) 296
D.4 Linear Problem (Type II) 296
D.5 Singularities 297
References 299
Author Index 323
Subject Index 327
END
