Preface ...................................................... xiii
1 Biological Systems and Dynamics .............................. 1
1.1 In the Beginning ........................................ 1
1.2 The Hemodynamic System .................................. 4
1.3 Cheyne-Stokes Respiration .............................. 12
1.4 Summary ................................................ 15
2 Population Dynamics of a Single Species ..................... 19
2.1 Fibonacci, Malthus and Nicholson's Blowflies ........... 19
2.2 Fixed Points and Stability of a One-Dimensional
First-Order Difference Equation ........................ 25
2.3 The Cobweb Diagram ..................................... 28
2.4 An Example: Prostate Cancer ............................ 33
2.5 Higher-Dimensional Maps ................................ 35
2.6 Period-Doubling Bifurcation in Infant Respiration ...... 41
2.7 Summary ................................................ 41
3 Observability of Dynamic Variables .......................... 47
3.1 Bioelectric Phenomena Measurement ...................... 47
3.2 ECG, EEG, EMG, EOG and All That ........................ 48
3.3 Measuring Movement ..................................... 54
3.4 Measuring Temperature .................................. 58
3.5 Measuring Oxygen Concentration ......................... 59
3.6 Biomedical Imaging ..................................... 60
3.7 The Importance of Measurement .......................... 65
3.8 Summary ................................................ 67
4 Biomedical Signal Processing ................................ 75
4.1 Automatic Analysis of Electroencephalograms ............ 75
4.2 Electrocardiographic Signal Processing ................. 81
4.3 Vector Cardiography .................................... 87
4.4 Embedology and State Space Representation .............. 89
4.5 Fractals, Chaos and Non-Linear Dynamics ................ 95
4.6 Prediction ............................................. 97
4.7 Summary ............................................... 104
5 Computational Neurophysiology .............................. 111
5.1 The Cell .............................................. 111
5.2 Action Potentials and Ion Channels .................... 114
5.3 Fick's Law, Ohm's Law and the Einstein Relation ....... 116
5.4 Cellular Equilibrium: Nernst and Goldman .............. 117
5.5 Equivalent Circuits ................................... 120
5.6 Dendrites ............................................. 126
5.7 Summary ............................................... 131
6 Mathematical Neurodynamics ................................. 139
6.1 Hodgkin, Huxley and the Squid Giant Axon .............. 139
6.2 FitzHugh-Nagumo Model ................................. 144
6.3 Fixed Points and Stability of a One-Dimensional
Differential Equation ................................. 145
6.4 Nullclines and Phase-Planes ........................... 146
6.5 Pitchfork and Hopf Bifurcations in Two Dimensions ..... 148
6.6 Excitability .......................................... 154
6.7 Summary ............................................... 158
7 Population Dynamics ........................................ 163
7.1 Predator-Prey Interactions ............................ 163
7.2 Fixed Points and Stability of Two-Dimensional
Differential Equations ................................ 164
7.3 Disease Models: SIS, SIR and SEIR ..................... 169
7.4 SARS in Hong Kong ..................................... 173
7.5 Summary ............................................... 176
8 Action, Reaction and Diffusion ............................. 183
8.1 Black Death and Spatial Disease Transmission .......... 183
8.2 Reaction-Diffusion .................................... 187
8.3 Cardiac Dynamics ...................................... 189
8.4 Summary ............................................... 195
9 Autonomous Agents .......................................... 199
9.1 Flocking .............................................. 199
9.2 Celluloid Penguins and Roosting Starlings ............. 200
9.3 Evaluating Crowd Simulations .......................... 207
9.4 Summary ............................................... 208
10 Complex Networks ........................................... 211
10.1 Human Networks: Growing Complex Networks .............. 211
10.2 Small-World Networks and the Spread of SARS ........... 224
10.3 Global Spread of Avian Influenza ...................... 232
10.4 Complex Disease Transmission and Immunisation ......... 239
10.5 Complex Networks Constructed from Musical
Composition ........................................... 240
10.6 Interaction of Grazing Herbivores ..................... 242
10.7 Neuronal Networks Are Complex Networks ................ 243
10.8 Summary ............................................... 245
11 Conclusion ................................................. 249
11.1 Models Are a Reflection of Reality .................... 249
Bibliography .................................................. 253
Index ......................................................... 257
|
