Mathematical Approaches to Biological Systems (eBook)

Preface 6
Contents 10
1 Human Balance Control: Dead Zones, Intermittency, and Micro-chaos 11
1.1 Introduction 12
1.2 Historical Background 14
1.2.1 An Ecological Example 14
1.2.2 Micro-chaos 16
1.3 Human Postural Sway 19
1.4 Stabilizing the Upright Position 22
1.4.1 First-Order Models 23
1.4.2 Propagation of Threshold Effects 26
1.4.3 Transient Stabilization 27
1.5 Stick Balancing at the Fingertip 29
1.6 Dead Zone Benefits 31
1.7 Concluding Remarks 33
References 34
2 Dynamical Robustness of Complex Biological Networks 39
2.1 Network Robustness 39
2.2 Coupled Oscillator Networks 41
2.2.1 Globally Coupled Networks 43
2.2.2 Homogeneously Coupled Networks 44
2.2.3 Heterogeneously Coupled Networks 45
Random Inactivation 45
Targeted Inactivation 48
Weighted Coupling 50
Heterogeneity of Oscillator Units 50
2.2.4 Other Network Structures 52
2.3 Application to Biological Networks 53
2.3.1 A Neuronal Network Model 53
2.3.2 Robustness of Firing Activity 56
Inactivation of Neurons 56
Targeted Inactivation 56
Effects of Chemical Synapses 59
2.4 Summary 60
References 61
3 Hardware-Oriented Neuron Modeling Approach by Reconfigurable Asynchronous Cellar Automaton 64
3.1 Introduction 64
3.2 Concepts of Asynchronous Sequential Logic Neuron Model 66
3.3 Examples of the Asynchronous Sequential Logic Neuron Models 68
3.4 Bifurcation Analysis and On-Chip Learning of the Fourth-Generation Model 70
3.5 Future Plans and Potential Applications 81
3.6 Conclusions 83
References 83
4 Entrainment Limit of Weakly Forced Nonlinear Oscillators 85
4.1 Introduction 85
4.2 Entrainment Modeled by the Phase Equation 86
4.3 Entrainment Design Under Practical Constrains 87
4.4 Fundamental Limits of Entrainment 88
4.4.1 1:1 Entrainment for 1< p<
4.4.2 1:1 Entrainment for p=1 and p=? 91
4.4.3 General m:n Entrainment 92
4.5 An Example of Efficient Injection Locking: The Hodgkin–Huxley Neuron Model 92
4.6 Conclusion and Discussion 93
Appendix 1 Assumptions on the Phase Response Function and Outlines of the Presented Proofs 94
Appendix 2 Derivation of the Nonlinear Equations Determining Optimal Forcings 96
Appendix 3 Detailed Information Regarding Optimal Forcings 98
Appendix 4 Derivation of Optimal Forcings in Two Limits 99
References 100
5 A Universal Mechanism of Determining the Robustness of Evolving Systems 102
5.1 Introduction 102
5.2 A Simple Model of the Transition in Robustness of Open Systems 104
5.2.1 The Model 104
5.2.2 Transition in the Growth Behavior 106
Transition in Network Topology at Between m=4 and 5 106
The Novel Transition 107
5.3 A Mean-Field Approach for the Transition 107
5.3.1 Fitness Distribution Function and the Convolution-and-Cut Process 107
5.3.2 Determination of the Transition Point 108
5.3.3 Negative Drift During the Link-Deletion Event 109
5.3.4 Analytical Approach for the Estimation of E 111
Convolution Without Drift 111
Calculating Older Generations 112
5.3.5 Analytical Estimation of E Using Fokker-Plank Equation 116
Solution for Neutral Diffusion Process 116
Solution for the System with Negative Drift 118
5.3.6 Numerical Calculation of E 120
5.4 Discussion 122
References 123
6 Switching of Primarily Relied Information by Ants: A Combinatorial Study of Experiment and Modeling 125
6.1 Introduction 125
6.2 Experiment for the Foraging Path Selection by Ants 127
6.2.1 Preparation 127
6.2.2 Measurement 127
6.3 Qualitative Analysis of Experiment 130
6.4 Model 134
6.4.1 Basic Setup 134
6.4.2 Walking Rule 135
6.4.3 Time Development of Pheromone Field 137
6.4.4 Initial Condition and Values of Parameters 138
6.4.5 Simulation and Analysis 139
6.5 Conclusion and Perspectives 141
References 142
7 Chases and Escapes: From Singles to Groups 144
7.1 Introduction 144
7.2 Simple Chase-and-Escape Problems 145
7.3 Theories of Collective Motions 148
7.3.1 Vicsek Model 148
7.3.2 Optimal Velocity Model 149
7.4 Group Chase and Escape 151
7.4.1 Basic Model 151
7.4.2 Simulation Results 153
Lifetimes of Targets 153
7.4.3 Quantitative Analysis of Catching Process 157
7.4.4 Extensions 164
Issues of Range of Each Chaser 164
Issues of Long-Range Chaser Doping 165
Issues of Hopping Fluctuations 165
7.5 Recent Developments on Group Chase and Escape 167
7.5.1 Reactions 167
7.5.2 Motions 168
7.6 Discussion 170
References 171