Title Page
abstract
Contents
1. Introduction 12
2. Neural dynamics on complex networks 17
2.1. Background 19
2.1.1. Neuron and action potential 19
2.1.2. Synapse plasticity 20
2.2. SOC in Neural system 25
2.2.1. Self-organized criticality 25
2.2.2. Integrate firing model with short term depression 30
2.2.3. Simulation in complex networks 36
2.2.4. Summary 45
2.3. Oscillation and phase transition of cellular automatic neural model 47
2.3.1. Cellular automata model with neural state 47
2.3.2. Phase diagram of neural activity 51
2.3.2. Phase diagram with short term synapse depression 53
3. Population dynamics with Allee effect 57
3.1. Background 59
3.1.1. Population dynamics and logistic equation 59
3.1.2. Population dynamics with Allee effect 61
3.2. Logistic map and Chaotic behavior 62
3.2.1. Logistic map 62
3.1.2. Numerical Methods 63
3.2.2. Phase diagram with Allee effect 68
3.3. Dispersal of population 75
3.3.1. Population diffusion model 75
3.3.2. Species diffusion and invasion 76
3.3.3. Invasion pinning and absorbing 78
4. Conclusions 82
References 84
Fig. 1. Membrane potential of single neuron. (a)-(g) represent (a) polarization, (b) threshold, (c) depolarization, (d) spike, (e) repolarization, (f) hyperpolarization, (g) ion re-distribution. 19
Fig. 2. Function of spike timing dependence plasticity in a neuron. 22
Fig. 3. Dynamics of neurotransmitter J and fraction of utilization of neurotransmitter u. The parameters are τJ = 750,τu = 50, U = 0.45(left), U = 0.15(right)[이미지참조] 24
Fig. 4. A toppling dynamics in BTW model. The avalanche starts at top-left panel and terminate to bottom-right panel. 27
Fig. 5. Avalanche size distribution (a), (c) and life time distribution (b), (d) in 2D and 3D lattice. 29
Fig. 6. Neuron and synapse dynamics. Top panel represents the membrane potential over time. Bottom panel shows the corresponding time dependence of neurotransmitter. 31
Fig. 7. Raster plot of membrane potential in the critical state. Gray color: relaxation time, orange color: avalanche life time. 33
Fig. 8. Size of the avalanche over time at the subcritical state α < αc, critical state α = αc, and super critical state α > αc. We plot the corresponding distribution of the avalanche size.[이미지참조] 34
Fig. 9. The probability distribution of avalanche size with the number of nodes N=300. 35
Fig. 10. Distribution functions observed in neural dynamics an fully connected networks, (a) avalanche size distribution, (b) life time distribution, and (c) relaxation time distribution. 37
Fig. 11. Characteristic patterns of avalanche size distribution. (a) subcritical state, α =0.8, (b) near critical state, α =1.0, (c) critical state, α =1.4, (d) super critical state, α =2.0. 38
Fig. 12. Complex networks using in simulation. Random networks with removal rate (a) 88%, (b) 96%. Scale-free networks with degree exponents (c) γ = 2.5, (d) γ = 4.0. small-world... 40
Fig. 13. Avalanche size distribution[(a), (d), (g)], lifetime distribution[(b), (e), (h)], relaxation time distribution[(c), (f), (i)] on random networks, scale-free networks, and small-world... 41
Fig. 14. The power law of the avalanche size distribution was observed at (a) fully connected network and sparse random network, (b) scale-free network, and (c) small-world network.... 42
Fig. 15. Firing rate over potential. We set parameters as fa = 25, βa = 1.1, and V0 = 24.7.[이미지참조] 49
Fig. 16. Three states with control parameters. We set the control parameters as α = 0.01. We observed (a) a calm state, 〈n〉=5, (b) periodic state, 〈n〉=20, and (c) active state, 〈n〉... 50
Fig. 17. Neural activity phase diagram in the plane of the control parameter α and 〈n〉. Black: inactive, red : oscillation, yellow : active. 52
Fig. 18. Four typical states on phase diagram at α = 0.2, 0.4, 0.6, 0.8. Bottom plots represent each state's oscillation patterns. 55
Fig. 19. Average activities with synapse depression. We represent the heatmap in a plane (Pd, τR) with 〈n〉=15. We plot the activities varying the synaptic strength (a) α =0.2, (b) α = 0.4, (c)...[이미지참조] 56
Fig. 20. Population growth of logistic equation(left). Population growth rate Differential equation of logistic equation(right). We set the parameters as r = 0.2, K = 10, N0 = 0.2.[이미지참조] 60
Fig. 21. Population growth with Allee effect (left) and the population growth rate (right). We set the parameters as A = 0.5, K = 1, r = 0.2. 61
Fig. 22. Population dynamics of logistic map with the different initial population N0 (left). Population dynamics of logistic map with the different grow rate r (right).[이미지참조] 62
Fig. 23. Cobweb diagram with (a) a single fixed point, (b) period-2, (c) chaotic behavior. 63
Fig. 24. Bifurcation diagram in logistic map. For the growth rate r, we plot fixed points of the logistic map. 64
Fig. 25. Lyapunov exponent in logistic map. The region with the positive Lyapunov exponents (red) indicates the chaotic state. 67
Fig. 26. Population growth in Allee effect. We represent the population density over the time (a) for different initial values, (b) for different values of the growth rate r, (c) for the... 68
Fig. 27. Bifurcation diagram in a plane (μ,x*) with selected growth rate (a) r = 2.0, (b) r = 5.0, and (c) r = 1.0. We show the bifurcation diagram in a plane (r,x*) with the threshold (d) μ = 0.2, (e) μ = 0.4,...[이미지참조] 71
Fig. 28. Dependence of Lyapunov exponents on r and μ. (left) dependence of μ at fixed r. (top : r = 2, mid : r = 5, bottom : r = 10). (right) dependence of r at fixed μ. (top : μ = 0.2, mid : μ = 0.4, bottom :... 72
Fig. 28. Diffusion process on 1D patch. The population diffuse to the nearest patch at each time. 76
Fig. 29. Invasion distance and invasion velocity with logistic equation at 1D patch 77
Fig. 30. Invasion distance and invasion velocity for the population diffusion model with Allee effect on 1D patch. 78
Fig. 31. Invasion velocity with diffusion rate d. 79
Fig. 32. Invasion velocity of ecological population dynamics with Allee effect on one dimensional patchy structure. We show the invasion velocity in a plane of (A, d) with (a) r =... 81