UNSTABLE ATTRACTORS AND IRREGULAR TRANSIENTS IN NETWORKS OF PULSE-COUPLED OSCILLATORS ZOU HAILIN (M.E., Xi’an Jiaotong University) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF PHYSICS NATIONAL UNIVERSITY OF SINGAPORE 2011 I would like to dedicate this thesis to my family members for their long term support. Meanwhile I also would like to dedicate this thesis to my wife Wang HuaLei and our little child Zou YanHao. Acknowledgements Foremost, I would like to thank my advisor Prof. Lai Choy Heng for his guidance and enthusiastic support. Without his encouragement and help, this thesis would have been impossible. His scientific advice, wisdom and insight into problems are always invaluable to me. Meanwhile, I would want to thank my group members, Guang Shuguang, Wang Xinggang, Wang Jiao, Gong Xiaofeng, Li Menghui, Li Kun, Zhou Jie, Yan Gang, Zhao Ming and Chung NingNing, for the many discussions with them. I have benefited much from their feedback on my work and manuscripts. Additionally, I learn much from their research in the group meetings. I also want to thank Prof. Lai Ying-Cheng, Prof. Liu Zonghua, Prof. Zheng Zhi-gang, and Prof. Wang Bing-Hong for their guidance and various discussions. Special thanks to the officers in the computational centers at CCSE and HPC for their quick response to the problems I encountered. ii Table of Contents Acknowledgements ii Abstract vi Publications viii List of Figures ix Introduction 1.1 Complex Interacting Systems . . . . . . . . . . . . . . . . . . . . . 1.2 Different Concepts of Attractors . . . . . . . . . . . . . . . . . . . . 1.2.1 Attractors with Stability . . . . . . . . . . . . . . . . . . . . 1.2.2 Milnor Attractors . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Unstable Attractors as a Novel Type of Milnor Attractors . 1.3 Transients and Their Sensitivity under Perturbations . . . . . . . . 1.3.1 Chaotic Irregular Transients . . . . . . . . . . . . . . . . . . 11 1.3.2 Stable Irregular Transients . . . . . . . . . . . . . . . . . . . 13 1.4 Random Directed Networks . . . . . . . . . . . . . . . . . . . . . . 15 1.5 Motivation and Outline of the Dissertation . . . . . . . . . . . . . . 17 1.5.1 Motivation of the Dissertation . . . . . . . . . . . . . . . . . 17 iii 1.5.2 Outline of the Dissertation . . . . . . . . . . . . . . . . . . . Pulse-Coupled Networks with Delay 19 21 2.1 General Pulse-Coupled Oscillators . . . . . . . . . . . . . . . . . . . 22 2.2 Peskin’s All-to-All Pulse-Coupled Oscillators . . . . . . . . . . . . . 23 2.3 Mirollo-Strogatz Model . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4 Event Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4.1 Event Driven Simulation . . . . . . . . . . . . . . . . . . . . 27 2.4.2 Return Maps . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Unstable Attractors with Active Simultaneous Firing in PulseCoupled Oscillators 30 3.1 Networks of Excitatory Pulse-Coupled Oscillators . . . . . . . . . . 31 3.2 Active Firing Events . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3 Unstable Attractors with Active Simultaneous Firing Events . . . . 34 3.4 Separation of Oscillators with Active Simultaneous Firing by General Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.5 Bifurcation as the Failure of Establishing Active Simultaneous Firing 50 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Chaotic Irregular Transients near the Phase Boundary in Networks of Excitatory Pulse-Coupled Oscillators 56 4.1 Attractors with Sequential Active Firing (SAF) . . . . . . . . . . . 57 4.2 Chaotic Transients near the Phase Boundary . . . . . . . . . . . . . 61 4.3 Dynamics near the Phase Boundary: Bifurcation of SAF Attractors 65 4.4 The Effect of Bifurcation on Long Transients . . . . . . . . . . . . . 72 iv 4.4.1 Bifurcation near the Phase Boundary can Induce Irregular Transients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 The Effect of Delay . . . . . . . . . . . . . . . . . . . . . . . 75 4.5 The Temporal Structures of Long Transients . . . . . . . . . . . . . 77 4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.4.2 Dynamical Formation of Stable Irregular Transients in Inhibitory Networks 80 5.1 Introduction to Discontinuous Map Systems . . . . . . . . . . . . . 81 5.2 The Distance Sequence of Transients to the Basin Boundaries . . . 83 5.3 The Regular Pattern Accompanying Stable Irregular Transients . . 84 5.3.1 Inhibitory Pulse-Coupled Oscillator . . . . . . . . . . . . . . 84 5.3.2 Discontinuous Maps Systems . . . . . . . . . . . . . . . . . . 87 5.4 Dynamical Formation of Stable Irregular Transients . . . . . . . . . 89 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Summary and Outlook 96 6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 References 100 v Abstract We explore the mechanisms of some novel collective dynamical behaviors in networks of pulse-coupled oscillators. The model possesses three main characteristics: individual threshold dynamics to generate pulses which mediate the interactions, a delay time in the pulse transmission, and random disorder in the coupling structures. Specifically, we investigate unstable attractors and long irregular transients, whose mechanisms are unknown. We mainly use the event approach focusing on the microscopic events such as firing and receiving of pulses to study these collective behaviors. We first investigate the source of instability for unstable attractors in networks of excitatory pulse-coupled oscillators. Unstable attractors are a type of attractors whose nearby points within a neighborhood will almost leave this neighborhood. An oscillator fires and sends out a pulse when reaching the threshold. In terms of these firing events, we find that the unstable attractors have a simple property hidden in the event sequences. They coexist with active simultaneous firing events. That is, at least two oscillators reach the threshold simultaneously, which is not directly caused by the receiving pulses. We show that the split of the active simultaneous firing events by general perturbations can make the nearby points vi leave the unstable attractors. Furthermore, this structure can be applied to study the bifurcation of unstable attractors. Unstable attractors can bifurcate due to the failure of establishing active simultaneous firing events. We then study the dynamical mechanism of long chaotic irregular transients in networks of excitatory pulse-coupled oscillators by the event approach. We introduce a type of attractors with certain event structure: sequential active firing (SAF). By using the fraction of SAF attractors in phase space as an order parameter, a phase boundary between SAF and non-SAF attractors is located. Interestingly, the long chaotic transients occur near the phase boundary. The bifurcations of SAF attractors tend to induce irregular transients because passive firings are easier to be converted into active firings near the phase boundary. In addition, many SAF attractors bifurcate near the phase boundary. The above two facts can greatly enhance the average transient time near the phase boundary. Lastly, we investigate the long irregular transients in networks of inhibitory pulsecoupled oscillators, which are insensitive to infinitesimal perturbations. We focus on the dynamical formation of these irregular transients. Interestingly, it is found that the transient dynamics has a hidden pattern in phase space: it repeatedly approaches a basin boundary and then jumps from the boundary to a remote region in phase space. This pattern can be clearly visualized by measuring the distance sequences between the trajectory and the basin boundaries. The dynamical formation of these stable irregular transients originates from the intersection points of the discontinuous boundaries and their images. We carry out numerical experiments to verify this mechanism. vii Publications [1] H.L. Zou, S.G. Guan, and C. H. Lai, Dynamical formation of stable irregular transients in discontinuous map systems , Phys. Rev. E 80, 046214, (2009) [2] H.L. Zou, X.F. Gong and C. H. Lai, Unstable attractors with active simultaneous firing in pulse-coupled oscillators, Phys. Rev. E 82, 046209 (2010). [3] H.L. Zou, M.H. Li, C.H. Lai, and Y.C. Lai, Origin of chaotic irregular transients in networks of pulse-coupled oscillators, submitted viii List of Figures 1.1 Schematic representation of an undirected and a directed network . 16 2.1 The schematic representation of transform function U 25 . . . . . . . 3.1 The schematic representation of phase changes for an oscillator induced by receiving pulses . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2 Firing times for an unstable attractor. . . . . . . . . . . . . . . . . 35 3.3 Unstable attractors with active simultaneous firing for random networks on the parameter plane. . . . . . . . . . . . . . . . . . . . . . 3.4 The time evolution of phase-variables under different perturbations 38 40 3.5 The departure from an unstable attractor under a general perturbation 45 3.6 Analysis of bifurcation of unstable attractors . . . . . . . . . . . . . 53 4.1 A typical long chaotic irregular transient trajectory . . . . . . . . . 62 4.2 Average transient time and fraction of SAF attractors for a network with density of links p = 0.6 . . . . . . . . . . . . . . . . . . . . . . 63 4.3 Average transient time and fraction of SAF attractors for a network with density of links p = 0.8 . . . . . . . . . . . . . . . . . . . . . . 66 ix Chapter 5. Dynamical Formation of Stable Irregular Transients in Inhibitory Networks (a) 0.9 x1 0.6 0.3 0.0 50 100 150 200 n (b) 0.025 0.020 d 0.015 0.010 0.005 0.000 20 40 60 80 100 120 140 160 n Figure 5.5: (a) A typical transient trajectory for Eq. (5.4) and (b) its corresponding sequence of distance to the basin boundaries. 93 Chapter 5. Dynamical Formation of Stable Irregular Transients in Inhibitory Networks points are not the same, which is the major difference with an UPO. The regular patterns occurring in the high dimensional systems Eq. (5.6) and Eq. (5.4) imply that the guiding paths are clustered in phase space, which is similar to the dense UPOs. Here, the clustered guiding paths are generally associated with the large number of discontinuous boundaries in the high dimensional systems. For ( ) ( ) Eq. (5.6), the number of discontinuous boundaries is N + N2 + N3 + · · ·. It is expected that this fast growing of the number of discontinuous boundaries with dimension will make the guiding paths more clustered in high dimensional systems, somewhat similar to the attractor crowding effect [100]. In turn, it will make stable chaos more easily observed in high dimensional systems. 1.0 25 0.9 20 21 31 0.8 35 0.7 27 0.6 32 x2 0.5 0.3 18 33 0.4 26 23 28 19 34 22 29 0.2 0.1 30 24 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 x1 Figure 5.6: Part of transient trajectory shown in plus and a guiding path shown in circle from n = 18 to n = 35. The two lines represent two discontinuous boundaries. The number represents the the time step n. 94 Chapter 5. Dynamical Formation of Stable Irregular Transients in Inhibitory Networks From the microscopic structure, we can also understand why stable chaos is unstable against finite small perturbation (stable against infinitesimally small perturbation) [85]. During the long transient time, stable chaos takes places near many guiding paths whose starting and ending points belong to the basin boundaries. 5.5 Summary In this chapter, we investigate the dynamical formation of long stable irregular transients, directly based on the dynamical equations. We show that these irregular transients actually have certain structure which can be illustrated by the distance sequence to the basin boundaries. The transients repeatedly approaches the basin boundaries and then jumps from the boundaries to a remote region in the phase space. Through numerical simulations, it is shown that there exists a guiding path whose ending point is an image of the starting point and both of them are on the discontinuous boundaries. It is these guiding pathes that connect different points on the basin boundaries, making the dynamics of the system exhibit long irregular behavior before it goes to the final stable attractor. Thus the present work reveals a mechanism for the formation of stable chaos in coupled discontinuous map systems. 95 Chapter Summary and Outlook 6.1 Summary In this thesis, we study the collective dynamics in networks of pulse-coupled oscillators. We are particularly interested in understanding the collective dynamical behaviors, such as unstable attractors and irregular transients, through the microscopic events. There are two types of events: firing and receiving pulses. We further classify firing into active and passive firing. The main difference is that active firings can keep the effect of local perturbations into their firing times. We show that the sequence of events occurring at different time provide much information about the underlying network dynamics. Unstable attractors, a novel type of Milnor attractors, occur in networks of excitatory pulse-coupled oscillators. An unstable attractor possesses a novel local 96 Chapter 6. Summary and Outlook property that almost all points within a neighborhood will leave this neighborhood. Chapter is devoted to finding the source of this kind of instability. By carefully examining the event sequences associated with attractors, we show that unstable attractors coexist with a special type of events: active simultaneous firing. In the presence of general perturbations, these active simultaneous firing events can be split, i.e., the firing events now occur at two different times. This split can further amplify the perturbations and make nearby points leave the unstable attractors. We further study the bifurcation of unstable attractors using the active simultaneous firing. In some cases, the saddle points associated with unstable attractors always exist. Therefore we usually can not understand the bifurcation of unstable attractors through the linear stability analysis, which is widely applied for conventional stable attractors. We show that the active simultaneous firing events can be used to predict the bifurcation of unstable attractors, which is due to the failure of establishing the active simultaneous firing events. The chaotic irregular transients can occur in excitatory pulse-coupled networks, which are sensitive to infinitesimal perturbations. We study the mechanism for these transients in Chapter by using the event approach. The event sequences associated with attractors provide much information about their behavior after bifurcation. Therefore we introduce a type of attractors with sequential active firing. Then we use the fraction of this type of attractors in phase space as an order parameter. We can find a phase boundary, and chaotic transients occur near this phase boundary. Interestingly, the long chaotic transients also occur near this phase boundary. 97 Chapter 6. Summary and Outlook The bifurcation of this type of attractors near phase boundary tends to induce long irregular transients. Furthermore, the fact that many attractors bifurcate near the phase boundary can significantly enhance the average transient time. Lastly, we study in the stable irregular transients in inhibitory pulse-coupled networks in Chapter 5. These transients are also called stable chaos which incorporates local stability and global irregularity. These transients also occur in some coupled discontinuous map systems. We are particularly interested in maps with contracting pieces, as the the transients are insensitive to infinitesimal perturbations except at discontinuous boundaries. Therefore, we also study the networks of inhibitory pulse-coupled oscillators in return maps. In order to understand where these transients occur in phase space, we measure the distance of each point in a long transient trajectory to the basin boundaries. The distance sequence, surprisingly, shows similar regular patterns for both inhibitory pulse-coupled networks and coupled discontinuous maps despite the irregular transients. During transients, the system repeatedly approaches a basin boundary and then jumps from the boundary to a remote region in the phase space. This allows us to find that stable irregular transients occur near a set determined by the set of pre-image and image of discontinuous boundaries, which are composed by many guiding paths. 98 Chapter 6. Summary and Outlook 6.2 Outlook Our work in this thesis can be extended in several ways by taking more realistic features of the systems. For some real systems, excitatory and inhibitory couplings can coexist. The cerebral cortex, for example, has more than 80% excitatory synapses and with much less inhibitory synapses. When considering networks of both inhibitory and excitatory couplings, it is reasonable to first study the simple cases of excitatory (inhibitory) networks with additional weak inhibitory (excitatory) couplings. Then we can study networks with balanced excitatory and inhibitory couplings [110, 111]. It will be interesting to see how the collective dynamical behaviors studied in thesis arise in these networks. The delta pulse coupling treated here is of course an ideal approximation for realistic interactions. The synaptic dynamics, for example, which mediates the interactions among neurons, usually has rising and decaying phases. How the synaptic dynamics affects the collective behaviors is worth investigating. We hope that the understanding of collective dynamics in the simple pulse-coupled framework could help us to design neural networks with expected spiking behaviors which are observed in experiments. 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Sompolinsky, Neural Comput. 10, 1321 (1998). 107 UNSTABLE ATTRACTORS AND IRREGULAR TRANSIENTS IN NETWORKS OF PULSE-COUPLED OSCILLATORS ZOU HAILIN NATIONAL UNIVERSITY OF SINGAPORE 2011 [...]... excitatory pulse- coupled oscillators is not well understood In this study, we investigate the unstable attractors through the events including the sending and receiving of pulses We focus on understanding what event structures bring about the occurrence of unstable attractors For chaotic irregular transients in the networks of excitatory pulse- coupled oscillators, the mechanism for these transients is... Ashwin and M Timme [49] We can distinguish two types of unstable attractors For the first type of unstable attractors [49], almost all of the nearby points will go to other attractors eventually; and for the second type of unstable attractors [49], a positive measure of nearby points will first leave the attractor and come back later Interestingly, the unstable attractors are usually of period one in. .. stable irregular transients) arising in networks of pulse- coupled oscillators For unstable attractors, the source of instability is not clear For the excitatory pulse- coupled oscillators, many stable attractors can also exist which have very 17 Chapter 1 Introduction different response behaviors to perturbations when compared with unstable attractors Thus how unstable attractors arise in the excitatory pulse- coupled. .. tractable Finally, we also discuss the event approach and return maps in studying the dynamics of network pulse- coupled oscillators 21 Chapter 2 Pulse- Coupled Networks with Delay 2.1 General Pulse- Coupled Oscillators Pulse- coupled oscillators are typically used to model dynamical phenomena in networks of agents such as fireflies and neurons whose individual behavior is periodic The oscillators interact... bifurcation of chaos The irregular transients in networks of inhibitory pulse- coupled oscillators are stable to in nitesimal perturbations To understand the dynamical mechanism of these irregular transients is challenging due to the large phase space We tackle this problem by first analyzing where these long irregular transients happen in phase space, i.e., the dynamical formation of these irregular transients. .. mechanism of chaotic transients [104] We introduce a type of attractors according to their event sequences Then we can find a phase boundary Interestingly, the chaotic transients occur near this phase boundary The behaviors of attractors and temporal structures are investigated in detail In chapter 5, the dynamical mechanism of stable irregular transients are directly studied in both pulse- coupled oscillators. .. number of links attached to a node In directed networks, the degree can be divided into the in- degree and the out-degree, which are two useful local measurements The in- degree denotes the number of incoming links for a node, and the out-degree refers to the number of outgoing links for a node The node 6 in Fig 1.1 (b), for example, is with out-degree 3 and in- degree 1 In the last decade, much interest... behaviors of oscillators The advantage of the event approach is discussed In Chapter 3, the source of instability of unstable attractors is investigated through the event approach [102] First, we classify the firing events into two types: active and passive firing After that, we show that source of instability of unstable attractors is due to the appearance of active simultaneous firings In Chapter 4,... further shown as the instability in the associated linear map applying linear stability analysis to an unstable attractor [48] How this property arise in the pulse- coupled oscillators and what is the main source of this type of instability are unclear 1.3 Transients and Their Sensitivity under Perturbations Before settling down onto attractors, the system stays in transient states Transients may provide... work 20 Chapter 2 Pulse- Coupled Networks with Delay In this chapter, we describe the general pulse- coupled oscillators with delay For this type of models, the interactions among the oscillators take place at discrete times Peskin presented an early realization of pulse- coupled integrate -and- fire oscillators [24] One of the widely studied pulse- coupled models is presented by Mirollo and Strogatz [25], . simultane- ous firing in pulse- coupled oscillators, Phys. Rev. E 82, 046209 (2010). [3] H.L. Zou, M.H. Li, C.H. Lai, and Y.C. Lai, Origin of chaotic irregular transients in networks of pulse- coupled oscillators, . UNSTABLE ATTRACTORS AND IRREGULAR TRANSIENTS IN NETWORKS OF PULSE- COUPLED OSCILLATORS ZOU HAILIN (M.E., Xi’an Jiaotong University) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT. mediate the interactions, a delay time in the pulse transmission, and random disorder in the coupling struc- tures. Specifically, we investigate unstable attractors and long irregular transients, whose