Dynamics of Complex Autonomous Boolean Networks

Abstract

Network science provides a powerful framework for analyzing complex systems found in physics, biology, and social sciences. One way of studying the dynamics of networks is to engineer and measure them in the laboratory, which is particularly difficult with established approaches. In this thesis, I approach this problem using a hardware device with time-delay elements executing Boolean functions that can be connected to autonomous Boolean networks with chaotic, periodic, or excitable dynamics. I am able to make scientific discoveries for networks with each of these three different node dynamics, driven by the large flexibility and the non-ideal effects of the experiment complemented by analytical and numerical investigations.Using network realizations with periodic Boolean oscillators, I study socalled chimera states and find that they can disappear and reappear-the resurgence of chimera states. I measure the transient times of chimera states and find a power-law relationship between the average transient time and the phase space volume with an exponent of κ = −0.28 ± 0.10.I also study cluster synchronization in networks of coupled excitable systems. In these artificial neural networks, I find a breakdown of an established theoretical tool when the heterogeneity of the link time delays is greater than the neural refractory period. This phenomenon is used to derive a control scheme for spiking patterns generated by neural networks.Experimental implementations of these systems take advantage of the fast timescale of electronic logic gates, large scalability, and low price. These properties make the system attractive for technological applications, as I demonstrate by realizing a physical random number generator that has an ultra-high bitrate of 12.8 Gbit/s and a silicon neuron that is a thousand times faster than the fastest preceding silicon neuron. For the study of coupled oscillator networks, I develop a phase-locked loop allowing for multiple drivers that may be advantageous for clock synchronization. Instead of the common topologies with one driver per oscillator, it allows for heavily connected clock networks to increase robustness against failure.iii Z U S A M M E N FA S S U N G • D. P. Rosin, D. Rontani, and D. J. Gauthier. Ultrafast physical generation of random numbers using hybrid Boolean networks. Phys. Rev. E 87, 040902(R) (2013).• D. P. Rosin, D. Rontani, D. J. Gauthier, and E. Schöll. Excitability in autonomous Boolean networks. Europhys. Lett. 100, 30003 (2012).• D. P. Rosin, D. Rontani, D. J. Gauthier, and E. Schöll. Experiments on autonomous Boolean networks. Chaos 23, 025102 (2013).• D. P. Rosin, D. Rontani, and D. J. Gauthier. Synchronization of coupled Boolean phase oscillators. Phys. Rev. E 89, 042907 (2014).• D. P. Rosin, D. Rontani, E. Schöll, and D. J. Gauthier. Transient scaling and resurgence of chimera states in coupled Boolean phase oscillators.

Super-transient scaling in time-delay autonomous Boolean network motifs

“…Field-programmable gate arrays (FPGAs) are a type of programmable integrated circuit that can be used to synthesize large, experimental complex networks with arbitrary topologies, asynchronous (i.e., unclocked) update rules, and time-delay links. 26 Nodes of these experimental autonomous Boolean networks are built from asynchronous logic elements on the FPGA, which can be configured to perform arbitrary Boolean functions. By wiring a specific number of "delay elements" consisting of pairs of inverter gates in series, the finite rise and fall times of the logic elements can be exploited to create links with a specific amount of time delay.…”

“…By wiring a specific number of "delay elements" consisting of pairs of inverter gates in series, the finite rise and fall times of the logic elements can be exploited to create links with a specific amount of time delay. 26,27 We use an Altera Cyclone IV FPGA to construct two paradigmatic network motifs-the toggle switch and the repressilator, which consist of two and three unidirectionally coupled inverters, respectively-and vary the amount of delay between each of the nodes to study the transient evolution of these systems to their stable states. The average delay of a single delay element on the Cyclone IV is experimentally measured to be 520 ps, though heterogeneity between logic elements due to chip architecture and manufacturing imperfections causes the possible delay to vary between 250 and 750 ps.…”

“…Moreover, the gates can show a difference in response time between the upward transition and the downward transition. 26,28 These rise/fall asymmetries are as well specific for each node, and combine to a rise/fall asymmetry over the whole delay line. The effect is theoretically modeled in Section III D.…”

Controlling Synchronization Patterns in Complex Networks

“…Besides zero-lag synchronization, a state where all nodes undergo the same dynamics without a phase shift, group and cluster synchronization have received growing interest both in theory [Sorrentino and Ott, 2007;Kestler et al, 2007Kestler et al, , 2008Dahms et al, 2012;Kanter et al, 2011b,a;Golubitsky and Stewart, 2002;Lücken and Yanchuk, 2012;Sorrentino, 2014;Pecora et al, 2014;Poel et al, 2015] and in experiments [Illing et al, 2011;Aviad et al, 2012;Blaha et al, 2013;Williams et al, 2012Williams et al, , 2013aRosin, 2015]. In the case of group synchrony, the network consists of several groups where the nodes within one group are in zero-lag synchrony Dahms et al, 2012].…”

“…Recently, more complex synchronization patterns, including cluster and group synchronization, have received growing interest both in theory [Sorrentino and Ott, 2007;Kestler et al, 2007;Ashwin et al, 2007;Kestler et al, 2008;Kori and Kuramoto, 2001;Lücken and Yanchuk, 2012;Dahms et al, 2012;Kanter et al, 2011b,a;Golubitsky and Stewart, 2002;Sorrentino, 2014;Pecora et al, 2014;Poel et al, 2015] and in experiments [Illing et al, 2011;Aviad et al, 2012;Blaha et al, 2013;Rosin et al, 2013;Williams et al, 2012Williams et al, , 2013aRosin, 2015]. These scenarios appear in many biological systems, examples include dynamics of nephrons [Mosekilde et al, 2002], central pattern generation in animal locomotion [Ijspeert, 2008], or population dynamics [Blasius et al, 1999].…”