We study networks of nonlocally coupled electronic oscillators that can be described approximately by a Kuramoto-like model. The experimental networks show long complex transients from random initial conditions on the route to network synchronization. The transients display complex behaviors, including resurgence of chimera states, which are network dynamics where order and disorder coexists. The spatial domain of the chimera state moves around the network and alternates with desynchronized dynamics. The fast time scale of our oscillators (on the order of 100 ns) allows us to study the scaling of the transient time of large networks of more than a hundred nodes, which has not yet been confirmed previously in an experiment and could potentially be important in many natural networks. We find that the average transient time increases exponentially with the network size and can be modeled as a Poisson process in experiment and simulation. This exponential scaling is a result of a synchronization rate that follows a power law of the phase-space volume.
Biochemical systems with switch-like interactions, such as gene regulatory networks, are well modeled by autonomous Boolean networks. Specifically, the topology and logic of gene interactions can be described by systems of continuous piecewise-linear differential equations, enabling analytical predictions of the dynamics of specific networks. However, most models do not account for time delays along links associated with spatial transport, mRNA transcription, and translation. To address this issue, we we have developed an experimental testbed to realize a time-delay autonomous Boolean network with three inhibitory nodes, known as a repressilator, and use it to study the dynamics that arise as time delays along the links vary. We observe various nearly-periodic oscillatory transient patterns with extremely long lifetime, which emerge in small network motifs due to the delay, and which are distinct from the eventual asymptotically stable periodic attractors. For repeated experiments with a given network, we find that stochastic processes give rise to a broad distribution of transient times with an exponential tail. In some cases, the transients are so long that it is doubtful the attractors will ever be approached in a biological system that has a finite lifetime. To counteract the long transients, we show experimentally that small, occasional perturbations applied to the time delays can force the trajectories to rapidly approach the attractors.
Autonomous Boolean networks are commonly used to model the dynamics of gene regulatory networks and allow for the prediction of stable dynamical attractors. However, most models do not account for time delays along the network links and noise, which are crucial features of real biological systems. Concentrating on two paradigmatic motifs, the toggle switch and the repressilator, we develop an experimental testbed that explicitly includes both inter-node time delays and noise using digital logic elements on field-programmable gate arrays. We observe transients that last millions to billions of characteristic time scales and scale exponentially with the amount of time delays between nodes, a phenomenon known as super-transient scaling. We develop a hybrid model that includes time delays along network links and allows for stochastic variation in the delays. Using this model, we explain the observed super-transient scaling of both motifs and recreate the experimentally measured transient distributions.
In this Chapter, we consider realizing a reservoir computer on an electronic chip that allows for many tens of network nodes whose connection topology can be quickly reconfigured. The reservoir computer displays analog-like behavior and has the potential to perform computations beyond that of a classic Turning machine. In detail, we present our preliminary results of using a physical reservoir computer for performing the task of identifying written digits. The reservoir is realized on a commercially-available electronic device known as a field-programmable gate array on which we create an autonomous Boolean network for information processing. Even though the network nodes are Boolean logic elements, they display analog behavior because their is no master clock that controls the nodes. In addition, the electronic signals related to the written-digit images are injected into the reservoir at high speed, leading to the possibility of full image classification on the nanosecond time scale. We explore the dynamics of the autonomous Boolean networks in response to injected signals and, based on these results, investigate the performance of the reservoir computer on the written-digit task. For a wide range of
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