Transient dynamics and their control in time-delay autonomous Boolean ring networks

Abstract

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.

Structured light entities, chaos and nonlocal maps

“…In this case evolution of spatial structure E n ( r) follows to point transformations [23]. The other case is a single spatial mode limit N f ∼ 1 when spatial filtering during each passage through the cavity is strong enough so E n ( r) has a predefined single mode shape, say TEM oo whose amplitude evolve in time as (8-9) [27] or as (11)(12) [9] in a presence of self-phase modulation. The most interesting case is the intermediate one N f > 1 when mode interactions are mediated by nonlinearities, diffraction and dispersion.…”

“…Wide aperture solid state lasers emit coherent multivortex phase-locked beams of rectangular symmetry [7][8][9]. Delayed feedback systems of coupled oscillators perform reservoir computations [10] with Boolean ring networks [11].…”

Challenges in the analysis of complex systems: introduction and overview

“…[137,138]), networks of Boolean switching circuits can readily incorporate time delays difficult to incorporate in differential equation models. This can lead to long-term transients and chimera states [139][140][141].…”

Mathematical Model of Strong Physically Unclonable Functions Based on Hybrid Boolean Networks

“…Current HBN models only consider a subset of the realistic experimental conditions present on FPGAs -namely noise on the signal, time-delays on the wires, and differences in logic operation time [3], [4], [8]. Each of these prior works on HBNs have emphasized the importance of variations to arXiv:2207.10816v1 [cs.CR] 20 Jul 2022 these parameters on the resulting dynamics.…”