Noise-aided logic in an electronic analog of synthetic genetic networks

Theoretical description of logical stochastic resonance and its enhancement: Fast Fourier transform filtering method

Over the past decade, dynamic schemes have been proposed for the use of bistable systems in the design of logic devices. A bistable system in a noisy background can operate as a reliable logic gate in a moderate noise level, which is called a logical stochastic resonance (LSR). In this paper, we theoretically explore the emergence of LSR in general bistable systems and identify the dynamical mechanisms of LSR. The timescale relationship between the measured time and the mean first-pass time of two-state transitions is a key condition in determining whether the system is reliable. Furthermore, we demonstrate that the stability of the logic operation can be significantly improved by choosing the appropriate filtering method. Low-pass filtered noise-driven systems are more stable than Gaussian white noise. However, band-pass and high-pass filtered noise are harmful to the stability of the system due to the filtering of low-frequency components. Our theoretical and numerical simulation results offer perspectives for the development of logic devices.

Emergent noise-aided logic through synchronization

In this article, we present a dynamical scheme to obtain a reconfigurable noise-aided logic gate that yields all six fundamental two-input logic operations, including the xor operation. The setup consists of two coupled bistable subsystems that are each driven by one subthreshold logic input signal, in the presence of a noise floor. The synchronization state of their outputs robustly maps to two-input logic operations of the driving signals, in an optimal window of noise and coupling strengths. Thus the interplay of noise, nonlinearity, and coupling leads to the emergence of logic operations embedded within the collective state of the coupled system. This idea is manifested using both numerical simulations and proof-of-principle circuit experiments. The regions in parameter space that yield reliable logic operations were characterized through a stringent measure of reliability, using both numerical and experimental data.

Effect of time delay in a bistable synthetic gene network

The essence of logical stochastic resonance is the dynamic manipulation of potential wells. The effect of time delay on the depth of potential wells and the width of a bistable region can be inferred by logic operations in the bistable system with time delay. In a time-delayed synthetic gene network, time delay in the synthesis process can increase the depth of the potential wells, while that in the degradation process, it can reduce the depth of the potential wells, which will result in a decrease in the width of the bistable region (the reason for time delay to induce logic operations without external driving force) and the instability of the system (oscillation). These two opposite effects imply stretching and folding, leading to complex dynamical behaviors of the system, including period, chaos, bubble, chaotic bubble, forward and reverse period doubling bifurcation, intermittency, and coexisting attractors.

Construction of logic gates exploiting resonance phenomena in nonlinear systems

A two-state system driven by two inputs has been found to consistently produce a response mirroring a logic function of the two inputs, in an optimal window of moderate noise. This phenomenon is called logical stochastic resonance (LSR). We extend the conventional LSR paradigm to implement higher-level logic architecture or typical digital electronic structures via carefully crafted coupling schemes. Further, we examine the intriguing possibility of obtaining reliable logic outputs from a noise-free bistable system, subject only to periodic forcing, and show that this system also yields a phenomenon analogous to LSR, termed Logical Vibrational Resonance (LVR), in an appropriate window of frequency and amplitude of the periodic forcing. Lastly, this approach is extended to realize morphable logic gates through the Logical Coherence Resonance (LCR) in excitable systems under the influence of noise. The results are verified with suitable circuit experiments, demonstrating the robustness of the LSR, LVR and LCR phenomena. This article is part of the theme issue 'Vibrational and stochastic resonance in driven nonlinear systems (part 1)'.

Periodic and aperiodic force-induced logical stochastic resonance in a bistable system

It was demonstrated recently that there are optimal windows of noise intensity or frequency and amplitude of the periodic driving force, which let a bistable system operate reliably as logic gates. These phenomena are called logical stochastic resonance (LSR). Given that the driving force is not always perfect regular, there may be phase disturbance in driving force; therefore, the Wiener process is used here to model phase disturbance of driving force, and then the effects of phase disturbance on reliability and agility of logic gates are explored in detail. Comparing with the periodic force, the aperiodic force with appropriate intensity phase disturbance can drive a bistable system to yield phenomena similar to LSR in a wider reliable region and can reduce mean switching time to obtain a faster response of logic devices to the input signal. On the other hand, depending on the amplitude and average angular frequency, moderate-intensity phase disturbance may also reduce success probability and increase mean switching time and thus lead to the instability and the slower response of logic devices.

Set-reset latch logic operation in a bistable system under suprathreshold and subthreshold signals

A set-reset latch is a basic building block of computers and can be used to store state information. Here, by testing the influence of the two logical input signals on the reliable set-reset latch logic operation in the bistable system, we found that there are two types of input signals, namely, suprathreshold and subthreshold signals. For the suprathreshold signals, reliable set-reset logic operation can be achieved without any driving forces and exhibits certain anti-interference ability; for the subthreshold signals, a single harmonic could induce correct set-reset latch logic operation but with a narrow optimal parameter region. The introduction of biharmonic-induced set-reset latch logic operation (logical vibrational resonance) could greatly expand the parameter region. Explanations for the above results were provided by taking the logical inputs as the dynamic bias to analyze the dynamic changes in the system. Finally, the results were further verified by circuit simulation and actual hardware circuit.

Chaotic attractor hopping yields logic operations

Certain nonlinear systems can switch between dynamical attractors occupying different regions of phase space, under variation of parameters or initial states. In this work we exploit this feature to obtain reliable logic operations. With logic output 0/1 mapped to dynamical attractors bounded in distinct regions of phase space, and logic inputs encoded by a very small bias parameter, we explicitly demonstrate that the system hops consistently in response to an external input stream, operating effectively as a reliable logic gate. This system offers the advantage that very low-amplitude inputs yield highly amplified outputs. Additionally, different dynamical variables in the system yield complementary logic operations in parallel. Further, we show that in certain parameter regions noise aids the reliability of logic operations, and is actually necessary for obtaining consistent outputs. This leads us to a generalization of the concept of Logical Stochastic Resonance to attractors more complex than fixed point states, such as periodic or chaotic attractors. Lastly, the results are verified in electronic circuit experiments, demonstrating the robustness of the phenomena. So we have combined the research directions of Chaos Computing and Logical Stochastic Resonance here, and this approach has potential to be realized in wide-ranging systems.

Identifying nodal properties that are crucial for the dynamical robustness of multistable networks

We investigate the collective dynamics of bistable elements connected in different network topologies and estimate the network response to localized perturbations on different classes of nodes by introducing a variant of the concept of multinode basin stability. We show that perturbations on nodes with high closeness and betweeness centrality drastically reduces the capacity of the system to return to the original state. This effect is most pronounced for a star network, where perturbation of the single hub node can destroy the collective state, while the system manages to recover even when a majority of the peripheral nodes are strongly perturbed. This demonstrates the extreme effect of the centrality of the perturbed node on the stability of the network. Further, we exploit the difference in centrality distributions in random scale-free networks with m=1 and m=2 to probe which property most influences the collective dynamics in heterogeneous networks. Significantly, we find clear evidence that the betweeness centrality of the perturbed node is more crucial for dynamical robustness than closeness centrality or degree of the node. This allows us to decide which nodes to safeguard in order to maintain the collective state of a network against targeted localized attacks.

Adaptive logical stochastic resonance in time-delayed synthetic genetic networks

In the paper, the concept of logical stochastic resonance is applied to implement logic operation and latch operation in time-delayed synthetic genetic networks derived from a bacteriophage λ. Clear logic operation and latch operation can be obtained when the network is tuned by modulated periodic force and time-delay. In contrast with the previous synthetic genetic networks based on logical stochastic resonance, the proposed system has two advantages. On one hand, adding modulated periodic force to the background noise can increase the length of the optimal noise plateau of obtaining desired logic response and make the system adapt to varying noise intensity. On the other hand, tuning time-delay can extend the optimal noise plateau to larger range. The result provides possible help for designing new genetic regulatory networks paradigm based on logical stochastic resonance.

Information-based measures for logical stochastic resonance in a synthetic gene network under Lévy flight superdiffusion

We investigate the logical information transmission of a synthetic gene network under Lévy flight superdiffusion by an information-based methodology. We first present the stochastic synthetic gene network model driven by a square wave signal under Lévy noise caused by Lévy flight superdiffusion. Then, to quantify the potential of logical information transmission and logical stochastic resonance, we theoretically obtain an information-based methodology of the symbol error rate, the noise entropy, and the mutual information of the logical information transmission. Consequently, based on the complementary "on" and "off" states shown in the logical information transmission for the repressive proteins, we numerically calculate the symbol error rate for logic gates, which demonstrate that the synthetic gene network under Lévy noise can achieve some logic gates as well as logical stochastic resonance. Furthermore, we calculate the noise entropy and the mutual information between the square wave signal and the logical information transmission, which reveal and quantify the potential of logical information transmission and logical stochastic resonance. In addition, we analyze the synchronization degree of the mutual information for the accomplished logical stochastic resonance of two repressive proteins of the synthetic gene network by synchronization variances, which shows that those mutual information changes almost synchronously.

Enhanced Logical Stochastic Resonance in Synthetic Genetic Networks

In this brief, the concept of logical stochastic resonance is applied to implement the Set-Reset latch in a synthetic gene network derived from a bacteriophage λ . Clear Set-Reset latch operation is obtained when the network is only subjected to periodic forcing. The correct probability of obtaining the desired logic operation first increases to unity and then decreases as the amplitude of the periodic forcing increases. In addition, the output logic operation can be easily morphed by tuning the frequency and the amplitude of the periodic forcing. At the same time, we indicate that adding moderate periodic forcing to the background Gaussian noise may increase the length of the optimal plateau of getting the desired logic operation in genetic regulatory network. We also point out that robust Set-Reset latch operation can be obtained using the interplay of periodic forcing and background noise when the noise strength is lower than what is required.

The Switch in a Genetic Toggle System with Lévy Noise

A bistable toggle switch is a paradigmatic model in the field of biology. The dynamics of the system induced by Gaussian noise has been intensively investigated, but Gaussian noise cannot incorporate large bursts typically occurring in real experiments. This paper aims to examine effects of variations from one protein imposed by a non-Gaussian Lévy noise, which is able to describe even large jumps, on the coherent switch and the on/off switch via the steady-state probability density, the joint steady-state probability density, and the mean first passage time. We find that a large burst of one protein due to the Lévy noises can induce coherent switches even with small noise intensities in contrast to the Gaussian case which requires large intensities for this. The influences of the stability index, skewness parameter and noise intensity on the on/off switch are analyzed, leading to an adjustment of the concentrations of both proteins and a decision which stable point to stay most. The mean first passage times show complex effects under Lévy noise, especially the stability index and skewness parameter. Our results also imply that the presence of non-Gaussian Lévy noises has fundamentally changed the escape mechanism in such a system compared with Gaussian noise.

A bio-inspired spatial patterning circuit

Lateral Inhibition (LI) is a widely conserved patterning mechanism in biological systems across species. Distinct from better-known Turing patterns, LI depend on cell-cell contact rather than diffusion. We built an in silico genetic circuit model to analyze the dynamic properties of LI. The model revealed that LI amplifies differences between neighboring cells to push them into opposite states, hence forming stable 2-D patterns. Inspired by this insight, we designed and implemented an electronic circuit that recapitulates LI patterning dynamics. This biomimetic system serve as a physical model to elucidate the design principle of generating robust patterning through spatial feedback, regardless of the underlying devices being biological or electrical.

Electronic implementation of a repressilator with quorum sensing feedback

We investigate the dynamics of a synthetic genetic repressilator with quorum sensing feedback. In a basic genetic ring oscillator network in which three genes inhibit each other in unidirectional manner, an additional quorum sensing feedback loop stimulates the activity of a chosen gene providing competition between inhibitory and stimulatory activities localized in that gene. Numerical simulations show several interesting dynamics, multi-stability of limit cycle with stable steady-state, multi-stability of different stable steady-states, limit cycle with period-doubling and reverse period-doubling, and infinite period bifurcation transitions for both increasing and decreasing strength of quorum sensing feedback. We design an electronic analog of the repressilator with quorum sensing feedback and reproduce, in experiment, the numerically predicted dynamical features of the system. Noise amplification near infinite period bifurcation is also observed. An important feature of the electronic design is the accessibility and control of the important system parameters.