Lyapunov exponents, noise-induced synchronization, and Parrondo's paradox

Parrondo's paradox reveals counterintuitive wins in biology and decision making in society

Parrondo's paradox refers to the paradoxical phenomenon of combining two losing strategies in a certain manner to obtain a winning outcome. It has been applied to uncover unexpected outcomes across various disciplines, particularly at different spatiotemporal scales within ecosystems. In this article, we provide a comprehensive review of recent developments in Parrondo's paradox within the interdisciplinary realm of the physics of life, focusing on its significant applications across biology and the broader life sciences. Specifically, we examine its relevance from genetic pathways and phenotypic regulation, to intercellular interaction within multicellular organisms, and finally to the competition between populations and species in ecosystems. This phenomenon, spanning multiple biological domains and scales, enhances our understanding of the unified characteristics of life and reveals that adaptability in a drastically changing environment, rather than the inherent excellence of a trait, underpins survival in the process of evolution. We conclude by summarizing our findings and discussing future research directions that hold promise for advancing the field.

Noise-induced Parrondo's paradox in discrete-time quantum walks

Parrondo's paradox refers to the apparently paradoxical effect whereby two or more dynamics in which a given quantity decreases are combined in such a way that the same quantity increases in the resulting dynamics. We show that noise can induce Parrondo's paradox in one-dimensional discrete-time quantum walks with deterministic periodic as well as aperiodic sequences of two-state quantum coins where this paradox does not occur in the absence of noise. Moreover, we show how the noise-induced Parrondo's paradox affects the time evolution of quantum entanglement for such quantum walks.

Taming non-stationary chimera states in locally coupled oscillators

The imperfect traveling chimera (ITC) state is a novel non-stationary chimera pattern in which the incoherent domain of oscillators spreads into the coherent domain. We investigate the ITC state in locally coupled pendulum oscillators with heterogeneous driving forces. We introduce the heterogeneous phase value in the driving forces by two different ways, namely, the random phase from uniform distribution and random phase directions with identical amplitude. We discover two transition mechanisms from ITC to coherent state through traveling chimera-like state by taking the two different phase heterogeneity. The transition phenomena are investigated using cylindrical and polar coordinate phase spaces. In the numerical study, we propose a quantitative measurement named "spatiotemporal consistency" strength for distinguishing the ITC from the traveling one. Our research facilitates the exploration of potential applications of heterogeneous interactions in neuroscience.

Parrondo's paradox in quantum walks with three coins

Parrondo's paradox refers to the apparently paradoxical effect whereby a certain combination of biased random walks displays a counterintuitive reversal of the bias direction. We show that Parrondo's paradox can occur not only in the case of one-dimensional discrete-time quantum walks with a deterministic sequence of two quantum coins but also in the case of one-dimensional discrete-time quantum walks with a deterministic sequence of three quantum coins. Moreover, we show how Parrondo's paradox affects the time evolution of quantum entanglement for such quantum walks.

Parrondo's paradox in quantum walks with deterministic aperiodic sequence of coins

Parrondo's effect is a well-known apparent paradox where a combination of biased random walks displays a counterintuitive reversal of the bias direction. We show that Parrondo's effect can occur not only in the case of one-dimensional discrete quantum walks with random or deterministic periodic sequence of two- or multistate quantum coins but also in the case of one-dimensional discrete quantum walks with deterministic aperiodic sequence of two-state quantum coins. Moreover, we show how Parrondo's effect affects the time evolution of the walker-coin quantum entanglement.

Relieving Cost of Epidemic by Parrondo's Paradox: A COVID-19 Case Study

COVID-19, also known as SARS-CoV-2, is a coronavirus that is highly pathogenic and virulent. It spreads very quickly through close contact, and so in response to growing numbers of cases, many countries have imposed lockdown measures to slow its spread around the globe. The purpose of a lockdown is to reduce reproduction, that is, the number of people each confirmed case infects. Lockdown measures have worked to varying extents but they come with a massive price. Nearly every individual, community, business, and economy has been affected. In this paper, switching strategies that take into account the total "cost" borne by a community in response to COVID-19 are proposed. The proposed cost function takes into account the health and well-being of the population, as well as the economic impact due to the lockdown. The model allows for a comparative study to investigate the effectiveness of various COVID-19 suppression strategies. It reveals that both the strategy to implement a lockdown and the strategy to maintain an open community are individually losing in terms of the total "cost" per day. However, switching between these two strategies in a certain manner can paradoxically lead to a winning outcome-a phenomenon attributed to Parrondo's paradox.

Stochastic master stability function for noisy complex networks

In this paper, we broaden the master stability function approach to study the stability of the synchronization manifold in complex networks of stochastic dynamical systems. We provide necessary and sufficient conditions for exponential stability that allow us to discriminate the impact of noise. We observe that noise can be beneficial for synchronization when it diffuses evenly in the network. On the contrary, an excessively large amount of noise only acting on a subset of the node state variables might have disruptive effects on the network synchronizability. To demonstrate our findings, we complement our theoretical derivations with extensive simulations on paradigmatic examples of networks of noisy systems.

Parrondo effect in quantum coin-toss simulations

Game A + Game B = Game C. Parrondo's games follow this basic structure where A and B are losing games and C is a winning game-a phenomenon called Parrondo's paradox. These games can take on a wider class of definitions and exhibit these paradoxical results. In this paper, we show three paradoxical cases. (i) The successive "tossing" of a single fair quantum coin gives a biased result, a previously known result. (ii) The random tossing of two quantum coins, each with successive biased expectations, gives an average random walk position of approximately zero. (iii) The sequential periodic tossing of two quantum coins, each with successive negative biased expectations, gives an average random walk with positive expectation. Using these results, we then propose a protocol for identifying and classifying quantum operations that span the same Hilbert space for a two-level quantum system.

Paradoxical Survival: Examining the Parrondo Effect across Biology

Parrondo's paradox, in which losing strategies can be combined to produce winning outcomes, has received much attention in mathematics and the physical sciences; a plethora of exciting applications has also been found in biology at an astounding pace. In this review paper, the authors examine a large range of recent developments of Parrondo's paradox in biology, across ecology and evolution, genetics, social and behavioral systems, cellular processes, and disease. Intriguing connections between numerous works are identified and analyzed, culminating in an emergent pattern of nested recurrent mechanics that appear to span the entire biological gamut, from the smallest of spatial and temporal scales to the largest-from the subcellular to the complete biosphere. In analyzing the macro perspective, the pivotal role that the paradox plays in the shaping of biological life becomes apparent, and its identity as a potential universal principle underlying biological diversity and persistence is uncovered. Directions for future research are also discussed in light of this new perspective.

Extended Parrondo's game and Brownian ratchets: strong and weak Parrondo effect

Inspired by the flashing ratchet, Parrondo's game presents an apparently paradoxical situation. Parrondo's game consists of two individual games, game A and game B. Game A is a slightly losing coin-tossing game. Game B has two coins, with an integer parameter M. If the current cumulative capital (in discrete unit) is a multiple of M, an unfavorable coin p(b) is used, otherwise a favorable p(g) coin is used. Paradoxically, a combination of game A and game B could lead to a winning game, which is the Parrondo effect. We extend the original Parrondo's game to include the possibility of M being either M(1) or M(2). Also, we distinguish between strong Parrondo effect, i.e., two losing games combine to form a winning game, and weak Parrondo effect, i.e., two games combine to form a better-performing game. We find that when M(2) is not a multiple of M(1), the combination of B(M(1)) and B(M(2)) has strong and weak Parrondo effect for some subsets in the parameter space (p(b),p(g)), while there is neither strong nor weak effect when M(2) is a multiple of M(1). Furthermore, when M(2) is not a multiple of M(1), a stochastic mixture of game A may cancel the strong and weak Parrondo effect. Following a discretization scheme in the literature of Parrondo's game, we establish a link between our extended Parrondo's game with the analysis of discrete Brownian ratchet. We find a relation between the Parrondo effect of our extended model to the macroscopic bias in a discrete ratchet. The slope of a ratchet potential can be mapped to the fair game condition in the extended model, so that under some conditions, the macroscopic bias in a discrete ratchet can provide a good predictor for the game performance of the extended model. On the other hand, our extended model suggests a design of a ratchet in which the potential is a mixture of two periodic potentials.

Parrondo's games based on complex networks and the paradoxical effect

Parrondo’s games were first constructed using a simple tossing scenario, which demonstrates the following paradoxical situation: in sequences of games, a winning expectation may be obtained by playing the games in a random order, although each game (game A or game B) in the sequence may result in losing when played individually. The available Parrondo’s games based on the spatial niche (the neighboring environment) are applied in the regular networks. The neighbors of each node are the same in the regular graphs, whereas they are different in the complex networks. Here, Parrondo’s model based on complex networks is proposed, and a structure of game B applied in arbitrary topologies is constructed. The results confirm that Parrondo’s paradox occurs. Moreover, the size of the region of the parameter space that elicits Parrondo’s paradox depends on the heterogeneity of the degree distributions of the networks. The higher heterogeneity yields a larger region of the parameter space where the strong paradox occurs. In addition, we use scale-free networks to show that the network size has no significant influence on the region of the parameter space where the strong or weak Parrondo’s paradox occurs. The region of the parameter space where the strong Parrondo’s paradox occurs reduces slightly when the average degree of the network increases.

Paradoxical persistence through mixed-system dynamics: towards a unified perspective of reversal behaviours in evolutionary ecology

Counterintuitive dynamics of various biological phenomena occur when composite system dynamics differ qualitatively from that of their component systems. Such composite systems typically arise when modelling situations with time-varying biotic or abiotic conditions, and examples range from metapopulation dynamics to population genetic models. These biological, and related physical, phenomena can often be modelled as simple financial games, wherein capital is gained and lost through gambling. Such games have been developed and used as heuristic devices to elucidate the processes at work in generating seemingly paradoxical outcomes across a spectrum of disciplines, albeit in a field-specific, ad hoc fashion. Here, we propose that studying these simple games can provide a much deeper understanding of the fundamental principles governing paradoxical behaviours in models from a diversity of topics in evolution and ecology in which fluctuating environmental effects, whether deterministic or stochastic, are an essential aspect of the phenomenon of interest. Of particular note, we find that, for a broad class of models, the ecological concept of equilibrium reactivity provides an intuitive necessary condition that must be satisfied in order for environmental variability to promote population persistence. We contend that further investigations along these lines promise to unify aspects of the study of a range of topics, bringing questions from genetics, species persistence and coexistence and the evolution of bet-hedging strategies, under a common theoretical purview.

Scaling of noisy fluctuations in complex networks and applications to network prediction

We study the collective dynamics of oscillator-network systems in the presence of noise. By focusing on the time-averaged fluctuation of dynamical variable of interest about the mean field, we discover a scaling law relating the average fluctuation to the node degree. The scaling law is quite robust as it holds for a variety of network topologies and node dynamics. Analyses and numerical support for different types of networks and node dynamics are provided. We also point out an immediate application of the scaling law: predicting complex networks based on time series only, and we articulate how this can be done.

Optimal sequence for Parrondo games

An algorithm based on backward induction is devised in order to compute the optimal sequence of games to be played in Parrondo games. The algorithm can be used to find the optimal sequence for any finite number of turns or in the steady state, showing that ABABB... is the sequence with the highest steady state average gain. The algorithm can also be generalized to find the optimal adaptive strategy in a multiplayer version of the games, where a finite number of players may choose, at every turn, the game the whole ensemble should play.

Effect of noise on generalized chaotic synchronization

When two characteristically different chaotic oscillators are coupled, generalized synchronization can occur. Motivated by the phenomena that common noise can induce and enhance complete synchronization or phase synchronization in chaotic systems, we investigate the effect of noise on generalized chaotic synchronization. We develop a phase-space analysis, which suggests that the effect can be system dependent in that common noise can either induce/enhance or destroy generalized synchronization. A prototype model consisting of a Lorenz oscillator coupled with a dynamo system is used to illustrate these phenomena.

Minimal Brownian ratchet: an exactly solvable model

We develop an analytically solvable three-state discrete-time minimal Brownian ratchet (MBR), where the transition probabilities between states are asymmetric. By solving the master equations, we obtain the steady-state probabilities. Generally, the steady-state solution does not display detailed balance, giving rise to an induced directional motion in the MBR. For a reduced two-dimensional parameter space, we find the null curve on which the net current vanishes and detailed balance holds. A system on this curve is said to be balanced. On the null curve, an additional source of external random noise is introduced to show that a directional motion can be induced under the zero overall driving force.

Noise-stabilized long-distance synchronization in populations of model neurons

Rhythmic, synchronous firing of groups of neurons is associated with behaviorally relevant states, and it is thus of interest to understand the mechanisms by which synchronization may be achieved. In hippocampal slice preparations, networks of excitatory and inhibitory neurons have been seen to synchronize when strong stimulation is applied at separated sites between which any coupling must be subject to a significant axonal delay. We extend previous work on synchronization in a model system based on the network architecture of these hippocampal slices. Our new analysis addresses the effects of heterogeneous populations and noisy inputs on the stability of synchronous solutions in the system. We find that, with experimentally motivated constraints on the coupling strength, sufficiently large heterogeneity in the input currents renders synchrony unstable. The addition of noise, however, restores stable near-synchrony. We analytically reduce the high-dimensional biophysical equations for the full population to a simple three-dimensional map, and show that the map's stability properties correctly predict both the loss of stability and the restabilizing effect of the noise.