Multiple Attractors Research Articles

Chaotic dynamics of allele frequencies in condition-dependent mating systems

I study the dynamics of allele frequencies in sexually reproducing populations where the choosy sex has a preference for condition-dependent displays of the opposite sex. The condition of an individual is assumed to be shaped by frequency-dependent selection. For sufficiently strong preferences the dynamics becomes increasingly complex, and periodic orbits and chaos are observed. Moreover, multiple attractors can exist simultaneously. The results hold also when the choosy sex is allowed to maintain a moderate level of assortative mating. Complex dynamics, a well studied phenomenon in a purely ecological setting, has been rarely observed in ecologically motivated population genetic models.

Landscapes of non-gradient dynamics without detailed balance: Stable limit cycles and multiple attractors

Landscape is one of the key notions in literature on biological processes and physics of complex systems with both deterministic and stochastic dynamics. The large deviation theory (LDT) provides a possible mathematical basis for the scientists' intuition. In terms of Freidlin-Wentzell's LDT, we discuss explicitly two issues in singularly perturbed stationary diffusion processes arisen from nonlinear differential equations: (1) For a process whose corresponding ordinary differential equation has a stable limit cycle, the stationary solution exhibits a clear separation of time scales: an exponential terms and an algebraic prefactor. The large deviation rate function attains its minimum zero on the entire stable limit cycle, while the leading term of the prefactor is inversely proportional to the velocity of the non-uniform periodic oscillation on the cycle. (2) For dynamics with multiple stable fixed points and saddles, there is in general a breakdown of detailed balance among the corresponding attractors. Two landscapes, a local and a global, arise in LDT, and a Markov jumping process with cycle flux emerges in the low-noise limit. A local landscape is pertinent to the transition rates between neighboring stable fixed points; and the global landscape defines a nonequilibrium steady state. There would be nondifferentiable points in the latter for a stationary dynamics with cycle flux. LDT serving as the mathematical foundation for emergent landscapes deserves further investigations.

Control of attractor preference by low-pass filtered noise in a multistable fiber laser

A diode-pumped erbium-doped fiber laser with pump modulation shows a large dynamic behavior including the coexistence of multiple attractors. In this work it is demonstrated numerically and experimentally that a low-pass noise filtering can control the probability for the appearance of a particular state. The results of numerical simulations with the use of a three-level laser model display good agreement with experimental results. We construct three-dimensional bifurcation diagrams of the probability using the noise amplitude and cutting frequency as control parameters. Finally, it is found the existence of one noise amplitude that cause one periodic orbit to appears more frequently compared to different noise amplitudes.

Attractors near grazing–sliding bifurcations

In this paper we prove, for the first time, that multistability can occur in three-dimensional Fillipov type flows due to grazing–sliding bifurcations. We do this by reducing the study of the dynamics of Filippov type flows around a grazing–sliding bifurcation to the study of appropriately defined one-dimensional maps. In particular, we prove the presence of three qualitatively different types of multiple attractors born in grazing–sliding bifurcations. Namely, a period-two orbit with a sliding segment may coexist with a chaotic attractor, two stable, period-two and period-three orbits with a segment of sliding each may coexist, or a non-sliding and period-three orbit with two sliding segments may coexist.

Transitions between multiple attractors in a granular experiment

We report on a granular experiment that produces multiple quasiperiodic pattern solutions in a rotating flat container filled with a bidisperse mixture of grains. The observed dynamics show a pronounced spatiotemporally periodic drift of segregated patterns. In the course of long experiment durations, the system switches between different states that appear to be stable oscillatory solutions over many periods. In the long-term limit, the system does not tend toward a stationary state. The results complement and extend previous observations in cylindrical geometries, and represent a challenge for modeling and theoretical description.

Neural Network Mechanisms Underlying Stimulus Driven Variability Reduction

It is well established that the variability of the neural activity across trials, as measured by the Fano factor, is elevated. This fact poses limits on information encoding by the neural activity. However, a series of recent neurophysiological experiments have changed this traditional view. Single cell recordings across a variety of species, brain areas, brain states and stimulus conditions demonstrate a remarkable reduction of the neural variability when an external stimulation is applied and when attention is allocated towards a stimulus within a neuron's receptive field, suggesting an enhancement of information encoding. Using an heterogeneously connected neural network model whose dynamics exhibits multiple attractors, we demonstrate here how this variability reduction can arise from a network effect. In the spontaneous state, we show that the high degree of neural variability is mainly due to fluctuation-driven excursions from attractor to attractor. This occurs when, in the parameter space, the network working point is around the bifurcation allowing multistable attractors. The application of an external excitatory drive by stimulation or attention stabilizes one specific attractor, eliminating in this way the transitions between the different attractors and resulting in a net decrease in neural variability over trials. Importantly, non-responsive neurons also exhibit a reduction of variability. Finally, this reduced variability is found to arise from an increased regularity of the neural spike trains. In conclusion, these results suggest that the variability reduction under stimulation and attention is a property of neural circuits.

Dynamic noise, chaos and parameter estimation in population biology

We revisit the parameter estimation framework for population biological dynamical systems, and apply it to calibrate various models in epidemiology with empirical time series, namely influenza and dengue fever. When it comes to more complex models such as multi-strain dynamics to describe the virus-host interaction in dengue fever, even the most recently developed parameter estimation techniques, such as maximum likelihood iterated filtering, reach their computational limits. However, the first results of parameter estimation with data on dengue fever from Thailand indicate a subtle interplay between stochasticity and the deterministic skeleton. The deterministic system on its own already displays complex dynamics up to deterministic chaos and coexistence of multiple attractors.

Quasiperiodic, periodic, and slowing-down states of coupled heteroclinic cycles

We investigate two coupled oscillators, each of which shows an attracting heteroclinic cycle in the absence of coupling. The two heteroclinic cycles are nonidentical. Weak coupling can lead to the elimination of the slowing-down state that asymptotically approaches the heteroclinic cycle for a single cycle, giving rise to either quasiperiodic motion with separate frequencies from the two cycles or periodic motion in which the two cycles are synchronized. The synchronization transition, which occurs via a Hopf bifurcation, is not induced by the commensurability of the two cycle frequencies but rather by the disappearance of the weaker frequency oscillation. For even larger coupling the motion changes via a resonant heteroclinic bifurcation to a slowing-down state corresponding to a single attracting heteroclinic orbit. Coexistence of multiple attractors can be found for some parameter regions. These results are of interest in ecological, sociological, neuronal, and other dynamical systems, which have the structure of coupled heteroclinic cycles.

Evolution before genes

BackgroundOur current understanding of evolution is so tightly linked to template-dependent replication of DNA and RNA molecules that the old idea from Oparin of a self-reproducing 'garbage bag' ('coacervate') of chemicals that predated fully-fledged cell-like entities seems to be farfetched to most scientists today. However, this is exactly the kind of scheme we propose for how Darwinian evolution could have occurred prior to template replication.ResultsWe cannot confirm previous claims that autocatalytic sets of organic polymer molecules could undergo evolution in any interesting sense by themselves. While we and others have previously imagined inhibition would result in selectability, we found that it produced multiple attractors in an autocatalytic set that cannot be selected for. Instead, we discovered that if general conditions are satisfied, the accumulation of adaptations in chemical reaction networks can occur. These conditions are the existence of rare reactions producing viable cores (analogous to a genotype), that sustains a molecular periphery (analogous to a phenotype).ConclusionsWe conclude that only when a chemical reaction network consists of many such viable cores, can it be evolvable. When many cores are enclosed in a compartment there is competition between cores within the same compartment, and when there are many compartments, there is between-compartment competition due to the phenotypic effects of cores and their periphery at the compartment level. Acquisition of cores by rare chemical events, and loss of cores at division, allows macromutation, limited heredity and selectability, thus explaining how a poor man's natural selection could have operated prior to genetic templates. This is the only demonstration to date of a mechanism by which pre-template accumulation of adaptation could occur.ReviewersThis article was reviewed by William Martin and Eugene Koonin.

A Periodically Forced Wilson–Cowan System with Multiple Attractors

Using symmetry and the Poincare map, it is shown that a periodically forced cubic-like differential equation (single-neuron equation) can have more than three periodic solutions. It is then possible to locate regions of parameter space where a related periodically forced Wilson–Cowan system can have three or more periodic attractors. This result is used to show that certain Wilson–Cowan systems, with low levels of inhibition and receiving periodic excitatory input of moderate amplitude, will oscillate if and only if they are initially in a relatively active state.

How anatomy shapes dynamics: a semi-analytical study of the brain at rest by a simple spin model

Resting state networks (RSNs) show a surprisingly coherent and robust spatiotemporal organization. Previous theoretical studies demonstrated that these patterns can be understood as emergent on the basis of the underlying neuroanatomical connectivity skeleton. Integrating the biologically realistic DTI/DSI-(Diffusion Tensor Imaging/Diffusion Spectrum Imaging)based neuroanatomical connectivity into a brain model of Ising spin dynamics, we found a system with multiple attractors, which can be studied analytically. The multistable attractor landscape thus defines a functionally meaningful dynamic repertoire of the brain network that is inherently present in the neuroanatomical connectivity. We demonstrate that the more entropy of attractors exists, the richer is the dynamical repertoire and consequently the brain network displays more capabilities of computation. We hypothesize therefore that human brain connectivity developed a scale free type of architecture in order to be able to store a large number of different and flexibly accessible brain functions.

Life Stages: Interactions and Spatial Patterns

In many stage-structured species, different life stages often occupy separate spatial niches in a heterogeneous environment. Life stages of the giant flour beetle Tribolium brevicornis (Leconte), in particular adults and pupae, occupy different locations in a homogeneous habitat. This unique spatial pattern does not occur in the well-studied stored grain pests T. castaneum (Herbst) and T. confusum (Duval). We propose density dependent dispersal as a causal mechanism for this spatial pattern. We model and explore the spatial dynamics of T. brevicornis with a set of four density dependent integrodifference and difference equations. The spatial model exhibits multiple attractors: a spatially uniform attractor and a patchy attractor with pupae and adults spatially separated. The model attractors are consistent with experimental observations.

Representational Switching by Dynamical Reorganization of Attractor Structure in a Network Model of the Prefrontal Cortex

The prefrontal cortex (PFC) plays a crucial role in flexible cognitive behavior by representing task relevant information with its working memory. The working memory with sustained neural activity is described as a neural dynamical system composed of multiple attractors, each attractor of which corresponds to an active state of a cell assembly, representing a fragment of information. Recent studies have revealed that the PFC not only represents multiple sets of information but also switches multiple representations and transforms a set of information to another set depending on a given task context. This representational switching between different sets of information is possibly generated endogenously by flexible network dynamics but details of underlying mechanisms are unclear. Here we propose a dynamically reorganizable attractor network model based on certain internal changes in synaptic connectivity, or short-term plasticity. We construct a network model based on a spiking neuron model with dynamical synapses, which can qualitatively reproduce experimentally demonstrated representational switching in the PFC when a monkey was performing a goal-oriented action-planning task. The model holds multiple sets of information that are required for action planning before and after representational switching by reconfiguration of functional cell assemblies. Furthermore, we analyzed population dynamics of this model with a mean field model and show that the changes in cell assemblies' configuration correspond to those in attractor structure that can be viewed as a bifurcation process of the dynamical system. This dynamical reorganization of a neural network could be a key to uncovering the mechanism of flexible information processing in the PFC.

Structural perturbations to population skeletons: transient dynamics, coexistence of attractors and the rarity of chaos.

BackgroundSimple models of insect populations with non-overlapping generations have been instrumental in understanding the mechanisms behind population cycles, including wild (chaotic) fluctuations. The presence of deterministic chaos in natural populations, however, has never been unequivocally accepted. Recently, it has been proposed that the application of chaos control theory can be useful in unravelling the complexity observed in real population data. This approach is based on structural perturbations to simple population models (population skeletons). The mechanism behind such perturbations to control chaotic dynamics thus far is model dependent and constant (in size and direction) through time. In addition, the outcome of such structurally perturbed models is [almost] always equilibrium type, which fails to commensurate with the patterns observed in population data.Methodology/Principal FindingsWe present a proportional feedback mechanism that is independent of model formulation and capable of perturbing population skeletons in an evolutionary way, as opposed to requiring constant feedbacks. We observe the same repertoire of patterns, from equilibrium states to non-chaotic aperiodic oscillations to chaotic behaviour, across different population models, in agreement with observations in real population data. Model outputs also indicate the existence of multiple attractors in some parameter regimes and this coexistence is found to depend on initial population densities or the duration of transient dynamics. Our results suggest that such a feedback mechanism may enable a better understanding of the regulatory processes in natural populations.

Weak Allee effects and species coexistence

In this article, we study the population dynamics of a two-species discrete-time competition model where each species suffers from either predator saturation induced Allee effects and/or mate limitation induced Allee effects. We focus on the following two possible outcomes of the competition: 1. one species goes to extinction; 2. the system is permanent. Our results indicate that, even if one species’ intra-specific competition is less than its inter-specific competition, weak Allee effects induced by predation saturation can promote coexistence of the two competing species. This is supported by the outcome of two-species competition models without Allee effects. Also, we discuss our results and future work on multiple attractors in competition models with Allee effects.

Inference and learning in sparse systems with multiple states

We discuss how inference can be performed when data are sampled from the nonergodic phase of systems with multiple attractors. We take as a model system the finite connectivity Hopfield model in the memory phase and suggest a cavity method approach to reconstruct the couplings when the data are separately sampled from few attractor states. We also show how the inference results can be converted into a learning protocol for neural networks in which patterns are presented through weak external fields. The protocol is simple and fully local, and is able to store patterns with a finite overlap with the input patterns without ever reaching a spin-glass phase where all memories are lost.

Nonlinear stochastic dynamics of mesoscopic homogeneous biochemical reaction systems—an analytical theory

The nonlinear dynamics of biochemical reactions in a small-sized system on the order of a cell are stochastic. Assuming spatial homogeneity, the populations of n molecular species follow a multi-dimensional birth-and-death process on . We introduce the Delbrück–Gillespie process, a continuous-time Markov jump process, whose Kolmogorov forward equation has been known as the chemical master equation, and whose stochastic trajectories can be computed via the Gillespie algorithm. Using simple models, we illustrate that a system of nonlinear ordinary differential equations on emerges in the infinite system size limit. For finite system size, transitions among multiple attractors of the nonlinear dynamical system are rare events with exponentially long transit times. There is a separation of time scales between the deterministic ODEs and the stochastic Markov jumps between attractors. No diffusion process can provide a global representation that is accurate on both short and long time scales for the nonlinear, stochastic population dynamics. On the short time scale and near deterministic stable fixed points, Ornstein–Uhlenbeck Gaussian processes give linear stochastic dynamics that exhibit time-irreversible circular motion for open, driven chemical systems. Extending this individual stochastic behaviour-based nonlinear population theory of molecular species to other biological systems is discussed.

Occurrence of multiple attractor bifurcations in the two-dimensional piecewise linear normal form map

Multiple attractor bifurcations occurring in piecewise smooth dynamical systems may lead to potentially damaging situations. In order to avoid these in physical systems, it is necessary to know their conditions of occurrence. Using the piecewise-linear 2D normal form, we investigate which types of multiple attractor bifurcations may occur and where in the parameter space they can be expected. For piecewise smooth maps, multiple attractor bifurcations will be expected to occur if the condition we identified for the piecewise-linear 2D normal form are satisfied in the close neighborhood of the border.

Complex Dynamics of Discrete SEIS Models with Simple Demography

We investigate bifurcations and dynamical behaviors of discrete SEIS models with exogenous reinfections and a variety of treatment strategies. Bifurcations identified from the models include period doubling, backward, forward‐backward, and multiple backward bifurcations. Multiple attractors, such as bistability and tristability, are observed. We also estimate the ultimate boundary of the infected regardless of initial status. Our rigorously mathematical analysis together with numerical simulations show that epidemiological factors alone can generate complex dynamics, though demographic factors only support simple equilibrium dynamics. Our model analysis supports and urges to treat a fixed percentage of exposed individuals.

Web toolkit: an agent based scalable search engine using cellular automata based classification for duly ranked retrieved data

Web page classification is a major issue for categorising web documents to facilitate indexing, search and retrieval of web pages for search engine. Different crawling techniques have been utilised to accumulate web pages of different domains under separate databases depending on practical scenario. Downloaded web pages are being parsed for further processing. A classifier is designed dynamically using single cycle multiple attractor cellular automata for mapping downloaded web pages of different domains into specific structure. This paper proposes alternate technique for automatic categorisation of web pages into different domains. Retrieved web pages have been ranked automatically at the time of classifier formation. Typically, our system consists of crawling, ranking and storage parts created in a different way. Hierarchical concept has been used over parallel crawler. GF(2P) concept is introduced in ranking. The concept of SMACA has been utilised in indexing storage. Overall, a search engine module has been created using agent-based method.