Point Attractor Research Articles

Amplitude death phenomena in delay-coupled Hamiltonian systems

Hamiltonian systems, when coupled via time-delayed interactions, do not remain conservative. In the uncoupled system, the motion can typically be periodic, quasiperiodic, or chaotic. This changes drastically when delay coupling is introduced since now attractors can be created in the phase space. In particular, for sufficiently strong coupling there can be amplitude death (AD), namely, the stabilization of point attractors and the cessation of oscillatory motion. The approach to the state of AD or oscillation death is also accompanied by a phase flip in the transient dynamics. A discussion and analysis of the phenomenology is made through an application to the specific cases of harmonic as well as anharmonic coupled oscillators, in particular the Hénon-Heiles system.

COEXISTENCE OF POINT, PERIODIC AND STRANGE ATTRACTORS

For a dynamical system described by a set of autonomous ordinary differential equations, an attractor can be a point, a periodic cycle, or even a strange attractor. Recently, a new chaotic system with only one stable equilibrium was described, which locally converges to the stable equilibrium but is globally chaotic. This paper further shows that for certain parameters, besides the point attractor and chaotic attractor, this system also has a coexisting stable limit cycle, demonstrating that this new system is truly complicated and interesting.

Evidence of alternate attractors from a whole-ecosystem regime shift experiment

Ecosystems sometimes shift between different states or dynamic regimes. Theory attributes these shifts to multiple ecosystem attractors. However, documenting multiple ecosystem attractors is difficult, particularly at spatial and temporal scales relevant to ecosystem management. We manipulated the fish community of a lake with the goal of causing trophic cascades and shifting the food web from a planktivore-dominated state to an alternate piscivore-dominated state. We evaluated evidence that the shifts in the fish community comprise alternate attractors using two complementary approaches. First, we calculated phase space trajectories to visualize the shift between attractors. Second, we computed generalized autoregressive conditional heteroskedasticity (GARCH) models and the Brock–Dechert–Scheinkman (BDS) test for linearity. The reconstructed phase space trajectories show the system departing a point attractor, entering a limit cycle, and then shifting to a new point attractor. The GARCH and BDS results indicate that linear explanations are not sufficient to explain the observed patterns. The results provide evidence for alternate attractors based on high-frequency time series of field measurements.

Network Class Superposition Analyses

Networks are often used to understand a whole system by modeling the interactions among its pieces. Examples include biomolecules in a cell interacting to provide some primary function, or species in an environment forming a stable community. However, these interactions are often unknown; instead, the pieces' dynamic states are known, and network structure must be inferred. Because observed function may be explained by many different networks (e.g., for the yeast cell cycle process [1]), considering dynamics beyond this primary function means picking a single network or suitable sample: measuring over all networks exhibiting the primary function is computationally infeasible. We circumvent that obstacle by calculating the network class ensemble. We represent the ensemble by a stochastic matrix , which is a transition-by-transition superposition of the system dynamics for each member of the class. We present concrete results for derived from Boolean time series dynamics on networks obeying the Strong Inhibition rule, by applying to several traditional questions about network dynamics. We show that the distribution of the number of point attractors can be accurately estimated with . We show how to generate Derrida plots based on . We show that -based Shannon entropy outperforms other methods at selecting experiments to further narrow the network structure. We also outline an experimental test of predictions based on . We motivate all of these results in terms of a popular molecular biology Boolean network model for the yeast cell cycle, but the methods and analyses we introduce are general. We conclude with open questions for , for example, application to other models, computational considerations when scaling up to larger systems, and other potential analyses.

Attractor behaviour in ELKO cosmology

We study the dynamics of ELKO in the context of accelerated phase of our universe. To avoid the fine tuning problem associated with the initialconditions, it is required that the dynamical equations lead to an early-time attractor. In the earlier works, it was shown that the dynamical equations containing ELKO fields do not lead to early-time stable fixed points. In this work, using redefinition of variables, we show that ELKO cosmology admits early-time stable fixed points. More interestingly, we show that ELKO cosmology admit two sets of attractor points corresponding to slow and fast-roll inflation. The fast-roll inflation attractor point is unique for ELKO as it is independent of the form of the potential. We also discuss the plausible choice of interaction terms in these two sets of attractor points and constraints on the coupling constant.

Dynamical Movement Primitives: Learning Attractor Models for Motor Behaviors

Nonlinear dynamical systems have been used in many disciplines to model complex behaviors, including biological motor control, robotics, perception, economics, traffic prediction, and neuroscience. While often the unexpected emergent behavior of nonlinear systems is the focus of investigations, it is of equal importance to create goal-directed behavior (e.g., stable locomotion from a system of coupled oscillators under perceptual guidance). Modeling goal-directed behavior with nonlinear systems is, however, rather difficult due to the parameter sensitivity of these systems, their complex phase transitions in response to subtle parameter changes, and the difficulty of analyzing and predicting their long-term behavior; intuition and time-consuming parameter tuning play a major role. This letter presents and reviews dynamical movement primitives, a line of research for modeling attractor behaviors of autonomous nonlinear dynamical systems with the help of statistical learning techniques. The essence of our approach is to start with a simple dynamical system, such as a set of linear differential equations, and transform those into a weakly nonlinear system with prescribed attractor dynamics by means of a learnable autonomous forcing term. Both point attractors and limit cycle attractors of almost arbitrary complexity can be generated. We explain the design principle of our approach and evaluate its properties in several example applications in motor control and robotics.

Rapid processing of chemosensor transients in a neuromorphic implementation of the insect macroglomerular complex

We present a biologically-constrained neuromorphic spiking model of the insect antennal lobe macroglomerular complex that encodes concentration ratios of chemical components existing within a blend, implemented using a set of programmable logic neuronal modeling cores. Depending upon the level of inhibition and symmetry in its inhibitory connections, the model exhibits two dynamical regimes: fixed point attractor (winner-takes-all type), and limit cycle attractor (winnerless competition type) dynamics. We show that, when driven by chemosensor input in real-time, the dynamical trajectories of the model's projection neuron population activity accurately encode the concentration ratios of binary odor mixtures in both dynamical regimes. By deploying spike timing-dependent plasticity in a subset of the synapses in the model, we demonstrate that a Hebbian-like associative learning rule is able to organize weights into a stable configuration after exposure to a randomized training set comprising a variety of input ratios. Examining the resulting local interneuron weights in the model shows that each inhibitory neuron competes to represent possible ratios across the population, forming a ratiometric representation via mutual inhibition. After training the resulting dynamical trajectories of the projection neuron population activity show amplification and better separation in their response to inputs of different ratios. Finally, we demonstrate that by using limit cycle attractor dynamics, it is possible to recover and classify blend ratio information from the early transient phases of chemosensor responses in real-time more rapidly and accurately compared to a nearest-neighbor classifier applied to the normalized chemosensor data. Our results demonstrate the potential of biologically-constrained neuromorphic spiking models in achieving rapid and efficient classification of early phase chemosensor array transients with execution times well beyond biological timescales.

Nonlinear Analysis of Guillain Barré Time Series to Elucidate Its Epidemiology

The etiology of Guillain Barré Syndrome (GBS) is not fully clarified, and there is a lack of agreement concerning its putative epidemic character. The low incidence rate of this disease is a disadvantage for employing the traditional statistical methods used in the analysis of epidemics. The objective of this paper is to clarify the GBS epidemic behavior applying a nonlinear time series identification approach. The authors obtained one time series of GBS and nine series of classical infectious epidemics (5 national and 4 international). These data were processed with advanced techniques of statistical time series analysis. This paper shows that GBS behaves similar to the other time series of classical epidemic studied. It corresponds to a nonlinear dynamics, with a point attractor. The spectral analysis pointed to an annual periodicity, and preference for the warmest month of the year was found. These results might suggest that Guillain Barré Syndrome has an epidemic behavior. The adequacy of nonlinear methods for analyzing the dynamics of epidemics, particularly those with low incidence rate, such as GBS was revealed.

Model-based bifurcation and power spectral analyses of thalamocortical alpha rhythm slowing in Alzheimer's Disease

The focus of this paper is to correlate the bifurcation behaviour of a thalamocortical neural mass model with the power spectral alpha (8–13Hz) oscillatory activity in Electroencephalography (EEG). The aim is to understand the neural correlates of alpha rhythm slowing (decrease in mean frequency of oscillation), a hallmark in the EEG of Alzheimer's Disease (AD) patients. The neural mass model used, referred to herein as the modARm, is a modified version of Lopes da Silva's alpha rhythm model (ARm). Previously, the power spectral behaviour of the modARm was analysed in context to AD. In this work, we revisit the modARm to make a combined study of the dynamical behaviour of the model and its power spectral behaviour within the alpha band while simulating the hallmark neuropathological condition of ‘synaptic depletion’ in AD. The results show that the modARm exhibits two ‘operating modes’ in the time-domain i.e. a point attractor and a limit cycle mode; the alpha rhythmic content in the model output is maximal at the vicinity of the point of bifurcation. Furthermore, the inhibitory synaptic connectivity from the cells of the Thalamic Reticular Nucleus to the Thalamo-Cortical Relay cells significantly influence bifurcation behaviour—while a decrease in the inhibition can induce limit-cycle behaviour corresponding to abnormal brain states such as seizures, an increase in inhibition in awake state corresponding to a point attractor mode may result in the slowing of the alpha rhythms as observed in AD. These observations help emphasise the importance of bifurcation analysis of model behaviour in inferring the biological relevance of results obtained from power-spectral analysis of the neural models in the context of understanding neurodegeneration.

THE INFLUENCE OF INCREASING LIFE EXPECTANCY ON THE DYNAMICS OF SIRS SYSTEMS WITH IMMUNE BOOSTING

AbstractEndemic infectious diseases constantly circulate in human populations, with prevalence fluctuating about a (theoretical and unobserved) time-independent equilibrium. For diseases for which acquired immunity is not lifelong, the classic susceptible–infectious–recovered–susceptible (SIRS) model provides a framework within which to consider temporal trends in the observed epidemiology. However, in some cases (notably pertussis), sustained multiannual fluctuations are observed, whereas the SIRS model is characterized by damped oscillatory dynamics for all biologically meaningful choices of model parameters. We show that a model that allows for “boosting” of immunity may naturally give rise to undamped oscillatory behaviour for biologically realistic parameter choices. The life expectancy of the population is critical in determining the characteristic dynamics of the system. For life expectancies up to approximately$50$years, we find that, even with boosting, damped oscillatory dynamics persist. For increasing life expectancy, the system may sustain oscillatory dynamics, or even exhibit bistable behaviour, in which both stable point attractor and limit cycle dynamics may coexist. Our results suggest that rising life expectancy may induce changes in the characteristic dynamics of infections for which immunity is not lifelong, with potential implications for disease control strategies.

Cosmology of axions and moduli: A dynamical systems approach

This paper is concerned with string cosmology and the dynamics of multiple scalar fields in potentials that can become negative, and their features as (early) dark energy models. Our point of departure is the ``string axiverse,'' a scenario that motivates the existence of cosmologically light axion fields as a generic consequence of string theory. We couple such an axion to its corresponding modulus. We give a detailed presentation of the rich cosmology of such a model, ranging from the setting of initial conditions on the fields during inflation, to the asymptotic future. We present some simplifying assumptions based on the fixing of the axion decay constant ${f}_{a}$, and on the effective field theory when the modulus trajectory is adiabatic, and find the conditions under which these assumptions break down. As a by-product of our analysis, we find that relaxing the assumption of fixed ${f}_{a}$ leads to the appearance of a new metastable de Sitter region for the modulus without the need for uplifting by an additional constant. A dynamical systems analysis reveals the existence of many fixed point attractors, repellers and saddle points, which we analyze in detail. We also provide geometric interpretations of the phase space. The fixed points can be used to bound the couplings in the model. A systematic scan of certain regions of parameter space reveals that the future evolution of the Universe in this model can be rich, containing multiple epochs of accelerated expansion.

A Chaotic Feature Extracting BAM and Its Application in Implementing Memory Search

An unsupervised trained, chaotic BAM composed of units with saturated limits logistic function can extract intrinsic features of data and exhibit various associative dynamics for different values of transmission parameters of the neurons' activation functions during recall. The output behavior of its units can be a fixed point or periodic attractor, a constrained aperiodic attractor consisting of one or more stored patterns, or a chaotic attractor. This characteristic indicates that the model is a promising technique that can be applied to information processing, such as pattern recognition and memory recall. However, controlling the amount of output variability and stabilizing it in a desired attractor is a crucial issue in practice. In this work it is shown that the transmission parameters of the units' activation functions play a significant role in identifying the output behavior. Using different time-series generated by the trained network, Largest Lyapunov Exponent is computed for different values of transmission parameter. Then, critical values of this parameter that lead to the highest chaotic behavior for each unit are stored and used to set the network during recall. Interaction between some chaotic feature units and some fixed-point ones produces desired behaviors with various degrees of uncertainty. An evolutionary algorithm is then introduced to find the units that should work chaotically to generate the desired behavior. Achievement of this method implies that such a controlled chaotic feature extracting BAM can be feasibly applied to information processing.

Attractor Dynamics of Spatially Correlated Neural Activity in the Limbic System

Attractor networks are a popular computational construct used to model different brain systems. These networks allow elegant computations that are thought to represent a number of aspects of brain function. Although there is good reason to believe that the brain displays attractor dynamics, it has proven difficult to test experimentally whether any particular attractor architecture resides in any particular brain circuit. We review models and experimental evidence for three systems in the rat brain that are presumed to be components of the rat's navigational and memory system. Head-direction cells have been modeled as a ring attractor, grid cells as a plane attractor, and place cells both as a plane attractor and as a point attractor. Whereas the models have proven to be extremely useful conceptual tools, the experimental evidence in their favor, although intriguing, is still mostly circumstantial.

Modified post-bifurcation dynamics and routes to chaos from double-Hopf bifurcations in a hyperchaotic system

In order to understand the onset of hyperchaotic behavior recently observed in many systems, we study bifurcations in the modified Chen system leading from simple dynamics into chaotic regimes. In particular, we demonstrate that the existence of only one fixed point of the system in all regions of parameter space implies that this simple point attractor may only be destabilized via a Hopf or double Hopf bifurcation as system parameters are varied. Saddle-node, transcritical and pitchfork bifurcations are precluded. The normal form immediately following double Hopf bifurcations is constructed analytically by the method of multiple scales. Analysis of this generalized double Hopf normal form along standard lines reveals possible regimes of periodic solutions, two-period tori, and three-period tori in parameter space. However, considering these more carefully, we find that only certain combinations or sequences of these dynamical regimes are possible, while others derived and considered in earlier work are in fact mathematically impossible. We also discuss the post-bifurcation dynamics in the context of two intermittent routes to chaos (routes following either (i) subcritical or (ii) supercritical Hopf or double Hopf bifurcations). In particular, the route following supercritical bifurcations is somewhat subtle. Such behavior following repeated Hopf bifurcations is well-known and widely observed, including in the classical Ruelle–Takens and quasiperiodic routes to chaos. However, to the best of our knowledge, it has not been considered in the context of the double-Hopf normal form, although it has been numerically observed and tracked in the post-double Hopf regime. Numerical simulations are employed to corroborate these various predictions from the normal form. They reveal the existence of stable periodic and toroidal attractors in the post-supercritical-Hopf cases, and either attractors at infinity or bounded chaotic dynamics following subcritical Hopf bifurcations. Future work will map out the remainder of the routes into the chaotic regimes, including further bifurcations of the post-supercritical-Hopf two- and three-tori via either torus doubling or breakdown.

Stimulus and Network Dynamics Collide in a Ratiometric Model of the Antennal Lobe Macroglomerular Complex

Time is considered to be an important encoding dimension in olfaction, as neural populations generate odour-specific spatiotemporal responses to constant stimuli. However, during pheromone mediated anemotactic search insects must discriminate specific ratios of blend components from rapidly time varying input. The dynamics intrinsic to olfactory processing and those of naturalistic stimuli can therefore potentially collide, thereby confounding ratiometric information. In this paper we use a computational model of the macroglomerular complex of the insect antennal lobe to study the impact on ratiometric information of this potential collision between network and stimulus dynamics. We show that the model exhibits two different dynamical regimes depending upon the connectivity pattern between inhibitory interneurons (that we refer to as fixed point attractor and limit cycle attractor), which both generate ratio-specific trajectories in the projection neuron output population that are reminiscent of temporal patterning and periodic hyperpolarisation observed in olfactory antennal lobe neurons. We compare the performance of the two corresponding population codes for reporting ratiometric blend information to higher centres of the insect brain. Our key finding is that whilst the dynamically rich limit cycle attractor spatiotemporal code is faster and more efficient in transmitting blend information under certain conditions it is also more prone to interference between network and stimulus dynamics, thus degrading ratiometric information under naturalistic input conditions. Our results suggest that rich intrinsically generated network dynamics can provide a powerful means of encoding multidimensional stimuli with high accuracy and efficiency, but only when isolated from stimulus dynamics. This interference between temporal dynamics of the stimulus and temporal patterns of neural activity constitutes a real challenge that must be successfully solved by the nervous system when faced with naturalistic input.

A phenomenological model of seizure initiation suggests network structure may explain seizure frequency in idiopathic generalised epilepsy

We describe a phenomenological model of seizure initiation, consisting of a bistable switch between stable fixed point and stable limit-cycle attractors. We determine a quasi-analytic formula for the exit time problem for our model in the presence of noise. This formula--which we equate to seizure frequency--is then validated numerically, before we extend our study to explore the combined effects of noise and network structure on escape times. Here, we observe that weakly connected networks of 2, 3 and 4 nodes with equivalent first transitive components all have the same asymptotic escape times. We finally extend this work to larger networks, inferred from electroencephalographic recordings from 35 patients with idiopathic generalised epilepsies and 40 controls. Here, we find that network structure in patients correlates with smaller escape times relative to network structures from controls. These initial findings are suggestive that network structure may play an important role in seizure initiation and seizure frequency.

Numerical Solutions of a Variable‐Order Fractional Financial System

The numerical solution of a variable‐order fractional financial system is calculated by using the Adams‐Bashforth‐Moulton method. The derivative is defined in the Caputo variable‐order fractional sense. Numerical examples show that the Adams‐Bashforth‐Moulton method can be applied to solve such variable‐order fractional differential equations simply and effectively. The convergent order of the method is also estimated numerically. Moreover, the stable equilibrium point, quasiperiodic trajectory, and chaotic attractor are found in the variable‐order fractional financial system with proper order functions.

Time Scale Hierarchies in the Functional Organization of Complex Behaviors

Traditional approaches to cognitive modelling generally portray cognitive events in terms of ‘discrete’ states (point attractor dynamics) rather than in terms of processes, thereby neglecting the time structure of cognition. In contrast, more recent approaches explicitly address this temporal dimension, but typically provide no entry points into cognitive categorization of events and experiences. With the aim to incorporate both these aspects, we propose a framework for functional architectures. Our approach is grounded in the notion that arbitrary complex (human) behaviour is decomposable into functional modes (elementary units), which we conceptualize as low-dimensional dynamical objects (structured flows on manifolds). The ensemble of modes at an agent’s disposal constitutes his/her functional repertoire. The modes may be subjected to additional dynamics (termed operational signals), in particular, instantaneous inputs, and a mechanism that sequentially selects a mode so that it temporarily dominates the functional dynamics. The inputs and selection mechanisms act on faster and slower time scales then that inherent to the modes, respectively. The dynamics across the three time scales are coupled via feedback, rendering the entire architecture autonomous. We illustrate the functional architecture in the context of serial behaviour, namely cursive handwriting. Subsequently, we investigate the possibility of recovering the contributions of functional modes and operational signals from the output, which appears to be possible only when examining the output phase flow (i.e., not from trajectories in phase space or time).

Quantum-behaved particle swarm optimization with Gaussian distributed local attractor point

This paper proposes a novel variant of quantum-behaved particle swarm optimization (QPSO) algorithm with the local attractor point subject to a Gaussian probability distribution (GAQPSO). The local attractor point in QPSO plays an important in that determining the convergence behavior of an individual particle. As such, the mean value and standard deviation of the proposed Gaussian probability distribution in GAQPSO are carefully selected. The distributions and diversities of the local attractor points in GAQPSO and QPSO are evaluated and compared. For the purpose of comparison, two variants of the GAQPSO algorithm are proposed by using a mutation probability and other types of probability distribution. The GAQPSO has been comprehensively evaluated on the suite of CEC2005 benchmark functions, and the experimental results are compared with those of the PSO and QPSO algorithms based on different probability distributions. It is shown by the results that the GAQPSO algorithm is an effective approach that can improve the QPSO performance considerably, that is, the GAQPSO algorithm is less likely to be stuck in local optima and hence it can achieve better solutions in most cases.

Generalized attractor points in gauged supergravity

The attractor mechanism governs the near-horizon geometry of extremal black holes in ungauged four-dimensional $N=2$ supergravity theories and in Calabi-Yau compactifications of string theory. In this paper, we study a natural generalization of this mechanism to solutions of arbitrary 4D $N=2$ gauged supergravities. We define generalized attractor points as solutions of an ansatz which reduces the Einstein, gauge field, and scalar equations of motion to algebraic equations. The simplest generalized attractor geometries are characterized by nonvanishing constant anholonomy coefficients in an orthonormal frame. Basic examples include Lifshitz and Schr\"odinger solutions, as well as anti-de Sitter and de Sitter vacua. There is a generalized attractor potential whose critical points are the attractor points, and its extremization explains the algebraic nature of the equations governing both supersymmetric and nonsupersymmetric attractors.