The behavior of measures of non-linear dynamics in 24 hours heart rate variability analysis.
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Cardiomyopathy, Dilated↗
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Computing with dynamic attractors in neural networks.
In this paper we report on some new architectures for neural computation, motivated in part by biological considerations. One of our goals is to demonstrate that it is just as easy for a neural net to compute with arbitrary attractors--oscillatory or chaotic--as with the more usual asymptotically stable fixed points. The advantages (if any) of such architectures are currently being investigated; but it seems reasonable that the much richer dynamics of recurrent networks, so obvious in recordings of brain activity, must be useful for something. On the other hand, the constraints of computing with biological wet-ware may make chaotic dynamics unavoidable in complex nervous systems. We hypothesize also that the as yet unrivaled capabilities of the human brain derive from an ability to integrate both analog intuitive pattern recognition operations, and digital symbolic logical operations at the ground level of its hardware. To investigate these possibilities, we have constructed a parallel distributed processing architecture inspired by the structure and dynamics of cerebral cortex. The construction assumes that cortex is a set of coupled associative memories with dynamic attractors. It is guided also by a particular concept of the physical structure required of macroscopic computational systems in general for reliable computation in the presence of noise. Our challenge is to accomplish real tasks that brains can do, using ordinary differential equations, in networks that are as faithful as possible to the known anatomy and dynamics of cortex.
Algorithms↗
Dynamics of the neural discharge in snail neurons.
Spike trains recorded under weak sinusoidal driving from central neurons of Lymnaea stagnalis appear quite irregular and envisage the possibility of an underlying chaotic process. Therefore, the sequences of interspike intervals are analyzed in the framework of non-linear dynamics. Since, for several reasons, these sequences are rather short, the analysis is performed by using methods of non-linear forecasting. To reject the null hypothesis that the original time series is a realization of a linear stochastic process with the same autocorrelation function, the results obtained on the original data are compared with those from surrogate data sets. Some 'non-linear' predictability occurs only in narrow regions of the space of stimulus parameters and the frequency of perturbation is critical in determining it. Moreover, it is shown that such behavior can be qualitatively mimicked by the FitzHugh-Nagumo model driven by a weak sinusoidal signal plus noise. It is argued that the narrowness of the non-linear predictability regions renders quite unlikely the detection of deterministic dynamics in the activity of these neurons.
Action Potentials↗
Competing populations in flows with chaotic mixing.
We investigate the effects of spatial heterogeneity on the coexistence of competing species in the case when the heterogeneity is dynamically generated by environmental flows with chaotic mixing properties. We show that one effect of chaotic advection on the passively advected species (such as phytoplankton, or self-replicating macro-molecules) is the possibility of coexistence of more species than that limited by the number of niches they occupy. We derive a novel set of dynamical equations for competing populations.
Competitive Behavior↗
Quantification of the spatial aspect of chaotic dynamics in biological and chemical systems.
The need to study spatio-temporal chaos in a spatially extended dynamical system which exhibits not only irregular, initial-value sensitive temporal behavior but also the formation of irregular spatial patterns, has increasingly been recognized in biological science. While the temporal aspect of chaotic dynamics is usually characterized by the dominant Lyapunov exponent, the spatial aspect can be quantified by the correlation length. In this paper, using the diffusion-reaction model of population dynamics and considering the conditions of the system stability with respect to small heterogeneous perturbations, we derive an analytical formula for an 'intrinsic length' which appears to be in a very good agreement with the value of the correlation length of the system. Using this formula and numerical simulations, we analyze the dependence of the correlation length on the system parameters. We show that our findings may lead to a new understanding of some well-known experimental and field data as well as affect the choice of an adequate model of chaotic dynamics in biological and chemical systems.
Animals↗
A computational perspective on the neural basis of multisensory spatial representations.
We argue that current theories of multisensory representations are inconsistent with the existence of a large proportion of multimodal neurons with gain fields and partially shifting receptive fields. Moreover, these theories do not fully resolve the recoding and statistical issues involved in multisensory integration. An alternative theory, which we have recently developed and review here, has important implications for the idea of 'frame of reference' in neural spatial representations. This theory is based on a neural architecture that combines basis functions and attractor dynamics. Basis function units are used to solve the recoding problem, whereas attractor dynamics are used for optimal statistical inferences. This architecture accounts for gain fields and partially shifting receptive fields, which emerge naturally as a result of the network connectivity and dynamics.
Animals↗
Observation of a continuous interior crisis in the Hindmarsh-Rose neuron model.
Interior crises are understood as discontinuous changes of the size of a chaotic attractor that occur when an unstable periodic orbit collides with the chaotic attractor. We present here numerical evidence and theoretical reasoning which prove the existence of a chaos-chaos transition in which the change of the attractor size is sudden but continuous. This occurs in the Hindmarsh-Rose model of a neuron, at the transition point between the bursting and spiking dynamics, which are two different dynamic behaviors that this system is able to present. Moreover, besides the change in attractor size, other significant properties of the system undergoing the transitions do change in a relevant qualitative way. The mechanism for such transition is understood in terms of a simple one-dimensional map whose dynamics undergoes a crossover between two different universal behaviors.
Animals↗
Computation in gene networks.
Genetic regulatory networks have the complex task of controlling all aspects of life. Using a model of gene expression by piecewise linear differential equations we show that this process can be considered as a process of computation. This is demonstrated by showing that this model can simulate memory bounded Turing machines. The simulation is robust with respect to perturbations of the system, an important property for both analog computers and biological systems. Robustness is achieved using a condition that ensures that the model equations, that are generally chaotic, follow a predictable dynamics.
Algorithms↗
Dynamic phase separation: from coarsening to turbulence via structure formation.
We investigate some new two-dimensional evolution models belonging to the class of convective Cahn-Hilliard models: (i) a local model with a scalar order parameter, (ii) a nonlocal model with a scalar order parameter, and (iii) a model with a vector order parameter. These models are applicable to phase-separating system where concentration gradients cause hydrodynamic motion due to buoyancy or Marangoni effect. The numerical study of the models shows transition from coarsening, typical of Cahn-Hilliard systems, to spatiotemporally irregular behavior (turbulence), typical of the Kuramoto-Sivashinsky equation, which is obtained in the limit of very strong driving. The transition occurs not in a straightforward way, but through the formation of spatial patterns that emerge for intermediate values of the driving intensity. As in driven one-dimensional models studied before, the mere presence of the driving force, however small, breaks the symmetry between the two separating phases, as well as increases the coarsening rate. With increasing driving, coarsening stops. The dynamics is generally irregular at strong driving, but exhibits specific structural features.
Algorithms↗
On Jacobian matrices for flows.
We present a general method for constructing numerical Jacobian matrices for flows discretized on a Poincaré surface of section. Special attention is given to Hamiltonian flows where the additional constraint of energy conservation is explicitly taken into account. We demonstrate the approach for a conservative dynamical flow and apply the technique for the general detection of periodic orbits.
Algorithms↗
Constructing dynamical systems with specified symbolic dynamics.
In this paper we demonstrate how to construct signals (time series) of continuous-time dynamical systems that exhibit a given symbolic dynamics. This is achieved without construction of the ordinary differential equations that generate the flow. This construction is of theoretical interest and is useful as a source of dynamical data that can be used to test various data analysis algorithms.
Algorithms↗
A simple chaotic neuron model: stochastic behavior of neural networks.
We have briefly reviewed the occurrence of the post-synaptic potentials between neurons, the relationship between EEG and neuron dynamics, as well as methods of signal analysis. We propose a simple stochastic model representing electrical activity of neuronal systems. The model is constructed using the Monte Carlo simulation technique. The results yielded EEG-like signals with their phase portraits in three-dimensional space. The Lyapunov exponent was positive, indicating chaotic behavior. The correlation of the EEG-like signals was.92, smaller than those reported by others. It was concluded that this neuron model may provide valuable clues about the dynamic behavior of neural systems.
Electroencephalography↗
Characteristics of period adding bifurcation without chaos in firing pattern transitions in an experimental neural pacemaker.
Time variant signals received by a neuron were encoded into and output in the changing firing patterns. Recent investigations revealed that the firing patterns changed according to certain bifurcation principles. Thus, the bifurcation scenarios among various firing patterns were fundamental to the study of neural coding. In this study one of the previous theoretically proposed scenarios, period adding bifurcation without chaos, was experimentally confirmed to exist in real neuronal activities. The realistic bifurcation process possessed integer multiple patterns not found in previous theoretical reports. The observed bifurcation process was simulated in the stochastic neuronal model and the generation of the integer multiple patterns were related to the interplay between the internal noise and the dynamics of the neuron.
Action Potentials↗
Intermittency and transient chaos from simple frequency-dependent selection.
Frequency-dependent selection is an important determinant of the evolution of gametophytic self-incompatibility systems in plants, aposematic (warning) and cryptic coloration, systems of mimicry, competitive interactions among members of a population, mating preferences, predator-prey and host-parasite interactions, aggression and other behavioural traits. Past theoretical studies of frequency-dependent selection have shown it to be a plausible mechanism for the maintenance of genetic variability in natural populations. Here, through an analysis of a simple deterministic model for frequency-dependent selection, we demonstrate that complex dynamic behaviour is possible under a broad range of parameter values. In particular we show that the model exhibits not only cycles and chaos but also, for a more restricted set of parameters, transient chaos and intermittency: alterations between an apparently deterministic behaviour and apparently chaotic fluctuations. This behaviour, which has not been stressed within the population genetics literature, provides an explanation for erratic dynamics of gene frequencies.
Alleles↗
A non-linear model of growth in the first year of life.
Weight gain curves in infancy and childhood show valleys and hills which cannot be simplistically ascribed to random variations, so that they are referred to as pulsatile or pulse growth. Classical mathematical models of human growth, upon which the statistical percentiles are based, do not account for these oscillations. Moreover, statistical curves can hide individual patterns of growth. In the present paper a system dynamics approach to modelling the first year weight gain is suggested which simultaneously accounts for the oscillations of weight gain, for its decreasing trend, and for a more or less marked initial spurt observed approximately in the second month of life. It is suggested that the overall weight growth in the first year is the output of complex hidden non-linear dynamics.
Anthropometry↗
Analysis of tensioned membrane structures considering cable sliding.
In routine design of tensioned membrane structures, the membrane is generally modeled using space membrane elements and the cables by space cable elements, with no sliding allowed between the membrane and the cables. On the other hand, large deflections are expected and sliding between the membrane and the cables is inevitable. In the present paper, the general finite element code ABAQUS was employed to investigate the influence of cable sliding on membrane surface on the structural behavior. Three analysis models were devised to fulfill this purpose: (1) The membrane element shares nodes with the cable element; (2) The cable can slide on the membrane surface freely (without friction) and (3) The cable can slide on the membrane surface, but with friction between the cable and the membrane. The sliding problem is modeled using a surface-based contact algorithm. The results from three analysis models are compared, showing that cable sliding has only little influence on the structure shape and on the stress distributions in the membrane. The main influence of cable sliding may be its effect on the dynamic behavior of tensioned membrane structures.
Computer Simulation↗
Dynamics of oscillator chains from high frequency initial conditions: comparison of phi4 and FPU-beta models.
The dynamics of oscillator chains are studied, starting from high frequency initial conditions (h.f.i.c.). In particular, the formation and evolution of chaotic breathers (CB's) of the Klein-Gordon chain with quartic nonlinearity in the Hamiltonian (the phi(4) model) are compared to the results of the previously studied Fermi-Pasta-Ulam (FPU-beta) chain. We find an important difference for h.f.i.c. is that the quartic nonlinearity, which drives the high frequency phenomena, being a self-force on each individual oscillator in the phi(4) model is significantly weaker than the quartic term in the FPU-beta model, which acts between neighboring oscillators that are nearly out-of-phase. The addition of a self-force breaks the translational invariance and adds a parameter. We compare theoretical results, using the envelope approximation to reduce the discrete coupled equations to a partial differential equation for each chain, indicating that various scalings can be used to predict the relative energies at which the basic phenomena of parametric instability, breather formation and coalescence, and ultimately breather decay to energy equipartition, will occur. Detailed numerical results, comparing the two chains, are presented to verify the scalings.
Models, Statistical↗
Breakdown of a conservation law in incommensurate systems.
We show that invariance properties of the Lagrangian of an incommensurate system, as described by the Frenkel-Kontorova model, imply the existence of a generalized angular momentum that is an integral of motion if the system remains floating. The behavior of this quantity can therefore monitor the character of the system as floating (when it is conserved) or locked (when it is not). We find that, during the dynamics, the nonlinear couplings of our model cause parametric phonon excitations that lead to the appearance of Umklapp terms and to a sudden deviation of the generalized momentum from a constant value, signaling a dynamical transition from a floating to a pinned state. We point out that this transition is related but does not coincide with the onset of sliding friction, which can take place when the system is still floating.
Journal Article↗