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Concept: Attractor


Restricting our scope to the dynamical motion of the leaflets, we present a computational model for a symmetric, tri-leaflet, bioprosthetic heart valve (BHV) at the end of five complete cardiac pressure cycles, reaching the steady state of deformation during both closing and opening phases. To this end, we utilized a highly anisotropic material model for the large deformation behavior of the tissue material, for which an experimental validation was provided. The important findings are: (1) material anisotropy has significant effect on the valve opening/closing; (2) the asymmetric deformations, especially in the fully closed configuration, justify the use of cyclic symmetry; (3) adopting the fully-open position as an initial/reference configuration has the advantage of completely bypassing any complications arising from the need to assume the size and shape of the contact area in the coaptation regions of the leaflets that is necessary when the alternative, commonly-used, approach of selecting the fully-closed position is used as a reference; and (4) with proper treatments for both material anisotropy and tissue-to-tissue contact, the overall BHV model provide realistic results in conformity with the ex vivo/in vitro experiments.

Concepts: Thermodynamics, Symmetry, Geometry, Young's modulus, Attractor, Artificial heart valve, Asymmetry, Point groups in three dimensions


Many ecological systems exhibit multi-year cycles. In such systems, invasions have a complicated spatiotemporal structure. In particular, it is common for unstable steady states to exist as long-term transients behind the invasion front, a phenomenon known as dynamical stabilisation. We combine absolute stability theory and computation to predict how the width of the stabilised region depends on parameter values. We develop our calculations in the context of a model for a cyclic predator-prey system, in which the invasion front and spatiotemporal oscillations of predators and prey are separated by a region in which the coexistence steady state is dynamically stabilised.

Concepts: Mathematics, Predation, Ecology, Systems theory, Model theory, Attractor, Lyapunov stability, Structural stability


The characterization of heart dynamics with a view to distinguish abnormal from normal behavior is an interesting topic in clinical sciences. Here we present an analysis of the Electro-cardiogram (ECG) signals from several healthy and unhealthy subjects using the framework of dynamical systems approach to multifractal analysis. Our analysis differs from the conventional nonlinear analysis in that the information contained in the amplitude variations of the signal is being extracted and quantified. The results thus obtained reveal that the attractor underlying the dynamics of the heart has multifractal structure and the variations in the resultant multifractal spectra can clearly separate healthy subjects from unhealthy ones. We use supervised machine learning approach to build a model that predicts the group label of a new subject with very high accuracy on the basis of the multifractal parameters. By comparing the computed indices in the multifractal spectra with that of beat replicated data from the same ECG, we show how each ECG can be checked for variations within itself. The increased variability observed in the measures for the unhealthy cases can be a clinically meaningful index for detecting the abnormal dynamics of the heart.

Concepts: Scientific method, Psychology, Behavior, Normality, Attractor, Dynamical system, Supervised learning, Lorenz attractor


Stability and performance during rhythmic motor behaviors such as locomotion are critical for survival across taxa: falling down would bode well for neither cheetah nor gazelle. Little is known about how haptic feedback, particularly during discrete events such as the heel-strike event during walking, enhances rhythmic behavior. To determine the effect of haptic cues on rhythmic motor performance, we investigated a virtual paddle juggling behavior, analogous to bouncing a table tennis ball on a paddle. Here, we show that a force impulse to the hand at the moment of ball-paddle collision categorically improves performance over visual feedback alone, not by regulating the rate of convergence to steady state (e.g. via higher gain feedback or modifying the steady-state hand motion), but rather by reducing cycle-to-cycle variability. This suggests that the timing and state cues afforded by haptic feedback decreases the nervous system’s uncertainty of the ball’s state to enable more accurate control, but that the feedback gain itself is unaltered. This decrease in variability leads to a substantial increase in the mean first passage time, a measure of the long-term metastability of a stochastic dynamical system. Rhythmic tasks such as locomotion and juggling involve intermittent contact with the environment (i.e. hybrid transitions), and the timing of such transitions is generally easy to sense via haptic feedback. This timing information may improve metastability, equating to less frequent falls or other failures depending on the task.

Concepts: Nervous system, Psychology, Thermodynamics, Systems theory, Probability theory, Attractor, Steady state, Tennis


Landscape topography is the expression of the dynamic equilibrium between external forcings (for example, climate and tectonics) and the underlying lithology. The magnitude and spatial arrangement of erosional and depositional fluxes dictate the evolution of landforms during both statistical steady state (SS) and transient state (TS) of major landscape reorganization. For SS landscapes, the common expectation is that any point of the landscape has an equal chance to erode below or above the landscape median erosion rate. We show that this is not the case. Afforded by a unique experimental landscape that provided a detailed space-time recording of erosional fluxes and by defining the so-called E50-area curve, we reveal for the first time that there exists a hierarchical pattern of erosion. Specifically, hillslopes and fluvial channels erode more rapidly than the landscape median erosion rate, whereas intervening parts of the landscape in terms of upstream contributing areas (colluvial regime) erode more slowly. We explain this apparent paradox by documenting the dynamic nature of SS landscapes-landscape locations may transition from being a hillslope to being a valley and then to being a fluvial channel due to ridge migration, channel piracy, and small-scale landscape dynamics through time. Under TS conditions caused by increased precipitation, we show that the E50-area curve drastically changes shape during landscape reorganization. Scale-dependent erosional patterns, as observed in this study, suggest benchmarks in evaluating numerical models and interpreting the variability of sampled erosional rates in field landscapes.

Concepts: Time, Thermodynamics, Erosion, Attractor, Steady state, Topography, Geomorphology, Dynamic equilibrium


Most natural odors have sparse molecular composition. This makes the principles of compressed sensing potentially relevant to the structure of the olfactory code. Yet, the largely feedforward organization of the olfactory system precludes reconstruction using standard compressed sensing algorithms. To resolve this problem, recent theoretical work has shown that signal reconstruction could take place as a result of a low dimensional dynamical system converging to one of its attractor states. However, the dynamical aspects of optimization slowed down odor recognition and were also found to be susceptible to noise. Here we describe a feedforward model of the olfactory system that achieves both strong compression and fast reconstruction that is also robust to noise. A key feature of the proposed model is a specific relationship between how odors are represented at the glomeruli stage, which corresponds to a compression, and the connections from glomeruli to third-order neurons (neurons in the olfactory cortex of vertebrates or Kenyon cells in the mushroom body of insects), which in the model corresponds to reconstruction. We show that should this specific relationship hold true, the reconstruction will be both fast and robust to noise, and in particular to the false activation of glomeruli. The predicted connectivity rate from glomeruli to third-order neurons can be tested experimentally.

Concepts: Mathematics, Olfactory bulb, Olfaction, Olfactory system, Odor, Attractor, Dynamical system, Lorenz attractor


Game theory is the standard tool used to model strategic interactions in evolutionary biology and social science. Traditionally, game theory studies the equilibria of simple games. However, is this useful if the game is complicated, and if not, what is? We define a complicated game as one with many possible moves, and therefore many possible payoffs conditional on those moves. We investigate two-person games in which the players learn based on a type of reinforcement learning called experience-weighted attraction (EWA). By generating games at random, we characterize the learning dynamics under EWA and show that there are three clearly separated regimes: (i) convergence to a unique fixed point, (ii) a huge multiplicity of stable fixed points, and (iii) chaotic behavior. In case (iii), the dimension of the chaotic attractors can be very high, implying that the learning dynamics are effectively random. In the chaotic regime, the total payoffs fluctuate intermittently, showing bursts of rapid change punctuated by periods of quiescence, with heavy tails similar to what is observed in fluid turbulence and financial markets. Our results suggest that, at least for some learning algorithms, there is a large parameter regime for which complicated strategic interactions generate inherently unpredictable behavior that is best described in the language of dynamical systems theory.

Concepts: Game theory, Evolution, Mathematics, Chaos theory, Economics, Complexity, Machine learning, Attractor


Experimental records of active bundle motility are used to demonstrate the presence of a low-dimensional chaotic attractor in hair cell dynamics. Dimensionality tests from dynamic systems theory are applied to estimate the number of independent variables sufficient for modelling the hair cell response. PoincarĂ© maps are constructed to observe a quasiperiodic transition from chaos to order with increasing amplitudes of mechanical forcing. The onset of this transition is accompanied by a reduction of Kolmogorov entropy in the system and an increase in transfer entropy between the stimulus and the hair bundle, indicative of signal detection. A simple theoretical model is used to describe the observed chaotic dynamics. The model exhibits an enhancement of sensitivity to weak stimuli when the system is poised in the chaotic regime. We propose that chaos may play a role in the hair cell’s ability to detect low-amplitude sounds.

Concepts: Scientific method, Mathematics, Physics, Chaos theory, Philosophy of science, Inner ear, Attractor, Dynamical system


Electro-cortical activity in patients with epilepsy may show abnormal rhythmic transients in response to stimulation. Even when using the same stimulation parameters in the same patient, wide variability in the duration of transient response has been reported. These transients have long been considered important for the mapping of the excitability levels in the epileptic brain but their dynamic mechanism is still not well understood. To investigate the occurrence of abnormal transients dynamically, we use a thalamo-cortical neural population model of epileptic spike-wave activity and study the interaction between slow and fast subsystems. In a reduced version of the thalamo-cortical model, slow wave oscillations arise from a fold of cycles (FoC) bifurcation. This marks the onset of a region of bistability between a high amplitude oscillatory rhythm and the background state. In vicinity of the bistability in parameter space, the model has excitable dynamics, showing prolonged rhythmic transients in response to suprathreshold pulse stimulation. We analyse the state space geometry of the bistable and excitable states, and find that the rhythmic transient arises when the impending FoC bifurcation deforms the state space and creates an area of locally reduced attraction to the fixed point. This area essentially allows trajectories to dwell there before escaping to the stable steady state, thus creating rhythmic transients. In the full thalamo-cortical model, we find a similar FoC bifurcation structure. Based on the analysis, we propose an explanation of why stimulation induced epileptiform activity may vary between trials, and predict how the variability could be related to ongoing oscillatory background activity. We compare our dynamic mechanism with other mechanisms (such as a slow parameter change) to generate excitable transients, and we discuss the proposed excitability mechanism in the context of stimulation responses in the epileptic cortex.

Concepts: Thermodynamics, Systems theory, Oscillation, Epilepsy, Rhythm, State, Attractor, Parameter


Infectious diseases are notorious for their complex dynamics, which make it difficult to fit models to test hypotheses. Methods based on state-space reconstruction have been proposed to infer causal interactions in noisy, nonlinear dynamical systems. These “model-free” methods are collectively known as convergent cross-mapping (CCM). Although CCM has theoretical support, natural systems routinely violate its assumptions. To identify the practical limits of causal inference under CCM, we simulated the dynamics of two pathogen strains with varying interaction strengths. The original method of CCM is extremely sensitive to periodic fluctuations, inferring interactions between independent strains that oscillate with similar frequencies. This sensitivity vanishes with alternative criteria for inferring causality. However, CCM remains sensitive to high levels of process noise and changes to the deterministic attractor. This sensitivity is problematic because it remains challenging to gauge noise and dynamical changes in natural systems, including the quality of reconstructed attractors that underlie cross-mapping. We illustrate these challenges by analyzing time series of reportable childhood infections in New York City and Chicago during the pre-vaccine era. We comment on the statistical and conceptual challenges that currently limit the use of state-space reconstruction in causal inference.

Concepts: Disease, Infectious disease, Causality, Infection, Attractor, Dynamical system, Dynamical systems, Lorenz attractor