Simple Dynamical System Research Articles

Calibration of Visually Guided Reaching Is Driven by Error-Corrective Learning and Internal Dynamics

The sensorimotor calibration of visually guided reaching changes on a trial-to-trial basis in response to random shifts in the visual feedback of the hand. We show that a simple linear dynamical system is sufficient to model the dynamics of this adaptive process. In this model, an internal variable represents the current state of sensorimotor calibration. Changes in this state are driven by error feedback signals, which consist of the visually perceived reach error, the artificial shift in visual feedback, or both. Subjects correct for > or =20% of the error observed on each movement, despite being unaware of the visual shift. The state of adaptation is also driven by internal dynamics, consisting of a decay back to a baseline state and a "state noise" process. State noise includes any source of variability that directly affects the state of adaptation, such as variability in sensory feedback processing, the computations that drive learning, or the maintenance of the state. This noise is accumulated in the state across trials, creating temporal correlations in the sequence of reach errors. These correlations allow us to distinguish state noise from sensorimotor performance noise, which arises independently on each trial from random fluctuations in the sensorimotor pathway. We show that these two noise sources contribute comparably to the overall magnitude of movement variability. Finally, the dynamics of adaptation measured with random feedback shifts generalizes to the case of constant feedback shifts, allowing for a direct comparison of our results with more traditional blocked-exposure experiments.

On differential equations with nonstandard coefficients

The classical distribution theory turns out to be insufficient for treating certain differential equations involving nonlinear operations and distributions. This kind of problems appear, e.g., in a rather simple dynamical system which describes the evolution of the population densities of predators and preys. In the theory of distributions there are two complementary points of view: (1) A distribution f ∈ D(R) is a continuous linear functional on the space D(R) of compactly supported smooth functions with an appropriate convergence, see, [8]. Here we have a linear action

Observation of matter wave interference on the macro-scale for charged particles in a magnetic field and the manifestation of Landau levels

Matter wave interference effects in the macrodomain for charged particles in a magnetic field are reported for different scatterer positions between the electron gun and collector plate. The results obtained serve to validate and establish the important premises of the formalism that it is the scattering, (inter alia off the grid wires in this case) which generates the transition amplitude waves resulting in the novel quantum effects observed on the macro-scale associated with the transition amplitude wave. The variation of the position of origin of the grid-generated transition amplitude wave then leads to the variation of interference characteristics accordingly in the plate and grid currents. A Fourier decomposition of the interference peaks reveal also the presence of harmonics which are a manifestation of the Landau level structure. A rich class of quantum phenomena are thus discovered in the simple dynamical system of charged particles in a magnetic field in the ostensibly classical domain of operation, with the transition amplitude wave generated as a consequence of scattering being the new element of quantum description on the macro-scale.

Impact of the surface temperature and vertical shear of zonal wind on the dynamics of a simple two-layer model of the atmosphere

The paper presents and analyzes, from the point of view of smooth dynamic systems theory, a two-layer baroclinic model of the troposphere in geostrophic approximation. The model describes airflow in β-channel within the tropospheric part of the main Hadley circulation cell. It enables to obtain, after application of the Galerkin method, a fairly simple low-parametric dynamic system describing the phenomena of non-linear interactions, bifurcations and blocking in the atmosphere. This enables to take into consideration such basic factors influencing the atmospheric dynamics like the heat exchange within the surface, orography, vertical variability of zonal wind and hydrostatic stability.

Critical properties and stability of stationary solutions in multitransonic pseudo-Schwarzschild accretion

For inviscid, rotational accretion flows, both isothermal and polytropic, a simple dynamical system analysis of the critical points has given a very accurate mathematical scheme to understand the nature of these points, for any pseudo-potential by which the flow may be driven on to a Schwarzschild black hole. This allows us for a complete classification of the critical points for a wide range of flow parameters, and shows that the only possible critical points for this kind of flow are saddle points and centre-type points. A restrictive upper bound on the angular momentum of critical solutions has been established. A time-dependent perturbative study reveals that the form of the perturbation equation, for both isothermal and polytropic flows, is invariant under the choice of any particular pseudo-potential. Under generically true outer boundary conditions, the inviscid flow has been shown to be stable under an adiabatic and radially propagating perturbation. The perturbation equation has also served the dual purpose of enabling and understanding the acoustic geometry for inviscid and rotational flows.

AN IDEAL DISASSORTATIVE NETWORK AND SYNCHRONIZATION

This paper proposes a novel complex network with disassortative property based on multicenter networks. The average path length and clustering coefficient of the network are calculated, and the impact on the network topology is investigated. A simple dynamic system established on the proposed network is used to analyze how the disassortative property of the network affects synchronization.

Transient Resetting: A Novel Mechanism for Synchrony and Its Biological Examples

The study of synchronization in biological systems is essential for the understanding of the rhythmic phenomena of living organisms at both molecular and cellular levels. In this paper, by using simple dynamical systems theory, we present a novel mechanism, named transient resetting, for the synchronization of uncoupled biological oscillators with stimuli. This mechanism not only can unify and extend many existing results on (deterministic and stochastic) stimulus-induced synchrony, but also may actually play an important role in biological rhythms. We argue that transient resetting is a possible mechanism for the synchronization in many biological organisms, which might also be further used in the medical therapy of rhythmic disorders. Examples of the synchronization of neural and circadian oscillators as well as a chaotic neuron model are presented to verify our hypothesis.

Probing Selectivity in Recognition-Mediated Dynamic Covalent Processes

[reaction: see text] Two simple recognition-mediated dynamic Diels-Alder systems are used to probe the role of kinetics and thermodynamics in determining the equilibrium position in exchanging libraries and the time taken to reach that equilibrium. The selectivity expressed by recognition-driven dynamic processes is demonstrated to be less than the free-energy difference between the components as a result of compensatory effects arising from the extent of conversion to products within the library.

Stratospheric wave-mean flow interaction: Simple modeling

We show that the interaction between planetary waves and the stratospheric zonal mean flow results in bi-modal (direct-reverse flow) or unimodal state depending on wave number of the waves. First we demonstrate this using a simple non-linear dynamic system of the wave-flow interaction, which has two stable equilibrium states and one unstable state (attractors) in its phase space. Then we compare this model dynamics with the stratosphere dynamics using the same dynamical variables and a similar parameter range in the National Centers for Environmental Prediction (NCEP) Reanalysis data. This comparison supports the tendency for the states of planetary wave–zonal mean flow in the upper stratosphere to be bi-modal for wave number 2 and unimodal for wave number 1.

Modeling Sensorimotor Learning with Linear Dynamical Systems

Recent studies have employed simple linear dynamical systems to model trial-by-trial dynamics in various sensorimotor learning tasks. Here we explore the theoretical and practical considerations that arise when employing the general class of linear dynamical systems (LDS) as a model for sensorimotor learning. In this framework, the state of the system is a set of parameters that define the current sensorimotor transformation-the function that maps sensory inputs to motor outputs. The class of LDS models provides a first-order approximation for any Markovian (state-dependent) learning rule that specifies the changes in the sensorimotor transformation that result from sensory feedback on each movement. We show that modeling the trial-by-trial dynamics of learning provides a substantially enhanced picture of the process of adaptation compared to measurements of the steady state of adaptation derived from more traditional blocked-exposure experiments. Specifically, these models can be used to quantify sensory and performance biases, the extent to which learned changes in the sensorimotor transformation decay over time, and the portion of motor variability due to either learning or performance variability. We show that previous attempts to fit such models with linear regression have not generally yielded consistent parameter estimates. Instead, we present an expectation-maximization algorithm for fitting LDS models to experimental data and describe the difficulties inherent in estimating the parameters associated with feedback-driven learning. Finally, we demonstrate the application of these methods in a simple sensorimotor learning experiment: adaptation to shifted visual feedback during reaching.

Modeling Sensorimotor Learning with Linear Dynamical Systems

Recent studies have employed simple linear dynamical systems to model trial-by-trial dynamics in various sensorimotor learning tasks. Here we explore the theoretical and practical considerations that arise when employing the general class of linear dynamical systems (LDS) as a model for sensorimotor learning. In this framework, the state of the system is a set of parameters that define the current sensorimotor transformation— the function that maps sensory inputs to motor outputs. The class of LDS models provides a first-order approximation for any Markovian (state-dependent) learning rule that specifies the changes in the sensorimotor transformation that result from sensory feedback on each movement. We show that modeling the trial-by-trial dynamics of learning provides a sub-stantially enhanced picture of the process of adaptation compared to measurements of the steady state of adaptation derived from more traditional blocked-exposure experiments. Specifically, these models can be used to quantify sensory and performance biases, the extent to which learned changes in the sensorimotor transformation decay over time, and the portion of motor variability due to either learning or performance variability. We show that previous attempts to fit such models with linear regression have not generally yielded consistent parameter estimates. Instead, we present an expectation-maximization algorithm for fitting LDS models to experimental data and describe the difficulties inherent in estimating the parameters associated with feedback-driven learning. Finally, we demonstrate the application of these methods in a simple sensorimotor learning experiment: adaptation to shifted visual feedback during reaching.

Stack-like and queue-like dynamics in recurrent neural networks

What dynamics do simple recurrent networks (SRNs) develop to represent stack-like and queue-like memories? SRNs have been widely used as models in cognitive science. However, they are interesting in their own right as non-symbolic computing devices from the viewpoints of analogue computing and dynamical systems theory. In this paper, SRNs are trained on two prototypical formal languages with recursive structures that need stack-like or queue-like memories for processing, respectively. The evolved dynamics are analysed, then interpreted in terms of simple dynamical systems, and the different ease with which SRNs aquire them is related to the properties of these simple dynamical systems. Within the dynamical systems framework, it is concluded that the stack-like language is simpler than the queue-like language, without making use of arguments from symbolic computation theory.

Pulse replications and spatially differentiated structure formation in one-dimensional lattice dynamical system

Replication and differentiation of pulses are studied through a simple dynamical system on a one-dimensional lattice. Depending on the parameter values, our proposed system exhibits the following types of pulse replications: (I) each pulse replicates by itself, (II) each pulse replicates if two or more pulses collide each other, and (III) pulses replicate by the organization of pulse generators. Moreover, replicated pulses construct the spatial–temporal pattern when the number of pulses is large. By selecting local objects in such spatial–temporal patterns and replanting them numerically, it is found that there are two types differentiated local structures: one that continues the production of pulses and the other that does not.

Crossover between Ballistic and Normal Diffusion

Crossover between ballistic motion and normal diffusion is studied based on the continuous-time random walk (CTRW) approach in order to analyze universal properties of strongly correlated motion and the decay process of correlation in deterministic diffusion. There exists a characteristic time scale r. For the time region t « T, ballistic motion is observed, which is followed by normal diffusion for t >> T. Higher-order moments are analytically obtained, and it is found that they obey scaling relations that are reminiscent of the generalized extended self-similarity (GESS) found in turbulent systems. As a simple dynamical system for numerical simulations, the climbing sine map in the vicinity of band crisis is considered. Good agreement between the theory and the numerical simulations is observed.

A new route to non invasive diagnosis in neurodegenerative diseases?

We propose a simple dynamical system to interpret the gait time series from patients affected by three neurodegenerative diseases: Parkinson, Huntington and Amyotrophic Lateral Sclerosis. The model is shown to reproduce the main aspects of the experimental time series. Within this scenario, quantitative differences in specific indicators are detected thus opening up the perspective for innovative, non invasive, diagnosis procedures from direct measurements of gait dynamics.

Streamer saturation: a dynamical systems approach

Small-scale turbulence in toroidal magnetoplasmas driven by electron or ion temperature gradient modes is known for its tendency to form strongly anisotropic structures, namely radially elongated vortices called streamers or (mostly) poloidal E × B shear flows called zonal flows. The interaction of such patterns is studied here in the framework of a simple dynamical system, focusing in particular on the initial saturation phase of a streamer. It is shown that the saturation amplitude, in this reduced description, is determined by a balance between the streamer's linear growth rate and the zonal mode's nonlinear growth rate.

A formally grounded software specification method

One of the goals of software engineering is to provide what is necessary to write relevant, legible, useful descriptions of the systems to be developed, which will be the basis of successful developments. This goal was addressed both from informal approaches (providing in particular visual notations) and formal ones (providing a formal sound semantic basis). Informal approaches are often driven by a software development method, and, while formal approaches sometimes provide a user method, it is usually aimed at helping to use the proposed formalism when writing a specification. Our goal here is to provide a companion method that helps the user to understand the system to be developed, and to write the corresponding formal specifications. We also aim at supporting visual presentations of formal specifications, so as to “make the best of both formal and informal worlds”. We developed this method for the (logical-algebraic) specification languages CASL (Common Algebraic Specification Language, developed within the joint initiative CoFI) and for an extension for dynamic systems C asl-L tl, and we believe it is general enough to be adapted to other paradigms. Another challenge is that a method that is too general does not encompass the different kinds of systems to be studied, while too many different specialized methods result in partial views that may be difficult to integrate in a single global one. We deal with this issue by providing a limited number of instances of our method, fitted for three different kinds of software items, while keeping a common “meta”-structure and way of thinking. More precisely, we consider here that a software item may be a simple dynamic system, a structured dynamic system, or a data structure, and we show here how to support property-oriented (axiomatic) specifications. We are thus providing support for the “building-bricks” tasks of specifying software artifacts that in our experience are needed for the development process. Our approach is illustrated with a lift case study.

Origin of Non-Gaussian Statistics in Hydrodynamic Turbulence

Turbulent flows are notoriously difficult to describe and understand based on first principles. One reason is that turbulence contains highly intermittent bursts of vorticity and strain rate with highly non-Gaussian statistics. Quantitatively, intermittency is manifested in highly elongated tails in the probability density functions of the velocity increments between pairs of points. A long-standing open issue has been to predict the origins of intermittency and non-Gaussian statistics from the Navier-Stokes equations. Here we derive, from the Navier-Stokes equations, a simple nonlinear dynamical system for the Lagrangian evolution of longitudinal and transverse velocity increments. From this system we are able to show that the ubiquitous non-Gaussian tails in turbulence have their origin in the inherent self-amplification of longitudinal velocity increments, and cross amplification of the transverse velocity increments.

The transition from regular to irregular motions, explained as travel on Riemann surfaces

We introduce and discuss a simple Hamiltonian dynamical system, interpretable as a 3-body problem in the complex plane and providing the prototype of a mechanism explaining the transition from regular to irregular motions as travel on Riemann surfaces. The interest of this phenomenology -- illustrating the onset in a deterministic context of irregular motions -- is underlined by its generality, suggesting its eventual relevance to understand natural phenomena and experimental investigations. Here only some of our main findings are reported, without detailing their proofs: a more complete presentation will be published elsewhere.

Principal bundle structure of quantum adiabatic dynamics with a Berry phase which does not commute with the dynamical phase

A geometric model is proposed to describe the Berry phase phenomenon when the geometric phase does not commute with the dynamical phase. The structure used is a principal composite bundle in which the adiabatic transport appears as a horizontal lift. The formulation is applied to a simple quantum dynamical system controlled by two lasers.