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PubMed2006

V4 receptive field dynamics as predicted by a systems-level model of visual attention using feedback from the frontal eye field.

Visual attention is generally considered to facilitate the processing of the attended stimulus. Its mechanisms, however, are still under debate. We have developed a systems-level model of visual attention which predicts that attentive effects emerge by the interactions between different brain areas. Recent physiological studies have provided evidence that attention also alters the receptive field structure. For example, V4 receptive fields typically shrink and shift towards the saccade target around saccade onset. We show that receptive field dynamics are inherently predicted by the mechanism of feedback in our model. According to the model an oculomotor feedback signal from an area involved in the competition for the saccade target location, e.g. the frontal eye field, enhances the gain of V4 cells. V4 receptive field dynamics can be observed after pooling the gain modulated responses to obtain a certain degree of spatial invariance. The time course of the receptive field dynamics in the model resemble those obtained from macaque V4.

Animals↗
PubMed2003

An efficient numerical method for distributed-loop models of the urine concentrating mechanism.

In this study we describe an efficient numerical method, based on the semi-Lagrangian (SL) semi-implicit (SI) method and Newton's method, for obtaining steady-state (SS) solutions of equations arising in distributed-loop models of the urine concentrating mechanism. Dynamic formulations of these models contain large systems of coupled hyperbolic partial differential equations (PDEs). The SL method advances the solutions of these PDEs in time by integrating backward along flow trajectories, thus allowing large time steps while maintaining stability. The SI approach controls stiffness arising from transtubular transport terms by averaging these terms in time along flow trajectories. An approximate SS solution of a dynamic formulation obtained via the SLSI method can be used as an initial guess for a Newton-type solver, which rapidly converges to a highly accurate numerical approximation to the solution of the ordinary differential equations that arise in the corresponding SS model formulation. In general, it is difficult to specify a priori for a Newton-type solver an initial guess that falls within the radius of convergence; however, the initial guess generated by solving the dynamic formulation via the SLSI method can be made sufficiently close to the SS solution to avoid numerical instability. The combination of the SLSI method and the Newton-type solver generates stable and accurate solutions with substantially reduced computation times, when compared to previously applied dynamic methods.

Animals↗
PubMed2003

Chaotic itinerancy in coupled dynamical recognizers.

We argue that chaotic itinerancy in interaction between humans originates in the fluctuation of predictions provided by the nonconvergent nature of learning dynamics. A simple simulation model called the coupled dynamical recognizer is proposed to study this phenomenon. Daily cognitive phenomena provide many examples of chaotic itinerancy, such as turn taking in conversation. It is therefore an interesting problem to bridge two chaotic itinerant phenomena. A clue to solving this is the fluctuation of prediction, which can be translated as "hot prediction" in the context of cognitive theory. Hot prediction is simply defined as a prediction based on an unstable model. If this approach is correct, the present simulation will reveal some dynamic characteristics of cognitive interactions.

Cognition↗
PubMed2005

Vortex core identification in viscous hydrodynamics.

We describe a software package designed for the investigation of topological fluid dynamics with a novel algorithm for locating and tracking vortex cores. The package is equipped with modules for generating desired vortex knots and links and evolving them according to the Navier-Stokes equations, while tracking and visualizing them. The package is parallelized using a message passing interface for a multiprocessor environment and makes use of a computational steering library for dynamic user intervention.

Algorithms↗
PubMed2004

Evaluation of different measures of functional connectivity using a neural mass model.

We use a neural mass model to address some important issues in characterising functional integration among remote cortical areas using magnetoencephalography or electroencephalography (MEG or EEG). In a previous paper [Neuroimage (in press)], we showed how the coupling among cortical areas can modulate the MEG or EEG spectrum and synchronise oscillatory dynamics. In this work, we exploit the model further by evaluating different measures of statistical dependencies (i.e., functional connectivity) among MEG or EEG signals that are mediated by neuronal coupling. We have examined linear and nonlinear methods, including phase synchronisation. Our results show that each method can detect coupling but with different sensitivity profiles that depended on (i) the frequency specificity of the interaction (broad vs. narrow band) and (ii) the nature of the coupling (linear vs. nonlinear). Our analyses suggest that methods based on the concept of generalised synchronisation are the most sensitive when interactions encompass different frequencies (broadband analyses). In the context of narrow-band analyses, mutual information was found to be the most sensitive way to disclose frequency-specific couplings. Measures based on generalised synchronisation and phase synchronisation are the most sensitive to nonlinear coupling. These different sensitivity profiles mean that the choice of coupling measures can have dramatic effects on the cortical networks identified. We illustrate this using a single-subject MEG study of binocular rivalry and highlight the greater recovery of statistical dependencies among cortical areas in the beta band when mutual information is used.

Beta Rhythm↗
PubMed2003

Disentangling the dynamic core: a research program for a neurodynamics at the large-scale.

My purpose in this paper is to sketch a research direction based on Francisco Varela's pioneering work in neurodynamics (see also Rudrauf et al. 2003, in this issue). Very early on he argued that the internal coherence of every mental-cognitive state lies in the global self-organization of the brain activities at the large-scale, constituting a fundamental pole of integration called here a "dynamic core". Recent neuroimaging evidence appears to broadly support this hypothesis and suggests that a global brain dynamics emerges at the large scale level from the cooperative interactions among widely distributed neuronal populations. Despite a growing body of evidence supporting this view, our understanding of these large-scale brain processes remains hampered by the lack of a theoretical language for expressing these complex behaviors in dynamical terms. In this paper, I propose a rough cartography of a comprehensive approach that offers a conceptual and mathematical framework to analyze spatio-temporal large-scale brain phenomena. I emphasize how these nonlinear methods can be applied, what property might be inferred from neuronal signals, and where one might productively proceed for the future. This paper is dedicated, with respect and affection, to the memory of Francisco Varela.

Brain↗
PubMed1992

Dynamic properties of cardiovascular systems.

A recently derived mathematical model of an isolated heart is extended here to a closed-loop cardiovascular system. Taking the end-diastolic volume as state variable, the authors show that the closed-loop cardiovascular system can be described by a one-dimensional nonlinear discrete dynamical system that depends on parameters describing the systolic and diastolic properties of the heart, heart rate, total peripheral resistance, and arterial capacitance. Studies of this model show that the system possesses a rich spectrum of dynamical behavior, from stable points through stable cycles to a "chaotic" behavior. It is shown that such an analysis of dynamic behavior yields those domains in the parameter space that correspond to a normal and abnormal beating heart, when the heart ejects time-invariant and time-variant (periodic or aperiodic) stable stroke volumes, respectively. Determination of such domains may lead to better understanding of the specific pathologic mechanism involved in the evolution of an abnormal beating heart.

Cardiovascular Physiological Phenomena↗
PubMed1998

Quantification of effects of global ischemia on dynamics of ventricular fibrillation in isolated rabbit heart.

BACKGROUND: Ventricular fibrillation (VF) leads to global ischemia of the heart. After 1 to 2 minutes of onset, the VF rate decreases and appears more organized. The objectives of this study were to determine the effects of no-flow global ischemia on nonlinear wave dynamics and establish the mechanism of ischemia-induced slowing of the VF rate. METHODS AND RESULTS: Activation patterns of VF in the Langendorff-perfused rabbit heart were studied with the use of 2 protocols: (1) 15 minutes of no-flow global ischemia followed by reperfusion (n=7) and (2) decreased excitability induced by perfusion with 5 micromol/L of tetrodotoxin (TTX) followed by washout (n=3). Video imaging ( approximately 7500 pixels per frame; 240 frames per second) with a voltage-sensitive dye, ECG, and signal processing (fast Fourier transform) were used for analysis. The dominant frequency of VF decreased from 13.5+/-1.3 during control to 9.3+/-1.4 Hz at 5 minutes of global ischemia (P<0.02). The dominant frequency decreased from 13.9+/-1.1 during control to 7.0+/-0.3 Hz at 2 minutes of TTX infusion (P<0.001). The rotation period of rotors on the epicardial surface (n=27) strongly correlated with the inverse dominant frequency of the corresponding episode of VF (R2=0. 93). The core area, measured for 27 transiently appearing rotors, was 5.3+/-0.7 mm2 during control. A remarkable increase in core area was observed both during global ischemia (13.6+/-1.7 mm2; P<0.001) and TTX perfusion (16.8+/-3.6 mm2; P<0.001). Density of wave fronts decreased during both global ischemia (P<0.002) and TTX perfusion (P<0.002) compared with control. CONCLUSIONS: This study suggests that rotating spiral waves are most likely the underlying mechanism of VF and contribute to its frequency content. Ischemia-induced decrease in the VF rate results from an increase in the rotation period of spiral waves that occurs secondary to an increase in their core area. Remarkably, similar findings in the TTX protocol suggest that reduced excitability during ischemia is an important underlying mechanism for the changes seen.

Animals↗
PubMed2000

Full instability behavior of N-dimensional dynamical systems with a one-directional nonlinear vector field

We show how certain N-dimensional dynamical systems are able to exploit the full instability capabilities of their fixed points to do Hopf bifurcations and how such a behavior produces complex time evolutions based on the nonlinear combination of the oscillation modes that emerged from these bifurcations. For really different oscillation frequencies, the evolutions describe robust wave form structures, usually periodic, in which self-similarity with respect to both the time scale and system dimension is clearly appreciated. For closer frequencies, the evolution signals usually appear irregular but are still based on the repetition of complex wave form structures. The study is developed by considering vector fields with a scalar-valued nonlinear function of a single variable that is a linear combination of the N dynamical variables. In this case, the linear stability analysis can be used to design N-dimensional systems in which the fixed points of a saddle-node pair experience up to N-1 Hopf bifurcations with preselected oscillation frequencies. The secondary processes occurring in the phase region where the variety of limit cycles appear may be rather complex and difficult to characterize, but they produce the nonlinear mixing of oscillation modes with relatively generic features.

Journal Article↗
PubMed2003

Deriving reaction-diffusion models in ecology from interacting particle systems.

We use a scaling procedure based on averaging Poisson distributed random variables to derive population level models from local models of interactions between individuals. The procedure is suggested by using the idea of hydrodynamic limits to derive reaction-diffusion models for population interactions from interacting particle systems. The scaling procedure is formal in the sense that we do not address the issue of proving that it converges; instead we focus on methods for computing the results of the scaling or deriving properties of rescaled systems. To that end we treat the scaling procedure as a transform, in analogy with the Laplace or Fourier transform, and derive operational formulas to aid in the computation of rescaled systems or the derivation of their properties. Since the limiting procedure is adapted from work by Durrett and Levin, we refer to the transform as the Durrett-Levin transform. We examine the effects of rescaling in various standard models, including Lotka-Volterra models, Holling type predator-prey models, and ratio-dependent models. The effects of scaling are mostly quantitative in models with smooth interaction terms, but ratio-dependent models are profoundly affected by the scaling. The scaling transforms ratio-dependent terms that are singular at the origin into smooth terms. Removing the singularity at the origin eliminates some of the unique dynamics that can arise in ratio-dependent models.

Animals↗
PubMed1994

The brain as a dynamic physical system.

The brain is a dynamic system that is non-linear at multiple levels of analysis. Characterization of its non-linear dynamics is fundamental to our understanding of brain function. Identifying families of attractors in phase space analysis, an approach which has proven valuable in describing non-linear mechanical and electrical systems, can prove valuable in describing a range of behaviors and associated neural activity including sensory and motor repertoires. Additionally, transitions between attractors may serve as useful descriptors for analysing state changes in neurons and neural ensembles. Recent observations of synchronous neural activity, and the emerging capability to record the spatiotemporal dynamics of neural activity by voltage-sensitive dyes and electrode arrays, provide opportunities for observing the population dynamics of neural ensembles within a dynamic systems context. New developments in the experimental physics of complex systems, such as the control of chaotic systems, selection of attractors, attractor switching and transient states, can be a source of powerful new analytical tools and insights into the dynamics of neural systems.

Action Potentials↗
PubMed2006

Awaking and sleeping of a complex network.

A network with a logistic-like local dynamics is considered. We implement a mean-field multiplicative coupling among first-neighbor nodes. When the coupling parameter is small, the dynamics is dissipated and there is no activity: the network is turned off. For a critical value of the coupling, a non-null stable synchronized state, which represents a turned on network, emerges. This global bifurcation is independent of the network topology. We characterize the bistability of the system by studying how to perform the transition, which is now topology dependent, from the active state to that with no activity, for the particular case of a scale-free network. This could be a naive model for the wakening and sleeping of a brain-like system, i.e., a multi-component system with two different dynamical behaviors.

Animals↗
PubMed2002

Software sensor for on-line estimation of the microbial activity in activated sludge systems.

This paper considers the design of a software sensor (or soft-sensor) for the on-line estimation of the biological activities of a colony of aerobic micro-organisms acting on activated sludge processes, where the carbonaceous waste degradation and nitrification processes are taken into account. These bioactivities are intimately related to the dissolved oxygen concentration. Two factors that affect the dynamics of the dissolved oxygen are the respiration rate or the oxygen uptake rate (OUR) and the oxygen transfer function (K(l)a). These items are challenging topics for the application of recursive identification due the nonlinear characteristic of the oxygen transfer function, and to the time-varying feature of the respiration rate. In this work, OUR and the oxygen transfer function are estimated through a software sensor, which is based on a modified version of the discrete extended Kalman filter. Numerical simulations are carried out in a predenitrifying activated sludge process benchmark and the obtained results demonstrate the applicability and efficiency of the proposed methodology, which should provide a valuable tool to supervise and control activated sludge processes.

Algorithms↗
PubMed2003

Emergence of complex behaviour from simple circuit structures.

The set of (feedback) circuits of a complex system is the machinery that allows the system to be aware of the levels of its crucial constituents. Circuits can be identified without ambiguity from the elements of the Jacobian matrix of the system. There are two types of circuits: positive if they comprise an even number of negative interactions, negative if this number is odd. The two types of circuits play deeply different roles: negative circuits are required for homeostasis, with or without oscillations, positive circuits are required for multistationarity, and hence, in biology, for differentiation and memory. In non-linear systems, a circuit can positive or negative (an 'ambiguous circuit', depending on the location in phase space. Full circuits are those circuits (or unions of disjoint circuits) that imply all the variables of the system. There is a tight relation between circuits and steady states. Each full circuit, if isolated, generates steady state(s) whose nature (eigenvalues) is determined by the structure of the circuit. Multistationarity requires the presence of at least two full circuits of opposite Eisenfeld signs, or else, an ambiguous circuit. We show how a significant part of the dynamical behaviour of a system can be predicted by a mere examination of its Jacobian matrix. We also show how extremely complex dynamics can be generated by such simple logical structures as a single (full and ambiguous) circuit.

Feedback↗
PubMed2006

Jumping solitary waves in an autonomous reaction-diffusion system with subcritical wave instability.

We describe a new type of solitary waves, which propagate in such a manner that the pulse periodically disappears from its original position and reemerges at a fixed distance. We find such jumping waves as solutions to a reaction-diffusion system with a subcritical short-wavelength instability. We demonstrate closely related solitary wave solutions in the quintic complex Ginzburg-Landau equation. We study the characteristics of and interactions between these solitary waves and the dynamics of related wave trains and standing waves.

Algorithms↗
PubMed2003

Conditions for efficient chaos-based communication.

We find the conditions for a chaotic system to transmit a general source of information efficiently. Transmission of information with very low probability of error is possible if the topological entropy of the transmitted wave signal is greater than or equal to the Shannon entropy of the source message minus the conditional entropy coming from the limitations of the channel (such as equivocation by the noise). This condition may not be always satisfied both due to dynamical constraints and due to the nonoptimal use of the dynamical partition. In both cases, we describe strategies to overcome these limitations.

Communication↗