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PubMed2001

Global identifiability of nonlinear models of biological systems.

A prerequisite for well-posedness of parameter estimation of biological and physiological systems is a priori global identifiability, a property which concerns uniqueness of the solution for the unknown model parameters. Assessing a priori global identifiability is particularly difficult for nonlinear dynamic models. Various approaches have been proposed in the literature but no solution exists in the general case. In this paper, we present a new algorithm for testing global identifiability of nonlinear dynamic models, based on differential algebra. The characteristic set associated to the dynamic equations is calculated in an efficient way and computer algebra techniques are used to solve the resulting set of nonlinear algebraic equations. The algorithm is capable of handling many features arising in biological system models, including zero initial conditions and time-varying parameters. Examples of usage of the algorithm for analyzing a priori global identifiability of nonlinear models of biological and physiological systems are presented.

Algorithms↗

PubMed1992

Chaotic dynamics in autonomic nervous system activity of a patient during a psychotherapy session.

Chaos theory and nonlinear dynamical modeling have been applied successfully in a variety of scientific fields ranging from the physical sciences to population biology. More recently, investigators have modeled long-term psychological processes by applying this paradigm to subjective measures. Yet, mathematically rigorous dynamical modeling of transient psychodynamic processes remains uncharted. Human behaviors and intrapsychic processes often defy prediction, yet psychodynamic theory views these as fundamentally deterministic. We postulate that a nonlinear dynamical analysis of mental states and mental control processes may help reconcile the paradox that seemingly random mental states are thought ultimately to result from deterministic, rather than stochastic, psychological processes. As an initial empirical approach, we examined spontaneously occurring autonomic activity of a patient during 1 hr of psychotherapy and illustrate that cardiac responses associated with psychologically meaningful events possess nonlinear characteristics indicative of chaos. Such nonlinear analyses may shed light on fundamental issues of psychological functioning.

Adult↗

PubMedDate not supplied

[A new method of identifying a systems based on the minimal quadratic discrepancy criterion for biophysics problems].

A wide range of biophysical systems are described by nonlinear dynamic models mathematically presented as a set of ordinary differential equations in the Cauchy explicit form: [formula: see text] Fij(X1(t),..,XN(t),t), (i = 1,...,N, j = 1,...,M), where Fij (X1(t), ..., XN(t), t) is a set of basis functions satisfying the Lipschitz condition. We investigate the problem of evaluation of model constants aij (the system identification) using experimental data about the time dependence of the dynamic parameters of the system Xi(t). A new method of system identification for the class of similar nonlinear dynamic models is proposed. It is shown that the problem of identifying an initial nonlinear model can be reduced to the solution of a system of linear equations for the matrix of the dynamic model constants [aj]i. It is proposed to determine the set of dynamic model constants aij using the criterion of minimal quadratic discrepancy for the time dependence of the set of dynamic parameters Xi(t). An important special case of the nonlinear model, the quadratic model, is considered. Test problems of identification using this method are presented for two nonlinear systems: the Van der Pol type multiparametric nonlinear oscillator and the strange attractor of Ressler, a widely known example of dynamic systems showing the stochastic behavior.

Algorithms↗

PubMed2006

Nonlinear fingering dynamics of reaction-diffusion acidity fronts: self-similar scaling and influence of differential diffusion.

Nonlinear interactions between chemical reactions and buoyancy-driven Rayleigh-Taylor instability of reaction-diffusion acidity fronts of the chlorite-tetrathionate (CT) reaction are studied theoretically in a vertical Hele-Shaw cell or a porous medium. To do so, we perform a numerical integration of a two-variable reaction-diffusion model of the CT system coupled through an advection term to Darcy's law ruling the evolution of the velocity field of the fluid. The fingering dynamics of these chemical fronts is characterized by the appearance of several fingers at onset. These fingers then undergo coarsening and eventually merge to form one single symmetric finger. We study this asymptotic dynamics as a function of the three dimensionless parameters of the problem, i.e., the Damkohler number Da, the diffusivity ratio delta of the two chemical species, and the Rayleigh number Ra constructed here on the basis of the width L(y) of the system. For moderate values of Ra, the asymptotic single finger is shown to have self-similar scaling properties while above a given value of Ra, which depends on the other values of the parameters, tip splitting comes into play. Increasing the difference of diffusivities of the two chemical species (i.e., increasing delta) leads to more efficient coarsening and smaller asymptotic fingers. Experimental procedures to verify our predictions are proposed.

Journal Article↗

PubMed2003

Output regulation of a class of unstructured models of continuous bioreactors: steady-state approaches.

This article deals with the output regulation of continuous bioreactors in the face of constant disturbances and inverse dynamics. Nonlinear controllers developed on the basis of approximate equilibrium manifolds can almost attenuate measurable or unmeasurable disturbances on the output. This nonlinear feed-forward/feedback control framework without any tuning parameters can be directly implemented to strictly nonlinear systems. Under dynamic actuator constraints and the availability of only output signals for use in the control law, closed-loop simulations demonstrate that the proposed control techniques are superior to a nonlinear PI control scheme based on the identified Hammerstein model.

Journal Article↗

PubMed2004

Angular velocity: a new method to improve prediction of ventricular fibrillation duration.

Ventricular fibrillation (VF) is a leading cause of sudden death. Electrical defibrillation is the primary modality of treatment, but evidence is accumulating that its use in the late stage of VF prior to providing ventilation, chest compressions and the administration of appropriate medication is detrimental. In VF of <5 min duration a 'shock first' strategy is effective. In VF of 5> min duration a 'perfuse first' approach is more effective. Because of the difficulty in determining the duration of VF in the clinical setting we have sought to develop method which analyze 5 s intervals of VF waveform and quickly provide an estimate of duration. Such methods would be useful in directing clinical interventions. Using methods of nonlinear dynamics and fractal geometry we have previously derived a quantitative measure of VF duration, namely the scaling exponent (ScE). In this study we report on a novel method also based on nonlinear dynamics, the angular velocity (AV). By constructing a flat, circular disk-shaped structure in a three-dimensional phase space and measuring the velocity of rotation of the position vector over time, a statistic is developed which rises from 58 rad/s at 1 min to 79 rad/s at 4 min and then decreases in a linear manner to 32 rad/s at 12.5 min. Using ScE and AV probability density estimated, VF of <5 min duration can be identified with 90% sensitivity on the basis of a single 5 s recording of the waveform. The combination of ScE and AV can be used in developing strategies for the treatment of VF during the different clinical phases of VF.

Algorithms↗

PubMed1997

[Contribution of non-linear mathematics (chaos theory) to EEG analysis].

Since 1963, nonlinear dynamics or "chaos theory" were widely used in various area of physics. The first application to the analysis of the human electroencephalogram (EEG) was performed in 1985. A tutorial revue of some concepts of nonlinear dynamics is presented with the various results obtained since 1985. The dimensional complexity of the EEG (DC) seems a good descriptor of the "desynchronisation" or decorrelation of the EEG: DC is high with eyes open, during a mental task and also during a tonic epileptic discharge. DC is low with eyes closed, during slow wave sleep, and during a clonic epileptic discharge. The measure of DC could allow a more objective comparison between various states of electrical cerebral activity.

Electroencephalography↗

PubMed2006

Nonlinear ripple dynamics on amorphous surfaces patterned by ion beam sputtering.

Erosion by ion-beam sputtering (IBS) of amorphous targets at off-normal incidence frequently produces a (nanometric) rippled surface pattern, strongly resembling macroscopic ripples on aeolian sand dunes. A suitable generalization of continuum descriptions of the latter allows us to describe theoretically for the first time the main nonlinear features of ripple dynamics by IBS, namely, wavelength coarsening and nonuniform translation velocity, that agree with similar results in experiments and discrete models. These properties are seen to be the anisotropic counterparts of in-plane ordering and (interrupted) pattern coarsening in IBS experiments on rotating substrates and at normal incidence.

Journal Article↗

PubMed2003

[Nonlinear analysis on the EEG information of rat epileptic model].

The aim of this study was to develop a new method of epileptic prediction using nonlinear dynamic theory. When rat was falling sickness, its EEG was researched by using approximate entropy and correlation dimension. The results showed the approximate entropy and correlation dimension during epileptic seizure are obviously lower than those before seizure and after seizure. The span of time before seizure is a special phase. Before the seizure symptom appeared, the complexity of EEG had begun declining. Thus, the outbreak of epilepsy could be predicted in short time using nonlinear dynamic methods.

Algorithms↗

PubMed1998

Nonlinear interdependencies of EEG signals in human intracranially recorded temporal lobe seizures.

The degree of interdependence between intracranial EEG channels was investigated in four epileptic patients with complex partial seizures of mesial temporal lobe origin. With a new method to characterize nonlinear dynamical interdependence-the mutual nonlinear prediction-we demonstrated here a possibility to quantify, during epileptic seizures, the relationships between EEG signals of electrode contacts in the epileptogenic area. During the interictal period, the degree of nonlinear interdependences were very low or absent. In contrast, it was found that transient patterns of nonlinear interdependences emerge at the initial spread of the seizure, during essential parts of its development, and at seizure end, but the maintenance of these interactions are not observed throughout the seizure activity. These results suggest that the nonlinear associations plays an important role in epileptogenesis, and that the process of neuronal entrainment during seizure onset involves a transient interaction between a distributed network of neuronal aggregates, but the maintenance of this interaction is not required for sustained seizure activity. Furthermore, this technique can describe properly the spatio-temporal organisation of the seizures of medio-temporal lobe origin and could become a very useful tool to aid the localization of the epileptogenic regions at the origin of epileptic seizures and their pathways of propagation.

Algorithms↗

PubMed1999

Detection of the cardiac function by fractal dimension analysis.

Nonlinearity in circulation control attracts attention because nonlinearity is thought to be essential in the function of the living body. Many investigators have pointed out that the analysis of heart rate variability in particular is important in the analysis of autonomic nerve and cardiac function evaluation. Heart rate variability shows nonlinear behavior. However, until the present, many reports have been premised on linearity; linear correlation by frequency analysis has been used by many studies. However, in terms of this methodology, there is a problem applying it to the nonlinear living body. Therefore, fractal and chaos methodology has been used. The ascertainment of cardiac function has become important in allowing the clinical stage of a ventricular assist system to be successful. The purpose of this study was cardiac function evaluation by a methodology that was premised on nonlinearity. Chaos and fractal theory was used as a nonlinear dynamic theory. As a methodology of measurement, the volume of the left ventricle was used rather than an electrocardiogram, the waveform of arterial blood pressure. The volume was measured using acoustic quantification (AQ) ultrasonic echocardiography. Using these methodologies, the time series of many patients were analyzed. For example, drug administration was attempted in this study, and it was found that some drugs like ACE inhibitors showed a significant effect upon nonlinear dynamics in the cardiovascular system. The result, which attempted cardiac function evaluation by these various methodologies, is reported.

Angiotensin-Converting Enzyme Inhibitors↗

PubMed2001

Predicting the responses of mechanoreceptor neurons to physiological inputs by nonlinear system identification.

The nonlinear dynamic properties of action potential encoding were studied in mechanosensory neurons innervating the slits of a slit-sense organ in the tropical wandering spider, Cupiennius salei. The organ contains two types of neurons that are morphologically similar but have different dynamic properties. Type A neurons produce only one or two action potentials in response to a mechanical or electrical stimulus of any suprathreshold amplitude, while type B neurons can fire prolonged bursts of action potentials in response to similar stimuli. Neurons were stimulated with pseudorandomly modulated intracellular current while recording the resultant fluctuations in membrane potential and action potentials. A parallel cascade method was used to estimate a third-order Volterra series to describe the nonlinear dynamic relationship between membrane potential and action potentials. Kernels measured for the two types of neurons had reproducible forms that showed differences between the two neuron types. The measured kernels were able to predict the responses of the neurons to novel pseudorandomly modulated inputs with reasonable fidelity. However, the Volterra series did not adequately predict the difference in responses to step depolarizations.

Action Potentials↗

PubMed2001

Fractal analysis of eye movements during reading.

We present a new method for the analysis of reading eye movements based on the methods of nonlinear dynamics. In this preliminary study, the eye movements of normal and abnormal readers were analyzed for evidence of chaotic, nonlinear dynamical behavior. Both power spectral density analysis and fractal dimension determination showed evidence of nonlinearity as manifest in chaotic behavior. The computed fractal dimension of the system's presumed attractor seemed directly related to qualitative assessment of reading ability. Representative subjects did not differ in a similar analysis of pursuit movements. Although fractal analysis did not distinguish unconditionally between normal and abnormal reading in this preliminary study, it nevertheless offers a promising hitherto unused approach to the analysis of eye movements and the modeling of oculomotor behavior.

Adult↗

PubMed2004

Cross-scale interactions, nonlinearities, and forecasting catastrophic events.

Catastrophic events share characteristic nonlinear behaviors that are often generated by cross-scale interactions and feedbacks among system elements. These events result in surprises that cannot easily be predicted based on information obtained at a single scale. Progress on catastrophic events has focused on one of the following two areas: nonlinear dynamics through time without an explicit consideration of spatial connectivity [Holling, C. S. (1992) Ecol. Monogr. 62, 447-502] or spatial connectivity and the spread of contagious processes without a consideration of cross-scale interactions and feedbacks [Zeng, N., Neeling, J. D., Lau, L. M. & Tucker, C. J. (1999) Science 286, 1537-1540]. These approaches rarely have ventured beyond traditional disciplinary boundaries. We provide an interdisciplinary, conceptual, and general mathematical framework for understanding and forecasting nonlinear dynamics through time and across space. We illustrate the generality and usefulness of our approach by using new data and recasting published data from ecology (wildfires and desertification), epidemiology (infectious diseases), and engineering (structural failures). We show that decisions that minimize the likelihood of catastrophic events must be based on cross-scale interactions, and such decisions will often be counterintuitive. Given the continuing challenges associated with global change, approaches that cross disciplinary boundaries to include interactions and feedbacks at multiple scales are needed to increase our ability to predict catastrophic events and develop strategies for minimizing their occurrence and impacts. Our framework is an important step in developing predictive tools and designing experiments to examine cross-scale interactions.

Animals↗

PubMed2000

Nonlinearly saturated dynamical state of a three-wave mode-coupled dissipative system with linear instability

Linearly unstable dissipative systems with quadratic nonlinearity occurring in plasma physics, optics, fluid mechanics, etc. are often modeled by a general set of three-wave mode-coupled ordinary differential equations for complex variables. Bounded attractors of the set approximate nonlinearly saturated turbulent states of real physical systems. Exact criteria for boundedness of the attractors are found. Fundamentally different kinds of asymptotic behavior of the wave triad are classified in the parameter space and quantitatively assessed.

Journal Article↗

PubMed1992

Complex dynamics underlying the human electrocardiogram.

Sequences of different human cardiac rhythms terminating in ventricular fibrillation have been studied, both qualitatively and quantitatively, with methods of nonlinear dynamics. The analysis has been applied to ECG epochs belonging to rhythms of increasing electrocardiographic irregularity: from sinus rhythm to prefibrillatory rhythms and then to ventricular fibrillation. The phase portraits of these rhythms have been reconstructed from the ECG recording with the time-delay technique, and their correlation dimensions have been estimated with the algorithm of Grassberger and Procaccia (1983a, b). Different cardiac rhythms exhibit different correlation dimensions that describe the corresponding degrees of complexity. The correlation dimension increases as one proceeds from sinus rhythm to fully developed ventricular fibrillation via intermediate rhythms. The fully developed ventricular fibrillation shows the highest degree of complexity. The dimensional analysis supports the existence of complex dynamics underlying different cardiac rhythms and reveals an increase in dimensional complexity corresponding to an increase in electrocardiographic irregularity. Our results indicate that nonlinear dynamics may be used to assess various dynamic states of the heart and may offer a non-invasive tool to investigate the complex dynamic phenomena occurring during arrhythmia.

Algorithms↗

PubMed2000

Information transmission and recovery in neural communications channels.

Biological neural communications channels transport environmental information from sensors through chains of active dynamical neurons to neural centers for decisions and actions to achieve required functions. These kinds of communications channels are able to create information and to transfer information from one time scale to the other because of the intrinsic nonlinear dynamics of the component neurons. We discuss a very simple neural information channel composed of sensory input in the form of a spike train that arrives at a model neuron, then moves through a realistic synapse to a second neuron where the information in the initial sensory signal is read. Our model neurons are four-dimensional generalizations of the Hindmarsh-Rose neuron, and we use a model of chemical synapse derived from first-order kinetics. The four-dimensional model neuron has a rich variety of dynamical behaviors, including periodic bursting, chaotic bursting, continuous spiking, and multistability. We show that, for many of these regimes, the parameters of the chemical synapse can be tuned so that information about the stimulus that is unreadable at the first neuron in the channel can be recovered by the dynamical activity of the synapse and the second neuron. Information creation by nonlinear dynamical systems that allow chaotic oscillations is familiar in their autonomous oscillations. It is associated with the instabilities that lead to positive Lyapunov exponents in their dynamical behavior. Our results indicate how nonlinear neurons acting as input/output systems along a communications channel can recover information apparently "lost" in earlier junctions on the channel. Our measure of information transmission is the average mutual information between elements, and because the channel is active and nonlinear, the average mutual information between the sensory source and the final neuron may be greater than the average mutual information at an earlier neuron in the channel. This behavior is strikingly different than the passive role communications channels usually play, and the "data processing theorem" of conventional communications theory is violated by these neural channels. Our calculations indicate that neurons can reinforce reliable transmission along a chain even when the synapses and the neurons are not completely reliable components. This phenomenon is generic in parameter space, robust in the presence of noise, and independent of the discretization process. Our results suggest a framework in which one might understand the apparent design complexity of neural information transduction networks. If networks with many dynamical neurons can recover information not apparent at various waystations in the communications channel, such networks may be more robust to noisy signals, may be more capable of communicating many types of encoded sensory neural information, and may be the appropriate design for components, neurons and synapses, which can be individually imprecise, inaccurate "devices."

Action Potentials↗

PubMed2002

An efficient method for studying short-term plasticity with random impulse train stimuli.

In this article, we introduce an efficient method that models quantitatively nonlinear dynamics associated with short-term plasticity (STP) in biological neural systems. It is based on the Voterra-Wiener modeling approach adapted for special stimulus/response datasets. The stimuli are random impulse trains (RITs) of fixed amplitude and Poisson distributed, variable interimpulse intervals. The class of stimuli, we use can be viewed as a hybrid between the paired impulse approach (variable interimpulse interval between two input impulses) and the fixed frequency approach (impulses repeated at fixed intervals, varying in frequency from one stimulus dataset to the next). The responses are sequences of population spike amplitudes of variable size and are assumed to be contemporaneous with the corresponding impulses in the RITs they are evoked by. The nonlinear dynamics of the mechanisms underlying STP are captured by kernels used to create compact STP models with predictive capabilities. Compared to similar methods in the literature, the method presented in this article provides a comprehensive model of STP with considerable improvement in prediction accuracy and requires shorter experimental data collection time.

Action Potentials↗