Nonlinear time series analysis of jerk congenital nystagmus.

Nonlinear dynamics provides a complementary framework to control theory for the quantitative analysis of the oculomotor control system. This paper presents a number of findings relating to the aetiology and mechanics of the pathological ocular oscillation jerk congenital nystagmus (jerk CN). A range of time series analysis techniques were applied to recorded jerk CN waveforms, and also to simulated jerk waveforms produced by an established model in which the oscillations are a consequence of an unstable neural integrator. The results of the analysis were then interpreted within the framework of a generalised model of the unforced oculomotor system.This work suggests that for jerk oscillations, the origin of the instability lies in one of the five oculomotor subsystems, rather than in the final common pathway (the neural integrator and muscle plant). Additionally, experimental estimates of the linearised foveation dynamics imply that a refixating fast phase induced by a near-homoclinic trajectory will result in periodic oscillations. Local dimension calculations show that the dimension of the experimental jerk CN data increases during the fast phase, indicating that the oscillations are not periodic, and hence that the refixation mechanism is of greater complexity than a homoclinic reinjection. The dimension increase is hypothesised to result either from a signal-dependent noise process in the saccadic system, or the activation of additional oculomotor components at the beginning of the fast phase. The modification of a recent saccadic system model to incorporate biologically realistic signal-dependent noise is suggested, in order to test the first of these hypotheses.

Eye movement instabilities and nystagmus can be predicted by a nonlinear dynamics model of the saccadic system.

The study of eye movements and oculomotor disorders has, for four decades, greatly benefitted from the application of control theoretic concepts. This paper is an example of a complementary approach based on the theory of nonlinear dynamical systems. Recently, a nonlinear dynamics model of the saccadic system was developed, comprising a symmetric piecewise-smooth system of six first-order autonomous ordinary differential equations. A preliminary numerical investigation of the model revealed that in addition to generating normal saccades, it could also simulate inaccurate saccades, and the oscillatory instability known as congenital nystagmus (CN). By varying the parameters of the model, several types of CN oscillations were produced, including jerk, bidirectional jerk and pendular nystagmus. The aim of this study was to investigate the bifurcations and attractors of the model, in order to obtain a classification of the simulated oculomotor behaviours. The application of standard stability analysis techniques, together with numerical work, revealed that the equations have a rich bifurcation structure. In addition to Hopf, homoclinic and saddlenode bifurcations organised by a Takens-Bogdanov point, the equations can undergo nonsmooth pitchfork bifurcations and nonsmooth gluing bifurcations. Evidence was also found for the existence of Hopf-initiated canards. The simulated jerk CN waveforms were found to correspond to a pair of post-canard symmetry-related limit cycles, which exist in regions of parameter space where the equations are a slow-fast system. The slow and fast phases of the simulated oscillations were attributed to the geometry of the corresponding slow manifold. The simulated bidirectional jerk and pendular waveforms were attributed to a symmetry invariant limit cycle produced by the gluing of the asymmetric cycles. In contrast to control models of the oculomotor system, the bifurcation analysis places clear restrictions on which kinds of behaviour are likely to be associated with each other in parameter space, enabling predictions to be made regarding the possible changes in the oscillation type that may be observed upon changing the model parameters. The analysis suggests that CN is one of a range of oculomotor disorders associated with a pathological saccadic braking signal, and that jerk and pendular nystagmus are the most probable oscillatory instabilities. Additionally, the transition from jerk CN to bidirectional jerk and pendular nystagmus observed experimentally when the gaze angle or attention level is changed is attributed to a gluing bifurcation. This suggests the possibility of manipulating the waveforms of subjects with jerk CN experimentally to produce waveforms with an extended foveation period, thereby improving visual resolution.

Synergistic control of oscillations in the NF-kappaB signalling pathway.

In previous work, we studied the behaviour of a model of part of the NF-kappaB signalling pathway. The model displayed oscillations that varied both in number, amplitude and frequency when its parameters were varied. Sensitivity analysis showed that just nine of the 64 reaction parameters were mainly responsible for the control of the oscillations when these parameters were varied individually. However, the control of the properties of any complex system is distributed, and, as many of these reactions are highly non-linear, we expect that their interactions will be too. Pairwise modulation of these nine parameters gives a search space some 50 times smaller (81 against 4096) than that required for the pairwise modulation of all 64 reactions, and this permitted their study (which would otherwise have been effectively intractable). Strikingly synergistic effects were observed, in which the effect of one of the parameters was strongly (and even qualitatively) dependent on the values of another parameter. Regions of parameter space could be found in which the amplitude, but not the frequency (timing), of oscillations varied, and vice versa. Such modelling will permit the design and performance of experiments aimed at disentangling the role of the dynamics of oscillations, rather than simply their amplitude, in determining cell fate. Overall, the analyses reveal a level of complexity in these dynamic models that is not apparent from study of their individual parameters alone and point to the value of manipulating multiple elements of complex networks to achieve desired physiological effects.

Sensitivity analysis of parameters controlling oscillatory signalling in the NF-kappaB pathway: the roles of IKK and IkappaBalpha.

Analysis of cellular signalling interactions is expected to create an enormous informatics challenge, perhaps even greater than that of analysing the genome. A key step in the evolution towards a more quantitative understanding of signalling is to specify explicitly the kinetics of all chemical reaction steps in a pathway. We have reconstructed a model of the nuclear factor, kappaB (NF-kappaB) signalling pathway, containing 64 parameters and 26 variables, including steps in which the activation of the NF-kappaB transcription factor is intimately associated with the phosphorylation and ubiquitination of its inhibitor kappaB by a membrane-associated kinase, and its translocation from the cytoplasm to the nucleus. We apply sensitivity analysis to the model. This identifies those parameters in this (IkappaB)/NF-kappaB signalling system (containing only induced IkappaBalpha isoform) that most affect the oscillatory concentration of nuclear NF-kappaB (in terms of both period and amplitude). The intention is to provide guidance on which proteins are likely to be most significant as drug targets or should be exploited for further, more detailed experiments. The sensitivity coefficients were found to be strongly dependent upon the magnitude of the parameter change studied, indicating the highly non-linear nature of the system. Of the 64 parameters in the model, only eight to nine exerted a major control on nuclear NF-kappaB oscillations, and each of these involved as reaction participants either the IkappaB kinase (IKK) or IkappaBalpha, directly. This means that the dominant dynamics of the pathway can be reflected, in addition to that of nuclear NF-kappaB itself, by just two of the other pathway variables. This is conveniently observed in a phase-plane plot.

The Whitney reduction network: a method for computing autoassociative graphs.

This article introduces a new architecture and associated algorithms ideal for implementing the dimensionality reduction of an m-dimensional manifold initially residing in an n-dimensional Euclidean space where n >> m. Motivated by Whitney's embedding theorem, the network is capable of training the identity mapping employing the idea of the graph of a function. In theory, a reduction to a dimension d that retains the differential structure of the original data may be achieved for some d < or = 2m + 1. To implement this network, we propose the idea of a good-projection, which enhances the generalization capabilities of the network, and an adaptive secant basis algorithm to achieve it. The effect of noise on this procedure is also considered. The approach is illustrated with several examples.

Ants and agents: a process algebra approach to modelling ant colony behaviour.

Process algebras are widely used in the analysis of distributed computer systems. They allow formal reasoning about how the various components of a system contribute to its overall behaviour. In this paper we show how process algebras can be usefully applied to understanding social insect biology, in particular to studying the relationship between algorithmic behaviour of individual insects and the dynamical behaviour of their colony. We argue that process algebras provide a useful formalism for understanding this relationship, since they combine computer simulation, Markov chain analysis and mean-field methods of analysis. Indeed, process algebras can provide a framework for relating these three methods of analysis to each other and to experiments. We illustrate our approach with a series of graded examples of modelling activity in ant colonies.

Relating individual behaviour to population dynamics.

How do the behavioural interactions between individuals in an ecological system produce the global population dynamics of that system? We present a stochastic individual-based model of the reproductive cycle of the mite Varroa jacobsoni, a parasite of honeybees. The model has the interesting property in that its population level behaviour is approximated extremely accurately by the exponential logistic equation or Ricker map. We demonstrated how this approximation is obtained mathematically and how the parameters of the exponential logistic equation can be written in terms of the parameters of the individual-based model. Our procedure demonstrates, in at least one case, how study of animal ecology at an individual level can be used to derive global models which predict population change over time.

Modelling of congenital nystagmus waveforms produced by saccadic system abnormalities.

Models of the mechanisms of normal eye movements are typically described in terms of the block diagrams which are used in control theory. An alternative approach to understanding the mechanisms of normal eye movements involves describing the eye movement behaviour in terms of smooth changes in state variables. The latter approach captures the burst cell firing against motor error (difference between target gaze angle and current gaze angle) phase plane behaviour which is found experimentally and facilitates the modelling of variations in burst cell behaviour. A novel explanation of several types of congenital nystagmus waveforms is given in terms of a saccadic termination abnormality.

Shape and dynamics of thermoregulating honey bee clusters.

A model of simple algorithmic "agents" acting in a discrete temperature field is used to investigate the movement of individuals in thermoregulating honey bee (Apis mellifera) clusters. Thermoregulation in over-wintering clusters is thought to be the result of individual bees attempting to regulate their own body temperatures. At ambient temperatures above 0( degrees )C, a clustering bee will move relative to its neighbours so as to put its local temperature within some ideal range. The proposed model incorporates this behaviour into an algorithm for bee agents moving on a two-dimensional lattice. Heat transport on the lattice is modelled by a discrete diffusion process. Computer simulation of this model demonstrates qualitative behaviour which agrees with that of real honey bee clusters. In particular, we observe the formation of both disc- and ring-like cluster shapes. The simulation also suggests that at lower ambient temperatures, clusters do not always have a stable shape but can oscillate between insulating rings of different sizes and densities.

Scaling and interleaving of subsystem Lyapunov exponents for spatio-temporal systems.

The computation of the entire Lyapunov spectrum for extended dynamical systems is a very time consuming task. If the system is in a chaotic spatio-temporal regime it is possible to approximately reconstruct the Lyapunov spectrum from the spectrum of a subsystem by a suitable rescaling in a very cost effective way. We compute the Lyapunov spectrum for the subsystem by truncating the original Jacobian without modifying the original dynamics and thus taking into account only a portion of the information of the entire system. In doing so we notice that the Lyapunov spectra for consecutive subsystem sizes are interleaved and we discuss the possible ways in which this may arise. We also present a new rescaling method, which gives a significantly better fit to the original Lyapunov spectrum. We evaluate the performance of our rescaling method by comparing it to the conventional rescaling (dividing by the relative subsystem volume) for one- and two-dimensional lattices in spatio-temporal chaotic regimes. Finally, we use the new rescaling to approximate quantities derived from the Lyapunov spectrum (largest Lyapunov exponent, Lyapunov dimension, and Kolmogorov-Sinai entropy), finding better convergence as the subsystem size is increased than with conventional rescaling. (c) 1999 American Institute of Physics.

Dynamical systems analysis: a new method of analysing congenital nystagmus waveforms.

Congenital nystagmus is an oculomotor disorder in which fixation is disrupted by rhythmical, bilateral involuntary oscillations. Clinically these eye movements have been described with some degree of success in terms of their peak-to-peak amplitude, frequency, mean velocity and waveform shape. However, it has not proved possible to diagnose any underlying pathology from the nystagmus characteristics. Here, we propose a new approach to understanding the nystagmus using dynamical systems theory. Our approach is based on the use of delay embedding techniques, which allow one to relate a time series of scalar observations to the state space dynamics of the underlying dynamical system. Using this approach we quantify the dynamics of the nystagmus in the region of foveation and present evidence to suggest that it is low-dimensional and deterministic. Our results put new constraints on acceptable models of nystagmus and suggest a way to make a closer link between data analysis and model development. This approach raises the hope that techniques originally developed to stabilise chaotic systems, by using small perturbations, may prove useful in the control of nystagmus.