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Abstract: 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.

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, κB (NF-κB) signalling pathway, containing 64 parameters and 26 variables, including steps in which the activation of the NF-κB transcription factor is intimately associated with the phosphorylation and ubiquitination of its inhibitor κB 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 (IκB)/NF-κB signalling system (containing only induced IκBα isoform) that most affect the oscillatory concentration of nuclear NF-κB (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-κB oscillations, and each of these involved as reaction participants either the IκB kinase (IKK) or IκBα, directly. This means that the dominant dynamics of the pathway can be reflected, in addition to that of nuclear NF-κB itself, by just two of the other pathway variables. This is conveniently observed in a phase-plane plot.

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°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.

This paper proposes a novel technique for the prediction of solar-terrestrial data sets that contain a significant proportion of missing data points. A nonlinear interpolation technique is employed to assign values to gaps in a time series. Each missing point is interpolated such that the error introduced into any specific predictive function is minimised. Radial Basis Function Neural Networks are adopted for the purpose of prediction, and their advantages over their Multi-Layer Perceptron counterparts are outlined. This technique has general application in any instance where the effects of interpolation upon a given analysis process need to be minimised or a complete time series needs to be constructed from incomplete data.

A fully systolic network is proposed for rapid and efficient multidimensional interpolation using radial basis functions (RBFs). The resulting processor, which constitutes a form of nonlinear adaptive filter, resembles a neural network based on the nonlinear, feedforward, multilayer perceptron model.