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

Abadi, Richard V.; Broomhead, D. S.; Clement, Richard A.; Whittle, J. P.; Worfolk, Ralph

doi:10.1007/s002210050229

Cited by 27 publications

(29 citation statements)

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“…A cycle was reset by a simulated saccade using a minimum jerk profile [15]. The three-dimensional projection of the reconstructed phase space trajectory, calculated using Matlab's Tstool toolbox [16], shows a similar profile as observed empirically [13] [ Figure 5(c) and (d)].…”

Section: Timing Limitations and Extended Extremalsmentioning

confidence: 76%

“…Real CN sometimes shows waveforms similar to the global optimum waveform predicted by Equations (10) and (11) (e.g., [13]), but it is not the most common. More realism emerges when we consider the limitations imposed by the eye movement system, which we now outline.…”

Section: Optimal Image Motionmentioning

confidence: 91%

“…This will influence the next cycle either directly or stochastically, adding to the complexity of the sequence of cycles. The possibility that CN may be a nonlinear dynamical system has been explored empirically by Abadi and colleagues, who have reported low dimensionality and determinism for real CN data taken from a female subject with typical idiopathic congenital nystagmus [13]. Using a similar approach based on the use of delay embedding techniques, we quantified the dynamics of simulated cycles in the region of foveation.…”

Section: Timing Limitations and Extended Extremalsmentioning

confidence: 99%

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A Distal Model of Congenital Nystagmus as Nonlinear Adaptive Oscillations

Harris

Berry

2006

Nonlinear Dyn

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Abstract. Congenital nystagmus (CN) is an incurable pathological spontaneous oscillation of the eyes with an onset in the first few months of life. The pathophysiology of CN is mysterious. There is no consistent neurological abnormality, but the majority of patients have a wide range of unrelated congenital visual abnormalities affecting either the cornea, lens, retina or optic nerve. In this theoretical study, we show that these eye oscillations could develop as an adaptive response to maximize visual contrast with poor foveal function in the infant visuomotor system, at a time of peak neural plasticity. We argue that in a visual system with abnormally poor high spatial frequency sensitivity, image contrast is not only maintained by keeping the image on the fovea (or its remnant) but also by some degree of image motion. Using the calculus of variations, we show that the optimal trade-off between these conflicting goals is to generate oscillatory eye movements with increasing velocity waveforms, as seen in real CN. When we include a stochastic component to the start of each epoch (quick-phase inaccuracy) various observed waveforms (including pseudo-cycloid) emerge as optimal strategies. Using the delay embedding technique, we find a low fractional dimension as reported in real data. We further show that, if a velocity command-based pre-motor circuitry (neural integrator) is harnessed to generate these waveforms, the emergence of a null region is inevitable. We conclude that CN could emerge paradoxically as an 'optimal' adaptive response in the infant visual system during an early critical period. This can explain why CN does not emerge later in life and why CN is so refractory to treatment. It also implies that any therapeutic intervention would need to be very early in life.

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Section: Timing Limitations and Extended Extremalsmentioning

confidence: 76%

Section: Optimal Image Motionmentioning

confidence: 91%

Section: Timing Limitations and Extended Extremalsmentioning

confidence: 99%

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A Distal Model of Congenital Nystagmus as Nonlinear Adaptive Oscillations

Harris

Berry

2006

Nonlinear Dyn

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“…In the last few years, however, there has been increasing interest in the use of nonlinear dynamics techniques to model oculomotor control, and to analyse eye movement time series [3], [37], [25], [18], [7], [20].…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2005

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The study of eye movement control and oculomotor disorders has, for four decades, relied on control theoretic concepts for its theoretical foundation. 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. * Now at: Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK 1The 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.

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“…The last few years have seen the development of differential equation models (Gancarz and Grossberg, 1998;Broomhead et al, 2000) together with the application of time series analysis to eye movement recordings (Shelhamer, 1997;Abadi et al, 1997;Clement et al, 2002b,c).…”

Section: Introductionmentioning

confidence: 99%

Nonlinear time series analysis of jerk congenital nystagmus

Akman

Broomhead

Clement³

et al. 2006

J Comput Neurosci

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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 both recorded jerk CN waveforms and simulated waveforms produced by an established model in which the oscillations are a consequence of an unstable neural integrator. The results of the time series 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.

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

Cited by 27 publications

References 0 publications

A Distal Model of Congenital Nystagmus as Nonlinear Adaptive Oscillations

A Distal Model of Congenital Nystagmus as Nonlinear Adaptive Oscillations

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

Nonlinear time series analysis of jerk congenital nystagmus

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