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

The School of Mathematics, The University of Manchester, P.O. Box 88, Manchester M60 1QD, UK.
Journal of Mathematical Biology (Impact Factor: 1.85). 01/2006; 51(6):661-94. DOI: 10.1007/s00285-005-0336-4
Source: PubMed


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.

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Available from: David S Broomhead, Sep 30, 2015
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    • "A previous paper showed that one advantage of the nonlinear dynamics approach is the capacity of bifurcation analysis to reveal the full range of behaviour that a model is capable of producing (Akman et al., 2005). "
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    • "Possible implications of the unstable time constant for understanding the normal eye movement control system Various models have been proposed to explain how congenital nystagmus might arise; the sawtooth nystagmus waveform can be generated by an abnormality of the saccadic system (Akman et al. 2005), pursuit system (Wang and Dell'Osso 2011) or neural integrator (Optican and Zee 1984; Barreiro et al. 2009), whilst the pendular waveform can be generated by an abnormality of the optokinetic (Huang et al. 2006) or pursuit systems (Harris 1995; Jacobs and Dell'Osso 2004). Although such models can produce plausible nystagmus waveforms, this does not imply that the local linear dynamics of the models matches that found experimentally (Akman et al. 2006). "
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    ABSTRACT: Congenital nystagmus is an involuntary bilateral horizontal oscillation of the eyes that develops soon after birth. In this study, the time constants of each of the components of the neural signal underlying congenital nystagmus were obtained by time series analysis and interpreted by comparison with those of the normal oculomotor system. In the neighbourhood of the fixation position, the system generating the neural signal is approximately linear with 3 degrees of freedom. The shortest time constant was in the range of 7-9 ms and corresponds to a normal saccadic burst signal. The other stable time constant was in the range of 22-70 ms and corresponds to the slide signal. The final time constant characterises the unidentified neural mechanism underlying the unstable drift component of the oscillation cycle and ranges between 31 and 32 ms across waveforms. The characterisation of this unstable time constant poses a challenge for the modelling of both the normal and abnormal oculomotor control system. We tentatively identify the unstable component with the eye position signal supplied to the superior colliculus in the normal eye movement system and explore some of the implications of this hypothesis.
    Experimental Brain Research 05/2012; 220(3-4):213-21. DOI:10.1007/s00221-012-3130-8 · 2.04 Impact Factor
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    • "In the case of any misalignment of the visual axis (e.g., squint) two classes of pause cells could presumably develop: one which fires whenever the right eye is fixating and one when the left eye is fixating — each exhibiting a different offset. The effect of incorporating pause cells with offsets into models such as those presented by Broomhead et al. (2000), Akman et al. (2005), Laptev et al. (2006), and Clement et al. (2008), is to generate MLN type waveforms which switch beat direction according to which class of pause cell are activated. Although this could explain the sensori-motor behaviour during exogenous orienting (Fig. 4a), endogenous orienting (Fig. 4b "
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    ABSTRACT: Latent nystagmus (LN) and manifest latent nystagmus (MLN) are closely associated with early visual deprivation and strabismus. In both cases, the eyes oscillate horizontally in an involuntary manner and the fast phases always beat towards the attending or fixing eye. By simultaneously recording eye movements during the dichoptic viewing of dissimilar visual stimuli we present evidence that MLN offers a unique opportunity to examine the nature of sensori-motor switching. In particular, we show how the nystagmus beat direction is strongly influenced by endogenous and exogenous attention. A model describing the possible mechanisms underpinning the sensori-motor switching is proposed.
    Progress in brain research 02/2008; 171:497-502. DOI:10.1016/S0079-6123(08)00671-7 · 2.83 Impact Factor
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