Multifractal characterization of nystagmus eye movements.

In this work, we investigate the multifractal properties of eye movement dynamics of children with infantile nystagmus, particularly the fluctuations of its velocity. The eye movements of three children and one adult with infantile nystagmus were evaluated in a simple task in comparison with 28 children with no ocular pathologies. Four indices emerge from the analysis: the classical Hurst exponent, the singularity strength corresponding to the maximum of the singularity spectrum, the asymmetry of the singularity spectrum, and the multifractal strength, each of which characterizes a particular aspect of eye movement dynamics. Our findings indicate that, when compared to children with no ocular pathologies, patients with infantile nystagmus present lower values of all indices. Except for the multifractal strength, the difference in the remaining indices is statistically significant. To test whether the characterization of patients with infantile nystagmus in terms of multifractality indices allows them to be distinguished from children without ocular pathologies, we performed an unsupervised clustering analysis and classified the subjects using supervised clustering techniques. The results indicate that these indices do, indeed, distinctively characterize the eye movements of patients with infantile nystagmus.

Modelling the eye movements of dyslexic children during reading as a continuous time random walk.

The study of eye movements during reading is considered a valuable tool for understanding the underlying cognitive processes and for its ability to detect alterations that could be associated with neurocognitive deficiencies or visual conditions. During reading, the gaze moves from one position to the next on the text performing a saccade-fixation sequence. This dynamics resembles processes usually described as continuous time random walk, where the jumps are the saccadic movements and waiting times are the duration of fixations. The time between jumps (intersaccadic time) consists of stochastic waiting time and flight time, which is a function of the jump length (the amplitude of the saccade). This motivates the present proposal of a model of eye movements during reading in the framework of the intermittent random walk but considering the time between jumps as a combined stochastic-deterministic process. The parameters used in this model were obtained from records of eye movements of children with dyslexia and typically developed for children performing a reading task. The jump lengths arise from the characteristics of the selected text. The time required for the flights was obtained based on a previously proposed model. Synthetic signals were generated and compared with actual eye movement signals in a complexity-entropy plane.

Ordinal pattern transition networks in eye tracking reading signals.

Eye tracking is an emerging technology with a wide spectrum of applications, including non-invasive neurocognitive diagnosis. An advantage of the use of eye trackers is in the improved assessment of indirect latent information about several aspects of the subjects' neurophysiology. The path to uncover and take advantage of the meaning and implications of this information, however, is still in its very early stages. In this work, we apply ordinal patterns transition networks as a means to identify subjects with dyslexia in simple text reading experiments. We registered the tracking signal of the eye movements of several subjects (either normal or with diagnosed dyslexia). The evolution of the left-to-right movement over time was analyzed using ordinal patterns, and the transitions between patterns were analyzed and characterized. The relative frequencies of these transitions were used as feature descriptors, with which a classifier was trained. The classifier is able to distinguish typically developed vs dyslexic subjects with almost 100% accuracy only analyzing the relative frequency of the eye movement transition from one particular permutation pattern (plain left to right) to four other patterns including itself. This characterization helps understand differences in the underlying cognitive behavior of these two groups of subjects and also paves the way to several other potentially fruitful analyses applied to other neurocognitive conditions and tests.

Information-theoretic characterization of eye-tracking signals with relation to cognitive tasks.

Eye tracking is being increasingly used as a more powerful diagnosis instrument when compared with traditional pen-and-paper tests in psychopedagogy and psychology. This technology may significantly improve neurocognitive assessments in gathering indirect latent information about the subjects' performance. However, the meaning and implications of these data are far from being fully understood. In this work, we present a comprehensive study of eye tracking time series in terms of statistical complexity measures. We registered the eye tracking movements of several subjects solving the two parts of the commonly applied Trail Making Test (TMT-A and TMT-B) and studied their Shannon entropy, disequilibrium, statistical complexity, and Fisher information with respect to three different probability distributions. The results show that these quantifiers reveal information about different features of the gaze depending on the distribution considered. As a meaningful result, we found that Fisher information in the position distribution reflects the difficulties encountered by the subject when solving the task. Such a characterization may be of interest to understand the underlying cognitive tasks performed by the subjects, and, additionally, it can serve as a source of valuable parameters to quantitatively assess how and why the subjects budget their attention, providing psychologists and psychopedagogues with more refined neuropsychological evaluation features and tools.

Models for saccadic motion and postsaccadic oscillations.

In a recent letter [S. Bouzat et al., Phys. Rev. Lett. 120, 178101 (2018)10.1103/PhysRevLett.120.178101], a mathematical model for eyeball and pupil motion was developed allowing for the understanding of the postsaccadic oscillations (PSO) as inertial effects. The model assumes that the inner part of the iris, which defines the pupil, moves driven by inertial forces induced by the eyeball rotation, in addition to viscous and elastic forces. Among other achievements, the model correctly reproduces eye-tracking experiments concerning PSO profiles and their dependence on the saccade size. In this paper we propose various extensions of the mentioned model, we provide analytical solutions, and we perform an exhaustive analysis of the dynamics. In particular, we consider a more general time dependence for the eyeball velocity enabling the description of saccades with vanishing initial acceleration. Moreover, we give the analytical solution in terms of hypergeometric functions for the constant parameter version of the model and we provide particular expressions for some cases of interest. We also introduce a new version of the model with inhomogeneous viscosity that can improve the fitting of the experimental results. Our analysis of the solutions explores the dependence of the PSO profiles on the system parameters for varying saccade sizes. We show that the PSO emerge in critical-like ways when parameters such as the elasticity of the iris, the global eyeball velocity, or the saccade size vary. Moreover, we find that the PSO profiles with the first overshoot smaller than the second one, which are usually observed in experiments, can be associated to parameter regions close to criticality.

Inertial Movements of the Iris as the Origin of Postsaccadic Oscillations.

Recent studies on the human eye indicate that the pupil moves inside the eyeball due to deformations of the iris. Here we show that this phenomenon can be originated by inertial forces undergone by the iris during the rotation of the eyeball. Moreover, these forces affect the iris in such a way that the pupil behaves effectively as a massive particle. To show this, we develop a model based on the Newton equation on the noninertial reference frame of the eyeball. The model allows us to reproduce and interpret several important findings of recent eye-tracking experiments on saccadic movements. In particular, we get correct results for the dependence of the amplitude and period of the postsaccadic oscillations on the saccade size and also for the peak velocity. The model developed may serve as a tool for characterizing eye properties of individuals.

Theory of hyperspherical Sturmians for three-body reactions.

In this paper we present a theory to describe three-body reactions. Fragmentation processes are studied by means of the Schrodinger equation in hyperspherical coordinates. The three-body wave function is written as a sum of two terms. The first one defines the initial channel of the collision while the second one describes the scattered wave, which contains all the information about the collision process. The dynamics is ruled by an nonhomogeneous equation with a driven term related to the initial channel and to the three-body interactions. A basis set of functions with outgoing behavior at large values of hyperradius is introduced as products of angular and radial hyperspherical Sturmian functions. The scattered wave is expanded on this basis and the nonhomogeneous equation is transformed into an algebraic problem that can be solved by standard matrix methods. To be able to deal with general systems, discretization schemes are proposed to solve the angular and radial Sturmian equations. This procedure allows these discrete functions to be connected with the hyperquatization algorithm. Finally, the fragmentation transition amplitude is derived from the asymptotic limit of the scattered wave function.