Searching for Chaos Evidence in Eye Movement Signals

Abstract

Most naturally-occurring physical phenomena are examples of nonlinear dynamic systems, the functioning of which attracts many researchers seeking to unveil their nature. The research presented in this paper is aimed at exploring eye movement dynamic features in terms of the existence of chaotic nature. Nonlinear time series analysis methods were used for this purpose. Two time series features were studied: fractal dimension and entropy, by utilising the embedding theory. The methods were applied to the data collected during the experiment with "jumping point" stimulus. Eye movements were registered by means of the Jazz-novo eye tracker. One thousand three hundred and ninety two (1392) time series were defined, based on the horizontal velocity of eye movements registered during imposed, prolonged fixations. In order to conduct detailed analysis of the signal and identify differences contributing to the observed patterns of behaviour in time scale, fractal dimension and entropy were evaluated in various time series intervals. The influence of the noise contained in the data and the impact of the utilized filter on the obtained results were also studied. The low pass filter was used for the purpose of noise reduction with a 50 Hz cut-off frequency, estimated by means of the Fourier transform and all concerned methods were applied to time series before and after noise reduction. These studies provided some premises, which allow perceiving eye movements as observed chaotic data: characteristic of a space-time separation plot, low and non-integer time series dimension, and the time series entropy characteristic for chaotic systems.

An Analysis of Entropy-Based Eye Movement Events Detection

“…Collating the outcomes from this research with those obtained in [26], one may notice differences in the values of calculated entropy. This may stem from several reasons.…”

“…Analysis of this indicator in all the aforementioned studies demonstrated chaos in the parts of eye movement under consideration. Additionally, in [26], fractal dimension and sample entropy were evaluated for eye movement velocity during fixations. These factors, as well as the LLE, were examined, taking different eye movement signal segments into account.…”

“…The dimension of a hidden dynamical system can be estimated from the data by the use of the False Nearest Neighbors method [40], and may differ for different data series. Such a solution was utilized in [26]. In the current studies, one value, m = 2 for all analyzed EM series was used, as recommended in [27] for approximate entropy usage.…”

Application of Time-Scale Decomposition of Entropy for Eye Movement Analysis

“…The results presented in Figure 4 showed that measures calculated for one EM series, including three eye movement events-saccadic latency, saccade and fixation-have different values in various time scopes of the EM series. The previous studies [18,19,21,22] already provided some evidence in this regard for ApEn, SampEn, and the LLE; however, only during independent analyses. One of the advantages of the current studies is a comparison of these measures and introducing the additional signal description through the FuzEn usage.…”

“…However, its dynamics in this scope was not so widely explored as in other biological signals. Nonetheless, in several studies, it was discovered, based on the LLE [18][19][20] and entropy measures: ApEn and SampEn ( [21,22]), that various phases of eye movement signal reveal different dynamic characteristics.…”