Synchronicity Rectangle for temporal gait analysis: Application to Parkinson's Disease. (September 2020) - Marino, R.; Verrelli, C.M.; Gnucci, M.
Abstract: Human walking and running gaits are represented by the same temporal model. The golden ratio ϕ - namely, the largest solution to the equation x 2 = 1 + x - is associated with all of the four eigenvalues of such a model. An eigenspace of dimension two, named symmetry plane, is determined. It contains an eigenspace of dimension one, named golden line, on which the ratio between stance and swing durations is equal to ϕ for walking and to ϕ − 1 for running. The Synchronicity Rectangle is defined, whose sides are: the normalized distance of the gait from the symmetry plane and the normalized distance from the golden line of the gait projection on the symmetry plane . It constitutes a new comprehensive geometric representation of walking and running gaits. The area of such a rectangle computed for an averaged gait is illustratively shown to discriminate between patients affected by Parkinson's Disease and healthy subjects: its mean value is significantly higher for patients affected by Parkinson's Disease; high Sensitivity and moderate Specificity, along with relatively large likelihood and odds ratios, are obtained. Highlights: Asymmetric human walking and running have been represented by the same linear model. Such a linear model involves the time intervals between foot strike and foot lift off of right and left legs. The related symmetric matrix exhibits eigenvalues that are all related to the golden ratio. Two orthogonal eigenspaces of dimension two, namely the symmetry and asymmetry planes, have been defined. The symmetry plane contains a special eigenspace of dimension one, named golden line. The spectral analysis has led to the definition of the Synchronicity Rectangle as a comprehensive geometric representation of human walking and running gaits. The area of such a rectangle has been illustratively tested on a public walking data set of healthy control subjects and patients affected by Parkinson's Disease. The mean value for the area of the Synchronicity Rectangle is significantly higher in patients affected by Parkinson's Disease. The ROC analysis displays a moderate accuracy. The contingency table shows high Sensitivity and moderate Specificity, along with relatively large likelihood and odds ratio.