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We have read with great interest the paper published by the Journal of Anatomy [244(5), 2024, 861-872] on Is human height based on a Lucas sequence relationship between the foot height, tibial length, femur length and upper body length? by Paley et al. The authors show that foot height, tibial length, femur length and upper body length follow a generalized Lucas sequence. Our letter demonstrates that their result is indeed stronger, as their data follow the original, homogeneous Lucas sequence. 

Terrestrial locomotion is a complex phenomenon that is often linked to the survival of an individual and of an animal species. Mathematical models seek to express in quantitative terms how animals move, but this is challenging because the ways in which the nervous and musculoskeletal systems interact to produce body movement is not completely understood. Models with many variables tend to lack biological interpretability and describe the motion of an animal with too many independent degrees of freedom. Instead, reductionist models aim to describe the essential features of a gait with the smallest number of variables, often concentrating on the center of mass dynamics. In particular, spring–mass models have been successful in extracting and describing important characteristics of running. In this paper, we consider the spring loaded inverted pendulum model under the regime of constant angular velocity, small compression, and small angle swept during stance. We provide conditions for the asymptotic stability of periodic trajectories for the full range of parameters. The hypothesis of linear angular dynamics during stance is successfully tested on publicly available human data of individuals running on a treadmill at different velocities. Our analysis highlights a novel bifurcation phenomenon for varying Froude number: there are periodic trajectories of the spring loaded inverted pendulum model that are stable only in a restricted range of Froude numbers, while they become unstable for smaller or larger Froude numbers. 

In this paper, we discuss a selection of tools from dynamical systems and order statistics, which are most often utilized separately, and combine them into an algorithm to estimate the parameters of mathematical models for infectious diseases in the case of small sample sizes and left censoring, which is relevant in the case of rapidly evolving infectious diseases and remote populations. The proposed method relies on the analogy between survival functions and the dynamics of the susceptible compartment in SIR-type models, which are both monotone decreasing in time and are both determined by a dual variable: the hazard function in survival prediction and the number of infected people in SIR-type models. We illustrate the methodology in the case of a continuous model in the presence of noisy measurements with different distributions (Normal, Poisson, Negative Binomial) and in a discrete model, reminiscent of the Ricker map, which admits chaotic dynamics. This estimation procedure shows stable results in experiments based on a popular benchmark dataset for SIR-type models and small samples. This manuscript illustrates how classical theoretical statistical methods and dynamical systems can be merged in interesting ways to study problems ranging from more fundamental small sample situations to more complex infectious disease and survival models, with the potential that this tools can be applied in the presence of a large number of covariates and different types of censored data. 

Typically, animal locomotion studies involve consecutive strides, which are frequently assumed to be independent with parameters that do not vary across strides. This assumption is often not tested. However, failing in particular to account for dependence across strides may cause an incorrect estimate of the uncertainty of the measurements and thereby lead to either missing (overestimating variance) or over-evaluating (underestimating variance) biological signals. In turn, this impacts replicability of the results because variability is accounted for differently across experiments. In this paper, we analyse the changes of a couple of measures of human leg stiffness across strides during running experiments, using a publicly available dataset. A major finding of this analysis is that the time series of these measurements of stiffness show autocorrelation even at large lags and so there is dependence between individual strides, even when separated by many intervening strides. Our results question the practice in biomechanics research of using each stride as an independent observation or of sub-selecting strides at small lags. Following the outcome of our analysis, we strongly recommend caution in doing so without first confirming the independence of the measurements across strides and without confirming that sub-selection does not produce spurious results.