Application of interval iterations to the entrainment problem in respiratory physiology

Abstract

We present here some theoretical and numerical results about interval iterations. We consider first an application of the interval iterations theory to the problem of entrainment in respiratory physiology for which the classical point iterations theory fails. Then, after a brief review of some of the main aspects of point iterations, we explain what is meant by the term 'interval iterations'. It consists essentially in replacing in the point iterations the function to iterate by a set-valued map. We present both theoretical and numerical aspects of this new type of iterations and we observe the dynamical behaviours encountered, such as fixed intervals and interval limit cycles. The comparison between point and interval iterations is carried out with respect to a parameter ε, which determines the thickness of a neighbourhood around the function to iterate. We will finally focus our attention on the Verhulst and Ricker functions largely used in population dynamics, which exhibit various asymptotic behaviours.

Smart Homes and Sensors for Surveillance and Preventive Education at Home: Example of Obesity

“…This fusion could be done through a score (coming for instance from a principal or independent component analysis), which is a scalar variable triggering the alert, or through a multidimensional vector that has to remain within a certain value domain, e.g., between some threshold values. In this last case, the passages of the boundaries of a "confident" domain have to be detected, and "viability" methods could be used [142,144].…”

“…After clustering and personalized calibration, different so-called "comfort" zones are defined and "viable" behaviors ( Figure 1) are proposed, e.g., corresponding to a physiologic respiratory and cardiac rhythm. The "viable" tubes are calculated after clustering collective real data by using a combination of statistical and dynamical systems techniques [142,144], providing scenarios of daily activity (Figure 13), which can be considered as "normal". These scenarios allow the personalization of the alimentation necessary to fulfill the energetic and glycemic charges needed by this activity (Figure 12).…”

From biological and clinical experiments to mathematical models

“…Article 5, 'Application of interval iterations to the entrainment problem in respiratory physiology' by Demongeot & Waku (2009), yields novel use of one-dimensional discrete dynamical systems to investigate frequency locking phenomena in respiratory physiology. It relies on a rich collection of clinical data accumulated for years in Grenoble hospitals.…”