S-Propagators: A Formalism for the Hierarchical Organization of Physiological Systems. Application to the Nervous and the Respiratory Systems

Despite the large number of studies in experimental and theoretical physiology, no formalism has yet been proposed to mathematically describe the dynamics in hierarchical systems, taking into account their geometry. We formulate the passage from one hierarchical level to another on the basis of a non-local field theory (Chauvet, 1993e, Journal of Mathematical Biology, 31, 475-486), in which the geometry of the system as well as the functional interactions play an essential role. A 3D diagrammatic representation of a biological system is proposed in terms of (I) the structural organization determined by the space scales, (ii) the functional organizations determined by the time scales of the dynamics, and (iii) the spaces of the structural units, i.e. the structures which give rise to the global variables in the hierarchy. The dynamics of the structural organization is presented in terms of the structural propagators (S-propagators), i.e. non-local operators which describe the propagation of field variables at each level of the structure, together with the kernels specific to the interactions. The general equations for the dynamics are established and used to obtain: (1) the neural field equations that describe the propagation of activity in the nervous system, and (2) the respiratory field equations that describe the transport of 02 molecules in the lungs.