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The S-Propagator formalism describes the dynamics in the structural organization
Fields and functional interactions

With the theoretical hierarchical framework described in the section theory, I represent a physiological process, expressed by functional interactions related to the geometry of the structure, in terms of the transport of a field variable submitted to the action of a field operator. Let (r,t) be the field variable defined in the r-space, e. g. membrane potential, and let H  be the field operator which depends on and on successive derivatives with respect to time and space coordinates. The general form of the field equation is given by :

           (3)

where Г is the source term. In this equation, H  describes the propagation of the field variable from r’ to r, and the local transformation in r is represented by Г(r,t).  Since the operator acts from one point in space on another, it must take into account the distance between these two points, and thus include an interaction operator. More generally, the influence of the location of the points, i.e. the role of geometry on the dynamical processes, may be studied by means of a field theory. The dynamical processes that express the behavior of the related functional interactions occur continuously in space and time with a finite velocity. Thus, what is observed at point (r,t) results from what was emitted at point (r’,t’), where  and  v_r is the velocity of the interaction.

The finite value of the velocity v_r of the transport of the interaction, i.e. the transport of molecules, potentials, currents, or parametric effects depending on the elementary physiological function, has a major effect on the behavior of the biological system. This is particularly true of the delay in the response between units. These effects are included directly in the field interaction operator. Let us now determine the specific operator that describes a physiological mechanism.

S-Propagator dynamics

The units u_i and u are assumed to be at level r in the structural organization (space scale k), and at level T in the functional organization (time scale T). The couple (k,T) in the 3-D representation (Fig. 4) defines the organization of the physiological function .

There is a structural discontinuity between the two units. Because of the hierarchy, u_i and u are associated with a non-local functional interaction represented by the field (r,t), where r(x,y,z) is the coordinate in the space of units refered to coordinates (x,y,z) in the physical space. Using operators, the local time-variation may be expressed as:

       (4)                                                             

where H_I is the non-local operator. What are these operators? As shown in Figure 3, in going from u_i at r’ to u at r, the functional interaction must cross the structural discontinuity at the lower level, i.e. it must use processes "outside" the level.

figure 3

In table A, the S-propagator formalism has been summarized which leads from Eq.(4) to the local time and non-local space equation (A-8) for the dynamics of the field variable :

(5)

                                      (i)                                        (ii)                                                 (iii)                              

where the sumation is on the domain D_r(r) of the u-units connected with the units at r. Here, D^r need not be constant, as the medium may not be heterogeneous, in which case the term may be space-dependent; the time scale is T, and d(r’,r) is the distance between r’ and r in the space of units u. The S-propagator describes the functional action of u’ at r’ onto u at r per unit time, because the field variable y^r is emitted by u’ at r’ and is transported to u at r. Locally, the field variable depends on the lower levels and is under three influences, which are shown by the three terms in Eq.(5) : (i) a local process of diffusion between units through the extra-unit space, i.e. transport through the medium in which the units are located, as defined by the diffusion constant D^r; (ii) the S-propagator  represents the transport of the field variable through "homogeneous" structures at the lower level inside u_i or u, i.e. structures that are homogeneous relative to the processes in a medium with locally identical properties, without structural discontinuities; and (iii) the generation of the field variable at r as a result of local processes in physical space, represented by the source term Г_r, and possibly due to the higher levels.

Finally, the determination of the dynamics of physiological functions results from the determination of the propagators P in the above Eq.(5). The linear case may be explicited. In particular, this formalism is used for the dynamics of the nervous system. These results are valid whatever the level of organization. Because the same formalism applies to each level of the hierarchy, it provides a tool for the rigorous study of coupled biological systems in terms of elementary mechanisms. As shown below, the mechanisms included in Eq.5 provide the neural field equations.