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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 .

*figure 4*

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.