The goal of
an integrative biology

Gilbert Chauvet website

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Functional organisation and n-levels field theory
- Theory
- Formalism

 Integration of physiological functions
- Cerebellum
(movement memory)
- Hippocampus
(cognitive memory)
- Organisation of biological
- Evolution
- Development
- Ageing
- Consciousness

- Respiratory, cardio-

vascular and renal systems
- Resuscitation

Drug discovery
 Journal of Integrative Neuroscience
Fundamental basis - Theory

Hierarchical representation for a biological theory of functional organization
(MTIP : Mathematical Theory of Integrative Physiology)

The conceptuel framework

The hierarchy and its consequences : Functional interactions

In the course of my work on physiological models, ranging from the molecular to the organismal levels (See Chauvet, "Theoretical systems in Biology", page 143, Vol 1, Pergamon Press, 1996), some novel ideas specific to the study of biology have been introduced, in particular the concepts of non-symmetric and non-local functional interactions in hierarchical space. These basic concepts emerged from a 'bottom-up' approach to living systems, i.e. from a systematic study of isolated physiological functions, followed by the integration of these functions at the level of the organism. A significant consequence of this theory is that living organisms can be given not only a double organizational representation, simultaneously structural and functional, but also a double mathematical representation, simultaneously geometrical and topological.

But what exactly is a physiological function? We may compare it to a mathematical function in the sense that the action of one structure on another results in a certain product. The physiological function would then be the action (the application, in mathematical terms) and the product would be the result of the function (the value of the function, in mathematical terms) that is often identified with the physiological function itself. Although this definition is general, it is unfortunately not operational. It is relatively easy to describe particular physiological functions such as vision, digestion, memorization and so on, but it is far more difficult to give an operational definition of a physiological function in general. One possibility may be to define a physiological function in terms of a combinatorial set of functional interactions between structures. Such functional interactions are evidently specific since they describe the action (whatever its nature) of one structure on another or, more precisely, the action of a source on a sink, after the action has undergone a transformation in the source. This action clearly possesses the property of non-symmetry. In addition, it has another very important property, that of non-locality, a notion somewhat more difficult to appreciate since it stems from the structural hierarchy of the system (see Chauvet, "Hierarchical functional organization of formal biological systems", 1993), i.e. certain structures are included in others. It may be explained as follows. (i) From a mathematical point of view, in a continuous representation, the action of one structure on another is necessarily the action of one point on another. This does not correspond to the action of one cell on another in physical space since a cell contains regions with specialized functions and therefore cannot be reduced to a point. (ii) The interaction between one structure and another has to operate across other structures, which we have called structural discontinuities, within which the processes follow a different course. Thus, other levels of organisation in the hierarchical system contribute to the working of a given structure at a given level in the hierarchy. This is non-locality, due to the choice of the representation, here the hierarchical representation. Equations that represent processes have then a different structure and must include non-local terms.

The same reasoning applies to the dynamic processes of functional interactions operating, for example, between neural groups or between endocrine glands. In more general terms, this can be extended to the entire activity of the organism, provided that all the functional interactions involved are correctly represented. We may then formulate a hierarchical theory of functional organization as follows: in a multiple-level hierarchical system, each functional interaction is described by the transport of an activating and/or inhibiting signal (in the form of an action potential, a hormone or some other type of interaction) between a source and a sink, and each physiological function results from a combination of such interactions. This idea can be conveniently expressed in terms of a field theory according to which an operator transmits an interaction at a certain rate from a source to a sink situated in the space of units, with the source and the sink each being reduced to a point. This representation constitutes the basis for the definition of a physiological function as the overall behavior of a group of structural units within a hierarchical system. 

figure 1

From the mathematical point of view :

(i) A functional interaction is defined as the interaction between two of the p structural units ui and uj  (i,j = 1,p) of a formal biological system (FBS). One of the units, for example ui, emits a signal that acts on the other, uj, which in turn emits a substance, after an eventual transformation f :


This interaction, called an elementary function, is represented by yij (Figure 1) and constitutes an element of the mathematical graph representing the organization of the formal biological system (O-FBS). The dynamics of the functional interactions is then described by a system of equations of the type:


where the r 's are specific physical or geometrical parameters.

(ii) The structural unit is defined as the set of anatomical or physical elements intervening in the physiological function.

Thus, from a functional point of view, a system made up of a set of elements, such as molecules, cellular organelles, cells, tissues and organs, is represented by functional interactions and structural units. This structural hierarchy is shown in Figure 2.

figure 2

Functional interactions are identified by structural discontinuities

Functional interactions may be identified by the presence of structural discontinuities. Suppose we have two structural units separated by a structural discontinuity. The interaction is propagated from one unit to the other across the discontinuity, which could for example be a membrane allowing active transport. The membrane is at a lower level in the structural hierarchy compared to the two interacting units. From the point of view of the dynamics of the functional interaction, we may say that this interaction consists of a certain physiological process operating in the two units (located at r’ and r in the space of units, i.e. the r-space, refered to r’(x’,y’,z’) and r(x,y,z) in the physical three-dimensional space), with a different physiological process being executed at a lower level in the structural discontinuity. A functional interaction may be represented in the form of a diagram as shown in Figure 3. The equation governing the transport of the interaction applies to a continuous medium and explains why the equation of the process is different at the lower level of organization. This observation constitutes the basis of a new formalism (see Chauvet, 1999, 2002) involving what we have called structural propagators (S-propagators).

figure 3

A three-dimensional representation of a biological system

As we have seen, a physiological function may be represented by a mathematical graph in which the nodes correspond to the structural units and the edges correspond to the oriented, non-symmetric interactions. All physiological functions are intricately linked in a hierarchical fashion. They are linked relatively to space, which is evident, but also to time, which represents a different evolution rate with physiological fonctions.. Probably the best way to realize this aspect of the hierarchy is to consider the intricated time loops of the algorithm that represents the working of the function. We have therefore to consider not only the structural hierarchy but also the functional hierarchy of the system. Then, each level of the functional organization will correspond to a particular physiological function, i.e. a process that occurs on a certain time scale. How do we define these two types of hierarchy? It is convenient to consider the structural hierarchy as being organized along the space scales of a physiological process while the functional hierarchy is organized according to the corresponding time scales. Moreover, it offers the advantage of clearly separating the structural and functional organizations, i.e. the structure and the function of the biological system studied.

This “separation” may be viewed as follows. Using axes for the space scales, the time scales and the space of structural units, we have a three-dimensional representation of a physiological function (Figure 4), showing: 

The structural units in space for a given function; and the hierarchical organization of physiological functions for a given space scale.

The integration of physiological functions, i.e. the identification of the couplings between the functions, requires determination of the functional interactions at the different hierarchical levels involved.

For example, the interactions at the molecular level between angiotensin and renin will be situated at the lowest level of the hierarchical organization representing blood circulation, and will themselves be coupled with the neural network. This complex task can only be undertaken using the highly abstract and technically advanced mathematical methods.

figure 4