Hierarchical functional organization of formal biological systems: a dynamical approach. III. The concept of non-locality leads to a field theory describing the dynamics at each level of organization of the (d-fbs) sub-system

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Abstract
In paper I, the construction of the graph of interactions, called (o-fbs), was deduced from the ‘selfassociation hypothesis’. In paper II, a criterion of evolution during development for the (o-fbs), which represents the topology of the biological system , was deduced from an optimum principle leading to specific dynamics. Experimental verification of the proposed extremum hypothesis is possible because precise knowledge of the dynamics is not necessary; only knowledge of the monotonic variation of the number of sinks is required for given initial conditions. Essentially, the properties o f the (o-fbs) are based on the concept o f non-symmetry of functional interactions, as shown by the ‘orgatropy’ function (paper II). In this paper, a field theory is proposed to describe the (d-fbs), i.e. the physiological processes expressed by functional interactions: (i) physiological processes are conceived as the transport of a field variable submitted to the action of a field operator; (ii) because of hierarchy, this field theory is based on the concept of non-locality, and includes a non-local and non-symmetric interaction operator, (iii) the geometry of the structure contributes to the dynamics via the densities of structural units; and (iv) because a physiological process evolves on a particular timescale, it is possible to classify the levels of organization according to distinct timescales, and, therefore, to obtain a ‘decoupling’ of dynamics at each level. Thus, a property of structurality for a biological system is proposed, which is based on the finiteness of the velocity of the interaction, thus, with distinct values of timescales for the construction of the hierarchy of the system. Three axioms are introduced to define the fields associated with the topology of the system: (i) the existence of the fields; (ii) the decoupling of the dynamics; and (iii) the ability of activation-inhibition. This formulation leads to a self-coherent definition of auto-organization: an fbs is self-organized if it goes from one stable state for the (d-fbs) to another under the influence of certain modifications of its topology, i.e. a modification of the (o-fbs). It is shown that properties deduced with this formalism give the relationship between topology and geometry in an fbs, and particularly, the geometrical re-distribution of units. In the framework of this field theory, a statistical distribution function of the states of the field is introduced, which shows that the collective behavior of the population of units is not a simple summation of the individual elements, and gives a solution to the problem of the passage from one level to another. Two examples are given: a justification of the self-association hypothesis in the case of field variables, and a method to determine the 2-level neural field equations. Finally, the concepts of complexity and autonomy are discussed, and we show that the autonomy of a biological system increases with the potential of organization. The proposed principle of functional order from hierarchy, which describes the natural trend towards time decoupling of the physiological function, leads, in that sense, towards a simplification of the dynamics.

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