On the mathematical integration of the nervous tissue based on the S-propagator formalism

The integration of physiological functions in living organisms corresponds to the reconstruction of a biological system from its components. This calls for a sound theoretical framework based on the rigorous definition of the elementary physiological function within the context of multiple levels of biological organization. One of the main problems encountered in the neurosciences is that of extending the current theory of automata, as used in the study of artificial neural networks, to real neural networks. The difficulty arises because the theory of automata fails to take into account the various levels of biological organization involved in nervous activity. This article recalls the main elements of G. A. Chauvet's novel n-level field theory, i.e., the properties of non-symmetry and non-locality of functional interactions, and the S-propagator formalism that governs the propagation of a functional interaction across the different levels of the structural organization of a biological system. The neural field equations derived from this theory allow the inclusion of multiple organizational levels of a biological system into the analysis by incorporating specific local models into a global non-local model. The main advantage of the method presented here is the simplification obtained by breaking down the physiological function into its components according to the time scales and space scales of operation. Moreover, the method takes into account the non-locality of the functional interaction, assuming it to be propagated at finite velocity in a continuous and hierarchical space. Finally, this approach allows the systematic study of physiological functions within a single theoretical framework, the complexity of which could be progressively increased by integrating specific local models as new findings become available.