An n-level field theory of biological neural networks.

An n-level field theory, based on the concept of "functional interaction", is proposed for a description of the continuous dynamics of biological neural networks. A "functional interaction" describes the action from one substructure of a network to another at several levels of organization, molecular, synaptic, and neural. Because of the continuous representation of neurons and synapses, which constitute a hierarchical system, it is shown that the property of non-locality leads to a non-local field operator in the field equations. In a hierarchical continuous system, the finite velocity of the functional interaction at the lower level implies non-locality at the higher level. Two other properties of the functional interaction are introduced in the formulation: the non-symmetry between sources and sinks, and the non-uniformity of the medium. Thus, it is shown that: (i) The coupling between topology and geometry can be introduced via two functions, the density of neurons at the neuronal level of organization, and the density-connectivity of synapses between two points of the neural space at the synaptic level of organization. With densities chosen as Dirac functions at regularly spaced points, the dynamics of a discrete network becomes a particular case of the n-level field theory. (ii) The dynamics at each of the molecular and synaptic lower level are introduced, at the next upper level, both in the source and in the non-local interaction of the field to integrate the dynamics at the neural level. (iii) New learning rules are deduced from the structure of the field equations: Hebbian rules result from strictly local activation; non-Hebbian rules result from homosynaptic activation with strict heterosynaptic effects, i.e., when an activated synaptic pathway affects the efficacy of a non-activated one; non-Hebbian rules and/or non-linearities result from the structure of the interaction operator and/or the internal biochemical kinetics.