Calculus relations in spatial summation and determination of macro connection in neuronal networks

A concept of macro connection as an analytic representation for the interactions between any two neurons or neuronal groups is proposed. Based on this concept, the neuronal interactions are investigated at two levels, i.e., the synaptic connections and macro connections. The efficacy of macro connection is explicitly represented and quantitatively determined by the efficacies of the constitutent synapses. This issue is addressed through a calculus approach to obtain an analytical expression for spatial summation properties of a single cell. It is shown that the spatial summation processes in a single cell are, to a large extent, governed by the rules of differential calculus, and that the total differential form is meaningful from the range of subcellular and cellular levels through the level of local networks. As an application of the authors´ result, a computational method is proposed for the receptive field and projective field of single cells which, in the sense of medium-term average, can quantitatively give these macro functionally significant units a constructive explanation in terms of the contributive micro components