Multi-scale modeling of complex neuronal networks: a view towards striatal cholinergic pattern formations.

The phenomena related to brain function occur as the interplay of various modules at different spatial and temporal scales. Particularly, the integration of the dynamical behavior of cells within the complex brain topology reveals a heterogeneous multi-scale problem, which has, to date, mainly been addressed by methods of statistical physics such as mean-field approximations. In contrast, the present study introduces an abstract mathematical model of a deterministic nature that provides a robust integral transformation of the microscopic activities into macroscopic spatiotemporal patterns. The existence of the transformation operator is guaranteed by the convergence of a repetitive patching of the network domain with its fundamental domains that express the local topologies of the tissue. Depending on the choice of the local connectivity function, this framework represents a computationally efficient generalization of the classical Kirchhoff's, Hebbian, and Hopfield's approaches. The capabilities of this multi-scale method have been evaluated within the structure of the dorsal striatum of rats, a brain region with major involvement in motor and cognitive information processing. Numerical simulations suggest the formation of characteristic spatiotemporal patterns due to the activation of cholinergic interneurons.