Artificial neural systems and nervous graph theoretical analysis rely upon the stance that the neural code is endowed in logic circuits, e.g., spatio-temporal sequences of ON/OFF spiking neurons. Nevertheless, this assumption does not fully explain complex brain functions. Here we show how nervous activity, other than logic circuits, could instead depend on topological transformations and symmetry constraints occurring at the micro-level of the cortical microcolumn, i.e., the embryological, anatomical and functional basic unit of the brain. Tubular microcolumns can be flattened in guise of a fullerene-like two-dimensional lattices, equipped with about 80 nodes, standing for pyramidal neurons, where neural computations take place. We show how the countless possible combinations of activated neurons embedded in the lattice resemble a barcode. Different assemblies of firing neurons might stand for diverse codes, each one responsible for a single mental activity. A two-dimensional fullerene-like lattice not just simulates the real microcolumn's microcircuitry, but also allows us to build artificial networks equipped with robustness, plasticity and fastness, because they are grounded on simple topological changes corresponding to pyramidal neurons' activation.