Factorization and spatial encodings: a hypothesis about the foundations of the genomic code

The genomic mechanisms that efficiently encode the initial architecture and synaptic connectivity of neural circuits remain poorly understood. We hypothesise that two primary mechanisms -- spatial encoding and factorisation -- enable a limited genome to initialise networks of billions of neurons. Spatial encoding, a form of indirect representation, compresses neural network parameters while enforcing structural continuity. Complementarily, we introduce a factorisation mechanism inspired by reaction-diffusion models, comprising a spatially invariant reaction rule and spatially variant diffusion dynamics. Based on this insight, we can efficiently abstract a neural network layer as a spatially invariant weight kernel (or filter) and its spatially variant transformations along the spatial dimensions. Thus, coupling this variant-invariant decomposition with spatial encodings can substantially reduce the size of the solution space explored by the genome. In addition, we show that this coupling leads to efficient itialisation of cortical maps, such as V1 orientation maps, and neural networks with the ability to generalise. In summary, coupling factorisation and spatial encodings can offer functional advantages to the evolving genome.