In the intricate architecture of the mammalian central nervous system,
neurons form populations. Axonal bundles communicate between these clusters
using spike trains as their medium. However, these neuron populations' precise
encoding and operations have yet to be discovered. In our analysis, the
starting point is a state-of-the-art mechanistic model of a generic neuron
endowed with plasticity. From this simple framework emerges a profound
mathematical construct: The representation and manipulation of information can
be precisely characterized by an algebra of finite convex cones. Furthermore,
these neuron populations are not merely passive transmitters. They act as
operators within this algebraic structure, mirroring the functionality of a
low-level programming language. When these populations interconnect, they
embody succinct yet potent algebraic expressions. These networks allow them to
implement many operations, such as specialization, generalization, novelty
detection, dimensionality reduction, inverse modeling, prediction, and
associative memory. In broader terms, this work illuminates the potential of
matrix embeddings in advancing our understanding in fields like cognitive
science and AI. These embeddings enhance the capacity for concept processing
and hierarchical description over their vector counterparts.Comment: 34 pages, 12 figure