  In the intricate architecture of the mammalian central nervous system,
neurons form populations. Axonal bundles communicate between these clusters
using spike trains. 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 subtle mathematical construct: The
representation and manipulation of information can be precisely characterized
by an algebra of 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.
