Nilsson's cone algebra theory reveals neurons as operators for AI

In a major revision of his 2023 preprint, Martin N. P. Nilsson rigorously formalizes how neuron populations in the mammalian central nervous system encode and transform information. The paper, "Information Processing by Neuron Populations in the Central Nervous System," builds a mechanistic model of a single plastic neuron and derives an algebra of convex cones to describe population-level activity. This algebra reinterprets neuron populations not as passive signal transmitters but as operators acting within a structured mathematical space. Interconnected populations then realize compact algebraic expressions that support a wide repertoire of functions: specialization, generalization, novelty detection, dimensionality reduction, inverse modeling, prediction, and associative memory. The framework assumes familiarity with functional analysis and operator algebras, and comes with 60 pages and 12 figures of detailed proofs.

A key advance is connecting neuron-local learning to online projected-gradient optimization for non-negative least squares (NNLS), with new results showing how sparse, activity-selected mappings implement conic projection and rejection. Moreau-based proofs strengthen the cone algebra, and the paper characterizes approximate invariance under sparse embeddings, adding a sensorimotor application to ground the theory. Most provocatively, Nilsson highlights matrix embeddings as a way to extend representational capacity beyond vector-based models, enabling hierarchical concept formation and structured information processing. This suggests that brains may compute with operators rather than static vectors, a departure from typical neural network assumptions. The implications span cognitive neuroscience and AI, potentially inspiring new architectures that treat neurons as algebraic units rather than simple weighted sums.