Information Processing by Neuron Populations in the Central Nervous System: Mathematical Structure of Data and Operations

TL;DR

The work argues that CNS neuron populations implement an algebra of invariants that can be precisely modeled as convex cones in a real Hilbert space. Starting from a mechanistic, plastic neuron model, it shows how population activity over time forms invariant cones that are preserved under non-negative matrix transformations and conic operations (sum, intersection, projection, rejection, and reflection). Neuron populations act as primitive operators capable of executing these cone operations, enabling conditionals, memory write/read, and adaptive filtering akin to classical adaptive filters. The framework generalizes subspace algebras, enabling richer partial ordering and hierarchical concept representations via matrix embeddings, with implications for cognitive science and AI in terms of concept processing and compact, scalable neural architectures.

Abstract

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.

Figures (11)

• Figure 1: Interconnected neuron populations. We often envision neuron populations as orderly connected, as depicted in the image on the left. While this may characterize the peripheral nervous system (PNS), the image on the right portrays the central nervous system (CNS) more adequately. The absence of correlation between neighboring neurons evidences the irregularity of connections in the CNS.
• Figure 2: Messages and invariants. Geometrically, instantaneous spike rates are points in a subspace, where the points represent messages, and the subspace represents an invariant. Messages are not invariants.
• Figure 3: Basic operations on subspaces. The five basic operations on subspaces are illustrated in left-to-right and top-down order: orthogonal complement, orthogonal projection, sum, intersection, and orthogonal rejection.
• Figure 4: The activation function. The activation function $\varphi(\cdot)$ operates in three different ranges: cutoff, where it is zero; rectifying, where it operates as a soft-threshold function; and linear, where it is a purely linear function of the argument. Although there is a fourth range, saturation, it is irrelevant in practice due to the potential damage it causes to biological neurons through overexcitation (excitotoxicity).
• Figure 5: A generic neuron with plasticity. The neuron operates as an adaptive filter or combiner with internal feedback, where weights are non-negative. The mechanistic model above derives from experiments and known properties of ion channels Nilsson.2023meo. The decay factor $\lambda$ characterizes low-pass filtering of the output $z$. The proximal compartment comprises the soma and proximal sections of the dendrites.

Theorems & Definitions (5)

• proof : Example 1 (Neuron inputs as points in Hilbert space)
• proof : Example 2 (Non-negative weights)
• proof : Example 3 (Plasticity)
• proof : Example 4 (Populations)
• proof : Example 5 (Operations on cones)