A Possible Neural Representation of Mathematical Group Structures

Area of Science:

• Neuroscience
• Cognitive Science
• Abstract Algebra
• Computational Neuroscience

Background:

• Cognitive activities possess neural representations in the brain.
• Formal neurocognitive theories require accounting for all brain activities and their neural correlates.
• Associative memories offer a potential framework for universal cognitive phenomena representation.

Purpose of the Study:

• To present a novel neural representation for mathematical group structures.
• To utilize associative memory models for storing finite groups via their Cayley graphs.
• To establish a link between neurocognitive processes and abstract mathematical concepts.

Main Methods:

• Employing context-dependent associative memory to store group element transitions.
• Mapping group elements to neural activity vectors.
• Utilizing generator inputs to transform memory matrices into permutation matrices.

Main Results:

• The associative memory model generates virtual permutation matrices representing group generators.
• This neural representation aligns with the group's regular representation.
• The process is analogous to dissecting monochromatic subgraphs of the group's Cayley graph.

Conclusions:

• Associative memory models provide a viable neural representation for finite group structures.
• The proposed model bridges abstract algebra and neurocognitive theories.
• This framework offers insights into the neural basis of abstract mathematical thought.