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A Possible Neural Representation of Mathematical Group Structures

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Abstract

Every cognitive activity has a neural representation in the brain. When humans deal with abstract mathematical structures, for instance finite groups, certain patterns of activity are occurring in the brain that constitute their neural representation. A formal neurocognitive theory must account for all the activities developed by our brain and provide a possible neural representation for them. Associative memories are neural network models that have a good chance of achieving a universal representation of cognitive phenomena. In this work, we present a possible neural representation of mathematical group structures based on associative memory models that store finite groups through their Cayley graphs. A context-dependent associative memory stores the transitions between elements of the group when multiplied by each generator of a given presentation of the group. Under a convenient election of the vector basis mapping the elements of the group in the neural activity, the input of a vector corresponding to a generator of the group collapses the context-dependent rectangular matrix into a virtual square permutation matrix that is the matrix representation of the generator. This neural representation corresponds to the regular representation of the group, in which to each element is assigned a permutation matrix. This action of the generator on the memory matrix can also be seen as the dissection of the corresponding monochromatic subgraph of the Cayley graph of the group, and the adjacency matrix of this subgraph is the permutation matrix corresponding to the generator.

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Notes

  1. An exception of this assertion is, perhaps, the advancement in the knowledge of the cerebral networks involved in different arithmetic operations and brain mechanisms that take active part in the processing and representation of numbers. The work of Stanislas Dehaene has been particularly relevant in this topic. See, for example, Dehaene et al. (2003) and Nieder and Dehaene (2009).

  2. Quasi-orthogonality of neural patterns may be natural in brain processing (Kohonen 1997). Indeed, for non-orthogonal but linearly independent vectors, associative memories perform equally well, the matrix memory constructed with the Moore–Penrose pseudoinverse instead of transpose input vectors. Moreover, for patterns that are not linearly independent it is possible to employ regularization techniques [in beim Graben and Potthast (2009), the Moore–Penrose psudoinverse is given as a limit case of Tikhonov regularization].

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Acknowledgments

I am grateful to my colleagues Eduardo Mizraji and Marcelo Lanzilotta for helpful discussions and important suggestions, and to Matilde Pomi for invaluable help with the preparation of the figures.

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  1. Group of Cognitive Systems Modeling, Biophysics Section, Facultad de Ciencias, Universidad de la República, Iguá 4225, Montevideo, 11400, Uruguay

    Andrés Pomi

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Pomi, A. A Possible Neural Representation of Mathematical Group Structures. Bull Math Biol 78, 1847–1865 (2016). https://doi.org/10.1007/s11538-016-0202-0

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