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
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).
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].
References
Anderson JA (1972) A simple neural network generating an interactive memory. Math Biosci 14:197–220
Anderson JA (1995) An introduction to neural networks. MIT Press, Cambridge
Anderson JA, Cooper LN, Nass MN, Freiberger W, Grenander U (1972) Some properties of a neural model for memory. In: AAAS symposium on theoretical biology and biomathematics, Washington
Ashby WR, Von Foerster H, Walker CC (1962) Instability of pulse activity in a net with threshold. Nature 196:561–562
Bear MF, Cooper LN (1998) From molecules to mental states. Daedalus 127:131–144
beim Graben P, Potthast R (2009) Inverse problems in dynamic cognitive modeling. Chaos 19:015103
Boole G (1854) An investigation of the laws of thought. Macmillan, London
Bressler SL, Menon V (2010) Large-scale brain networks in cognition: emerging methods and principles. Trends Cognit Sci 14:277–290
Cayley A (1878) The theory of groups: graphical representations. Am J Math 1:174–176
Cooper LN (1973) A possible organization of animal memory and learning. In: Lundquist B, Lundqvist S (eds) Proceedings of the nobel symposium on collective properties of physical systems. Academic, New York, pp 252–264
Cooper LN (2000) Memories and memory: a physicist’s approach to the brain. Int J Mod Phys A 15:4069–4082
Dehaene S, Piazza M, Pinel P, Cohen L (2003) Three parietal circuits for number processing. Cognit Neuropsychol 20:487–506
Dong J, Xu S, Chen Z, Wu B (2001) On permutation symmetries of Hopfield model neural network. Discrete Dyn Nat Soc 6:129–136
Fiori S (2008) Lie-group-type neural system learning by manifold retractions. Neural Netw 21:1524–1529
Friston KJ (1995) Functional and effective connectivity in neuroimaging: a synthesis. Hum Brain Mapp 2:56–78
Friston KJ (2011) Functional and effective connectivity: a review. Brain Connect 1:13–36
Gayler RW (2006) Vector symbolic architectures are a viable alternative for Jackendoff’s challenges. Behav Brain Sci 29:78–79
Gradin VB, Pomi A (2008) The role of hippocampal atrophy in depression: a neurocomputational approach. J Biol Phys 34:107–120
Griffith JS (1963) A field theory of neural nets: I. Derivation of field equations. Bull Math Biophys 25:111–120
Griffith JS (1971) Mathematical neurobiology: an introduction to the mathematics of the nervous system. Academic Press, London chapter 5
Grossman I, Magnus W (1975) Groups and their graphs. New Mathematical Library. Mathematical Association of America, Washington
Hinton GE, Sejnowsky TJ (1983) Optimal perceptual inference. In: Proceedings of the IEEE conference on computer vision and pattern recognition, IEEE, New York, pp 448–453
Hopcroft JE, Ullman JD (1979) Introduction to automata theory, languages, and computation, 1st edn. Addison-Wesley, Reading
Inhelder B, Piaget J (1955) De la logique de l’enfant a la logique de l’adolescent. Presses Universitaires de France, Paris
Kanerva P (1988) Sparse distributed memory. MIT Press, Cambridge
Kohonen T (1972) Correlation matrix memories. IEEE Trans Comput C–21:353–359
Kohonen T (1977) Associative memory: a system theoretical approach. Springer, New York
Kohonen T (1997) Self-organizing maps. Springer, New York
Kosmann-Schwartzbach Y (2010) Groups and symmetries. Springer, London
Loos HG (1992) Group structure of Hadamard memories. In: International joint conference on neural networks, IJCNN, IEEE, vol 1, pp 505–510
Lourenço O, Machado A (1996) In defense of Piaget’s theory: a reply to 10 common criticisms. Psychol Rev 103:143–164
Maass W (2011) Liquid state machines: motivation, theory, and applications. In: Cooper SB, Sorbi A (eds) Computability in context: computation and logic in the real world. Imperial College Press, London, pp 275–296
Maass W, Natschläger T, Markram H (2002) Real-time computing without stable states: a new framework for neural computation based on perturbations. Neural Comput 14:2531–2560
McCulloch WS, Pitts W (1943) A logical calculus of the ideas immanent in nervous activity. Bull Math Biophys 5:115–133
Mizraji E (1989) Context-dependent associations in linear distributed memories. Bull Math Biol 51:195–205
Mizraji E (1992) Vector logics: the matrix-vector representation of logical calculus. Fuzzy Sets Syst 50:179–185
Mizraji E (2008a) Vector logic: a natural algebraic representation of the fundamental logical gates. J Logic Comput 18:97–121
Mizraji E (2008b) Neural memories and search engines. Int J Gen Syst 37:715–738
Mizraji E, Lin J (2011) Logic in a dynamic brain. Bull Math Biol 73:373–397
Mizraji E, Pomi A, Valle-Lisboa JC (2009) Dynamic searching in the brain. Cogn Neurodyn 3:401–414
Neusel MD (2007) Invariant theory. Student Mathematical Library, vol 36. AMS, Providence
Nieder A, Dehaene S (2009) Representation of number in the brain. Annu Rev Neurosci 32:185–208
Nielsen MA, Chuang IL (2000) Quantum computation and quantum information. Cambridge University Press, Cambridge Appendix 2: Group theory
Penrose R (2004) The road to reality: a complete guide to the laws of the universe. Jonathan Cape, London
Pitts W, McCulloch WS (1947) How we know universals: the perception of auditory and visual forms. Bull Math Biophys 9:127–147
Pomi-Brea A, Mizraji E (1999) Memories in context. BioSystems 50:173–188
Pomi A, Mizraji E (2001) A cognitive architecture that solves a problem stated by Minsky. IEEE Trans Syst Man Cybern Part B Cybern 31:729–734
Pomi A, Mizraji E (2004) Semantic graphs and associative memories. Phys Rev E 70:066136
Pomi A, Olivera F (2006) Context-sensitive autoassociative memories as expert systems in medical diagnosis. BMC Med Inf Decis Mak 6(1):39
Rabinovich MI, Huerta R, Varona P, Afraimovich VS (2008) Transient cognitive dynamics, metastability, and decision making. PLoS Comput Biol 4:e1000072
Rapoport A (1952) “Ignition” phenomena in random nets. Bull Math Biophys 14:35–44
Ricciardi LM (1994) Diffusion models of single neurons activity. In: Ventriglia F (ed) Neural modeling and neural networks. Pergamon, Oxford, pp 129–162
Rumelhart DE, McClelland JL, the PDP Research Group (1986) Parallel distributed processing: explorations in the microstructure of cognition. Volume 1: foundations. MIT Press, Cambridge
Sagan BE (2001) The symmetric group. Springer, New York
Salam A (1957) On parity conservation and neutrino mass. Il Nuovo Cimento 5:299–301
Sporns O (2010) Networks of the brain. MIT Press, Cambridge
Stewart I (2008) Why beauty is truth: the history of symmetry. Basic Books, New York
Valle-Lisboa JC, Reali F, Anastasía H, Mizraji E (2005) Elman topology with sigma-pi units: an application to the modeling of verbal hallucinations in schizophrenia. Neural Netw 18:863–877
Valle-Lisboa JC, Pomi A, Cabana A, Elvevåg B, Mizraji E (2014) A modular approach to language production: models and facts. Cortex 55:61–76
Vollmer G (1984) Mesocosm and objective knowledge: on problems solved by evolutionary epistemology. In: Wuketits FM (ed) Concepts and approaches in evolutionary epistemology: towards an evolutionary theory of knowledge. D Reidel Publishing Company, Dordrecht, pp 69–121
Weyl H (1952) Symmetry. Princeton University Press, Princeton
Wigner E (1959) Group theory and its application to the quantum mechanics of atomic spectra. Academic Press, New York
Wilson HR, Cowan JD (1973) A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Kybernetik 13:55–80
Wood J, Shawe-Taylor J (1996) Representation theory and invariant neural networks. Discrete Appl Math 69:33–60
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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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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DOI: https://doi.org/10.1007/s11538-016-0202-0