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Logic in a Dynamic Brain

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Abstract

The ability of the human brain to carry out logical reasoning can be interpreted, in general, as a by-product of adaptive capacities of complex neural networks. Thus, we seek to base abstract logical operations in the general properties of neural networks designed as learning modules. We show that logical operations executable by McCulloch–Pitts binary networks can also be programmed in analog neural networks built with associative memory modules that process inputs as logical gates. These modules can interact among themselves to generate dynamical systems that extend the repertoire of logical operations. We demonstrate how the operations of the exclusive-OR or the implication appear as outputs of these interacting modules. In particular, we provide a model of the exclusive-OR that succeeds in evaluating an odd number of options (the exclusive-OR of classical logic fails in his case), thus paving the way for a more reasonable biological model of this important logical operator. We propose that a brain trained to compute can associate a complex logical operation to an orderly structured but temporary contingent episode by establishing a codified association among memory modules. This explanation offers an interpretation of complex logical processes (eventually learned) as associations of contingent events in memorized episodes. We suggest, as an example, a cognitive model that describes these “logical episodes”.

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References

  • Anderson, J. A. (1972). A simple neural network generating an interactive memory. Math. Biosci., 14, 197–220.

    Article  MATH  Google Scholar 

  • Anderson, J. A. (1995). An introduction to neural networks. Cambridge: MIT Press.

    MATH  Google Scholar 

  • Anderson, J. A., & Rosenfeld, E. (Eds.) (1988). Neurocomputing. Cambridge: MIT Press.

    Google Scholar 

  • Ashby, W. R. (1956). An introduction to cybernetics. New York: Wiley.

    MATH  Google Scholar 

  • Ashby, W. R. (1960). Design for a brain (2nd ed.). New York: Wiley.

    MATH  Google Scholar 

  • Arbib, M. A. (Ed.) (1995). The handbook of brain theory and neural networks. Cambridge: MIT Press.

    Google Scholar 

  • Baddeley, A. (2003). Working memory: looking back and looking forward. Nat. Rev. Neurosci., 4, 829–839.

    Article  Google Scholar 

  • Balkenius, C., & Gärdenfors, P. (1991). Nonmonotonic inferences in neural networks. In R. Fikes & E. Sandewall (Eds.), Principles of knowledge representation and reasoning (pp. 29–32). San Mateo: Morgan Kaufmann.

    Google Scholar 

  • beim Graben, P., & Potthast, R. (2009). Inverse problems in dynamic cognitive modeling. Chaos, 19, 015103.

    Article  MathSciNet  Google Scholar 

  • beim Graben, P., Pinotsis, D., Saddy, D., & Potthast, R. (2008a). Language processing with dynamic fields. Cogn. Neurodyn., 2, 79–88.

    Article  Google Scholar 

  • beim Graben, P., Gerth, S., & Vasishth, S. (2008b). Towards dynamical system models of language-related brain potentials. Cogn. Neurodyn., 2, 229–255.

    Article  Google Scholar 

  • Besnard, P., Fanselow, G., & Schaub, T. (2003). Optimality theory as a family of cumulative logics. J. Logic, Lang. Inf., 12, 153–182.

    Article  MathSciNet  MATH  Google Scholar 

  • Blutner, R. (2004). Nonmonotonic inferences and neural networks. Synthese, 142, 143–174.

    Article  MathSciNet  MATH  Google Scholar 

  • Cannon, W. B. (1932). The wisdom of the body. New York: Norton.

    Google Scholar 

  • Cooper, L. N. (1973). A possible organization of animal memory and learning. In Proceedings of the Nobel symposium on collective properties of physical systems, Aspensagarden, Sweden.

  • Cooper, L. N. (2000). Memories and memory: a physicist’s approach to the brain. Int. J. Modern Phys. A, 15(26), 4069–4082.

    Article  Google Scholar 

  • Graham, A. (1981). Kronecker products and matrix calculus with applications. Chichester: Ellis Horwood.

    MATH  Google Scholar 

  • Hebb, D. O. (1949). The organization of behavior. New York: Wiley.

    Google Scholar 

  • Humphreys, M. S., Bain, J. D., & Pike, R. (1989). Different ways to cue a coherent memory system: a theory for episodic, semantic, and procedural tasks. Psychol. Rev., 96, 208–233.

    Article  Google Scholar 

  • James, W. (1911). Some problems of philosophy. New York: Longmans and Green.

    Google Scholar 

  • Jonides, J. R., Lewis, R. L., Nee, D. E., Lustig, C. A., Berman, M. G., & Moore, K. S. (2008). The mind and brain of short-term memory. Ann. Rev. Psychol., 59, 193–224.

    Article  Google Scholar 

  • Kandel, E. R., & Schwartz, J. H. (1985). Principles of neural science. Amsterdam: Elsevier.

    Google Scholar 

  • Koch, C., & Poggio, T. (1992). Multiplying with synapses and neurons. In T. McKenna, J. Davis, & S. F. Zornetzer (Eds.), Single neuron computation (pp. 315–345). San Diego: Academic Press.

    Google Scholar 

  • Kohonen, T. (1972). Correlation matrix memories. IEEE Trans. Comput., C-21, 353–359.

    Article  Google Scholar 

  • Kohonen, T. (1977). Associative memory: a system-theoretical approach. New York: Springer.

    MATH  Google Scholar 

  • Lashley, K. S. (1950). In search of the engram. In Society of experimental biology 4: psychological mechanisms in animal behavior (pp. 454–482). Cambridge: Cambridge University Press.

    Google Scholar 

  • Lewis, C. I., & Langford, C. H. (1959). Symbolic logic. New York: Dover.

    MATH  Google Scholar 

  • Lotka, A. (1956). Elements of mathematical biology. New York: Dover.

    MATH  Google Scholar 

  • McCulloch, W. S., & Pitts, W. (1943). A logical calculus of the ides immanent in nervous activity. Bull. Math. Biophys., 5, 115–133.

    Article  MathSciNet  MATH  Google Scholar 

  • Mel, B. W. (1992). NMDA-based pattern discrimination in a modeled cortical neuron. Neural Comput., 4, 502–517.

    Article  Google Scholar 

  • Minsky, M. L., & Papert, S. A. (1988). Perceptrons. Cambridge: MIT Press. Expanded Ed.

    MATH  Google Scholar 

  • Mizraji, E. (1989). Context-dependent associations in linear distributed memories. Bull. Math. Biol., 51, 195–205.

    MATH  Google Scholar 

  • Mizraji, E. (1992). Vector logics: the matrix-vector representation of logical calculus. Fuzzy Sets Syst., 50, 179–185.

    Article  MathSciNet  Google Scholar 

  • Mizraji, E. (2008a). Vector logic: a natural algebraic representation of the fundamental logical gates. J. Logic Comput., 18, 97–121.

    Article  MathSciNet  MATH  Google Scholar 

  • Mizraji, E. (2008b). Neural memories and search engines. Int. J. Gen. Syst., 37, 715–732.

    Article  MathSciNet  MATH  Google Scholar 

  • Mizraji, E., & Lin, J. (1997). A dynamical approach to logical decisions. Complexity, 2, 56–63.

    Article  MathSciNet  Google Scholar 

  • Mizraji, E., & Lin, J. (2001). Fuzzy decisions in modular neural networks. Int. J. Bifurc. Chaos, 11, 155–167.

    Article  Google Scholar 

  • Mizraji, E., & Lin, J. (2002). The dynamics of logical decisions. Physica D, 168–169, 386–396.

    Article  MathSciNet  Google Scholar 

  • Mizraji, E., Pomi, A., Reali, F., & Valle-Lisboa, J. C. (2003). Disyunciones dinámicas. In J. A. Hernández & A. Pomi (Eds.), Procesos biofísicos complejos (pp. 29–48). Montevideo: Dirac.

    Google Scholar 

  • Mizraji, E., Pomi, A., & Valle-Lisboa, J. C. (2009). Dynamic searching in the brain. Cogn. Neurodyn., 3, 401–414.

    Article  Google Scholar 

  • Monod, J. (1967). Leçon inaugurale. Paris: Collège de France.

    Google Scholar 

  • Monod, J., Changeux, J. P., & Jacob, F. (1963). Allosteric proteins and cellular control systems. J. Mol. Biol., 6, 306–329.

    Article  Google Scholar 

  • Nass, M. M., & Cooper, L. N. (1975). A theory for the development of feature detecting cells in visual cortex. Biol. Cybern., 19, 1–18.

    Article  Google Scholar 

  • Pomi, A. (2001). Estructuras cognitivas en modelos de memorias distribuidas. Ph.D. thesis, PEDECIBA-Universidad de la República, Montevideo, Uruguay.

  • Pomi, A., & Mizraji, E. (2004). Semantic graphs and associative memories. Phys. Rev. E, 70, 0666136(1-6).

    Article  Google Scholar 

  • Pomi, A., & Olivera, F. (2006). Context-sensitive autoassociative memories as expert systems in medical diagnosis. BMC Med. Inform. Decis. Mak., 6(39), 1–11.

    Google Scholar 

  • Poggio, T. (1990). A theory of how the brain might work. In The brain, Cold Spring Harbor symposia on quantitative biology (Vol. LV, pp. 390–431). Cold Spring Harbor: The Cold Spring Harbor Laboratory Press.

    Google Scholar 

  • Potthast, R., & beim Graben, P. (2009). Inverse problems in neural field theory. SIAM J. Appl. Dyn. Syst., 8, 1405–1433.

    Article  MathSciNet  MATH  Google Scholar 

  • Rapoport, A. (1948). Cycle distributions in random nets. Bull. Math. Biophys., 10, 145–157.

    Article  Google Scholar 

  • Repovs, G., & Baddeley, A. (2006). The multi-component model of working memory: explorations in experimental cognitive psychology. Neuroscience, 139, 5–21.

    Article  Google Scholar 

  • Rieke, F., Warland, D., van Steveninck, R., & Bialek, W. (1997). Spikes. Cambridge: MIT Press.

    Google Scholar 

  • Rosenblatt, F. (1958). The perceptron: a probabilistic model for information storage and organization in the brain. Psychol. Rev., 65, 386–408.

    Article  MathSciNet  Google Scholar 

  • Rumelhart, D. E., Hinton, G. E., & McClelland, J. L. (1986a). A general framework for parallel distributing processing. In D. E. Rumelhart & J. L. McClelland (Eds.), Parallel distributing processing Cambridge: MIT Press.

    Google Scholar 

  • Rumelhart, D. E., Hinton, G. E., & Williams, R. J. (1986b). Learning representations by back-propagating errors. Nature, 323, 533–536.

    Article  Google Scholar 

  • Russell, B. (1948). Human knowledge, its scope and limits. London: Allen & Unwin.

    Google Scholar 

  • Salinas, E., & Abbott, L. F. (1996). A model of multiplicative neural responses in parietal cortex. Proc. Natl. Acad. Sci., 93, 11956–11961.

    Article  Google Scholar 

  • Shaw, G. L., & Palm, G. (Eds.) (1988). Brain theory. Singapore: World Scientific.

    Google Scholar 

  • Shimbel, A., & Rapoport, A. (1948). A statistical approach to the theory of the central nervous system. Bull. Math. Biophys., 10, 41–55.

    Article  MathSciNet  Google Scholar 

  • Smolensky, P. (1990). Tensor product variable binding and the representation of symbolic structures in connectionist systems. Artif. Intell., 46, 159–216.

    Article  MathSciNet  MATH  Google Scholar 

  • Srinivasan, M. V., & Bernard, G. D. (1976). A proposed mechanism for multiplication of neural signals. Biol. Cybern., 21, 227–236.

    Article  Google Scholar 

  • Tsukada, M., & Fukushima, Y. (2010). A context sensitive mechanism in hyppocampal CA1 networks. BMB, this special issue.

  • Valle-Lisboa, J. C., 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.

    Article  Google Scholar 

  • von Neumann, J. (1945). First draft of a report on the EDVAC. Posted by M. D. Godfrey, in http://qss.stanford.edu/~godfrey/vonNeumann/vnedvac.pdf. Acceded 20 April 2009.

  • von Neumann, J. (1958). The computer and the brain. New Haven: Yale University Press.

    MATH  Google Scholar 

  • Watts, D. J. (1999). Small worlds. Princeton: Princeton University Press.

    Google Scholar 

  • Watts, D. J., & Strogatz, S. H. (1998). Collective dynamics of ‘small-world’ networks. Nature, 393, 440–442.

    Article  Google Scholar 

  • Wright, J. J. (2010). Attractor dynamics and thermodynamic analogies in the cerebral cortex: synchronous oscillation, the background EEG, and the regulation of attention. BMB, this special issue.

  • Wright, J. J., Rennie, C. J., Lees, G. J., Robinson, P. A., Bourke, P. D., Chapman, C. L., Gordon, E., & Rowe, D. L. (2004). Simulated electrocortical activity at microscopic, mesoscopic and global scales. Int. J. Bifurc. Chaos, 14, 853–872.

    Article  MathSciNet  MATH  Google Scholar 

  • Wolfram, S. (1985). Origins of randomness in physical systems. Phys. Rev. Lett., 55, 449–452.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

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

    Eduardo Mizraji

  2. Department of Physics, Washington College, Chestertown, MD, 21620, USA

    Juan Lin

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  1. Eduardo Mizraji
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Correspondence to Eduardo Mizraji.

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Mizraji, E., Lin, J. Logic in a Dynamic Brain. Bull Math Biol 73, 373–397 (2011). https://doi.org/10.1007/s11538-010-9561-0

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