×

Logic in a dynamic brain. (English) Zbl 1213.92006

Summary: 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 on 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 implications 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”.

MSC:

92C20 Neural biology
92B20 Neural networks for/in biological studies, artificial life and related topics
91E10 Cognitive psychology
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Anderson, J. A. (1972). A simple neural network generating an interactive memory. Math. Biosci., 14, 197–220. · Zbl 0255.92001 · doi:10.1016/0025-5564(72)90075-2
[2] Anderson, J. A. (1995). An introduction to neural networks. Cambridge: MIT Press. · Zbl 0850.68263
[3] Anderson, J. A., & Rosenfeld, E. (Eds.) (1988). Neurocomputing. Cambridge: MIT Press.
[4] Ashby, W. R. (1956). An introduction to cybernetics. New York: Wiley. · Zbl 0071.12303
[5] Ashby, W. R. (1960). Design for a brain (2nd ed.). New York: Wiley. · Zbl 0202.19105
[6] Arbib, M. A. (Ed.) (1995). The handbook of brain theory and neural networks. Cambridge: MIT Press.
[7] Baddeley, A. (2003). Working memory: looking back and looking forward. Nat. Rev. Neurosci., 4, 829–839. · doi:10.1038/nrn1201
[8] 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. · Zbl 0765.68165
[9] beim Graben, P., & Potthast, R. (2009). Inverse problems in dynamic cognitive modeling. Chaos, 19, 015103. · Zbl 1311.92025
[10] beim Graben, P., Pinotsis, D., Saddy, D., & Potthast, R. (2008a). Language processing with dynamic fields. Cogn. Neurodyn., 2, 79–88. · doi:10.1007/s11571-008-9042-4
[11] beim Graben, P., Gerth, S., & Vasishth, S. (2008b). Towards dynamical system models of language-related brain potentials. Cogn. Neurodyn., 2, 229–255. · doi:10.1007/s11571-008-9041-5
[12] Besnard, P., Fanselow, G., & Schaub, T. (2003). Optimality theory as a family of cumulative logics. J. Logic, Lang. Inf., 12, 153–182. · Zbl 1014.03032 · doi:10.1023/A:1022362118915
[13] Blutner, R. (2004). Nonmonotonic inferences and neural networks. Synthese, 142, 143–174. · Zbl 1073.68082 · doi:10.1007/s11229-004-1929-y
[14] Cannon, W. B. (1932). The wisdom of the body. New York: Norton.
[15] 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.
[16] Cooper, L. N. (2000). Memories and memory: a physicist’s approach to the brain. Int. J. Modern Phys. A, 15(26), 4069–4082.
[17] Graham, A. (1981). Kronecker products and matrix calculus with applications. Chichester: Ellis Horwood. · Zbl 0497.26005
[18] Hebb, D. O. (1949). The organization of behavior. New York: Wiley.
[19] 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. · doi:10.1037/0033-295X.96.2.208
[20] James, W. (1911). Some problems of philosophy. New York: Longmans and Green.
[21] 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. · doi:10.1146/annurev.psych.59.103006.093615
[22] Kandel, E. R., & Schwartz, J. H. (1985). Principles of neural science. Amsterdam: Elsevier.
[23] 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.
[24] Kohonen, T. (1972). Correlation matrix memories. IEEE Trans. Comput., C-21, 353–359. · Zbl 0232.68027 · doi:10.1109/TC.1972.5008975
[25] Kohonen, T. (1977). Associative memory: a system-theoretical approach. New York: Springer. · Zbl 0354.68115
[26] 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.
[27] Lewis, C. I., & Langford, C. H. (1959). Symbolic logic. New York: Dover. · Zbl 0087.00802
[28] Lotka, A. (1956). Elements of mathematical biology. New York: Dover. · Zbl 0074.14404
[29] McCulloch, W. S., & Pitts, W. (1943). A logical calculus of the ides immanent in nervous activity. Bull. Math. Biophys., 5, 115–133. · Zbl 0063.03860 · doi:10.1007/BF02478259
[30] Mel, B. W. (1992). NMDA-based pattern discrimination in a modeled cortical neuron. Neural Comput., 4, 502–517. · doi:10.1162/neco.1992.4.4.502
[31] Minsky, M. L., & Papert, S. A. (1988). Perceptrons. Cambridge: MIT Press. Expanded Ed. · Zbl 0794.68104
[32] Mizraji, E. (1989). Context-dependent associations in linear distributed memories. Bull. Math. Biol., 51, 195–205. · Zbl 0662.92009
[33] Mizraji, E. (1992). Vector logics: the matrix-vector representation of logical calculus. Fuzzy Sets Syst., 50, 179–185. · doi:10.1016/0165-0114(92)90216-Q
[34] Mizraji, E. (2008a). Vector logic: a natural algebraic representation of the fundamental logical gates. J. Logic Comput., 18, 97–121. · Zbl 1138.03047 · doi:10.1093/logcom/exm057
[35] Mizraji, E. (2008b). Neural memories and search engines. Int. J. Gen. Syst., 37, 715–732. · Zbl 1178.68542 · doi:10.1080/03081070802037738
[36] Mizraji, E., &amp; Lin, J. (1997). A dynamical approach to logical decisions. Complexity, 2, 56–63. · Zbl 05473638 · doi:10.1002/(SICI)1099-0526(199701/02)2:3<56::AID-CPLX12>3.0.CO;2-S
[37] Mizraji, E., &amp; Lin, J. (2001). Fuzzy decisions in modular neural networks. Int. J. Bifurc. Chaos, 11, 155–167. · doi:10.1142/S0218127401002043
[38] Mizraji, E., &amp; Lin, J. (2002). The dynamics of logical decisions. Physica D, 168–169, 386–396. · Zbl 0999.68168 · doi:10.1016/S0167-2789(02)00526-2
[39] Mizraji, E., Pomi, A., Reali, F., &amp; Valle-Lisboa, J. C. (2003). Disyunciones dinámicas. In J. A. Hernández &amp; A. Pomi (Eds.), Procesos biofísicos complejos (pp. 29–48). Montevideo: Dirac.
[40] Mizraji, E., Pomi, A., &amp; Valle-Lisboa, J. C. (2009). Dynamic searching in the brain. Cogn. Neurodyn., 3, 401–414. · doi:10.1007/s11571-009-9084-2
[41] Monod, J. (1967). Leçon inaugurale. Paris: Collège de France.
[42] Monod, J., Changeux, J. P., &amp; Jacob, F. (1963). Allosteric proteins and cellular control systems. J. Mol. Biol., 6, 306–329. · doi:10.1016/S0022-2836(63)80091-1
[43] Nass, M. M., &amp; Cooper, L. N. (1975). A theory for the development of feature detecting cells in visual cortex. Biol. Cybern., 19, 1–18. · doi:10.1007/BF00319777
[44] Pomi, A. (2001). Estructuras cognitivas en modelos de memorias distribuidas. Ph.D. thesis, PEDECIBA-Universidad de la República, Montevideo, Uruguay.
[45] Pomi, A., &amp; Mizraji, E. (2004). Semantic graphs and associative memories. Phys. Rev. E, 70, 0666136(1-6). · doi:10.1103/PhysRevE.70.066136
[46] Pomi, A., &amp; Olivera, F. (2006). Context-sensitive autoassociative memories as expert systems in medical diagnosis. BMC Med. Inform. Decis. Mak., 6(39), 1–11. · doi:10.1186/1472-6947-6-39
[47] 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.
[48] Potthast, R., &amp; beim Graben, P. (2009). Inverse problems in neural field theory. SIAM J. Appl. Dyn. Syst., 8, 1405–1433. · Zbl 1402.92026 · doi:10.1137/080731220
[49] Rapoport, A. (1948). Cycle distributions in random nets. Bull. Math. Biophys., 10, 145–157. · doi:10.1007/BF02477489
[50] Repovs, G., &amp; Baddeley, A. (2006). The multi-component model of working memory: explorations in experimental cognitive psychology. Neuroscience, 139, 5–21. · doi:10.1016/j.neuroscience.2005.12.061
[51] Rieke, F., Warland, D., van Steveninck, R., &amp; Bialek, W. (1997). Spikes. Cambridge: MIT Press. · Zbl 0912.92004
[52] Rosenblatt, F. (1958). The perceptron: a probabilistic model for information storage and organization in the brain. Psychol. Rev., 65, 386–408. · doi:10.1037/h0042519
[53] Rumelhart, D. E., Hinton, G. E., &amp; McClelland, J. L. (1986a). A general framework for parallel distributing processing. In D. E. Rumelhart &amp; J. L. McClelland (Eds.), Parallel distributing processing Cambridge: MIT Press.
[54] Rumelhart, D. E., Hinton, G. E., &amp; Williams, R. J. (1986b). Learning representations by back-propagating errors. Nature, 323, 533–536. · Zbl 1369.68284 · doi:10.1038/323533a0
[55] Russell, B. (1948). Human knowledge, its scope and limits. London: Allen &amp; Unwin.
[56] Salinas, E., &amp; Abbott, L. F. (1996). A model of multiplicative neural responses in parietal cortex. Proc. Natl. Acad. Sci., 93, 11956–11961. · doi:10.1073/pnas.93.21.11956
[57] Shaw, G. L., &amp; Palm, G. (Eds.) (1988). Brain theory. Singapore: World Scientific.
[58] Shimbel, A., &amp; Rapoport, A. (1948). A statistical approach to the theory of the central nervous system. Bull. Math. Biophys., 10, 41–55. · doi:10.1007/BF02478329
[59] Smolensky, P. (1990). Tensor product variable binding and the representation of symbolic structures in connectionist systems. Artif. Intell., 46, 159–216. · Zbl 0717.68095 · doi:10.1016/0004-3702(90)90007-M
[60] Srinivasan, M. V., &amp; Bernard, G. D. (1976). A proposed mechanism for multiplication of neural signals. Biol. Cybern., 21, 227–236. · doi:10.1007/BF00344168
[61] Tsukada, M., &amp; Fukushima, Y. (2010). A context sensitive mechanism in hyppocampal CA1 networks. BMB, this special issue.
[62] Valle-Lisboa, J. C., Reali, F., Anastasía, H., &amp; Mizraji, E. (2005). Elman topology with sigma-pi units: an application to the modeling of verbal hallucinations in schizophrenia. Neural Netw., 18, 863–877. · Zbl 02212679 · doi:10.1016/j.neunet.2005.03.009
[63] von Neumann, J. (1945). First draft of a report on the EDVAC. Posted by M. D. Godfrey, in http://qss.stanford.edu/\(\sim\)godfrey/vonNeumann/vnedvac.pdf . Acceded 20 April 2009.
[64] von Neumann, J. (1958). The computer and the brain. New Haven: Yale University Press. · Zbl 0085.14106
[65] Watts, D. J. (1999). Small worlds. Princeton: Princeton University Press. · Zbl 0940.82029
[66] Watts, D. J., &amp; Strogatz, S. H. (1998). Collective dynamics of ’small-world’ networks. Nature, 393, 440–442. · Zbl 1368.05139 · doi:10.1038/30918
[67] 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.
[68] Wright, J. J., Rennie, C. J., Lees, G. J., Robinson, P. A., Bourke, P. D., Chapman, C. L., Gordon, E., &amp; Rowe, D. L. (2004). Simulated electrocortical activity at microscopic, mesoscopic and global scales. Int. J. Bifurc. Chaos, 14, 853–872. · Zbl 1064.92012 · doi:10.1142/S0218127404009569
[69] Wolfram, S. (1985). Origins of randomness in physical systems. Phys. Rev. Lett., 55, 449–452. · doi:10.1103/PhysRevLett.55.449
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.