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The golden mean as clock cycle of brain waves. (English) Zbl 1129.91353
Summary: The principle of information coding by the brain seems to be based on the golden mean. For decades psychologists have claimed memory span to be the missing link between psychometric intelligence and cognition. By applying Bose-Einstein statistics to learning experiments, Pascual-Leone obtained a fit between predicted and tested span. Multiplying span by mental speed (bits processed per unit time) and using the entropy formula for bosons, we obtain the same result. If we understand span as the quantum number $$n$$ of a harmonic oscillator, we obtain this result from the EEG. The metric of brain waves can always be understood as a superposition of $$n$$ harmonics times $$2\varPhi$$, where half of the fundamental is the golden mean $$\varPhi$$ (=1.618) as the point of resonance. Such wave packets scaled in powers of the golden mean have to be understood as numbers with directions, where bifurcations occur at the edge of chaos, i.e. $$2\varPhi=3+\phi^3$$. Similarities with El Naschie’s theory for high energy particle’s physics are also discussed.

##### MSC:
 91E45 Measurement and performance in psychology 81V99 Applications of quantum theory to specific physical systems
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##### References:
 [1] Ilachinski, A., Cellular automata: a discrete universe, (2001), World Scientific Singapore · Zbl 1031.68081 [2] Wolfram, S., A new kind of science, (2002), Wolfram Media Champaign · Zbl 1022.68084 [3] Kac, M., Can one hear the shape of a drum, Am. math. mon., 73, part II, 1-23, (1966) · Zbl 0139.05603 [4] Fogleman, G., Quantum strings, Am. J. phys., 55, 330-336, (1987) [5] Eysenck, H.J., The theory of intelligence and the psychophysiology of cognition, (), 1-34 [6] Kawai, N.; Matsuzawa, T., Numerical memory span in a chimpanzee, Nature, 403, 39-40, (2000) [7] Miller, G.A., The magical number seven, plus or minus two: some limits on our capacity for processing information, Psychol. rev., 63, 81-97, (1956) [8] Piaget, J., The theory of stages in cognitive development, (), 1-11 [9] Pascual-Leone, J., A mathematical model for the transition rule in piaget’s developmental stages, Acta psychol., 32, 301-345, (1970) [10] Halford, G., The development of intelligence includes capacity to process relations of greater complexity, (), 193-213 [11] Humphreys, L.G.; Rich, S.A.; Davey, T.C., A Piagetian test of general intelligence, Dev. psychol., 21, 872-877, (1985) [12] Weiss, V., The relationship between short-term memory capacity and EEG power spectral density, Biol. cyber., 68, 165-172, (1992) [13] Lehrl, S.; Gallwitz, A.; Blaha, L.; Fischer, B., Geistige leistungsfähigkeit. theorie und messung der biologischen intelligenz mit dem kurztest KAI, (1991), Vless Ebersberg [14] Schröder, M., Fractals, chaos, power laws, (1991), W.H. Freeman New York [15] Bianconi, G.; Barabási, A.L., Bose – einstein condensation in complex networks, Phys. rev. lett., 86, 5632-5635, (2001) [16] Albert, R.; Barabási, A.L., Statistical mechanics of complex networks, Rev. mod. phys., 74, 47-97, (2002) · Zbl 1205.82086 [17] Wyner, A.D.; Ziv, J.; Wyner, A.J., On the role of pattern matching in information theory, IEEE trans. inf. theory, 44, 2045-2056, (1998) · Zbl 0943.94003 [18] Frank, H.G., Bildungskybernetik, (1996), Kopäd München [19] Limpert, E.; Stahel, W.A.; Abbt, M., Lognormal distributions across the sciences: keys and clues, Bioscience, 51, 341-352, (2001) [20] Naylor, G.F.K., Perception times and rates as a function of the qualitative and quantitative structure of the stimulus, Aust. J. psychol., 20, 165-172, (1968) [21] Lehrl, S.; Fischer, B., A basic information psychological parameter (BIP) for the reconstruction of concepts of intelligence, Eur. J. personality, 4, 259-286, (1990) [22] Baddeley, A.D.; Thomson, N.; Buchanan, N., Word length and the structure of short-term memory, J. verbal learning behav., 14, 575-589, (1975) [23] Weiss, V., Memory span as the quantum of action of thought, Cah. psychol. cogn., 14, 387-408, (1995) [24] Zipf, G.K., Human behavior and the principle of least effort, (1949), Addison-Wesley Cambridge, MA [25] Gibbs, F.A.; Davis, H.; Lennox, W.G., The electroencephalogram in epilepsy and in conditions of impaired consciousness, Arch. neurol. psychiat., 34, 1133-1148, (1935) [26] Giannitrapani, D., The electrophysiology of intellectual functions, (1985), Karger Basel, p. 153-76 [27] Gershenfeld, N., Signal entropy and the thermodynamics of computation, IBM syst. J., 35, 577-586, (1996) [28] Goebel, P.R., The mathematics of mental rotations, J. math. psychol., 34, 435-444, (1990) · Zbl 0707.92025 [29] Bennett, E.R., The Fourier transform of evoked responses, Nature, 239, 407-408, (1974) [30] Ertl, J.P.; Schafer, E.W.P., Brain response correlate of psychometric intelligence, Nature, 223, 421-422, (1969) [31] Flinn, J.M.; Kirsch, A.D.; Flinn, E.A., Correlations between intelligence and the frequency content of the evoked potential, Physiol. psychol., 5, 11-15, (1977) [32] Harwood, E.; Naylor, G.F.K., Rates and information transfer in elderly subjects, Aust. J. psychol., 21, 127-136, (1969) [33] Wentzel, G., Eine verallgemeinerung der quantenbedingungen für die zwecke der wellenmechanik, Z. phys., 38, 518-529, (1926) · JFM 52.0969.03 [34] Saltzberg, B.; Burch, N.R., Periodic analytic estimates of the power spectrum: a simplified EEG domain procedure, Electroen. clin. neurophysiol., 30, 568-570, (1971) [35] Bath, M., Spectral analysis in geophysics, (1974), Elsevier Amsterdam [36] Reiss, R.F., Neural theory and modelling, (1964), Stanford University Press Stanford, p. 105-37 [37] Glassman, R.B., Hypothesized neural dynamics of working memory: several chunks might be marked simultaneously by harmonic frequencies within an octave band of brain waves, Brain res. bull., 50, 77-94, (1999) [38] de Spinadel, V.W., From the Golden Mean to chaos, (1998), Nueva Libreria Buenos Aires · Zbl 0997.00500 [39] Datta, D.P., A new class of scale free solutions to linear ordinary differential equations and the universality of the Golden Mean $$(5−1)/2$$, Chaos solitons fractals, (2002), arXiv: nlin. CD/0209023, v1, 11 September 2002 [40] Koch, C., Computation and the single neuron, Nature, 385, 207-210, (1997) [41] Papa, A.R.R.; Da, L., Earthquakes in the brain, Theor. biosci., 116, 321-327, (1997) [42] Frieden, B.R., Physics from Fisher information: a unification, (1998), Cambridge University Press Cambridge · Zbl 0881.60016 [43] Planat M. Modular functions and Ramanujan sums for the analysis of 1/f noise in electronic circuits. arXiv: hep-th/0209243, v1, 27 September 2002 [44] Gilden, D.L.; Thornton, D.; Mallon, M.W., 1/f noise in human cognition, Science, 267, 1837-1839, (1995) [45] Frougny, C.; Sakarovitch, J., Automatic conversion from Fibonacci representation to representation in base phi, and a generalization, Int. J. algebra comput., 9, 351-384, (1999) · Zbl 1040.68061 [46] Kennedy, J.W.; Christopher, P.R., Binomial graphs and their spectra, Fibonacci quart., 35, 48-53, (1997) · Zbl 0878.05058 [47] Schürmann, T., Scaling behavior of entropy estimates, J. phys. A: math. gen., 35, 1589-1596, (2002) · Zbl 1008.94010 [48] El Naschie, M.S., Quantum groups and Hamiltonian sets on a nuclear space – time Cantorian manifold, Chaos, solitons & fractals, 10, 1251-1256, (1999) · Zbl 0985.37066 [49] Weimann, C.; Chaitin, G., Logarithmic spiral grids for image processing and display, Comput. graphics image process., 11, 197-226, (1979) [50] Kapraff, J.; Blackmore, D.; Adamson, G., Phyllotaxis as a dynamical system: a study in number, (), 409-458 [51] Mojsilovic, A.; Soljanin, E., Color quantization and processing by Fibonacci lattices, IEEE trans. image process., 10, 1712-1725, (2001) · Zbl 1037.68811 [52] Kimberling, C., A self-generating set and the Golden Mean, J. integer sequences, 3, (2000), Article 00.2.8 · Zbl 0993.11010 [53] () [54] Mignosi, F.; Restivo, A.; Salemi, S., Periodicity and the Golden ratio, Theor. comput. sci., 204, 153-167, (1998) · Zbl 0913.68162 [55] Diez, E.; Dominguez-Adame, F.; Maciá, E.; Sánchez, A., Dynamical phenomena in Fibonacci semiconductor superlattices, Phys. rev. B, 54, 792-798, (1996) [56] Cuesta IG, Satija II. Dimer-type correlations and band crossings in Fibonacci lattices 1999. arXiv: cond-mat/9904022 [57] Srinivasan, T.P., Fibonacci sequence, Golden ratio, and a network of resistors, Am. J. phys., 60, 461-462, (1992) [58] El Naschie, M.S., On the unification of heterotic strings, M theory and E(∞) theory, Chaos, solitons & fractals, 11, 2397-2408, (2000) · Zbl 1008.81511 [59] Tsuda, I., Towards an interpretation of dynamic neural activity in terms of chaotic dynamical systems, Behav. brain sci., 24, 793-847, (2001) [60] El Naschie, M.S., Modular groups in Cantorian E(∞) high-energy physics, Chaos, solitons & fractals, 16, 353-366, (2003) · Zbl 1035.83503 [61] El Naschie, M.S., Kleinian groups in E(∞) and their connection to particle physics and cosmology, Chaos, solitons & fractals, 16, 637-649, (2003) · Zbl 1035.83509
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