A bidirectional associative memory model with almost periodic endogenous and exogenous stimuli

Document Type : Research Article

Authors

1 Facultad de Ciencias Físico-Matemáticas, Universidad Michoacana. Edif. Alfa, Ciudad Universitaria, C.P. 58040. Morelia, Michoacán, México

2 Instituto de Física y Matemáticas, Universidad Michoacana. Ciudad Universitaria, C.P. 58040. Morelia, Michoacán, México

3 División de Ciencias Biológicas y de la Salud, Depto. de Ciencias Ambientales, Universidad Autónoma Metropolitana Unidad Lerma, Av. Hidalgo Poniente No. 46, Col. La Estación, 52006 Lerma de Villada, Edo. de México, México

Abstract

In this work, a two-neuron model that describes a module of a neuronal network is analyzed. Unlike other studies, all the rates involved in the model are asumed to be almost periodic functions. Assuming an almost periodicity in the neuronal mechanisms offers advantages because the endogenous and exogenous stimuli received by the neuron are not necesarilly periodic or constant. Analysis of the model showed that it is associated with a unique stable almost periodic solution when some conditions on the parameters of the model are satisfied. Numerical simulations of the solutions of the model show that the neuronal state variable of both neurons can be underestimated or overestimated depending on whether the neuronal dynamics is modeled by periodic or almost periodic functions. Such estimation errors can lead to failure in forecasting the time in which neurons must synchronize. 

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Main Subjects


[1] M.A. Arbib, Brains, Machines, and Mathematics, Second Ed., Springer-Verlag, New York, 1987.
[2] H. Ando, H. Suetani, J. Kurths, K. Aihara, Chaotic phase synchronization in bursting-neuron
models driven by a weak periodic force, Phys. Rev. E. 86 (2012) 016205.
[3] M. Bohner, G.T. Stamov, I.M. Stamova, Almost periodic solutions of CohenGrossberg neural
networks with time-varying delay and variable impulsive perturbations, Commun. Nonlinear
Sci. Numer. Simul. 80 (2020) 104952.
[4] H. Bohr, Almost Periodic Functions, Chelsea Publishing Company, New York, N.Y., 1947.
[5] R. Brette. Computing with neural synchronicity, PLoS Comput. Biol. 8 (2012) e1002561.
[6] M.A. Cohen, S. Grossberg, Absolute stability of global pattern formation and parallel memory
storage by competitive neural networks, IEEE Trans. Syst. Man. Cybern. 13 (1983) 815–826.
[7] M.A. Cohen, S. Grossberg, Absolute stability of global pattern formation and parallel memory
storage by competitive neural networks, Adv. Psychol. 42 (1987) 288–308.
[8] C. Corduneanu , N. Gheorghiu, V. Barbu, Translated from the Romanian Ed. by Gitta Bernstein
and Eugene Tomer, Almost Periodic Functions, New York, Interscience Publishers, 1968.
[9] H.G. D´ıaz-Mar´ın, F. L´opez-Hern´andez and O. Osuna, Almost periodic solutions for seasonal
cooperative systems, Ann. Pol. Math. 128 (2022) 1–14.
[10] I. R. Epstein, J. A. Pojman, An Introduction to Nonlinear Chemical Dynamics, Oxford Universite
Press, New York, 1998.
[11] R. Fitzhugh, Impulses and physiological states in theoretical models of nerve membrane, Bio-
phys. J. 1 (1961) 445–466.
[12] R. Fitzhugh, R. Fitzhugh, Thresholds and plateaus in the Hodgkin Huxley nerve equations, J.
Gen. Physiol. 43 (1960) 867–896.
[13] A. Goldbeter, Biochemical Oscillations and Cellular Rhythms, Cambridge University Press,
1996.
[14] K. Gopalsamy, X.Z. He, Stability in asymmetric Hopfield nets with transmission delays, Phys.
D. 76 (1994) 344–358.
[15] K. Gopalsamy, X.Z. He, Delay-independent stability in bidirectional associative memory net-
works, IEEE Transactions on neural networks, 5 (1994) 998–1002.
[16] L. Hartwell, J. Hopfield, S. Leiblerand, A. Murray, From molecular to modular cell biology,
Nature 402 (1999) 47–52.
[17] S.O. Haykin, Neural Networks and Learning Machines, Third Ed., Pearson, 2009.
[18] A.L. Hodgkin, A.F. Huxley, A quantitative description of membrane currents and its application
to conduction and excitation in nerve, J. Physiol. 117 (1952) 500–544.
[19] J.J. Hopfield, Neurons with graded response have collective computational properties like those
of two-state neurons, Biophysics 81 (1984) 3088–3092.
[20] J.J. Hopfield, Neural computation of decisions in optimization problems, Biol. Cybern. 52 (1985)
141–152.
[21] M.W. Kirschner, J.C. Gerhart, The plausibility of life: Resolving Darwins dilemma, New Haven
and London: Yale University Press, 2005.
[22] B. Kosko, Adaptive bidirectional associative memories, Appl. Optics 26 (1987) 4947–4960.
[23] B. Kosko, Bidirectional associative memories, IEEE Trans. Syst. Man. Cybern. SMC-18 (1988)
49–60.
[24] D.A. Lauffenburger, Cell signaling pathways as control modules: Complexity for simplicity?,
Proc. Natl. Acad. Sci. 97 (2000) 5031–5033.
[25] R. Lefever, G. Nicolis, I. Prigonine, On the occurrence of oscillations around the steady state in
systems of chemical reactions far from equilibrium, J. Chem. Phys. 47 (1967) 1045–1047.
[26] Y. Li, X. Fan, Existence and globally exponential stability of almost periodic solution for Cohen-
Grossberg BAM neural networks with variable coefficients, Appl. Math. Model. 33 (2009) 2114–
2120.
[27] B. Li, Q. Song, Some new results on periodic solution of Cohen-Grossberg neural network with
impulses, Neurocomputing 177 (2016) 401–408.
[28] B. Liu, L. Huang, Existence and exponential stability of almost periodic solutions for Hopfield
neural networks with delays, Neurocomputing 68 (2005) 196–207.
[29] A.J. Lotka, Contribution to the theory of periodic reactions, J. Phys. Chem 14 (1910) 271–274.
[30] A.J. Lotka, Undamped oscillations derived from the law mass action, J. Am. Chem. Soc. 42
(1920) 1595.
[31] A.J. Lotka, Elements of Physical Biology, The Williams & Wilkins Co, Baltimore, MD, 1925.
[32] J. Nagumo, S. Arimoto, S. Yoshizawa , An active pulse transmission line simulating nerve axon,
Proc. IRE. 50 (1962) 2061–2070.
[33] G. Pappas, I.J. Kiriazane, M.E. Falagas, Insights into infectious disease in the era of Hippocrates,
Int. J. Infect. Dis. 12 (2008) 347–350.
[34] H.L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and
Cooperative Systems, AMS, Vol. 41, 1995.
[35] H.L. Smith, Mathematics Inspired by Biology: Lectures given at the 1st Session of the Centro
Internazionale Matematico Estivo (C.I.M.E.) held in Martina Franca, Italy, June 13-20, 1997,
editor Capasso, Vincenzo Springer Berlin Heidelberg, pp. 191-240, 1999.
[36] P. van den Driessche, X. Zou, Global attractivity in delayed Hopfield neural network models,
SIAM J. Appl. Math. 58 (1998) 1798–1890.
[37] V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature 118
(1926) 558–560.
[38] W. Xie, F. Kong, H. Qiu, X. Fu, Stability analysis of almost periodic solutions for discontinuous
bidirectional associative memory (BAM) neural networks with discrete and distributed delays,
Int. J. Nonlinear Sci. Num. Simul. 22 (2021) 873–895.
[39] H. Zhao, L. Chen, Z. Mao, Existence and stability of almost periodic solution for Cohen-
Grossberg neural networks with variable coefficients, Nonlinear Anal. Real World Appl. 9 (2008)
663–673.