Dynamics of a random Hopfield neural lattice model with adaptive synapses and delayed Hebbian learning

Xiaoying Han; Peter E. Kloeden

doi:10.3842/umzh.v75i12.7594

Xiaoying Han Department of Mathematics and Statistics, Auburn University, USA
Peter E. Kloeden Mathematisches Institut, Universität Tübingen, Germany

DOI: https://doi.org/10.3842/umzh.v75i12.7594

Keywords: Hopfield neural network, Hebian learning, neural lattice model with delay, lattice dynamical system, sigmoidal function, random dynamical system, random attractor

Abstract

UDC 517.9

A Dong–Hopfield neural lattice model with random external forcing and delayed response to the evolution of interconnection weights is developed and studied. The interconnection weights evolve according to the Hebbian learning rule with a decay term and contribute to changes in the states after a short delay. The lattice system is first reformulated as a coupled functional-ordinary differential equation system on an appropriate product space. Then the solution of the system is shown to exist and be unique. Furthermore it is shown that the system of equations generates a continuous random dynamical system. Finally, the existence of random attractors for the random dynamical system generated by the Dong–Hopfield model is established.

References

S. I. Amari, Dynamics of pattern formation in lateral-inhibition type neural fields, Biol. Cybernet., 27, 77–87 (1977). DOI: https://doi.org/10.1007/BF00337259

L. Arnold, Random dynamical system, Springer-Verlag, Berlin (1998). DOI: https://doi.org/10.1007/978-3-662-12878-7

P. W. Bates, K. Lu, B. Wang, Attractors for lattice dynamical systems, Inter. J. Bifur. and Chaos, 11, 143–153 (2001). DOI: https://doi.org/10.1142/S0218127401002031

T. Caraballo, F. Morillas, J. Valero, On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems, Discrete and Contin. Dyn. Syst., 34, 51–77 (2014).

H. Crauel, P. E. Kloeden, Nonautonomous and random attractors, Jahresbericht der Deutschen Mathematiker-Vereinigung, 117, 173–206 (2015). DOI: https://doi.org/10.1365/s13291-015-0115-0

D. W. Dong, J. J. Hopfield, Dynamical properties of neural networks with adapting synapses, Network, 3, 267–283 (1992). DOI: https://doi.org/10.1088/0954-898X/3/3/002

Xiaoying Han, Basiru Usman, Peter Kloeden, Long term behavior of a random Hopfield neural lattice model, Commun. Pure and Appl. Anal., 18, 809–824 (2019). DOI: https://doi.org/10.3934/cpaa.2019039

Xiaoying Han, P. E. Kloeden, Asymptotic behaviour of a neural field lattice model with a Heaviside operator, Physica~D, 389, 1–12 (2019). DOI: https://doi.org/10.1016/j.physd.2018.09.004

Xiaoying Han, P. E. Kloeden, Sigmoidal approximations of Heaviside functions in neural lattice models, J. Different. Equat., 268, № 9, 5283–5300 (2020); Corrigendum J. Different. Equat., 274, 1214–1220 (2020).

Xiaoying Han, P. E. Kloeden, Dissipative lattice dynamical systems, World Sci. Publ. Co., Singapore (2023).

Xiaoying Han, P. E. Kloeden, Basiru Usman, Upper semi-continuous convergence of attractors for a Hopfield-type lattice model, Nonlinearity, 33, 1881–1906 (2020). DOI: https://doi.org/10.1088/1361-6544/ab6813

J. J. Hopfield, Neurons with graded response have collective computational properties like those of two-stage neurons, Proc. Nat. Acad. Sci. USA, 81, 3088–3092 (1984). DOI: https://doi.org/10.1073/pnas.81.10.3088

N. B. Janson, P. E. Kloeden, Mathematical consistency and long-term behaviour of a dynamical system with a self-organising vector field, J. Dynam. and Different. Equat., 34, 63–78 (2022). DOI: https://doi.org/10.1007/s10884-020-09834-7

N. B. Janson, P. E. Kloeden, Robustness of a dynamical systems model with a plastic self-organising vector field to noisy input signals, Eur. Phys. J. Plus, 136, 720 (2021). DOI: https://doi.org/10.1140/epjp/s13360-021-01662-y

N. B. Janson, C. J. Marsden, Dynamical system with plastic self-organized velocity field as an alternative conceptual model of a cognitive system, Sci. Rep., 7, Article~17007 (2017) plus supplementary note. DOI: https://doi.org/10.1038/s41598-017-16994-y

P. E. Kloeden, V. M. Villarragut, Sigmoidal approximations of a nonautonomous neural network with infinite delay and Heaviside function, J. Dynam. and Different. Equat., 34, 721–745 (2022). DOI: https://doi.org/10.1007/s10884-020-09899-4

S. Shinomoto, Memory maintenance in neural networks, J. Phys. A, 18, L1305–L1309 (1987). DOI: https://doi.org/10.1088/0305-4470/20/18/015

Xiaoli Wang, P. E. Kloeden, Xiaoying Han, Attractors of Hopfield-type lattice models with increasing neuronal input, Discrete and Contin. Dyn. Syst. Ser. B, 25, 799–813 (2020). DOI: https://doi.org/10.3934/dcdsb.2019268

Xiaoli Wang, P. E. Kloeden, Meihua Yang, Asymptotic behaviour of a neural field lattice model with delays, Election. Res. Arch., 28, 1037–1048 (2020). DOI: https://doi.org/10.3934/era.2020056

S. Zhou, Attractors for first order dissipative lattice dynamical systems, Physica D, 178, 51–61 (2003). DOI: https://doi.org/10.1016/S0167-2789(02)00807-2