Abstract
All neural networks, both natural and artificial, are characterized by two kinds of dynamics. The first one is concerned with what we would call “learning dynamics”, in fact the sequential (discrete time) dynamics of the choice of synaptic weights. The second one is the intrinsic dynamics of the neural network viewed as a dynamical system after the weights have been established via learning. The paper deals with the second kind of dynamics. Since the emergent computational capabilities of a recurrent neural network can be achieved provided it has suitable dynamical properties when viewed as a system with several equilibria, the paper deals with those qualitative properties connected to the achievement of such dynamical properties, more precisely the gradient like behavior. In the case of the neural networks with delays, these aspects are reformulated in accordance with the state of the art of the theory of delay dynamical systems.
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References
Bélair, J., Campbell, S.A., van den Driessche, P.: Frustration, stability and delay induced oscillations in a neural network model. SIAM J. Appl. Math. 56, 254–265 (1996)
Danciu, D.: Qualitative behavior of the time delay Hopfield type neural networks with time varying stimulus. Annals of The University of Craiova, Series: Electrical Engineering (Automatics, Computers, Electronics) 26, 72–82 (2002)
Danciu, D., Răsvan, V.: On Popov-type Stability Criteria for Neural Networks. Electronic Journal on Qualitative Theory of Differential Equations (Proc. 6th Coll. Qualitative Theory of Differential Equations QTDE) 23 (2000), http://www.math.u-szeged.hu/ejqtde/6/623.pdf
Danciu, D., Răsvan, V.: Gradient-like behaviour for Hopfield-type neural networks with delay. In: Proc. of The 3rd Int. Workshop on Intelligent Control Systems ICS’2001, pp. 20–24. Printech, Bucharest (2001)
Driessche, P.: Global attractivity in delayed Hopfield neural networks. SIAM J. Appl. Math. 58, 1878–1890 (1998)
Fink, W.: Neural attractor network for application in visual field data classification. Phys. Med. Biol. 49, 2799–2809 (2004)
Fortuna, L., Balya, D., Zarandy, A.: Cellular Neural Networks. IEEE Circuits and Systems Magazine 4, 6–21 (2001)
Gelig, A.K.: Dynamics of pulse systems and neural networks (in Russian). Leningrad Univ. Publishing House, Leningrad (1982)
Gelig, A.K., Leonov, G.A., Yakubovich, V.A.: Stability of the systems with non-unique equilibrium points (in Russian). Nauka, Moscow (1978)
Gopalsamy, K., He, X.: Stability in asymmetric Hopfield nets with transmission delays. Physica D 76, 344–358 (1994)
Halanay, A.: Invariant manifolds for systems with time lag. In: Hale, J.K., La Salle, J.P. (eds.) Differential and dynamical systems, pp. 199–213. Acad. Press, New York (1967)
Halanay, A., Răsvan, V.: Applications of Liapunov Methods to Stability. Kluwer Academic Publishers, Dordrecht (1993)
Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer, Heidelberg (1993)
Kharitonov, V.L., Zhabko, A.P.: Lyapunov-Krasovskii approach to the robust stability analysis of time-delay systems. Automatica 39, 15–20 (2003)
Kitamura, S., Hirai, K., Nishimura, M.: Stability of a Control System with Several Nonlinear Elements and Time Lags. Technology Repts. Osaka Univ. 17, 93–102 (1967)
König, P., Schillen, J.B.: Stimulus dependent assembly formation of oscillatory responses: I. Synchronization. Neural Computation 3, 155–166 (1991)
Kopell, N.: We got the rhythm: dynamical systems of the nervous system. Notices AMS 47, 6–16 (2000)
Leonov, G.A., Reitmann, V., Smirnova, V.B.: Pendulum like Feedback Systems. Teubner Verlag, Leipzig (1992)
Nishimura, M., Kitamura, S.: A Lyapunov Functional for Systems with Multiple Non-linearities and Time Lags. Technology Repts. Osaka Univ. 19, 83–88 (1969)
Osareh, A., Mirmehdi, M., Thomas, B., Markham, R.: Automatic Identification of Diabetic Retinal Exudates in Digital Colour Images. British Journal of Ophthalmology 87(10), 1220–1223 (2003)
Osareh, A.: Automated Identification of Diabetic Retinal Exudates and the Optic Disc. PhD Thesis, Bristol University, Faculty of Engineering, Department of Computer Science (2004)
Popov, V.M.: Monotonicity and Mutability. Journ. Diff. Eqs. 31, 337–358 (1979)
Răsvan, V.: Dynamical Systems with Several Equilibria and Natural Liapunov Functions. Archivum mathematicum 34, 207–215 (1998)
Răsvan, V., Danciu, D.: Neural networks - global behavior versus delay. Periodica Politechnica. Trans. Autom. Control and Comp. Sci. 49(63), 11–14 (2004)
Saperstone, S.: Semidynamical Systems in Infinite Dimensional Spaces. Springer, Heidelberg (1981)
Yi, Z.: Global exponential stability and periodic solutions of delay Hopfield neural netorks. Int. Journ. Syst. Sci. 27, 227–231 (1996)
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Danciu, D., Răsvan, V. (2007). Dynamics of Neural Networks - Some Qualitative Properties. In: Sandoval, F., Prieto, A., Cabestany, J., Graña, M. (eds) Computational and Ambient Intelligence. IWANN 2007. Lecture Notes in Computer Science, vol 4507. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73007-1_2
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DOI: https://doi.org/10.1007/978-3-540-73007-1_2
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