Necessary and sufficient condition for multistability of neural networks evolving on a closed hypercube

Neural Networks

Volumn 54, Issue , 2014, Pages 38-48

  Di Marco, Mauro ^a   Forti, Mauro ^a   Grazzini, Massimo ^a   Pancioni, Luca ^a  

^a UNIVERSITY OF SIENA   (Italy)

Author keywords

Cooperative dynamical systems; Differential variational inequalities; Multistability; Neural networks; Nonsmooth dynamical systems

Indexed keywords

DYNAMICAL SYSTEMS; LINEAR SYSTEMS; MATHEMATICAL TECHNIQUES;

ASYMPTOTICALLY STABLE; COOPERATIVE DYNAMICAL SYSTEMS; DIFFERENTIAL INCLUSIONS; DIFFERENTIAL VARIATIONAL INEQUALITY; MULTI STABILITIES; MULTIPLE PATTERNS; NON NEGATIVES; NON-SMOOTH DYNAMICAL SYSTEMS;

NEURAL NETWORKS;

ARTICLE; ARTIFICIAL NEURAL NETWORK; CLOSED HYPERCUBE; CONTROLLED STUDY; DIFFERENTIAL VARIATIONAL INEQUALITY; LINEAR SYSTEM; MATHEMATICAL PARAMETERS; MATHEMATICAL PHENOMENA; MULTISTABILITY; NONSMOOTH NEURAL NETWORK; PRIORITY JOURNAL; SIMULATION; THEORY; ALGORITHM; NERVE CELL; THEORETICAL MODEL;

ALGORITHMS; MODELS, THEORETICAL; NEURAL NETWORKS (COMPUTER); NEURONS;

EID: 84896290101     PISSN: 08936080     EISSN: 18792782     Source Type: Journal    
DOI: 10.1016/j.neunet.2014.02.010     Document Type: Article

References (51)

1. Angeli D., Sontag E.D. Multi-stability in monotone input/output systems. Systems and Control Letters 2004, 51:185-202.
2. Aubin J.P., Cellina A. Differential inclusions 1984, Springer-Verlag, Berlin.
3. Berman A., Plemmons R.J. Nonnegative matrices in the mathematical sciences 1979, Academic, New York.
4. Calvert B. Linear systems in a saturated mode and convergence as gain becomes large of asymptotically stable equilibrium points of neural nets. Circuits, Systems, and Signal Processing 1999, 18:241-267.
5. Chen W., Zheng W. A new method for complete stability analysis of cellular neural networks with time delay. IEEE Transactions on Neural Networks 2010, 21:1126-1139.
6. Cheng C., Lin K., Shih C. Multistability and convergence in delayed neural networks. Physica D: Nonlinear Phenomena 2007, 225:61-74.
7. Chu T., Zhang Z., Wang Z. A decomposition approach to analysis of competitive-cooperative neural networks with delay. Physics Letters A 2003, 312:339-347.
8. Chua L.O., Yang L. Cellular neural networks: theory. IEEE Transactions on Circuits Systems 1988, 35:1257-1272.
9. Cohen M.A. Sustained oscillations in a symmetric cooperative-competitive neural network: disproof of a conjecture about content addressable memory. Neural Networks 1988, 1:217-221.
10. Cohen M.A., Grossberg S. Absolute stability of global pattern formation and parallel memory storage by competitive neural networks. IEEE Transactions on Systems, Man, and Cybernetics 1983, 13:815-825.
11. De Sandre G., Forti M., Nistri P., Premoli A. Dynamical analysis of full-range cellular neural networks by exploiting differential variational inequalities. IEEE Transactions on Circuits and Systems I 2007, 54:1736-1749.
12. Di Marco M., Forti M., Grazzini M., Nistri P., Pancioni L. Lyapunov method and convergence of the full-range model of CNNs. IEEE Transactions on Circuits Systems I 2008, 55:3528-3541.
13. Di Marco M., Forti M., Grazzini M., Pancioni L. Limit set dichotomy and convergence of cooperative piecewise linear neural networks. IEEE Transactions on Circuits Systems I 2011, 58:1052-1062.
14. Di Marco M., Forti M., Grazzini M., Pancioni L. Limit set dichotomy and multistability for a class of cooperative neural networks with delays. IEEE Transactions on Neural Networks and Learning Systems 2012, 23:1473-1485.
15. Di Marco M., Forti M., Grazzini M., Pancioni L. A study on semiflows generated by cooperative full-range CNNs. International Journal of Circuit Theory and Applications 2012, 40:1191-1208.
16. Di Marco M., Forti M., Grazzini M., Pancioni L. Convergent dynamics of nonreciprocal differential variational inequalities modeling neural networks. IEEE Transactions on Circuits Systems I 2013, 60:3227-3238.
17. Di Marco M., Forti M., Tesi A. Bifurcations and oscillatory behavior in a class of competitive cellular neural networks. International Journal of Bifurcation and Chaos 2000, 10:1267-1293.
18. Driessche P.V.D., Wu J., Zou X. Stabilization role of inhibitory self-connections in a delayed neural network. Physica D: Nonlinear Phenomena 2001, 150:84-90.
19. Forti M., Tesi A. New conditions for global stability of neural networks with application to linear and quadratic programming problems. IEEE Transactions on Circuits Systems I 1995, 42:354-366.
20. Haykin S. Neural networks: a comprehensive foundation 1999, Prentice-Hall, New Jersey.
21. Hens C., Banerjee R., Feudel U., Dana S. How to obtain extreme multistability in coupled dynamical systems. Physical Review E-Statistical, Nonlinear, and Soft Matter Physics 2012, 85.
22. Hershkowitz D. Recent directions in matrix stability. Linear Algebra and its Applications 1992, 171:161-186.
23. Hirsch M. Convergent activation dynamics in continuous time networks. Neural Networks 1989, 2:331-349.
24. Hirsch M., Smith H. Monotone dynamical systems. Handbook of differential equations, ordinary differential equations (second volume) 2005, Elsevier, Available at: http://math.la.asu.edu/halsmith/MonotoneDynamics.pdf. A. Canada, P. Drabek, A. Fonda (Eds.).
25. Hirsch M.W. The dynamical systems approach to differential equations. American Mathematical Society. Bulletin 1984, 11:1-64.
26. Hirsch M.W. Systems of differential equations which are competitive or cooperative. II: convergence almost everywhere. SIAM Journal on Mathematical Analysis 1985, 16:423-439.
27. Hirsch M.W. Stability and convergence in strongly monotone dynamical systems. Journal für die Reine und Angewandte Mathematik 1988, 383:1-53.
28. Hopfield J.J. Neurons with graded response have collective computational properties like those of two-state neurons. Proceedings of the National Academy of Sciences 1984, 81:3088-3092.
29. Huang G., Cao J. Delay-dependent multistability in recurrent neural networks. Neural Networks 2010, 23:201-209.
30. Huang Y., Zhang H., Wang Z. Dynamical stability analysis of multiple equilibrium points in time-varying delayed recurrent neural networks with discontinuous activation functions. Neurocomputing 2012, 91:21-28.
31. Huang Y., Zhang H., Wang Z. Multistability and multiperiodicity of delayed bidirectional associative memory neural networks with discontinuous activation functions. Applied Mathematics and Computation 2012, 219:899-910.
32. Li J., Michel A.N., Porod W. Analysis and synthesis of a class of neural networks: linear systems operating on a closed hypercube. IEEE Transactions on Circuits Systems 1989, 36:1405-1422.
33. Liu D., Michel A. Dynamical systems with saturation nonlinearities. Lecture notes in control and information sciences 1994, Vol. 195. Springer, London.
34. Liz E., Ruiz-Herrera A. Attractivity, multistability, and bifurcation in delayed Hopfield's model with non-monotonic feedback. Journal of Differential Equations 2013, 255:4244-4266.
35. Lu W., Wang L., Chen T. Coexistence and local stability of multiple equilibria in neural networks with piecewise linear nondecreasing activation functions. Neural Networks 2010, 23:189-200.
36. Lu W., Wang L., Chen T. On attracting basins of multiple equilibria of a class of cellular neural networks. IEEE Transactions on Neural Networks 2011, 22:381-394.
37. Nie X., Huang Z. Multistability and multiperiodicity of high-order competitive neural networks with a general class of activation functions. Neurocomputing 2012, 82:1-13.
38. Oh C., Zak S. Large-scale pattern storage and retrieval using generalized brain-state-in-a-box neural networks. IEEE Transactions on Neural Networks 2010, 21:633-643.
39. Ozcan, N., Arik, S., & Tavsanoglu, V. (2003). New criteria for the existence of stable equilibrium points in nonsymmetric cellular neural networks, In Proc. ISCAS 2003 (pp. 753-756). Vol. 5.
40. Pang J., Stewart D. Differential variational inequalities. Mathematical Programming, Series A 2008, 113:345-424.
41. Roska T., Wu C.W., Balsi M., Chua L.O. Stability and dynamics of delay-type general and cellular neural networks. IEEE Transactions on Circuits Systems I 1992, 39:487-490.
42. Amari S., Arbib M.A. Competition and cooperation in neural networks 1982, Springer-Verlag, Berlin.
43. Shih C.W. Complete stability for a class of cellular neural networks. International Journal of Bifurcation and Chaos 2001, 11:169-177.
44. Smith H. Convergent and oscillatory activation dynamics for cascades of neural nets with nearest neighbor competitive or cooperative interactions. Neural Networks 1991, 4:41-46.
45. Takahashi N. A new sufficient condition for complete stability of cellular neural networks with delay. IEEE Transactions on Circuits Systems I 2000, 47:793-799.
46. Walter W. On strongly monotone flows. Annales Polonici Mathematici 1997, LXVI:269-274.
47. Wang L. Dynamical analysis on the multistability of high-order neural networks. Neurocomputing 2013, 110:137-144.
48. Wang L., Chen T. Multistability of neural networks with Mexican-hat-type activation functions. IEEE Transactions on Neural Networks and Learning Systems 2012, 23:1816-1826.
49. Zeng Z., Huang T., Zheng W. Multistability of recurrent neural networks with time-varying delays and the piecewise linear activation function. IEEE Transactions on Neural Networks 2010, 21:1371-1377.
50. Zeng Z., Wang J. Complete stability of cellular neural networks with time-varying delays. IEEE Transactions on Circuits Systems I 2006, 53:944-955.
51. Zeng Z., Zheng W.X. Multistability of neural networks with time-varying delays and concave-convex characteristics. IEEE Transactions on Neural Networks and Learning Systems 2012, 23:293-305.