Convergence in networks with counterclockwise neural dynamics

IEEE Transactions on Neural Networks

Volumn 20, Issue 5, 2009, Pages 794-804

  Angeli, David ^a,b  

^a IMPERIAL COLLEGE LONDON   (United Kingdom)

^b UNIVERSITY OF FLORENCE   (Italy)

Author keywords

Cellular nonlinear networks (CNNs); Complete stability; Counterclockwise (ccw) input output (I O) dynamics; Fitzhugh Nagumo circuit; Hopfield CNN; Passivity theory

Indexed keywords

CELLULAR NONLINEAR NETWORKS (CNNS); COMPLETE STABILITY; COUNTERCLOCKWISE (CCW) INPUT-OUTPUT (I-O) DYNAMICS; FITZHUGH-NAGUMO CIRCUIT; HOPFIELD CNN; PASSIVITY THEORY;

CELLULAR NEURAL NETWORKS; CHANNEL ESTIMATION; DYNAMICAL SYSTEMS; DYNAMICS; FEEDBACK; NEURONS; OPTICAL RESONATORS; PASSIVATION; SYSTEM STABILITY; TELECOMMUNICATION REPEATERS;

NONLINEAR NETWORKS;

EID: 67349207868     PISSN: 10459227     EISSN: None     Source Type: Journal    
DOI: 10.1109/TNN.2009.2013341     Document Type: Article

References (26)

1. D. Angeli, "Systems with counterclockwise input-output dynamics," IEEE Trans. Autom. Control, vol. 51, no. 7, pp. 1130-1143, Jul. 2006.
2. D. Angeli, "Multistability in systems with counterclockwise input-output dynamics," IEEE Trans. Autom. Control, vol. 52, no. 4, pp. 596-609, Apr. 2007.
3. D. Angeli and E. D. Sontag, "Multi-stability in monotone input/output systems," Syst. Control Lett., vol. 51, no. 3-4, pp. 185-202, 2004.
4. N. P. Bathia and G. P. Szegö, Stability Theory of Dynamical Systems. Berlin, Germany: Springer-Verlag, 1970.
5. L. O. Chua, "Passivity and complexity," IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 46, no. 1, pp. 71-82, Jan. 1999.
6. L. O. Chua and L. Yang, "Cellular neural networks: Theory," IEEE Trans. Circuits Syst., vol. CAS-35, no. 10, pp. 1257-1272, Oct. 1988.
7. L. O. Chua and L. Yang, "Cellular neural networks: Applications," IEEE Trans. Circuits Syst., vol. CAS-35, no. 10, pp. 1273-1290, Oct. 1988.
8. C. A. Eschenbach and C. R. Johnson, "Sign patterns that require real, nonreal or pure imaginary eigenvalues," Linear Multilinear Algebra vol. 29, pp. 299-311, 1991.
9. R. FitzHugh, Mathematical Models of Excitation and Propagation in Nerve, ser. Biological Engineering, I. Schwan, Ed. New York: McGraw-Hill, 1969.
10. M. Forti and A. Tesi, "A new method to analyze complete stability of PWL cellular neural networks," Int. J. Bifurc. Chaos, vol. 11, pp. 655-676, 2001.
11. M. Forti, "Some extensions of a new method to analyze complete stability of neural networks," IEEE Trans. Neural Netw., vol. 13, no. 5, pp. 1230-1238, Sep. 2002.
12. M. Forti and A. Tesi, "New conditions for global stability of neural networks with application to linear and quadratic programming problems," IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 42, no. 7, pp. 354-366, Jul. 1995.
13. M. Forti, S. Manetti, and M. Marini, "Necessary and sufficient condition for absolute stability of neural networks," IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 41, no. 7, pp. 491-494, Jul. 1994.
14. J. L. Gouze, "Positive and negative circuits in dynamical systems," J. Biol. Syst., vol. 6, pp. 11-15, 1998.
15. M. Hirsch, "Systems of differential equations that are competitive or cooperative II: Convergence almost everywhere," SIAM J. Math. Anal. vol. 16, pp. 423-439, 1985.
16. M. W. Hirsch, "Convergent activation dynamics in continuous time networks," Neural Netw., vol. 2, pp. 331-349, 1989.
17. E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. Cambridge, MA: MIT Press, 2006.
18. M. Laurent and N. Kellershohn, "Multistability: A major means of differentiation and evolution in biological systems," Trends Biochem. Sci., vol. 24, pp. 418-422, 1999.
19. G. A. Rovithakis and M. A. Christodoulou, Adaptive Control With Recurrent High-Order Neural Networks: Theory and Industrial Applications ser. Advances in Industrial Control. New York: Springer- Verlag, 2000.
20. H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems. Providence, RI: AMS, 1995, vol. 41, Mathematical Surveys and Monographs.
21. J. J. Steil, "Input-output stability of recurrent neural networks," Ph.D. dissertation, Der Technischen Fakultät der Universität Bielefeld, Bielefeld, Germany, 1999.
22. X.-Y. Wu, Y.-S. Xia, J. Li, and W.-K. Chen, "A high-performance neural network for solving linear and quadratic programming problems," IEEE Trans. Neural Netw., vol. 7, no. 3, pp. 643-651, May 1996.
23. W. Yu and X. Li, "Some stability properties of dynamic neural networks," IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 48, no. 2, pp. 256-259, Feb. 2001.
24. W. Yu and X. Li, "Passivity analysis of dynamic neural networks with different time-scales," Neural Process. Lett., vol. 25, no. 2, pp. 143-155, 2007.
25. J. C. Willems, "Dissipative dynamical systems, part I: General theory," Arch. Rat. Mech. Anal., vol. 45, pp. 321-351, 1972.
26. J. C. Willems, "Dissipative dynamical systems, part II: Linear systems with quadratic supply rates," Arch. Rat. Mech Anal., vol. 45, pp. 352-393, 1972.