Barone, E.; Tebaldi, C. Stability of equilibria in a neural network model. (English) Zbl 0959.92002 Math. Methods Appl. Sci. 23, No. 13, 1179-1193 (2000). The following neural network model is investigated: \[ X_i'(t)= X_i(t) \left[1-cX_i(t) +\sum^n_{j=1} \delta_{ij}A_{ij} X_j(t)\right], \quad 1\leq i\leq n, \] where \(X_i(t)\) is a positive continuous function which represents the activity level of cell \(i\) at time \(t\). The rate of change of activity level at the \(i\)th cell is described by a non-constant Lotka-Volterra system. \(\delta_{i j}=0\) if \(i=j\) or is cell \(j\) is not connected to cell \(i\); \(\delta_{ij}=1\) \((-1)\) if cell \(j\) has an excitatory connection to cell \(i\). \(A_{ij}\) is the weight of the connection from cell \(j\) to cell \(i\). The authors gives a complete account of existence and stability of the fixed points of the system. Moreover, they prove that of the \(2n+1\) critical points, \(n+1\) of them, namely \[ (r,r, \dots, r),(b_1,s_1, \dots, s_1), \dots, (s_1, \dots, s_1,b_1), \] are asymptotically stable for \(T\) small enough. The other \(n\) points \[ (b_2,s_2, \dots, s_2), \dots, (s_2, \dots, s_2,b_2) \] are unstable for all \(T\). Thus the authors prove a conjecture in the references. Reviewer: Chen Lan Sun (Beijing) Cited in 6 Documents MSC: 92B20 Neural networks for/in biological studies, artificial life and related topics 34D05 Asymptotic properties of solutions to ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems 92C20 Neural biology PDFBibTeX XMLCite \textit{E. Barone} and \textit{C. Tebaldi}, Math. Methods Appl. Sci. 23, No. 13, 1179--1193 (2000; Zbl 0959.92002) Full Text: DOI References: [1] Arneodo, J. Math. Biol. 14 pp 153– (1982) [2] Bortone, Dynamics of Continuous, Discrete and Impulsive Systems 4 pp 379– (1998) · Zbl 0918.34014 [3] (eds). Proceedings of the IEEE First International Conference on Neural Networks. Institute of Electrical and Electronic Engineers: New York, 1978. [4] Coste, SIAM J. Appl. Math. 36 pp 516– (1979) [5] Elements of Physical Biology. William and Wilkins: Baltimore, 1925. · JFM 51.0416.06 [6] May, SIAM J. Appl. Math. 29 pp 243– (1975) [7] Noonburg, J. Math. Biol. 15 pp 239– (1982) [8] Noonburg, SIAM J. Appl. Math. 49 pp 1779– (1989) [9] Phillipson, SIAM J. Appl. Math. 45 pp 541– (1985) [10] Schuster, SIAM J. Appl. Math. 37 pp 49– (1979) [11] Volterra, Mem. Accad. Nazionale Lincei 2 pp 31– (1926) [12] Population Dynamics from Game Theory, Global Theory of Dynamical Systems Proceedings. Springer: New York, 1979. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.