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Stability of equilibria in a neural network model. (English) Zbl 0959.92002

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.

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
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