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Towards a classification of networks with asymmetric inputs. (English) Zbl 1472.34069

Summary: Coupled cell systems associated with a coupled cell network are determined by (smooth) vector fields that are consistent with the network structure. Here, we follow the formalisms of I. Stewart et al. [SIAM J. Appl. Dyn. Syst. 2, No. 4, 609–646 (2003; Zbl 1089.34032)], M. Golubitsky et al. [ibid. 4, No. 1, 78–100 (2005; Zbl 1090.34030)] and M. Field [Dyn. Syst. 19, No. 3, 217–243 (2004; Zbl 1058.37008)]. It is known that two non-isomorphic \(n\)-cell coupled networks can determine the same sets of vector fields – these networks are said to be ordinary differential equation (ODE)-equivalent. The set of all \(n\)-cell coupled networks is so partitioned into classes of ODE-equivalent networks. With no further restrictions, the number of ODE-classes is not finite and each class has an infinite number of networks. Inside each ODE-class we can find a finite subclass of networks that minimize the number of edges in the class, called minimal networks. In this paper, we consider coupled cell networks with asymmetric inputs. That is, if \(k\) is the number of distinct edges types, these networks have the property that every cell receives \(k\) inputs, one of each type. Fixing the number \(n\) of cells, we prove that: the number of ODE-classes is finite; restricting to a maximum of \(n(n-1)\) inputs, we can cover all the ODE-classes; all minimal \(n\)-cell networks with \(n(n-1)\) asymmetric inputs are ODE-equivalent. We also give a simple criterion to test if a network is minimal and we conjecture lower estimates for the number of distinct ODE-classes of \(n\)-cell networks with any number \(k\) of asymmetric inputs. Moreover, we present a full list of representatives of the ODE-classes of networks with three cells and two asymmetric inputs.

MSC:

34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
05C90 Applications of graph theory
05C30 Enumeration in graph theory
92B20 Neural networks for/in biological studies, artificial life and related topics
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References:

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