Boolean networks (BNs) have been extensively used as mathematical models of genetic regulatory networks. The number of fixed points of a BN is a key feature of its dynamical behavior. Here, we study the maximum number of fixed points in a particular class of BNs called regulatory Boolean networks, where each interaction between the elements of the network is either an activation or an inhibition. We find relationships between the positive and negative cycles of the interaction graph and the number of fixed points of the network. As our main result, we exhibit an upper bound for the number of fixed points in terms of minimum cardinality of a set of vertices meeting all positive cycles of the network, which can be applied in the design of genetic regulatory networks.

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We study the relationships between the positive and negative circuits of the connection graph and the fixed points of discrete neural networks (DNNs). As main results, we give necessary conditions and sufficient conditions for the existence of fixed points in a DNN. Moreover, we exhibit an upper bound for the number of fixed points in terms of the structure and number of positive circuits in the connection graph. This allows the determination of the maximum capacity for storing vectors in DNNs as fixed points, depending on the architecture of the network.

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Given a digraph G, a lot of attention has been deserven on the maximum number φ(G) of fixed points in a Boolean network f : {0, 1} n → {0, 1} n with G as interaction graph. In particular, a central problem in network coding consists in studying the optimality of the feedback bound φ(G) ≤ 2 τ , where τ is the minimum size of a feedback vertex set of G. In this paper, we study the maximum number φ m (G) of fixed points in a monotone Boolean network with interaction graph G. We establish new upper and lower bounds on φ m (G) that depends on the cycle structure of G. In addition to τ , the involved parameters are the maximum number ν of vertex-disjoint cycles, and the maximum number ν * of vertex-disjoint cycles verifying some additional technical conditions. We improve the feedback bound 2 τ by proving that φ m (G) is at most the largest sub-lattice of {0, 1} τ without chain of size ν + 2, and without another forbidden pattern described by two disjoint antichains of size ν * + 1. Then, we prove two optimal lower bounds: φ m (G) ≥ ν + 1 and φ m (G) ≥ 2 ν * . As a consequence, we get the following characterization: φ m (G) = 2 τ if and only if ν * = τ . As another consequence, we get that if c is the maximum length of a chordless cycle of G then 2cν . Finally, with the techniques introduced, we establish an upper bound on the number of fixed points of any Boolean network according to its signed interaction graph.