Akian, M., Gaubert, S., & Hochart, A. (2020). A Game Theory Approach To The Existence And Uniqueness Of Nonlinear Perron-Frobenius Eigenvectors. Discret. Contin. Dyn. Syst., 40(1), 207–231.
Abstract: We establish a generalized Perron-Frobenius theorem, based on a combinatorial criterion which entails the existence of an eigenvector for any nonlinear order-preserving and positively homogeneous map f acting on the open orthant R->0(n). This criterion involves dominions, i.e., sets of states that can be made invariant by one player in a two-person game that only depends on the behavior of f “at infinity”. In this way, we characterize the situation in which for all alpha, beta > 0, the “slice space” S-alpha(beta) :={x is an element of R->0(n) vertical bar alpha x <= f(x) <= beta x} is bounded in Hilbert's projective metric, or, equivalently, for all uniform perturbations g of f, all the orbits of g are bounded in Hilbert's projective metric. This solves a problem raised by Gaubert and Gunawardena (Trans. AMS, 2004). We also show that the uniqueness of an eigenvector is characterized by a dominion condition, involving a different game depending now on the local behavior of f near an eigenvector. We show that the dominion conditions can be verified by directed hypergraph methods. We finally illustrate these results by considering specific classes of nonlinear maps, including Shapley operators, generalized means and nonnegative tensors.
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Goles, E., Montealegre, P., & Rios-Wilson, M. (2020). On The Effects Of Firing Memory In The Dynamics Of Conjunctive Networks. Discret. Contin. Dyn. Syst., 40(10), 5765–5793.
Abstract: A boolean network is a map F : {0, 1}(n) -> {0, 1}(n) that defines a discrete dynamical system by the subsequent iterations of F. Nevertheless, it is thought that this definition is not always reliable in the context of applications, especially in biology. Concerning this issue, models based in the concept of adding asynchronicity to the dynamics were propose. Particularly, we are interested in a approach based in the concept of delay. We focus in a specific type of delay called firing memory and it effects in the dynamics of symmetric (non-directed) conjunctive networks. We find, in the caseis in which the implementation of the delay is not uniform, that all the complexity of the dynamics is somehow encapsulated in the component in which the delay has effect. Thus, we show, in the homogeneous case, that it is possible to exhibit attractors of non-polynomial period. In addition, we study the prediction problem consisting in, given an initial condition, determinate if a fixed coordinate will eventually change its state. We find again that in the non-homogeneous case all the complexity is determined by the component that is affected by the delay and we conclude in the homogeneous case that this problem is PSPACE-complete.
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Jerez-Hanckes, C., Pettersson, I., & Rybalko, V. (2020). Derivation Of Cable Equation By Multiscale Analysis For A Model Of Myelinated Axons. Discrete Contin. Dyn. Syst.-Ser. B, 25(3), 815–839.
Abstract: We derive a one-dimensional cable model for the electric potential propagation along an axon. Since the typical thickness of an axon is much smaller than its length, and the myelin sheath is distributed periodically along the neuron, we simplify the problem geometry to a thin cylinder with alternating myelinated and unmyelinated parts. Both the microstructure period and the cylinder thickness are assumed to be of order epsilon, a small positive parameter. Assuming a nonzero conductivity of the myelin sheath, we find a critical scaling with respect to epsilon which leads to the appearance of an additional potential in the homogenized nonlinear cable equation. This potential contains information about the geometry of the myelin sheath in the original three-dimensional model.
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