An Algebraic Algorithm for the Identification of Glass Networks with Periodic Orbits Along Cyclic Attractors

Abstract

Glass piecewise linear ODE models are frequently used for simulation of neural and gene regulatory networks. Efficient computational tools for automatic synthesis of such models are highly desirable. However, the existing algorithms for the identification of desired models are limited to four-dimensional networks, and rely on numerical solutions of eigenvalue problems. We suggest a novel algebraic criterion to detect the type of the phase flow along network cyclic attractors that is based on a corollary of the Perron-Frobenius theorem. We show an application of the criterion to the analysis of bifurcations in the networks. We propose to encode the identification of models with periodic orbits along cyclic attractors as a propositional formula, and solving it using stateof-the-art SAT-based tools for real linear arithmetic. New lower bounds for the number of equivalence classes are calculated for cyclic attractors in six-dimensional networks. Experimental results indicate that the runtime of our algorithm increases slower than the size of the search space of the problem.

Computing Binary Combinatorial Gray Codes Via Exhaustive Search With SAT Solvers

“…The lower bounds for the number of the classes in dimension 6 has been obtained in [16]. The presented classification algorithm is a modified ALL-SAT procedure as described in [16].…”

Finding Lean Induced Cycles in Binary Hypercubes

“…The classification of induced cycles with respect to symmetries of a hypercube is of interest in Glass models for neural and gene regulatory networks, because the number of the equivalence classes of the codes indicates how many different types of cells can be regulated by a set of genes [21,10].…”

Periodic Orbits and Equilibria in Glass Models for Gene Regulatory Networks

“…This paper is an extension of two conference papers [1], [2]. A part of this work was presented at the 7 th Australia-New Zealand Mathematics Convention, Christchurch, New Zealand, December 11, 2008 where x i denotes the concentration of the product of gene i.…”

“…The results have been summarized as conjectures that link the equilibria and the periodic behavior of autonomous ODE to inhibiting and activating patterns of wiring schemes [20]. 1 Every edge in the graph that joins two vertices of the cycle is an edge of this cycle.…”

“…,ỹ n )| O k . Thus, the Glass network can be fully specified by a choice of focal points f (1) , . .…”