On the conversion of multivalued to Boolean dynamics

Abstract

Results and tools on discrete interaction networks are often concerned with Boolean variables, whereas considering more than two levels is sometimes useful. Multivalued networks can be converted to partial Boolean maps, in a way that preserves the asynchronous dynamics. We investigate the problem of extending these maps to non-admissible states, i.e. states that do not have a multivalued counterpart. We observe that attractors are preserved if a stepwise version of the original function is considered for conversion. Different extensions of the Boolean conversion affect the structure of the interaction graphs in different ways. A particular technique for extending the partial Boolean conversion is identified, that ensures that feedback cycles are preserved. This property, combined with the conservation of the asymptotic behaviour, can prove useful for the application of results and analyses defined in the Boolean setting to multivalued networks, and vice versa. As a first application, by considering the conversion of a known example for the discrete multivalued case, we create a Boolean map showing that the existence of a cyclic attractor and the absence of fixed points are compatible with the absence of local negative cycles. We then state a multivalued version of a result connecting mirror states and local feedback cycles.

Computing Integrated Information (Φ) in Discrete Dynamical Systems with Multi-Valued Elements

“…We chose this particular example, since discrete network models, as well as the binarization of models with multi-valued elements, are common tools in the study of biological regulatory networks. In particular, we will compare the original model to its Boolean versions that were obtained by means of different binarization methods [ 11 , 12 , 13 ].…”

“…Boolean states that do not have a counterpart in the original model with multi-valued elements are considered to be “non-admissible” under Van Ham’s one-to-one mapping [ 12 ]. This is problematic, as many tools and results concerning Boolean networks, including IIT’s causal analysis, require fully specified transition probability matrices [ 12 , 13 , 14 ].…”

“…Because the majority of tools and theorems available for the analysis of logical networks are restricted to the Boolean case, binarization methods for converting systems with multi-valued elements into Boolean models have been developed as a way to extend the utility of the available methods [ 5 , 11 , 12 , 13 ]. However, these binarization approaches mainly focus on maintaining a model’s asymptotic dynamics, rather than preserving the causal structure of the original non-binary system.…”

Local Negative Circuits and Cyclic Attractors in Boolean Networks with at most Five Components

“…Boolean networks with a cyclic attractor and no local negative circuits were presented in [12], with a method to create maps with antipodal attractive cycles and no local negative circuits, for n ≥ 7. Tonello [17] and Fauré and Kaji [3] exhibited maps with cyclic attractors and no local negative circuits, for n = 6. Maps with antipodal attractive cycles and no local negative circuits also exist for n = 6; a procedure that extends the one in [12] is presented, for completeness, in appendix A.…”

“…Ruet [12] exhibited a procedure to create counterexamples in the Boolean case, for every n ≥ 7, n being the number of variables; these are maps admitting an antipodal attractive cycle and no local negative circuits in the regulatory graph. Tonello [17] and Fauré and Kaji [3] identified different Boolean versions of Richard's discrete example, that provide counterexamples to Question 1 for n = 6. A map with an antipodal attractive cycle and no local regulatory circuits also exists for n = 6 (we present such a map in appendix A).…”