Abstract
Integrated information theory (IIT) provides a mathematical framework to characterize the cause-effect structure of a physical system and its amount of integrated information (). An accompanying Python software package (“PyPhi”) was recently introduced to implement this framework for the causal analysis of discrete dynamical systems of binary elements. Here, we present an update to PyPhi that extends its applicability to systems constituted of discrete, but multi-valued elements. This allows us to analyze and compare general causal properties of random networks made up of binary, ternary, quaternary, and mixed nodes. Moreover, we apply the developed tools for causal analysis to a simple non-binary regulatory network model (p53-Mdm2) and discuss commonly used binarization methods in light of their capacity to preserve the causal structure of the original system with multi-valued elements.
Highlights
Discrete models of biological systems often rely exclusively on binary, or “Boolean”variables with two functional states (“active/inactive”, “present/absent”, or “firing/not firing”)
While we found a correlation between the Φ values of the original non-binary system and its Fauré–Kaji binarization in all tested samples, the variability is quite large and non-binary systems with Φ = 0 may map onto Boolean systems with Φ > 0 and vice versa
We have introduced an extension of information theory (IIT)’s PyPhi toolbox for causal analysis [31] to discrete dynamical systems that are constituted of multi-valued elements
Summary
Discrete models of biological systems often rely exclusively on binary, or “Boolean”variables with two functional states (“active/inactive”, “present/absent”, or “firing/not firing”). In some situations, two functional states are insufficient for capturing an element’s behavior adequately, for instance, when an element specifies various effects, depending on different levels of activity [4,5]. This is the case in neuroscience, where neurons, in their simplest representation, can be viewed as logical elements that either fire or not [6]. Information between neurons (or groups of neurons) may be conveyed based on different modes of firing, which requires models of neural networks with multiple functional states per element (e.g., 0: low firing, 1: high firing, 2: bursting) [7,8,9,10]
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