Minimum Equivalence in Random Boolean Networks, Elementary Cellular Automata, and Beyond

Although deep learning has recently increased in popularity, it suffers from various problems including high computational complexity, energy greedy computation, and lack of scalability, to mention a few. In this paper, we investigate an alternative brain-inspired method for data analysis that circumvents the deep learning drawbacks by taking the actual dynamical behavior of biological neural networks into account. For this purpose, we develop a general framework for dynamical systems that can evolve and model a variety of substrates that possess computational capacity. Therefore, dynamical systems can be exploited in the reservoir computing paradigm, i.e., an untrained recurrent nonlinear network with a trained linear readout layer. Moreover, our general framework, called EvoDynamic, is based on an optimized deep neural network library. Hence, generalization and performance can be balanced. The EvoDynamic framework contains three kinds of dynamical systems already implemented, namely cellular automata, random Boolean networks, and echo state networks. The evolution of such systems towards a dynamical behavior, called criticality, is investigated because systems with such behavior may be better suited to do useful computation. The implemented dynamical systems are stochastic and their evolution with genetic algorithm mutates their update rules or network initialization. The obtained results are promising and demonstrate that criticality is achieved. In addition to the presented results, our framework can also be utilized to evolve the dynamical systems connectivity, update and learning rules to improve the quality of the reservoir used for solving computational tasks and physical substrate modeling.

Random Boolean networks have been proposed as a basic model of genetic regulatory networks for more than four decades. Attractors have been considered as the best way to represent the long-term behaviors of random Boolean networks. Most studies on attractors are made with random topologies. However, the real regulatory networks have been found to be modular or more complex topologies. In this work, we extend classical robustness and entropy analysis of random Boolean networks to random modular Boolean networks. We firstly focus on the robustness of the attractor to perturbations with different parameters. Then, we investigate and calculate how the amount of information propagated between the nodes when on an attractor, as quantified by the average pairwise mutual information. The results can be used to study the capability of genetic information propagation of different types of genetic regulatory networks.

A genetic regulatory network (GRN) is a dynamic system to describe interactions among a large number of different substances in a living biological cell. There have been numerous attempts to model the dynamical behaviour of genetic regulatory networks. Boolean models are a class of discrete models for modelling genetic regulatory networks, which are conditioned on the premise that genes interact with each other through Boolean logic. As mentioned previously, Boolean models include Boolean Networks, Instantaneously Random Probabilistic Boolean Networks and Context-Sensitive Boolean Networks. There are a large number of issues surrounding Boolean models, including Boolean Networks and Probabilistic Boolean Networks. The model inference is the procedure of determining whether the modelled networks are consistent with the given data sample, and choosing one model, from among many, that makes more sense for the data. This paper has reviewed the research into modelling gene regulatory networks, especially upon Boolean models.

This paper explores the concept of isomorphism in cellular automata (CAs), focusing on identifying and understanding isomorphic relationships between distinct CAs. A cellular automaton (CA) is said to be isomorphic to another CA if the configuration transition diagrams of the two CAs are identical. To illustrate how isomorphism entails the preservation of structural properties while allowing for variations in state representations, we exhaustively explore the isomorphic reversible rules in the first degree CAs (FDCAs), that is, the simple three-neighborhood CAs represented by eight parameters. The experimental study includes reversible FDCAs for states two to 10 with cell lengths of four to seven under the null boundary condition and states two to seven with cell lengths of four to seven under the periodic boundary condition. Given a CA, we also provide two synthesis methods for identifying the set of CAs that are isomorphic to it. The first method operates by permuting the states of a group of CA cell locations, whereas the second method involves exchanging cells at various points. We demonstrate that, depending on the range of exchanged cell positions, the isomorphic CAs produced in this way may have a bigger neighborhood than the input CA. Finally, we provide an exhaustive classification of the elementary CAs (ECAs) using the first synthesis method and compare it with existing results.

We propose quantum Boolean networks, which can be classified as deterministic reversible asynchronous Boolean networks. This model is based on the previously developed concept of quantum Boolean functions. A quantum Boolean network is a Boolean network where the functions associated with the nodes are quantum Boolean functions. We study some properties of this novel model and, using a quantum simulator, we study how the dynamics change in function of connectivity of the network and the set of operators we allow. For some configurations, this model resembles the behavior of reversible Boolean networks, while for other configurations a more complex dynamic can emerge. For example, cycles larger than 2N were observed. Additionally, using a scheme akin to one used previously with random Boolean networks, we computed the average entropy and complexity of the networks. As opposed to classic random Boolean networks, where “complex” dynamics are restricted mainly to a connectivity close to a phase transition, quantum Boolean networks can exhibit stable, complex, and unstable dynamics independently of their connectivity.

We propose quantum Boolean networks, which can be classified as deterministic reversible asynchronous Boolean networks. This model is based on the previously developed concept of quantum Boolean functions. A quantum Boolean network is a Boolean network where the functions associated with the nodes are quantum Boolean functions. We study some properties of this novel model and, using a quantum simulator, we study how the dynamics change in function of connectivity of the network and the set of operators we allow. For some configurations, this model resembles the behavior of reversible Boolean networks, while for other configurations a more complex dynamic can emerge. For example, cycles larger than 2N were observed. Additionally, using a scheme akin to one used previously with random Boolean networks, we computed the average entropy and complexity of the networks. As opposed to classic random Boolean networks, where “complex” dynamics are restricted mainly to a connectivity close to a phase transition, quantum Boolean networks can exhibit stable, complex, and unstable dynamics independently of their connectivity.

Mutual information between the time series of two dynamical elements measures how well their activities are coordinated. In a network of interacting elements, the average mutual information over all pairs of elements I is a global measure of the correlation between the elements' dynamics. Local topological features in the network have been shown to affect I . Here we define a generalized clustering coefficient C_{p} and show that this quantity captures the effects of local structures on the global dynamics of networks. Using random Boolean networks (RBNs) as models of networks of interacting elements, we show that the variation of I ( I averaged over an ensemble of RBNs with the number of nodes N and average connectivity k ) with N and k is caused by the variation of C_{p} . Also, the variability of I between RBNs with equal N and k is due to their distinct values of C_{p} . Consequently, we propose a rewiring method to generate ensembles of BNs, from ordinary RBNs, with fixed values of C_{p} up to order 5, while maintaining in- and out-degree distributions. Using this methodology, the dependency of C_{p} on N and k and the variability of I for RBNs with equal N and k are shown to disappear in RBNs with C_{p} set to zero. The I of ensembles of RBNs with fixed, nonzero C_{p} values, also becomes almost independent of N and k . In addition, it is shown that C_{p} exhibits a power-law dependence on N in ordinary RBNs, suggesting that the C_{p} affects even relatively large networks. The method of generating networks with fixed C_{p} values is useful to generate networks with small N whose dynamics have the same properties as those of large scale networks, or to generate ensembles of networks with the same C_{p} as some specific network, and thus comparable dynamics. These results show how a system's dynamics is constrained by its local structure, suggesting that the local topology of biological networks might be shaped by selection, for example, towards optimizing the coordination between its components.

Cellular Automata (CA) have a long history being employed as pseudo-random number generators (PRNG), especially for cryptographic applications such as keystream generation in stream ciphers. Initially starting from the study of rule 30 of elementary CA, multiple rules where the objects of investigation and were shown to be able to pass most of the rigorous statistical tests used to assess the quality of PRNG. In all cases, the CA employed where of the classical, synchronous kind. This assumes a global clock regulating all CA updates which can be a weakness if an attacker is able to tamper it. Here we study how much asynchrony is necessary to make a CA-based PRNG ineffective. We have found that elementary CA are subdivided into three class: (1) there is a “state transition” where, after a certain level of asynchrony, the CA loses the ability to generate strong random sequences, (2) the randomness of the sequences increases with a limited level of asynchrony, or (3) CA normally unable to be used as PRNG exhibit a much stronger ability to generate random sequences when asynchrony is introduced.

The phase transition between frozen and chaotic behavior in Kauffman's cellular automata on a nearest neighbor square lattice does not agree with the percolation threshold of the forcing functions.

Genetic network models and statistical properties of gene expression data in knock-out experiments

We propose a method of constructing differential equations directly from cellular automata. Any elementary cellular automata (ECAs) introduced by Wolfram can be transformed into a partial differential equation (PDE) which preserves the time evolution patterns of the ECAs. In particular, some PDEs constructed by the method reproduce complex fractal and self-organizing patterns found in their corresponding ECAs. It is expected that the method can be extended for general cellular automata with more complex rules and/or in higher dimensions.

Evolution-in-Materio (EIM) is a method of using artificial evolution to exploit physical properties of materials for computation. It has previously been successfully used to evolve a multitude of different computational devices implemented in physical materials. One of the main problems has been the stability of solutions due to physical changes in the substrate materials, e.g. topological or electrical, leading to devices sensitive to the evolutionary history, i.e. instability. This paper presents stable linearly and non-linearly separable logic gates, and elementary cellular automata (CA) transition functions, successfully evolved in single-walled carbon nanotube (SWCNT) and polymer composite materials. The results herein show the feasibility of building stand-alone in-materio devices and outline a novel computational abstraction for EIM based on CA. This work is done within the European Project NASCENCE.

Wherever we see life, we see different kinds of complex networks, reason why they are studied across various fields of science. Random Boolean Networks (RBNs) form a special class in which the links between the nodes and the boolean functions are specified at random. Whereas synchronous RBNs were investigated in detail, there has little been done around their asynchronous counterpart, although there is evidence that most living systems are governed by asynchronous updating. Derrida’s annealed approximation predicts a critical connectivity value of K=2 for synchronous RBNs. We present a similar and original approach for asynchronous RBNs and show that they do not possess such a critical connectivity value. The asynchronous and nondeterministic updating scheme introduces perturbations that reach about 25% of the nodes and thus prevents the networks to become stable. Further, our numerical simulations show that asynchronous RBN tend to amplify small and to reduce big perturbations.

Random Boolean networks (RBNs) have been widely explored as models for understanding complex systems, in particular gene regulatory networks (GRNs). Stability (order) and instability (chaos), critical characteristics of any dynamic system, have naturally been a central focus in RBN research. Several measures have previously been introduced to assess stability, e.g., via damage spreading or via statistical properties of attractors such as their number, lengths or basin sizes. Undoubtedly, network topology plays an important role in shaping the dynamics of RBN. Another key factor influencing the dynamics is the Boolean functions (BFs) assigned to each node. In this work, we conduct a systematic investigation to examine the influence of five different classes of BFs on the dynamics of RBNs. By employing various dynamical stability measures, we show that biologically meaningful BFs consistently drive the dynamics toward the stable regime compared to random BFs. This observation holds across networks with varying size, connectivity, and degree distributions. Additionally, we show that the change in the values of our stability measures with increasing connectivity or size of RBNs is different across the BF classes. For most observables, random BFs change the values significantly, pushing the dynamics toward a more chaotic regime with increasing connectivity or network size. In contrast, biologically meaningful BFs show very minimal fluctuations for most stability measures. These findings emphasize the advantage of restricting to biologically meaningful classes in the reconstruction and modeling of biological systems within Boolean framework.

Image scrambling is the process of converting an image to an unintelligible format, mainly for security reasons. The scrambling is considered as a pre-process or a post-process of security related applications such as watermarking, information hiding, fingerprinting, and encryption. Cellular automata are parallel models of computation that prove an interesting concept where a simple configuration can lead to a complex behavior. Since there are a lot of parameters to configure, cellular automata have many types and these types differ in terms of complexity and behavior. Cellular automata were previously used in scrambling different types of multimedia, but only complex two-dimensional automata were explored. We propose a scheme where the simplest type of cellular automata is used that is the elementary type. We test the scrambling degree for different cellular automata rules that belong to classes three and four of Wolfram's classification which correspond to complex and chaotic behavior; we also check the effect of other parameters such as the number of generations and the boundary condition. Experimental results show that our proposed scheme outperforms other schemes based on cellular automata in terms of scrambling degree.