Dynamical regimes and learning properties of evolved Boolean networks
Introduction
Models of neural networks can be roughly divided into two classes, i.e., those where there is a directional flow of activation, such as feed-forward layered networks [1], and those which are truly dynamical systems, like, for example, those proposed by Elman [2] and by Hopfield [3].
This is particularly clear in the case of the Boolean Hopfield model, whose attractors are fixed points (in the usual case with symmetric synaptic weights). Another well-known Boolean network model is that of Random Boolean Networks (briefly, RBNs), which display a much richer dynamics than that of the symmetric Hopfield model. The attractors of finite RBNs are cycles, so the dynamics are fairly trivial; however, it has been possible to introduce a notion of ordered vs. disordered attractors, which represents the analogue (in a finite discrete system) of the distinction between regular and chaotic attractors in continuous systems.
Interestingly, this distinction holds for many properties usually associated to continuous chaotic systems, as, for example, the stability of dynamical attractors with respect to small perturbations: in the case of ordered systems, small perturbations usually die out, while in disordered ones they tend to grow. In RBNs it has been observed that ordered systems usually have fairly regular basins on attraction, so that two nearby states often evolve to the same attractor, while in disordered systems they often go to different attractors. This behaviour is reminiscent of the “butterfly effect” and this provides a reason why disordered RBNs are often called “chaotic” (in spite of the fact that, since the attractors are cycles, the term “pseudo-chaotic” would be more appropriate). The reason for this choice of terms is the following: a deterministic discrete system composed by a finite number of nodes N, each node taking one of M possible values, owns a finite number of different states, and, evidently, sooner or later reaches an already visited state: from that moment on the system starts to repeat the same sequence of states. Nevertheless, the period of a cycle can range from 1 to MN, and for large systems the maximum value is so high that such a cycle could be covered only in a period of time greater than the age of the universe. For any purposes, a system owning cycles that long is called “pseudo-chaotic”, or simply “chaotic” [4].
It turns out that the value of the so-called Derrida parameter [4], which is the discrete analogue of the Lyapunov exponent of continuous dynamical systems, determines whether a given family of RBNs tends to display ordered or chaotic behaviours— corresponding to ordered networks and to chaotic ones. Particular interest has been raised by those networks which are in a critical state with equal to (or close to) 1 , i.e., an intermediate state between order and chaos.
It has been proposed in the past that biological systems should operate in critical states (or close to the boundary between ordered and disordered regions, slightly into the ordered region), on the basis of heuristic arguments which can be summarised as follows. Biological systems need a certain level of stability, in order not to be disrupted by fluctuations which can take place either in the system or in the environment, and they need at the same time to provide flexible responses to changes in the environment. While a chaotic system would be poor at satisfying the first need, a system deeply in the ordered region would fail to meet the second requirement. A critical system should allow for an optimal trade-off between the two, therefore natural evolution should drive biological systems towards critical states [4].
These very same reasons should hold as well for an artificial learning system. A question may arise as to what are the conditions for critical networks to outperform ordered and chaotic ones. A sound, theory-based approach to the design of effective learning networks could try to answer this question. Nevertheless, such a theory is still missing. In this work, we make a first step towards it: we investigate whether and under what conditions the network's dynamical regime influences the learning performance, as long as static environments (i.e., not changing in time) are concerned. Future work will be aimed at investigating the case of changing environments.
Another reason of interest on learning Boolean networks comes from progress in optimisation methods. Note that there have been some attempts in the past to devise learning algorithms for RBNs, which have met limited success [5], [6]. However, recent advances in the development of effective metaheuristics offer new tools to tackle the problem of devising RBNs which are able to perform well in difficult tasks through learning by examples. In this paper, we propose a principled approach for training Boolean networks and we show effective performance in some selected tasks.
This work is structured as follows: Section 2 provides a brief summary of the main concepts related to Boolean networks; Section 3 illustrates the optimisation method we apply to train our networks, whereas 4 Target state-controlled Boolean networks, 5 Density classification problem describe the two applications we focus on, namely, the problem of controlling the networks' trajectory to reach a target and the Density Classification Problem. In Section 6 we draw our conclusions and indicate some promising directions for future work.
Section snippets
Boolean networks
Boolean networks (BNs) have been introduced by Kauffman [7], [4] as a genetic regulatory network model. BNs have been proven to reproduce very important phenomena in genetics and they have also received considerable attention in the research communities on complex systems [8], [4]. A BN is a discrete-state and discrete-time dynamical system whose structure is defined by a directed graph of N nodes, each associated to a Boolean variable xi, i=1, …, N, and a Boolean function ), where
Training Boolean networks by metaheuristics
BNs have been mainly considered as genetic regulatory network models, enabling researchers to achieve prominent results in the field of complex systems biology [19], [20], [12], [9]. Nevertheless, in spite of their similarities with neural networks, their potential as learning systems has not yet been fully investigated and exploited. In this section, we first summarise the works in the literature that concern training or automatic designing BNs (Section 3.1) and then, in Section 3.2, we
Target state-controlled Boolean networks
In this section, we describe the experiments in which we train a BN in such a way that some requirements on its trajectory are fulfilled. The problem of designing a dynamical system such that its trajectory in the state space satisfies specific constraints is a typical control problem. In the case of BNs, which in the general case exhibit complex dynamics, this task is not trivial for an automatic procedure because an assignment of Boolean functions must be found such that the resulting BN
Density classification problem
The Density Classification Problem (DCP), also known as Density Classification Task, first introduced by Packard, is a simple counting problem [44] born within the area of cellular automata (CA), as paradigmatic example of a problem hardly solvable for decentralised systems. Informally, it requires that a binary CA (or more generally a discrete dynamical system—DDS) recognise whether an initial binary string contains more 0 s or more 1 s. In its original formulation, the nodes (or cells) are
Conclusions and future work
BNs have been mainly considered as genetic regulatory network models and are the subject of notable works in the complex systems biology literature. Nevertheless, in spite of their similarities with neural networks, their potential as learning systems has not yet been fully investigated and exploited.
In this work we use BNs as flexible objects, which can evolve by means of suitable optimisation processes, to deal with two notable issues: the problem of controlling the BN's trajectory to reach a
Stefano Benedettini is a Ph.D. student at Alma Mater Studiorum Università di Bologna (Italy) where he received his Bachelor and Master Degrees in Computer Science Engineering (2005). His main current research interests include application of metaheuristic algorithms to bioinformatics problems, complex systems and software engineering.
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Stefano Benedettini is a Ph.D. student at Alma Mater Studiorum Università di Bologna (Italy) where he received his Bachelor and Master Degrees in Computer Science Engineering (2005). His main current research interests include application of metaheuristic algorithms to bioinformatics problems, complex systems and software engineering.
Andrea Roli received the Ph.D. degree in Computer Science and Electronic Engineering from Alma Mater Studiorum Università di Bologna, where he currently is a assistant professor. He teaches subjects in artificial intelligence, complex systems and computer science basics. His main current research interests include metaheuristics and complex systems, with applications to swarm intelligence, bioinformatics and genetic regulatory network models. Andrea Roli is a member of the steering committee of the Italian Association for Artificial Intelligence (AI⁎IA).
Mattia Manfroni is a software engineer and collaborates with Alma Mater Studiorum Università di Bologna (Italy) as a free researcher since September 2010. He received his Bachelor and Master degrees in Computer Science Engineering, both from Università di Bologna, in January 2008 and July 2010, respectively. His research interests are in artificial intelligence and swarm robotics.
Marco Villani received his honours degree in Physics in 1992; in 1993 his thesis obtained the “Franco Viaggi” awards of the SFI (the Italian Physic Society). He worked at ENEA and then at the Environmental Research Centre of Montacatini S.p.A (Montedison Group); from 2005 is a researcher and professor in Engineering and Computer Science at the University of Modena and Reggio Emilia, where he also leads the Modelling and Simulation Laboratory. Winner of the best paper awards at ACRI2006 and ECCS2010, the two major European conferences, respectively, on cellular automata and complex systems, Villani applies complex systems concepts in areas that require strong interdisciplinary interactions.
Roberto Serra graduated in Physics at Alma Mater Studiorum Università di Bologna, and later performed research activities in the industrial groups Eni and Montedison, where he served as a director of the Environmental Research Centre until 2003. Since 2004 he is a full professor of Computer Science and Engineering at the Modena and Reggio Emilia University. His research interests concern several aspects of the dynamics of complex systems, paying particular attention to biological and social systems, and to the dynamical approach to Artificial Intelligence. He published more than 130 papers in international journals and refereed conference proceedings, and is a co-author of four books. He has been responsible of several research projects, funded by companies, by the Italian Ministry for Scientific Research (Miur), by the National Research Council (CNR) and by the European Union. He has served as a member of the program committee of several international conferences, and has delivered various invited talks and seminars. He recently chaired two international conferences, one on Artificial Life and Evolutionary Computation (Venice, 2008) and one on Artificial Intelligence (Reggio Emilia, 2009). Roberto Serra has also been president of AI⁎IA (Associazione Italiana per l'Intelligenza Artificiale) and is currently chairman of the Science Board of the European Centre for Living Technologies.
Antonio Gagliardi started out as an analyst and programmer for TELECOMSpA and TecnostMael SpA Olivetti Group. In 1998 he began his activities as a lecturer for computer courses in public and private education; he worked for Modena Formazione on courses funded by the European Community (2001–2003). In 2007 he graduated in Hypermedia Communications at Parma University, followed by a post-graduate degree in Economics and Complex Systems at the University of Modena and Reggio Emilia. Currently he works as a lab technician and collaborates with the University of Modena and Reggio Emilia. His research interests are in machine learning and Boolean networks.
Carlo Pinciroli is a Ph.D. student at IRIDIA, CoDE, Université Libre de Bruxelles in Belgium. Before joining IRIDIA, in 2005 he obtained a Master's degree in Computer Engineering at Politecnico di Milano, Milan, Italy and a second Master's degree in Computer Science at University of Illinois at Chicago, IL, USA. In 2007 he also obtained a Diplôme d'études approfondies from the Université Libre de Bruxelles. The focus of his research is computer simulation and swarm robotics.
Mauro Birattari received his Master's degree in Electronic Engineering from Politecnico di Milano, Italy, in 1997; and his Doctoral degree in Information Technologies from Université Libre de Bruxelles, Belgium, in 2004. He is currently with IRIDIA, Université Libre de Bruxelles, as a research associate of the fund for scientific research F.R.S.-FNRS of Belgium's French Community. Dr. Birattari co-authored about 100 peer-reviewed scientific publications in the field of computational intelligence. Dr. Birattari is an associate editor for the journal Swarm Intelligence and an area editor for the journal Computers & Industrial Engineering.
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Mauro Birattari acknowledges support from the F.R.S.-FNRS of Belgium's Wallonia-Brussels Federation.