Abstract This paper considers a simple Boolean network with N nodes, each node’s state at time t being determined by a certain number k of parent nodes, which is fixed for all nodes. The nodes, with randomly assigned neighborhoods, are updated based on various asynchronous schemes. We make use of a Boolean rule that is a generalization of rule 126 of elementary cellular automata. We provide formulae for the probability of finding a node in state 1 at a time t for the class of Asynchronous Random Boolean Networks (ARBN) in which only one node is updated at every time step, and for the class of Generalized ARBNs (GARBN) in which a random number of nodes can be updated at each time point. We use simulation methods to generate consecutive states of the network for both the real system and the models under the various schemes. The results match well. We study the dynamics of the models through sensitivity of the orbits to initial values, bifurcation diagrams, and fixed point analysis. We show, both theoretically and by example, that the ARBNs generate an ordered behavior regardless of the updating scheme used, whereas the GARBNs have behaviors that range from order to chaos depending on the type of random variable used to determine the number of nodes to be updated and the parameter combinations. 1.

Abstract—External control of a genetic regulatory network is used for the purpose of avoiding undesirable states, such as those associated with disease. Heretofore, intervention has focused on finite-horizon control, i.e., control over a small number of stages. This paper considers the design of optimal infinite-horizon control for context-sensitive probabilistic Boolean networks (PBNs). It can also be applied to instantaneously random PBNs. The stationary policy obtained is independent of time and dependent on the current state. This paper concentrates on discounted problems with bounded cost per stage and on average-cost-per-stage problems. These formulations are used to generate stationary policies for a PBN constructed from melanoma gene-expression data. The results show that the stationary policies obtained by the two different formulations are capable of shifting the probability mass of the stationary distribution from undesirable states to desirable ones. Index Terms—Altering steady state, genetic network intervention, infinite-horizon control, optimal control of probabilistic Boolean networks. I.

Abstract. Many complex systems can be modeled as multiagent systems in which the constituent entities (agents) interact with each other. The global dynamics of such a system is determined by the nature of the local interactions among the agents. Since it is difficult to formally analyze complex multiagent systems, they are often studied through computer simulations. While computer simulations can be very useful, results obtained through simulations do not formally validate the observed behavior. Thus, there is a need for a mathematical framework which one can use to represent multiagent systems and formally establish their properties. This work contains a brief exposition of some known mathematical frameworks that can model multiagent systems. The focus is on one such framework, namely that of finite dynamical systems. Both, deterministic and stochastic versions of this framework are discussed. The paper contains a sampling of the mathematical results from the literature to show how finite dynamical systems can be used to carry out a rigorous study of the properties of multiagent systems and it is shown how the framework

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We address the task of inducing explanatory models from observations and knowledge about candidate biological processes, using the illustrative problem of modeling photosynthesis regulation. We cast both models and background knowledge in terms of processes that interact to account for behavior. We also describe IPM, an algorithm for inducing quantitative process models from such input, and we demonstrate its use both on photosynthesis and on a second domain, biochemical kinetics. In closing, we consider the generality of our approach, discuss related research on biological modeling, and suggest directions for future work.

Abstract Context is an essential part of all cognitive function. However, neural network models have only considered this issue in limited ways, focusing primarily on the conditioning of a system’s response by its recent history. This type of context, which we term Type I, is clearly relevant in many situations, but in other cases, the system’s response for an extended period must be conditioned by stimuli encountered at a specific earlier time. For example, the decision to turn left or right at an intersection point in a navigation task depends on the goal set at the beginning of the task. We term this type of context, which sets the “frame of reference ” for an entire episode, Type II context. The prefrontal cortex in mammals has been hypothesized to perform this function, but it has been difficult to incorporate this into neural network models. In the present chapter, we describe an approach called latent attractors that allows self-organizing neural systems to simultaneously incorporate both Type I and Type II context dependency. We demonstrate this by applying the approach to a series of problems requiring one or both types of context. We also argue that the latent attractor approach is a general and flexible method for incorporating multi-scale temporal dependence into neural systems, and possibly other self-organized network models. 2

It is acknowledged that the presence of positive or negative circuits in regulatory networks such as genetic networks is linked to the emergence of significant dynamical properties such as multistability (involved in differentiation) and periodic oscillations (involved in homeostasis). Rules proposed by the biologist R. Thomas assert that these circuits are necessary for such dynamical properties. These rules have been studied by several authors. Their obvious interest is that they relate the rather simple information contained in the structure of the network (signed circuits) to its much more complex dynamical behaviour. We prove in this article a nontrivial converse of these rules, namely that certain positive or negative circuits in a regulatory graph are actually sufficient for the observation of a restricted form of the corresponding dynamical property, differentiation or homeostasis. More precisely, the crucial property that we require is that the circuit be globally minimal. We then apply these results to the vertebrate immune system, and show that the 2 minimal functional positive circuits of the model indeed behave as modules which combine to explain the presence of the 3 stable states corresponding to the Th0, Th1 and Th2 cells.

In conjunction with experimental investigation, appropriate computational tools can substantially help researchers to uncover the mechanism underlying gene regulation and understand gene functionality. Several computational approaches with different levels of modeling detail have been developed to investigate the dynamics of gene networks [1]–[3]. With limited knowledge of network structure and experimental data, a Boolean network provides a coarse model, capable of predicting certain dynamic behavior of a gene network [4], [5]. The approach based on deterministic differential equations (DDEs) provides a fine model [1], [3] but only reflects the deterministic dynamics of a gene network averaged over many cells. The finest stochastic approach [6], [7], based on stochastic kinetics [8]–[10], can capture the stochasticity inherent in gene expression in a single cell. The power of the stochastic model lies in its completeness and attention to detail [2]. Stochasticity in gene expression is mainly due to a series of events that involve a small number of molecules of DNA, RNA, and proteins. As each of these molecular events is subject to significant thermal fluctuations, the amount of mRNA and protein expressed from a gene is a stochastic process, which is called noise by biologists. Although gene expression noise was noticed more than more than four decades ago [11], only recently it received much attention, since recent advances in technology

Dedicated to Avner Friedman, on the occasion of his 75th birthday. Discrete dynamical systems defined on the state space Π = {0, 1,...,p − 1} n have been used in multiple applications, most recently for the modeling of gene and protein networks. In this paper we study to what extent well-known theorems by Smale and Hirsch, which form part of the theory of (continuous) monotone dynamical systems, generalize or fail to do so in the discrete case. We show that that arbitrary m-dimensional systems cannot necessarily be embedded into n-dimensional cooperative systems for n = m + 1, as in the Smale theorem for the continuous case, but we show that this is possible for n = m+2 as long as p is sufficiently large. We also prove that strict cooperativity, a natural weakening of the notion of strong cooperativity, implies nontrivial bounds on the lengths of periodic orbits in discrete systems and imposes a condition akin to Lyapunov stability on all attractors. Finally, we explore several natural candidates for definitions of irreducibility of a discrete system. While some of these notions imply the strict cooperativity of a given cooperative system and impose even tighter bounds on the lengths of periodic orbits than strict cooperativity alone, other plausible definitions allow the existence of exponentially long periodic orbits.

The Boolean network is a mathematical model of biological systems, and has attracted much attention as a qualitative tool for analyzing the regulatory system. The stable states and dynamics of Boolean networks are characterized by their attractors, whose properties have been analyzed computationally, yet not much work has been done from the viewpoint of logical inference systems. In this paper, we show direct translations of Boolean networks into logic programs, and propose new methods to compute their trajectories and attractors based on inference on such logic programs. In particular, point attractors of both synchronous and asynchronous Boolean networks are characterized as supported models of logic programs so that SAT techniques can be applied to compute them. Investigation of these relationships suggests us to view Boolean networks as logic programs and vice versa. 1