Volatility of Boolean functions

Journal of Theoretical Biology, 2010

The asymptotic dynamics of random Boolean networks subject to random fluctuations is investigated. Under the influence of noise, the system can escape from the attractors of the deterministic model, and a thorough study of these transitions is presented. We show that the dynamics is more properly described by sets of attractors rather than single ones. We generalize here a previous notion of ergodic sets, and we show that the Threshold Ergodic Sets so defined are robust with respect to noise and, at the same time, that they do not suffer from a major drawback of ergodic sets. The system jumps from one attractor to another of the same Threshold Ergodic Set under the influence of noise, never leaving it. By interpreting random Boolean networks as models of genetic regulatory networks, we also propose to associate cell types to Threshold Ergodic Sets rather than to deterministic attractors or to ergodic sets, as it had been previously suggested. We also propose to associate cell differentiation to the process whereby a Threshold Ergodic Set composed by several attractors gives rise to another one composed by a smaller number of attractors. We show that this approach accounts for several interesting experimental facts about cell differentiation, including the possibility to obtain an induced pluripotent stem cell from a fully differentiated one by overexpressing some of its genes.

Physical review. E, 2016

We present a characterization of short-term stability of Kauffman's NK (random) Boolean networks under arbitrary distributions of transfer functions. Given such a Boolean network where each transfer function is drawn from the same distribution, we present a formula that determines whether short-term chaos (damage spreading) will happen. Our main technical tool which enables the formal proof of this formula is the Fourier analysis of Boolean functions, which describes such functions as multilinear polynomials over the inputs. Numerical simulations on mixtures of threshold functions and nested canalyzing functions demonstrate the formula's correctness.

Physica A: Statistical Mechanics and its Applications, 2000

A new order parameter approximation to random boolean networks (RBN) is introduced, based on the concept of Boolean derivative. A statistical argument involving an annealed approximation is used, allowing to measure the order parameter in terms of the statistical properties of a random matrix. Using the same formalism, a Lyapunov exponent is calculated, allowing to provide the onset of damage spreading through the network and how sensitive it is to minimal perturbations. Finally, the Lyapunov exponents are obtained by means of di erent approximations: through distance method and a discrete variant of the Wolf's method for continuous systems.

Physica A-statistical Mechanics and Its Applications, 2004

In the classical model of Random Boolean Networks (RBN) the number of incoming connections is the same for every node, while the distribution of outgoing links is Poissonian. These RBN are known to display two major dynamical behaviours, depending upon the value of some model parameters: an "ordered" and a "chaotic" regime. We introduce a modiÿcation of the classical way of building a RBN, which maintains the property that all the nodes have the same number of incoming links, but which gives rise to a scale-free distribution of outgoing connections. Because of this modiÿcation, the dynamical properties are deeply modiÿed: the number of attractors is much smaller than in classical RBN, their length and the duration of the transients are shorter. Moreover, the number of di erent attractors is almost independent of the network size, over almost three orders of magnitudes (while in classical RBN this number grows with the size of the network). These results are based upon a detailed study of networks where each node has two input connections. A limited study of networks with three input connections per node shows that also in this case the number of attractors is almost independent of the network size.

Random boolean networks are a model of genetic regulatory networks that has proven able to describe experimental data in biology. They not only reproduce important phenomena in cell dynamics, but they are also extremely interesting from a theoretical viewpoint, since it is possible to tune their asymptotic behaviour from order to disorder. The usual approach characterizes network families as a whole, either by means of static or dynamic measures. We show here that a more detailed study, based on the properties of system's attractors, can provide information that makes it possible to predict with higher precision important properties, such as system's response to gene knock-out. A new set of principled measures is introduced, that explains some puzzling behaviours of these networks. These results are not limited to random Boolean network models, but they are general and hold for any discrete model exhibiting similar dynamical characteristics.

2008

Abstract—Boolean networks are considered generic models for a large class of asymmetric networks, such as neural networks or gene regulatory networks. Previous work has shown that such networks undergo a phase transition from an ordered to a'chaotic'phase as ...

Physical Review E, 2004

This paper considers a simple Boolean network with N nodes, each node's state at time t being determined by a certain number of parent nodes, which may vary from one node to another. This is an extension of a model studied by Andrecut and Ali [Int. J. Mod. Phys. B 15, 17 (2001)], who consider the same number of parents for all nodes. We make use of the same Boolean rule as Andrecut and Ali, provide a generalization of the formula for the probability of finding a node in state 1 at a time t, and use simulation methods to generate consecutive states of the network for both the real system and the model. The results match well. We study the dynamics of the model through sensitivity of the orbits to initial values, bifurcation diagrams, and fixed point analysis. We show that the route to chaos is due to a cascade of period-doubling bifurcations which turn into reversed (period-halving) bifurcations for certain combinations of parameter values.

2018

We consider continuous-time Markov chains on integers which allow transitions to adjacent states only, with alternating rates. We give explicit formulas for probability generating functions, and also for means, variances and state probabilities of the random variables of the process. Moreover we study independent random time-changes with the inverse of the stable subordinator, the stable subordinator and the tempered stable subodinator. We also present some asymptotic results in the fashion of large deviations. These results give some generalizations of those presented in Di Crescenzo A., Macci C., Martinucci B. (2014).

International Journal of Modern Physics B, 1998

We introduce a measure of complexity in terms of the average number of bits per time unit necessary to specify the sequence generated by the system. In random dynamical system, this indicator coincides with the rate K of divergence of nearby trajectories evolving under two different noise realizations. The meaning of K is discussed in the context of the information theory, and it is shown that it can be determined from real experimental data. In presence of strong dynamical intermittency, the value of K is very different from the standard Lyapunov exponent λ σ computed considering two nearby trajectories evolving under the same randomness. However, the former is much more relevant than the latter from a physical point of view as illustrated by some numerical computations for noisy maps and sandpile models.

Perspectives and Problems in Nolinear Science, 2003

This paper reviews a class of generic dissipative dynamical systems called N -K models. In these models, the dynamics of N elements, defined as Boolean variables, develop step by step, clocked by a discrete time variable. Each of the N Boolean elements at a given time is given a value which depends upon K elements in the previous time step. We review the work of many authors on the behavior of the models, looking particularly at the structure and lengths of their cycles, the sizes of their basins of attraction, and the flow of information through the systems. In the limit of infinite N , there is a phase transition between a chaotic and an ordered phase, with a critical phase in between. We argue that the behavior of this system depends significantly on the topology of the network connections. If the elements are placed upon a lattice with dimension d, the system shows correlations related to the standard percolation or directed percolation phase transition on such a lattice. On the other hand, a very different behavior is seen in the Kauffman net in which all spins are equally likely to be coupled to a given spin. In this situation, coupling loops are mostly suppressed, and the behavior of the system is much more like that of a mean field theory. We also describe possible applications of the models to, for example, genetic networks, cell differentiation, evolution, democracy in social systems and neural networks. Once these quantities have been specified, equation (1.1) fully determines the dynamics of the system. In the most general case, the connectivities K i may vary from one element to another. However, throughout this work we will consider only the case in which the connectivity is the same for all the nodes: K i = K, i = 1, 2, . . . , N . In doing so, it is possible to talk about the connectivity K of the whole system, which is an integer parameter by definition. It is worth mentioning though that when K i varies from one element to another, the important parameter is the mean connectivity of the system, K , defined as In this way, the mean connectivity might acquire non-integer values. Scalefree networks ; Albert and ), which have a very broad (power-law) distribution of K i , can also be defined and characterized. Of fundamental importance is the way the linkages are assigned to the elements, as the dynamics of the system both qualitatively and quantitatively depend strongly on this assignment. Throughout this paper, we distinguish between two different kinds of assignment: In a lattice assignment all the bonds are arranged on some regular lattice. For example, the K control elements σ j1(i) , σ j2(i) , . . . , σ j K (i) may be picked from among the 2d nearest neighbors on a d dimensional hyper-cubic lattice. Alternatively, in a uniform assignment each and every element has an equal chance of appearing in this list. We shall call a Boolean system with such a uniform assignment a Kauffman net. (See Figure .1.) Of course, intermediate cases are possible, for example, one may consider systems with some linkages to far-away elements and others to neighboring elements. Small-world networks ) are of this type. For convenience, we will denote the whole set of Boolean elements {σ 1 (t), σ 2 (t), . . . , σ N (t)} by the symbol Σ t : (1.2) Σ t represents then the state of the system at time t. We can think of Σ t as an integer number which is the base-10 representation of the binary chain {σ 1 (t), σ 2 (t), . . . , σ N (t)}. Since every variable σ i has only two possible values, 0 and 1, the number of all the possible configurations is Ω = 2 N , so that Σ t can be thought of as an integer satisfying 0 ≤ Σ t < 2 N . This collection of integers is the base-10 representation of the state space of the Version Jan.

Physical Review Letters, 2012

Boolean networks, widely used to model gene regulation, exhibit a phase transition between regimes in which small perturbations either die out or grow exponentially. We show and numerically verify that this phase transition in the dynamics can be mapped onto a static percolation problem which predicts the long-time average Hamming distance between perturbed and unperturbed orbits.

Journal of Physics: Conference Series, 2010

Random Boolean formulae, generated by a growth process of noisy logical gates are analyzed using the generating functional methodology of statistical physics. We study the type of functions generated for different input distributions, their robustness for a given level of gate error and its dependence on the formulae depth and complexity and the gates used. Bounds on their performance, derived in the information theory literature for specific gates, are straightforwardly retrieved, generalized and identified as the corresponding typical-case phase transitions. Results for error-rates, function-depth and sensitivity of the generated functions are obtained for various gate-type and noise models.

Acta Applicandae Mathematicae, 2007

1997

Boolean functions are discrete-valued functions that provide iterative models in different areas of science. They appear in theoretical computer science as automata networks, cellular, and threshold automata [2–4], and also in biomathematics as input-output functions of perceptrons and Hopfield networks [1, 5, 10]. We can also find them in physics as discrete models for disordered matter such as the spin glass problem [6]. In this paper we consider a class of n-valued boolean functions, designated hard-threshold, that represent the activity of discrete recurrent neural network (NN) models. RecurrentNNs are mathematical models consisting of a large number of parallel-operating and interconnected processing units. Information is encoded as a binary sequence (designated state vector) and is distributed among the units composing the network. This permits a high speed more economical machine implementation and avoids the expensive maintenance of a central database unit [12]. The activity...

Applications of Mathematics

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