CHAOTIC MEAN FIELD DYNAMICS OF A BOOLEAN NETWORK WITH RANDOM CONNECTIVITY

2007 ◽  
Vol 18 (09) ◽  
pp. 1459-1473 ◽  
Author(s):  
MALIACKAL POULO JOY ◽  
DONALD E. INGBER ◽  
SUI HUANG
Keyword(s):  
Regulatory Networks ◽  
Boolean Network ◽  
Mean Field ◽  
Period Doubling ◽  
Structural Heterogeneity ◽  
Boolean Networks ◽  
Analytical Treatment ◽  
Random Boolean Networks ◽  
The Mean ◽  
Parameter Values

Random Boolean networks have been used as simple models of gene regulatory networks, enabling the study of the dynamic behavior of complex biological systems. However, analytical treatment has been difficult because of the structural heterogeneity and the vast state space of these networks. Here we used mean field approximations to analyze the dynamics of a class of Boolean networks in which nodes have random degree (connectivity) distributions, characterized by the mean degree k and variance D. To achieve this we generalized the simple cellular automata rule 126 and used it as the Boolean function for all nodes. The equation for the evolution of the density of the network state is presented as a one-dimensional map for various input degree distributions, with k and D as the control parameters. The mean field dynamics is compared with the data obtained from the simulations of the Boolean network. Bifurcation diagrams and Lyapunov exponents for different parameter values were computed for the map, showing period doubling route to chaos with increasing k. Onset of chaos was delayed (occurred at higher k) with the increase in variance D of the connectivity. Thus, the network tends to be less chaotic when the heterogeneity, as measured by the variance of connectivity, was higher.

scholarly journals Adaptive Local Information Transfer in Random Boolean Networks

2017 ◽  
Vol 23 (1) ◽  
pp. 105-118 ◽  
Author(s):  
Taichi Haruna
Keyword(s):  
Information Processing ◽  
Information Transfer ◽  
Regulatory Networks ◽  
Mean Field Theory ◽  
Mean Field ◽  
Boolean Networks ◽  
Local Information ◽  
System Level ◽  
Adjustment Process ◽  
Random Boolean Networks

Living systems such as gene regulatory networks and neuronal networks have been supposed to work close to dynamical criticality, where their information-processing ability is optimal at the whole-system level. We investigate how this global information-processing optimality is related to the local information transfer at each individual-unit level. In particular, we introduce an internal adjustment process of the local information transfer and examine whether the former can emerge from the latter. We propose an adaptive random Boolean network model in which each unit rewires its incoming arcs from other units to balance stability of its information processing based on the measurement of the local information transfer pattern. First, we show numerically that random Boolean networks can self-organize toward near dynamical criticality in our model. Second, the proposed model is analyzed by a mean-field theory. We recognize that the rewiring rule has a bootstrapping feature. The stationary indegree distribution is calculated semi-analytically and is shown to be close to dynamical criticality in a broad range of model parameter values.

scholarly journals Coherent oscillations in balanced neural networks driven by endogenous fluctuations

2021 ◽  
Author(s):  
Matteo Di Volo ◽  
Marco Segneri ◽  
Denis Goldobin ◽  
Antonio Politi ◽  
Alessandro Torcini
Keyword(s):  
Mean Field ◽  
Planck Equation ◽  
Structural Heterogeneity ◽  
Mean Field Model ◽  
Self Consistent ◽  
Consistent Solution ◽  
The Mean ◽  
Collective Oscillations ◽  
Low Dimensional ◽  
Parameter Values

We present a detailed analysis of the dynamical regimes observed in a balanced network of identical Quadratic Integrate-and-Fire (QIF) neurons with a sparse connectivity for homogeneous and heterogeneous in-degree distribution. Depending on the parameter values, either an asynchronous regime or periodic oscillations spontaneously emerge. Numerical simulations are compared with a mean field model based on a self-consistent Fokker-Planck equation (FPE). The FPE reproduces quite well the asynchronous dynamics in the homogeneous case by either assuming a Poissonian or renewal distribution for the incoming spike trains. An exact self consistent solution for the mean firing rate obtained in the limit of infinite in-degree allows identifying balanced regimes that can be either mean- or fluctuation-driven. A low-dimensional reduction of the FPE in terms of circular cumulants is also considered. Two cumulants suffice to reproduce the transition scenario observed in the network. The emergence of periodic collective oscillations is well captured both in the homogeneous and heterogeneous setups by the mean field models upon tuning either the connectivity, or the input DC current. In the heterogeneous situation we analyze also the role of structural heterogeneity.

INFERENCE OF GENE REGULATORY NETWORKS USING BOOLEAN-NETWORK INFERENCE METHODS

2009 ◽  
Vol 07 (06) ◽  
pp. 1013-1029 ◽  
Author(s):  
GRAHAM J. HICKMAN ◽  
T. CHARLIE HODGMAN
Keyword(s):  
Regulatory Networks ◽  
Boolean Network ◽  
Network Inference ◽  
Genetic Network ◽  
Boolean Networks ◽  
Genetic Networks ◽  
Related Data ◽  
Boolean Network Model ◽  
Gene Regulatory ◽  
Inference Methods

The modeling of genetic networks especially from microarray and related data has become an important aspect of the biosciences. This review takes a fresh look at a specific family of models used for constructing genetic networks, the so-called Boolean networks. The review outlines the various different types of Boolean network developed to date, from the original Random Boolean Network to the current Probabilistic Boolean Network. In addition, some of the different inference methods available to infer these genetic networks are also examined. Where possible, particular attention is paid to input requirements as well as the efficiency, advantages and drawbacks of each method. Though the Boolean network model is one of many models available for network inference today, it is well established and remains a topic of considerable interest in the field of genetic network inference. Hybrids of Boolean networks with other approaches may well be the way forward in inferring the most informative networks.

scholarly journals Asynchronous and Coherent Dynamics in Balanced Excitatory-Inhibitory Spiking Networks

2021 ◽  
Vol 15 ◽  
Author(s):  
Hongjie Bi ◽  
Matteo di Volo ◽  
Alessandro Torcini
Keyword(s):  
Spatial Scales ◽  
Spatial Coherence ◽  
Mean Field ◽  
Period Doubling ◽  
Structural Heterogeneity ◽  
Neural Mass Model ◽  
Current Fluctuations ◽  
Mass Model ◽  
Frequency Locking ◽  
Neural Mass

Dynamic excitatory-inhibitory (E-I) balance is a paradigmatic mechanism invoked to explain the irregular low firing activity observed in the cortex. However, we will show that the E-I balance can be at the origin of other regimes observable in the brain. The analysis is performed by combining extensive simulations of sparse E-I networks composed of N spiking neurons with analytical investigations of low dimensional neural mass models. The bifurcation diagrams, derived for the neural mass model, allow us to classify the possible asynchronous and coherent behaviors emerging in balanced E-I networks with structural heterogeneity for any finite in-degree K. Analytic mean-field (MF) results show that both supra and sub-threshold balanced asynchronous regimes are observable in our system in the limit N >> K >> 1. Due to the heterogeneity, the asynchronous states are characterized at the microscopic level by the splitting of the neurons in to three groups: silent, fluctuation, and mean driven. These features are consistent with experimental observations reported for heterogeneous neural circuits. The coherent rhythms observed in our system can range from periodic and quasi-periodic collective oscillations (COs) to coherent chaos. These rhythms are characterized by regular or irregular temporal fluctuations joined to spatial coherence somehow similar to coherent fluctuations observed in the cortex over multiple spatial scales. The COs can emerge due to two different mechanisms. A first mechanism analogous to the pyramidal-interneuron gamma (PING), usually invoked for the emergence of γ-oscillations. The second mechanism is intimately related to the presence of current fluctuations, which sustain COs characterized by an essentially simultaneous bursting of the two populations. We observe period-doubling cascades involving the PING-like COs finally leading to the appearance of coherent chaos. Fluctuation driven COs are usually observable in our system as quasi-periodic collective motions characterized by two incommensurate frequencies. However, for sufficiently strong current fluctuations these collective rhythms can lock. This represents a novel mechanism of frequency locking in neural populations promoted by intrinsic fluctuations. COs are observable for any finite in-degree K, however, their existence in the limit N >> K >> 1 appears as uncertain.

The Robustness and Mutual Information Entropy of Random Modular Boolean Networks

2014 ◽  
Vol 989-994 ◽  
pp. 4417-4420 ◽  
Author(s):  
Nan Zhao ◽  
Bing Hui Guo ◽  
Fan Chao Meng
Keyword(s):  
Mutual Information ◽  
Regulatory Networks ◽  
Boolean Networks ◽  
Genetic Regulatory Networks ◽  
Information Propagation ◽  
Entropy Analysis ◽  
Random Boolean Networks ◽  
Basic Model ◽  
Different Types

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.

scholarly journals Modular Random Boolean Networks

2011 ◽  
Vol 17 (4) ◽  
pp. 331-351 ◽  
Author(s):  
Rodrigo Poblanno-Balp ◽  
Carlos Gershenson
Keyword(s):  
Chaotic Dynamics ◽  
Regulatory Networks ◽  
Boolean Networks ◽  
Genetic Regulatory Networks ◽  
Random Boolean Networks ◽  
Analytical Results ◽  
Statistical Experiments

Random Boolean networks (RBNs) have been a popular model of genetic regulatory networks for more than four decades. However, most RBN studies have been made with random topologies, while real regulatory networks have been found to be modular. In this work, we extend classical RBNs to define modular RBNs. Statistical experiments and analytical results show that modularity has a strong effect on the properties of RBNs. In particular, modular RBNs have more attractors, and are closer to criticality when chaotic dynamics would be expected, than classical RBNs.

NOTE ON ANALYTICAL STUDIES OF ONE-DIMENSIONAL HOLOGRAPHIC SUPERCONDUCTORS

2012 ◽  
Vol 27 (02) ◽  
pp. 1250001 ◽  
Author(s):  
RAN LI
Keyword(s):  
Equations Of Motion ◽  
Mean Field ◽  
Study Phase ◽  
Coupling Term ◽  
Analytical Treatment ◽  
One Dimensional ◽  
Holographic Superconductors ◽  
Scalar Operator ◽  
Sturm Liouville ◽  
The Mean

We employ the variational method for the Sturm–Liouville eigenvalue problem to analytically study phase transition of one-dimensional holographic superconductors. It is shown that this method is not a very powerful method to analytically calculate the properties of holographic superconductors. From the analytical treatment of scalar operator condensate at critical temperature, we also show that the mean-field critical exponent 1/2 results from the coupling term between scalar field and vector field, which may be an universal property of holographic superconductors with a similar coupling term in their equations of motion.

ASYNCHRONOUS RANDOM BOOLEAN NETWORK MODEL WITH VARIABLE NUMBER OF PARENTS BASED ON ELEMENTARY CELLULAR AUTOMATA RULE 126

2006 ◽  
Vol 20 (08) ◽  
pp. 897-923 ◽  
Author(s):  
MIHAELA T. MATACHE
Keyword(s):  
Cellular Automata ◽  
Network Model ◽  
Boolean Network ◽  
Random Variable ◽  
Variable Number ◽  
Boolean Networks ◽  
Random Boolean Networks ◽  
Elementary Cellular Automata ◽  
Boolean Network Model ◽  
Point Analysis

A Boolean network with N nodes, each node's state at time t being determined by a certain number of parent nodes, which can vary from one node to another, is considered. This is a generalization of previous results obtained for a constant number of parent nodes, by Matache and Heidel in "Asynchronous Random Boolean Network Model Based on Elementary Cellular Automata Rule 126", Phys. Rev. E71, 026 232, 2005. The nodes, with randomly assigned neighborhoods, are updated based on various asynchronous schemes. The Boolean rule is a generalization of rule 126 of elementary cellular automata, and is assumed to be the same for all the nodes. We provide a model for the probability of finding a node in state 1 at a time t for the class of generalized asynchronous random Boolean networks (GARBN) in which a random number of nodes can be updated at each time point. We generate consecutive states of the network for both the real system and the models under the various schemes, and use simulation algorithms to show that the results match well. We use the model to study the dynamics of the system through sensitivity of the orbits to initial values, bifurcation diagrams, and fixed point analysis. We show that the GARBN's dynamics range from order to chaos, depending on the type of random variable generating the asynchrony and the parameter combinations.

scholarly journals Random Networks with Quantum Boolean Functions

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 792 ◽  
Author(s):  
Mario Franco ◽  
Octavio Zapata ◽  
David A. Rosenblueth ◽  
Carlos Gershenson
Keyword(s):  
Boolean Functions ◽  
Boolean Network ◽  
Complex Dynamics ◽  
Boolean Networks ◽  
Stable Complex ◽  
Complex Dynamic ◽  
Random Boolean Networks ◽  
Quantum Simulator ◽  
Novel Model ◽  
Unstable Dynamics

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.

Encoding Threshold Boolean Networks into Reaction Systems for the Analysis of Gene Regulatory Networks

2021 ◽  
Vol 179 (2) ◽  
pp. 205-225
Author(s):  
Roberto Barbuti ◽  
Pasquale Bove ◽  
Roberta Gori ◽  
Damas Gruska ◽  
Francesca Levi ◽  
...  
Keyword(s):  
Gene Regulatory Networks ◽  
Regulatory Networks ◽  
Boolean Network ◽  
Reaction System ◽  
Boolean Networks ◽  
Specific Cell ◽  
Reaction Systems ◽  
Gene Regulatory ◽  
Closed Reaction System

Gene regulatory networks represent the interactions among genes regulating the activation of specific cell functionalities and they have been successfully modeled using threshold Boolean networks. In this paper we propose a systematic translation of threshold Boolean networks into reaction systems. Our translation produces a non redundant set of rules with a minimal number of objects. This translation allows us to simulate the behavior of a Boolean network simply by executing the (closed) reaction system we obtain. This can be very useful for investigating the role of different genes simply by “playing” with the rules. We developed a tool able to systematically translate a threshold Boolean network into a reaction system. We use our tool to translate two well known Boolean networks modelling biological systems: the yeast-cell cycle and the SOS response in Escherichia coli. The resulting reaction systems can be used for investigating dynamic causalities among genes.

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