Boolean Delay Equations Research Articles

Boolean Delay Equations: a dynamical approach to modeling complex systems

Boolean delay equations (BDEs) are equations with discrete variables evolving in continuous time. They serve as exploratory tools in the study of nonlinear and complex systems. In this review paper, we outline their formulation and illustrate their properties by application to a solid-earth problem and a climate one. The first problem is the seismotectonic description and prediction of earthquakes and of their clustering. The second one is that of the coupled atmosphere-ocean phenomenon of the El Niño-Southern Oscillation (ENSO). Both involve irregular behavior that is hard to predict, although some form of cyclicity is present in both, especially in ENSO. The paper concludes with broad perspectives on the further use of BDEs in the geosciences and elsewhere.

BDEtools: A MATLAB Package for Boolean Delay Equation Modeling.

Boolean Delay Equations (BDEs) can simulate surprisingly complex behavior, despite their relative simplicity. In addition to steady-state dynamics, BDEs can also generate periodic and quasiperiodic oscillations, m:n frequency locking, and even chaos. Further, the enumerability of Boolean update functions and their compact parametrization means that BDEs can be leveraged to generate low-level descriptions of biological networks, from which more detailed formulations (e.g., differential equation models) can be constructed. However, although several studies have demonstrated the utility of BDE modeling in computational biology, a current barrier to the wider adoption of the BDE approach is the absence of freely available simulation software. In this work, we present BDEtools-an open-source MATLAB package for numerically solving BDE models. After giving a brief introduction to BDE modeling, we describe the package's solver algorithms, together with several utility functions that can be used to provide solver inputs and to process solver outputs. We also demonstrate the functionality of BDEtools by illustrating its application to an established model of a gene regulatory network that controls circadian rhythms. BDEtools makes it straightforward for researchers to quickly build reliable BDE models of biological networks. We hope that its ease of use and free availability will encourage more researchers to explore BDE formulations of their systems of interest. Through the continued use of BDEs by the computational biology community, we will, no doubt, discover their potential applicability to a broader class of biological networks.

Modelling Oscillatory Patterns in the Bovine Estrous Cycle with Boolean Delay Equations

Boolean delay equations (BDEs), with their relatively simple and intuitive mode of modelling, have been used in many research areas including, for example, climate dynamics and earthquake propagation. Their application to biological systems has been scarce and limited to the molecular level. Here, we derive and present two BDE models. One is directly derived from a previously published ordinary differential equation (ODE) model for the bovine estrous cycle, whereas the second model includes a modification of a particular biological mechanism. We not only compare the simulation results from the BDE models with the trajectories of the ODE model, but also validate the BDE models with two additional numerical experiments. One experiment induces a switch in the oscillatory pattern upon changes in the model parameters, and the other simulates the administration of a hormone that is known to shift the estrous cycle in time. The models presented here are the first BDE models for hormonal oscillators, and the first BDE models for drug administration. Even though automatic parameter estimation still remains challenging, our results support the role of BDEs as a framework for the systematic modelling of complex biological oscillators.

Economic networks: Heterogeneity-induced vulnerability and loss of synchronization.

Interconnected systems are prone to propagation of disturbances, which can undermine their resilience to external perturbations. Propagation dynamics can clearly be affected by potential time delays in the underlying processes. We investigate how such delays influence the resilience of production networks facing disruption of supply. Interdependencies between economic agents are modeled using systems of Boolean delay equations (BDEs); doing so allows us to introduce heterogeneity in production delays and in inventories. Complex network topologies are considered that reproduce realistic economic features, including a network of networks. Perturbations that would otherwise vanish can, because of delay heterogeneity, amplify and lead to permanent disruptions. This phenomenon is enabled by the interactions between short cyclic structures. Difference in delays between two interacting, and otherwise resilient, structures can in turn lead to loss of synchronization in damage propagation and thus prevent recovery. Finally, this study also shows that BDEs on complex networks can lead to metastable relaxation oscillations, which are damped out in one part of a network while moving on to another part.

BOOLEAN DELAY EQUATIONS ON NETWORKS IN ECONOMICS AND THE GEOSCIENCES

We study damage propagation in networks, with an emphasis on production-chain models. The models are formulated as systems of Boolean delay equations. This formalism helps take into account the complexity of the interactions between firms; it turns out to be well adapted to investigating propagation of an initial damage due to a climatic or other natural disaster.We consider in detail the effects of distinct delays and forcing, which represent external intervention to prevent economic collapse. We also account for the possible presence of randomness in the links and the delays. The paper concentrates on two different network structures, periodic and random, respectively; their study allows one to understand the effects of multiple, concurrent production paths, and the role played by the network topology in damage propagation. Applications to the recent network modeling of climate variability are discussed.

A survey on piecewise-linear models of regulatory dynamical systems

Recent developments in understanding the various regulatory systems, especially the developments in biology and genomics, stimulated an interest in modelling such systems. Hybrid systems, originally developed for process control applications, provide advances in modelling such systems. A particular class of hybrid systems which are relatively simpler to analyze mathematically but still capable of demonstrating the essential features of many non-linear dynamical systems is piecewise-linear systems. Implementation of piecewise-linear systems for modelling of regulatory dynamical systems requires different considerations depending on the status of the problem. In this work we considered three different cases. Firstly, we consider the inferential modelling problem based on the empirical observations and study the discrete piecewise-linear system, whose inverse problem is solvable under some assumptions. Secondly, we considered the problem of obtaining some complex regulatory systems by tractable piecewise-linear formulations and study the qualitative dynamic features of the systems and their piecewise-linear models. Finally, we considered Boolean delay equations for building abstract models of regulatory systems, which might be the simplest models demonstrating the essential qualitative features of our interest underlying adaption, learning and memorization.

A Boolean Delay Equation Model of Colliding Cascades. Part II: Prediction of Critical Transitions

We consider here prediction of abrupt overall changes (“critical transitions”) in the behavior of hierarchical complex systems, using the model developed in the first part of this study. The model merges the physical concept of colliding cascades with the mathematical framework of Boolean delay equations. It describes critical transitions that are due to the interaction between direct cascades of loading and inverse cascades of failures in a hierarchical system. This interaction is controlled by distinct delays between switching of elements from one state to another: loaded vs. unloaded and intact vs. failed. We focus on the earthquake prediction problem; accordingly, the model's heuristic constraints are taken from the dynamics of seismicity. The model exhibits four major types of premonitory seismicity patterns (PSPs), which have been previously identified in seismic observations: (i) rise of earthquake clustering; (ii) rise of the earthquakes' intensity; (iii) rise of the earthquake correlation range; and (iv) certain changes in the size distribution of earthquakes (Gutenberg–Richter relation). The model exhibits new features of individual PSPs and their collective behavior, to be tested in turn on observations. There are indications that the premonitory phenomena considered are not seismicity-specific, but may be common to hierarchical systems of a more general nature.

A Boolean delay equation model of ENSO variability

Boolean delay equations (BDEs) provide a mathematical framework to formulate and analyze conceptual models of complex multi-component systems. This framework is used here to construct a simple conceptual model for the El-Niño/Southern Oscillation (ENSO) phenomenon. ENSO involves the coupling of atmospheric and oceanic processes that are far from being completely understood. Our BDE model uses Boolean variables to represent key atmospheric and oceanic quantities and equations that involve logical operators to describe their evolution. Two distinct time-delay parameters, one for the local atmosphere–ocean coupling effects and the other for oceanic wave propagation, are introduced. Over a range of physically relevant delay values, this truly minimal model captures two essential features of ENSO’s interannual variability — its regularity and its tendency to phase-lock to the annual cycle. Oscillations with average cycle length that is an integer multiple of the seasonal cycle are prevalent and range from 2 to 7 years. Transition zones — where the average period lengths are noninteger rational multiples of the forcing period — exhibit Devil’s staircases, a signature of the quasi-periodic (QP) route to chaos. Our BDE model thus validates results from previous studies of the interaction of the seasonal cycle with ENSO’s “delayed oscillator”. It gives therewith support to the view that the observed irregularity results predominantly from low-order chaotic processes rather than from stochastic weather noise. Moreover, in the transition zone between the two integer periodicities of 2 and 3 years, a heretofore unsuspected, self-similar “fractal sunburst” pattern emerges in phase-parameter space. This pattern provides a distinct and more complex scenario than the QP route to chaos found in earlier, more detailed ENSO models. Period selection in this 2–3-year transitional region seems to play a key role in ENSO’s irregularity, as well as in the appearance of the observed quasi-biennial mode of variability.

A note on quaternary climate modelling using Boolean delay equations

The Boolean delay equation model of Ghil et al. for the study of Quaternary ice ages has been re-examined and also extended to include a hydrological feedback mechanism that directly influences the thermohaline circulation. For the basic Ghil et al. model (with the original and with corrected time delays), we show that the maximum duration for a high ice volume state depends continuously but not monotonically on the time delays of the problem. In particular, both short spiky glacials and longer glacials can be obtained from identical parameter values by simply choosing different initial conditions. In the extended model, we find that an additional temperature-hydrology-ocean feedback mechanism tends to generate longer glacials, but in neither case do the average time scales of model variability compare favourably with those of the major Quarternary glacials.

Boolean delay equations, debruijn oscillations and intermittency

We prove the PLP-tauberian theorem, a generalization of the Hardy-Littlewood tauberian theorem for polynomial log-periodic growth laws. We use this to show that the intermittency present in linear nonresonant boolean delay equations is due to log-periodic DeBruijn oscillations. In another paper we use this fact to construct deterministic white noise. We have also used the PLP-tauberian theorem in models of homogeneous fluid turbulence.

Intermittency and De Bruijn's oscillations

Intermittency in fluid turbulence has been associated with log-periodic oscillations in moments of the velocity. These oscillations may have their origin in a self-similar, fractal description of the flow. The author has recently rigorously proved that such oscillations appear in the measurement of fractal dimension in boolean delay equations using a theorem of De Bruijn and a new tauberian theorem. This note applies this theory to a simple fractal to demonstrate the generality of this new technique.

Deep water formation and Quaternary glaciations

A large fraction of climatic varability on the Quaternary time scale can be explained by nonlinear interactions between the radiation balance of the global atmosphere-ocean system and the mass balance of Northern Hemisphere ice sheets. Recent analyses of paleoclimatic proxy data indicate a further important contribution to this variability from changes in deep-water formation occurring in the North Atlantic Subpolar Sea. We study the effects of these changes on variations in global temperature and ice volume characteristic of the late Quaternary. The novel framework of Boolean delay equations (BDEs) is used to formulate a conceptual model of the climatic system under study, and to analyze this formal model. Selfsustained oscillations in the intensity of the Atlantic Ocean's thermohaline circulation result from the interaction of sea-ice formation with the waxing and waning of continental ice sheets. The comparison of model results with paleoclimatic records suggests a considerable slowing down of the abyssal circulation during glacial episodes.

Boolean delay equations. II. Periodic and aperiodic solutions

Boolean delay equations (BDEs) areevolution equations for a vector of discrete variables x(t). The value of each componentX i (t), 0 or 1. depends on previous values of all componentsx j (t− t ij ), x i (t)=f i (x1(t−t i1),...,x n (t −t in )). BDEs model the evolution of biological and physical systems with threshold behavior and nonlinear feedbacks. The delays model distinct interaction times between pairs of variables. In this paper, BDEs are studied by algebraic, analytic, and numerical methods. It is shown that solutions depend continuously on the initial data and on the delays. BDEs are classified intoconservative anddissipative. All BDEs with rational delays only haveperiodic solutions only. But conservative BDEs with rationally unrelated delays haveaperiodic solutions of increasing complexity. These solutions can be approximated arbitrarily well by periodic solutions of increasing period.Self-similarity andintermittency of aperiodic solutions is studied as a function of delay values, and certain number-theoretic questions related toresonances and diophantine approximation are raised. Period length is shown to be a lower semicontinuous function of the delays for a given BDE, and can be evaluated explicitly for linear equations. We prove that a BDE isstructurable stable if and only if it has eventually periodic solutions of bounded period, and if the length of initial transients is bounded. It is shown that, for dissipative BDEs, asymptotic solution behavior is typically governed by areduced BDE. Applications toclimate dynamics and other problems are outlined.

Boolean Difference Equations, I: Formulation and Dynamic Behavior

In many biological and physical systems, feedback mechanisms depend on a set of thresholds associated with the state variables. Each feedback has a characteristic time scale. We suggest that delay-difference equations for Boolean-valued variables are an appropriate mathematical framework for such situations: the feedback thresholds result in the discrete, on–off character of the variables, and the interaction time scales of the feedbacks are expressed as delays.The initial-value problem for Boolean delay equations (BΔEs) is formulated and shown to have unique solutions for all times. Examples of periodic and aperiodic solutions are given. Aperiodic solutions have increasing complexity which depends on time t roughly as $t^{l - 1} ,l$ being the number of delays. Stability of solutions is defined and some examples of stability analysis are given; additional stability questions are raised. The present formulation of BΔEs is compared with related work and generalizations are suggested. A classification of BΔE...