Journal articles on the topic 'Complex systems, networks, dynamical models on networks, stochastic models'
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Rozum, Jordan C., Jorge Gómez Tejeda Zañudo, Xiao Gan, Dávid Deritei, and Réka Albert. "Parity and time reversal elucidate both decision-making in empirical models and attractor scaling in critical Boolean networks." Science Advances 7, no. 29 (July 2021): eabf8124. http://dx.doi.org/10.1126/sciadv.abf8124.
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We present new applications of parity inversion and time reversal to the emergence of complex behavior from simple dynamical rules in stochastic discrete models. Our parity-based encoding of causal relationships and time-reversal construction efficiently reveal discrete analogs of stable and unstable manifolds. We demonstrate their predictive power by studying decision-making in systems biology and statistical physics models. These applications underpin a novel attractor identification algorithm implemented for Boolean networks under stochastic dynamics. Its speed enables resolving a long-standing open question of how attractor count in critical random Boolean networks scales with network size and whether the scaling matches biological observations. Via 80-fold improvement in probed network size (N = 16,384), we find the unexpectedly low scaling exponent of 0.12 ± 0.05, approximately one-tenth the analytical upper bound. We demonstrate a general principle: A system’s relationship to its time reversal and state-space inversion constrains its repertoire of emergent behaviors.
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Morrison, Megan, and Lai-Sang Young. "Chaotic heteroclinic networks as models of switching behavior in biological systems." Chaos: An Interdisciplinary Journal of Nonlinear Science 32, no. 12 (December 2022): 123102. http://dx.doi.org/10.1063/5.0122184.
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Key features of biological activity can often be captured by transitions between a finite number of semi-stable states that correspond to behaviors or decisions. We present here a broad class of dynamical systems that are ideal for modeling such activity. The models we propose are chaotic heteroclinic networks with nontrivial intersections of stable and unstable manifolds. Due to the sensitive dependence on initial conditions, transitions between states are seemingly random. Dwell times, exit distributions, and other transition statistics can be built into the model through geometric design and can be controlled by tunable parameters. To test our model’s ability to simulate realistic biological phenomena, we turned to one of the most studied organisms, C. elegans, well known for its limited behavioral states. We reconstructed experimental data from two laboratories, demonstrating the model’s ability to quantitatively reproduce dwell times and transition statistics under a variety of conditions. Stochastic switching between dominant states in complex dynamical systems has been extensively studied and is often modeled as Markov chains. As an alternative, we propose here a new paradigm, namely, chaotic heteroclinic networks generated by deterministic rules (without the necessity for noise). Chaotic heteroclinic networks can be used to model systems with arbitrary architecture and size without a commensurate increase in phase dimension. They are highly flexible and able to capture a wide range of transition characteristics that can be adjusted through control parameters.
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Safdari, Hadiseh, Martina Contisciani, and Caterina De Bacco. "Reciprocity, community detection, and link prediction in dynamic networks." Journal of Physics: Complexity 3, no. 1 (February 28, 2022): 015010. http://dx.doi.org/10.1088/2632-072x/ac52e6.
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Abstract Many complex systems change their structure over time, in these cases dynamic networks can provide a richer representation of such phenomena. As a consequence, many inference methods have been generalized to the dynamic case with the aim to model dynamic interactions. Particular interest has been devoted to extend the stochastic block model and its variant, to capture community structure as the network changes in time. While these models assume that edge formation depends only on the community memberships, recent work for static networks show the importance to include additional parameters capturing structural properties, as reciprocity for instance. Remarkably, these models are capable of generating more realistic network representations than those that only consider community membership. To this aim, we present a probabilistic generative model with hidden variables that integrates reciprocity and communities as structural information of networks that evolve in time. The model assumes a fundamental order in observing reciprocal data, that is an edge is observed, conditional on its reciprocated edge in the past. We deploy a Markovian approach to construct the network’s transition matrix between time steps and parameters’ inference is performed with an expectation-maximization algorithm that leads to high computational efficiency because it exploits the sparsity of the dataset. We test the performance of the model on synthetic dynamical networks, as well as on real networks of citations and email datasets. We show that our model captures the reciprocity of real networks better than standard models with only community structure, while performing well at link prediction tasks.
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KNOPOFF, D. "ON A MATHEMATICAL THEORY OF COMPLEX SYSTEMS ON NETWORKS WITH APPLICATION TO OPINION FORMATION." Mathematical Models and Methods in Applied Sciences 24, no. 02 (December 12, 2013): 405–26. http://dx.doi.org/10.1142/s0218202513400137.
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This paper presents a development of the so-called kinetic theory for active particles to the modeling of living, hence complex, systems localized in networks. The overall system is viewed as a network of interacting nodes, mathematical equations are required to describe the dynamics in each node and in the whole network. These interactions, which are nonlinearly additive, are modeled by evolutive stochastic games. The first conceptual part derives a general mathematical structure, to be regarded as a candidate towards the derivation of models, suitable to capture the main features of the said systems. An application on opinion formation follows to show how the theory can generate specific models.
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Jirsa, Viktor, and Hiba Sheheitli. "Entropy, free energy, symmetry and dynamics in the brain." Journal of Physics: Complexity 3, no. 1 (February 3, 2022): 015007. http://dx.doi.org/10.1088/2632-072x/ac4bec.
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Abstract Neuroscience is home to concepts and theories with roots in a variety of domains including information theory, dynamical systems theory, and cognitive psychology. Not all of those can be coherently linked, some concepts are incommensurable, and domain-specific language poses an obstacle to integration. Still, conceptual integration is a form of understanding that provides intuition and consolidation, without which progress remains unguided. This paper is concerned with the integration of deterministic and stochastic processes within an information theoretic framework, linking information entropy and free energy to mechanisms of emergent dynamics and self-organization in brain networks. We identify basic properties of neuronal populations leading to an equivariant matrix in a network, in which complex behaviors can naturally be represented through structured flows on manifolds establishing the internal model relevant to theories of brain function. We propose a neural mechanism for the generation of internal models from symmetry breaking in the connectivity of brain networks. The emergent perspective illustrates how free energy can be linked to internal models and how they arise from the neural substrate.
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Penfold, Christopher A., and David L. Wild. "How to infer gene networks from expression profiles, revisited." Interface Focus 1, no. 6 (August 10, 2011): 857–70. http://dx.doi.org/10.1098/rsfs.2011.0053.
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Inferring the topology of a gene-regulatory network (GRN) from genome-scale time-series measurements of transcriptional change has proved useful for disentangling complex biological processes. To address the challenges associated with this inference, a number of competing approaches have previously been used, including examples from information theory, Bayesian and dynamic Bayesian networks (DBNs), and ordinary differential equation (ODE) or stochastic differential equation. The performance of these competing approaches have previously been assessed using a variety of in silico and in vivo datasets. Here, we revisit this work by assessing the performance of more recent network inference algorithms, including a novel non-parametric learning approach based upon nonlinear dynamical systems. For larger GRNs, containing hundreds of genes, these non-parametric approaches more accurately infer network structures than do traditional approaches, but at significant computational cost. For smaller systems, DBNs are competitive with the non-parametric approaches with respect to computational time and accuracy, and both of these approaches appear to be more accurate than Granger causality-based methods and those using simple ODEs models.
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Parham, Paul E., and Neil M. Ferguson. "Space and contact networks: capturing the locality of disease transmission." Journal of The Royal Society Interface 3, no. 9 (December 19, 2005): 483–93. http://dx.doi.org/10.1098/rsif.2005.0105.
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While an arbitrary level of complexity may be included in simulations of spatial epidemics, computational intensity and analytical intractability mean that such models often lack transparency into the determinants of epidemiological dynamics. Although numerous approaches attempt to resolve this complexity–tractability trade-off, moment closure methods arguably offer the most promising and robust frameworks for capturing the role of the locality of contact processes on global disease dynamics. While a close analogy may be made between full stochastic spatial transmission models and dynamic network models, we consider here the special case where the dynamics of the network topology change on time-scales much longer than the epidemiological processes imposed on them; in such cases, the use of static network models are justified. We show that in such cases, static network models may provide excellent approximations to the underlying spatial contact process through an appropriate choice of the effective neighbourhood size. We also demonstrate the robustness of this mapping by examining the equivalence of deterministic approximations to the full spatial and network models derived under third-order moment closure assumptions. For systems where deviation from homogeneous mixing is limited, we show that pair equations developed for network models are at least as good an approximation to the underlying stochastic spatial model as more complex spatial moment equations, with both classes of approximation becoming less accurate only for highly localized kernels.
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Warne, David J., Ruth E. Baker, and Matthew J. Simpson. "Simulation and inference algorithms for stochastic biochemical reaction networks: from basic concepts to state-of-the-art." Journal of The Royal Society Interface 16, no. 151 (February 2019): 20180943. http://dx.doi.org/10.1098/rsif.2018.0943.
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Stochasticity is a key characteristic of intracellular processes such as gene regulation and chemical signalling. Therefore, characterizing stochastic effects in biochemical systems is essential to understand the complex dynamics of living things. Mathematical idealizations of biochemically reacting systems must be able to capture stochastic phenomena. While robust theory exists to describe such stochastic models, the computational challenges in exploring these models can be a significant burden in practice since realistic models are analytically intractable. Determining the expected behaviour and variability of a stochastic biochemical reaction network requires many probabilistic simulations of its evolution. Using a biochemical reaction network model to assist in the interpretation of time-course data from a biological experiment is an even greater challenge due to the intractability of the likelihood function for determining observation probabilities. These computational challenges have been subjects of active research for over four decades. In this review, we present an accessible discussion of the major historical developments and state-of-the-art computational techniques relevant to simulation and inference problems for stochastic biochemical reaction network models. Detailed algorithms for particularly important methods are described and complemented with Matlab ® implementations. As a result, this review provides a practical and accessible introduction to computational methods for stochastic models within the life sciences community.
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Cardelli, Luca, Isabel Cristina Perez-Verona, Mirco Tribastone, Max Tschaikowski, Andrea Vandin, and Tabea Waizmann. "Exact maximal reduction of stochastic reaction networks by species lumping." Bioinformatics 37, no. 15 (February 3, 2021): 2175–82. http://dx.doi.org/10.1093/bioinformatics/btab081.
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Abstrtact Motivation Stochastic reaction networks are a widespread model to describe biological systems where the presence of noise is relevant, such as in cell regulatory processes. Unfortunately, in all but simplest models the resulting discrete state-space representation hinders analytical tractability and makes numerical simulations expensive. Reduction methods can lower complexity by computing model projections that preserve dynamics of interest to the user. Results We present an exact lumping method for stochastic reaction networks with mass-action kinetics. It hinges on an equivalence relation between the species, resulting in a reduced network where the dynamics of each macro-species is stochastically equivalent to the sum of the original species in each equivalence class, for any choice of the initial state of the system. Furthermore, by an appropriate encoding of kinetic parameters as additional species, the method can establish equivalences that do not depend on specific values of the parameters. The method is supported by an efficient algorithm to compute the largest species equivalence, thus the maximal lumping. The effectiveness and scalability of our lumping technique, as well as the physical interpretability of resulting reductions, is demonstrated in several models of signaling pathways and epidemic processes on complex networks. Availability and implementation The algorithms for species equivalence have been implemented in the software tool ERODE, freely available for download from https://www.erode.eu. Supplementary information Supplementary data are available at Bioinformatics online.
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Bombieri, Nicola, Silvia Scaffeo, Antonio Mastrandrea, Simone Caligola, Tommaso Carlucci, Franco Fummi, Carlo Laudanna, Gabriela Constantin, and Rosalba Giugno. "SystemC Implementation of Stochastic Petri Nets for Simulation and Parameterization of Biological Networks." ACM Transactions on Embedded Computing Systems 20, no. 4 (June 2021): 1–20. http://dx.doi.org/10.1145/3427091.
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Model development and simulation of biological networks is recognized as a key task in Systems Biology. Integrated with in vitro and in vivo experimental data, network simulation allows for the discovery of the dynamics that regulate biological systems. Stochastic Petri Nets (SPNs) have become a widespread and reference formalism to model metabolic networks thanks to their natural expressiveness to represent metabolites, reactions, molecule interactions, and simulation randomness due to system fluctuations and environmental noise. In the literature, starting from the network model and the complete set of system parameters, there exist frameworks that allow for dynamic system simulation. Nevertheless, they do not allow for automatic model parameterization, which is a crucial task to identify, in silico, the network configurations that lead the model to satisfy specific temporal properties. To cover such a gap, this work first presents a framework to implement SPN models into SystemC code. Then, it shows how the framework allows for automatic parameterization of the networks. The user formally defines the network properties to be observed and the framework automatically extrapolates, through Assertion-based Verification (ABV), the parameter configurations that satisfy such properties. We present the results obtained by applying the proposed framework to model the complex metabolic network of the purine metabolism. We show how the automatic extrapolation of the system parameters allowed us to simulate the model under different conditions, which led to the understanding of behavioral differences in the regulation of the entire purine network. We also show the scalability of the approach through the modeling and simulation of four biological networks, each one with different structural characteristics.
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KIRKILIONIS, MARKUS, and LUCA SBANO. "AN AVERAGING PRINCIPLE FOR COMBINED INTERACTION GRAPHS — CONNECTIVITY AND APPLICATIONS TO GENETIC SWITCHES." Advances in Complex Systems 13, no. 03 (June 2010): 293–326. http://dx.doi.org/10.1142/s0219525910002669.
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Time-continuous dynamical systems defined on graphs are often used to model complex systems with many interacting components in a non-spatial context. In the reverse sense attaching meaningful dynamics to given "interaction diagrams" is a central bottleneck problem in many application areas, especially in cell biology where various such diagrams with different conventions describing molecular regulation are presently in use. In most situations these diagrams can only be interpreted by the use of both discrete and continuous variables during the modelling process, corresponding to both deterministic and stochastic hybrid dynamics. The conventions in genetics are well known, and therefore we use this field for illustration purposes. In [25] and [26] the authors showed that with the help of a multi-scale analysis stochastic systems with both continuous variables and finite state spaces can be approximated by dynamical systems whose leading order time evolution is given by a combination of ordinary differential equations (ODEs) and Markov chains. The leading order term in these dynamical systems is called average dynamics and turns out to be an adequate concept to analyze a class of simplified hybrid systems. Once the dynamics is defined the mutual interaction of both ODEs and Markov chains can be analyzed through the (reverse) introduction of the so-called Interaction Graph, a concept originally invented for time-continuous dynamical systems, see [5]. Here we transfer this graph concept to the average dynamics, which itself is introduced as a heuristic tool to construct models of reaction or contact networks. The graphical concepts introduced form the basis for any subsequent study of the qualitative properties of hybrid models in terms of connectivity and (feedback) loop formation.
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Stutz, Timothy C., Alfonso Landeros, Jason Xu, Janet S. Sinsheimer, Mary Sehl, and Kenneth Lange. "Stochastic simulation algorithms for Interacting Particle Systems." PLOS ONE 16, no. 3 (March 2, 2021): e0247046. http://dx.doi.org/10.1371/journal.pone.0247046.
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Interacting Particle Systems (IPSs) are used to model spatio-temporal stochastic systems in many disparate areas of science. We design an algorithmic framework that reduces IPS simulation to simulation of well-mixed Chemical Reaction Networks (CRNs). This framework minimizes the number of associated reaction channels and decouples the computational cost of the simulations from the size of the lattice. Decoupling allows our software to make use of a wide class of techniques typically reserved for well-mixed CRNs. We implement the direct stochastic simulation algorithm in the open source programming language Julia. We also apply our algorithms to several complex spatial stochastic phenomena. including a rock-paper-scissors game, cancer growth in response to immunotherapy, and lipid oxidation dynamics. Our approach aids in standardizing mathematical models and in generating hypotheses based on concrete mechanistic behavior across a wide range of observed spatial phenomena.
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Antonacci, Yuri, Laura Astolfi, Giandomenico Nollo, and Luca Faes. "Information Transfer in Linear Multivariate Processes Assessed through Penalized Regression Techniques: Validation and Application to Physiological Networks." Entropy 22, no. 7 (July 1, 2020): 732. http://dx.doi.org/10.3390/e22070732.
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The framework of information dynamics allows the dissection of the information processed in a network of multiple interacting dynamical systems into meaningful elements of computation that quantify the information generated in a target system, stored in it, transferred to it from one or more source systems, and modified in a synergistic or redundant way. The concepts of information transfer and modification have been recently formulated in the context of linear parametric modeling of vector stochastic processes, linking them to the notion of Granger causality and providing efficient tools for their computation based on the state–space (SS) representation of vector autoregressive (VAR) models. Despite their high computational reliability these tools still suffer from estimation problems which emerge, in the case of low ratio between data points available and the number of time series, when VAR identification is performed via the standard ordinary least squares (OLS). In this work we propose to replace the OLS with penalized regression performed through the Least Absolute Shrinkage and Selection Operator (LASSO), prior to computation of the measures of information transfer and information modification. First, simulating networks of several coupled Gaussian systems with complex interactions, we show that the LASSO regression allows, also in conditions of data paucity, to accurately reconstruct both the underlying network topology and the expected patterns of information transfer. Then we apply the proposed VAR-SS-LASSO approach to a challenging application context, i.e., the study of the physiological network of brain and peripheral interactions probed in humans under different conditions of rest and mental stress. Our results, which document the possibility to extract physiologically plausible patterns of interaction between the cardiovascular, respiratory and brain wave amplitudes, open the way to the use of our new analysis tools to explore the emerging field of Network Physiology in several practical applications.
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Бурыкин, Yu Burykin, Даянова, D. Dayanova, Козупица, G. Kozupitsa, Берестин, and D. Berestin. "Compartment-cluster modeling of uncertainties according to determinism." Complexity. Mind. Postnonclassic 3, no. 2 (May 21, 2014): 68–80. http://dx.doi.org/10.12737/5520.
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Transition from determinism to stochastic sand further to chaos (self-organization) in the study of biomechanical systems leads to the problem of chaotic dynamics modeling of a post- ural tremor. In general, there is a problem of identifying the voluntary human movements. In other words biophysics of complex systems has approached the global challenges of voluntary and involuntary performance of any motor functions. The possibility of modeling these processes qualitatively and quantitativelyisdiscussed. Specific models demonstrate the effectiveness of the compartment-cluster modeling of biosystems and possibilities of controlof such models from the neural networks of the brain. Comparative analysis of the simulated and real recorded signals has shown a high consistent dynamics of simulated and real signals of complex biological systems. In particular, changes in tremor parameters can be described by the change in quasi-attractors which essentially depend on the mental state of a person. In experiments it is shown in the form of sight effects, which are considered in the report as a test model on experimental data.
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Cocucci, Tadeo Javier, Manuel Pulido, Juan Pablo Aparicio, Juan Ruíz, Mario Ignacio Simoy, and Santiago Rosa. "Inference in epidemiological agent-based models using ensemble-based data assimilation." PLOS ONE 17, no. 3 (March 4, 2022): e0264892. http://dx.doi.org/10.1371/journal.pone.0264892.
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To represent the complex individual interactions in the dynamics of disease spread informed by data, the coupling of an epidemiological agent-based model with the ensemble Kalman filter is proposed. The statistical inference of the propagation of a disease by means of ensemble-based data assimilation systems has been studied in previous works. The models used are mostly compartmental models representing the mean field evolution through ordinary differential equations. These techniques allow to monitor the propagation of the infections from data and to estimate several parameters of epidemiological interest. However, there are many important features which are based on the individual interactions that cannot be represented in the mean field equations, such as social network and bubbles, contact tracing, isolating individuals in risk, and social network-based distancing strategies. Agent-based models can describe contact networks at an individual level, including demographic attributes such as age, neighborhood, household, workplaces, schools, entertainment places, among others. Nevertheless, these models have several unknown parameters which are thus difficult to prescribe. In this work, we propose the use of ensemble-based data assimilation techniques to calibrate an agent-based model using daily epidemiological data. This raises the challenge of having to adapt the agent populations to incorporate the information provided by the coarse-grained data. To do this, two stochastic strategies to correct the model predictions are developed. The ensemble Kalman filter with perturbed observations is used for the joint estimation of the state and some key epidemiological parameters. We conduct experiments with an agent based-model designed for COVID-19 and assess the proposed methodology on synthetic data and on COVID-19 daily reports from Ciudad Autónoma de Buenos Aires, Argentina.
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Antonacci, Yuri, Ludovico Minati, Luca Faes, Riccardo Pernice, Giandomenico Nollo, Jlenia Toppi, Antonio Pietrabissa, and Laura Astolfi. "Estimation of Granger causality through Artificial Neural Networks: applications to physiological systems and chaotic electronic oscillators." PeerJ Computer Science 7 (May 18, 2021): e429. http://dx.doi.org/10.7717/peerj-cs.429.
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One of the most challenging problems in the study of complex dynamical systems is to find the statistical interdependencies among the system components. Granger causality (GC) represents one of the most employed approaches, based on modeling the system dynamics with a linear vector autoregressive (VAR) model and on evaluating the information flow between two processes in terms of prediction error variances. In its most advanced setting, GC analysis is performed through a state-space (SS) representation of the VAR model that allows to compute both conditional and unconditional forms of GC by solving only one regression problem. While this problem is typically solved through Ordinary Least Square (OLS) estimation, a viable alternative is to use Artificial Neural Networks (ANNs) implemented in a simple structure with one input and one output layer and trained in a way such that the weights matrix corresponds to the matrix of VAR parameters. In this work, we introduce an ANN combined with SS models for the computation of GC. The ANN is trained through the Stochastic Gradient Descent L1 (SGD-L1) algorithm, and a cumulative penalty inspired from penalized regression is applied to the network weights to encourage sparsity. Simulating networks of coupled Gaussian systems, we show how the combination of ANNs and SGD-L1 allows to mitigate the strong reduction in accuracy of OLS identification in settings of low ratio between number of time series points and of VAR parameters. We also report how the performances in GC estimation are influenced by the number of iterations of gradient descent and by the learning rate used for training the ANN. We recommend using some specific combinations for these parameters to optimize the performance of GC estimation. Then, the performances of ANN and OLS are compared in terms of GC magnitude and statistical significance to highlight the potential of the new approach to reconstruct causal coupling strength and network topology even in challenging conditions of data paucity. The results highlight the importance of of a proper selection of regularization parameter which determines the degree of sparsity in the estimated network. Furthermore, we apply the two approaches to real data scenarios, to study the physiological network of brain and peripheral interactions in humans under different conditions of rest and mental stress, and the effects of the newly emerged concept of remote synchronization on the information exchanged in a ring of electronic oscillators. The results highlight how ANNs provide a mesoscopic description of the information exchanged in networks of multiple interacting physiological systems, preserving the most active causal interactions between cardiovascular, respiratory and brain systems. Moreover, ANNs can reconstruct the flow of directed information in a ring of oscillators whose statistical properties can be related to those of physiological networks.
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Choi, Boseung, Yu-Yu Cheng, Selahattin Cinar, William Ott, Matthew R. Bennett, Krešimir Josić, and Jae Kyoung Kim. "Bayesian inference of distributed time delay in transcriptional and translational regulation." Bioinformatics 36, no. 2 (July 26, 2019): 586–93. http://dx.doi.org/10.1093/bioinformatics/btz574.
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Abstract Motivation Advances in experimental and imaging techniques have allowed for unprecedented insights into the dynamical processes within individual cells. However, many facets of intracellular dynamics remain hidden, or can be measured only indirectly. This makes it challenging to reconstruct the regulatory networks that govern the biochemical processes underlying various cell functions. Current estimation techniques for inferring reaction rates frequently rely on marginalization over unobserved processes and states. Even in simple systems this approach can be computationally challenging, and can lead to large uncertainties and lack of robustness in parameter estimates. Therefore we will require alternative approaches to efficiently uncover the interactions in complex biochemical networks. Results We propose a Bayesian inference framework based on replacing uninteresting or unobserved reactions with time delays. Although the resulting models are non-Markovian, recent results on stochastic systems with random delays allow us to rigorously obtain expressions for the likelihoods of model parameters. In turn, this allows us to extend MCMC methods to efficiently estimate reaction rates, and delay distribution parameters, from single-cell assays. We illustrate the advantages, and potential pitfalls, of the approach using a birth–death model with both synthetic and experimental data, and show that we can robustly infer model parameters using a relatively small number of measurements. We demonstrate how to do so even when only the relative molecule count within the cell is measured, as in the case of fluorescence microscopy. Availability and implementation Accompanying code in R is available at https://github.com/cbskust/DDE_BD. Supplementary information Supplementary data are available at Bioinformatics online.
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Grace, Adam W., Dirk P. Kroese, and Werner Sandmann. "Automated State-Dependent Importance Sampling for Markov Jump Processes via Sampling from the Zero-Variance Distribution." Journal of Applied Probability 51, no. 3 (September 2014): 741–55. http://dx.doi.org/10.1239/jap/1409932671.
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Many complex systems can be modeled via Markov jump processes. Applications include chemical reactions, population dynamics, and telecommunication networks. Rare-event estimation for such models can be difficult and is often computationally expensive, because typically many (or very long) paths of the Markov jump process need to be simulated in order to observe the rare event. We present a state-dependent importance sampling approach to this problem that is adaptive and uses Markov chain Monte Carlo to sample from the zero-variance importance sampling distribution. The method is applicable to a wide range of Markov jump processes and achieves high accuracy, while requiring only a small sample to obtain the importance parameters. We demonstrate its efficiency through benchmark examples in queueing theory and stochastic chemical kinetics.
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Grace, Adam W., Dirk P. Kroese, and Werner Sandmann. "Automated State-Dependent Importance Sampling for Markov Jump Processes via Sampling from the Zero-Variance Distribution." Journal of Applied Probability 51, no. 03 (September 2014): 741–55. http://dx.doi.org/10.1017/s0021900200011645.
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Many complex systems can be modeled via Markov jump processes. Applications include chemical reactions, population dynamics, and telecommunication networks. Rare-event estimation for such models can be difficult and is often computationally expensive, because typically many (or very long) paths of the Markov jump process need to be simulated in order to observe the rare event. We present a state-dependent importance sampling approach to this problem that is adaptive and uses Markov chain Monte Carlo to sample from the zero-variance importance sampling distribution. The method is applicable to a wide range of Markov jump processes and achieves high accuracy, while requiring only a small sample to obtain the importance parameters. We demonstrate its efficiency through benchmark examples in queueing theory and stochastic chemical kinetics.
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Karagiannis, Georgios A., and Athanasios D. Panagopoulos. "Dynamic Lognormal Shadowing Framework for the Performance Evaluation of Next Generation Cellular Systems." Future Internet 11, no. 5 (May 2, 2019): 106. http://dx.doi.org/10.3390/fi11050106.
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Performance evaluation tools for wireless cellular systems are very important for the establishment and testing of future internet applications. As the complexity of wireless networks keeps growing, wireless connectivity becomes the most critical requirement in a variety of applications (considered also complex and unfavorable from propagation point of view environments and paradigms). Nowadays, with the upcoming 5G cellular networks the development of realistic and more accurate channel model frameworks has become more important since new frequency bands are used and new architectures are employed. Large scale fading known also as shadowing, refers to the variations of the received signal mainly caused by obstructions that significantly affect the available signal power at a receiver’s position. Although the variability of shadowing is considered mostly spatial for a given propagation environment, moving obstructions may significantly impact the received signal’s strength, especially in dense environments, inducing thus a temporal variability even for the fixed users. In this paper, we present the case of lognormal shadowing, a novel engineering model based on stochastic differential equations that models not only the spatial correlation structure of shadowing but also its temporal dynamics. Based on the proposed spatio-temporal shadowing field we present a computationally efficient model for the dynamics of shadowing experienced by stationary or mobile users. We also present new analytical results for the average outage duration and hand-offs based on multi-dimensional level crossings. Numerical results are also presented for the validation of the model and some important conclusions are drawn.
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Read, Mark, Paul S. Andrews, Jon Timmis, and Vipin Kumar. "Modelling biological behaviours with the unified modelling language: an immunological case study and critique." Journal of The Royal Society Interface 11, no. 99 (October 6, 2014): 20140704. http://dx.doi.org/10.1098/rsif.2014.0704.
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We present a framework to assist the diagrammatic modelling of complex biological systems using the unified modelling language (UML). The framework comprises three levels of modelling, ranging in scope from the dynamics of individual model entities to system-level emergent properties. By way of an immunological case study of the mouse disease experimental autoimmune encephalomyelitis, we show how the framework can be used to produce models that capture and communicate the biological system, detailing how biological entities, interactions and behaviours lead to higher-level emergent properties observed in the real world. We demonstrate how the UML can be successfully applied within our framework, and provide a critique of UML's ability to capture concepts fundamental to immunology and biology more generally. We show how specialized, well-explained diagrams with less formal semantics can be used where no suitable UML formalism exists. We highlight UML's lack of expressive ability concerning cyclic feedbacks in cellular networks, and the compounding concurrency arising from huge numbers of stochastic, interacting agents. To compensate for this, we propose several additional relationships for expressing these concepts in UML's activity diagram. We also demonstrate the ambiguous nature of class diagrams when applied to complex biology, and question their utility in modelling such dynamic systems. Models created through our framework are non-executable, and expressly free of simulation implementation concerns. They are a valuable complement and precursor to simulation specifications and implementations, focusing purely on thoroughly exploring the biology, recording hypotheses and assumptions, and serve as a communication medium detailing exactly how a simulation relates to the real biology.
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Alameddine, Abdallah K., Frederick Conlin, and Brian Binnall. "An Introduction to the Mathematical Modeling in the Study of Cancer Systems Biology." Cancer Informatics 17 (January 2018): 117693511879975. http://dx.doi.org/10.1177/1176935118799754.
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Background: Frequently occurring in cancer are the aberrant alterations of regulatory onco-metabolites, various oncogenes/epigenetic stochasticity, and suppressor genes, as well as the deficient mismatch repair mechanism, chronic inflammation, or those deviations belonging to the other cancer characteristics. How these aberrations that evolve overtime determine the global phenotype of malignant tumors remains to be completely understood. Dynamic analysis may have potential to reveal the mechanism of carcinogenesis and can offer new therapeutic intervention. Aims: We introduce simplified mathematical tools to model serial quantitative data of cancer biomarkers. We also highlight an introductory overview of mathematical tools and models as they apply from the viewpoint of known cancer features. Methods: Mathematical modeling of potentially actionable genomic products and how they proceed overtime during tumorigenesis are explored. This report is intended to be instinctive without being overly technical. Results: To date, many mathematical models of the common features of cancer have been developed. However, the dynamic of integrated heterogeneous processes and their cross talks related to carcinogenesis remains to be resolved. Conclusions: In cancer research, outlining mathematical modeling of experimentally obtained data snapshots of molecular species may provide insights into a better understanding of the multiple biochemical circuits. Recent discoveries have provided support for the existence of complex cancer progression in dynamics that span from a simple 1-dimensional deterministic system to a stochastic (ie, probabilistic) or to an oscillatory and multistable networks. Further research in mathematical modeling of cancer progression, based on the evolving molecular kinetics (time series), could inform a specific and a predictive behavior about the global systems biology of vulnerable tumor cells in their earlier stages of oncogenesis. On this footing, new preventive measures and anticancer therapy could then be constructed.
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LIN, CHENG-JIAN. "A FUZZY ADAPTIVE LEARNING CONTROL NETWORK WITH ON-LINE STRUCTURE AND PARAMETER LEARNING." International Journal of Neural Systems 07, no. 05 (November 1996): 569–90. http://dx.doi.org/10.1142/s0129065796000567.
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This paper addresses a general connectionist model, called Fuzzy Adaptive Learning Control Network (FALCON), for the realization of a fuzzy logic control system. An on-line supervised structure/parameter learning algorithm is proposed for constructing the FALCON dynamically. It combines the backpropagation learning scheme for parameter learning and the fuzzy ART algorithm for structure learning. The supervised learning algorithm has some important features. First of all, it partitions the input state space and output control space using irregular fuzzy hyperboxes according to the distribution of training data. In many existing fuzzy or neural fuzzy control systems, the input and output spaces are always partitioned into “grids”. As the number of input/output variables increase, the number of partitioned grids will grow combinatorially. To avoid the problem of combinatorial growing of partitioned grids in some complex systems, the proposed learning algorithm partitions the input/output spaces in a flexible way based on the distribution of training data. Second, the proposed learning algorithm can create and train the FALCON in a highly autonomous way. In its initial form, there is no membership function, fuzzy partition, and fuzzy logic rule. They are created and begin to grow as the first training pattern arrives. The users thus need not give it any a priori knowledge or even any initial information on these. In some real-time applications, exact training data may be expensive or even impossible to obtain. To solve this problem, a Reinforcement Fuzzy Adaptive Learning Control Network (RFALCON) is further proposed. The proposed RFALCON is constructed by integrating two FALCONs, one FALCON as a critic network, and the other as an action network. By combining temporal difference techniques, stochastic exploration, and a proposed on-line supervised structure/parameter learning algorithm, a reinforcement structure/parameter learning algorithm is proposed, which can construct a RFALCON dynamically through a reward/penalty signal. The ball and beam balancing system is presented to illustrate the performance and applicability of the proposed models and learning algorithms.
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Pinotti, Francesco, Fakhteh Ghanbarnejad, Philipp Hövel, and Chiara Poletto. "Interplay between competitive and cooperative interactions in a three-player pathogen system." Royal Society Open Science 7, no. 1 (January 2020): 190305. http://dx.doi.org/10.1098/rsos.190305.
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In ecological systems, heterogeneous interactions between pathogens take place simultaneously. This occurs, for instance, when two pathogens cooperate, while at the same time, multiple strains of these pathogens co-circulate and compete. Notable examples include the cooperation of human immunodeficiency virus with antibiotic-resistant and susceptible strains of tuberculosis or some respiratory infections with Streptococcus pneumoniae strains. Models focusing on competition or cooperation separately fail to describe how these concurrent interactions shape the epidemiology of such diseases. We studied this problem considering two cooperating pathogens, where one pathogen is further structured in two strains. The spreading follows a susceptible-infected-susceptible process and the strains differ in transmissibility and extent of cooperation with the other pathogen. We combined a mean-field stability analysis with stochastic simulations on networks considering both well-mixed and structured populations. We observed the emergence of a complex phase diagram, where the conditions for the less transmissible, but more cooperative strain to dominate are non-trivial, e.g. non-monotonic boundaries and bistability. Coupled with community structure, the presence of the cooperative pathogen enables the coexistence between strains by breaking the spatial symmetry and dynamically creating different ecological niches. These results shed light on ecological mechanisms that may impact the epidemiology of diseases of public health concern.
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Gupta, Abhinav, and Pierre F. J. Lermusiaux. "Neural closure models for dynamical systems." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 477, no. 2252 (August 2021): 20201004. http://dx.doi.org/10.1098/rspa.2020.1004.
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Complex dynamical systems are used for predictions in many domains. Because of computational costs, models are truncated, coarsened or aggregated. As the neglected and unresolved terms become important, the utility of model predictions diminishes. We develop a novel, versatile and rigorous methodology to learn non-Markovian closure parametrizations for known-physics/low-fidelity models using data from high-fidelity simulations. The new neural closure models augment low-fidelity models with neural delay differential equations (nDDEs), motivated by the Mori–Zwanzig formulation and the inherent delays in complex dynamical systems. We demonstrate that neural closures efficiently account for truncated modes in reduced-order-models, capture the effects of subgrid-scale processes in coarse models and augment the simplification of complex biological and physical–biogeochemical models. We find that using non-Markovian over Markovian closures improves long-term prediction accuracy and requires smaller networks. We derive adjoint equations and network architectures needed to efficiently implement the new discrete and distributed nDDEs, for any time-integration schemes and allowing non-uniformly spaced temporal training data. The performance of discrete over distributed delays in closure models is explained using information theory, and we find an optimal amount of past information for a specified architecture. Finally, we analyse computational complexity and explain the limited additional cost due to neural closure models.
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House, Thomas, and Matt J. Keeling. "Insights from unifying modern approximations to infections on networks." Journal of The Royal Society Interface 8, no. 54 (June 10, 2010): 67–73. http://dx.doi.org/10.1098/rsif.2010.0179.
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Networks are increasingly central to modern science owing to their ability to conceptualize multiple interacting components of a complex system. As a specific example of this, understanding the implications of contact network structure for the transmission of infectious diseases remains a key issue in epidemiology. Three broad approaches to this problem exist: explicit simulation; derivation of exact results for special networks; and dynamical approximations. This paper focuses on the last of these approaches, and makes two main contributions. Firstly, formal mathematical links are demonstrated between several prima facie unrelated dynamical approximations. And secondly, these links are used to derive two novel dynamical models for network epidemiology, which are compared against explicit stochastic simulation. The success of these new models provides improved understanding about the interaction of network structure and transmission dynamics.
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Wu, Xu, Guo-Ping Jiang, and Xinwei Wang. "A New Model for Complex Dynamical Networks Considering Random Data Loss." Entropy 21, no. 8 (August 15, 2019): 797. http://dx.doi.org/10.3390/e21080797.
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Model construction is a very fundamental and important issue in the field of complex dynamical networks. With the state-coupling complex dynamical network model proposed, many kinds of complex dynamical network models were introduced by considering various practical situations. In this paper, aiming at the data loss which may take place in the communication between any pair of directly connected nodes in a complex dynamical network, we propose a new discrete-time complex dynamical network model by constructing an auxiliary observer and choosing the observer states to compensate for the lost states in the coupling term. By employing Lyapunov stability theory and stochastic analysis, a sufficient condition is derived to guarantee the compensation values finally equal to the lost values, namely, the influence of data loss is finally eliminated in the proposed model. Moreover, we generalize the modeling method to output-coupling complex dynamical networks. Finally, two numerical examples are provided to demonstrate the effectiveness of the proposed model.
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Uthamacumaran, A. "A Review of Complex Systems Approaches to Cancer Networks." Complex Systems 29, no. 4 (December 15, 2020): 779–835. http://dx.doi.org/10.25088/complexsystems.29.4.779.
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Cancers remain the leading cause of disease-related pediatric death in North America. The emerging field of complex systems has redefined cancer networks as a computational system. Herein, a tumor and its heterogeneous phenotypes are discussed as dynamical systems having multiple strange attractors. Machine learning, network science and algorithmic information dynamics are discussed as current tools for cancer network reconstruction. Deep learning architectures and computational fluid models are proposed for better forecasting gene expression patterns in cancer ecosystems. Cancer cell decision-making is investigated within the framework of complex systems and complexity theory.
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Rahimi-Majd, M., J. G. Restrepo, and M. N. Najafi. "Stochastic and deterministic dynamics in networks with excitable nodes." Chaos: An Interdisciplinary Journal of Nonlinear Science 33, no. 2 (February 2023): 023134. http://dx.doi.org/10.1063/5.0103806.
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Networks of excitable systems provide a flexible and tractable model for various phenomena in biology, social sciences, and physics. A large class of such models undergo a continuous phase transition as the excitability of the nodes is increased. However, models of excitability that result in this continuous phase transition are based implicitly on the assumption that the probability that a node gets excited, its transfer function, is linear for small inputs. In this paper, we consider the effect of cooperative excitations, and more generally the case of a nonlinear transfer function, on the collective dynamics of networks of excitable systems. We find that the introduction of any amount of nonlinearity changes qualitatively the dynamical properties of the system, inducing a discontinuous phase transition and hysteresis. We develop a mean-field theory that allows us to understand the features of the dynamics with a one-dimensional map. We also study theoretically and numerically finite-size effects by examining the fate of initial conditions where only one node is excited in large but finite networks. Our results show that nonlinear transfer functions result in a rich effective phase diagram for finite networks, and that one should be careful when interpreting predictions of models that assume noncooperative excitations.
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Alekseeva, I. V., O. I. Lysenko, and O. M. Tachinina. "Necessary optimality conditions of control of stochastic compound dynamic system in case of full in-formation about state vector." Mathematical machines and systems 4 (2020): 136–47. http://dx.doi.org/10.34121/1028-9763-2020-4-136-147.
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Stochastic Compound Dynamic Systems (CDS) are complex technical systems that are created through the use of precision mechanics in combination with modern telecommunications and computer technologies. Incertitude in these CDS shows up under the influence of external and internal stochastic perturbations. The constituent elements of CDS are combined into a single system because these elements perform a single complex mission. The information exchange is wireless, there is no mechanical connection between the elements of the CDS. The paper considers groups of unmanned aerial vehicles (UAVs), which are equipped with sensors or multisensors that are able to perform a mobile sensor network. The trajectories of individual elements of the mobile sensor network are trajectories formed under the influence of stochastic perturbations. This fact means that the nature of the mobile sensor network can be classified as a stochastic compound dynamic system and for the mathematical description and optimization of the movement of this system is adequate to use models and methods for optimizing stochastic CDS. The model of CDS motion is considered to be a branching trajectory or, as they say, a branched trajectory. A stochastic mathematical model of the motion of a mobile sensor network, which performs the combined mission of surveying an emergency zone, can be classified as a model of the motion of a stochastic compound dynamic system. This approach is an adequate for mathematical model creation to the mobile sensor network physical condition, for its operation in the zone of natural disaster by natural or anthropogenic origin. This paper is devoted to solving a theoretical problem related to the formulation and proof of the necessary conditions for stochastic CDS moving optimal control along a branched trajectory with an arbitrary branching scheme.
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Arellano-Delgado, A., C. Cruz-Hernández, R. M. López Gutiérrez, and C. Posadas-Castillo. "Outer Synchronization of Simple Firefly Discrete Models in Coupled Networks." Mathematical Problems in Engineering 2015 (2015): 1–14. http://dx.doi.org/10.1155/2015/895379.
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Synchronization is one of the most important emerging collective behaviors in nature, which results from the interaction in groups of organisms. In this paper, network synchronization of discrete-time dynamical systems is studied. In particular, network synchronization with fireflies oscillators like nodes is achieved by using complex systems theory. Different cases of interest on network synchronization are studied, including for a large number of fireflies oscillators; we consider synchronization in small-world networks and outer synchronization among different coupled networks topologies; for all presented cases, we provide appropriate ranges of values for coupling strength and extensive numerical simulations are included. In addition, for illustrative purposes, we show the effectiveness of network synchronization by means of experimental implementation of coupled nine electronics fireflies in different topologies.
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MUEZZINOGLU, MEHMET K., IRMA TRISTAN, RAMON HUERTA, VALENTIN S. AFRAIMOVICH, and MIKHAIL I. RABINOVICH. "TRANSIENTS VERSUS ATTRACTORS IN COMPLEX NETWORKS." International Journal of Bifurcation and Chaos 20, no. 06 (June 2010): 1653–75. http://dx.doi.org/10.1142/s0218127410026745.
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Understanding and predicting the behavior of complex multiagent systems like brain or ecological food net requires new approaches and paradigms. Traditional analyses based on just asymptotic results of behavior as time goes to infinity, or on straightforward mathematical images that can accommodate only fixed points or limit cycles do not tell much about these systems. To obtain sensible dynamical models of natural phenomena, such as the reproducible order observed in ecological, cognitive or behavioral experiments, one cannot afford to neglect the transient dynamics of the underlying complex network. In disclosing such dynamical mechanisms, the focus of interest must be on reproducible or, even, structurally stable transients. In this tutorial, we formulate the Winnerless Competition (WLC) principle that induces robust transient dynamics in open complex networks. The main point of WLC principle is the transformation of the acquired information into ensemble (spatio)-temporal output via intrinsic transient dynamics of the network. Such encoding provides a reproducible transient response, whose geometrical image in phase space is a stable heteroclinic sequence. We compile a diverse list of natural phenomena which can be rigorously modeled by the WLC. Together with the experimental and numerical results of the networks with different levels of complexity, we evaluate the robustness and reproducibility of the WLC dynamics and discuss the advantages of future possible application of the discussed approach.
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Wittenstein, Timon, Nava Leibovich, and Andreas Hilfinger. "Quantifying biochemical reaction rates from static population variability within incompletely observed complex networks." PLOS Computational Biology 18, no. 6 (June 22, 2022): e1010183. http://dx.doi.org/10.1371/journal.pcbi.1010183.
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Quantifying biochemical reaction rates within complex cellular processes remains a key challenge of systems biology even as high-throughput single-cell data have become available to characterize snapshots of population variability. That is because complex systems with stochastic and non-linear interactions are difficult to analyze when not all components can be observed simultaneously and systems cannot be followed over time. Instead of using descriptive statistical models, we show that incompletely specified mechanistic models can be used to translate qualitative knowledge of interactions into reaction rate functions from covariability data between pairs of components. This promises to turn a globally intractable problem into a sequence of solvable inference problems to quantify complex interaction networks from incomplete snapshots of their stochastic fluctuations.
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Tanzi, Matteo, and Lai-Sang Young. "Existence of physical measures in some excitation–inhibition networks*." Nonlinearity 35, no. 2 (December 22, 2021): 889–915. http://dx.doi.org/10.1088/1361-6544/ac3eb6.
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Abstract In this paper we present a rigorous analysis of a class of coupled dynamical systems in which two distinct types of components, one excitatory and the other inhibitory, interact with one another. These network models are finite in size but can be arbitrarily large. They are inspired by real biological networks, and possess features that are idealizations of those in biological systems. Individual components of the network are represented by simple, much studied dynamical systems. Complex dynamical patterns on the network level emerge as a result of the coupling among its constituent subsystems. Appealing to existing techniques in (nonuniform) hyperbolic theory, we study their Lyapunov exponents and entropy, and prove that large time network dynamics are governed by physical measures with the SRB property.
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Xinghua, Hu, and Zhang Yu. "A Parameters Calibration Method in Simulated Complex Traffic Network." Open Construction and Building Technology Journal 9, no. 1 (October 27, 2015): 262–65. http://dx.doi.org/10.2174/1874836801509010262.
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Traffic simulation models have been extensively used because of their ability to model the dynamic stochastic nature of transportation systems. Parameter calibration is very complex and does not give optimal results easily. Besides, it is also time-consuming especially for large and complex networks. Initially, the procedure of traffic micro-simulation parameter calibration was put forward. A Vehicle Intelligent Simulation Software Model (VISSIM) models were selected for parameter calibration in complex-network, and, the role of Simultaneous Perturbation Genetic Algorithm (SPGA) was examined in the optimization of component. Moreover an automatic calibration methodology for micro-simulation models was developed in order to select the best parameter set based on the observed Intelligent Transportation Systems (ITS) data which proved effective for different networks. Finally, the methodology was applied to calibrate the Beijing city VISSIM models, followed by the comparison of convergence rate of Genetic Algorithm (GA), Simultaneous Perturbation Stochastic Approximation (SPSA) and SPGA algorithm. The results show that the SPGA was effective and had good performance.
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Halevi, Y., and A. Ray. "Performance Analysis of Integrated Communication and Control System Networks." Journal of Dynamic Systems, Measurement, and Control 112, no. 3 (September 1, 1990): 365–71. http://dx.doi.org/10.1115/1.2896153.
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This paper presents statistical analysis of delays in Integrated Communication and Control System (ICCS) networks [1–4] that are based on asynchronous time-division multiplexing. The models are obtained in closed form for analyzing control systems with randomly varying delays. The results of this research are applicable to ICCS design for complex dynamical processes like advanced aircraft and spacecraft, autonomous manufacturing plants, and chemical and processing plants.
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ROSSELLÓ, JOSEP L., VINCENT CANALS, ANTONI MORRO, and ANTONI OLIVER. "HARDWARE IMPLEMENTATION OF STOCHASTIC SPIKING NEURAL NETWORKS." International Journal of Neural Systems 22, no. 04 (July 25, 2012): 1250014. http://dx.doi.org/10.1142/s0129065712500141.
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Spiking Neural Networks, the last generation of Artificial Neural Networks, are characterized by its bio-inspired nature and by a higher computational capacity with respect to other neural models. In real biological neurons, stochastic processes represent an important mechanism of neural behavior and are responsible of its special arithmetic capabilities. In this work we present a simple hardware implementation of spiking neurons that considers this probabilistic nature. The advantage of the proposed implementation is that it is fully digital and therefore can be massively implemented in Field Programmable Gate Arrays. The high computational capabilities of the proposed model are demonstrated by the study of both feed-forward and recurrent networks that are able to implement high-speed signal filtering and to solve complex systems of linear equations.
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Asllani, Malbor, Renaud Lambiotte, and Timoteo Carletti. "Structure and dynamical behavior of non-normal networks." Science Advances 4, no. 12 (December 2018): eaau9403. http://dx.doi.org/10.1126/sciadv.aau9403.
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We analyze a collection of empirical networks in a wide spectrum of disciplines and show that strong non-normality is ubiquitous in network science. Dynamical processes evolving on non-normal networks exhibit a peculiar behavior, as initial small disturbances may undergo a transient phase and be strongly amplified in linearly stable systems. In addition, eigenvalues may become extremely sensible to noise and have a diminished physical meaning. We identify structural properties of networks that are associated with non-normality and propose simple models to generate networks with a tunable level of non-normality. We also show the potential use of a variety of metrics capturing different aspects of non-normality and propose their potential use in the context of the stability of complex ecosystems.
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Niu, Haoyu, YangQuan Chen, and Bruce J. West. "Why Do Big Data and Machine Learning Entail the Fractional Dynamics?" Entropy 23, no. 3 (February 28, 2021): 297. http://dx.doi.org/10.3390/e23030297.
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Fractional-order calculus is about the differentiation and integration of non-integer orders. Fractional calculus (FC) is based on fractional-order thinking (FOT) and has been shown to help us to understand complex systems better, improve the processing of complex signals, enhance the control of complex systems, increase the performance of optimization, and even extend the enabling of the potential for creativity. In this article, the authors discuss the fractional dynamics, FOT and rich fractional stochastic models. First, the use of fractional dynamics in big data analytics for quantifying big data variability stemming from the generation of complex systems is justified. Second, we show why fractional dynamics is needed in machine learning and optimal randomness when asking: “is there a more optimal way to optimize?”. Third, an optimal randomness case study for a stochastic configuration network (SCN) machine-learning method with heavy-tailed distributions is discussed. Finally, views on big data and (physics-informed) machine learning with fractional dynamics for future research are presented with concluding remarks.
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Kolářová, Edita. "Applications of second order stochastic integral equations to electrical networks." Tatra Mountains Mathematical Publications 63, no. 1 (June 1, 2015): 163–73. http://dx.doi.org/10.1515/tmmp-2015-0028.
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The theory of stochastic differential equations is used in various fields of science and engineering. This paper deals with vector-valued stochastic integral equations. We show some applications of the presented theory to the problem of modelling RLC electrical circuits by noisy parameters. From practical point of view, the second-order RLC circuits are of major importance, as they are the building blocks of more complex physical systems. The mathematical models of such circuits lead to the second order differential equations. We construct stochastic models of the RLC circuit by replacing a coefficient in the deterministic system with a noisy one. In this paper we present the analytic solution of these equations using the Itô calculus and compute confidence intervals for the stochastic solutions. Numerical simulations in the examples are performed using Matlab.
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Nikolic, Sasa S., Dragan S. Antic, Nikola B. Dankovic, Aleksandra A. Milovanovic, Darko B. Mitic, Miroslav B. Milovanovic, and Petar S. Djekic. "Generalized Quasi-Orthogonal Functional Networks Applied in Parameter Sensitivity Analysis of Complex Dynamical Systems." Elektronika ir Elektrotechnika 28, no. 4 (August 24, 2022): 19–26. http://dx.doi.org/10.5755/j02.eie.31110.
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This paper presents one possible application of generalized quasi-orthogonal functional networks in the sensitivity analysis of complex dynamical systems. First, a new type of first order (k = 1) generalized quasi-orthogonal polynomials of Legendre type via classical quasi-orthogonal polynomials was introduced. The short principle to design generalized quasi-orthogonal polynomials and filters was also shown. A generalized quasi-orthogonal functional network represents an extension of classical orthogonal functional networks and neural networks, which deal with general functional models. A sequence of the first order (k = 1) generalized quasi-orthogonal polynomials was used as a new basis in the proposed generalized quasi-orthogonal functional networks. The proposed method for determining the parameter sensitivity of complex dynamical systems is also given, and an example of a complex industrial system in the form of a tower crane was considered. The results obtained have been compared with different methods for parameter sensitivity analysis.
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PLESA, TOMISLAV, RADEK ERBAN, and HANS G. OTHMER. "Noise-induced mixing and multimodality in reaction networks." European Journal of Applied Mathematics 30, no. 5 (September 18, 2018): 887–911. http://dx.doi.org/10.1017/s0956792518000517.
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We analyse a class of chemical reaction networks under mass-action kinetics involving multiple time scales, whose deterministic and stochastic models display qualitative differences. The networks are inspired by gene-regulatory networks and consist of a slow subnetwork, describing conversions among the different gene states, and fast subnetworks, describing biochemical interactions involving the gene products. We show that the long-term dynamics of such networks can consist of a unique attractor at the deterministic level (unistability), while the long-term probability distribution at the stochastic level may display multiple maxima (multimodality). The dynamical differences stem from a phenomenon we call noise-induced mixing, whereby the probability distribution of the gene products is a linear combination of the probability distributions of the fast subnetworks which are ‘mixed’ by the slow subnetworks. The results are applied in the context of systems biology, where noise-induced mixing is shown to play a biochemically important role, producing phenomena such as stochastic multimodality and oscillations.
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Bone, Rebecca A., and Jason R. Green. "Optimizing dynamical functions for speed with stochastic paths." Journal of Chemical Physics 157, no. 22 (December 14, 2022): 224101. http://dx.doi.org/10.1063/5.0125479.
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Living systems are built from microscopic components that function dynamically; they generate work with molecular motors, assemble and disassemble structures such as microtubules, keep time with circadian clocks, and catalyze the replication of DNA. How do we implement these functions in synthetic nanostructured materials to execute them before the onset of dissipative losses? Answering this question requires a quantitative understanding of when we can improve performance and speed while minimizing the dissipative losses associated with operating in a fluctuating environment. Here, we show that there are four modalities for optimizing dynamical functions that can guide the design of nanoscale systems. We analyze Markov models that span the design space: a clock, ratchet, replicator, and self-assembling system. Using stochastic thermodynamics and an exact expression for path probabilities, we classify these models of dynamical functions based on the correlation of speed with dissipation and with the chosen performance metric. We also analyze random networks to identify the model features that affect their classification and the optimization of their functionality. Overall, our results show that the possible nonequilibrium paths can determine our ability to optimize the performance of dynamical functions, despite ever-present dissipation, when there is a need for speed.
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Abrevaya, Germán, Guillaume Dumas, Aleksandr Y. Aravkin, Peng Zheng, Jean-Christophe Gagnon-Audet, James Kozloski, Pablo Polosecki, et al. "Learning Brain Dynamics With Coupled Low-Dimensional Nonlinear Oscillators and Deep Recurrent Networks." Neural Computation 33, no. 8 (July 26, 2021): 2087–127. http://dx.doi.org/10.1162/neco_a_01401.
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Many natural systems, especially biological ones, exhibit complex multivariate nonlinear dynamical behaviors that can be hard to capture by linear autoregressive models. On the other hand, generic nonlinear models such as deep recurrent neural networks often require large amounts of training data, not always available in domains such as brain imaging; also, they often lack interpretability. Domain knowledge about the types of dynamics typically observed in such systems, such as a certain type of dynamical systems models, could complement purely data-driven techniques by providing a good prior. In this work, we consider a class of ordinary differential equation (ODE) models known as van der Pol (VDP) oscil lators and evaluate their ability to capture a low-dimensional representation of neural activity measured by different brain imaging modalities, such as calcium imaging (CaI) and fMRI, in different living organisms: larval zebrafish, rat, and human. We develop a novel and efficient approach to the nontrivial problem of parameters estimation for a network of coupled dynamical systems from multivariate data and demonstrate that the resulting VDP models are both accurate and interpretable, as VDP's coupling matrix reveals anatomically meaningful excitatory and inhibitory interactions across different brain subsystems. VDP outperforms linear autoregressive models (VAR) in terms of both the data fit accuracy and the quality of insight provided by the coupling matrices and often tends to generalize better to unseen data when predicting future brain activity, being comparable to and sometimes better than the recurrent neural networks (LSTMs). Finally, we demonstrate that our (generative) VDP model can also serve as a data-augmentation tool leading to marked improvements in predictive accuracy of recurrent neural networks. Thus, our work contributes to both basic and applied dimensions of neuroimaging: gaining scientific insights and improving brain-based predictive models, an area of potentially high practical importance in clinical diagnosis and neurotechnology.
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Hu, Yiwen, and Markus J. Buehler. "Deep language models for interpretative and predictive materials science." APL Machine Learning 1, no. 1 (March 1, 2023): 010901. http://dx.doi.org/10.1063/5.0134317.
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Machine learning (ML) has emerged as an indispensable methodology to describe, discover, and predict complex physical phenomena that efficiently help us learn underlying functional rules, especially in cases when conventional modeling approaches cannot be applied. While conventional feedforward neural networks are typically limited to performing tasks related to static patterns in data, recursive models can both work iteratively based on a changing input and discover complex dynamical relationships in the data. Deep language models can model flexible modalities of data and are capable of learning rich dynamical behaviors as they operate on discrete or continuous symbols that define the states of a physical system, yielding great potential toward end-to-end predictions. Similar to how words form a sentence, materials can be considered as a self-assembly of physically interacted building blocks, where the emerging functions of materials are analogous to the meaning of sentences. While discovering the fundamental relationships between building blocks and function emergence can be challenging, language models, such as recurrent neural networks and long-short term memory networks, and, in particular, attention models, such as the transformer architecture, can solve many such complex problems. Application areas of such models include protein folding, molecular property prediction, prediction of material failure of complex nonlinear architected materials, and also generative strategies for materials discovery. We outline challenges and opportunities, especially focusing on extending the deep-rooted kinship of humans with symbolism toward generalizable artificial intelligence (AI) systems using neuro-symbolic AI, and outline how tools such as ChatGPT and DALL·E can drive materials discovery.
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Ghahramani, Zoubin, and Geoffrey E. Hinton. "Variational Learning for Switching State-Space Models." Neural Computation 12, no. 4 (April 1, 2000): 831–64. http://dx.doi.org/10.1162/089976600300015619.
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We introduce a new statistical model for time series that iteratively segments data into regimes with approximately linear dynamics and learns the parameters of each of these linear regimes. This model combines and generalizes two of the most widely used stochastic time-series models—hidden Markov models and linear dynamical systems—and is closely related to models that are widely used in the control and econometrics literatures. It can also be derived by extending the mixture of experts neural network (Jacobs, Jordan, Nowlan, & Hinton, 1991) to its fully dynamical version, in which both expert and gating networks are recurrent. Inferring the posterior probabilities of the hidden states of this model is computationally intractable, and therefore the exact expectation maximization (EM) algorithm cannot be applied. However, we present a variational approximation that maximizes a lower bound on the log-likelihood and makes use of both the forward and backward recursions for hidden Markov models and the Kalman filter recursions for linear dynamical systems. We tested the algorithm on artificial data sets and a natural data set of respiration force from a patient with sleep apnea. The results suggest that variational approximations are a viable method for inference and learning in switching state-space models.
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ANDREYEV, YU V., A. S. DMITRIEV, D. A. KUMINOV, L. O. CHUA, and C. W. WU. "1-D MAPS, CHAOS AND NEURAL NETWORKS FOR INFORMATION PROCESSING." International Journal of Bifurcation and Chaos 06, no. 04 (April 1996): 627–46. http://dx.doi.org/10.1142/s021812749600031x.
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An application of complex dynamics and chaos in neural networks to information processing is studied. Mathematical models based on piecewise-linear maps implementing basic functions of information processing via complex dynamics and chaos are discussed. Realizations of these models by neural networks are presented. In contrast to other methods of using neural networks and associative memory to store information, the information is stored in dynamical attractors such as limit cycles, rather than equilibrium points. Retrieval of information corresponds to getting the state into the basin of attraction of the attractor. We show that noise-corrupted information or partial information are sufficient to drive the state into the basin of attraction of the attractor, thus these systems exhibit the property of associative memory.
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Hasani, Ramin, Mathias Lechner, Alexander Amini, Lucas Liebenwein, Aaron Ray, Max Tschaikowski, Gerald Teschl, and Daniela Rus. "Closed-form continuous-time neural networks." Nature Machine Intelligence 4, no. 11 (November 15, 2022): 992–1003. http://dx.doi.org/10.1038/s42256-022-00556-7.
Full textAbstract:
AbstractContinuous-time neural networks are a class of machine learning systems that can tackle representation learning on spatiotemporal decision-making tasks. These models are typically represented by continuous differential equations. However, their expressive power when they are deployed on computers is bottlenecked by numerical differential equation solvers. This limitation has notably slowed down the scaling and understanding of numerous natural physical phenomena such as the dynamics of nervous systems. Ideally, we would circumvent this bottleneck by solving the given dynamical system in closed form. This is known to be intractable in general. Here, we show that it is possible to closely approximate the interaction between neurons and synapses—the building blocks of natural and artificial neural networks—constructed by liquid time-constant networks efficiently in closed form. To this end, we compute a tightly bounded approximation of the solution of an integral appearing in liquid time-constant dynamics that has had no known closed-form solution so far. This closed-form solution impacts the design of continuous-time and continuous-depth neural models. For instance, since time appears explicitly in closed form, the formulation relaxes the need for complex numerical solvers. Consequently, we obtain models that are between one and five orders of magnitude faster in training and inference compared with differential equation-based counterparts. More importantly, in contrast to ordinary differential equation-based continuous networks, closed-form networks can scale remarkably well compared with other deep learning instances. Lastly, as these models are derived from liquid networks, they show good performance in time-series modelling compared with advanced recurrent neural network models.
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Hanel, Rudolf, Manfred Pöchacker, and Stefan Thurner. "Living on the edge of chaos: minimally nonlinear models of genetic regulatory dynamics." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 368, no. 1933 (December 28, 2010): 5583–96. http://dx.doi.org/10.1098/rsta.2010.0267.
Full textAbstract:
Linearized catalytic reaction equations (modelling, for example, the dynamics of genetic regulatory networks), under the constraint that expression levels, i.e. molecular concentrations of nucleic material, are positive, exhibit non-trivial dynamical properties, which depend on the average connectivity of the reaction network. In these systems, an inflation of the edge of chaos and multi-stability have been demonstrated to exist. The positivity constraint introduces a nonlinearity, which makes chaotic dynamics possible. Despite the simplicity of such minimally nonlinear systems , their basic properties allow us to understand the fundamental dynamical properties of complex biological reaction networks. We analyse the Lyapunov spectrum, determine the probability of finding stationary oscillating solutions, demonstrate the effect of the nonlinearity on the effective in- and out-degree of the active interaction network , and study how the frequency distributions of oscillatory modes of such a system depend on the average connectivity.
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WANG, HONGCHUN, KEQING HE, BING LI, and JINHU LÜ. "ON SOME RECENT ADVANCES IN COMPLEX SOFTWARE NETWORKS: MODELING, ANALYSIS, EVOLUTION AND APPLICATIONS." International Journal of Bifurcation and Chaos 22, no. 02 (February 2012): 1250024. http://dx.doi.org/10.1142/s0218127412500241.
Full textAbstract:
Complex software networks, as a typical kind of man-made complex networks, have attracted more and more attention from various fields of science and engineering over the past ten years. With the dramatic increase of scale and complexity of software systems, it is essential to develop a systematic approach to further investigate the complex software systems by using the theories and methods of complex networks and complex adaptive systems. This paper attempts to briefly review some recent advances in complex software networks and also develop some novel tools to further analyze complex software networks, including modeling, analysis, evolution, measurement, and some potential real-world applications. More precisely, this paper first describes some effective modeling approaches for characterizing various complex software systems. Based on the above theoretical and practical models, this paper introduces some recent advances in analyzing the static and dynamical behaviors of complex software networks. It is then followed by some further discussions on potential real-world applications of complex software networks. Finally, this paper outlooks some future research topics from an engineering point of view.