Journal articles: 'Dynamical models on networks'

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Innocenti, Giacomo, and Paolo Paoletti. "Embedding dynamical networks into distributed models." Communications in Nonlinear Science and Numerical Simulation 24, no. 1-3 (July 2015): 21–39. http://dx.doi.org/10.1016/j.cnsns.2014.12.009.

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Piqueira, José R. C., and Felipe Barbosa Cesar. "Dynamical Models for Computer Viruses Propagation." Mathematical Problems in Engineering 2008 (2008): 1–11. http://dx.doi.org/10.1155/2008/940526.

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Nowadays, digital computer systems and networks are the main engineering tools, being used in planning, design, operation, and control of all sizes of building, transportation, machinery, business, and life maintaining devices. Consequently, computer viruses became one of the most important sources of uncertainty, contributing to decrease the reliability of vital activities. A lot of antivirus programs have been developed, but they are limited to detecting and removing infections, based on previous knowledge of the virus code. In spite of having good adaptation capability, these programs work just as vaccines against diseases and are not able to prevent new infections based on the network state. Here, a trial on modeling computer viruses propagation dynamics relates it to other notable events occurring in the network permitting to establish preventive policies in the network management. Data from three different viruses are collected in the Internet and two different identification techniques, autoregressive and Fourier analyses, are applied showing that it is possible to forecast the dynamics of a new virus propagation by using the data collected from other viruses that formerly infected the network.

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Malyshev, V. A. "Networks and dynamical systems." Advances in Applied Probability 25, no. 01 (March 1993): 140–75. http://dx.doi.org/10.1017/s0001867800025210.

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A new approach to the problem of classification of (deflected) random walks inor Markovian models for queueing networks with identical customers is introduced. It is based on the analysis of the intrinsic dynamical system associated with the random walk. Earlier results for small dimensions are presented from this novel point of view. We give proofs of new results for higher dimensions related to the existence of a continuous invariant measure for the underlying dynamical system. Two constants are shown to be important: the free energyM< 0 corresponds to ergodicity, the Lyapounov exponentL< 0 defines recurrence. General conjectures, examples, unsolved problems and surprising connections with ergodic theory, classical dynamical systems and their random perturbations are largely presented. A useful notion naturally arises, the so-called scaled random perturbation of a dynamical system.

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Malyshev, V. A. "Networks and dynamical systems." Advances in Applied Probability 25, no. 1 (March 1993): 140–75. http://dx.doi.org/10.2307/1427500.

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A new approach to the problem of classification of (deflected) random walks in or Markovian models for queueing networks with identical customers is introduced. It is based on the analysis of the intrinsic dynamical system associated with the random walk. Earlier results for small dimensions are presented from this novel point of view. We give proofs of new results for higher dimensions related to the existence of a continuous invariant measure for the underlying dynamical system. Two constants are shown to be important: the free energy M < 0 corresponds to ergodicity, the Lyapounov exponent L < 0 defines recurrence. General conjectures, examples, unsolved problems and surprising connections with ergodic theory, classical dynamical systems and their random perturbations are largely presented. A useful notion naturally arises, the so-called scaled random perturbation of a dynamical system.

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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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Yeung, Enoch, Jongmin Kim, Ye Yuan, Jorge Gonçalves, and Richard M. Murray. "Data-driven network models for genetic circuits from time-series data with incomplete measurements." Journal of The Royal Society Interface 18, no. 182 (September 2021): 20210413. http://dx.doi.org/10.1098/rsif.2021.0413.

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Synthetic gene networks are frequently conceptualized and visualized as static graphs. This view of biological programming stands in stark contrast to the transient nature of biomolecular interaction, which is frequently enacted by labile molecules that are often unmeasured. Thus, the network topology and dynamics of synthetic gene networks can be difficult to verify in vivo or in vitro , due to the presence of unmeasured biological states. Here we introduce the dynamical structure function as a new mesoscopic, data-driven class of models to describe gene networks with incomplete measurements of state dynamics. We develop a network reconstruction algorithm and a code base for reconstructing the dynamical structure function from data, to enable discovery and visualization of graphical relationships in a genetic circuit diagram as time-dependent functions rather than static, unknown weights. We prove a theorem, showing that dynamical structure functions can provide a data-driven estimate of the size of crosstalk fluctuations from an idealized model. We illustrate this idea with numerical examples. Finally, we show how data-driven estimation of dynamical structure functions can explain failure modes in two experimentally implemented genetic circuits, a previously reported in vitro genetic circuit and a new E. coli -based transcriptional event detector.

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CESSAC, B. "A VIEW OF NEURAL NETWORKS AS DYNAMICAL SYSTEMS." International Journal of Bifurcation and Chaos 20, no. 06 (June 2010): 1585–629. http://dx.doi.org/10.1142/s0218127410026721.

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We present some recent investigations resulting from the modeling of neural networks as dynamical systems, and deal with the following questions, adressed in the context of specific models. (i) Characterizing the collective dynamics; (ii) Statistical analysis of spike trains; (iii) Interplay between dynamics and network structure; (iv) Effects of synaptic plasticity.

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CAO, QI, GUILHERME RAMOS, PAUL BOGDAN, and SÉRGIO PEQUITO. "THE ACTUATION SPECTRUM OF SPATIOTEMPORAL NETWORKS WITH POWER-LAW TIME DEPENDENCIES." Advances in Complex Systems 22, no. 07n08 (November 2019): 1950023. http://dx.doi.org/10.1142/s0219525919500231.

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The ability to steer the state of a dynamical network towards a desired state within a time horizon is intrinsically dependent on the number of driven nodes considered, as well as the network’s topology. The trade-off between time-to-control and the minimum number of driven nodes is captured by the notion of the actuation spectrum (AS). We study the actuation spectra of a variety of artificial and real-world networked systems, modeled by fractional-order dynamics that are capable of capturing non-Markovian time properties with power-law dependencies. We find evidence that, in both types of networks, the actuation spectra are similar when the time-to-control is less or equal to about 1/5 of the size of the network. Nonetheless, for a time-to-control larger than the network size, the minimum number of driven nodes required to attain controllability in networks with fractional-order dynamics may still decrease in comparison with other networks with Markovian properties. These differences suggest that the minimum number of driven nodes can be used to determine the true dynamical nature of the network. Furthermore, such differences also suggest that new generative models are required to reproduce the actuation spectra of real fractional-order dynamical networks.

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Hasani, Ramin, Mathias Lechner, Alexander Amini, Daniela Rus, and Radu Grosu. "Liquid Time-constant Networks." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 9 (May 18, 2021): 7657–66. http://dx.doi.org/10.1609/aaai.v35i9.16936.

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We introduce a new class of time-continuous recurrent neural network models. Instead of declaring a learning system's dynamics by implicit nonlinearities, we construct networks of linear first-order dynamical systems modulated via nonlinear interlinked gates. The resulting models represent dynamical systems with varying (i.e., liquid) time-constants coupled to their hidden state, with outputs being computed by numerical differential equation solvers. These neural networks exhibit stable and bounded behavior, yield superior expressivity within the family of neural ordinary differential equations, and give rise to improved performance on time-series prediction tasks. To demonstrate these properties, we first take a theoretical approach to find bounds over their dynamics, and compute their expressive power by the trajectory length measure in a latent trajectory space. We then conduct a series of time-series prediction experiments to manifest the approximation capability of Liquid Time-Constant Networks (LTCs) compared to classical and modern RNNs.

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WANG, XIAO FAN. "COMPLEX NETWORKS: TOPOLOGY, DYNAMICS AND SYNCHRONIZATION." International Journal of Bifurcation and Chaos 12, no. 05 (May 2002): 885–916. http://dx.doi.org/10.1142/s0218127402004802.

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Dramatic advances in the field of complex networks have been witnessed in the past few years. This paper reviews some important results in this direction of rapidly evolving research, with emphasis on the relationship between the dynamics and the topology of complex networks. Basic quantities and typical examples of various complex networks are described; and main network models are introduced, including regular, random, small-world and scale-free models. The robustness of connectivity and the epidemic dynamics in complex networks are also evaluated. To that end, synchronization in various dynamical networks are discussed according to their regular, small-world and scale-free connections.

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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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Doya, Kenji, and Allen I. Selverston. "Dimension Reduction of Biological Neuron Models by Artificial Neural Networks." Neural Computation 6, no. 4 (July 1994): 696–717. http://dx.doi.org/10.1162/neco.1994.6.4.696.

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An artificial neural network approach to dimension reduction of dynamical systems is proposed and applied to conductance-based neuron models. Networks with bottleneck layers of continuous-time dynamical units could make a two-dimensional model from the trajectories of the Hodgkin-Huxley model and a three-dimensional model from the trajectories of a six-dimensional bursting neuron model. Nullcline analysis of these reduced models revealed the bifurcations of the dynamical system underlying firing and bursting behaviors.

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Costa, Felipe Xavier, Jordan C. Rozum, Austin M. Marcus, and Luis M. Rocha. "Effective Connectivity and Bias Entropy Improve Prediction of Dynamical Regime in Automata Networks." Entropy 25, no. 2 (February 18, 2023): 374. http://dx.doi.org/10.3390/e25020374.

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Biomolecular network dynamics are thought to operate near the critical boundary between ordered and disordered regimes, where large perturbations to a small set of elements neither die out nor spread on average. A biomolecular automaton (e.g., gene, protein) typically has high regulatory redundancy, where small subsets of regulators determine activation via collective canalization. Previous work has shown that effective connectivity, a measure of collective canalization, leads to improved dynamical regime prediction for homogeneous automata networks. We expand this by (i) studying random Boolean networks (RBNs) with heterogeneous in-degree distributions, (ii) considering additional experimentally validated automata network models of biomolecular processes, and (iii) considering new measures of heterogeneity in automata network logic. We found that effective connectivity improves dynamical regime prediction in the models considered; in RBNs, combining effective connectivity with bias entropy further improves the prediction. Our work yields a new understanding of criticality in biomolecular networks that accounts for collective canalization, redundancy, and heterogeneity in the connectivity and logic of their automata models. The strong link we demonstrate between criticality and regulatory redundancy provides a means to modulate the dynamical regime of biochemical networks.

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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.

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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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KLESSE, ROCHUS, and MARCUS METZLER. "MODELING DISORDERED QUANTUM SYSTEMS WITH DYNAMICAL NETWORKS." International Journal of Modern Physics C 10, no. 04 (June 1999): 577–606. http://dx.doi.org/10.1142/s0129183199000449.

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It is the purpose of the present article to show that so-called network models, originally designed to describe static properties of disordered electronic systems, can be easily generalized to quantum-dynamical models, which then allow for an investigation of dynamical and spectral aspects. This concept is exemplified by the Chalker–Coddington model for the quantum Hall effect and a three-dimensional generalization of it. We simulate phase coherent diffusion of wave packets and consider spatial and spectral correlations of network eigenstates as well as the distribution of (quasi-)energy levels. Apart from that, it is demonstrated how network models can be used to determine two-point conductances. Our numerical calculations for the three-dimensional model at the Metal-Insulator transition point delivers, among others, an anomalous diffusion exponent of η=3-D2=1.7±0.1. The methods presented here in detail have been used partially in earlier work.

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Afraimovich, V. S., L. A. Bunimovich, and S. V. Moreno. "Dynamical networks: Continuous time and general discrete time models." Regular and Chaotic Dynamics 15, no. 2-3 (April 27, 2010): 127–45. http://dx.doi.org/10.1134/s1560354710020036.

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Abou-Jaoudé, Wassim, Denis Thieffry, and Jérôme Feret. "Formal derivation of qualitative dynamical models from biochemical networks." Biosystems 149 (November 2016): 70–112. http://dx.doi.org/10.1016/j.biosystems.2016.09.001.

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DeFord, Daryl R., and Scott D. Pauls. "A new framework for dynamical models on multiplex networks." Journal of Complex Networks 6, no. 3 (September 1, 2017): 353–81. http://dx.doi.org/10.1093/comnet/cnx041.

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SCHMIDT, G., G. ZAMORA-LÓPEZ, and J. KURTHS. "SIMULATION OF LARGE SCALE CORTICAL NETWORKS BY INDIVIDUAL NEURON DYNAMICS." International Journal of Bifurcation and Chaos 20, no. 03 (March 2010): 859–67. http://dx.doi.org/10.1142/s0218127410026149.

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Understanding the functional dynamics of the mammalian brain is one of the central aims of modern neuroscience. Mathematical modeling and computational simulations of neural networks can help in this quest. In recent publications, a multilevel model has been presented to simulate the resting-state dynamics of the cortico-cortical connectivity of the mammalian brain. In the present work we investigate how much of the dynamical behavior of the multilevel model can be reproduced by a strongly simplified model. We find that replacing each cortical area by a single Rulkov map recreates the patterns of dynamical correlation of the multilevel model, while the outcome of other models and setups mainly depends on the local network properties, e.g. the input degree of each vertex. In general, we find that a simple simulation whose dynamics depends on the global topology of the whole network is far from trivial. A systematic analysis of different dynamical models and coupling setups is required.

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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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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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Gates, Alexander J., Rion Brattig Correia, Xuan Wang, and Luis M. Rocha. "The effective graph reveals redundancy, canalization, and control pathways in biochemical regulation and signaling." Proceedings of the National Academy of Sciences 118, no. 12 (March 18, 2021): e2022598118. http://dx.doi.org/10.1073/pnas.2022598118.

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The ability to map causal interactions underlying genetic control and cellular signaling has led to increasingly accurate models of the complex biochemical networks that regulate cellular function. These network models provide deep insights into the organization, dynamics, and function of biochemical systems: for example, by revealing genetic control pathways involved in disease. However, the traditional representation of biochemical networks as binary interaction graphs fails to accurately represent an important dynamical feature of these multivariate systems: some pathways propagate control signals much more effectively than do others. Such heterogeneity of interactions reflects canalization—the system is robust to dynamical interventions in redundant pathways but responsive to interventions in effective pathways. Here, we introduce the effective graph, a weighted graph that captures the nonlinear logical redundancy present in biochemical network regulation, signaling, and control. Using 78 experimentally validated models derived from systems biology, we demonstrate that 1) redundant pathways are prevalent in biological models of biochemical regulation, 2) the effective graph provides a probabilistic but precise characterization of multivariate dynamics in a causal graph form, and 3) the effective graph provides an accurate explanation of how dynamical perturbation and control signals, such as those induced by cancer drug therapies, propagate in biochemical pathways. Overall, our results indicate that the effective graph provides an enriched description of the structure and dynamics of networked multivariate causal interactions. We demonstrate that it improves explainability, prediction, and control of complex dynamical systems in general and biochemical regulation in particular.

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Cardelli, Luca, Mirco Tribastone, Max Tschaikowski, and Andrea Vandin. "Maximal aggregation of polynomial dynamical systems." Proceedings of the National Academy of Sciences 114, no. 38 (September 6, 2017): 10029–34. http://dx.doi.org/10.1073/pnas.1702697114.

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Ordinary differential equations (ODEs) with polynomial derivatives are a fundamental tool for understanding the dynamics of systems across many branches of science, but our ability to gain mechanistic insight and effectively conduct numerical evaluations is critically hindered when dealing with large models. Here we propose an aggregation technique that rests on two notions of equivalence relating ODE variables whenever they have the same solution (backward criterion) or if a self-consistent system can be written for describing the evolution of sums of variables in the same equivalence class (forward criterion). A key feature of our proposal is to encode a polynomial ODE system into a finitary structure akin to a formal chemical reaction network. This enables the development of a discrete algorithm to efficiently compute the largest equivalence, building on approaches rooted in computer science to minimize basic models of computation through iterative partition refinements. The physical interpretability of the aggregation is shown on polynomial ODE systems for biochemical reaction networks, gene regulatory networks, and evolutionary game theory.

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Levin, Simon A., Kirk Moloney, Linda Buttel, and Carlos Castillo-Chavez. "Dynamical models of ecosystems and epidemics." Future Generation Computer Systems 5, no. 2-3 (September 1989): 265–74. http://dx.doi.org/10.1016/0167-739x(89)90046-0.

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Beiran, Manuel, Alexis Dubreuil, Adrian Valente, Francesca Mastrogiuseppe, and Srdjan Ostojic. "Shaping Dynamics With Multiple Populations in Low-Rank Recurrent Networks." Neural Computation 33, no. 6 (May 13, 2021): 1572–615. http://dx.doi.org/10.1162/neco_a_01381.

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An emerging paradigm proposes that neural computations can be understood at the level of dynamic systems that govern low-dimensional trajectories of collective neural activity. How the connectivity structure of a network determines the emergent dynamical system, however, remains to be clarified. Here we consider a novel class of models, gaussian-mixture, low-rank recurrent networks in which the rank of the connectivity matrix and the number of statistically defined populations are independent hyperparameters. We show that the resulting collective dynamics form a dynamical system, where the rank sets the dimensionality and the population structure shapes the dynamics. In particular, the collective dynamics can be described in terms of a simplified effective circuit of interacting latent variables. While having a single global population strongly restricts the possible dynamics, we demonstrate that if the number of populations is large enough, a rank R network can approximate any R-dimensional dynamical system.

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Fehr, David A., Joanna E. Handzlik, Manu, and Yen Lee Loh. "Classification-Based Inference of Dynamical Models of Gene Regulatory Networks." G3: Genes|Genomes|Genetics 9, no. 12 (October 17, 2019): 4183–95. http://dx.doi.org/10.1534/g3.119.400603.

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Voelker, Aaron R., and Chris Eliasmith. "Improving Spiking Dynamical Networks: Accurate Delays, Higher-Order Synapses, and Time Cells." Neural Computation 30, no. 3 (March 2018): 569–609. http://dx.doi.org/10.1162/neco_a_01046.

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Researchers building spiking neural networks face the challenge of improving the biological plausibility of their model networks while maintaining the ability to quantitatively characterize network behavior. In this work, we extend the theory behind the neural engineering framework (NEF), a method of building spiking dynamical networks, to permit the use of a broad class of synapse models while maintaining prescribed dynamics up to a given order. This theory improves our understanding of how low-level synaptic properties alter the accuracy of high-level computations in spiking dynamical networks. For completeness, we provide characterizations for both continuous-time (i.e., analog) and discrete-time (i.e., digital) simulations. We demonstrate the utility of these extensions by mapping an optimal delay line onto various spiking dynamical networks using higher-order models of the synapse. We show that these networks nonlinearly encode rolling windows of input history, using a scale invariant representation, with accuracy depending on the frequency content of the input signal. Finally, we reveal that these methods provide a novel explanation of time cell responses during a delay task, which have been observed throughout hippocampus, striatum, and cortex.

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Iigaya, Kiyohito, and Stefano Fusi. "Dynamical Regimes in Neural Network Models of Matching Behavior." Neural Computation 25, no. 12 (December 2013): 3093–112. http://dx.doi.org/10.1162/neco_a_00522.

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The matching law constitutes a quantitative description of choice behavior that is often observed in foraging tasks. According to the matching law, organisms distribute their behavior across available response alternatives in the same proportion that reinforcers are distributed across those alternatives. Recently a few biophysically plausible neural network models have been proposed to explain the matching behavior observed in the experiments. Here we study systematically the learning dynamics of these networks while performing a matching task on the concurrent variable interval (VI) schedule. We found that the model neural network can operate in one of three qualitatively different regimes depending on the parameters that characterize the synaptic dynamics and the reward schedule: (1) a matching behavior regime, in which the probability of choosing an option is roughly proportional to the baiting fractional probability of that option; (2) a perseverative regime, in which the network tends to make always the same decision; and (3) a tristable regime, in which the network can either perseverate or choose the two targets randomly approximately with the same probability. Different parameters of the synaptic dynamics lead to different types of deviations from the matching law, some of which have been observed experimentally. We show that the performance of the network depends on the number of stable states of each synapse and that bistable synapses perform close to optimal when the proper learning rate is chosen. Because our model provides a link between synaptic dynamics and qualitatively different behaviors, this work provides us with insight into the effects of neuromodulators on adaptive behaviors and psychiatric disorders.

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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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Banerjee, Arunava, Peggy Seriès, and Alexandre Pouget. "Dynamical Constraints on Using Precise Spike Timing to Compute in Recurrent Cortical Networks." Neural Computation 20, no. 4 (April 2008): 974–93. http://dx.doi.org/10.1162/neco.2008.05-06-206.

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Several recent models have proposed the use of precise timing of spikes for cortical computation. Such models rely on growing experimental evidence that neurons in the thalamus as well as many primary sensory cortical areas respond to stimuli with remarkable temporal precision. Models of computation based on spike timing, where the output of the network is a function not only of the input but also of an independently initializable internal state of the network, must, however, satisfy a critical constraint: the dynamics of the network should not be sensitive to initial conditions. We have previously developed an abstract dynamical system for networks of spiking neurons that has allowed us to identify the criterion for the stationary dynamics of a network to be sensitive to initial conditions. Guided by this criterion, we analyzed the dynamics of several recurrent cortical architectures, including one from the orientation selectivity literature. Based on the results, we conclude that under conditions of sustained, Poisson-like, weakly correlated, low to moderate levels of internal activity as found in the cortex, it is unlikely that recurrent cortical networks can robustly generate precise spike trajectories, that is, spatiotemporal patterns of spikes precise to the millisecond timescale.

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Kinney, Adrienne C., Sean Current, and Joceline Lega. "Aedes-AI: Neural network models of mosquito abundance." PLOS Computational Biology 17, no. 11 (November 19, 2021): e1009467. http://dx.doi.org/10.1371/journal.pcbi.1009467.

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We present artificial neural networks as a feasible replacement for a mechanistic model of mosquito abundance. We develop a feed-forward neural network, a long short-term memory recurrent neural network, and a gated recurrent unit network. We evaluate the networks in their ability to replicate the spatiotemporal features of mosquito populations predicted by the mechanistic model, and discuss how augmenting the training data with time series that emphasize specific dynamical behaviors affects model performance. We conclude with an outlook on how such equation-free models may facilitate vector control or the estimation of disease risk at arbitrary spatial scales.

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GIACOMETTI, A., A. MARITAN, and A. L. STELLA. "RENORMALIZATION GROUP APPROACH TO DYNAMICAL PROPERTIES OF HIERARCHICAL NETWORKS." International Journal of Modern Physics B 05, no. 05 (March 1991): 709–50. http://dx.doi.org/10.1142/s0217979291000407.

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Diffusion processes in the presence of hierarchical distributions of transition rates or waiting times are investigated by Renormalization Group (RG) techniques. Diffusion on one-dimensional chains, loop-less fractals and fully ultrametric spaces are considered. RG techniques are shown to be most natural and powerful to apply when infinitely many time scales are simultaneously involved in a problem. Generalizations and extensions of existing models and results are easily accomplished in the RG context. Wherever possible, heuristic scaling arguments are also presented in order to give an easier physical interpretation of the analytical results. Two relevant applications of ultradiffusion models are reviewed in detail. One of them concerns breakdown of dynamic scaling in a one-dimensional hierarchical Glauber chain. The other one is in the context of tethered random surface models.

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El Hamidi, Khadija, Mostafa Mjahed, Abdeljalil El Kari, and Hassan Ayad. "Adaptive Control Using Neural Networks and Approximate Models for Nonlinear Dynamic Systems." Modelling and Simulation in Engineering 2020 (August 26, 2020): 1–13. http://dx.doi.org/10.1155/2020/8642915.

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In this research, a comparative study of two recurrent neural networks, nonlinear autoregressive with exogenous input (NARX) neural network and nonlinear autoregressive moving average (NARMA-L2), and a feedforward neural network (FFNN) is performed for their ability to provide adaptive control of nonlinear systems. Three dynamical nonlinear systems of different complexity are considered. The aim of this work is to make the output of the plant follow the desired reference trajectory. The problem becomes more challenging when the dynamics of the plants are assumed to be unknown, and to tackle this problem, a multilayer neural network-based approximate model is set up which will work in parallel to the plant and the control scheme. The network parameters are updated using the dynamic backpropagation (BP) algorithm.

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Aledo, Juan A., Luis G. Diaz, Silvia Martinez, and Jose C. Valverde. "Dynamical attraction in parallel network models." Applied Mathematics and Computation 361 (November 2019): 874–88. http://dx.doi.org/10.1016/j.amc.2019.05.048.

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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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Summers, Tyler H., and Iman Shames. "Active influence in dynamical models of structural balance in social networks." EPL (Europhysics Letters) 103, no. 1 (July 1, 2013): 18001. http://dx.doi.org/10.1209/0295-5075/103/18001.

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Gilman, Alex, and Adam P. Arkin. "GENETIC“CODE”: Representations and Dynamical Models of Genetic Components and Networks." Annual Review of Genomics and Human Genetics 3, no. 1 (September 2002): 341–69. http://dx.doi.org/10.1146/annurev.genom.3.030502.111004.

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Geißler, Björn, Oliver Kolb, Jens Lang, Günter Leugering, Alexander Martin, and Antonio Morsi. "Mixed integer linear models for the optimization of dynamical transport networks." Mathematical Methods of Operations Research 73, no. 3 (April 21, 2011): 339–62. http://dx.doi.org/10.1007/s00186-011-0354-5.

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Orsatti, F. M., R. Carareto, and J. R. C. Piqueira. "Mutually connected phase-locked loop networks: dynamical models and design parameters." IET Circuits, Devices & Systems 2, no. 6 (2008): 495. http://dx.doi.org/10.1049/iet-cds:20080116.

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Aguirre, Luis Antonio, Gladstone Barbosa Alves, and Marcelo Vieira Corrêa. "Steady-state performance constraints for dynamical models based on RBF networks." Engineering Applications of Artificial Intelligence 20, no. 7 (October 2007): 924–35. http://dx.doi.org/10.1016/j.engappai.2006.11.021.

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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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Pineda, Fernando J. "Recurrent Backpropagation and the Dynamical Approach to Adaptive Neural Computation." Neural Computation 1, no. 2 (June 1989): 161–72. http://dx.doi.org/10.1162/neco.1989.1.2.161.

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Error backpropagation in feedforward neural network models is a popular learning algorithm that has its roots in nonlinear estimation and optimization. It is being used routinely to calculate error gradients in nonlinear systems with hundreds of thousands of parameters. However, the classical architecture for backpropagation has severe restrictions. The extension of backpropagation to networks with recurrent connections will be reviewed. It is now possible to efficiently compute the error gradients for networks that have temporal dynamics, which opens applications to a host of problems in systems identification and control.

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Schilling, Christian, Marcelo Forets, and Sebastián Guadalupe. "Verification of Neural-Network Control Systems by Integrating Taylor Models and Zonotopes." Proceedings of the AAAI Conference on Artificial Intelligence 36, no. 7 (June 28, 2022): 8169–77. http://dx.doi.org/10.1609/aaai.v36i7.20790.

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We study the verification problem for closed-loop dynamical systems with neural-network controllers (NNCS). This problem is commonly reduced to computing the set of reachable states. When considering dynamical systems and neural networks in isolation, there exist precise approaches for that task based on set representations respectively called Taylor models and zonotopes. However, the combination of these approaches to NNCS is non-trivial because, when converting between the set representations, dependency information gets lost in each control cycle and the accumulated approximation error quickly renders the result useless. We present an algorithm to chain approaches based on Taylor models and zonotopes, yielding a precise reachability algorithm for NNCS. Because the algorithm only acts at the interface of the isolated approaches, it is applicable to general dynamical systems and neural networks and can benefit from future advances in these areas. Our implementation delivers state-of-the-art performance and is the first to successfully analyze all benchmark problems of an annual reachability competition for NNCS.

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Linden, Nathaniel J., Boris Kramer, and Padmini Rangamani. "Bayesian parameter estimation for dynamical models in systems biology." PLOS Computational Biology 18, no. 10 (October 21, 2022): e1010651. http://dx.doi.org/10.1371/journal.pcbi.1010651.

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Dynamical systems modeling, particularly via systems of ordinary differential equations, has been used to effectively capture the temporal behavior of different biochemical components in signal transduction networks. Despite the recent advances in experimental measurements, including sensor development and ‘-omics’ studies that have helped populate protein-protein interaction networks in great detail, modeling in systems biology lacks systematic methods to estimate kinetic parameters and quantify associated uncertainties. This is because of multiple reasons, including sparse and noisy experimental measurements, lack of detailed molecular mechanisms underlying the reactions, and missing biochemical interactions. Additionally, the inherent nonlinearities with respect to the states and parameters associated with the system of differential equations further compound the challenges of parameter estimation. In this study, we propose a comprehensive framework for Bayesian parameter estimation and complete quantification of the effects of uncertainties in the data and models. We apply these methods to a series of signaling models of increasing mathematical complexity. Systematic analysis of these dynamical systems showed that parameter estimation depends on data sparsity, noise level, and model structure, including the existence of multiple steady states. These results highlight how focused uncertainty quantification can enrich systems biology modeling and enable additional quantitative analyses for parameter estimation.

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Stewart, Ian, and Dinis Gökaydin. "Symmetries of Quotient Networks for Doubly Periodic Patterns on the Square Lattice." International Journal of Bifurcation and Chaos 29, no. 10 (September 2019): 1930026. http://dx.doi.org/10.1142/s021812741930026x.

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Patterns of synchrony in networks of coupled dynamical systems can be represented as colorings of the nodes, in which nodes of the same color are synchronous. Balanced colorings, where nodes of the same color have color-isomorphic input sets, correspond to dynamically invariant subspaces, which can have a significant effect on the typical bifurcations of network dynamical systems. Orbit colorings for subgroups of the automorphism (symmetry) group of the network are always balanced, although the converse is false. We compute the automorphism groups of all doubly periodic quotient networks of the square lattice with nearest-neighbor coupling, and classify the “exotic” cases where this quotient network has extra automorphisms not induced by automorphisms of the square lattice. These comprise five isolated exceptions and two infinite families with wreath product symmetry. We also comment briefly on implications for bifurcations to doubly periodic patterns in square lattice models.

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Liu, Yuan Sophie, Angela Yu, and Philip Holmes. "Dynamical Analysis of Bayesian Inference Models for the Eriksen Task." Neural Computation 21, no. 6 (June 2009): 1520–53. http://dx.doi.org/10.1162/neco.2009.03-07-495.

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The Eriksen task is a classical paradigm that explores the effects of competing sensory inputs on response tendencies and the nature of selective attention in controlling these processes. In this task, conflicting flanker stimuli interfere with the processing of a central target, especially on short reaction time trials. This task has been modeled by neural networks and more recently by a normative Bayesian account. Here, we analyze the dynamics of the Bayesian models, which are nonlinear, coupled discrete time dynamical systems, by considering simplified, approximate systems that are linear and decoupled. Analytical solutions of these allow us to describe how posterior probabilities and psychometric functions depend on model parameters. We compare our results with numerical simulations of the original models and derive fits to experimental data, showing that agreements are rather good. We also investigate continuum limits of these simplified dynamical systems and demonstrate that Bayesian updating is closely related to a drift-diffusion process, whose implementation in neural network models has been extensively studied. This provides insight into how neural substrates can implement Bayesian computations.

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Zhou, Mingcong, and Zhaoyan Wu. "Structure Identification of Fractional-Order Dynamical Network with Different Orders." Mathematics 9, no. 17 (August 30, 2021): 2096. http://dx.doi.org/10.3390/math9172096.

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Topology structure and system parameters have a great influence on the dynamical behavior of dynamical networks. However, they are sometimes unknown or uncertain in advance. How to effectively identify them has been investigated in various network models, from integer-order networks to fractional-order networks with the same order. In the real world, many systems consist of subsystems with different fractional orders. Therefore, the structure identification of a dynamical network with different fractional orders is investigated in this paper. Through designing proper adaptive controllers and parameter updating laws, two network estimators are well constructed. One is for identifying only the unknown topology structure. The other is for identifying both the unknown topology structure and system parameters. Based on the Lyapunov function method and the stability theory of fractional-order dynamical systems, the theoretical results are analytically proved. The effectiveness is verified by three numerical examples as well. In addition, the designed estimators have a good performance in monitoring switching topology. From the practical viewpoint, the designed estimators can be used to monitor the change of current and voltage in the fractional-order circuit systems.

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Lotito, Quintino Francesco, Davide Zanella, and Paolo Casari. "Realistic Aspects of Simulation Models for Fake News Epidemics over Social Networks." Future Internet 13, no. 3 (March 17, 2021): 76. http://dx.doi.org/10.3390/fi13030076.

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The pervasiveness of online social networks has reshaped the way people access information. Online social networks make it common for users to inform themselves online and share news among their peers, but also favor the spreading of both reliable and fake news alike. Because fake news may have a profound impact on the society at large, realistically simulating their spreading process helps evaluate the most effective countermeasures to adopt. It is customary to model the spreading of fake news via the same epidemic models used for common diseases; however, these models often miss concepts and dynamics that are peculiar to fake news spreading. In this paper, we fill this gap by enriching typical epidemic models for fake news spreading with network topologies and dynamics that are typical of realistic social networks. Specifically, we introduce agents with the role of influencers and bots in the model and consider the effects of dynamical network access patterns, time-varying engagement, and different degrees of trust in the sources of circulating information. These factors concur with making the simulations more realistic. Among other results, we show that influencers that share fake news help the spreading process reach nodes that would otherwise remain unaffected. Moreover, we emphasize that bots dramatically speed up the spreading process and that time-varying engagement and network access change the effectiveness of fake news spreading.

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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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Cao, Hongjun, and Borja Ibarz. "Hybrid discrete-time neural networks." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 368, no. 1930 (November 13, 2010): 5071–86. http://dx.doi.org/10.1098/rsta.2010.0171.

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Hybrid dynamical systems combine evolution equations with state transitions. When the evolution equations are discrete-time (also called map-based), the result is a hybrid discrete-time system. A class of biological neural network models that has recently received some attention falls within this category: map-based neuron models connected by means of fast threshold modulation (FTM). FTM is a connection scheme that aims to mimic the switching dynamics of a neuron subject to synaptic inputs. The dynamic equations of the neuron adopt different forms according to the state (either firing or not firing) and type (excitatory or inhibitory) of their presynaptic neighbours. Therefore, the mathematical model of one such network is a combination of discrete-time evolution equations with transitions between states, constituting a hybrid discrete-time (map-based) neural network. In this paper, we review previous work within the context of these models, exemplifying useful techniques to analyse them. Typical map-based neuron models are low-dimensional and amenable to phase-plane analysis. In bursting models, fast–slow decomposition can be used to reduce dimensionality further, so that the dynamics of a pair of connected neurons can be easily understood. We also discuss a model that includes electrical synapses in addition to chemical synapses with FTM. Furthermore, we describe how master stability functions can predict the stability of synchronized states in these networks. The main results are extended to larger map-based neural networks.

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Journal articles on the topic 'Dynamical models on networks'

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