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Voutsa V, Battaglia D, Bracken LJ, Brovelli A, Costescu J, Díaz Muñoz M, Fath BD, Funk A, Guirro M, Hein T, Kerschner C, Kimmich C, Lima V, Messé A, Parsons AJ, Perez J, Pöppl R, Prell C, Recinos S, Shi Y, Tiwari S, Turnbull L, Wainwright J, Waxenecker H, Hütt MT. Two classes of functional connectivity in dynamical processes in networks. J R Soc Interface 2021; 18:20210486. [PMID: 34665977 PMCID: PMC8526174 DOI: 10.1098/rsif.2021.0486] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/12/2021] [Accepted: 09/13/2021] [Indexed: 12/12/2022] Open
Abstract
The relationship between network structure and dynamics is one of the most extensively investigated problems in the theory of complex systems of recent years. Understanding this relationship is of relevance to a range of disciplines-from neuroscience to geomorphology. A major strategy of investigating this relationship is the quantitative comparison of a representation of network architecture (structural connectivity, SC) with a (network) representation of the dynamics (functional connectivity, FC). Here, we show that one can distinguish two classes of functional connectivity-one based on simultaneous activity (co-activity) of nodes, the other based on sequential activity of nodes. We delineate these two classes in different categories of dynamical processes-excitations, regular and chaotic oscillators-and provide examples for SC/FC correlations of both classes in each of these models. We expand the theoretical view of the SC/FC relationships, with conceptual instances of the SC and the two classes of FC for various application scenarios in geomorphology, ecology, systems biology, neuroscience and socio-ecological systems. Seeing the organisation of dynamical processes in a network either as governed by co-activity or by sequential activity allows us to bring some order in the myriad of observations relating structure and function of complex networks.
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Affiliation(s)
- Venetia Voutsa
- Department of Life Sciences and Chemistry, Jacobs University Bremen, 28759 Bremen, Germany
| | - Demian Battaglia
- Aix-Marseille Université, Inserm, Institut de Neurosciences des Systèmes (UMR 1106), Marseille, France
- University of Strasbourg Institute for Advanced Studies (USIAS), Strasbourg 67083, France
| | | | - Andrea Brovelli
- Aix-Marseille Université, CNRS, Institut de Neurosciences de la Timone (UMR 7289), Marseille, France
| | - Julia Costescu
- Department of Geography, Durham University, Durham DH1 3LE, UK
| | - Mario Díaz Muñoz
- Department of Sustainability, Governance and Methods, Modul University Vienna, 1190 Vienna, Austria
| | - Brian D. Fath
- Department of Biological Sciences, Towson University, Towson, Maryland 21252, USA
- Advancing Systems Analysis Program, International Institute for Applied Systems Analysis, Laxenburg 2361, Austria
- Department of Environmental Studies, Masaryk University, 60200 Brno, Czech Republic
| | - Andrea Funk
- Institute of Hydrobiology and Aquatic Ecosystem Management (IHG), University of Natural Resources and Life Sciences Vienna (BOKU), 1180 Vienna, Austria
- WasserCluster Lunz - Biologische Station GmbH, Dr. Carl Kupelwieser Promenade 5, 3293 Lunz am See, Austria
| | - Mel Guirro
- Department of Geography, Durham University, Durham DH1 3LE, UK
| | - Thomas Hein
- Institute of Hydrobiology and Aquatic Ecosystem Management (IHG), University of Natural Resources and Life Sciences Vienna (BOKU), 1180 Vienna, Austria
- WasserCluster Lunz - Biologische Station GmbH, Dr. Carl Kupelwieser Promenade 5, 3293 Lunz am See, Austria
| | - Christian Kerschner
- Department of Sustainability, Governance and Methods, Modul University Vienna, 1190 Vienna, Austria
- Department of Environmental Studies, Masaryk University, 60200 Brno, Czech Republic
| | - Christian Kimmich
- Department of Environmental Studies, Masaryk University, 60200 Brno, Czech Republic
- Regional Science and Environmental Research, Institute for Advanced Studies, 1080 Vienna, Austria
| | - Vinicius Lima
- Aix-Marseille Université, Inserm, Institut de Neurosciences des Systèmes (UMR 1106), Marseille, France
- Aix-Marseille Université, CNRS, Institut de Neurosciences de la Timone (UMR 7289), Marseille, France
| | - Arnaud Messé
- Department of Computational Neuroscience, University Medical Center Eppendorf, Hamburg University, Germany
| | | | - John Perez
- Department of Geography, Durham University, Durham DH1 3LE, UK
| | - Ronald Pöppl
- Department of Geography and Regional Research, University of Vienna, Universitätsstr. 7, 1010 Vienna, Austria
| | - Christina Prell
- Department of Cultural Geography, University of Groningen, 9747 AD, Groningen, The Netherlands
| | - Sonia Recinos
- Institute of Hydrobiology and Aquatic Ecosystem Management (IHG), University of Natural Resources and Life Sciences Vienna (BOKU), 1180 Vienna, Austria
| | - Yanhua Shi
- Department of Environmental Studies, Masaryk University, 60200 Brno, Czech Republic
| | - Shubham Tiwari
- Department of Geography, Durham University, Durham DH1 3LE, UK
| | - Laura Turnbull
- Department of Geography, Durham University, Durham DH1 3LE, UK
| | - John Wainwright
- Department of Geography, Durham University, Durham DH1 3LE, UK
| | - Harald Waxenecker
- Department of Environmental Studies, Masaryk University, 60200 Brno, Czech Republic
| | - Marc-Thorsten Hütt
- Department of Life Sciences and Chemistry, Jacobs University Bremen, 28759 Bremen, Germany
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Moretti P, Hütt MT. Link-usage asymmetry and collective patterns emerging from rich-club organization of complex networks. Proc Natl Acad Sci U S A 2020; 117:18332-18340. [PMID: 32690716 PMCID: PMC7414146 DOI: 10.1073/pnas.1919785117] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
In models of excitable dynamics on graphs, excitations can travel in both directions of an undirected link. However, as a striking interplay of dynamics and network topology, excitations often establish a directional preference. Some of these cases of "link-usage asymmetry" are local in nature and can be mechanistically understood, for instance, from the degree gradient of a link (i.e., the difference in node degrees at both ends of the link). Other contributions to the link-usage asymmetry are instead, as we show, self-organized in nature, and strictly nonlocal. This is the case for excitation waves, where the preferential propagation of excitations along a link depends on its orientation with respect to a hub acting as a source, even if the link in question is several steps away from the hub itself. Here, we identify and quantify the contribution of such self-organized patterns to link-usage asymmetry and show that they extend to ranges significantly longer than those ascribed to local patterns. We introduce a topological characterization, the hub-set-orientation prevalence of a link, which indicates its average orientation with respect to the hubs of a graph. Our numerical results show that the hub-set-orientation prevalence of a link strongly correlates with the preferential usage of the link in the direction of propagation away from the hub core of the graph. Our methodology is embedding-agnostic and allows for the measurement of wave signals and the sizes of the cores from which they originate.
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Affiliation(s)
- Paolo Moretti
- Institute of Materials Simulation, Department of Materials Science, Friedrich-Alexander-University Erlangen-Nürnberg, D-90762 Fürth, Germany;
| | - Marc-Thorsten Hütt
- Department of Life Sciences and Chemistry, Jacobs University Bremen, D-28759 Bremen, Germany
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3
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Sizemore Blevins A, Bassett DS. Reorderability of node-filtered order complexes. Phys Rev E 2020; 101:052311. [PMID: 32575295 DOI: 10.1103/physreve.101.052311] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/08/2019] [Accepted: 02/19/2020] [Indexed: 06/11/2023]
Abstract
Growing graphs describe a multitude of developing processes from maturing brains to expanding vocabularies to burgeoning public transit systems. Each of these growing processes likely adheres to proliferation rules that establish an effective order of node and connection emergence. When followed, such proliferation rules allow the system to properly develop along a predetermined trajectory. But rules are rarely followed. Here we ask what topological changes in the growing graph trajectories might occur after the specific but basic perturbation of permuting the node emergence order. Specifically, we harness applied topological methods to determine which of six growing graph models exhibit topology that is robust to randomizing node order, termed global reorderability, and robust to temporally local node swaps, termed local reorderability. We find that the six graph models fall upon a spectrum of both local and global reorderability, and furthermore we provide theoretical connections between robustness to node pair ordering and robustness to arbitrary node orderings. Finally, we discuss real-world applications of reorderability analyses and suggest possibilities for designing reorderable networks.
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Affiliation(s)
- Ann Sizemore Blevins
- Department of Bioengineering, School of Engineering and Applied Science, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Danielle S Bassett
- Department of Bioengineering, School of Engineering and Applied Science, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
- Department of Physics and Astronomy, College of Arts and Sciences, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
- Department of Electrical and Systems Engineering, School of Engineering and Applied Science, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
- Department of Neurology, Perelman School of Medicine, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
- Department of Psychiatry, Perelman School of Medicine, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
- Santa Fe Institute, Santa Fe, New Mexico 87501, USA
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Abstract
Concepts shape the interpretation of facts. One of the most popular concepts in systems neuroscience is that of ‘hierarchy’. However, this concept has been interpreted in many different ways, which are not well aligned. This observation suggests that the concept is ill defined. Using the example of the organization of the primate visual cortical system, we explore several contexts in which ‘hierarchy’ is currently used in the description of brain networks. We distinguish at least four different uses, specifically, ‘hierarchy’ as a topological sequence of projections, as a gradient of features, as a progression of scales, or as a sorting of laminar projection patterns. We discuss the interpretation and functional implications of the different notions of ‘hierarchy’ in these contexts and suggest that more specific terms than ‘hierarchy’ should be used for a deeper understanding of the different dimensions of the organization of brain networks. This article is part of the theme issue ‘Unifying the essential concepts of biological networks: biological insights and philosophical foundations’.
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Affiliation(s)
- Claus C Hilgetag
- Institute of Computational Neuroscience, University Medical Center Eppendorf, Hamburg University, Hamburg, Germany.,Department of Health Sciences, Boston University, Boston, MA, USA
| | - Alexandros Goulas
- Institute of Computational Neuroscience, University Medical Center Eppendorf, Hamburg University, Hamburg, Germany
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Damicelli F, Hilgetag CC, Hütt MT, Messé A. Topological reinforcement as a principle of modularity emergence in brain networks. Netw Neurosci 2019; 3:589-605. [PMID: 31157311 PMCID: PMC6542620 DOI: 10.1162/netn_a_00085] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/11/2018] [Accepted: 03/21/2019] [Indexed: 12/02/2022] Open
Abstract
Modularity is a ubiquitous topological feature of structural brain networks at various scales. Although a variety of potential mechanisms have been proposed, the fundamental principles by which modularity emerges in neural networks remain elusive. We tackle this question with a plasticity model of neural networks derived from a purely topological perspective. Our topological reinforcement model acts enhancing the topological overlap between nodes, that is, iteratively allowing connections between non-neighbor nodes with high neighborhood similarity. This rule reliably evolves synthetic random networks toward a modular architecture. Such final modular structure reflects initial "proto-modules," thus allowing to predict the modules of the evolved graph. Subsequently, we show that this topological selection principle might be biologically implemented as a Hebbian rule. Concretely, we explore a simple model of excitable dynamics, where the plasticity rule acts based on the functional connectivity (co-activations) between nodes. Results produced by the activity-based model are consistent with the ones from the purely topological rule in terms of the final network configuration and modules composition. Our findings suggest that the selective reinforcement of topological overlap may be a fundamental mechanism contributing to modularity emergence in brain networks.
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Affiliation(s)
- Fabrizio Damicelli
- Institute of Computational Neuroscience, University Medical Center Eppendorf, Hamburg University, Hamburg, Germany
| | - Claus C. Hilgetag
- Institute of Computational Neuroscience, University Medical Center Eppendorf, Hamburg University, Hamburg, Germany
- Department of Health Sciences, Boston University, Boston, Massachusetts, United States of America
| | - Marc-Thorsten Hütt
- Department of Life Sciences and Chemistry, Jacobs University, Bremen, Germany
| | - Arnaud Messé
- Institute of Computational Neuroscience, University Medical Center Eppendorf, Hamburg University, Hamburg, Germany
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Rodriguez N, Izquierdo E, Ahn YY. Optimal modularity and memory capacity of neural reservoirs. Netw Neurosci 2019; 3:551-566. [PMID: 31089484 PMCID: PMC6497001 DOI: 10.1162/netn_a_00082] [Citation(s) in RCA: 20] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/28/2018] [Accepted: 02/25/2019] [Indexed: 11/04/2022] Open
Abstract
The neural network is a powerful computing framework that has been exploited by biological evolution and by humans for solving diverse problems. Although the computational capabilities of neural networks are determined by their structure, the current understanding of the relationships between a neural network's architecture and function is still primitive. Here we reveal that a neural network's modular architecture plays a vital role in determining the neural dynamics and memory performance of the network of threshold neurons. In particular, we demonstrate that there exists an optimal modularity for memory performance, where a balance between local cohesion and global connectivity is established, allowing optimally modular networks to remember longer. Our results suggest that insights from dynamical analysis of neural networks and information-spreading processes can be leveraged to better design neural networks and may shed light on the brain's modular organization.
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Affiliation(s)
- Nathaniel Rodriguez
- School of Informatics, Computing, and Engineering, Indiana University, Bloomington, IN, USA
| | - Eduardo Izquierdo
- School of Informatics, Computing, and Engineering, Indiana University, Bloomington, IN, USA
- Cognitive Science Program, Indiana University, Bloomington, IN, USA
| | - Yong-Yeol Ahn
- School of Informatics, Computing, and Engineering, Indiana University, Bloomington, IN, USA
- Indiana University Network Science Institute, Bloomington, IN, USA
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Morgan SE, Achard S, Termenon M, Bullmore ET, Vértes PE. Low-dimensional morphospace of topological motifs in human fMRI brain networks. Netw Neurosci 2018; 2:285-302. [PMID: 30215036 PMCID: PMC6130546 DOI: 10.1162/netn_a_00038] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/01/2017] [Accepted: 12/04/2017] [Indexed: 11/20/2022] Open
Abstract
We present a low-dimensional morphospace of fMRI brain networks, where axes are defined in a data-driven manner based on the network motifs. The morphospace allows us to identify the key variations in healthy fMRI networks in terms of their underlying motifs, and we observe that two principal components (PCs) can account for 97% of the motif variability. The first PC of the motif distribution is correlated with efficiency and inversely correlated with transitivity. Hence this axis approximately conforms to the well-known economical small-world trade-off between integration and segregation in brain networks. Finally, we show that the economical clustering generative model proposed by Vértes et al. (2012) can approximately reproduce the motif morphospace of the real fMRI brain networks, in contrast to other generative models. Overall, the motif morphospace provides a powerful way to visualize the relationships between network properties and to investigate generative or constraining factors in the formation of complex human brain functional networks.
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Affiliation(s)
- Sarah E. Morgan
- Brain Mapping Unit, Psychiatry Department, Cambridge University, Cambridge, United Kingdom
| | - Sophie Achard
- Cambridgeshire and Peterborough NHS Foundation Trust, Huntingdon, PE29 3RJ, UK
| | - Maite Termenon
- Cambridgeshire and Peterborough NHS Foundation Trust, Huntingdon, PE29 3RJ, UK
| | - Edward T. Bullmore
- Brain Mapping Unit, Psychiatry Department, Cambridge University, Cambridge, United Kingdom
- Cambridgeshire and Peterborough NHS Foundation Trust, Huntingdon, PE29 3RJ, UK
- ImmunoPsychiatry, Immuno-Inflammation Therapeutic Area Unit, GlaxoSmithKline R&D, Stevenage, SG1 2NY, UK
| | - Petra E. Vértes
- Brain Mapping Unit, Psychiatry Department, Cambridge University, Cambridge, United Kingdom
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Fretter C, Lesne A, Hilgetag CC, Hütt MT. Topological determinants of self-sustained activity in a simple model of excitable dynamics on graphs. Sci Rep 2017; 7:42340. [PMID: 28186182 PMCID: PMC5301238 DOI: 10.1038/srep42340] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/16/2016] [Accepted: 01/09/2017] [Indexed: 11/08/2022] Open
Abstract
Simple models of excitable dynamics on graphs are an efficient framework for studying the interplay between network topology and dynamics. This topic is of practical relevance to diverse fields, ranging from neuroscience to engineering. Here we analyze how a single excitation propagates through a random network as a function of the excitation threshold, that is, the relative amount of activity in the neighborhood required for the excitation of a node. We observe that two sharp transitions delineate a region of sustained activity. Using analytical considerations and numerical simulation, we show that these transitions originate from the presence of barriers to propagation and the excitation of topological cycles, respectively, and can be predicted from the network topology. Our findings are interpreted in the context of network reverberations and self-sustained activity in neural systems, which is a question of long-standing interest in computational neuroscience.
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Affiliation(s)
- Christoph Fretter
- Department of Life Sciences and Chemistry, Jacobs University Bremen, D-28759 Bremen, Germany
- Department of Computational Neuroscience, Universitätsklinikum Hamburg-Eppendorf, D-20246 Hamburg, Germany
| | - Annick Lesne
- LPTMC, CNRS, UMR 7600, UPMC-Paris 6, Sorbonne Universités, 4 place Jussieu, F-75252, Paris, France
- Institut de Génétique Moléculaire de Montpellier, UMR 5535 CNRS, 1919 route de Mende, 34293 Montpellier cedex 5, France; Université de Montpellier, 163 rue Auguste Broussonnet, 34090 Montpellier, France
| | - Claus C. Hilgetag
- Department of Computational Neuroscience, Universitätsklinikum Hamburg-Eppendorf, D-20246 Hamburg, Germany
- Department of Health Sciences, Boston University, Boston, USA
| | - Marc-Thorsten Hütt
- Department of Life Sciences and Chemistry, Jacobs University Bremen, D-28759 Bremen, Germany
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