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Bagheri A, Dehshiri M, Bagheri Y, Akhondi-Asl A, Nadjar Araabi B. Brain effective connectome based on fMRI and DTI data: Bayesian causal learning and assessment. PLoS One 2023; 18:e0289406. [PMID: 37594972 PMCID: PMC10437876 DOI: 10.1371/journal.pone.0289406] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/20/2023] [Accepted: 07/18/2023] [Indexed: 08/20/2023] Open
Abstract
Neuroscientific studies aim to find an accurate and reliable brain Effective Connectome (EC). Although current EC discovery methods have contributed to our understanding of brain organization, their performances are severely constrained by the short sample size and poor temporal resolution of fMRI data, and high dimensionality of the brain connectome. By leveraging the DTI data as prior knowledge, we introduce two Bayesian causal discovery frameworks -the Bayesian GOLEM (BGOLEM) and Bayesian FGES (BFGES) methods- that offer significantly more accurate and reliable ECs and address the shortcomings of the existing causal discovery methods in discovering ECs based on only fMRI data. Moreover, to numerically assess the improvement in the accuracy of ECs with our method on empirical data, we introduce the Pseudo False Discovery Rate (PFDR) as a new computational accuracy metric for causal discovery in the brain. Through a series of simulation studies on synthetic and hybrid data (combining DTI from the Human Connectome Project (HCP) subjects and synthetic fMRI), we demonstrate the effectiveness of our proposed methods and the reliability of the introduced metric in discovering ECs. By employing the PFDR metric, we show that our Bayesian methods lead to significantly more accurate results compared to the traditional methods when applied to the Human Connectome Project (HCP) data. Additionally, we measure the reproducibility of discovered ECs using the Rogers-Tanimoto index for test-retest data and show that our Bayesian methods provide significantly more reliable ECs than traditional methods. Overall, our study's numerical and visual results highlight the potential for these frameworks to significantly advance our understanding of brain functionality.
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Affiliation(s)
- Abdolmahdi Bagheri
- School of Electrical and Computer Engineering, College of Engineering, University of Tehran, Tehran, Iran
| | - Mahdi Dehshiri
- School of Electrical and Computer Engineering, College of Engineering, University of Tehran, Tehran, Iran
| | - Yamin Bagheri
- Department of Psychology, Faculty of Psychology and Education, University of Tehran, Tehran, Iran
| | - Alireza Akhondi-Asl
- Department of Anaesthesia, Harvard Medical School, Boston, Massachusetts, United States of America
| | - Babak Nadjar Araabi
- School of Electrical and Computer Engineering, College of Engineering, University of Tehran, Tehran, Iran
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2
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Messé A, Hollensteiner KJ, Delettre C, Dell-Brown LA, Pieper F, Nentwig LJ, Galindo-Leon EE, Larrat B, Mériaux S, Mangin JF, Reillo I, de Juan Romero C, Borrell V, Engler G, Toro R, Engel AK, Hilgetag CC. Structural basis of envelope and phase intrinsic coupling modes in the cerebral cortex. Neuroimage 2023; 276:120212. [PMID: 37269959 PMCID: PMC10300241 DOI: 10.1016/j.neuroimage.2023.120212] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/09/2022] [Revised: 05/09/2023] [Accepted: 05/30/2023] [Indexed: 06/05/2023] Open
Abstract
Intrinsic coupling modes (ICMs) can be observed in ongoing brain activity at multiple spatial and temporal scales. Two families of ICMs can be distinguished: phase and envelope ICMs. The principles that shape these ICMs remain partly elusive, in particular their relation to the underlying brain structure. Here we explored structure-function relationships in the ferret brain between ICMs quantified from ongoing brain activity recorded with chronically implanted micro-ECoG arrays and structural connectivity (SC) obtained from high-resolution diffusion MRI tractography. Large-scale computational models were used to explore the ability to predict both types of ICMs. Importantly, all investigations were conducted with ICM measures that are sensitive or insensitive to volume conduction effects. The results show that both types of ICMs are significantly related to SC, except for phase ICMs when using measures removing zero-lag coupling. The correlation between SC and ICMs increases with increasing frequency which is accompanied by reduced delays. Computational models produced results that were highly dependent on the specific parameter settings. The most consistent predictions were derived from measures solely based on SC. Overall, the results demonstrate that patterns of cortical functional coupling as reflected in both phase and envelope ICMs are both related, albeit to different degrees, to the underlying structural connectivity in the cerebral cortex.
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Affiliation(s)
- Arnaud Messé
- Institute of Computational Neuroscience, Hamburg Center of Neuroscience, University Medical Center Hamburg-Eppendorf, Hamburg University, Martinistraße 52, Hamburg 20246, Germany.
| | - Karl J Hollensteiner
- Department of Neurophysiology and Pathophysiology, Hamburg Center of Neuroscience, University Medical Center Hamburg-Eppendorf, Hamburg University, Martinistraße 52, Hamburg 20246, Germany
| | - Céline Delettre
- Institute of Computational Neuroscience, Hamburg Center of Neuroscience, University Medical Center Hamburg-Eppendorf, Hamburg University, Martinistraße 52, Hamburg 20246, Germany; Unité de Neuroanatomie Appliquée et Théorique, Unité de Génétique Humaine et Fonctions Cognitives, Institut Pasteur, Université Paris Cité, 25-28 rue du Dr Roux, Paris 75015, France
| | - Leigh-Anne Dell-Brown
- Institute of Computational Neuroscience, Hamburg Center of Neuroscience, University Medical Center Hamburg-Eppendorf, Hamburg University, Martinistraße 52, Hamburg 20246, Germany
| | - Florian Pieper
- Department of Neurophysiology and Pathophysiology, Hamburg Center of Neuroscience, University Medical Center Hamburg-Eppendorf, Hamburg University, Martinistraße 52, Hamburg 20246, Germany
| | - Lena J Nentwig
- Institute of Computational Neuroscience, Hamburg Center of Neuroscience, University Medical Center Hamburg-Eppendorf, Hamburg University, Martinistraße 52, Hamburg 20246, Germany
| | - Edgar E Galindo-Leon
- Department of Neurophysiology and Pathophysiology, Hamburg Center of Neuroscience, University Medical Center Hamburg-Eppendorf, Hamburg University, Martinistraße 52, Hamburg 20246, Germany
| | - Benoît Larrat
- NeuroSpin, CEA, Paris-Saclay University, Centre d'études de Saclay, Bâtiment 145, Gif-sur-Yvette 91191, France
| | - Sébastien Mériaux
- NeuroSpin, CEA, Paris-Saclay University, Centre d'études de Saclay, Bâtiment 145, Gif-sur-Yvette 91191, France
| | - Jean-François Mangin
- NeuroSpin, CEA, Paris-Saclay University, Centre d'études de Saclay, Bâtiment 145, Gif-sur-Yvette 91191, France
| | - Isabel Reillo
- Developmental Neurobiology Unit, Instituto de Neurociencias, Consejo Superior de Investigaciones Científicas - Universidad Miguel Hernández, Sant Joan d'Alacant, Av. Santiago Ramón y Cajal s/n, Sant Joan d'Alacant 03550, Spain
| | - Camino de Juan Romero
- Developmental Neurobiology Unit, Instituto de Neurociencias, Consejo Superior de Investigaciones Científicas - Universidad Miguel Hernández, Sant Joan d'Alacant, Av. Santiago Ramón y Cajal s/n, Sant Joan d'Alacant 03550, Spain
| | - Víctor Borrell
- Developmental Neurobiology Unit, Instituto de Neurociencias, Consejo Superior de Investigaciones Científicas - Universidad Miguel Hernández, Sant Joan d'Alacant, Av. Santiago Ramón y Cajal s/n, Sant Joan d'Alacant 03550, Spain
| | - Gerhard Engler
- Department of Neurophysiology and Pathophysiology, Hamburg Center of Neuroscience, University Medical Center Hamburg-Eppendorf, Hamburg University, Martinistraße 52, Hamburg 20246, Germany
| | - Roberto Toro
- Unité de Neuroanatomie Appliquée et Théorique, Unité de Génétique Humaine et Fonctions Cognitives, Institut Pasteur, Université Paris Cité, 25-28 rue du Dr Roux, Paris 75015, France; Center for Research and Interdisciplinarity, Paris Descartes University, 24, rue du Faubourg Saint Jacques, Paris 75014, France
| | - Andreas K Engel
- Department of Neurophysiology and Pathophysiology, Hamburg Center of Neuroscience, University Medical Center Hamburg-Eppendorf, Hamburg University, Martinistraße 52, Hamburg 20246, Germany
| | - Claus C Hilgetag
- Institute of Computational Neuroscience, Hamburg Center of Neuroscience, University Medical Center Hamburg-Eppendorf, Hamburg University, Martinistraße 52, Hamburg 20246, Germany; Department of Health Sciences, Boston University, 635 Commonwealth Avenue, Boston, Massachusetts 02215, USA
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3
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Alexandersen CG, de Haan W, Bick C, Goriely A. A multi-scale model explains oscillatory slowing and neuronal hyperactivity in Alzheimer's disease. J R Soc Interface 2023; 20:20220607. [PMID: 36596460 PMCID: PMC9810432 DOI: 10.1098/rsif.2022.0607] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/05/2023] Open
Abstract
Alzheimer's disease is the most common cause of dementia and is linked to the spreading of pathological amyloid-β and tau proteins throughout the brain. Recent studies have highlighted stark differences in how amyloid-β and tau affect neurons at the cellular scale. On a larger scale, Alzheimer's patients are observed to undergo a period of early-stage neuronal hyperactivation followed by neurodegeneration and frequency slowing of neuronal oscillations. Herein, we model the spreading of both amyloid-β and tau across a human connectome and investigate how the neuronal dynamics are affected by disease progression. By including the effects of both amyloid-β and tau pathology, we find that our model explains AD-related frequency slowing, early-stage hyperactivation and late-stage hypoactivation. By testing different hypotheses, we show that hyperactivation and frequency slowing are not due to the topological interactions between different regions but are mostly the result of local neurotoxicity induced by amyloid-β and tau protein.
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Affiliation(s)
| | - Willem de Haan
- Alzheimer Center Amsterdam, Department of Neurology, Amsterdam Neuroscience, Vrije Universiteit Amsterdam, Amsterdam UMC, Amsterdam, The Netherlands
| | - Christian Bick
- Mathematical Institute, University of Oxford, Oxford, UK,Department of Mathematics, Vrije Universiteit Amsterdam, Amsterdam, The Netherlands,Amsterdam Neuroscience—Systems and Network Neuroscience, Amsterdam, The Netherlands
| | - Alain Goriely
- Mathematical Institute, University of Oxford, Oxford, UK
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4
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Lea-Carnall CA, Tanner LI, Montemurro MA. Noise-modulated multistable synapses in a Wilson-Cowan-based model of plasticity. Front Comput Neurosci 2023; 17:1017075. [PMID: 36817317 PMCID: PMC9931909 DOI: 10.3389/fncom.2023.1017075] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/12/2022] [Accepted: 01/10/2023] [Indexed: 02/05/2023] Open
Abstract
Frequency-dependent plasticity refers to changes in synaptic strength in response to different stimulation frequencies. Resonance is a factor known to be of importance in such frequency dependence, however, the role of neural noise in the process remains elusive. Considering the brain is an inherently noisy system, understanding its effects may prove beneficial in shaping therapeutic interventions based on non-invasive brain stimulation protocols. The Wilson-Cowan (WC) model is a well-established model to describe the average dynamics of neural populations and has been shown to exhibit bistability in the presence of noise. However, the important question of how the different stable regimes in the WC model can affect synaptic plasticity when cortical populations interact has not yet been addressed. Therefore, we investigated plasticity dynamics in a WC-based model of interacting neural populations coupled with activity-dependent synapses in which a periodic stimulation was applied in the presence of noise of controlled intensity. The results indicate that for a narrow range of the noise variance, synaptic strength can be optimized. In particular, there is a regime of noise intensity for which synaptic strength presents a triple-stable state. Regulating noise intensity affects the probability that the system chooses one of the stable states, thereby controlling plasticity. These results suggest that noise is a highly influential factor in determining the outcome of plasticity induced by stimulation.
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Affiliation(s)
- Caroline A Lea-Carnall
- School of Health Sciences, Manchester Academic Health Science Centre, Faculty of Biology, Medicine and Health, University of Manchester, Manchester, United Kingdom
| | - Lisabel I Tanner
- School of Health Sciences, Manchester Academic Health Science Centre, Faculty of Biology, Medicine and Health, University of Manchester, Manchester, United Kingdom
| | - Marcelo A Montemurro
- School of Mathematics and Statistics, Faculty of Science, Technology, Engineering and Mathematics, The Open University, Milton Keynes, United Kingdom
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5
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Tewarie P, Prasse B, Meier J, Mandke K, Warrington S, Stam CJ, Brookes MJ, Van Mieghem P, Sotiropoulos SN, Hillebrand A. Predicting time-resolved electrophysiological brain networks from structural eigenmodes. Hum Brain Mapp 2022; 43:4475-4491. [PMID: 35642600 PMCID: PMC9435022 DOI: 10.1002/hbm.25967] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/13/2022] [Revised: 04/25/2022] [Accepted: 05/16/2022] [Indexed: 01/20/2023] Open
Abstract
How temporal modulations in functional interactions are shaped by the underlying anatomical connections remains an open question. Here, we analyse the role of structural eigenmodes, in the formation and dissolution of temporally evolving functional brain networks using resting-state magnetoencephalography and diffusion magnetic resonance imaging data at the individual subject level. Our results show that even at short timescales, phase and amplitude connectivity can partly be expressed by structural eigenmodes, but hardly by direct structural connections. Albeit a stronger relationship was found between structural eigenmodes and time-resolved amplitude connectivity. Time-resolved connectivity for both phase and amplitude was mostly characterised by a stationary process, superimposed with very brief periods that showed deviations from this stationary process. For these brief periods, dynamic network states were extracted that showed different expressions of eigenmodes. Furthermore, the eigenmode expression was related to overall cognitive performance and co-occurred with fluctuations in community structure of functional networks. These results implicate that ongoing time-resolved resting-state networks, even at short timescales, can to some extent be understood in terms of activation and deactivation of structural eigenmodes and that these eigenmodes play a role in the dynamic integration and segregation of information across the cortex, subserving cognitive functions.
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Affiliation(s)
- Prejaas Tewarie
- Sir Peter Mansfield Imaging Centre, School of Physics and Astronomy, University of Nottingham, Nottingham, UK
| | - Bastian Prasse
- Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, Delft, The Netherlands
| | - Jil Meier
- Department of Neurology, Brain Simulation Section, Charité-Universitätsmedizin Berlin, Freie Universität Berlin, Humboldt-Universität zu Berlin, and Berlin Institute of Health, Berlin, Germany
| | - Kanad Mandke
- Centre for Neuroscience in Education, Department of Psychology, University of Cambridge, Cambridge, UK
| | - Shaun Warrington
- Sir Peter Mansfield Imaging Centre, School of Medicine, University of Nottingham, Nottingham, UK
| | - Cornelis J Stam
- Department of Clinical Neurophysiology and MEG Center, Amsterdam UMC, Vrije Universiteit Amsterdam, Amsterdam Neuroscience, Amsterdam, The Netherlands
| | - Matthew J Brookes
- Sir Peter Mansfield Imaging Centre, School of Physics and Astronomy, University of Nottingham, Nottingham, UK
| | - Piet Van Mieghem
- Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, Delft, The Netherlands
| | - Stamatios N Sotiropoulos
- Sir Peter Mansfield Imaging Centre, School of Medicine, University of Nottingham, Nottingham, UK.,Wellcome Centre for Integrative Neuroimaging (WIN-FMRIB), University of Oxford, Oxford, UK.,NIHR Biomedical Research Centre, University of Nottingham, Nottingham University Hospitals NHS Trust, Nottingham, UK
| | - Arjan Hillebrand
- Department of Clinical Neurophysiology and MEG Center, Amsterdam UMC, Vrije Universiteit Amsterdam, Amsterdam Neuroscience, Amsterdam, The Netherlands
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6
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FNS allows efficient event-driven spiking neural network simulations based on a neuron model supporting spike latency. Sci Rep 2021; 11:12160. [PMID: 34108523 PMCID: PMC8190312 DOI: 10.1038/s41598-021-91513-8] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/10/2021] [Accepted: 05/24/2021] [Indexed: 02/05/2023] Open
Abstract
Neural modelling tools are increasingly employed to describe, explain, and predict the human brain's behavior. Among them, spiking neural networks (SNNs) make possible the simulation of neural activity at the level of single neurons, but their use is often threatened by the resources needed in terms of processing capabilities and memory. Emerging applications where a low energy burden is required (e.g. implanted neuroprostheses) motivate the exploration of new strategies able to capture the relevant principles of neuronal dynamics in reduced and efficient models. The recent Leaky Integrate-and-Fire with Latency (LIFL) spiking neuron model shows some realistic neuronal features and efficiency at the same time, a combination of characteristics that may result appealing for SNN-based brain modelling. In this paper we introduce FNS, the first LIFL-based SNN framework, which combines spiking/synaptic modelling with the event-driven approach, allowing us to define heterogeneous neuron groups and multi-scale connectivity, with delayed connections and plastic synapses. FNS allows multi-thread, precise simulations, integrating a novel parallelization strategy and a mechanism of periodic dumping. We evaluate the performance of FNS in terms of simulation time and used memory, and compare it with those obtained with neuronal models having a similar neurocomputational profile, implemented in NEST, showing that FNS performs better in both scenarios. FNS can be advantageously used to explore the interaction within and between populations of spiking neurons, even for long time-scales and with a limited hardware configuration.
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7
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Jung K, Eickhoff SB, Popovych OV. Tractography density affects whole-brain structural architecture and resting-state dynamical modeling. Neuroimage 2021; 237:118176. [PMID: 34000399 DOI: 10.1016/j.neuroimage.2021.118176] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/16/2020] [Revised: 05/09/2021] [Accepted: 05/13/2021] [Indexed: 11/24/2022] Open
Abstract
Dynamical modeling of the resting-state brain dynamics essentially relies on the empirical neuroimaging data utilized for the model derivation and validation. There is however still no standardized data processing for magnetic resonance imaging pipelines and the structural and functional connectomes involved in the models. In this study, we thus address how the parameters of diffusion-weighted data processing for structural connectivity (SC) can influence the validation results of the whole-brain mathematical models informed by SC. For this, we introduce a set of simulation conditions including the varying number of total streamlines of the whole-brain tractography (WBT) used for extraction of SC, cortical parcellations based on functional and anatomical brain properties and distinct model fitting modalities. The main objective of this study is to explore how the quality of the model validation can vary across the considered simulation conditions. We observed that the graph-theoretical network properties of structural connectome can be affected by varying tractography density and strongly relate to the model performance. We also found that the optimal number of the total streamlines of WBT can vary for different brain atlases. Consequently, we suggest a way how to improve the model performance based on the network properties and the optimal parameter configurations from multiple WBT conditions. Furthermore, the population of subjects can be stratified into subgroups with divergent behaviors induced by the varying WBT density such that different recommendations can be made with respect to the data processing for individual subjects and brain parcellations.
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Affiliation(s)
- Kyesam Jung
- Institute of Neuroscience and Medicine, Brain and Behavior (INM-7), Research Center Jülich, Germany; Institute for Systems Neuroscience, Medical Faculty, Heinrich Heine University Düsseldorf, Germany.
| | - Simon B Eickhoff
- Institute of Neuroscience and Medicine, Brain and Behavior (INM-7), Research Center Jülich, Germany; Institute for Systems Neuroscience, Medical Faculty, Heinrich Heine University Düsseldorf, Germany.
| | - Oleksandr V Popovych
- Institute of Neuroscience and Medicine, Brain and Behavior (INM-7), Research Center Jülich, Germany; Institute for Systems Neuroscience, Medical Faculty, Heinrich Heine University Düsseldorf, Germany.
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8
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Ghaderi AH, Baltaretu BR, Andevari MN, Bharmauria V, Balci F. Synchrony and Complexity in State-Related EEG Networks: An Application of Spectral Graph Theory. Neural Comput 2020; 32:2422-2454. [DOI: 10.1162/neco_a_01327] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/04/2022]
Abstract
The brain may be considered as a synchronized dynamic network with several coherent dynamical units. However, concerns remain whether synchronizability is a stable state in the brain networks. If so, which index can best reveal the synchronizability in brain networks? To answer these questions, we tested the application of the spectral graph theory and the Shannon entropy as alternative approaches in neuroimaging. We specifically tested the alpha rhythm in the resting-state eye closed (rsEC) and the resting-state eye open (rsEO) conditions, a well-studied classical example of synchrony in neuroimaging EEG. Since the synchronizability of alpha rhythm is more stable during the rsEC than the rsEO, we hypothesized that our suggested spectral graph theory indices (as reliable measures to interpret the synchronizability of brain signals) should exhibit higher values in the rsEC than the rsEO condition. We performed two separate analyses of two different datasets (as elementary and confirmatory studies). Based on the results of both studies and in agreement with our hypothesis, the spectral graph indices revealed higher stability of synchronizability in the rsEC condition. The k-mean analysis indicated that the spectral graph indices can distinguish the rsEC and rsEO conditions by considering the synchronizability of brain networks. We also computed correlations among the spectral indices, the Shannon entropy, and the topological indices of brain networks, as well as random networks. Correlation analysis indicated that although the spectral and the topological properties of random networks are completely independent, these features are significantly correlated with each other in brain networks. Furthermore, we found that complexity in the investigated brain networks is inversely related to the stability of synchronizability. In conclusion, we revealed that the spectral graph theory approach can be reliably applied to study the stability of synchronizability of state-related brain networks.
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Affiliation(s)
- Amir Hossein Ghaderi
- Centre for Vision Research and Canada Vision: Science to Applications Program, York University, Toronto M3J 1P3, Canada, and Iranian Neuro-Wave Lab., No. 32, Vilashahr, Isfahan, Iran
| | | | | | - Vishal Bharmauria
- Centre for Vision Research, York University, Toronto M3J 1P3, Canada
| | - Fuat Balci
- Department of Psychology and Research Center for Translational Medicine, Koç University, Istanbul, Turkey
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9
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Allegra Mascaro AL, Falotico E, Petkoski S, Pasquini M, Vannucci L, Tort-Colet N, Conti E, Resta F, Spalletti C, Ramalingasetty ST, von Arnim A, Formento E, Angelidis E, Blixhavn CH, Leergaard TB, Caleo M, Destexhe A, Ijspeert A, Micera S, Laschi C, Jirsa V, Gewaltig MO, Pavone FS. Experimental and Computational Study on Motor Control and Recovery After Stroke: Toward a Constructive Loop Between Experimental and Virtual Embodied Neuroscience. Front Syst Neurosci 2020; 14:31. [PMID: 32733210 PMCID: PMC7359878 DOI: 10.3389/fnsys.2020.00031] [Citation(s) in RCA: 20] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/28/2019] [Accepted: 05/08/2020] [Indexed: 01/22/2023] Open
Abstract
Being able to replicate real experiments with computational simulations is a unique opportunity to refine and validate models with experimental data and redesign the experiments based on simulations. However, since it is technically demanding to model all components of an experiment, traditional approaches to modeling reduce the experimental setups as much as possible. In this study, our goal is to replicate all the relevant features of an experiment on motor control and motor rehabilitation after stroke. To this aim, we propose an approach that allows continuous integration of new experimental data into a computational modeling framework. First, results show that we could reproduce experimental object displacement with high accuracy via the simulated embodiment in the virtual world by feeding a spinal cord model with experimental registration of the cortical activity. Second, by using computational models of multiple granularities, our preliminary results show the possibility of simulating several features of the brain after stroke, from the local alteration in neuronal activity to long-range connectivity remodeling. Finally, strategies are proposed to merge the two pipelines. We further suggest that additional models could be integrated into the framework thanks to the versatility of the proposed approach, thus allowing many researchers to achieve continuously improved experimental design.
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Affiliation(s)
- Anna Letizia Allegra Mascaro
- Neuroscience Institute, National Research Council, Pisa, Italy.,European Laboratory for Non-Linear Spectroscopy, Sesto Fiorentino, Italy
| | - Egidio Falotico
- Department of Excellence in Robotics & AI, The BioRobotics Institute, Scuola Superiore Sant'Anna, Pontedera, Italy
| | - Spase Petkoski
- Aix-Marseille Université, Inserm, INS UMR_S 1106, Marseille, France
| | - Maria Pasquini
- Department of Excellence in Robotics & AI, The BioRobotics Institute, Scuola Superiore Sant'Anna, Pontedera, Italy
| | - Lorenzo Vannucci
- Department of Excellence in Robotics & AI, The BioRobotics Institute, Scuola Superiore Sant'Anna, Pontedera, Italy
| | - Núria Tort-Colet
- Paris-Saclay University, Institute of Neuroscience, CNRS, Gif-sur-Yvette, France
| | - Emilia Conti
- European Laboratory for Non-Linear Spectroscopy, Sesto Fiorentino, Italy.,Department of Physics and Astronomy, University of Florence, Florence, Italy
| | - Francesco Resta
- European Laboratory for Non-Linear Spectroscopy, Sesto Fiorentino, Italy.,Department of Physics and Astronomy, University of Florence, Florence, Italy
| | | | | | | | - Emanuele Formento
- Bertarelli Foundation Chair in Translational NeuroEngineering, Institute of Bioengineering, Swiss Federal Institute of Technology (EPFL), Lausanne, Switzerland
| | - Emmanouil Angelidis
- Fortiss GmbH, Munich, Germany.,Chair of Robotics, Artificial Intelligence and Embedded Systems, Department of Informatics, Technical University of Munich, Munich, Germany
| | | | | | - Matteo Caleo
- Neuroscience Institute, National Research Council, Pisa, Italy.,Department of Biomedical Sciences, University of Padua, Padua, Italy
| | - Alain Destexhe
- Paris-Saclay University, Institute of Neuroscience, CNRS, Gif-sur-Yvette, France
| | - Auke Ijspeert
- Biorobotics Laboratory, École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
| | - Silvestro Micera
- Department of Excellence in Robotics & AI, The BioRobotics Institute, Scuola Superiore Sant'Anna, Pontedera, Italy.,Bertarelli Foundation Chair in Translational NeuroEngineering, Institute of Bioengineering, Swiss Federal Institute of Technology (EPFL), Lausanne, Switzerland
| | - Cecilia Laschi
- Department of Excellence in Robotics & AI, The BioRobotics Institute, Scuola Superiore Sant'Anna, Pontedera, Italy
| | - Viktor Jirsa
- Aix-Marseille Université, Inserm, INS UMR_S 1106, Marseille, France
| | - Marc-Oliver Gewaltig
- Blue Brain Project (BBP), École Polytechnique Fédérale de Lausanne (EPFL), Geneva, Switzerland
| | - Francesco S Pavone
- European Laboratory for Non-Linear Spectroscopy, Sesto Fiorentino, Italy.,Department of Physics and Astronomy, University of Florence, Florence, Italy
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10
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Forrester M, Crofts JJ, Sotiropoulos SN, Coombes S, O'Dea RD. The role of node dynamics in shaping emergent functional connectivity patterns in the brain. Netw Neurosci 2020; 4:467-483. [PMID: 32537537 PMCID: PMC7286301 DOI: 10.1162/netn_a_00130] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/02/2019] [Accepted: 01/31/2020] [Indexed: 11/07/2022] Open
Abstract
The contribution of structural connectivity to functional brain states remains poorly understood. We present a mathematical and computational study suited to assess the structure–function issue, treating a system of Jansen–Rit neural mass nodes with heterogeneous structural connections estimated from diffusion MRI data provided by the Human Connectome Project. Via direct simulations we determine the similarity of functional (inferred from correlated activity between nodes) and structural connectivity matrices under variation of the parameters controlling single-node dynamics, highlighting a nontrivial structure–function relationship in regimes that support limit cycle oscillations. To determine their relationship, we firstly calculate network instabilities giving rise to oscillations, and the so-called ‘false bifurcations’ (for which a significant qualitative change in the orbit is observed, without a change of stability) occurring beyond this onset. We highlight that functional connectivity (FC) is inherited robustly from structure when node dynamics are poised near a Hopf bifurcation, whilst near false bifurcations, and structure only weakly influences FC. Secondly, we develop a weakly coupled oscillator description to analyse oscillatory phase-locked states and, furthermore, show how the modular structure of FC matrices can be predicted via linear stability analysis. This study thereby emphasises the substantial role that local dynamics can have in shaping large-scale functional brain states. Patterns of oscillation across the brain arise because of structural connections between brain regions. However, the type of oscillation at a site may also play a contributory role. We focus on an idealised model of a neural mass network, coupled using estimates of structural connections obtained via tractography on Human Connectome Project MRI data. Using a mixture of computational and mathematical techniques, we show that functional connectivity is inherited most strongly from structural connectivity when the network nodes are poised at a Hopf bifurcation. However, beyond the onset of this oscillatory instability a phase-locked network state can undergo a false bifurcation, and structural connectivity only weakly influences functional connectivity. This highlights the important effect that local dynamics can have on large-scale brain states.
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Affiliation(s)
- Michael Forrester
- Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham, UK
| | - Jonathan J Crofts
- Department of Physics and Mathematics, School of Science and Technology, Nottingham Trent University, Nottingham, UK
| | - Stamatios N Sotiropoulos
- Sir Peter Mansfield Imaging Centre, Queen's Medical Centre, University of Nottingham, Nottingham, UK
| | - Stephen Coombes
- Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham, UK
| | - Reuben D O'Dea
- Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham, UK
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11
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Mapping functional brain networks from the structural connectome: Relating the series expansion and eigenmode approaches. Neuroimage 2020; 216:116805. [PMID: 32335264 DOI: 10.1016/j.neuroimage.2020.116805] [Citation(s) in RCA: 30] [Impact Index Per Article: 7.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/19/2020] [Revised: 03/14/2020] [Accepted: 03/31/2020] [Indexed: 12/11/2022] Open
Abstract
Functional brain networks are shaped and constrained by the underlying structural network. However, functional networks are not merely a one-to-one reflection of the structural network. Several theories have been put forward to understand the relationship between structural and functional networks. However, it remains unclear how these theories can be unified. Two existing recent theories state that 1) functional networks can be explained by all possible walks in the structural network, which we will refer to as the series expansion approach, and 2) functional networks can be explained by a weighted combination of the eigenmodes of the structural network, the so-called eigenmode approach. To elucidate the unique or common explanatory power of these approaches to estimate functional networks from the structural network, we analysed the relationship between these two existing views. Using linear algebra, we first show that the eigenmode approach can be written in terms of the series expansion approach, i.e., walks on the structural network associated with different hop counts correspond to different weightings of the eigenvectors of this network. Second, we provide explicit expressions for the coefficients for both the eigenmode and series expansion approach. These theoretical results were verified by empirical data from Diffusion Tensor Imaging (DTI) and functional Magnetic Resonance Imaging (fMRI), demonstrating a strong correlation between the mappings based on both approaches. Third, we analytically and empirically demonstrate that the fit of the eigenmode approach to measured functional data is always at least as good as the fit of the series expansion approach, and that errors in the structural data lead to large errors of the estimated coefficients for the series expansion approach. Therefore, we argue that the eigenmode approach should be preferred over the series expansion approach. Results hold for eigenmodes of the weighted adjacency matrices as well as eigenmodes of the graph Laplacian. Taken together, these results provide an important step towards unification of existing theories regarding the structure-function relationships in brain networks.
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12
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Forrester M, Crofts JJ, Sotiropoulos SN, Coombes S, O'Dea RD. The role of node dynamics in shaping emergent functional connectivity patterns in the brain. Netw Neurosci 2020; 4:467-483. [PMID: 32537537 DOI: 10.1162/netn\_a_00130] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/02/2019] [Accepted: 01/31/2020] [Indexed: 05/21/2023] Open
Abstract
The contribution of structural connectivity to functional brain states remains poorly understood. We present a mathematical and computational study suited to assess the structure-function issue, treating a system of Jansen-Rit neural mass nodes with heterogeneous structural connections estimated from diffusion MRI data provided by the Human Connectome Project. Via direct simulations we determine the similarity of functional (inferred from correlated activity between nodes) and structural connectivity matrices under variation of the parameters controlling single-node dynamics, highlighting a nontrivial structure-function relationship in regimes that support limit cycle oscillations. To determine their relationship, we firstly calculate network instabilities giving rise to oscillations, and the so-called 'false bifurcations' (for which a significant qualitative change in the orbit is observed, without a change of stability) occurring beyond this onset. We highlight that functional connectivity (FC) is inherited robustly from structure when node dynamics are poised near a Hopf bifurcation, whilst near false bifurcations, and structure only weakly influences FC. Secondly, we develop a weakly coupled oscillator description to analyse oscillatory phase-locked states and, furthermore, show how the modular structure of FC matrices can be predicted via linear stability analysis. This study thereby emphasises the substantial role that local dynamics can have in shaping large-scale functional brain states.
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Affiliation(s)
- Michael Forrester
- Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham, UK
| | - Jonathan J Crofts
- Department of Physics and Mathematics, School of Science and Technology, Nottingham Trent University, Nottingham, UK
| | - Stamatios N Sotiropoulos
- Sir Peter Mansfield Imaging Centre, Queen's Medical Centre, University of Nottingham, Nottingham, UK
| | - Stephen Coombes
- Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham, UK
| | - Reuben D O'Dea
- Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham, UK
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13
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Papo D, Buldú JM. Brain synchronizability, a false friend. Neuroimage 2019; 196:195-199. [PMID: 30986500 DOI: 10.1016/j.neuroimage.2019.04.029] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/20/2018] [Revised: 03/28/2019] [Accepted: 04/08/2019] [Indexed: 01/20/2023] Open
Abstract
Synchronization plays a fundamental role in healthy cognitive and motor function. However, how synchronization depends on the interplay between local dynamics, coupling and topology and how prone to synchronization a network is, given its topological organization, are still poorly understood issues. To investigate the synchronizability of both anatomical and functional brain networks various studies resorted to the Master Stability Function (MSF) formalism, an elegant tool which allows analysing the stability of synchronous states in a dynamical system consisting of many coupled oscillators. Here, we argue that brain dynamics does not fulfil the formal criteria under which synchronizability is usually quantified and, perhaps more importantly, this measure refers to a global dynamical condition that never holds in the brain (not even in the most pathological conditions), and therefore no neurophysiological conclusions should be drawn based on it. We discuss the meaning of synchronizability and its applicability to neuroscience and propose alternative ways to quantify brain networks synchronization.
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Affiliation(s)
- D Papo
- SCALab UMR CNRS 9193, Université de Lille, Villeneuve d'Ascq, France.
| | - J M Buldú
- Laboratory of Biological Networks, Center for Biomedical Technology (UPM), 28223, Pozuelo de Alarcón, Madrid, Spain; Complex Systems Group & G.I.S.C., Universidad Rey Juan Carlos, 28933, Móstoles, Madrid, Spain
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14
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Popovych OV, Manos T, Hoffstaedter F, Eickhoff SB. What Can Computational Models Contribute to Neuroimaging Data Analytics? Front Syst Neurosci 2019; 12:68. [PMID: 30687028 PMCID: PMC6338060 DOI: 10.3389/fnsys.2018.00068] [Citation(s) in RCA: 16] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/27/2018] [Accepted: 12/17/2018] [Indexed: 01/12/2023] Open
Abstract
Over the past years, nonlinear dynamical models have significantly contributed to the general understanding of brain activity as well as brain disorders. Appropriately validated and optimized mathematical models can be used to mechanistically explain properties of brain structure and neuronal dynamics observed from neuroimaging data. A thorough exploration of the model parameter space and hypothesis testing with the methods of nonlinear dynamical systems and statistical physics can assist in classification and prediction of brain states. On the one hand, such a detailed investigation and systematic parameter variation are hardly feasible in experiments and data analysis. On the other hand, the model-based approach can establish a link between empirically discovered phenomena and more abstract concepts of attractors, multistability, bifurcations, synchronization, noise-induced dynamics, etc. Such a mathematical description allows to compare and differentiate brain structure and dynamics in health and disease, such that model parameters and dynamical regimes may serve as additional biomarkers of brain states and behavioral modes. In this perspective paper we first provide very brief overview of the recent progress and some open problems in neuroimaging data analytics with emphasis on the resting state brain activity. We then focus on a few recent contributions of mathematical modeling to our understanding of the brain dynamics and model-based approaches in medicine. Finally, we discuss the question stated in the title. We conclude that incorporating computational models in neuroimaging data analytics as well as in translational medicine could significantly contribute to the progress in these fields.
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Affiliation(s)
- Oleksandr V. Popovych
- Institute of Neuroscience and Medicine - Brain & Behaviour (INM-7), Research Centre Jülich, Jülich, Germany
- Institute of Systems Neuroscience, Medical Faculty, Heinrich-Heine University Düsseldorf, Düsseldorf, Germany
| | - Thanos Manos
- Institute of Neuroscience and Medicine - Brain & Behaviour (INM-7), Research Centre Jülich, Jülich, Germany
- Institute of Systems Neuroscience, Medical Faculty, Heinrich-Heine University Düsseldorf, Düsseldorf, Germany
| | - Felix Hoffstaedter
- Institute of Neuroscience and Medicine - Brain & Behaviour (INM-7), Research Centre Jülich, Jülich, Germany
- Institute of Systems Neuroscience, Medical Faculty, Heinrich-Heine University Düsseldorf, Düsseldorf, Germany
| | - Simon B. Eickhoff
- Institute of Neuroscience and Medicine - Brain & Behaviour (INM-7), Research Centre Jülich, Jülich, Germany
- Institute of Systems Neuroscience, Medical Faculty, Heinrich-Heine University Düsseldorf, Düsseldorf, Germany
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15
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O'Neill GC, Tewarie P, Vidaurre D, Liuzzi L, Woolrich MW, Brookes MJ. Dynamics of large-scale electrophysiological networks: A technical review. Neuroimage 2018; 180:559-576. [PMID: 28988134 DOI: 10.1016/j.neuroimage.2017.10.003] [Citation(s) in RCA: 110] [Impact Index Per Article: 18.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/09/2017] [Revised: 09/23/2017] [Accepted: 10/02/2017] [Indexed: 12/12/2022] Open
Abstract
For several years it has been argued that neural synchronisation is crucial for cognition. The idea that synchronised temporal patterns between different neural groups carries information above and beyond the isolated activity of these groups has inspired a shift in focus in the field of functional neuroimaging. Specifically, investigation into the activation elicited within certain regions by some stimulus or task has, in part, given way to analysis of patterns of co-activation or functional connectivity between distal regions. Recently, the functional connectivity community has been looking beyond the assumptions of stationarity that earlier work was based on, and has introduced methods to incorporate temporal dynamics into the analysis of connectivity. In particular, non-invasive electrophysiological data (magnetoencephalography/electroencephalography (MEG/EEG)), which provides direct measurement of whole-brain activity and rich temporal information, offers an exceptional window into such (potentially fast) brain dynamics. In this review, we discuss challenges, solutions, and a collection of analysis tools that have been developed in recent years to facilitate the investigation of dynamic functional connectivity using these imaging modalities. Further, we discuss the applications of these approaches in the study of cognition and neuropsychiatric disorders. Finally, we review some existing developments that, by using realistic computational models, pursue a deeper understanding of the underlying causes of non-stationary connectivity.
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Affiliation(s)
- George C O'Neill
- Sir Peter Mansfield Imaging Centre, School of Physics and Astronomy, University of Nottingham, Nottingham, United Kingdom
| | - Prejaas Tewarie
- Sir Peter Mansfield Imaging Centre, School of Physics and Astronomy, University of Nottingham, Nottingham, United Kingdom
| | - Diego Vidaurre
- Oxford Centre for Human Brain Activity, Department of Psychiatry, University of Oxford, Oxford, United Kingdom
| | - Lucrezia Liuzzi
- Sir Peter Mansfield Imaging Centre, School of Physics and Astronomy, University of Nottingham, Nottingham, United Kingdom
| | - Mark W Woolrich
- Oxford Centre for Human Brain Activity, Department of Psychiatry, University of Oxford, Oxford, United Kingdom
| | - Matthew J Brookes
- Sir Peter Mansfield Imaging Centre, School of Physics and Astronomy, University of Nottingham, Nottingham, United Kingdom.
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16
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Petkoski S, Palva JM, Jirsa VK. Phase-lags in large scale brain synchronization: Methodological considerations and in-silico analysis. PLoS Comput Biol 2018; 14:e1006160. [PMID: 29990339 PMCID: PMC6039010 DOI: 10.1371/journal.pcbi.1006160] [Citation(s) in RCA: 44] [Impact Index Per Article: 7.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/21/2017] [Accepted: 04/29/2018] [Indexed: 01/24/2023] Open
Abstract
Architecture of phase relationships among neural oscillations is central for their functional significance but has remained theoretically poorly understood. We use phenomenological model of delay-coupled oscillators with increasing degree of topological complexity to identify underlying principles by which the spatio-temporal structure of the brain governs the phase lags between oscillatory activity at distant regions. Phase relations and their regions of stability are derived and numerically confirmed for two oscillators and for networks with randomly distributed or clustered bimodal delays, as a first approximation for the brain structural connectivity. Besides in-phase, clustered delays can induce anti-phase synchronization for certain frequencies, while the sign of the lags is determined by the natural frequencies and by the inhomogeneous network interactions. For in-phase synchronization faster oscillators always phase lead, while stronger connected nodes lag behind the weaker during frequency depression, which consistently arises for in-silico results. If nodes are in anti-phase regime, then a distance π is added to the in-phase trends. The statistics of the phases is calculated from the phase locking values (PLV), as in many empirical studies, and we scrutinize the method’s impact. The choice of surrogates do not affects the mean of the observed phase lags, but higher significance levels that are generated by some surrogates, cause decreased variance and might fail to detect the generally weaker coherence of the interhemispheric links. These links are also affected by the non-stationary and intermittent synchronization, which causes multimodal phase lags that can be misleading if averaged. Taken together, the results describe quantitatively the impact of the spatio-temporal connectivity of the brain to the synchronization patterns between brain regions, and to uncover mechanisms through which the spatio-temporal structure of the brain renders phases to be distributed around 0 and π. Trial registration: South African Clinical Trials Register: http://www.sanctr.gov.za/SAClinicalbrnbspTrials/tabid/169/Default.aspx, then link to respiratory tract then link to tuberculosis, pulmonary; and TASK Applied Sciences Clinical Trials, AP-TB-201-16 (ALOPEXX): https://task.org.za/clinical-trials/. Functional connectivity, and in particular, phase coupling between distant brain regions may be fundamental in regulating neuronal processing and communication. However, phase relationships between the nodes of the brain and how they are confined by its spatio-temporal structure, have been mostly overlooked. We use a model of oscillatory dynamics superimposed on the space-time structure defined by the connectome, and we analyze the possible regimes of synchronization. Limitations of data analysis are also considered and we show that the choice of the significance threshold for coherence does not essentially impact the statistics of the observed phase lags, although it is crucial for the right detection of statistically significant coherence. Analytical insights are obtained for networks with heterogeneous time-delays, based on the empirical data from the connectome, and these are confirmed by numerical simulations, which show in- or anti-phase synchronization depending on the frequency and the distribution of time-delays. Phase lags are shown to result from inhomogeneous network interactions, so that stronger connected nodes generally phase lag behind the weaker.
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Affiliation(s)
- Spase Petkoski
- Aix-Marseille Université, Inserm, INS UMR_S 1106, Marseille, France
- * E-mail: (SP); (VKJ)
| | - J. Matias Palva
- Neuroscience Center, Helsinki Institute of Life Science, University of Helsinki, Helsinki, Finland
| | - Viktor K. Jirsa
- Aix-Marseille Université, Inserm, INS UMR_S 1106, Marseille, France
- * E-mail: (SP); (VKJ)
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17
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Montbrió E, Pazó D. Kuramoto Model for Excitation-Inhibition-Based Oscillations. PHYSICAL REVIEW LETTERS 2018; 120:244101. [PMID: 29956946 DOI: 10.1103/physrevlett.120.244101] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/04/2018] [Revised: 04/10/2018] [Indexed: 06/08/2023]
Abstract
The Kuramoto model (KM) is a theoretical paradigm for investigating the emergence of rhythmic activity in large populations of oscillators. A remarkable example of rhythmogenesis is the feedback loop between excitatory (E) and inhibitory (I) cells in large neuronal networks. Yet, although the EI-feedback mechanism plays a central role in the generation of brain oscillations, it remains unexplored whether the KM has enough biological realism to describe it. Here we derive a two-population KM that fully accounts for the onset of EI-based neuronal rhythms and that, as the original KM, is analytically solvable to a large extent. Our results provide a powerful theoretical tool for the analysis of large-scale neuronal oscillations.
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Affiliation(s)
- Ernest Montbrió
- Center for Brain and Cognition. Department of Information and Communication Technologies, Universitat Pompeu Fabra, 08018 Barcelona, Spain
| | - Diego Pazó
- Instituto de Física de Cantabria (IFCA), CSIC-Universidad de Cantabria, 39005 Santander, Spain
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18
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Daffertshofer A, Ton R, Pietras B, Kringelbach ML, Deco G. Scale-freeness or partial synchronization in neural mass phase oscillator networks: Pick one of two? Neuroimage 2018; 180:428-441. [PMID: 29625237 DOI: 10.1016/j.neuroimage.2018.03.070] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/29/2017] [Revised: 03/22/2018] [Accepted: 03/28/2018] [Indexed: 11/18/2022] Open
Abstract
Modeling and interpreting (partial) synchronous neural activity can be a challenge. We illustrate this by deriving the phase dynamics of two seminal neural mass models: the Wilson-Cowan firing rate model and the voltage-based Freeman model. We established that the phase dynamics of these models differed qualitatively due to an attractive coupling in the first and a repulsive coupling in the latter. Using empirical structural connectivity matrices, we determined that the two dynamics cover the functional connectivity observed in resting state activity. We further searched for two pivotal dynamical features that have been reported in many experimental studies: (1) a partial phase synchrony with a possibility of a transition towards either a desynchronized or a (fully) synchronized state; (2) long-term autocorrelations indicative of a scale-free temporal dynamics of phase synchronization. Only the Freeman phase model exhibited scale-free behavior. Its repulsive coupling, however, let the individual phases disperse and did not allow for a transition into a synchronized state. The Wilson-Cowan phase model, by contrast, could switch into a (partially) synchronized state, but it did not generate long-term correlations although being located close to the onset of synchronization, i.e. in its critical regime. That is, the phase-reduced models can display one of the two dynamical features, but not both.
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Affiliation(s)
- Andreas Daffertshofer
- Institute for Brain and Behavior Amsterdam & Amsterdam Movement Sciences, Faculty of Behavioural and Movement Sciences, Vrije Universiteit Amsterdam, van der Boechorststraat 9, 1081BT, Amsterdam, The Netherlands.
| | - Robert Ton
- Institute for Brain and Behavior Amsterdam & Amsterdam Movement Sciences, Faculty of Behavioural and Movement Sciences, Vrije Universiteit Amsterdam, van der Boechorststraat 9, 1081BT, Amsterdam, The Netherlands; Center for Brain and Cognition, Computational Neuroscience Group, Universitat Pompeu Fabra, Carrer Tanger 122-140, 08018, Barcelona, Spain
| | - Bastian Pietras
- Institute for Brain and Behavior Amsterdam & Amsterdam Movement Sciences, Faculty of Behavioural and Movement Sciences, Vrije Universiteit Amsterdam, van der Boechorststraat 9, 1081BT, Amsterdam, The Netherlands; Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK
| | - Morten L Kringelbach
- University Department of Psychiatry, University of Oxford, Oxford, OX3 7JX, UK; Center for Music in the Brain, Department of Clinical Medicine, Aarhus University, Denmark
| | - Gustavo Deco
- Center for Brain and Cognition, Computational Neuroscience Group, Universitat Pompeu Fabra, Carrer Tanger 122-140, 08018, Barcelona, Spain; Institució Catalana de la Recerca i Estudis Avanats (ICREA), Universitat Pompeu Fabra, Carrer Tanger 122-140, 08018, Barcelona, Spain
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19
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Abeysuriya RG, Hadida J, Sotiropoulos SN, Jbabdi S, Becker R, Hunt BAE, Brookes MJ, Woolrich MW. A biophysical model of dynamic balancing of excitation and inhibition in fast oscillatory large-scale networks. PLoS Comput Biol 2018; 14:e1006007. [PMID: 29474352 PMCID: PMC5841816 DOI: 10.1371/journal.pcbi.1006007] [Citation(s) in RCA: 47] [Impact Index Per Article: 7.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/28/2017] [Revised: 03/07/2018] [Accepted: 01/28/2018] [Indexed: 01/03/2023] Open
Abstract
Over long timescales, neuronal dynamics can be robust to quite large perturbations, such as changes in white matter connectivity and grey matter structure through processes including learning, aging, development and certain disease processes. One possible explanation is that robust dynamics are facilitated by homeostatic mechanisms that can dynamically rebalance brain networks. In this study, we simulate a cortical brain network using the Wilson-Cowan neural mass model with conduction delays and noise, and use inhibitory synaptic plasticity (ISP) to dynamically achieve a spatially local balance between excitation and inhibition. Using MEG data from 55 subjects we find that ISP enables us to simultaneously achieve high correlation with multiple measures of functional connectivity, including amplitude envelope correlation and phase locking. Further, we find that ISP successfully achieves local E/I balance, and can consistently predict the functional connectivity computed from real MEG data, for a much wider range of model parameters than is possible with a model without ISP.
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Affiliation(s)
- Romesh G. Abeysuriya
- Oxford Centre for Human Brain Activity, Wellcome Centre for Integrative Neuroimaging, Department of Psychiatry, University of Oxford, United Kingdom
| | - Jonathan Hadida
- Oxford Centre for Human Brain Activity, Wellcome Centre for Integrative Neuroimaging, Department of Psychiatry, University of Oxford, United Kingdom
- Oxford Centre for Functional Magnetic Resonance Imaging of the Brain, Wellcome Centre for Integrative Neuroimaging, University of Oxford, United Kingdom
| | - Stamatios N. Sotiropoulos
- Oxford Centre for Functional Magnetic Resonance Imaging of the Brain, Wellcome Centre for Integrative Neuroimaging, University of Oxford, United Kingdom
- Sir Peter Mansfield Imaging Centre, School of Physics and Astronomy, University of Nottingham, United Kingdom
- National Institute for Health Research (NIHR) Nottingham Biomedical Research Centre, Queens Medical Centre, Nottingham
| | - Saad Jbabdi
- Oxford Centre for Functional Magnetic Resonance Imaging of the Brain, Wellcome Centre for Integrative Neuroimaging, University of Oxford, United Kingdom
| | - Robert Becker
- Oxford Centre for Human Brain Activity, Wellcome Centre for Integrative Neuroimaging, Department of Psychiatry, University of Oxford, United Kingdom
| | - Benjamin A. E. Hunt
- Sir Peter Mansfield Imaging Centre, School of Physics and Astronomy, University of Nottingham, United Kingdom
- Department of Diagnostic Imaging, Neurosciences & Mental Health, Research Institute, The Hospital for Sick Children, Toronto, Ontario, Canada
| | - Matthew J. Brookes
- Sir Peter Mansfield Imaging Centre, School of Physics and Astronomy, University of Nottingham, United Kingdom
| | - Mark W. Woolrich
- Oxford Centre for Human Brain Activity, Wellcome Centre for Integrative Neuroimaging, Department of Psychiatry, University of Oxford, United Kingdom
- Oxford Centre for Functional Magnetic Resonance Imaging of the Brain, Wellcome Centre for Integrative Neuroimaging, University of Oxford, United Kingdom
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20
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Devalle F, Roxin A, Montbrió E. Firing rate equations require a spike synchrony mechanism to correctly describe fast oscillations in inhibitory networks. PLoS Comput Biol 2017; 13:e1005881. [PMID: 29287081 PMCID: PMC5764488 DOI: 10.1371/journal.pcbi.1005881] [Citation(s) in RCA: 43] [Impact Index Per Article: 6.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/25/2017] [Revised: 01/11/2018] [Accepted: 11/15/2017] [Indexed: 12/25/2022] Open
Abstract
Recurrently coupled networks of inhibitory neurons robustly generate oscillations in the gamma band. Nonetheless, the corresponding Wilson-Cowan type firing rate equation for such an inhibitory population does not generate such oscillations without an explicit time delay. We show that this discrepancy is due to a voltage-dependent spike-synchronization mechanism inherent in networks of spiking neurons which is not captured by standard firing rate equations. Here we investigate an exact low-dimensional description for a network of heterogeneous canonical Class 1 inhibitory neurons which includes the sub-threshold dynamics crucial for generating synchronous states. In the limit of slow synaptic kinetics the spike-synchrony mechanism is suppressed and the standard Wilson-Cowan equations are formally recovered as long as external inputs are also slow. However, even in this limit synchronous spiking can be elicited by inputs which fluctuate on a time-scale of the membrane time-constant of the neurons. Our meanfield equations therefore represent an extension of the standard Wilson-Cowan equations in which spike synchrony is also correctly described. Population models describing the average activity of large neuronal ensembles are a powerful mathematical tool to investigate the principles underlying cooperative function of large neuronal systems. However, these models do not properly describe the phenomenon of spike synchrony in networks of neurons. In particular, they fail to capture the onset of synchronous oscillations in networks of inhibitory neurons. We show that this limitation is due to a voltage-dependent synchronization mechanism which is naturally present in spiking neuron models but not captured by traditional firing rate equations. Here we investigate a novel set of macroscopic equations which incorporate both firing rate and membrane potential dynamics, and that correctly generate fast inhibition-based synchronous oscillations. In the limit of slow-synaptic processing oscillations are suppressed, and the model reduces to an equation formally equivalent to the Wilson-Cowan model.
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Affiliation(s)
- Federico Devalle
- Center for Brain and Cognition, Department of Information and Communication Technologies, Universitat Pompeu Fabra, Barcelona, Spain
- Department of Physics, Lancaster University, Lancaster, United Kingdom
| | - Alex Roxin
- Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, Bellaterra, Barcelona, Spain
| | - Ernest Montbrió
- Center for Brain and Cognition, Department of Information and Communication Technologies, Universitat Pompeu Fabra, Barcelona, Spain
- * E-mail:
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21
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de Pasquale F, Della Penna S, Sabatini U, Caravasso Falletta C, Peran P. The anatomical scaffold underlying the functional centrality of known cortical hubs. Hum Brain Mapp 2017; 38:5141-5160. [PMID: 28681960 DOI: 10.1002/hbm.23721] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/14/2016] [Revised: 05/17/2017] [Accepted: 06/27/2017] [Indexed: 11/10/2022] Open
Abstract
Cortical hubs play a fundamental role in the functional architecture of brain connectivity at rest. However, the anatomical scaffold underlying their centrality is still under debate. Certainly, the brain function and anatomy are significantly entwined through synaptogenesis and pruning mechanisms that continuously reshape structural and functional connections. Thus, if hubs are expected to exhibit a large number of direct anatomical connections with the rest of the brain, such a dense wiring is extremely inefficient in energetic terms. In this work, we investigate these aspects on fMRI and DTI data from a set of know resting-state networks, starting from the hypothesis that to promote integration, functional, and anatomical connections link different areas at different scales or hierarchies. Thus, we focused on the role of functional hubs in this hierarchical organization of functional and anatomical architectures. We found that these regions, from a structural point of view, are first linked to each other and successively to the rest of the brain. Thus, functionally central nodes seem to show few strong anatomical connections. These findings suggest an efficient strategy of the investigated cortical hubs in exploiting few direct anatomical connections to link functional hubs among each other that eventually reach the rest of the considered nodes through local indirect tracts. Hum Brain Mapp 38:5141-5160, 2017. © 2017 Wiley Periodicals, Inc.
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Affiliation(s)
- Francesco de Pasquale
- Faculty of Veterinary Medicine, University of Teramo, Italy.,Department of Radiology, IRCCS Santa Lucia Foundation, Rome, Italy
| | - Stefania Della Penna
- Department of Neuroscience Imaging and Clinical Science, University of Chieti, Italy
| | - Umberto Sabatini
- Neuroradiology Unit, Department of Medical and Surgical Sciences, University Magna Graecia of Catanzaro, Italy
| | | | - Patrice Peran
- ToNIC, Toulouse NeuroImaging Center, Université de Toulouse, Inserm, UPS, France
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22
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Structure Shapes Dynamics and Directionality in Diverse Brain Networks: Mathematical Principles and Empirical Confirmation in Three Species. Sci Rep 2017; 7:46606. [PMID: 28425500 PMCID: PMC5397857 DOI: 10.1038/srep46606] [Citation(s) in RCA: 27] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/10/2016] [Accepted: 03/21/2017] [Indexed: 01/19/2023] Open
Abstract
Identifying how spatially distributed information becomes integrated in the brain is essential to understanding higher cognitive functions. Previous computational and empirical studies suggest a significant influence of brain network structure on brain network function. However, there have been few analytical approaches to explain the role of network structure in shaping regional activities and directionality patterns. In this study, analytical methods are applied to a coupled oscillator model implemented in inhomogeneous networks. We first derive a mathematical principle that explains the emergence of directionality from the underlying brain network structure. We then apply the analytical methods to the anatomical brain networks of human, macaque, and mouse, successfully predicting simulation and empirical electroencephalographic data. The results demonstrate that the global directionality patterns in resting state brain networks can be predicted solely by their unique network structures. This study forms a foundation for a more comprehensive understanding of how neural information is directed and integrated in complex brain networks.
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23
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Ogawa Y, Yamaguchi I, Kotani K, Jimbo Y. Deriving theoretical phase locking values of a coupled cortico-thalamic neural mass model using center manifold reduction. J Comput Neurosci 2017; 42:231-243. [PMID: 28236135 DOI: 10.1007/s10827-017-0638-8] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/08/2016] [Revised: 09/18/2016] [Accepted: 02/19/2017] [Indexed: 11/28/2022]
Abstract
Cognitive functions such as sensory processing and memory processes lead to phase synchronization in the electroencephalogram or local field potential between different brain regions. There are a lot of computational researches deriving phase locking values (PLVs), which are an index of phase synchronization intensity, from neural models. However, these researches derive PLVs numerically. To the best of our knowledge, there have been no reports on the derivation of a theoretical PLV. In this study, we propose an analytical method for deriving theoretical PLVs from a cortico-thalamic neural mass model described by a delay differential equation. First, the model for generating neural signals is transformed into a normal form of the Hopf bifurcation using center manifold reduction. Second, the normal form is transformed into a phase model that is suitable for analyzing synchronization phenomena. Third, the Fokker-Planck equation of the phase model is derived and the phase difference distribution is obtained. Finally, the PLVs are calculated from the stationary distribution of the phase difference. The validity of the proposed method is confirmed via numerical simulations. Furthermore, we apply the proposed method to a working memory process, and discuss the neurophysiological basis behind the phase synchronization phenomenon. The results demonstrate the importance of decreasing the intensity of independent noise during the working memory process. The proposed method will be of great use in various experimental studies and simulations relevant to phase synchronization, because it enables the effect of neurophysiological changes on PLVs to be analyzed from a mathematical perspective.
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Affiliation(s)
- Yutaro Ogawa
- Research Center for Advanced Science and Technology, The University of Tokyo, Tokyo, Japan.
| | | | - Kiyoshi Kotani
- Research Center for Advanced Science and Technology, The University of Tokyo, Tokyo, Japan.,PRESTO, Japan Science and Technology Agency (JST), Kawaguchi, Japan
| | - Yasuhiko Jimbo
- Graduate School of Engineering, The University of Tokyo, Tokyo, Japan
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24
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Finger H, Bönstrup M, Cheng B, Messé A, Hilgetag C, Thomalla G, Gerloff C, König P. Modeling of Large-Scale Functional Brain Networks Based on Structural Connectivity from DTI: Comparison with EEG Derived Phase Coupling Networks and Evaluation of Alternative Methods along the Modeling Path. PLoS Comput Biol 2016; 12:e1005025. [PMID: 27504629 PMCID: PMC4978387 DOI: 10.1371/journal.pcbi.1005025] [Citation(s) in RCA: 68] [Impact Index Per Article: 8.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/17/2016] [Accepted: 06/17/2016] [Indexed: 11/19/2022] Open
Abstract
In this study, we investigate if phase-locking of fast oscillatory activity relies on the anatomical skeleton and if simple computational models informed by structural connectivity can help further to explain missing links in the structure-function relationship. We use diffusion tensor imaging data and alpha band-limited EEG signal recorded in a group of healthy individuals. Our results show that about 23.4% of the variance in empirical networks of resting-state functional connectivity is explained by the underlying white matter architecture. Simulating functional connectivity using a simple computational model based on the structural connectivity can increase the match to 45.4%. In a second step, we use our modeling framework to explore several technical alternatives along the modeling path. First, we find that an augmentation of homotopic connections in the structural connectivity matrix improves the link to functional connectivity while a correction for fiber distance slightly decreases the performance of the model. Second, a more complex computational model based on Kuramoto oscillators leads to a slight improvement of the model fit. Third, we show that the comparison of modeled and empirical functional connectivity at source level is much more specific for the underlying structural connectivity. However, different source reconstruction algorithms gave comparable results. Of note, as the fourth finding, the model fit was much better if zero-phase lag components were preserved in the empirical functional connectome, indicating a considerable amount of functionally relevant synchrony taking place with near zero or zero-phase lag. The combination of the best performing alternatives at each stage in the pipeline results in a model that explains 54.4% of the variance in the empirical EEG functional connectivity. Our study shows that large-scale brain circuits of fast neural network synchrony strongly rely upon the structural connectome and simple computational models of neural activity can explain missing links in the structure-function relationship. Brain imaging techniques are broadly divided into the two categories of structural and functional imaging. Structural imaging provides information about the static physical connectivity within the brain, while functional imaging provides data about the dynamic ongoing activation of brain areas. Computational models allow to bridge the gap between these two modalities and allow to gain new insights. Specifically, in this study, we use structural data from diffusion tractography recordings to model functional brain connectivity obtained from fast EEG dynamics occurring at the alpha frequency. First, we present a simple reference procedure which consists of several steps to link the structural to the functional empirical data. Second, we systematically compare several alternative methods along the modeling path in order to assess their impact on the overall fit between simulations and empirical data. We explore preprocessing steps of the structural connectivity and different levels of complexity of the computational model. We highlight the importance of source reconstruction and compare commonly used source reconstruction algorithms and metrics to assess functional connectivity. Our results serve as an important orienting frame for the emerging field of brain network modeling.
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Affiliation(s)
- Holger Finger
- Institute of Cognitive Science, University of Osnabrück, Osnabrück, Germany
- Department of Neurophysiology and Pathophysiology, University Medical Center Hamburg-Eppendorf, Hamburg, Germany
- * E-mail:
| | - Marlene Bönstrup
- Department of Neurology, University Medical Center Hamburg-Eppendorf, Hamburg, Germany
| | - Bastian Cheng
- Department of Neurology, University Medical Center Hamburg-Eppendorf, Hamburg, Germany
| | - Arnaud Messé
- Department of Computational Neuroscience, University Medical Center Hamburg-Eppendorf, Hamburg, Germany
| | - Claus Hilgetag
- Department of Computational Neuroscience, University Medical Center Hamburg-Eppendorf, Hamburg, Germany
| | - Götz Thomalla
- Department of Neurology, University Medical Center Hamburg-Eppendorf, Hamburg, Germany
| | - Christian Gerloff
- Department of Neurology, University Medical Center Hamburg-Eppendorf, Hamburg, Germany
| | - Peter König
- Institute of Cognitive Science, University of Osnabrück, Osnabrück, Germany
- Department of Neurophysiology and Pathophysiology, University Medical Center Hamburg-Eppendorf, Hamburg, Germany
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25
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Petkoski S, Spiegler A, Proix T, Aram P, Temprado JJ, Jirsa VK. Heterogeneity of time delays determines synchronization of coupled oscillators. Phys Rev E 2016; 94:012209. [PMID: 27575125 DOI: 10.1103/physreve.94.012209] [Citation(s) in RCA: 32] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/12/2015] [Indexed: 05/01/2023]
Abstract
Network couplings of oscillatory large-scale systems, such as the brain, have a space-time structure composed of connection strengths and signal transmission delays. We provide a theoretical framework, which allows treating the spatial distribution of time delays with regard to synchronization, by decomposing it into patterns and therefore reducing the stability analysis into the tractable problem of a finite set of delay-coupled differential equations. We analyze delay-structured networks of phase oscillators and we find that, depending on the heterogeneity of the delays, the oscillators group in phase-shifted, anti-phase, steady, and non-stationary clusters, and analytically compute their stability boundaries. These results find direct application in the study of brain oscillations.
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Affiliation(s)
- Spase Petkoski
- Aix-Marseille Université, Inserm, INS UMR_S 1106, 13005, Marseille, France
- Aix-Marseille Université, CNRS, ISM UMR 7287, 13288, Marseille, France
| | - Andreas Spiegler
- Aix-Marseille Université, Inserm, INS UMR_S 1106, 13005, Marseille, France
| | - Timothée Proix
- Aix-Marseille Université, Inserm, INS UMR_S 1106, 13005, Marseille, France
| | - Parham Aram
- Department of Automatic Control and Systems Engineering, University of Sheffield, Sheffield S10 2TN, United Kingdom
| | | | - Viktor K Jirsa
- Aix-Marseille Université, Inserm, INS UMR_S 1106, 13005, Marseille, France
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26
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Meier J, Tewarie P, Hillebrand A, Douw L, van Dijk BW, Stufflebeam SM, Van Mieghem P. A Mapping Between Structural and Functional Brain Networks. Brain Connect 2016; 6:298-311. [PMID: 26860437 DOI: 10.1089/brain.2015.0408] [Citation(s) in RCA: 88] [Impact Index Per Article: 11.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
The relationship between structural and functional brain networks is still highly debated. Most previous studies have used a single functional imaging modality to analyze this relationship. In this work, we use multimodal data, from functional MRI, magnetoencephalography, and diffusion tensor imaging, and assume that there exists a mapping between the connectivity matrices of the resting-state functional and structural networks. We investigate this mapping employing group averaged as well as individual data. We indeed find a significantly high goodness of fit level for this structure-function mapping. Our analysis suggests that a functional connection is shaped by all walks up to the diameter in the structural network in both modality cases. When analyzing the inverse mapping, from function to structure, longer walks in the functional network also seem to possess minor influence on the structural connection strengths. Even though similar overall properties for the structure-function mapping are found for different functional modalities, our results indicate that the structure-function relationship is modality dependent.
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Affiliation(s)
- Jil Meier
- 1 Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology , The Netherlands
| | - Prejaas Tewarie
- 2 Department of Neurology, Neuroscience Campus Amsterdam, VU University Medical Center , Amsterdam, The Netherlands
| | - Arjan Hillebrand
- 3 Department of Clinical Neurophysiology and Magnetoencephalography Center, Neuroscience Campus Amsterdam, VU University Medical Center , Amsterdam, The Netherlands
| | - Linda Douw
- 4 Department of Anatomy and Neurosciences, VU University Medical Center , Amsterdam, The Netherlands .,5 Department of Radiology, Athinoula A. Martinos Center for Biomedical Imaging/Massachusetts General Hospital , Boston, Massachusetts
| | - Bob W van Dijk
- 3 Department of Clinical Neurophysiology and Magnetoencephalography Center, Neuroscience Campus Amsterdam, VU University Medical Center , Amsterdam, The Netherlands
| | - Steven M Stufflebeam
- 5 Department of Radiology, Athinoula A. Martinos Center for Biomedical Imaging/Massachusetts General Hospital , Boston, Massachusetts.,6 Department of Radiology, Harvard Medical School , Boston, Massachusetts
| | - Piet Van Mieghem
- 1 Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology , The Netherlands
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27
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Erneux T, Weicker L, Bauer L, Hövel P. Short-time-delay limit of the self-coupled FitzHugh-Nagumo system. Phys Rev E 2016; 93:022208. [PMID: 26986332 DOI: 10.1103/physreve.93.022208] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/18/2015] [Indexed: 11/07/2022]
Abstract
We analyze the FitzHugh-Nagumo equations subject to time-delayed self-feedback in the activator variable. Parameters are chosen such that the steady state is stable independent of the feedback gain and delay τ. We demonstrate that stable large-amplitude τ-periodic oscillations can, however, coexist with a stable steady state even for small delays, which is mathematically counterintuitive. In order to explore how these solutions appear in the bifurcation diagram, we propose three different strategies. We first analyze the emergence of periodic solutions from Hopf bifurcation points for τ small and show that a subcritical Hopf bifurcation allows the coexistence of stable τ-periodic and stable steady-state solutions. Second, we construct a τ-periodic solution by using singular perturbation techniques appropriate for slow-fast systems. The theory assumes τ=O(1) and its validity as τ→0 is investigated numerically by integrating the original equations. Third, we develop an asymptotic theory where the delay is scaled with respect to the fast timescale of the activator variable. The theory is applied to the FitzHugh-Nagumo equations with threshold nonlinearity, and we show that the branch of τ-periodic solutions emerges from a limit point of limit cycles.
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Affiliation(s)
- Thomas Erneux
- Université Libre de Bruxelles, Optique Nonlinéaire théorique, Campus Plaine, C.P. 231, 1050 Bruxelles, Belgium
| | - Lionel Weicker
- LMOPS, CentraleSupélec, Université Paris-Saclay, 57070 METZ.,LMOPS, CentraleSupélec, Université de Lorraine, 57070 METZ
| | - Larissa Bauer
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Philipp Hövel
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany.,Bernstein Center for Computational Neuroscience Berlin, Humboldt-Universität zu Berlin, Philippstraße 13, 10115 Berlin, Germany
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