Synchronization of Phase Oscillators on the Hierarchical Lattice

Synchronization of neurons forming a network with a hierarchical structure is essential for the brain to be able to function optimally. In this paper we study synchronization of phase oscillators on the most basic example of such a network, namely, the hierarchical lattice. Each site of the lattice...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Veröffentlicht in:Journal of statistical physics 2019-01, Vol.174 (1), p.188-218
Hauptverfasser: Garlaschelli, D., den Hollander, F., Meylahn, J. M., Zeegers, B.
Format: Artikel
Sprache:eng
Schlagworte:
Online-Zugang:Volltext
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
container_end_page 218
container_issue 1
container_start_page 188
container_title Journal of statistical physics
container_volume 174
creator Garlaschelli, D.
den Hollander, F.
Meylahn, J. M.
Zeegers, B.
description Synchronization of neurons forming a network with a hierarchical structure is essential for the brain to be able to function optimally. In this paper we study synchronization of phase oscillators on the most basic example of such a network, namely, the hierarchical lattice. Each site of the lattice carries an oscillator that is subject to noise. Pairs of oscillators interact with each other at a strength that depends on their hierarchical distance, modulated by a sequence of interaction parameters. We look at block averages of the oscillators on successive hierarchical scales, which we think of as block communities. In the limit as the number of oscillators per community tends to infinity, referred to as the hierarchical mean-field limit, we find a separation of time scales, i.e., each block community behaves like a single oscillator evolving on its own time scale. We argue that the evolution of the block communities is given by a renormalized mean-field noisy Kuramoto equation, with a synchronization level that depends on the hierarchical scale of the block community. We find three universality classes for the synchronization levels on successive hierarchical scales, characterized in terms of the sequence of interaction parameters. What makes our model specifically challenging is the non-linearity of the interaction between the oscillators. The main results of our paper therefore come in three parts: (I) a conjecture about the nature of the renormalisation transformation connecting successive hierarchical scales; (II) a truncation approximation that leads to a simplified renormalization transformation; (III) a rigorous analysis of the simplified renormalization transformation. We provide compelling arguments in support of (I) and (II), but a full verification remains an open problem.
doi_str_mv 10.1007/s10955-018-2208-5
format Article
fullrecord <record><control><sourceid>gale_proqu</sourceid><recordid>TN_cdi_proquest_journals_2172720549</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><galeid>A571733726</galeid><sourcerecordid>A571733726</sourcerecordid><originalsourceid>FETCH-LOGICAL-c355t-bc5ddb124ce0ba4862f39f618410fcdacbffb2228198ef5a672bf5fb0331ddc83</originalsourceid><addsrcrecordid>eNp1kMFKAzEQhoMoWKsP4G3B8-ok2WyyN0tRKxQqqOeQzSZtynZTk_RQn96UFTzJHAaG_5sZPoRuMdxjAP4QMTSMlYBFSQiIkp2hCWaclE2N6TmaABBSVhyzS3QV4xYAGtGwCXp8Pw56E_zgvlVyfii8Ld42KppiFbXre5V8iEWep40pFs4EFfTGadUXS5WS0-YaXVjVR3Pz26fo8_npY74ol6uX1_lsWWrKWCpbzbquxaTSBlpViZpY2tgaiwqD1Z3SrbUtIUTgRhjLVM1Ja5ltgVLcdVrQKbob9-6D_zqYmOTWH8KQT0qCOeEEWNXk1P2YWqveSDdYn4LSuTqzc9oPxro8nzGOOaWc1BnAI6CDjzEYK_fB7VQ4SgzyZFaOZmU2K09mJcsMGZmYs8PahL9X_od-AE9oe14</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2172720549</pqid></control><display><type>article</type><title>Synchronization of Phase Oscillators on the Hierarchical Lattice</title><source>SpringerNature Journals</source><creator>Garlaschelli, D. ; den Hollander, F. ; Meylahn, J. M. ; Zeegers, B.</creator><creatorcontrib>Garlaschelli, D. ; den Hollander, F. ; Meylahn, J. M. ; Zeegers, B.</creatorcontrib><description>Synchronization of neurons forming a network with a hierarchical structure is essential for the brain to be able to function optimally. In this paper we study synchronization of phase oscillators on the most basic example of such a network, namely, the hierarchical lattice. Each site of the lattice carries an oscillator that is subject to noise. Pairs of oscillators interact with each other at a strength that depends on their hierarchical distance, modulated by a sequence of interaction parameters. We look at block averages of the oscillators on successive hierarchical scales, which we think of as block communities. In the limit as the number of oscillators per community tends to infinity, referred to as the hierarchical mean-field limit, we find a separation of time scales, i.e., each block community behaves like a single oscillator evolving on its own time scale. We argue that the evolution of the block communities is given by a renormalized mean-field noisy Kuramoto equation, with a synchronization level that depends on the hierarchical scale of the block community. We find three universality classes for the synchronization levels on successive hierarchical scales, characterized in terms of the sequence of interaction parameters. What makes our model specifically challenging is the non-linearity of the interaction between the oscillators. The main results of our paper therefore come in three parts: (I) a conjecture about the nature of the renormalisation transformation connecting successive hierarchical scales; (II) a truncation approximation that leads to a simplified renormalization transformation; (III) a rigorous analysis of the simplified renormalization transformation. We provide compelling arguments in support of (I) and (II), but a full verification remains an open problem.</description><identifier>ISSN: 0022-4715</identifier><identifier>EISSN: 1572-9613</identifier><identifier>DOI: 10.1007/s10955-018-2208-5</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Brain ; Communities ; Interaction parameters ; Linearity ; Mathematical and Computational Physics ; Neurons ; Oscillators ; Physical Chemistry ; Physics ; Physics and Astronomy ; Quantum Physics ; Statistical Physics and Dynamical Systems ; Structural hierarchy ; Synchronism ; Theoretical ; Transformations (mathematics)</subject><ispartof>Journal of statistical physics, 2019-01, Vol.174 (1), p.188-218</ispartof><rights>Springer Science+Business Media, LLC, part of Springer Nature 2018</rights><rights>COPYRIGHT 2019 Springer</rights><rights>Copyright Springer Nature B.V. 2019</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c355t-bc5ddb124ce0ba4862f39f618410fcdacbffb2228198ef5a672bf5fb0331ddc83</citedby><cites>FETCH-LOGICAL-c355t-bc5ddb124ce0ba4862f39f618410fcdacbffb2228198ef5a672bf5fb0331ddc83</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s10955-018-2208-5$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10955-018-2208-5$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Garlaschelli, D.</creatorcontrib><creatorcontrib>den Hollander, F.</creatorcontrib><creatorcontrib>Meylahn, J. M.</creatorcontrib><creatorcontrib>Zeegers, B.</creatorcontrib><title>Synchronization of Phase Oscillators on the Hierarchical Lattice</title><title>Journal of statistical physics</title><addtitle>J Stat Phys</addtitle><description>Synchronization of neurons forming a network with a hierarchical structure is essential for the brain to be able to function optimally. In this paper we study synchronization of phase oscillators on the most basic example of such a network, namely, the hierarchical lattice. Each site of the lattice carries an oscillator that is subject to noise. Pairs of oscillators interact with each other at a strength that depends on their hierarchical distance, modulated by a sequence of interaction parameters. We look at block averages of the oscillators on successive hierarchical scales, which we think of as block communities. In the limit as the number of oscillators per community tends to infinity, referred to as the hierarchical mean-field limit, we find a separation of time scales, i.e., each block community behaves like a single oscillator evolving on its own time scale. We argue that the evolution of the block communities is given by a renormalized mean-field noisy Kuramoto equation, with a synchronization level that depends on the hierarchical scale of the block community. We find three universality classes for the synchronization levels on successive hierarchical scales, characterized in terms of the sequence of interaction parameters. What makes our model specifically challenging is the non-linearity of the interaction between the oscillators. The main results of our paper therefore come in three parts: (I) a conjecture about the nature of the renormalisation transformation connecting successive hierarchical scales; (II) a truncation approximation that leads to a simplified renormalization transformation; (III) a rigorous analysis of the simplified renormalization transformation. We provide compelling arguments in support of (I) and (II), but a full verification remains an open problem.</description><subject>Brain</subject><subject>Communities</subject><subject>Interaction parameters</subject><subject>Linearity</subject><subject>Mathematical and Computational Physics</subject><subject>Neurons</subject><subject>Oscillators</subject><subject>Physical Chemistry</subject><subject>Physics</subject><subject>Physics and Astronomy</subject><subject>Quantum Physics</subject><subject>Statistical Physics and Dynamical Systems</subject><subject>Structural hierarchy</subject><subject>Synchronism</subject><subject>Theoretical</subject><subject>Transformations (mathematics)</subject><issn>0022-4715</issn><issn>1572-9613</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2019</creationdate><recordtype>article</recordtype><recordid>eNp1kMFKAzEQhoMoWKsP4G3B8-ok2WyyN0tRKxQqqOeQzSZtynZTk_RQn96UFTzJHAaG_5sZPoRuMdxjAP4QMTSMlYBFSQiIkp2hCWaclE2N6TmaABBSVhyzS3QV4xYAGtGwCXp8Pw56E_zgvlVyfii8Ld42KppiFbXre5V8iEWep40pFs4EFfTGadUXS5WS0-YaXVjVR3Pz26fo8_npY74ol6uX1_lsWWrKWCpbzbquxaTSBlpViZpY2tgaiwqD1Z3SrbUtIUTgRhjLVM1Ja5ltgVLcdVrQKbob9-6D_zqYmOTWH8KQT0qCOeEEWNXk1P2YWqveSDdYn4LSuTqzc9oPxro8nzGOOaWc1BnAI6CDjzEYK_fB7VQ4SgzyZFaOZmU2K09mJcsMGZmYs8PahL9X_od-AE9oe14</recordid><startdate>20190101</startdate><enddate>20190101</enddate><creator>Garlaschelli, D.</creator><creator>den Hollander, F.</creator><creator>Meylahn, J. M.</creator><creator>Zeegers, B.</creator><general>Springer US</general><general>Springer</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20190101</creationdate><title>Synchronization of Phase Oscillators on the Hierarchical Lattice</title><author>Garlaschelli, D. ; den Hollander, F. ; Meylahn, J. M. ; Zeegers, B.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c355t-bc5ddb124ce0ba4862f39f618410fcdacbffb2228198ef5a672bf5fb0331ddc83</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2019</creationdate><topic>Brain</topic><topic>Communities</topic><topic>Interaction parameters</topic><topic>Linearity</topic><topic>Mathematical and Computational Physics</topic><topic>Neurons</topic><topic>Oscillators</topic><topic>Physical Chemistry</topic><topic>Physics</topic><topic>Physics and Astronomy</topic><topic>Quantum Physics</topic><topic>Statistical Physics and Dynamical Systems</topic><topic>Structural hierarchy</topic><topic>Synchronism</topic><topic>Theoretical</topic><topic>Transformations (mathematics)</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Garlaschelli, D.</creatorcontrib><creatorcontrib>den Hollander, F.</creatorcontrib><creatorcontrib>Meylahn, J. M.</creatorcontrib><creatorcontrib>Zeegers, B.</creatorcontrib><collection>CrossRef</collection><jtitle>Journal of statistical physics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Garlaschelli, D.</au><au>den Hollander, F.</au><au>Meylahn, J. M.</au><au>Zeegers, B.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Synchronization of Phase Oscillators on the Hierarchical Lattice</atitle><jtitle>Journal of statistical physics</jtitle><stitle>J Stat Phys</stitle><date>2019-01-01</date><risdate>2019</risdate><volume>174</volume><issue>1</issue><spage>188</spage><epage>218</epage><pages>188-218</pages><issn>0022-4715</issn><eissn>1572-9613</eissn><abstract>Synchronization of neurons forming a network with a hierarchical structure is essential for the brain to be able to function optimally. In this paper we study synchronization of phase oscillators on the most basic example of such a network, namely, the hierarchical lattice. Each site of the lattice carries an oscillator that is subject to noise. Pairs of oscillators interact with each other at a strength that depends on their hierarchical distance, modulated by a sequence of interaction parameters. We look at block averages of the oscillators on successive hierarchical scales, which we think of as block communities. In the limit as the number of oscillators per community tends to infinity, referred to as the hierarchical mean-field limit, we find a separation of time scales, i.e., each block community behaves like a single oscillator evolving on its own time scale. We argue that the evolution of the block communities is given by a renormalized mean-field noisy Kuramoto equation, with a synchronization level that depends on the hierarchical scale of the block community. We find three universality classes for the synchronization levels on successive hierarchical scales, characterized in terms of the sequence of interaction parameters. What makes our model specifically challenging is the non-linearity of the interaction between the oscillators. The main results of our paper therefore come in three parts: (I) a conjecture about the nature of the renormalisation transformation connecting successive hierarchical scales; (II) a truncation approximation that leads to a simplified renormalization transformation; (III) a rigorous analysis of the simplified renormalization transformation. We provide compelling arguments in support of (I) and (II), but a full verification remains an open problem.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s10955-018-2208-5</doi><tpages>31</tpages></addata></record>
fulltext fulltext
identifier ISSN: 0022-4715
ispartof Journal of statistical physics, 2019-01, Vol.174 (1), p.188-218
issn 0022-4715
1572-9613
language eng
recordid cdi_proquest_journals_2172720549
source SpringerNature Journals
subjects Brain
Communities
Interaction parameters
Linearity
Mathematical and Computational Physics
Neurons
Oscillators
Physical Chemistry
Physics
Physics and Astronomy
Quantum Physics
Statistical Physics and Dynamical Systems
Structural hierarchy
Synchronism
Theoretical
Transformations (mathematics)
title Synchronization of Phase Oscillators on the Hierarchical Lattice
url https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2024-12-25T18%3A13%3A35IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-gale_proqu&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=Synchronization%20of%20Phase%20Oscillators%20on%20the%20Hierarchical%20Lattice&rft.jtitle=Journal%20of%20statistical%20physics&rft.au=Garlaschelli,%20D.&rft.date=2019-01-01&rft.volume=174&rft.issue=1&rft.spage=188&rft.epage=218&rft.pages=188-218&rft.issn=0022-4715&rft.eissn=1572-9613&rft_id=info:doi/10.1007/s10955-018-2208-5&rft_dat=%3Cgale_proqu%3EA571733726%3C/gale_proqu%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_pqid=2172720549&rft_id=info:pmid/&rft_galeid=A571733726&rfr_iscdi=true