?url_ver=Z39.88-2004&rft_id=arXiv%3A1703.02535&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rft.relation=http%3A%2F%2Feprints.imtlucca.it%2F3999%2F&rft.title=Synchronization+of+phase+oscillators+on+the+hierarchical+lattice&rft.creator=Garlaschelli%2C+Diego&rft.creator=den+Hollander%2C+Frank&rft.creator=Meylahn%2C+Janusz&rft.creator=Zeegers%2C+Benthen&rft.subject=QC+Physics&rft.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%2C+namely%2C+the+hierarchical+lattice.+Each+oscillator+has+a+natural+frequency%2C+drawn+independently+from+a+common+probability+distribution.+In+addition%2C+pairs+of+oscillators+interact+with+each+other+at+a+strength+that+depends+on+their+hierarchical+distance%2C+modulated+by+a+sequence+of+interaction+parameters.+We+look+at+block+averages+of+the+oscillators+on+successive+hierarchical+scales%2C+which+we+think+of+as+block+communities.+Also+these+block+communities+are+given+a+natural+frequency%2C+drawn+independently+from+a+common+probability+distribution+that+depends+on+their+hierarchical+scale.+In+the+limit+as+the+number+of+oscillators+per+community+tends+to+infinity%2C+referred+to+as+the+hierarchical+mean-field+limit%2C+we+find+a+separation+of+time+scales%2C+i.e.%2C+each+block+community+behaves+like+a+single+oscillator+evolving+on+its+own+time+scale.+We+show+that+the+evolution+of+the+block+communities+is+given+by+a+renormalized+mean-field+noisy+Kuramoto+equation%2C+with+a+synchronization+level+that+depends+on+the+hierarchical+scale+of+the+block+community.+We+identify+three+universality+classes+for+the+synchronization+levels+on+successive+hierarchical+scales%2C+with+explicit+characterizations+in+terms+of+the+sequence+of+interaction+parameters+and+the+sequence+of+natural+frequency+probability+distributions.+We+show+that+disorder+reduces+synchronization+when+the+natural+frequency+probability+distributions+are+symmetric+and+unimodal%2C+with+the+reduction+gradually+vanishing+as+the+hierarchical+scale+goes+up.&rft.publisher=arXiv&rft.date=2017&rft.type=Working+Paper&rft.type=NonPeerReviewed&rft.identifier=++Garlaschelli%2C+Diego+and+den+Hollander%2C+Frank+and+Meylahn%2C+Janusz+and+Zeegers%2C+Benthen++Synchronization+of+phase+oscillators+on+the+hierarchical+lattice.++Working+Paper+++arXiv+++++++(Submitted)+++&rft.relation=http%3A%2F%2Farxiv.org%2Fpdf%2F1703.02535&rft.relation=arXiv%3A1703.02535