Patterns of oscillations and synchronisation in networks Jonathan Dawes Dept of Mathematical Sciences.

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Presentation transcript:

Patterns of oscillations and synchronisation in networks Jonathan Dawes Dept of Mathematical Sciences

Outline Small networks: – Huygens’ pendulum clocks – Slime mold – Ball passing: football / basketball Patterns on grids: – Robust grid patterns forced by symmetry – Robust patterns without symmetry Large networks: – Fireflies and the Kuramoto model

Christiaan Huygens ( ): clocks

Huygens’ pendulum clocks

``Robustness’’ 1/2 If the system has `underlying symmetry operations’: – permutation (no preferred arrangement in space) – time invariance (no preferred origin t=0 in time) Then solutions have the following special property: – any `symmetry operation’ that is present initially, and which leaves the system unchanged... –... also remains present at all future times

``Robustness’’ 2/2 Often, symmetry operations combine spatial and temporal information: Reflect in midplane, then wait half a period Reflect in midplane

Physarum polycephalum (Slime mold) Collection of unicellular amoebae Aggregation via cAMP signalling - chemotaxis

Physarum polycephalum (Slime mold) A. Takamatsu, Physica D (2006)

3 observed oscillation patterns: Physarum polycephalum (Slime mold) A. Takamatsu, Physica D (2006) Rotation2 in phase 2 anti-phase 1 double freq

Rotation2 in phase2 anti-phase 1 double freq Physarum polycephalum (Slime mold) A. Takamatsu and collaborators. See

Football /Basketball PloS Computational Biology 7(10): e , October 2011

Football / Basketball Movie: – journal.pcbi s010.avi journal.pcbi s010.avi Rotation: – journal.pcbi s008.avi journal.pcbi s008.avi 2 anti-phase (and 1 lower amplitude): – journal.pcbi s009.avi journal.pcbi s009.avi

Robust grid patterns Symmetric: ‘Balanced colourings’: Every W has 2W and 2B neighbours Every B has 2W and 2B neighbours Switch colours on a diagonal: M. Golubitsky and I Stewart. The Symmetry Perspective. Birkhauser 2002.

`Asymmetric’ (i.e. Not forced by spatial symmetry): Robust grid patterns Every W has 2W and 2B neighbours Every B has 1W and 3B neighbours Every W has 2W and 2B neighbours Every B has 2W and 2B neighbours

Asymmetric surprise: Robust grid patterns Every W has 2W and 2B neighbours Every B has 1W and 3B neighbours Every W has 2W, 1B and 1R neighbour Every B has 2B, 1W and 1R neighbour Every R has 2R, 1W and 1B neighbour

Fireflies all-to-all coupling of similar but not identical oscillators Kuramoto model Wbo Wbo

Kuramoto model All to all coupling Oscillator k has phase φ k : Mean field: phase Θ and amplitude KDistribution of natural frequencies: A. Pikovsky and M. Rosemblum and J. Kurths, Synchronisation: a universal concept in nonlinear sciences. CUP (2001)

Kuramoto model ε small: no synchronisationε large: synchronisation arises

Recent extensions 1.We are trying to develop theory for larger numbers of oscs by working out what ‘scales’ from small numbers of oscs 2.Interestingly, some couplings in a network do not affect oscillation pattern, for example... 3.Instabilities within a scale-free network can promote ‘pattern formation’ by node degree.