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

2. 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

3. Christiaan Huygens (1629-1695): clocks

4. Huygens’ pendulum clocks

5. ``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

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

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

8. Physarum polycephalum (Slime mold) A. Takamatsu, Physica D 223 180-188 (2006)

9. 3 observed oscillation patterns: Physarum polycephalum (Slime mold) A. Takamatsu, Physica D 223 180-188 (2006) Rotation2 in phase 2 anti-phase 1 double freq

10. Rotation2 in phase2 anti-phase 1 double freq Physarum polycephalum (Slime mold) A. Takamatsu and collaborators. See www.f.waseda.jp/atsuko_ta/Atsuko_E.html

11. Football /Basketball PloS Computational Biology 7(10): e1002181, October 2011

12. Football / Basketball Movie: – journal.pcbi.1002181.s010.avi journal.pcbi.1002181.s010.avi Rotation: – journal.pcbi.1002181.s008.avi journal.pcbi.1002181.s008.avi 2 anti-phase (and 1 lower amplitude): – journal.pcbi.1002181.s009.avi journal.pcbi.1002181.s009.avi

13. 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.

14. `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

15. 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

16. Fireflies all-to-all coupling of similar but not identical oscillators Kuramoto model http://www.youtube.com/watch?v=sROKYela Wbo http://www.youtube.com/watch?v=sROKYela Wbo

17. 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)

18. Kuramoto model ε small: no synchronisationε large: synchronisation arises

19. 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.