2. Dynamics of Networks 1 Basic Formalism & Symmetry Ian Stewart Mathematics Institute University of Warwick UK-Japan Winter School Dynamics and Complexity

3. Examples of Network Dynamics nerve cell or neuron

4. Examples of Network Dynamics Neurons form networks that transmit and process signals

5. Examples of Network Dynamics Individual neurons can be modelled by an ODE Hodgkin-Huxley Equations Fitzhugh-Nagumo Equations Morris-Lecar Equations and many others...

6. Examples of Network Dynamics Fitzhugh-Nagumo Equations dv/dt = v(a-v)(v-1) - w + I a dw/dt = bv - w v = membrane potential w = substitute for ion channel variables I a = applied current a, b, are constants 0 < a < 1 b, 0

7. Examples of Network Dynamics Fitzhugh-Nagumo Equations dv/dt = v(a-v)(v-1) - w + I a dw/dt = bv - w

8. Examples of Network Dynamics Coupled Fitzhugh-Nagumo Equations for 2 neurons dv 1 /dt = v 1 (a-v 1 )(v 1 -1) - w 1 dw 1 /dt = bv 1 - w 1 I a = 0 dv 2 /dt = v 2 (a-v 2 )(v 2 -1) - w 2 dw 2 /dt = bv 2 - w 2

9. Examples of Network Dynamics Coupled Fitzhugh-Nagumo Equations for 2 neurons dv 1 /dt = v 1 (a-v 1 )(v 1 -1) - w 1 - cv 2 dw 1 /dt = bv 1 - w 1 I a = 0 dv 2 /dt = v 2 (a-v 2 )(v 2 -1) - w 2 - cv 1 dw 2 /dt = bv 2 - w 2

10. Examples of Network Dynamics Coupled Fitzhugh-Nagumo Equations for 2 neurons Identical waveforms half-period phase difference a = b = = 0.5 c = 1.1

11. Examples of Network Dynamics 3-cell bidirectional ring

12. Examples of Network Dynamics 3-cell bidirectional ring Identical waveforms 1/3-period phase difference

13. Examples of Network Dynamics 3-cell bidirectional ring 2 cells have identical waveforms half-period phase difference. Third cell has double frequency.

14. Synchrony and Phase Patterns Spatial symmetry distinct cells are synchronous Temporal symmetry cell state is time-periodic Spatio-temporal symmetry distinct cells are identical except for phase shift Multirhythms a cell is identical to itself with a nontrivial phase shift rational frequency relationships. This is a special type of resonance caused by symmetry

15. Example: Animal Gaits Eadweard Muybridge

16. Common Animal Gaits WALK LEFT rear LEFT front RIGHT rear RIGHT front

17. TROT LEFT rear + RIGHT front RIGHT rear + LEFT front

18. CANTER LEFT rear RIGHT rear + LEFT front RIGHT front

19. TRANSVERSE GALLOP LEFT rear + (delay) RIGHT rear LEFT front + (delay) RIGHT front

20. RACK or PACE LEFT rear + LEFT front RIGHT rear + RIGHT front

21. Pattern of Phases WALK Four legs hit the ground at equally spaced times from back to front: left, then right

22. Pattern of Phases TROT Diagonal pairs of legs hit the ground alternately

23. Pattern of Phases PACE Left legs hit the ground together; then right legs hit the ground together

24. Pattern of Phases BOUND Rear legs hit the ground together; then front legs hit the ground together

25. Pattern of Phases TRANSVERSE GALLOP Like a bound but with slight delays in each pair of legs

26. Pattern of Phases ROTARY GALLOP Like transverse gallop but one pair of legs uses opposite timing

27. Pattern of Phases CANTER One diagonal pair of legs is synchronised; other pair alternates

28. Pattern of Phases PRONK

29. Pronk ?

30. Pronking Alpaca

31. Pattern of Phases PRONK

32. WALKTROTPACEBOUND TRANSVERSE GALLOP ROTARY GALLOP CANTERPRONK

33. WHY are there so many? HOW are they produced?

34. WHY are there so many?

35. EFFICIENCY and EFFECTIVENESS

36. HOW are they produced?

37. CENTRAL PATTERN GENERATOR HOW are they produced?

38. Central Pattern Generator Network of nerve cells (neurons) in the spinal column, not in the brain

39. Coupled Oscillators

40. In phase Out of phase same state at all times state lags by half the period

41. Coupled Oscillators

46. pronktrot walk reversewalk But what about the others?

47. A more detailed model involving the main leg muscle groups uses all eight oscillators to drive the legs: these four to push and the other four to pull Four of the oscillators set the pattern of phase shifts An argument based on symmetry suggests an 8- oscillator network as the simplest possibility

48. WALK

49. TROT

50. BOUND

51. PACE

52. PRONK

53. ROTARY GALLOP TRANSVERSE GALLOP CANTER These also occur, as secondary patterns

54. BUCK

55. Classification of Phase Patterns H/K Theorem (Buono and Golubitsky) Let K be the set of all spatial symmetries those that leave the state of the system unchanged at every instant of time. Let H be the set of all spatio-temporal symmetries those that leave the state of the system unchanged except for a phase shift.

56. Example the PACE PACE Left legs hit the ground together; then right legs hit the ground together

57. Classification of Phase Patterns The set K of all spatial symmetries: Leave all legs unchanged Swap front and back Cyclic group Z 2 of order 2. The set K of all spatio-temporal symmetries: Leave all legs unchanged Swap front and back Swap left and right Swap front and back and left and right Dihedral group D 2 of order 4. Here K is normal in H and H/K is cyclic (of order 2).

58. H/K Theorem K is normal in H H/K is cyclic plus two more technical conditions Are necessary and sufficient for H and K to be the spatio-temporal and spatial symmetry groups of some periodic state (for some suitable ODE with the given symmetry)

59. H/K Theorem Essentially, the H/K theorem tells us which spatio-temporal symmetries are to be expected. Other Theorems (such as the Symmetric Hopf Bifurcation Theorem) provide sufficient conditions for various of these states to occur.

60. Patterns of Synchrony The spatial symmetry group K specifies which legs are in synchrony with which. It divides the legs into synchronous clusters

61. Networ k A network or directed graph consists of a set of: consists of a set of: nodes or vertices or cellsnodes or vertices or cells connected by directed edges or arrowsdirected edges or arrows

62. Networ k Each cell has a cell-type and each arrow has an arrow-type, allowing us to require the cells or arrows concerned to have the same structure. In effect these are labels on the cells and arrows. Abstractly they are specified by equivalence relations on the set of cells and the set of arrows.

63. Networ k Arrows may form loops (same head and tail), and there may be multiple arrows (connecting the same pair of cells). Special case: regular homogeneous networks. These have one type of cell, one type of arrow, and the number of arrows entering each cell is the same. This number is the valency of the network.

64. Regular Homogeneous Network This is a regular homogeneous network of valency 3.

65. Network Enumeration N v=1 v=2 v=3 v=4 v=5 v=6 1111111 23610152128 374418059015823724 4194756915634204122302080827 54768744447221040722682650761843405665412 613012675043242604556967721035590650168613508534834704 Number of topologically distinct regular homogeneous networks on N cells with valency v

66. Network Dynamics To any network we associate a class of admissible vector fields, defining admissible ODEs, which consists of those vector fields F(x) That respect the network structure, and the corresponding ODEs dx/dt = F(x) What does respect the network structure mean?

67. Admissible ODEs Admissible ODEs are defined in terms of the input structure of the network. The input set I(c) of a cell c is the set of all arrows whose head is c. This includes multiple arrows and loops.

68. Admissible ODEs Choose coordinates x c R k for each cell c. (We use R k for simplicity, and because we consider only local bifurcation). Then dx c /dt = f(x c,x T(I (c)) ) where T(I(c)) is the tuple of tail cells of I(c).

69. Admissible ODEs dx 1 /dt = f(x 1,x 1, x 2, x 3, x 3, x 4, x 5, x 5, x 5 ) dx c /dt = f(x c,x T(I (c)) )

70. Admissible ODEs Admissible ODEs for the example network: dx 1 /dt = f(x 1, x 2, x 2, x 3 ) dx 2 /dt = f(x 2, x 3, x 4, x 5 ) dx 3 /dt = f(x 3, x 1, x 3, x 4 ) dx 4 /dt = f(x 4, x 2, x 3, x 5 ) dx 5 /dt = f(x 5, x 2, x 4, x 4 ) Where f satisfies the symmetry condition f(x,u,v,w) is symmetric in u, v, w f(x,u,v,w) is symmetric in u, v, w

71. Admissible ODEs Because the network is regular and homogeneous, the condition respect the network structure implies that in any admissible ODE dx c /dt = f(x c,x T(I (c)) ) the same function f occurs in each equation. Moreover, f is symmetric in the variables x T(I (c)). However, the first variable is distinguished, so f is not required to be symmetric in that variable.

72. Admissible ODEs are those whose structure reflects the network topology and the types of the cells and arrows

73. dx 1 /dt = f 1 (x 1,x 2,x 4,x 5 ) dx 2 /dt = f 2 (x 2,x 1,x 3,x 5 ) dx 3 /dt = f 3 (x 3,x 1,x 4 ) dx 4 /dt = f 4 (x 4,x 2,x 4 ) dx 5 /dt = f 5 (x 5,x 4 ) domain condition

74. dx 1 /dt = f(x 1,x 2,x 4,x 5 ) dx 2 /dt = f(x 2,x 1,x 3,x 5 ) dx 3 /dt = g(x 3,x 1,x 4 ) dx 4 /dt = g(x 4,x 2,x 4 ) dx 5 /dt = h(x 5,x 4 ) pullback condition

75. dx 1 /dt = f(x 1,x 2,x 4,x 5 ) dx 2 /dt = f(x 2,x 1,x 3,x 5 ) dx 3 /dt = g(x 3,x 1,x 4 ) dx 4 /dt = g(x 4,x 2,x 4 ) dx 5 /dt = h(x 5,x 4 ) Vertex group symmetry

76. dx 1 /dt = f(x 1,x 2,x 4,x 5 ) dx 2 /dt = f(x 2,x 1,x 3,x 5 ) dx 3 /dt = g(x 3,x 1,x 4 ) dx 4 /dt = g(x 4,x 2,x 4 ) dx 5 /dt = h(x 5,x 4 ) How do synchronous states behave?

77. [with M.Golubitsky and M.Pivato] Symmetry groupoids and patterns of synchrony in coupled cell networks, SIAM J. Appl. Dyn. Sys. 2 (2003) 609-646. DOI: 10.1137/S1111111103419896 [with M.Golubitsky and M.Nicol] Some curious phenomena in coupled cell networks, J. Nonlin. Sci. 14 (2004) 207- 236. [with M.Golubitsky and A.Török] Patterns of synchrony in coupled cell networks with multiple arrows, SIAM J. Appl. Dyn. Sys. 4 (2005) 78-100. [DOI: 10.1137/040612634] [With F.A.M.Aldosray] Enumeration of homogeneous coupled cell networks, Internat. J. Bif. Chaos 15 (2005) 2361-2373. [with M.Golubitsky] Nonlinear dynamics of networks: the groupoid formalism, Bull. Amer. Math. Soc. 43 (2006) 305-364. References

78. Dynamics of Networks to be continued... Ian Stewart Mathematics Institute University of Warwick UK-Japan Winter School Dynamics and Complexity