Lecture 5: 2-d models and phase-plane analysis

1. Lecture 5: 2-d models and phase-plane analysis • references: Gerstner & Kistler, Ch 3 Koch, Ch 7

2. Reduced models In HH, m is much faster than n, h

3. Reduced models In HH, m is much faster than n, h

4. Reduced models In HH, m is much faster than n, h • Try 2-d system:

5. Reduced models In HH, m is much faster than n, h • Try 2-d system: membrane potential eqn:

6. Reduced models In HH, m is much faster than n, h • Try 2-d system: membrane potential eqn: Dynamics of w (like n):

7. Reduced models In HH, m is much faster than n, h • Try 2-d system: membrane potential eqn: Dynamics of w (like n): (from now on V -> u, C=1):

8. Nullclines

9. Nullclines u-nullcline w-nullcline

10. Nullclines u-nullcline w-nullcline

11. Nullclines u-nullcline w-nullcline intersections: fixed points

12. Stability of fixed points For a general system

13. Stability of fixed points For a general system Expand around FP:

14. Stability of fixed points For a general system Expand around FP:

15. Stability of fixed points For a general system Expand around FP: or

16. Stability of fixed points For a general system Expand around FP: or where

17. Stability of fixed points For a general system Expand around FP: or where

18. linearization set

19. linearization set Find eigenvectors v and eigenvalues l

20. linearization set Find eigenvectors v and eigenvalues l Stability if l1, l2 both < 0

21. linearization set Find eigenvectors v and eigenvalues l Stability if l1, l2 both < 0 i.e., l1+ l2 = trM< 0(Fu + Gw < 0) and l1l2 = det M > 0 (FuGw – FwGu > 0)

22. linearization set Find eigenvectors v and eigenvalues l Stability if l1, l2 both < 0 i.e., l1+ l2 = trM< 0(Fu + Gw < 0) and l1l2 = det M > 0 (FuGw – FwGu > 0) If det M < 0, both eigenvalues are real, one >0, the other <0:

23. linearization set Find eigenvectors v and eigenvalues l Stability if l1, l2 both < 0 i.e., l1+ l2 = trM< 0(Fu + Gw < 0) and l1l2 = det M > 0 (FuGw – FwGu > 0) If det M < 0, both eigenvalues are real, one >0, the other <0: Saddle point

24. Linearized equations: case A

25. Linearized equations: case A

26. Linearized equations: case A

27. Linearized equations: case A

28. Linearized equations: case A  stable FP

29. Case B:

30. Case B: Now make a > 0:

31. Case B: Now make a > 0: Lose stability if a > e or a > b

32. Case B: Now make a > 0: Lose stability if a > e or a > b Consider a < b:

33. Case B: Now make a > 0: Lose stability if a > e or a > b Consider a < b:

34. Case B: Now make a > 0: Lose stability if a > e or a > b Consider a < b: a< e: stable FP

35. Case B: Now make a > 0: Lose stability if a > e or a > b Consider a < b: a< e: stable FP a> e: unstable (both eigenvalues have positive real parts)

36. Case C: Now consider a > b > 0:

37. Case C: Now consider a > b > 0:

38. Case C: Now consider a > b > 0: det M < 0

39. Case C: Now consider a > b > 0: det M < 0 1 eigenvalue positive, 1 negative

40. Case C: Now consider a > b > 0: det M < 0 1 eigenvalue positive, 1 negative: saddle point

41. Case D:

42. Case D:

43. Case D:

44. Case D: det M < 0  saddle point

45. Poincare-Bendixson theorem If

46. Poincare-Bendixson theorem If • You have a repulsive fixed point • (both eigenvalues have positive real parts)

47. Poincare-Bendixson theorem If • You have a repulsive fixed point • (both eigenvalues have positive real parts) (2) There is a “bounding box” with the property that all flow across it is inward

48. Poincare-Bendixson theorem If • You have a repulsive fixed point • (both eigenvalues have positive real parts) (2) There is a “bounding box” with the property that all flow across it is inward

49. Poincare-Bendixson theorem If • You have a repulsive fixed point • (both eigenvalues have positive real parts) (2) There is a “bounding box” with the property that all flow across it is inward Then There must be a limit cycle in between

50. FitzHugh-Nagumo model