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Lecture 5: 2-d models and phase-plane analysis. references: Gerstner & Kistler, Ch 3 Koch, Ch 7. Reduced models. In HH, m is much faster than n, h. Reduced models. In HH, m is much faster than n, h. Reduced models. In HH, m is much faster than n, h. Try 2-d system:.

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


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    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