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This lecture delves into 2-D models and phase plane analysis in the context of reduced Hodgkin-Huxley (HH) models. We explore how membrane potential equations exhibit dynamics remarkably faster in recovery variables (m) compared to activation (n) and inactivation (h). Through the examination of nullclines and fixed points, we determine the stability of these systems by utilizing linearization techniques, investigating cases where eigenvalues provide insights into stability conditions. This foundational understanding is crucial for modeling neuronal dynamics and reaction patterns.
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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:
Reduced models In HH, m is much faster than n, h • Try 2-d system: membrane potential eqn:
Reduced models In HH, m is much faster than n, h • Try 2-d system: membrane potential eqn: Dynamics of w (like n):
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):
Nullclines u-nullcline w-nullcline
Nullclines u-nullcline w-nullcline
Nullclines u-nullcline w-nullcline intersections: fixed points
Stability of fixed points For a general system
Stability of fixed points For a general system Expand around FP:
Stability of fixed points For a general system Expand around FP:
Stability of fixed points For a general system Expand around FP: or
Stability of fixed points For a general system Expand around FP: or where
Stability of fixed points For a general system Expand around FP: or where
linearization set
linearization set Find eigenvectors v and eigenvalues l
linearization set Find eigenvectors v and eigenvalues l Stability if l1, l2 both < 0
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)
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:
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
Linearized equations: case A stable FP
Case B: Now make a > 0:
Case B: Now make a > 0: Lose stability if a > e or a > b
Case B: Now make a > 0: Lose stability if a > e or a > b Consider a < b:
Case B: Now make a > 0: Lose stability if a > e or a > b Consider a < b:
Case B: Now make a > 0: Lose stability if a > e or a > b Consider a < b: a< e: stable FP
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)
Case C: Now consider a > b > 0:
Case C: Now consider a > b > 0:
Case C: Now consider a > b > 0: det M < 0
Case C: Now consider a > b > 0: det M < 0 1 eigenvalue positive, 1 negative
Case C: Now consider a > b > 0: det M < 0 1 eigenvalue positive, 1 negative: saddle point
Case D: det M < 0 saddle point
Poincare-Bendixson theorem If • You have a repulsive fixed point • (both eigenvalues have positive real parts)
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
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
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
