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JEAN-MARC GINOUX BRUNO ROSSETTO ginoux@univ-tln.fr rossetto@univ-tln.fr

Slow Invariant Manifold of Heartbeat Model. JEAN-MARC GINOUX BRUNO ROSSETTO ginoux@univ-tln.fr rossetto@univ-tln.fr http://ginoux.univ-tln.fr http://rossetto.univ-tln.fr Laboratoire PROTEE, I.U.T. de Toulon Université du Sud, B.P. 20132, 83957, LA GARDE Cedex, France.

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JEAN-MARC GINOUX BRUNO ROSSETTO ginoux@univ-tln.fr rossetto@univ-tln.fr

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  1. Slow Invariant Manifold of Heartbeat Model JEAN-MARC GINOUX BRUNO ROSSETTO ginoux@univ-tln.frrossetto@univ-tln.fr http://ginoux.univ-tln.frhttp://rossetto.univ-tln.fr Laboratoire PROTEE, I.U.T. de Toulon Université du Sud, B.P. 20132, 83957, LA GARDE Cedex, France

  2. OUTLINE A. Flow Curvature Method (F.C.M.) 1. Flow curvature manifold 2. Slow invariant manifold B. Heartbeat Model 1. 4-D unforced heartbeat model 2. 6-D forced heartbeat model C. Discussion & Perspectives 1. Interpretation 2. Collaborations ROUEN BioMedicine 2009

  3. MODELING DYNAMICAL SYSTEMS Modeling: • Defining states variables of a system (predator, prey) • Describing their evolution with differential equations (O.D.E.) Dynamical System: Representation of a differential equation in phase space expresses variation of each state variable  Determining variables from their variation (velocity) ROUEN BioMedicine 2009

  4. n-dimensional Dynamical Systems velocity ROUEN BioMedicine 2009

  5. MANIFOLD DEFINTION A manifold isdefined as a set of points in satisfying a system of m scalar equations : In dimension 2 In dimension 3 curve surface ROUEN BioMedicine 2009

  6. LIE DERIVATIVE Let a manifold and the integral of any dynamical system. The Lie derivative is defined as: ROUEN BioMedicine 2009

  7. INVARIANT MANIFOLDS Darboux Theorem for Invariant Manifolds: A manifold is invariant if there exists a function denoted and called cofactor such for all This notion is due to Gaston Darboux (1878) ROUEN BioMedicine 2009

  8. FLOW CURVATURE METHOD Geometric Method Flow Curvature Method (Ginoux & Rossetto, 2005  2009) velocity velocity  acceleration  over-acceleration  etc. … position  ROUEN BioMedicine 2009

  9. FLOW CURVATURE METHOD “trajectory curve” n-Euclidean space curve plane or space curve curvatures ROUEN BioMedicine 2009

  10. FLOW CURVATURE METHOD Flow curvature manifold: The flow curvature manifold is defined as the location of the points where the curvature of the flow, i.e., the curvature of trajectory curve integral of the dynamical system vanishes. where represents the n-th derivative ROUEN BioMedicine 2009

  11. FLOW CURVATURE METHOD Flow Curvature Manifold: In dimension 2: curvature or 1st curvature In dimension 3: torsion ou 2nd curvature ROUEN BioMedicine 2009

  12. FLOW CURVATURE METHOD Flow Curvature Manifold: In dimension 4: 3rd curvature In dimension 5: 4th curvature ROUEN BioMedicine 2009

  13. Heartbeat Model ROUEN BioMedicine 2009

  14. Heartbeat Model: (AV) node Van der Pol [1928] ; ROUEN BioMedicine 2009

  15. Heartbeat Model: (AV) node Van der Pol [1928] ; di Bernardo [1998] ; Signorini [1998] Piecewise Linear Function ROUEN BioMedicine 2009

  16. Heartbeat Model ROUEN BioMedicine 2009

  17. 4D-Unforced Heartbeat Model di Bernardo et al., [1998] ; Signorini et al., [1998] ROUEN BioMedicine 2009

  18. 4D-Unforced Heartbeat Model di Bernardo et al., [1998] ; Signorini et al., [1998] ROUEN BioMedicine 2009

  19. 4D-Unforced Heartbeat Model di Bernardo et al., [1998] ; Signorini et al., [1998] ROUEN BioMedicine 2009

  20. 4D-Unforced Heartbeat Model Slow invariant manifold of Unforced heartbeat model ROUEN BioMedicine 2009

  21. 4D-Unforced Heartbeat Model Slow invariant manifold analytical equation ROUEN BioMedicine 2009

  22. 4D-Forced Heartbeat Model di Bernardo et al., [1998] ; Signorini et al., [1998] ROUEN BioMedicine 2009

  23. 6D-Forced Heartbeat Model ROUEN BioMedicine 2009

  24. 6D-Forced Heartbeat Model di Bernardo et al., [1998] ; Signorini et al., [1998] ROUEN BioMedicine 2009

  25. 6D-Unforced Heartbeat Model Slow invariant manifold of Forced heartbeat model ROUEN BioMedicine 2009

  26. DISCUSSION Flow Curvature Method: n-dimensional dynamical systems Autonomous or Non-autonomous • Slow invariant manifold analyticla equation of Unforced and Forced Heartbeat Models • « State Equation » linking all variables • Enabling to express one according to all others ROUEN BioMedicine 2009

  27. Publications Articles dans des revues de rang A • Chaos in a three dimensional Volterra-Gause model of predator-prey type, J.M. Ginoux, B. Rossetto & J.L. Jamet, International Journal of Bifurcation & Chaos, 5, Vol. 15, pp. 1689-1708, 2005 • Differential Geometry and Mechanics Applications to Chaotic Dynamical Systems, J.M. Ginoux & B. Rossetto, International Journal of Bifurcation & Chaos, 4, Vol. 16, pp. 887-910, 2006 • Dynamical Systems Analysis Using Differential Geometry,J.M. Ginoux & B. Rossetto Complex Computing-Networks, Series: Springer Proceedings in Physics, Vol. 104, 2006 • Slow Manifold of a Neuronal Bursting Model, J.M. Ginoux & B. Rossetto Emergent Properties in Natural and Artificial Dynamical Systems, Understanding Complex Systems. Springer-Verlag, Heidelberg, 2006 • Invariant Manifolds of Complex Systems,J.M. Ginoux & B. Rossetto Complex Systems and Self-organization Modelling, Understanding Complex Systems. Springer-Verlag, Heidelberg, in press, 2007 • Slow Invariant Manifolds as Curvature of the Flow of Dynamical Systems, J.M. Ginoux, B. Rossetto & L. Chua, International Journal of Bifurcation & Chaos, 11, Vol. 18, pp. 3409-3430, 2008 • Slow Invariant Manifolds as Curvature of the Flow of Dynamical Systems, J.M. Ginoux & C. Letellier, J. Phys. A: Math. Theor. 42 (2009) 285101 (17pp) ROUEN BioMedicine 2009

  28. Publications Book Differential Geometry Applied to Dynamical Systems World Scientific Series on Nonlinear Science, series A, 2009 ROUEN BioMedicine 2009

  29. Flow Curvature Method Dynamical Systems: Integrables or non-integrables analytically • Fixed Points • Local Bifurcations • Invariant manifolds •  center manifolds •  slow manifolds (local integrals) •  linear manifolds (global integrals) • Normal Forms ROUEN BioMedicine 2009

  30. Flow Curvature Method Meteorology Electronics ROUEN BioMedicine 2009

  31. Flow Curvature Method Neuronal Bursting Model Autocatalator ROUEN BioMedicine 2009

  32. Flow Curvature Method Forced Heartbeat Model Unforced Heartbeat Model ROUEN BioMedicine 2009

  33. Collaborations & Perspectives • Experiments & Protocols • Cardiology • Biodynamical Model of HIV-1 … ROUEN BioMedicine 2009

  34. Thanks for your attention. To be continued… ROUEN BioMedicine 2009

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