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  1. 1. From Cardiac Cells to Genetic Regulatory Networks Ezio Bartocci SUNY Stony Brook Joint work withR. Grosu, G. Batt, F. Fenton, J. Glimm, C. Le Guernic, and S. A. Smolka
  2. 2. Overview• Background – Cardiac Cells, Action Potential, Restitution• Biological Switching• Minimal Model – Resistor Model – Sigmoid Closure and Conductance Model• Piecewise Multi Affine Minimal Model – Optimal Polygonal Approximation – Model Comparison• Parameter Identification – RoverGene• Conclusion
  3. 3. Background
  4. 4. Emergent Behavior in Heart Cells EKG SurfaceArrhythmia afflicts more than 3 million Americans alone
  5. 5. Single Cell Reaction: Action Potential Schematic Action PotentialMembrane’s AP depends on:• Stimulus (voltage or current): – External / Neighboring cells• Cell’s state voltage Threshold Resting potential time
  6. 6. Single Cell Reaction: Action Potential Schematic Action PotentialMembrane’s AP depends on:• Stimulus (voltage or current): – External / Neighboring cells• Cell’s state voltage Threshold failed initiation Resting potential time
  7. 7. Single Cell Reaction: Action Potential Schematic Action PotentialMembrane’s AP depends on:• Stimulus (voltage or current): – External / Neighboring cells• Cell’s state voltage Threshold failed initiation Resting potential time
  8. 8. Single Cell Reaction: Action Potential Schematic Action Potential voltageAP has nonlinear behavior! Threshold• Reaction diffusion system: failed initiation u  R(u)  (Du) Resting potential t time
  9. 9. Single Cell Reaction: Action Potential Schematic Action Potential voltageAP has nonlinear behavior! Threshold• Reaction diffusion system: failed initiation u  R(u)  (Du) Resting potential t timeBehavior In time
  10. 10. Single Cell Reaction: Action Potential Schematic Action Potential voltageAP has nonlinear behavior! Threshold• Reaction diffusion system: failed initiation u  R(u)  (Du) Resting potential t time Reaction
  11. 11. Single Cell Reaction: Action Potential Schematic Action Potential voltageAP has nonlinear behavior! Threshold• Reaction diffusion system: failed initiation u  R(u)  (Du) Resting potential t time Diffusion
  12. 12. Frequency Response u (mV)APD APD DI t (ms) DI
  13. 13. Existing Models• Detailed ionic models: – Luo and Rudi: 14 variables – Tusher, Noble2 and Panfilov: 17 variables – Priebe and Beuckelman: 22 variables – Iyer, Mazhari and Winslow: 67 variables• Approximate models: – Cornell: 3 or 4 variables – SUNYSB: 2 or 3 variable
  14. 14. Biological Switching
  15. 15. Biological Switching 0 u  1   u  1R  (u, 1 , 2 , 0,1)   else 1 S  (u, , k,0,1)    2  1 1  e2 k(u  ) 1 u  2    0.5  0 u  k  16 H (u, ,0,1)   1  u 
  16. 16. Threshold-Based Switching Functions• Arithmetic Generalization of Boolean predicates u ≤ θ: - Step: H+(u,θ,0,1), H-(u,θ,0,1) = 1 - H+(u,θ,0,1) - Sigmoid: S+(u,θ,k,0,1), S-(u,θ,k,0,1) = 1 - S+(u,θ,k,0,1) - Ramp: R+(u,θ1,θ2,0,1), R-(u,θ1,θ2,0,1) = 1 - R+(u,θ1,θ2,0,1)• Boolean algebra generalizes to probability algebra: ~(u ≤ θ) : H-(u,θ,0,1) = 1 – H+(u,θ,0,1) (u ≤θ1) & (v ≤θ2) : H+(u,θ1,0,1) * H+(v,θ2,0,1) (u ≤θ1) | (u ≤θ2) : H+(u,θ1,0,1) + H+(v,θ2,0,1) - H+(u,θ1,0,1) * H+(v,θ2,0,1)• Generalization: H±(u,θ,um,uM), S± (u,θ,k,um,uM) , R± (u,θ1,θ2,um,uM) S± (u,θ,k,um,uM) = um+(uM-um)S+(u,k, θ)
  17. 17. Gene Regulatory Networks (GRN)• GRNs have the following general form: mi nm xi   amn s  (xmn, , mn , kmn ,umn ,vmn )  bi xi m1 n1 where: amn : are activation / inhibition constants bi : are decay constants s  (..) : are possibly complemented sigmoidal functions• Note: steps and ramps are sigmoid approximations
  18. 18. Minimal Model
  19. 19. Cornell’s Minimal Resistor Modelu  (Du)  (J fi  J si  J so )J fi  H  (u, v , 0,1) (u   v )(uu  u)v /  fiJ si  H  (u, w , 0,1) ws /  siJ so  H  (u, w , 0,1) u /  o  H  (u, w , 0,1) /  sov  H  (u, v , 0,1) (v  v) /  v  H  (u, v ,0,1)v /  v  w  H  (u, w , 0,1)(w  w) /  w  H  (u, w , 0,1)w /  w&  s  (S  (u,us , ks , 0,1)  s) /  s&J fi  H (u   v )(u   v )(uu  u)v /  fi
  20. 20. Cornell’s Minimal Resistor Model u  (Du)  (J fi  J si  J so ) J fi  H  (u, v , 0,1) (u   v )(uu  u)v /  fivoltage  H  (u, w , 0,1) ws /  si J si J so  H  (u, w , 0,1) u /  o  H  (u, w , 0,1) /  so v  H  (u, v , 0,1) (v  v) /  v  H  (u, v ,0,1)v /  v   w  H  (u, w , 0,1)(w  w) /  w  H  (u, w , 0,1)w /  w &   s  (S  (u,us , ks , 0,1)  s) /  s & J fi  H (u   v )(u   v )(uu  u)v /  fi
  21. 21. Cornell’s Minimal Resistor Modelu  (Du)  (J fi  J si  J so )J fi  H  (u, v , 0,1) (u   v )(uu  u)v /  fi DiffusionJ si  H  (u, w , 0,1) ws /  si Laplacia nJ so  H (u, w , 0,1) u /  o  H  (u, w , 0,1) /  so v  H  (u, v , 0,1) (v  v) /  v  H  (u, v ,0,1)v /  v  w  H  (u, w , 0,1)(w  w) /  w  H  (u, w , 0,1)w /  w&  s  (S  (u,us , ks , 0,1)  s) /  s&J fi  H (u   v )(u   v )(uu  u)v /  fi
  22. 22. Cornell’s Minimal Resistor Modelu  (Du)  (J fi  J si  J so )J fi  H  (u, v , 0,1) (u   v )(uu  u)v /  fi Fast inputJ si  H  (u, w , 0,1) ws /  si currentJ so  H  (u, w , 0,1) u /  o  H  (u, w , 0,1) /  sov  H  (u, v , 0,1) (v  v) /  v  H  (u, v ,0,1)v /  v  w  H  (u, w , 0,1)(w  w) /  w  H  (u, w , 0,1)w /  w&  s  (S  (u,us , ks , 0,1)  s) /  s&J fi  H (u   v )(u   v )(uu  u)v /  fi
  23. 23. Cornell’s Minimal Resistor Modelu  (Du)  (J fi  J si  J so )J fi  H  (u, v , 0,1) (u   v )(uu  u)v /  fiJ si  H  (u, w , 0,1) Slowinput ws / si currentJ so  H (u, w , 0,1) u /  o  H  (u, w , 0,1) /  so v  H  (u, v , 0,1) (v  v) /  v  H  (u, v ,0,1)v /  v  w  H  (u, w , 0,1)(w  w) /  w  H  (u, w , 0,1)w /  w&  s  (S  (u,us , ks , 0,1)  s) /  s&J fi  H (u   v )(u   v )(uu  u)v /  fi
  24. 24. Cornell’s Minimal Resistor Modelu  (Du)  (J fi  J si  J so )J fi  H  (u, v , 0,1) (u   v )(uu  u)v /  fiJ si  H  (u, w , 0,1) ws / Slow output si currentJ so  H (u, w , 0,1) u /  o  H  (u, w , 0,1) /  so v  H  (u, v , 0,1) (v  v) /  v  H  (u, v ,0,1)v /  v  w  H  (u, w , 0,1)(w  w) /  w  H  (u, w , 0,1)w /  w&  s  (S  (u,us , ks , 0,1)  s) /  s&J fi  H (u   v )(u   v )(uu  u)v /  fi
  25. 25. Cornell’s Minimal Resistor Modelu  (Du)  (J fi  J si  J so )J fi  H  (u, v , 0,1) (u   v )(uu  u)v /  fiJ si  H  (u, w , 0,1) ws /  siJ so  H  (u, w , 0,1) u /  o  H  (u, w , 0,1) /  sov  H  (u, v , 0,1) (v  v) /  v  H  (u, v ,0,1)v /  v  w  H  (u, w , 0,1)(w  w) /  w  H  (u, w , 0,1)w /  w&  s  (S  (u,us , ks , 0,1)  s) /  s&J fi  H (u   v )(u   v )(uu  u)v /  fi
  26. 26. Cornell’s Minimal Resistor Model Activationu  (Du)  (J fi  J si  J so ) Threshol dJ fi  H  (u, v , 0,1) (u   v )(uu  u)v /  fiJ si  H  (u, w , 0,1) ws /  siJ so  H  (u, w , 0,1) u /  o  H  (u, w , 0,1) /  sov  H  (u, v , 0,1) (v  v) /  v  H  (u, v ,0,1)v /  v  w  H  (u, w , 0,1)(w  w) /  w  H  (u, w , 0,1)w /  w&  s  (S  (u,us , ks , 0,1)  s) /  s&J fi  H (u   v )(u   v )(uu  u)v /  fi
  27. 27. Cornell’s Minimal Resistor Model Heavisideu  (Du)  (J fi  J si(step))  J soJ fi  H  (u, v , 0,1) (u   v )(uu  u)v /  fiJ si  H  (u, w , 0,1) ws /  siJ so  H  (u, w , 0,1) u /  o  H  (u, w , 0,1) /  sov  H  (u, v , 0,1) (v  v) /  v  H  (u, v ,0,1)v /  v  w  H  (u, w , 0,1)(w  w) /  w  H  (u, w , 0,1)w /  w&  s  (S  (u,us , ks , 0,1)  s) /  s&J fi  H (u   v )(u   v )(uu  u)v /  fi
  28. 28. Cornell’s Minimal Resistor Model Fast inputu  (Du)  (J fi  J si  J so ) GateJ fi  H  (u, v , 0,1) (u   v )(uu  u)v /  fiJ si  H  (u, w , 0,1) ws /  siJ so  H  (u, w , 0,1) u /  o  H  (u, w , 0,1) /  sov  H  (u, v , 0,1) (v  v) /  v  H  (u, v ,0,1)v /  v  w  H  (u, w , 0,1)(w  w) /  w  H  (u, w , 0,1)w /  w&  s  (S  (u,us , ks , 0,1)  s) /  s&J fi  H (u   v )(u   v )(uu  u)v /  fi
  29. 29. Cornell’s Minimal Resistor Modelu  (Du)  (J fi  J si  J so ) Resistance Time CstJ fi  H  (u, v , 0,1) (u   v )(uu  u)v /  fiJ si  H  (u, w , 0,1) ws /  siJ so  H  (u, w , 0,1) u /  o  H  (u, w , 0,1) /  sov  H  (u, v , 0,1) (v  v) /  v  H  (u, v ,0,1)v /  v  w  H  (u, w , 0,1)(w  w) /  w  H  (u, w , 0,1)w /  w&  s  (S  (u,us , ks , 0,1)  s) /  s&J fi  H (u   v )(u   v )(uu  u)v /  fi
  30. 30. Cornell’s Minimal Resistor Modelu  (Du)  (J fi  J si  J so ) Piecewise NonlinearJ fi  H  (u, v , 0,1) (u   v )(uu  u)v /  fiJ si  H  (u, w , 0,1) ws /  siJ so  H  (u, w , 0,1) u /  o  H  (u, w , 0,1) /  sov  H  (u, v , 0,1) (v  v) /  v  H  (u, v ,0,1)v /  v  w  H  (u, w , 0,1)(w  w) /  w  H  (u, w , 0,1)w /  w&  s  (S  (u,us , ks , 0,1)  s) /  s&J fi  H (u   v )(u   v )(uu  u)v /  fi
  31. 31. Cornell’s Minimal Resistor Modelu  (Du)  (J fi Input  J so ) Slow  J si GateJ fi  H (u, v , 0,1) (u   v )(uu  u)v /  fi J si  H  (u, w , 0,1) ws /  siJ so  H  (u, w , 0,1) u /  o  H  (u, w , 0,1) /  sov  H  (u, v , 0,1) (v  v) /  v  H  (u, v ,0,1)v /  v  w  H  (u, w , 0,1)(w  w) /  w  H  (u, w , 0,1)w /  w&  s  (S  (u,us , ks , 0,1)  s) /  s&J fi  H (u   v )(u   v )(uu  u)v /  fi
  32. 32. Cornell’s Minimal Resistor Modelu  (Du)  (J fi  J si  J so ) Output Slow GateJ fi  H (u, v , 0,1) (u   v )(uu  u)v /  fi J si  H  (u, w , 0,1) ws /  siJ so  H  (u, w , 0,1) u /  o  H  (u, w , 0,1) /  sov  H  (u, v , 0,1) (v  v) /  v  H  (u, v ,0,1)v /  v  w  H  (u, w , 0,1)(w  w) /  w  H  (u, w , 0,1)w /  w&  s  (S  (u,us , ks , 0,1)  s) /  s&J fi  H (u   v )(u   v )(uu  u)v /  fi
  33. 33. Cornell’s Minimal Resistor Modelu  (Du)  (J fi  J si  J so )J fi  H  (u, v , 0,1) (u   v )(uu  u)v /  fi Piecewise BilinearJ si  H (u, w , 0,1) ws /  si J so  H  (u, w , 0,1) u /  o  H  (u, w , 0,1) /  sov  H  (u, v , 0,1) (v  v) /  v  H  (u, v ,0,1)v /  v  w  H  (u, w , 0,1)(w  w) /  w  H  (u, w , 0,1)w /  w&  s  (S  (u,us , ks , 0,1)  s) /  s&J fi  H (u   v )(u   v )(uu  u)v /  fi
  34. 34. Cornell’s Minimal Resistor Modelu  (Du)  (J fi  J si  J so )J fi  H  (u, v , 0,1) (u   v )(uu  u)v /  fi PiecewiseJ si  H  (u, w , 0,1) ws /  si NonlinearJ so  H  (u, w , 0,1) u /  o  H  (u, w , 0,1) /  sov  H  (u, v , 0,1) (v  v) /  v  H  (u, v ,0,1)v /  v  w  H  (u, w , 0,1)(w  w) /  w  H  (u, w , 0,1)w /  w&  s  (S  (u,us , ks , 0,1)  s) /  s&J fi  H (u   v )(u   v )(uu  u)v /  fi
  35. 35. Cornell’s Minimal Resistor Modelu  (Du)  (J fi  J si  J so )J fi  H  (u, v , 0,1) (u   v )(uu  u)v /  fiJ si  H  (u, w , 0,1) ws /  siJ so  H  (u, w , 0,1) u /  o  H  (u, w , 0,1) /  sov  H  (u, v , 0,1) (v  v) /  v  H  (u, v ,0,1)v /  v  w  H  (u, w , 0,1)(w  w) /  w  H  (u, w , 0,1)w /  w&  s  (S  (u,us , ks , 0,1)  s) /  s&J fi  H (u   v )(u   v )(uu  u)v /  fi
  36. 36. Cornell’s Minimal Resistor Modelu  (Du)  (J fi  J si  J so )J fi  H  (u, v , 0,1) (u   v )(uu  u)v /  fiJ si  H  (u, w , 0,1) ws /  siJ so  H  (u, w , 0,1) u /  o  H  (u, w , 0,1) /  sov  H  (u, v , 0,1) (v  v) /  v  H  (u, v ,0,1)v /Piecewise  v Linearw  H (u, w , 0,1)(w  w) /   H (u, w , 0,1)w /  w&   w  s  (S  (u,us , ks , 0,1)  s) /  s&J fi  H (u   v )(u   v )(uu  u)v /  fi
  37. 37. Cornell’s Minimal Resistor Modelu  (Du)  (J fi  J si  J so )J fi  H  (u, v , 0,1) (u   v )(uu  u)v /  fiJ si  H  (u, w , 0,1) ws /  siJ so  H  (u, w , 0,1) u /  o  H  (u, w , 0,1) /  sov  H  (u, v , 0,1) (v  v) /  v  H  (u, v ,0,1)v /  v   Sigmoidw & H  (u,(s-step)   w) /  w  H  (u, w , 0,1)w /  w w , 0,1)(w  Nonlinear s  (S  (u,us , ks , 0,1)  s) /  s&J fi  H (u   v )(u   v )(uu  u)v /  fi
  38. 38. Cornell’s Minimal Resistor Modelu  (Du)  (J fi  J si  J so )J fi  H  (u, v , 0,1) (u   v )(uu  u)v /  fiJ si  H  (u, w , 0,1) ws /  siJ so  H  (u, w , 0,1) u /  o  H  (u, w , 0,1) /  sov  H  (u, v , 0,1) (v  v) /  v  H  (u, v ,0,1)v /  v  w  H  (u, w , 0,1)(w  w) /  w  H  (u, w , 0,1)w /  w&  s  (S  (u,us , ks , 0,1)  s) /  s&J fi  H (u   v )(u   v )(uu  u)v /  fi
  39. 39. Voltage-controlled resistances v  H  (u, v ,  v1 ,  v2 )    Piecewise Constant o  H  (u, v ,  o2 ,  o1 ) s  H  (u, w ,  s1 ,  s 2 ) w  S  (u, kw ,uw ,  w1 ,  w2 )      so  S  (u, kso ,uso ,  so1 ,  so2 )    v  h  (u, v , 0,1)w  h  (u, v , 0,1) (1  u /  w )  h  (u, v ,0,w* )   v  uw h(u, v v0,1) o1 uso  Hh(u   o )  o2 so   w , us uu u0.006 0.03 0.13 0.3 0.65 0.9087 1.55
  40. 40. Voltage-controlled resistances v  H  (u, v ,  v1 ,  v2 )    o  H  (u, v ,  o2 ,  o1 ) s  H  (u, w ,  s1 ,  s 2 ) w  S  (u, kw ,uw ,  w1 ,  w2 )      Sigmoidal so  S  (u, kso ,uso ,  so1 ,  so2 )    v  h  (u, v , 0,1)w  h  (u, v , 0,1) (1  u /  w )  h  (u, v ,0,w* )   v  uw h(u, v v0,1) o1 uso  Hh(u   o )  o2 so   w , us uu u0.006 0.03 0.13 0.3 0.65 0.9087 1.55
  41. 41. Voltage-controlled resistances v  H  (u, v ,  v1 ,  v2 )    o  H  (u, v ,  o2 ,  o1 ) s  H  (u, w ,  s1 ,  s 2 ) w  S  (u, kw ,uw ,  w1 ,  w2 )      so  S  (u, kso ,uso ,  so1 ,  so2 )     Piecewisev  h  (u, v , 0,1) Linearw  h  (u, v , 0,1) (1  u /  w )  h  (u, v ,0,w* )   v  uw h(u, v v0,1) o1 uso  Hh(u   o )  o2 so   w , us uu u0.006 0.03 0.13 0.3 0.65 0.9087 1.55
  42. 42. Cornell’s Minimal Resistance Modelu  v u   v  0.3u  w u   w  0. 13u  o u   o   v  0.006 
  43. 43. Cornell’s Minimal Resistance Modelu  v u   v  0.3 u  ou  w u   w  0. 13 u  (Du)  u /  o1 &u  o v  (1  v) /  v1 &  u   o   v  0.006  w  (1  u /  w  w) /  w &  s  (S(2ks (u  us ))  s) /  s &
  44. 44. Cornell’s Minimal Resistance Model o  u  w u  (Du)  u /  o2 & v  v /  v2 & u  v w  (w  w) /  w1 & *  u   v  0.3 u  o s  (S(2ks (u  us ))  s) /  s &u  w u   w  0. 13 u  (Du)  u /  o1 &u  o v  (1  v) /  v1 &  u   o   v  0.006  w  (1  u /  w  w) /  w &  s  (S(2ks (u  us ))  s) /  s &
  45. 45. Cornell’s Minimal Resistance Model w  u  v u  (Du)  ws /  si  1 /  so & v  v /  v2 &  o  u  w w  w /  w &  u  (Du)  u /  o2 & s  (S(2ks (u  us ))  s) /  s 2 & v  v /  v2 & u  v w  (w  w) /  w1 & *  u   v  0.3 u  o s  (S(2ks (u  us ))  s) /  s &u  w u   w  0. 13 u  (Du)  u /  o1 &u  o v  (1  v) /  v1 &  u   o   v  0.006  w  (1  u /  w  w) /  w &  s  (S(2ks (u  us ))  s) /  s &
  46. 46. Cornell’s Minimal Resistance Model v  u u  (Du)  (u   v )(uu  u)v /  fi  ws /  fi  1 /  so & v  v /  v &  w  w /  w &  s  (S(2ks (u  us ))  s) /  s 2 & w  u  v u  (Du)  ws /  si  1 /  so & v  v /  v2 &  o  u  w w  w /  w &  u  (Du)  u /  o2 & s  (S(2ks (u  us ))  s) /  s 2 & v  v /  v2 & u  v w  (w  w) /  w1 & *  u   v  0.3 u  o s  (S(2ks (u  us ))  s) /  s &u  w u   w  0. 13 u  (Du)  u /  o1 &u  o v  (1  v) /  v1 &  u   o   v  0.006  w  (1  u /  w  w) /  w &  s  (S(2ks (u  us ))  s) /  s &
  47. 47. Sigmoid ClosureFor ab > 0, scaled sigmoids are closed under multiplicative inverses (division): S  (u, k, ,a,b)1  S  (u, k,  ln(a / b) / 2k,b ,a )Proof S  (u, k, ,a,b) 1 1  e2 k (u  )S (u, k, , a,b)   1   ba b  ae2 k (u  ) a 1  e2 k u   1 1 1 a  b  b  ae 2 k (u   ) 1  b-a    a b  a b  ae2 k (u  ) a 1  a e2 k (u  ) b 1 1 a 1  ln 1 1 a a b b, , )  ln a  ln b  S (u, k,     a 2 k (u (  )) 2k b a 1 e 2k
  48. 48. Sigmoid ClosureFor ab > 0, scaled sigmoids are closed under multiplicative inverses (division): S  (u, k, ,a,b)1  S  (u, k,  ln(a / b) / 2k,b ,a )Proof 1 1  e2 k (u  )S (u, k, , a,b)   1   ba b  ae2 k (u  ) a 1  e2 k u   1 1 1 a  b  b  ae 2 k (u   ) 1     a b  a b  ae2 k (u  ) a 1  a e2 k (u  ) b 1 1 a 1  ln 1 1 a b b, , ) ln a  ln b  S (u, k,     a 2 k (u (  )) 2k b a 1 e 2k
  49. 49. Sigmoid ClosureFor ab > 0, scaled sigmoids are closed under multiplicative inverses (division): S  (u, k, ,a,b)1  S  (u, k,  ln(a / b) / 2k,b ,a )Proof 2 k (u   ) S  (u, k, ,a,b)1 1 1 eS  (u, k, , a,b)1    ba b  ae2 k (u  ) a 1  e2 k u   1/b-1/a 1 1 1 a  b  b  ae 2 k (u   ) 1     a b  1/a   ln(a / b) / 2k a b  ae2 k (u  ) a 1  a e2 k (u  ) b 1 1 a 1  ln 1 1 a b b, , ) ln a  ln b  S (u, k,     a 2 k (u (  )) 2k b a 1 e 2k
  50. 50. Resistances vs ConductancesRemoving Divisions using Sigmoid Closure w  S  (u, kw ,uw ,  w1 ,  w2 )      gw  1 /  w  S  (u, kw ,u  , gw1 , gw2 )    w   so  S  (u, kso ,uso ,  so1 ,  so2 ) gso  1 /  s0  S (u, kso ,u so , gso1 , gso2 ) Removing Divisions using H  (u,,a,b)1  H  (u,,b1,a1 )  H (u, ,  ,  )      gv  1 /  v  H  (u, v , gv1 , gv2 )     v v v1 v2 o  H (u, ,  o1 ,  o2 )   go  1 /  o  H  (u, v , go1 , go2 ) v s  H (u, w ,  s1 ,  s 2 )  gs  1 /  s  H  (u, w , gs1 , gs2 ) v  h  (u, v , 0,1) w w  h  (u, v , 0,1) (1  ugw )  h  (u, v ,0,w* )   v u w u   w  v uso us u  uu w so u 0.006 0.03 0.13 0.3 0.65 0.9087 1.55
  51. 51. Conductances Minimal Modelu  v u   v  0.3u  w u   w  0. 13u  o u   o   v  0.006 
  52. 52. Conductances Minimal Modelu  v u   v  0.3 u    vu  w u  (Du)  u go1 & u   w  0. 13u  o v  (1  v) g  & v1 u   o   v  0.006  w  (1  u gw  w) g  & w s  (s  (u,us , 2ks , 0,1)  s) gs1 &
  53. 53. Conductances Minimal Model  v  u   w u  (Du)  u g o2 &  v  v gv2 & w  (w  w) g  & *u  v w1 u   v  0.3 u    v  s  (s (u,us , 2ks , 0,1)  s) gs1 &u  w u  (Du)  u go1 & u   w  0. 13u  o v  (1  v) g  & v1 u   o   v  0.006  w  (1  u gw  w) g  & w s  (s  (u,us , 2ks , 0,1)  s) gs1 &
  54. 54. Conductances Minimal Model w  u  v u  (Du)  ws gsi  gso & v  v g  & v2  v  u   w  w  w gw & u  (Du)  u g o2 &  s  (Ss  (u,us , 2ks , 0,1)  s) gs 2 v  v gv2 & & w  (w  w) g  & *u  v w1 u   v  0.3 u    v  s  (s (u,us , 2ks , 0,1)  s) gs1 &u  w u  (Du)  u go1 & u   w  0. 13u  o v  (1  v) g  & v1 u   o   v  0.006  w  (1  u gw  w) g  & w s  (s  (u,us , 2ks , 0,1)  s) gs1 &
  55. 55. Conductances Minimal Model v  u u  (Du)  (u   v )(uu  u)v g fi  ws gsi  gso &  v  v gv &  w  w gw & s  (s  (u,us , 2ks , 0,1)  s) gs 2 & w  u  v u  (Du)  ws gsi  gso & v  v g  & v2  v  u   w  w  w gw & u  (Du)  u g o2 &  s  (Ss  (u,us , 2ks , 0,1)  s) gs 2 v  v gv2 & & w  (w  w) g  & *u  v w1 u   v  0.3 u    v  s  (s (u,us , 2ks , 0,1)  s) gs1 &u  w u  (Du)  u go1 & u   w  0. 13u  o v  (1  v) g  & v1 u   o   v  0.006  w  (1  u gw  w) g  & w s  (s  (u,us , 2ks , 0,1)  s) gs1 &
  56. 56. Piecewise Multi Affine Model
  57. 57. Our goals• Derive a Piecewise Multi Affine model: – This should facilitate analysis – We want to improve the computational efficiency• Identify the parameters based on: – Data generated by a detailed ionic model – Experimental, in-vivo data
  58. 58. Our goals• Identify the parameters based on: – Data generated by a detailed ionic model – Experimental, in-vivo data
  59. 59. Optimal Polygonal ApproximationProblem to solve:Given a nonlinear curve and the desired number of the segments return theoptimal polygonal approximation:Example:
  60. 60. Optimal Polygonal ApproximationProblem to solve:Given a nonlinear curve and the desired number of the segments return theoptimal polygonal approximation:Example: What is the optimal polygonal approximation of the blu curve with 3 segments ?
  61. 61. Optimal Polygonal ApproximationProblem to solve:Given a nonlinear curve and the desired number of the segments return theoptimal polygonal approximation:Example: What is the optimal polygonal approximation of the blu curve with 3 segments ?Dynamic Programming Algorithmwith Complexity O(P2S)P the number of points of the curveS the number of segmentsMarc Salotti, An efficient algorithm for the optimal polygonalapproximation of digitized curves, Pattern Recognition Letters22 (2001), Pag 215-221
  62. 62. Optimal Polygonal ApproximationProblem to solve:Given a nonlinear curve and the desired number of the segments return theoptimal polygonal approximation:Example: What is the optimal polygonal approximation of the blu curve with 3 segments ?Dynamic Programming Algorithmwith Complexity O(P2S)P the number of points of the curveS the number of segmentsMarc Salotti, An efficient algorithm for the optimal polygonalapproximation of digitized curves, Pattern Recognition Letters22 (2001), Pag 215-221
  63. 63. Optimal Polygonal ApproximationProblem to solve:Given a nonlinear curve and the desired number of the segments return theoptimal polygonal approximation:Example: What is the optimal polygonal approximation of the blu curve with 3 segments ?Dynamic Programming Algorithmwith Complexity O(P2S)P the number of points of the curveS the number of segmentsMarc Salotti, An efficient algorithm for the optimal polygonalapproximation of digitized curves, Pattern Recognition Letters22 (2001), Pag 215-221
  64. 64. Optimal Polygonal ApproximationProblem to solve:Given a nonlinear curve and the desired number of the segments return theoptimal polygonal approximation:Example: What is the optimal polygonal approximation of the blu curve with 3 segments ?Dynamic Programming Algorithmwith Complexity O(P2S)P the number of points of the curveS the number of segmentsMarc Salotti, An efficient algorithm for the optimal polygonalapproximation of digitized curves, Pattern Recognition Letters22 (2001), Pag 215-221
  65. 65. Optimal Polygonal ApproximationProblem to solve:Given a nonlinear curve and the desired number of the segments return theoptimal polygonal approximation:Example: What is the optimal polygonal approximation of the blu curve with 3 segments ?Dynamic Programming Algorithmwith Complexity O(P2S)P the number of points of the curveS the number of segmentsMarc Salotti, An efficient algorithm for the optimal polygonalapproximation of digitized curves, Pattern Recognition Letters22 (2001), Pag 215-221
  66. 66. Global Optimal Polygonal ApproximationOur problem:Given a set of nonlinear curves and the desired number of the segmentsreturn the optimal polygonal approximation:
  67. 67. Global Optimal Polygonal ApproximationOur problem:Given a set of nonlinear curves and the desired number of the segmentsreturn the optimal polygonal approximation:Example: What is the optimal polygonal approximation of the blu and the red curve with 5 segments ?
  68. 68. Global Optimal Polygonal ApproximationOur problem:Given a set of nonlinear curves and the desired number of the segmentsreturn the optimal polygonal approximation:Example: What is the optimal polygonal approximation of the blu and the red curve with 5 segments ? But combining the two we obtain 8 segments and not 5 segments
  69. 69. Global Optimal Polygonal ApproximationOur problem:Given a set of nonlinear curves and the desired number of the segmentsreturn the optimal polygonal approximation:Example: What is the optimal polygonal approximation of the blu and the red curve with 5 segments ? But combining the two we obtain 8 segments and not 5 segmentsOur solution: We modify the optimal polygonal approximation algorithm to perform the linearization on a set of curves trying to minimize the maximum error.
  70. 70. Global Optimal Polygonal ApproximationOur problem:Given a set of nonlinear curves and the desired number of the segmentsreturn the optimal polygonal approximation:Example: What is the optimal polygonal approximation of the blu and the red curve with 5 segments ?Our solution: We modify the optimal polygonal approximation algorithm to perform the linearization on a set of curves trying to minimize the maximum error.
  71. 71. Deriving the Piecewise Multi Affine Model
  72. 72. Deriving the Piecewise Multi Affine Model
  73. 73. Deriving the Piecewise Multi Affine Model
  74. 74. Deriving the Piecewise Multi Affine Model
  75. 75. Deriving the Piecewise Multi Affine Model
  76. 76. Deriving the Piecewise Multi Affine Model
  77. 77. 1D Cable Comparison Minimal Resistor Model Multi Affine Model (26 ramps) u (mV)APD APD DI t (ms) DI
  78. 78. 2D Comparison1024x1024 cells 100000 iterations 556 sec. vs. 810 sec. (GPU Tesla C1060) Simulation of the Multi Affine Model is computational efficient (1.43 x)
  79. 79. Parameter Identification
  80. 80. Genetic Regulatory Networks cross-inhibition network x: protein concentration  a , a ,b ,b 1 2 1 2 threshold concentrationG. Batt, C. Belta and R. Weiss (2008)
Temporal logic ka , kb ,  a ,  b : rate parametersanalysis of gene networks under parameteruncertainty• Find parameters such that network is bistable
  81. 81. Genetic Regulatory Networks• Partition of the state space: rectangles ka , kb
  82. 82. Specifications of dynamical properties
  83. 83. Embedding transition system
  84. 84. Iterative exploration of parameter space
  85. 85. Parameter Idenfication for APD
  86. 86. Encoding a Property on the APD Introducing a new state variable for the stimulus:1 q  R (u,0,0.1,0,1)  R (u,0.9,1,1,0) 0 1 Measuring the APD with another state variable: z  R (u,0.3,0.3,0,1) Property: starting with initial conditions u=0,w=0.8,v=1.0,s=0.0 the cell will fire an action potential with APD between 200 and 300 ms   (u  0  w  0.8  v 1.0  s  0.0) F(z  200  z  300)
  87. 87. Alternans
  88. 88. Future Works• Performing experiments with Rovergene• Encoding more sophisticated properties• Discriminate healthy/unhealthy tissues using model checking
  89. 89. Thanks forthe attention
  90. 90. function [e,a,b,xb] = optimalLinearApproximation(x,y,S)Input: x,y: Curves given as an x-points vector and a vector of y-points vectors S: Number >= 2 of desired segmentsOutput: e: Errors matrix a,b: Line-segment-coefficients matrix xb: x-coordinate at breaking point matrixInitializationz1 = size(x); P = z1 (2); Get number of points in each curvez2 = size(y); C = z2 (1); Get number of digitized curvesse = zeros(1,C); Initialize vector of errors, one error for each curveCost tablescost = ones(P,S) * inf; cost(30,4) = min cost to pt 30 with 4-segm polylineerror = ones(P,P) * inf; error(i, n) = cached error of line segment (i,n)cost(2,1) = 0; 1-segment-polyline cost of polyline (1,2) = 0Predecessor tablefather = ones(P,S) * inf; father(30,4) = pred of pt 30 on a 4-segm polylineComputation of optimal segmentationInitialize cost and father for 1-segment-polyline, from pt 1 to all other ptsfor p = 2:P Traverse all other points for c = 1:C Traverse all curves se(c) = segmentError(x(1:p), y(c,1:p)); (1,p)-line-segment appr error end; for c cost(p,1) = max(se); Maximum error among all curves father(p,1) = 1; All 1-segment polylines have father point 1end; for p
  91. 91. Compute s-segm-polyline cost from point 1 to all other pointsfor s = 2:S Number of segments in the polyline for p = 3:P Next-point-number to consider minErr = cost(p-1,s-1); minIndex = p-1; Error of (p-1,p) = 0 for i = s:p-2 Next-intermediate-point to consider if (error(i,p) == Inf) Error of line segment (i,n) not cashed for c = 1:C Next curve-number to consider se(c) = segmentError(x(i:p), y(k,i:p)); (i,p)-segment error end; for k error(i,p) = max(se); Maximum line segment error end; if currErr = cost(i,s-1) + error(i,p); s-segment-polyline error if (currErr < minErr) Smaller error? minErr = currErr; minIndex = i; Update error and parent end; if end; for i cost(p,s) = minErr; s-segment-polyline minimal cost father(p,s) = minIndex; Last points father on the polyline end; for pend; for s[e,a,b,xb] = ExtractAnswer;end
  92. 92. function [e,a,b] = segmentError(x,y)Input: x,y: Digitized curve-segment as an x-vector and an y-vectorOutput: e: Error of the line segment between the first and last point a,b: The coeficients defining this segmentInitialization:z = size(x); P = s(2); Find out the number of points of x,yCompute 1-segment linear-interpolation of (x,y) coefficientsa = (y(n) - y(1)) / (x(n) - x(1));b = (y(1) * x(n) - y(n) * x(1)) / (x(n) - x(1));Compute perpendicular-distance error for above line segmente = 0; Initialize Errorfor p = 1:P Compute error for the each point on the curve e = e + (y(p) - a * x(p) - b)2 / (a2 +1); Accumulate least squareend;end
  93. 93. function [e,a,b,xb] = ExtractAnswerOutput: e,a,b,xb: As in the output of optimalLinearApproximationInitialization:ib = zeros(S,S+1); xb = zeros(S,S+1); yb = zeros(C, S,S+1); Points matricesa = zeros(C, S,S); b = zeros(C, S,S); er = zeros(C, S,S); Coefficients/errorExtract error and coefficient matricesfor s = S:-1:1 Traverse polyline segments in inverse order ib(s,s+1) = P; Get last point number xb(s,s+1) = x(ib(s,s+1)); Get x-value for this point for c = 1:C Traverse all curves yb(c,s,s+1) = y(c,ib(s,s+1)); Get y-value for this point end; for i = s:-1:1 Traverse predecessor points in inverse order ib(s,i) = father(ib(j,i+1),i); Get predecessor point number xb(s,i) = x(ib(s,i)); Get x-value for this point for c = 1:C Traverse all curves yb(c,s,i) = y(c,ib(s,i)); Get y-value for this point [er(c,s,i), a(c,s,i), b(c,s,i)] = Compute err, a and b for segm (x,y) segmentError( x(ib(s,i):ib(s,i+1)), y(c,ib(s,i):ib(s,i+1)) ); end end;end;end

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