predicting emergent behavior in cardiac tissue a grand
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Predicting Emergent Behavior in Cardiac Tissue: A Grand Challenge Radu Grosu SUNY at Stony Brook Joint work with Ezio Bartocci, Gregory Batt, Flavio H. Fenton, James Glimm, Colas Le Guernic, and Scott A. Smolka Excitable Cells Generate

  1. MRM: Currents Equations Constant u ( u , v , w , s )   ( D  u )  ( J fi ( u , v )  J si ( u , w , s )  J so ( u )) Resistanc e   H  ( u ,  v ) ( u   v )( u u  u ) v /  fi J fi ( u , v ) J si ( u , w , s )   H  ( u ,  w ) ws /  si  H  ( u ,  w ) u /  o ( u )  H  ( u ,  w ) /  so ( u ) J so ( u )

  2. MRM: Currents Equations u ( u , v , w , s )   ( D  u )  ( J fi ( u , v )  J si ( u , w , s )  J so ( u ))   H  ( u ,  v ) ( u   v )( u u  u ) v /  fi J fi ( u , v ) Piecewise Nonlinear J si ( u , w , s )   H  ( u ,  w ) ws /  si  H  ( u ,  w ) u /  o ( u )  H  ( u ,  w ) /  so ( u ) J so ( u )

  3. MRM: Currents Equations u ( u , v , w , s )   ( D  u )  ( J fi ( u , v )  J si ( u , w , s )  J so ( u ))   H  ( u ,  v ) ( u   v )( u u  u ) v /  fi J fi ( u , v ) J si ( u , w , s )   H  ( u ,  w ) ws /  si Piecewise  H  ( u ,  w ) u /  o ( u )  H  ( u ,  w ) /  so ( u ) J so ( u ) Bilinear

  4. MRM: Currents Equations u ( u , v , w , s )   ( D  u )  ( J fi ( u , v )  J si ( u , w , s )  J so ( u ))   H  ( u ,  v ) ( u   v )( u u  u ) v /  fi J fi ( u , v ) J si ( u , w , s )   H  ( u ,  w ) ws /  si  H  ( u ,  w ) u /  o ( u )  H  ( u ,  w ) /  so ( u ) J so ( u ) Piecewise Sigmoidal Resistanc Resistanc e e

  5. MRM: Currents Equations u ( u , v , w , s )   ( D  u )  ( J fi ( u , v )  J si ( u , w , s )  J so ( u ))   H  ( u ,  v ) ( u   v )( u u  u ) v /  fi J fi ( u , v ) J si ( u , w , s )   H  ( u ,  w ) ws /  si  H  ( u ,  w ) u /  o ( u )  H  ( u ,  w ) /  so ( u ) J so ( u ) Piecewise Nonlinear

  6. MRM: Gates ODEs u ( u , v , w , s )   ( D  u )  ( J fi ( u , v )  J si ( u , w , s )  J so ( u ))   H  ( u ,  v ,0,1) ( u   v )( u u  u ) v /  fi J fi ( u , v ) J si ( u , w , s )   H  ( u ,  w ,0,1) ws /  si  H  ( u ,  w ,0,1) u /  o ( u )  H  ( u ,  w ,0,1) /  so ( u ) J so ( u )  H  ( u ,  v ) ( v   v ) /  v  ( u )  H  ( u ,  v ) v /  v  v ( u , v ) w ( u , w )  H  ( u ,  w )( w   w ) /  w  ( u )  H  ( u ,  w ) w /  w  &  ( S  ( u , u s , k s )  s ) /  s ( u ) & s ( u , s )   H ( u   v )( u   v )( u u  u ) v /  fi J fi

  7. MRM: Gates ODEs u ( u , v , w , s )   ( D  u )  ( J fi ( u , v )  J si ( u , w , s )  J so ( u ))   H  ( u ,  v ,0,1) ( u   v )( u u  u ) v /  fi J fi ( u , v ) J si ( u , w , s )   H  ( u ,  w ,0,1) ws /  si Piecewise  H  ( u ,  w ,0,1) u /  o ( u )  H  ( u ,  w ,0,1) /  so ( u ) J so ( u ) Resistance  H  ( u ,  v ) ( v   v ) /  v  ( u )  H  ( u ,  v ) v /  v  v ( u , v ) w ( u , w )  H  ( u ,  w )( w   w ) /  w  ( u )  H  ( u ,  w ) w /  w  &  ( S  ( u , u s , k s )  s ) /  s ( u ) & s ( u , s )   H ( u   v )( u   v )( u u  u ) v /  fi J fi Piecewise Resistance

  8. MRM: Gates ODEs u ( u , v , w , s )   ( D  u )  ( J fi ( u , v )  J si ( u , w , s )  J so ( u ))   H  ( u ,  v ,0,1) ( u   v )( u u  u ) v /  fi J fi ( u , v ) J si ( u , w , s )   H  ( u ,  w ,0,1) ws /  si  H  ( u ,  w ,0,1) u /  o ( u )  H  ( u ,  w ,0,1) /  so ( u ) J so ( u )  H  ( u ,  v ) ( v   v ) /  v  ( u )  H  ( u ,  v ) v /  v  v ( u , v ) w ( u , w )  H  ( u ,  w )( w   w ) /  w  ( u )  H  ( u ,  w ) w /  w  &  ( S  ( u , u s , k s )  s ) /  s ( u ) & s ( u , s ) Sigmoidal Resistance   H ( u   v )( u   v )( u u  u ) v /  fi J fi Sigmoid

  9. MRM: Voltage-Controlled Resistances/SSV   H  ( u ,  o )  v 2  v  ( u )  H  ( u ,  o )  v 1  Piecewis e  s ( u )  H  ( u ,  w )  s 1  H  ( u ,  w )  s 2 Constant  o ( u )  H  ( u ,  o )  o 1  H  ( u ,  o )  o 2 w  ( u )

  10. MRM: Voltage-Controlled Resistances/SSV   H  ( u ,  o )  v 2  v  ( u )  H  ( u ,  o )  v 1   s ( u )  H  ( u ,  w )  s 1  H  ( u ,  w )  s 2  o ( u )  H  ( u ,  o )  o 1  H  ( u ,  o )  o 2 w  ( u )    w 1  ) S  ( u , u s , k w Sigmoidal  w  ( u )   w 1   (  w 2  )  so ( u )   so 1  (  so 2   so 1 ) S  ( u , u s , k so ) w  ( u )

  11. MRM: Voltage-Controlled Resistances/SSV   H  ( u ,  o )  v 2  v  ( u )  H  ( u ,  o )  v 1   s ( u )  H  ( u ,  w )  s 1  H  ( u ,  w )  s 2  o ( u )  H  ( u ,  o )  o 1  H  ( u ,  o )  o 2 w  ( u )    w 1  ) S  ( u , u s , k w  w  ( u )   w 1   (  w 2  )  so ( u )   so 1  (  so 2   so 1 ) S  ( u , u s , k so ) Piecewis Piecewise w  ( u ) e v  ( u )  H  ( u ,  o ) Linear Constant w  ( u )  H  ( u ,  o ) (1  u /  w  )  H  ( u ,  o ) w  * H  ( u ,  o )  o 1  H  ( u ,  o )  o 2  so ( u ) 

  12. MRM: Scaled Steps and Sigmoids  ,  v 2  )  v  ( u )  H  ( u ,  o ,  v 1 Piecewis e  s ( u )  H  ( u ,  w ,  s 1 ,  s 2 ) Constant  o ( u )  H  ( u ,  o ,  o 1 ,  o 2 ) w  ( u )

  13. MRM: Scaled Steps and Sigmoids  ,  v 2  )  v  ( u )  H  ( u ,  o ,  v 1  s ( u )  H  ( u ,  w ,  s 1 ,  s 2 )  o ( u )  H  ( u ,  o ,  o 1 ,  o 2 ) w  ( u )  ,  w 2  )  w  ( u )  S  ( u , u s , k w  ,  w 1 Sigmoidal  so ( u )  S  ( u , u s , k so ,  so 1 ,  so 2 ) w  ( u )

  14. Minimal Resistance Model (MRM) u   v u   v  0.3 u   w  0. 13 u   w u   o  0.006 u   o

  15. Minimal Resistance Model (MRM) u   v u   v  0.3 0  u   o u   w  0. 13 u   w u   ( D  u )  u /  o 1 & u   o  0.006 u   o v  (1  v ) /  v 1  & w  (1  u /  w   w ) /  w &  ( u ) s  ( S  ( u , u s , k s )  s ) /  s 1 &

  16. Minimal Resistance Model (MRM)  o  u   w u   ( D  u )  u /  o 2 & v   v /  v 2  & *  w ) /  w w  ( w  &  ( u ) u   v u   v  0.3 0  u   o s  ( S  ( u , u s , k s )  s ) /  s 1 & u   w  0. 13 u   w u   ( D  u )  u /  o 1 & u   o  0.006 u   o v  (1  v ) /  v 1  & w  (1  u /  w   w ) /  w &  ( u ) s  ( S  ( u , u s , k s )  s ) /  s 1 &

  17. Minimal Resistance Model (MRM)  w  u   v u   ( D  u )  ws /  si  1/  so ( u ) &  o  u   w v   v /  v 2 &  u   ( D  u )  u /  o 2 &  w   w /  w & v   v /  v 2  s  ( S  ( u , u s , k s )  s ) /  s 2 & & *  w ) /  w w  ( w  &  ( u ) u   v u   v  0.3 0  u   o s  ( S  ( u , u s , k s )  s ) /  s 1 & u   w  0. 13 u   w u   ( D  u )  u /  o 1 & u   o  0.006 u   o v  (1  v ) /  v 1  & w  (1  u /  w   w ) /  w &  ( u ) s  ( S  ( u , u s , k s )  s ) /  s 1 &

  18. Minimal Resistance Model (MRM)  v  u < u s u   ( D  u )  ( u   v )( u u  u ) v /  fi  ws /  fi  1/  so ( u ) & v   v /  v &   w   w /  w & s  ( S  ( u , u s , k s )  s ) /  , s 2 &  w  u   v u   ( D  u )  ws /  si  1/  so ( u ) &  o  u   w v   v /  v 2 &  u   ( D  u )  u /  o 2 &  w   w /  w & v   v /  v 2  s  ( S  ( u , u s , k s )  s ) /  s 2 & & *  w ) /  w w  ( w  &  ( u ) u   v u   v  0.3 0  u   o s  ( S  ( u , u s , k s )  s ) /  s 1 & u   w  0. 13 u   w u   ( D  u )  u /  o 1 & u   o  0.006 u   o v  (1  v ) /  v 1  & w  (1  u /  w   w ) /  w &  ( u ) s  ( S  ( u , u s , k s )  s ) /  s 1 &

  19. Sigmoid Closure Property Theorem: For ab > 0, scaled sigmoids are closed under the reciprocal operation: ln( a b ) 2 k , 1 b , 1 S  ( u , k ,  , a , b )  1  S  ( u , k ,   a )

  20. Sigmoid Closure Property ln( a b ) 2 k , 1 b , 1 S  ( u , k ,  , a , b )  1  S  ( u , k ,   a ) S  ( u , k ,  , a , b ) Proof: b b  a S  ( u , k ,  , a , b )  1  ( a  b-a  )  1  1  e  2 k u   a 

  21. Sigmoid Reciprocal Closure ln( a b ) 2 k , 1 b , 1 S  ( u , k ,  , a , b )  1  S  ( u , k ,   a ) S  ( u , k ,  , a , b )  1 1 Proof: a 1 a  1 S  ( u , k ,  , a , b )  1  1 a  1 1 b a   2 k ( u  (   ln a  ln b b )) 1  e 2 k 1 b   ln( a / b ) / 2 k

  22. From Resistances to Conductances Removing Divisions using Sigmoid Reciprocal:   1/  w   S  ( u , k w   S  ( u , k w  ,  w 2  )  , u ' w  ,  w 1  1 ,  w 2  1 )  w  , u w  ,  w 1 g w  v u s u u   u w u ' w u 0.03 0.04 0.9087 0.3 1.55

  23. From Resistances to Conductances Removing Divisions using Sigmoid Reciprocal:   1/  w   S  ( u , k w   S  ( u , k w  ,  w 2  )  , u ' w  ,  w 1  1 ,  w 2  1 )  w  , u w  ,  w 1 g w g so  1/  s 0  S  ( u , k so , u ' so ,   1 so 1 ,   1  so  S  ( u , k so , u so ,  so 1 ,  so 2 ) so 2 ) u so  v u s u u   u ' so u w u ' w u 0.65 0.03 0.04 0.9087 1.48 0.3 1.55

  24. From Resistances to Conductances   1/  w   S  ( u , k w   S  ( u , k w  ,  w 2  )  , u ' w  ,  w 1  1 ,  w 2  1 )  w  , u w  ,  w 1 g w g so  1/  s 0  S  ( u , k so , u ' so ,   1 so 1 ,   1  so  S  ( u , k so , u so ,  so 1 ,  so 2 ) so 2 ) Removing Divisions using Step Reciprocal:   1/  v   H  ( u ,  o ,  v 1   H  ( u ,  o ,  v 1  ,  v 2  )  1 ,  v 2  1 )  v g v g o  1/  o  H  ( u ,  o ,   1 o 1 ,   1  o  H  ( u ,  o ,  o 1 ,  o 2 ) o 2 ) * ) v   H  ( u ,  o ,0,1) w   H  ( u ,  o ,0,1) (1  ug w  )  H  ( u ,  o ,0,w  u so  v  o u s u u   u ' so u w u ' w u 0.006 0.65 0.03 0.04 0.9087 1.48 0.3 1.55

  25. From Resistances to Conductances   1/  w   S  ( u , k w   S  ( u , k w  ,  w 2  )  , u ' w  ,  w 1  1 ,  w 2  1 )  w  , u w  ,  w 1 g w g so  1/  s 0  S  ( u , k so , u ' so ,   1 so 1 ,   1  so  S  ( u , k so , u so ,  so 1 ,  so 2 ) so 2 ) Removing Divisions using Step Reciprocal:   1/  v   H  ( u ,  o ,  v 1   H  ( u ,  o ,  v 1  ,  v 2  )  1 ,  v 2  1 )  v g v g o  1/  o  H  ( u ,  o ,   1 o 1 ,   1  o  H  ( u ,  o ,  o 1 ,  o 2 ) o 2 )  s  H  ( u ,  w ,  s 1 ,  s 2 ) g s  1/  s  H  ( u ,  w ,  s 1  1 ,  s 2  1 ) * ) v   H  ( u ,  o ,0,1) w   H  ( u ,  o ,0,1) (1  ug w  )  H  ( u ,  o ,0,w  u so  v  o  w u s u u   u ' so u w u ' w u 0.006 0.13 0.65 0.03 0.04 0.9087 1.48 0.3 1.55

  26. Minimal Conductance Model (MCM)  v  u < u s u   ( D  u )  ( u   v )( u u  u ) v g fi  ws g si  g so ( u ) & v   v g v  &  w   w g w & s  ( S  ( u , u s , k s ,0,1)  s ) g s 2 &  w  u   v u   ( D  u )  ws g si  g so ( u ) &  o  u   w v   v g v 2  & u   ( D  u )  u g o 2 &  w   w g w & v   v g v 2  s  ( S  ( u , u s , k s ,0,1)  s ) g s 2 & & *  w ) g w  ( u ) w  ( w  & u   v u   v  0.3 0  u   o s  ( S  ( u , u s , k s ,0,1)  s ) g s 1 & u   w  0. 13 u   w u   ( D  u )  u g o 1 &  v  (1  v ) g v 1 u   o  0.006 u   o &  ( u ) w  (1  u g w   w ) g w & s  ( S  ( u , u s , k s ,0,1)  s ) g s 1 &

  27. Gene Regulatory Networks (GRN) GRN canonical sigmoidal form: n j m i   S  ( u k , k k u i  ,  k , a k , b k )  b i u i a ij j  1 k  1

  28. Gene Regulatory Networks (GRN) GRN canonical sigmoidal form: n j m i   S  ( u k , k k u i  ,  k , a k , b k )  b i u i a ij j  1 k  1 where: a ij : are activation / inhibition constants b i : are decay constants S  (..) : are on / off sigmoidal functions

  29. Gene Regulatory Networks (GRN) GRN canonical sigmoidal form: n j m i   S  ( u k , k k u i  ,  k , a k , b k )  b i u i a ij j  1 k  1 where: a ij : are activation / inhibition constants b i : are decay constants S  (..) : are on / off sigmoidal functions Note: steps and ramps are sigmoid approximations

  30. Optimal Polygonal Approximation Given: One nonlinear curve and desired # segments Find: Optimal polygonal approximation

  31. Optimal Polygonal Approximation Example: What is the optimal polygonal approxima- tion of the blue curve with 3 segments ?

  32. Optimal Polygonal Approximation Example: What is the optimal polygonal approxima- tion of the blue curve with 3 segments ?

  33. Optimal Polygonal Approximation Example: What is the optimal polygonal approxima- tion of the blue curve with 3 segments ? 

  34. Optimal Polygonal Approximation Example: What is the optimal polygonal approxima- tion of the blue curve with 3 segments ? 

  35. Optimal Polygonal Approximation Example: What is the optimal polygonal approxima- tion of the blue curve with 3 segments ? 

  36. Optimal Polygonal Approximation  Dynamic Programming Algorithm • Complexity: O(P 2 ) • P: # points of the curve M. Salotti, Pattern Recognition Letters 22 (2001), Pag 215-221

  37. Optimal Polygonal Approximation  Dynamic Programming Algorithm • Complexity: O(P 2 ) • P: # points of the curve M. Salotti, Pattern Recognition Letters 22 (2001), Pag 215-221

  38. Optimal Polygonal Approximation  Dynamic Programming Algorithm • Complexity: O(P 2 ) • P: # points of the curve M. Salotti, Pattern Recognition Letters 22 (2001), Pag 215-221

  39. Optimal Polygonal Approximation  Dynamic Programming Algorithm • Complexity: O(P 2 ) • P: # points of the curve M. Salotti, Pattern Recognition Letters 22 (2001), Pag 215-221

  40. Optimal Polygonal Approximation  Dynamic Programming Algorithm • Complexity: O(P 2 ) • P: # points of the curve M. Salotti, Pattern Recognition Letters 22 (2001), Pag 215-221

  41. Globally-Optimal Polygonal Approximation Given: Set of nonlinear curves and desired # of segments Find: Globally optimal polygonal approximation

  42. Globally-Optimal Polygonal Approximation Example: What is the optimal polygonal approxima- tion of the curves below with 5 segments ?

  43. Globally-Optimal Polygonal Approximation Example: What is the optimal polygonal approxima- tion of the curves below with 5 segments ? Combining the two we obtain 8 segments and not 5 segments

  44. Globally-Optimal Polygonal Approximation Example: What is the optimal polygonal approxima- tion of the curves below with 5 segments ? Solution: modify the OPAA to minimize the maximum error of a set of curves simultaneously.

  45. Deriving the Piecewise Multi-Affine Model (  v  u < u u ) u  e  ( u   v )( u u  u ) v g fi  ws g si  g so ( u ) &  v   v g v &  w   w g w & s  S  ( u , k s , u s ,0,1) g s 2  s g s 2 &  w  u   v u   v u  e  ws g si  g so ( u ) &  v   v g v 2 & u   v w   w g w  & s  S  ( u , k s , u s ,0,1) g s 2  s g s 2 &  o  u   w u   w u  e  u g o 2 &  v   v g v 2 & u   w *  w ) g w  ( u ) w  ( w  & s  S  ( u , k s , u s ) g s 1  s g s 1 & 0  u   o u   o u  e  u g o 1 & v  (1  v ) g v 1  & u   o w  (1  u g w   w ) g w  ( u ) & s  S  ( u , k s , u s ) g s 1  s g s 1 &

  46. Deriving the Piecewise Multi-Affine Model (  v  u < u u ) u  e  ( u   v )( u u  u ) v g fi  ws g si  g so ( u ) &  v   v g v &  w   w g w & s  S  ( u , k s , u s ,0,1) g s 2  s g s 2 &  w  u   v u   v u  e  ws g si  g so ( u ) &  v   v g v 2 & u   v w   w g w  & s  S  ( u , k s , u s ,0,1) g s 2  s g s 2 &  o  u   w u   w u  e  u g o 2 &  v   v g v 2 & u   w *  w ) g w  ( u ) w  ( w  & s  S  ( u , k s , u s ) g s 1  s g s 1 & 0  u   o u   o u  e  u g o 1 & v  (1  v ) g v 1  & u   o w  (1  u g w   w ) g w  ( u ) & s  S  ( u , k s , u s ) g s 1  s g s 1 &

  47. Deriving the Piecewise Multi-Affine Model (  v  u < u u ) u  e  ( u   v )( u u  u ) v g fi  ws g si  g so ( u ) &  v   v g v &  w   w g w & s  S  ( u , k s , u s ,0,1) g s 2  s g s 2 &  w  u   v u   v u  e  ws g si  g so ( u ) &  v   v g v 2 & u   v w   w g w  & s  S  ( u , k s , u s ,0,1) g s 2  s g s 2 &  o  u   w u   w u  e  u g o 2 &  v   v g v 2 & u   w *  w ) g w  ( u ) w  ( w  & s  S  ( u , k s , u s ) g s 1  s g s 1 & 0  u   o u   o u  e  u g o 1 & v  (1  v ) g v 1  & u   o w  (1  u g w   w ) g w  ( u ) & s  S  ( u , k s , u s ) g s 1  s g s 1 &

  48. Deriving the Piecewise Multi-Affine Model (  v  u < u u ) u  e  ( u   v )( u u  u ) v g fi  ws g si  g so ( u ) &  v   v g v &  w   w g w & s  S  ( u , k s , u s ) g s 2  s g s 2 &  w  u   v u   v u  e  ws g si  g so ( u ) &  v   v g v 2 & u   v w   w g w  & s  S  ( u , k s , u s ) g s 2  s g s 2 &  o  u   w u   w u  e  u g o 2 &  v   v g v 2 & u   w *  w ) g w  ( u ) w  ( w  & s  S  ( u , k s , u s ) g s 1  s g s 1 & 0  u   o u   o u  e  u g o 1 & v  (1  v ) g v 1  & u   o w  (1  u g w   w ) g w  ( u ) & s  S  ( u , k s , u s ) g s 1  s g s 1 &

  49. Deriving the Piecewise Multi-Affine Model (  v  u < u u ) u  e  ( u   v )( u u  u ) v g fi  ws g si  g so ( u ) &  v   v g v &  w   w g w & s  S  ( u , k s , u s ) g s 2  s g s 2 &  w  u   v u   v u  e  ws g si  g so ( u ) &  v   v g v 2 & u   v w   w g w  & s  S  ( u , k s , u s ) g s 2  s g s 2 &  o  u   w u   w u  e  u g o 2 &  v   v g v 2 & u   w *  w ) g w  ( u ) w  ( w  & s  S  ( u , k s , u s ) g s 1  s g s 1 & 0  u   o u   o u  e  u g o 1 & v  (1  v ) g v 1  & u   o w  (1  u g w   w ) g w  ( u ) & s  S  ( u , k s , u s ) g s 1  s g s 1 &

  50. Deriving the Piecewise Multi-Affine Model  12   v < u  u u   26   25 25 u  e  v g fi  ws g si  & R ( u ,  i ,  i  1 , u fi i , u fi i  1 ) R ( u ,  i ,  i  1 , u so i , u so i  1 ) g so i  12 i  12 v   v g v  & w   w g w  &  25 s  (  s ) g s 2 & R ( u ,  i ,  i  1 , u s i , u s i  1 ) i  12  8   w  u   v   12  11 u  e  ws g si  & R ( u ,  i ,  i  1 , u so i , u so i  1 ) g so u   v i  8 v   v g v 2  & u   v w   w g w  &  11 s  (  s ) g s 2 & R ( u ,  i ,  i  1 , u s i , u s i  1 ) i  8  2   o  u   w   8 u  e  u g o 2 & u   w v   v g v 2  & *  w ) u   w  7 w  ( w  & R ( u ,  i ,  i  1 , u w i , u w i  1 ) g w b i  2  7 s  (  s ) g s 1 & R ( u ,  i ,  i  1 , u s i , u s i  1 ) i  2  0  0  u   o   2 u   o u  e  u g o 1 &  v  (1  v ) g v 1 & u   o   , u w i  1  )  , u w i  1  )) g w a 1 w  (  wR ( u ,  i ,  i  1 , u w i & ( R ( u ,  i ,  i  1 , u w i i  0  1 s  (  s ) g s 1 & R ( u ,  i ,  i  1 , u s i , u s i  1 ) i  0

  51. 2D Comparison

  52. Analysis Problem • Find parameter ranges reproducing non-excitability: – Restated as an LTL formula: G ( u   v )

  53. Analysis Problem G ( u   v ) • Initial region: u  [0,  1 ] s  [0,0.01] v  [0.95,1] w  [0.95,1]

  54. Analysis Problem G ( u   v ) u  [0,  1 ] s  [0,0.01] v  [0.95,1] w  [0.95,1] • Uncertain parameter ranges: g o 1  [1,180] g o 2  [0,10] g si  [0.1,100] g so  [0.9,50]

  55. Analysis Problem G ( u   v ) u  [0,  1 ] s  [0,0.01] v  [0.95,1] w  [0.95,1] g o 1  [1,180] g o 2  [0,10] g si  [0.1,100] g so  [0.9,50] e  1 • Stimulus:

  56. State Space Partition v 1.00 0.95 u 0.00  0  1  2  3  7  8  9  11  12  13  25  26 • Hyperrectangles: 4 dimensional (uv-projection) – Arrows: indicate the vector field

  57. Embedding Transition System T X (p) v 1.00 0.95 x '  x u 0.00  0  1  2  3  7  8  9  11  12  13  25  26 T X ( p )    x ' iff there is a solution  and time  such that: x   ( 0 )  x ,  (  )  x '   t  [ 0 ,  ].  ( t )  rect ( x )  rect ( x ')  rect ( x ) is adjacent to rect ( x ')

  58. The Discrete Abstraction T R (p) v 1.00 0.95 u 0.00  0  1  2  3  7  8  9  11  12  13  25  26 x : R ( p ) x ' iff rect ( x )  rect ( x ') T R ( p ) is the quotiont of T X ( p ) with respect to : R ( p )

  59. The Discrete Abstraction T R (p) v 1.00 0.95 u 0.00  0  1  2  3  7  8  9  11  12  13  25  26 x : R ( p ) x ' iff rect ( x )  rect ( x ') T R ( p ) is the quotiont of T X ( p ) with respect to : R ( p ) Theorem:  p . T X ( p )  T R ( p )

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