Predicting Emergent Behavior in Cardiac Tissue: A Grand Challenge - PowerPoint PPT Presentation

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 )

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 )

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

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

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

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

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

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

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 )

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 )

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 ) 

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 )

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 )

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

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 &

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 &

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 &

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 &

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 )

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 

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

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

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

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

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

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 &

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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 &

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 &

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 &

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 &

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 &

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

2D Comparison

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

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

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]

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:

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

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 ')

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 )

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 )