(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 16669, 531] NotebookOptionsPosition[ 14452, 457] NotebookOutlinePosition[ 15968, 510] CellTagsIndexPosition[ 15890, 505] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["\<\ Mathematical Modelling and Scientific Computing in the Biosciences II\ \>", "Title", CellChangeTimes->{{3.400869877645982*^9, 3.400869884970946*^9}, { 3.400869925353651*^9, 3.400869959090375*^9}}, FontSize->24], Cell[CellGroupData[{ Cell["Exercise 1 (Due in 2 weeks: Tues, 30 October)", "Section", CellChangeTimes->{{3.400916250886026*^9, 3.400916256519915*^9}, { 3.4014755231781*^9, 3.401475540317594*^9}, 3.401520941983456*^9, { 3.401522840631402*^9, 3.401522880686961*^9}}, FontSize->24], Cell["\<\ The ODE system for pituary gonadotroph ER oscillator in closed cell is:\ \>", "Text", CellChangeTimes->{{3.401475559671969*^9, 3.401475582451554*^9}, 3.401520941984051*^9, {3.401522921836328*^9, 3.401522972167018*^9}}, FontSize->16], Cell[BoxData[ RowBox[{ StyleBox[ FractionBox[ SubscriptBox[ RowBox[{"d", "[", SuperscriptBox["Ca", RowBox[{"2", "+"}]], "]"}], "i"], "dt"], FontSize->14], StyleBox["=", FontSize->14], RowBox[{ StyleBox[ FractionBox["fi", "Vi"], FontSize->18], StyleBox["*", FontSize->18], RowBox[{ StyleBox["(", FontSize->18], RowBox[{ RowBox[{ RowBox[{ StyleBox["(", FontSize->18], RowBox[{ StyleBox["L", FontSize->18], StyleBox[" ", FontSize->18], StyleBox["+", FontSize->18], StyleBox[" ", FontSize->18], StyleBox[ RowBox[{ StyleBox["P", FontSize->18], StyleBox["*", FontSize->18], SuperscriptBox[ StyleBox[ RowBox[{"(", FractionBox[ RowBox[{"I", "*", SubscriptBox[ RowBox[{"[", SuperscriptBox["Ca", RowBox[{"2", "+"}]], "]"}], "i"], "*", "h"}], RowBox[{ RowBox[{"(", RowBox[{"I", "+", "Ki"}], ")"}], "*", RowBox[{"(", RowBox[{ SubscriptBox[ RowBox[{"[", SuperscriptBox["Ca", RowBox[{"2", "+"}]], "]"}], "i"], "+", "Ka"}], ")"}]}]], ")"}], FontSize->18], "3"]}], FontSize->14]}], StyleBox[")", FontSize->18]}], StyleBox["*", FontSize->18], RowBox[{ StyleBox["(", FontSize->18], StyleBox[ RowBox[{ SubscriptBox[ RowBox[{"[", SuperscriptBox["Ca", RowBox[{"2", "+"}]], "]"}], "E"], "-", SubscriptBox[ RowBox[{"[", SuperscriptBox["Ca", RowBox[{"2", "+"}]], "]"}], "i"]}], FontSize->14], StyleBox[")", FontSize->18]}]}], StyleBox["-", FontSize->18], RowBox[{ StyleBox["Ve", FontSize->18], FractionBox[ SuperscriptBox[ SubscriptBox[ RowBox[{"[", SuperscriptBox["Ca", RowBox[{"2", "+"}]], "]"}], "i"], "2"], RowBox[{ SuperscriptBox["Ke", "2"], "+", SuperscriptBox[ SubscriptBox[ RowBox[{"[", SuperscriptBox["Ca", RowBox[{"2", "+"}]], "]"}], "i"], "2"]}]]}], "+", " ", RowBox[{"\[Epsilon]", "*", StyleBox[ RowBox[{"(", RowBox[{"Jin", " ", "-", " ", RowBox[{"Vp", FractionBox[ SuperscriptBox[ SubscriptBox[ RowBox[{"[", SuperscriptBox["Ca", RowBox[{"2", "+"}]], "]"}], "i"], "2"], RowBox[{ SuperscriptBox["Kp", "2"], "+", SuperscriptBox[ SubscriptBox[ RowBox[{"[", SuperscriptBox["Ca", RowBox[{"2", "+"}]], "]"}], "i"], "2"]}]]}]}], ")"}], FontSize->18]}]}], ")"}]}]}]], "Equation", CellChangeTimes->{{3.401475626043039*^9, 3.401475632108782*^9}, { 3.401475736817763*^9, 3.401475807482425*^9}, 3.401520941987854*^9, { 3.401521121713294*^9, 3.401521333408154*^9}, {3.401521481909734*^9, 3.401521495566641*^9}, {3.401521587416579*^9, 3.401521590977668*^9}, 3.401521652048453*^9, {3.402138623147098*^9, 3.402138710860927*^9}, 3.402138762219245*^9, 3.402138810647923*^9, {3.402138896323053*^9, 3.402138925121546*^9}}, FontSize->18], Cell[BoxData[ RowBox[{" ", RowBox[{ StyleBox[ FractionBox["dh", "dt"], FontSize->14], StyleBox["=", FontSize->14], RowBox[{ StyleBox["A", FontSize->14], StyleBox["*", FontSize->14], RowBox[{ StyleBox["(", FontSize->22], RowBox[{ StyleBox["Kd", FontSize->14], StyleBox["-", FontSize->14], RowBox[{ RowBox[{ StyleBox["(", FontSize->22], RowBox[{ StyleBox[ SubscriptBox[ RowBox[{"[", SuperscriptBox["Ca", RowBox[{"2", "+"}]], "]"}], "i"], FontSize->14], StyleBox["+", FontSize->22], StyleBox["Kd", FontSize->14]}], StyleBox[")", FontSize->14]}], StyleBox["*", FontSize->14], StyleBox["h", FontSize->14]}]}], StyleBox[")", FontSize->22]}]}]}]}]], "Equation", CellChangeTimes->{{3.401475859235036*^9, 3.401475908676152*^9}, { 3.401476752801277*^9, 3.401476765354664*^9}, 3.401520941991033*^9, { 3.401521377476187*^9, 3.401521391313315*^9}}, FontSize->18], Cell[BoxData[ RowBox[{ StyleBox[ FractionBox[ SubscriptBox[ RowBox[{"d", "[", SuperscriptBox["Ca", RowBox[{"2", "+"}]], "]"}], "T"], "dt"], FontSize->14], StyleBox["=", FontSize->14], RowBox[{ StyleBox[ FractionBox["fi", "Vi"], FontSize->18], StyleBox["*", FontSize->18], StyleBox["\[Epsilon]", FontSize->14], StyleBox["*", FontSize->14], RowBox[{ StyleBox["(", FontSize->18], StyleBox[ RowBox[{ StyleBox["Jin", FontSize->18], StyleBox[" ", FontSize->18], StyleBox["-", FontSize->18], RowBox[{ StyleBox["Vp", FontSize->18], StyleBox[" ", FontSize->18], FractionBox[ SuperscriptBox[ SubscriptBox[ RowBox[{"[", SuperscriptBox["Ca", RowBox[{"2", "+"}]], "]"}], "i"], "2"], RowBox[{"(", RowBox[{ SuperscriptBox["Kp", "2"], "+", SuperscriptBox[ SubscriptBox[ RowBox[{"[", SuperscriptBox["Ca", RowBox[{"2", "+"}]], "]"}], "i"], "2"]}], ")"}]]}]}], FontSize->14], StyleBox[")", FontSize->18]}]}]}]], "Equation", CellChangeTimes->{{3.401475626043039*^9, 3.401475632108782*^9}, { 3.401475736817763*^9, 3.401475807482425*^9}, 3.401520941987854*^9, { 3.401521121713294*^9, 3.401521333408154*^9}, {3.401521460832136*^9, 3.401521585810809*^9}, {3.401521649833961*^9, 3.401521659573902*^9}}, FontSize->18], Cell["where", "Text", CellChangeTimes->{{3.401521845418572*^9, 3.401521845579685*^9}}, FontSize->16], Cell[BoxData[ StyleBox[ RowBox[{ SubscriptBox[ RowBox[{"[", SuperscriptBox["Ca", RowBox[{"2", "+"}]], "]"}], "E"], " ", "=", " ", RowBox[{ RowBox[{"1", "/", "\[Sigma]"}], "*", RowBox[{"(", RowBox[{ SubscriptBox[ RowBox[{"[", SuperscriptBox["Ca", RowBox[{"2", "+"}]], "]"}], "T"], "-", SubscriptBox[ RowBox[{"[", SuperscriptBox["Ca", RowBox[{"2", "+"}]], "]"}], "i"]}], ")"}]}]}], FontSize->16]], "Equation", CellChangeTimes->{{3.401475626043039*^9, 3.401475632108782*^9}, { 3.401475736817763*^9, 3.401475807482425*^9}, 3.401520941987854*^9, { 3.401521121713294*^9, 3.401521333408154*^9}, {3.401521460832136*^9, 3.401521585810809*^9}, {3.401521649833961*^9, 3.401521659573902*^9}, { 3.401521745992267*^9, 3.401521780525757*^9}, {3.401521821183607*^9, 3.401521832410672*^9}}, FontSize->18], Cell["Take parameter values", "Text", CellChangeTimes->{{3.401521845418572*^9, 3.401521845579685*^9}, { 3.401522179748279*^9, 3.401522183656668*^9}}, FontSize->16], Cell[BoxData[ RowBox[{ RowBox[{"fi", "=", "0.01"}], ";", " ", RowBox[{"Vi", "=", "4"}], ";", " ", RowBox[{"L", "=", "0.37"}], ";", RowBox[{"P", "=", "26640"}], ";", " ", RowBox[{"I", "=", "0.9"}], ";", RowBox[{"Ki", "=", "1.0"}], ";", RowBox[{"Ka", "=", "0.4"}], ";", " ", RowBox[{"Ve", "=", "400"}], ";", RowBox[{"Ke", "=", "0.2"}], ";", RowBox[{"A", "=", "0.5"}], ";", RowBox[{"Kd", "=", "0.4"}], ";", RowBox[{"\[Sigma]", "=", "0.185"}], ";", RowBox[{"\[Epsilon]", "=", "0.01"}], ";", RowBox[{"Vp", "=", "2000"}], ";", RowBox[{"Kp", "=", "0.3"}], ";"}]], "Equation", CellChangeTimes->{{3.401475626043039*^9, 3.401475632108782*^9}, { 3.401475736817763*^9, 3.401475807482425*^9}, 3.401520941987854*^9, { 3.401521121713294*^9, 3.401521333408154*^9}, {3.401521460832136*^9, 3.401521585810809*^9}, {3.401521649833961*^9, 3.401521659573902*^9}, { 3.401521745992267*^9, 3.401521780525757*^9}, {3.401521821183607*^9, 3.401521832410672*^9}, {3.401522074544402*^9, 3.401522131083251*^9}, { 3.401522204669258*^9, 3.401522205024624*^9}}, FontSize->18], Cell["and initial conditions", "Text", CellChangeTimes->{{3.401521845418572*^9, 3.401521845579685*^9}, { 3.401522179748279*^9, 3.401522195803733*^9}}, FontSize->16], Cell[BoxData[ StyleBox[ RowBox[{ RowBox[{ RowBox[{ SubscriptBox[ RowBox[{"[", SuperscriptBox["Ca", RowBox[{"2", "+"}]], "]"}], "i"], RowBox[{"(", "0", ")"}]}], " ", "=", " ", "0.2"}], ",", " ", RowBox[{ RowBox[{"h", RowBox[{"(", "0", ")"}]}], "=", "0.8"}], ",", RowBox[{ RowBox[{ SubscriptBox[ RowBox[{"[", SuperscriptBox["Ca", RowBox[{"2", "+"}]], "]"}], "T"], RowBox[{"(", "0", ")"}]}], "=", " ", "4"}]}], FontSize->16]], "Equation", CellChangeTimes->{{3.401475626043039*^9, 3.401475632108782*^9}, { 3.401475736817763*^9, 3.401475807482425*^9}, 3.401520941987854*^9, { 3.401521121713294*^9, 3.401521333408154*^9}, {3.401521460832136*^9, 3.401521585810809*^9}, {3.401521649833961*^9, 3.401521659573902*^9}, { 3.401521745992267*^9, 3.401521780525757*^9}, {3.401521821183607*^9, 3.401521832410672*^9}, {3.401522074544402*^9, 3.401522131083251*^9}, { 3.401522209625878*^9, 3.401522257575192*^9}}, FontSize->18], Cell[CellGroupData[{ Cell[TextData[{ "Question 1 (8 points): integrate the ODE system numerically, with Jin = \ 1200 if t <= 40, Jin = 0 for t > 40. What do you observe in ", Cell[BoxData[ SubscriptBox[ RowBox[{"[", SuperscriptBox["Ca", RowBox[{"2", "+"}]], "]"}], "T"]], CellChangeTimes->{{3.401475626043039*^9, 3.401475632108782*^9}, { 3.401475736817763*^9, 3.401475807482425*^9}, 3.401520941987854*^9, { 3.401521121713294*^9, 3.401521333408154*^9}, {3.401521460832136*^9, 3.401521585810809*^9}, {3.401521649833961*^9, 3.401521659573902*^9}}], "? " }], "Item1", CellChangeTimes->{{3.401476898555368*^9, 3.401476965712787*^9}, { 3.40147707978175*^9, 3.401477124531934*^9}, 3.401520942002151*^9, { 3.401521456001343*^9, 3.401521457255784*^9}, {3.401521663083643*^9, 3.401521666323462*^9}, {3.401521859732197*^9, 3.401521860320522*^9}, { 3.401521894923743*^9, 3.401522037871579*^9}, {3.401522517052127*^9, 3.401522569104853*^9}, 3.401524162242133*^9}, FontSize->14], Cell[TextData[{ "Question 2 (2 points): for small ", Cell[BoxData[ FormBox["\[Epsilon]", TraditionalForm]]], ", what can be extracted as the slow variable of the system? " }], "Item1", CellChangeTimes->{{3.401476898555368*^9, 3.401476965712787*^9}, { 3.40147707978175*^9, 3.401477124531934*^9}, 3.401520942002151*^9, { 3.401521456001343*^9, 3.401521457255784*^9}, {3.401521663083643*^9, 3.401521666323462*^9}, {3.401521859732197*^9, 3.401521860320522*^9}, { 3.401522282635317*^9, 3.401522283472218*^9}, {3.401522543649347*^9, 3.401522544273459*^9}, {3.401522575486064*^9, 3.401522575773154*^9}}, FontSize->14], Cell["\<\ Question 3 (10 points): hence, plot bifurcation diagram with the slow \ variable as the bifurcation parameter. Super-impose the time-series on it to \ show the behavior can be captured in terms of the bifurcation diagram.\ \>", "Item1", CellChangeTimes->{{3.401476898555368*^9, 3.401476965712787*^9}, { 3.40147707978175*^9, 3.401477124531934*^9}, 3.401520942002151*^9, { 3.401521456001343*^9, 3.401521457255784*^9}, {3.401521663083643*^9, 3.401521719310853*^9}, {3.401522289718974*^9, 3.401522357310511*^9}, { 3.401522475769002*^9, 3.401522546458574*^9}, {3.401522580589768*^9, 3.401522615188992*^9}}, FontSize->14] }, Open ]], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowPrevSlideText"], ButtonFrame->"None"], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->FEPrivate`FrontEndResource[ "FEStrings", "SlideshowNextSlideText"], ButtonFrame->"None"] }], "PreviousNext", CellChangeTimes->{3.401520942002775*^9}] }, Open ]] }, Open ]] }, AutoGeneratedPackage->None, WindowSize->{1208, 831}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, PrintingPageRange->{Automatic, Automatic}, PrintingOptions->{"Magnification"->1., "PaperOrientation"->"Portrait", "PaperSize"->{611.25, 789.5625}, "PostScriptOutputFile"->"/home/jameslu/Teaching/NoteBooks/II_Lecture2/Ex1.\ pdf"}, ShowCellLabel->False, Magnification->1., FrontEndVersion->"6.0 for Linux x86 (64-bit) (April 20, 2007)", StyleDefinitions->Notebook[{ Cell[ StyleData[ StyleDefinitions -> FrontEnd`FileName[{"Book"}, "Textbook.nb", CharacterEncoding -> "iso8859-1"]]], Cell[ StyleData["Text"], FontSize -> 18], Cell[ StyleData["Text"], FontSize -> 16], Cell[ StyleData["Item1"], FontSize -> 16], Cell[ StyleData["Item1"], FontSize -> 18], Cell[ StyleData["Item2"], FontSize -> 16], Cell[ StyleData["Subsection"], FontSize -> 18], Cell[ StyleData["Section"], FontSize -> 18, FontColor -> RGBColor[0.3568627450980392, 0.09411764705882353, 0.8823529411764706]]}, Visible -> False, FrontEndVersion -> "6.0 for Linux x86 (64-bit) (April 20, 2007)", StyleDefinitions -> "PrivateStylesheetFormatting.nb"] ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{ "SlideShowHeader"->{ Cell[568, 21, 64, 1, 11, "SlideShowNavigationBar", CellTags->"SlideShowHeader"]} } *) (*CellTagsIndex CellTagsIndex->{ {"SlideShowHeader", 15780, 499} } *) (*NotebookFileOutline Notebook[{ Cell[568, 21, 64, 1, 11, "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell[657, 26, 224, 5, 87, "Title"], Cell[CellGroupData[{ Cell[906, 35, 265, 4, 69, "Section"], Cell[1174, 41, 248, 5, 30, "Text"], Cell[1425, 48, 3651, 125, 111, "Equation"], Cell[5079, 175, 1161, 47, 45, "Equation"], Cell[6243, 224, 1549, 58, 57, "Equation"], Cell[7795, 284, 102, 2, 30, "Text"], Cell[7900, 288, 915, 26, 35, "Equation"], Cell[8818, 316, 167, 3, 30, "Text"], Cell[8988, 321, 1108, 24, 67, "Equation"], Cell[10099, 347, 168, 3, 30, "Text"], Cell[10270, 352, 1032, 28, 35, "Equation"], Cell[CellGroupData[{ Cell[11327, 384, 996, 20, 23, "Item1"], Cell[12326, 406, 633, 12, 22, "Item1"], Cell[12962, 420, 644, 11, 42, "Item1"] }, Open ]], Cell[13621, 434, 803, 19, 29, "PreviousNext"] }, Open ]] }, Open ]] } ] *) (* End of internal cache information *)