(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 77821, 2676]*) (*NotebookOutlinePosition[ 78868, 2710]*) (* CellTagsIndexPosition[ 78824, 2706]*) (*WindowFrame->Normal*) Notebook[{ Cell["\<\ Input Resistance of a Nerve Cell \ \>", "Title", CellDingbat->"\[FilledSmallSquare]", Evaluatable->False, CellHorizontalScrolling->False, TextAlignment->Center], Cell["\<\ The input resistance of a nerve cell is typically defined as the DC \ resistance between an intracellular pipette in the soma, and the external \ bathing medium or ground. It is usually determined by inserting an \ intracellular electrode or establishing a whole-cell recording connection \ with the soma and passing a current pulse through the electrode to measure \ the transmembrane DC potential (Note: revised 2006).\ \>", "Section", Evaluatable->False, FontSize->24], Cell[BoxData[{ StyleBox[\(\(Off[General::spell1]\)\(\ \)\( (*\ to\ turn\ off\ spell\ error\ messages\ *) \)\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Off[General::spell]\), FontSize->24]}], "Input"], Cell["\<\ Make Sure you have the packages Units.m and Units2.m in the Miscellaneous \ Directory of the StandardPackages Directory of Mathematica\ \>", "Text", FontSize->24], Cell[CellGroupData[{ Cell[BoxData[{ \(Needs["\"]\), "\[IndentingNewLine]", \(Needs["\"]\)}], "Input"], Cell[BoxData[ RowBox[{\(Miscellaneous`SIUnits`Gray::"shdw"\), \(\(:\)\(\ \)\), \ "\<\"Symbol \\!\\(\\\"Gray\\\"\\) appears in multiple contexts \ \\!\\({\\\"Miscellaneous`SIUnits`\\\", \\\"System`\\\"}\\); definitions in \ context \\!\\(\\\"Miscellaneous`SIUnits`\\\"\\) may shadow or be shadowed by \ other definitions. \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"General::shdw\\\"]\\)\"\>"}]], "Message"] }, Open ]], Cell[TextData[{ "\tDo not evalute the notebook through the kernel option. This program \ needs to be executed incrementally. When in doubt go back to the \ initialization routine to reset the values and parameters. This program has \ initialization routines for each separate section. First is a MN-like \ structure followed by three different cylinder lengths of 1 ", StyleBox["m", FontFamily->"Symbol"], StyleBox["m with a higher value of Rm. An additional file \ Neuronmb_2006_Bar_Graph.nb summarizes the results in an illustrated bar graph \ form.", FontFamily->"Arial"] }], "Subtitle"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{\(Lngth =. \), " ", \( (*\ length\ of\ equivalent\ cylinder\ *) \)}], "\n", RowBox[{\(Lambda =. \), " ", \( (*\ length\ constant\ *) \)}], "\n", RowBox[{\(Gd =. \), " ", \( (*\ input\ conductance\ of\ dendrite\ *) \)}], "\[IndentingNewLine]", RowBox[{\(X1 =. \), " ", \( (*\ length\ in\ lambda\ *) \)}], "\[IndentingNewLine]", RowBox[{\(Dia =. \), \( (*\ diameter\ of\ equivalent\ cylinder\ *) \)}], "\[IndentingNewLine]", RowBox[{\(Rm =. \), RowBox[{"(*", " ", RowBox[{"membrane", " ", "resistance", " ", "in", " ", StyleBox["W", FontFamily->"Symbol"], StyleBox[" ", FontFamily->"Courier New"], StyleBox["cm2", FontFamily->"Courier New"]}], StyleBox[" ", FontFamily->"Courier New"], StyleBox["*)", FontFamily->"Courier New"]}]}], "\[IndentingNewLine]", RowBox[{\(Ri =. \), RowBox[{"(*", " ", RowBox[{"internal", " ", "resistance", " ", "in", " ", StyleBox["W", FontFamily->"Symbol"], StyleBox[" ", FontFamily->"Courier New"], StyleBox["cm", FontFamily->"Courier New"]}], StyleBox[" ", FontFamily->"Courier New"], StyleBox["*)", FontFamily->"Courier New"]}]}], "\[IndentingNewLine]", RowBox[{\(Bjo =. \), " ", \( (*\ terminal\ factor\ related\ to\ end\ boundary\ condition\ *) \)}], "\ \[IndentingNewLine]", RowBox[{\(Gn =. \), \( (*\ conductance\ of\ neuron\ at\ the\ soma\ *) \)}], \ "\[IndentingNewLine]", RowBox[{\(Gs =. \), " ", \( (*\ conductance\ of\ the\ soma\ *) \)}], "\[IndentingNewLine]", RowBox[{\(Rn =. \), " ", \( (*\ input\ resistance\ of\ neuron\ at\ soma\ *) \)}], \ "\[IndentingNewLine]", RowBox[{\(Rend =. \), " ", \( (*\ input\ resistance\ at\ end\ of\ cylinder\ *) \)}], "\ \[IndentingNewLine]", RowBox[{\(B1o =. \), " ", \( (*\ this\ is\ the\ B1\ value\ for\ calculating\ at\ the\ end\ of\ \ dendrites\ *) \)}], "\[IndentingNewLine]", RowBox[{\(Bjs =. \), " ", \( (*\ this\ is\ the\ Bjo\ value\ for\ the\ soma\ end\ of\ the\ dendrite\ \ when\ moving\ out\ *) \)}], "\[IndentingNewLine]", RowBox[{\(Gend =. \), " ", \( (*\ conductance\ at\ end\ of\ cylinder\ *) \)}], "\[IndentingNewLine]", RowBox[{\(Ginf =. \), " ", \( (*\ conductance\ of\ infinitely\ long\ extension\ of\ cylinder\ *) \)}], \ "\[IndentingNewLine]", \(Ginf = 1\/2\ N[\[Pi]]\ 1\/\@\(Rm\ Ri\)\ Dia\^\(3/2\)\), "\[IndentingNewLine]", RowBox[{\(Ssa = \[Pi]\ DiaSoma\^2\), " ", \( (*\ definition\ of\ soma\ surface\ area\ *) \)}], "\[IndentingNewLine]", RowBox[{\(Gs = Ssa\/Rm\), " ", \( (*\ input\ conductance\ of\ soma\ as\ a\ function\ of\ Rm\ and\ Ssa\ *) \ \)}], "\[IndentingNewLine]", RowBox[{\(Gd = B1\ Ginf\), " ", \( (*\ this\ defines\ the\ conductance\ of\ the\ real\ cylinder\ *) \)}], "\ \[IndentingNewLine]", RowBox[{\(Gend = B1o\ Ginf\), " ", \( (*\ this\ is\ for\ the\ end\ of\ the\ process\ when\ going\ out\ *) \)}], \ "\[IndentingNewLine]", RowBox[{\(B1 = \(Bjo + Tanh[X1]\)\/\(1 + Bjo\ Tanh[X1]\)\), " ", \( (*\ for\ going\ in\ *) \)}], "\[IndentingNewLine]", RowBox[{\(B1o = \(Bjs + Tanh[X1]\)\/\(1 + Bjs\ Tanh[X1]\)\), " ", \( (*\ \(for\ going\ out ... \) the\ equivalent\ to\ B1\ going\ in\ *) \)}], "\[IndentingNewLine]", RowBox[{\(Lambda = \@\(\(Rm\ Dia\)\/\(Ri\ 4\)\)\), " ", RowBox[{"(*", " ", RowBox[{"definition", " ", "of", " ", StyleBox["l", FontFamily->"Symbol"]}], StyleBox[" ", FontFamily->"Courier New"], StyleBox["*)", FontFamily->"Courier New"]}]}], "\[IndentingNewLine]", RowBox[{\(Gn = Gd + Gs\), " ", \( (*\ input\ conductance\ of\ neuron\ at\ cell\ body\ *) \)}], "\ \[IndentingNewLine]", RowBox[{\(Rn = 1\/Gn\), " ", \( (*\ input\ resistance\ of\ neuron\ at\ cell\ body\ *) \)}], "\ \[IndentingNewLine]", RowBox[{\(Rend = 1/Gend\), " ", \( (*\ input\ resistance\ at\ distal\ end\ of\ cylinder\ *) \)}], "\ \[IndentingNewLine]", RowBox[{\(X1 = Lngth\/Lambda\), " ", RowBox[{"(*", " ", RowBox[{"number", " ", "of", " ", StyleBox["l", FontFamily->"Symbol"], StyleBox[" ", FontFamily->"Courier New"], StyleBox["in", FontFamily->"Courier New"], StyleBox[" ", FontFamily->"Courier New"], StyleBox["the", FontFamily->"Courier New"], StyleBox[" ", FontFamily->"Courier New"], StyleBox["real", FontFamily->"Courier New"], StyleBox[" ", FontFamily->"Courier New"], StyleBox["cylinder", FontFamily->"Courier New"]}], StyleBox[" ", FontFamily->"Courier New"], StyleBox["*)", FontFamily->"Courier New"]}]}], "\[IndentingNewLine]", StyleBox[\(Bjs = \(\(Gs/Ginf\)\(\ \)\( (*\ this\ is\ the\ Bjo\ for\ the\ soma\ end\ of\ the\ cylinder\ *) \)\)\ \), FontFamily->"Courier New"]}], "Input", PageWidth->Infinity, CellDingbat->None, FontSize->24], Cell[BoxData[ \(\(1.5707963267948966`\ Dia\^\(3/2\)\)\/\@\(Ri\ Rm\)\)], "Output"], Cell[BoxData[ \(DiaSoma\^2\ \[Pi]\)], "Output"], Cell[BoxData[ \(\(DiaSoma\^2\ \[Pi]\)\/Rm\)], "Output"], Cell[BoxData[ \(\(1.5707963267948966`\ B1\ Dia\^\(3/2\)\)\/\@\(Ri\ Rm\)\)], "Output"], Cell[BoxData[ \(\(1.5707963267948966`\ B1o\ Dia\^\(3/2\)\)\/\@\(Ri\ Rm\)\)], "Output"], Cell[BoxData[ \(\(Bjo + Tanh[X1]\)\/\(1 + Bjo\ Tanh[X1]\)\)], "Output"], Cell[BoxData[ \(\(Bjs + Tanh[X1]\)\/\(1 + Bjs\ Tanh[X1]\)\)], "Output"], Cell[BoxData[ \(1\/2\ \@\(\(Dia\ Rm\)\/Ri\)\)], "Output"], Cell[BoxData[ \(\(DiaSoma\^2\ \[Pi]\)\/Rm + \(1.5707963267948966`\ Dia\^\(3/2\)\ \((Bjo \ + Tanh[X1])\)\)\/\(\@\(Ri\ Rm\)\ \((1 + Bjo\ Tanh[X1])\)\)\)], "Output"], Cell[BoxData[ \(1\/\(\(DiaSoma\^2\ \[Pi]\)\/Rm + \(1.5707963267948966`\ Dia\^\(3/2\)\ \ \((Bjo + Tanh[X1])\)\)\/\(\@\(Ri\ Rm\)\ \((1 + Bjo\ Tanh[X1])\)\)\)\)], \ "Output"], Cell[BoxData[ \(\(0.6366197723675814`\ \@\(Ri\ Rm\)\ \((1 + Bjs\ Tanh[X1])\)\)\/\(Dia\^\ \(3/2\)\ \((Bjs + Tanh[X1])\)\)\)], "Output"], Cell[BoxData[ \(\(2\ Lngth\)\/\@\(\(Dia\ Rm\)\/Ri\)\)], "Output"], Cell[BoxData[ \(\(2.`\ DiaSoma\^2\ \@\(Ri\ Rm\)\)\/\(Dia\^\(3/2\)\ Rm\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(cm = Centimeter\)], "Input"], Cell[BoxData[ TagBox[\(1\ Centimeter\), UnitsWrapper]], "Output"] }, Open ]], Cell["\<\ An \"Equivalent Cylinder\" is one in which the dendritic tree collapses into \ a single, linear cable. For a dendrite to conform to equivalent cylinder \ conditions, 3/2 power branching must be present and the dendrites terminals \ should be the same electrotonic length from the soma.\ \>", "Section", Evaluatable->False, FontSize->24], Cell["\<\ In order to create an equivalent cylinder model of a neuron, let's take some \ data obtained from motoneurons, where RN=1.5 megohms (GN=1/RN=.667 uS); \ DiaSoma=70 um; Electrotonic Length of Equivalent Cylinder=1.5 L; Dia of \ equivalent cylinder dendrite is 10 um. If we assume Rm= 5000 ohms cm2 and \ Ri=110 ohms cm, then, \ \>", "Subsection", Evaluatable->False, FontSize->24], Cell[CellGroupData[{ Cell[BoxData[{ \(Rm = 5000\ Ohm\ \(\(cm\)\(\ \)\)\^2\), "\n", \(Ri = 110\ Ohm\ cm\), "\n", \(Dia = 10\ 10^\(-4\)\ cm\ // N\), "\[IndentingNewLine]", \(DiaSoma = 70\/10\^4\ cm\ // N\), "\[IndentingNewLine]", \(Bjo = 0.0\)}], "Input", PageWidth->Infinity, FontSize->24], Cell[BoxData[ TagBox[\(5000\ \(Centimeter\^2\ Ohm\)\), UnitsWrapper]], "Output"], Cell[BoxData[ TagBox[\(110\ \(Centimeter\ Ohm\)\), UnitsWrapper]], "Output"], Cell[BoxData[ TagBox[\(0.001`\ Centimeter\), UnitsWrapper]], "Output"], Cell[BoxData[ TagBox[\(0.007`\ Centimeter\), UnitsWrapper]], "Output"], Cell[BoxData[ \(0.`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(N[Lambda]\)], "Input", PageWidth->Infinity, FontSize->24], Cell[BoxData[ TagBox[\(0.10660035817780522`\ Centimeter\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Convert[Lambda, \ Micron]\)], "Input"], Cell[BoxData[ TagBox[\(1066.0035817780522`\ Micron\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(N[Lambda]\)], "Input"], Cell[BoxData[ TagBox[\(0.10660035817780522`\ Centimeter\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Convert[Lambda, \ Micron]\)], "Input"], Cell[BoxData[ TagBox[\(1066.0035817780522`\ Micron\), UnitsWrapper]], "Output"] }, Open ]], Cell["\<\ \t\tNote: this value is in cm so to convert to mm, we need to multiply by \ 10^4; make sure any change is not permanent. Leave in cm\ \>", "Text", PageWidth->Infinity, FontSize->24], Cell["\<\ According to this calculation, Lambda for the equivalent cylinder is 1066 um, \ which means, that to have a 1.5 Lambda cylinder, we need a process about 1599 \ um long.\ \>", "Subsection", Evaluatable->False, FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[\(Lngth = Lambda\ 1.5\), FontSize->24]], "Input"], Cell[BoxData[ TagBox[\(0.15990053726670783`\ Centimeter\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(X1\)], "Input", PageWidth->Infinity, FontSize->24], Cell[BoxData[ \(1.5000000000000002`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(B1\)], "Input", PageWidth->Infinity, FontSize->24], Cell[BoxData[ \(0.9051482536448665`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Ginf", "=", RowBox[{ RowBox[{"Convert", "[", StyleBox[\(Ginf, \ Mho\), FontSize->24], StyleBox["]", FontSize->24]}], StyleBox["//", FontSize->24], StyleBox["N", FontSize->24]}]}]], "Input"], Cell[BoxData[ TagBox[\(6.697898042428672`*^-8\ Mho\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Gd", "=", RowBox[{"Convert", "[", StyleBox[\(Gd, \ Mho\), FontSize->24], StyleBox["]", FontSize->24]}]}]], "Input"], Cell[BoxData[ TagBox[\(6.062590716195683`*^-8\ Mho\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Gs", "=", RowBox[{ RowBox[{"Convert", "[", StyleBox[\(Gs, \ Mho\), FontSize->24], StyleBox["]", FontSize->24]}], StyleBox["//", FontSize->24], StyleBox["N", FontSize->24]}]}]], "Input"], Cell[BoxData[ TagBox[\(3.0787608005179975`*^-8\ Mho\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Gn", "=", RowBox[{ RowBox[{"Convert", StyleBox["[", FontSize->24], StyleBox[\(Gn, \ Mho\), FontSize->24], StyleBox["]", FontSize->24]}], StyleBox["//", FontSize->24], StyleBox["N", FontSize->24]}]}]], "Input"], Cell[BoxData[ TagBox[\(9.14135151671368`*^-8\ Mho\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ StyleBox["Rn", FontSize->24]], "Input"], Cell[BoxData[ TagBox[\(1.093930146074834`*^7\ Ohm\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Convert", "[", StyleBox[\(Rn, \ Mega\ Ohm\), FontSize->24], StyleBox["]", FontSize->24]}], StyleBox["//", FontSize->24], StyleBox["N", FontSize->24], StyleBox[" ", FontSize->24], RowBox[{ StyleBox["(*", FontSize->24], StyleBox[" ", FontSize->24], RowBox[{ StyleBox["convert", FontSize->24], StyleBox[" ", FontSize->24], StyleBox["to", FontSize->24], StyleBox[" ", FontSize->24], StyleBox[ RowBox[{ StyleBox["M", FontSize->24], StyleBox["W", FontFamily->"Symbol", FontSize->24]}]]}], StyleBox[" ", FontFamily->"Symbol", FontSize->24], StyleBox["*)", FontFamily->"Symbol", FontSize->24]}]}]], "Input"], Cell[BoxData[ TagBox[\(10.93930146074834`\ \(Mega\ Ohm\)\), UnitsWrapper]], "Output"] }, Open ]], Cell[TextData[{ "\t\tThis is the input resistance of the neuron in M", StyleBox["W ", FontFamily->"Symbol"] }], "Text", FontSize->36], Cell["\<\ \t\tWhat is the input resistance at the end of the dendritic cylinder?\ \>", "Text", FontSize->36], Cell[CellGroupData[{ Cell[BoxData[ StyleBox["Bjs", FontSize->24]], "Input"], Cell[BoxData[ \(0.4596607444626961`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ StyleBox["Ginf", FontSize->24]], "Input"], Cell[BoxData[ TagBox[\(6.697898042428672`*^-8\ Mho\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ StyleBox["B1o", FontSize->24]], "Input"], Cell[BoxData[ \(0.9638065608300717`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Gend", "=", RowBox[{"Convert", "[", StyleBox[\(Gend, \ Mho\), FontSize->24], StyleBox["]", FontSize->24]}]}]], "Input"], Cell[BoxData[ TagBox[\(6.455478077063647`*^-8\ Mho\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ StyleBox["Rend", FontSize->24]], "Input"], Cell[BoxData[ TagBox[\(1.5490719479832271`*^7\ Ohm\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Convert", "[", StyleBox[\(Rend, \ Mega\ Ohm\), FontSize->24], StyleBox["]", FontSize->24]}]], "Input"], Cell[BoxData[ TagBox[\(15.49071947983227`\ \(Mega\ Ohm\)\), UnitsWrapper]], "Output"] }, Open ]], Cell["\<\ Note: Value is in megohms. See how the Rn is higher for the cylinder end than \ the soma end (10.93 in soma and 15.5 at end of cylinder)\ \>", "SmallText", Evaluatable->False, FontSize->24], Cell["\<\ Sample Probem: Equivalent cylinder model with Lambda of 0.5, 1.0 & 3.0 ; \ Soma=25 um; Process=1 um dia; Rm=100,000 ohms cm2; Ri=110 ohms cm;\ \>", "Section", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[{ \(Lngth =. \), "\n", \(Lambda =. \), "\n", \(Gd =. \ (*\ input\ conductance\ of\ dendrite\ *) \), "\[IndentingNewLine]", \(X1 =. \), "\[IndentingNewLine]", \(Dia =. \), "\[IndentingNewLine]", \(Rm =. \), "\[IndentingNewLine]", \(Ri =. \), "\[IndentingNewLine]", \(Bjo =. \), "\[IndentingNewLine]", \(Gn =. \), "\[IndentingNewLine]", \(Gs =. \), "\[IndentingNewLine]", \(Rn =. \), "\[IndentingNewLine]", \(Rend =. \), "\[IndentingNewLine]", \(B1 =. \), "\[IndentingNewLine]", \(Ssa =. \), "\[IndentingNewLine]", \(DiaSoma =. \), "\[IndentingNewLine]", \(B1o =. \ (*\ this\ is\ the\ B1\ value\ for\ calculating\ at\ the\ end\ of\ dendrites\ \ *) \), "\[IndentingNewLine]", \(Bjs =. \ (*\ this\ is\ the\ Bjo\ value\ for\ the\ soma\ end\ of\ the\ dendrite\ when\ \ moving\ out\ *) \), "\[IndentingNewLine]", \(Gend =. \), "\[IndentingNewLine]", \(Ginf =. \), "\[IndentingNewLine]", \(Ginf = 1\/2\ N[\[Pi]]\ 1\/\@\(Rm\ Ri\)\ Dia\^\(3/2\)\), "\[IndentingNewLine]", \ \(Ssa = \(\(\[Pi]\)\(\ \)\(DiaSoma\^2\)\(\ \ \)\( (*\ definition\ of\ soma\ surface\ area\ *) \)\)\), \ "\[IndentingNewLine]", \(Gs = Ssa\/Rm\ (*\ input\ conductance\ of\ soma\ as\ a\ function\ of\ Rm\ and\ Ssa\ *) \), \ "\[IndentingNewLine]", \(Gd = B1\ Ginf\), "\[IndentingNewLine]", \(Gend = \(\(B1o\)\(\ \)\(Ginf\)\(\ \)\( (*\ this\ is\ for\ the\ end\ of\ the\ process\ when\ going\ out\ *) \ \)\)\), "\[IndentingNewLine]", \(B1 = \(Bjo + Tanh[X1]\)\/\(1 + Bjo\ Tanh[X1]\)\ (*\ for\ going\ in\ *) \), "\[IndentingNewLine]", \(B1o = \(Bjs + Tanh[X1]\)\/\(1 + Bjs\ Tanh[X1]\)\ (*\ for\ going\ out\ *) \), "\[IndentingNewLine]", \(Lambda = \@\(\(Rm\ Dia\)\/\(Ri\ 4\)\)\), "\[IndentingNewLine]", \(Gn = Gd + Gs\), "\[IndentingNewLine]", \(Rn = 1\/Gn\), "\[IndentingNewLine]", \(Rend = 1/Gend\), "\[IndentingNewLine]", \(X1 = Lngth\/Lambda\), "\[IndentingNewLine]", \(Bjs = Gs/Ginf\)}], "Input", PageWidth->Infinity, FontSize->24], Cell[BoxData[ \(\(1.5707963267948966`\ Dia\^\(3/2\)\)\/\@\(Ri\ Rm\)\)], "Output"], Cell[BoxData[ \(DiaSoma\^2\ \[Pi]\)], "Output"], Cell[BoxData[ \(\(DiaSoma\^2\ \[Pi]\)\/Rm\)], "Output"], Cell[BoxData[ \(\(1.5707963267948966`\ B1\ Dia\^\(3/2\)\)\/\@\(Ri\ Rm\)\)], "Output"], Cell[BoxData[ \(\(1.5707963267948966`\ B1o\ Dia\^\(3/2\)\)\/\@\(Ri\ Rm\)\)], "Output"], Cell[BoxData[ \(\(Bjo + Tanh[X1]\)\/\(1 + Bjo\ Tanh[X1]\)\)], "Output"], Cell[BoxData[ \(\(Bjs + Tanh[X1]\)\/\(1 + Bjs\ Tanh[X1]\)\)], "Output"], Cell[BoxData[ \(1\/2\ \@\(\(Dia\ Rm\)\/Ri\)\)], "Output"], Cell[BoxData[ \(\(DiaSoma\^2\ \[Pi]\)\/Rm + \(1.5707963267948966`\ Dia\^\(3/2\)\ \((Bjo \ + Tanh[X1])\)\)\/\(\@\(Ri\ Rm\)\ \((1 + Bjo\ Tanh[X1])\)\)\)], "Output"], Cell[BoxData[ \(1\/\(\(DiaSoma\^2\ \[Pi]\)\/Rm + \(1.5707963267948966`\ Dia\^\(3/2\)\ \ \((Bjo + Tanh[X1])\)\)\/\(\@\(Ri\ Rm\)\ \((1 + Bjo\ Tanh[X1])\)\)\)\)], \ "Output"], Cell[BoxData[ \(\(0.6366197723675814`\ \@\(Ri\ Rm\)\ \((1 + Bjs\ Tanh[X1])\)\)\/\(Dia\^\ \(3/2\)\ \((Bjs + Tanh[X1])\)\)\)], "Output"], Cell[BoxData[ \(\(2\ Lngth\)\/\@\(\(Dia\ Rm\)\/Ri\)\)], "Output"], Cell[BoxData[ \(\(2.`\ DiaSoma\^2\ \@\(Ri\ Rm\)\)\/\(Dia\^\(3/2\)\ Rm\)\)], "Output"] }, Open ]], Cell["Now calculate Lambda and the Bo factor", "Subsubsection", Evaluatable->False, FontSize->24], Cell[CellGroupData[{ Cell[BoxData[{ \(Rm = 100000\ Ohm\ \ cm\^2\), "\n", \(Ri = 110\ Ohm\ cm\), "\n", \(Bjo = 0\), "\[IndentingNewLine]", \(Dia = 1.0\/10\^4\ \ cm\), "\[IndentingNewLine]", \(DiaSoma = 25\/10\^4\ cm\), "\[IndentingNewLine]", \(Lambda\)}], "Input", PageWidth->Infinity, FontSize->24], Cell[BoxData[ TagBox[\(100000\ \(Centimeter\^2\ Ohm\)\), UnitsWrapper]], "Output"], Cell[BoxData[ TagBox[\(110\ \(Centimeter\ Ohm\)\), UnitsWrapper]], "Output"], Cell[BoxData[ \(0\)], "Output"], Cell[BoxData[ TagBox[\(0.0001`\ Centimeter\), UnitsWrapper]], "Output"], Cell[BoxData[ TagBox[\(1\/400\ Centimeter\), UnitsWrapper]], "Output"], Cell[BoxData[ TagBox[\(0.15075567228888181`\ Centimeter\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Convert[Lambda, \ Micron]\)], "Input"], Cell[BoxData[ TagBox[\(1507.556722888818`\ Micron\), UnitsWrapper]], "Output"] }, Open ]], Cell["\<\ \tThis is the length constant (Lambda) in cm, so multiply by 10^4 to get um \ or 1507.56 um\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Convert[Lambda, \ Micron]\)], "Input"], Cell[BoxData[ TagBox[\(1507.556722888818`\ Micron\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Lngth = Lambda\/2\)], "Input", PageWidth->Infinity, FontSize->24], Cell[BoxData[ TagBox[\(0.07537783614444091`\ Centimeter\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(X1\)], "Input", PageWidth->Infinity, FontSize->24], Cell[BoxData[ \(0.5000000000000001`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(B1\)], "Input", PageWidth->Infinity, FontSize->24], Cell[BoxData[ \(0.46211715726000985`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[\(Ginf = Convert[Ginf, \ Mho]\), FontSize->24]], "Input"], Cell[BoxData[ TagBox[\(4.736129125497416`*^-10\ Mho\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Gd = Convert[Gd, \ Mho]\)], "Input", PageWidth->Infinity, FontSize->24], Cell[BoxData[ TagBox[\(2.1886465278912023`*^-10\ Mho\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Gs = Convert[Gs, \ Mho] // N\)], "Input", PageWidth->Infinity, FontSize->24], Cell[BoxData[ TagBox[\(1.9634954084936207`*^-10\ Mho\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Gn = Convert[Gn, \ Mho] // N\)], "Input", PageWidth->Infinity, FontSize->24], Cell[BoxData[ TagBox[\(4.152141936384823`*^-10\ Mho\), UnitsWrapper]], "Output"] }, Open ]], Cell["\<\ \t\tSo Gn is the sum of Gs (soma) and Gd (real value of the dendritic \ conductance at the soma)\ \>", "Text", FontSize->36], Cell[CellGroupData[{ Cell[BoxData[ \(Rn\)], "Input", PageWidth->Infinity, FontSize->24], Cell[BoxData[ TagBox[\(2.408395510849703`*^9\ Ohm\), UnitsWrapper]], "Output"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Convert", "[", StyleBox[\(Rn, \ Giga\ Ohm\), FontSize->24], StyleBox["]", FontSize->24]}], StyleBox[\( (*\ convert\ to\ megohms\ *) \), FontSize->24]}]], "Input"], Cell[BoxData[ TagBox[\(2.408395510849703`\ \(Giga\ Ohm\)\), UnitsWrapper]], "Output"] }, Open ]] }, Open ]], Cell["\<\ \t\tThis is the input resistance at the soma for a 1.0 um dendrite 0.5 Lambda \ in length (2.408 Gohms) \t\tNow let's find the input resistance of this structure at the end of the \ 0.5 lambda equivalent cylinder\ \>", "Text", FontSize->36], Cell[CellGroupData[{ Cell[BoxData[{ StyleBox[\(\(Bjs = Gs/Ginf\)\(\ \)\), FontSize->24], "\[IndentingNewLine]", RowBox[{ StyleBox[\( (*\ the\ Bjs\ in\ this\ case\ is\ that\ for\ the\ soma\ end\ of\ the\ \ equivalent\ cylinder\ *) \), FontSize->24]}]}], "Input"], Cell[BoxData[ \(0.41457809879442487`\)], "Output"], Cell["\<\ \t\tIn this case, we use Bjs to equal the Bjo when going in, but now we are \ going out\ \>", "Text", FontSize->24] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[\(\(Ginf\)\(\ \)\( (*\ this\ remains\ the\ same\ *) \)\), FontSize->24]], "Input"], Cell[BoxData[ TagBox[\(4.736129125497416`*^-10\ Mho\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[\(\(B1o\)\(\ \)\( (*\ this\ will\ be\ the\ B1\ factor\ for\ calculating\ the\ conductance\ \ at\ the\ distal\ end\ *) \)\), FontSize->24]], "Input"], Cell[BoxData[ \(0.7357395800386364`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[\(\(Gend\)\(=\)\(Convert[Gend, \ Mho]\)\( (*\ gives\ us\ the\ conductance\ at\ the\ end\ of\ the\ 0.5\ Lambda\ \ Process\ *) \)\), FontSize->24]], "Input"], Cell[BoxData[ TagBox[\(3.484557653802223`*^-10\ Mho\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ StyleBox["Rend", FontSize->24]], "Input"], Cell[BoxData[ TagBox[\(2.869804719427835`*^9\ Ohm\), UnitsWrapper]], "Output"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ StyleBox["Convert", FontSize->24], "[", StyleBox[\(Rend, \ Giga\ Ohm\), FontSize->24], StyleBox["]", FontSize->24]}], "\[IndentingNewLine]"}]], "Input"], Cell[BoxData[ TagBox[\(2.8698047194278353`\ \(Giga\ Ohm\)\), UnitsWrapper]], "Output"] }, Open ]] }, Open ]], Cell[TextData[{ "\t\tHow does this compare to the resistance in the soma? (2870 at end 2408 \ M", StyleBox["W ", FontFamily->"Symbol"], StyleBox["soma)?", FontFamily->"Times New Roman"] }], "Text", FontSize->36], Cell[TextData[{ "\t", StyleBox["Now we have to re-initialize the parameters to carry out the next \ calculation", FontSize->24] }], "Subsection"], Cell["Now assume that the length of the process is 1 lambda", "Subsubsection", Evaluatable->False, FontSize->36], Cell[CellGroupData[{ Cell[BoxData[{ StyleBox[\(Lngth =. \), FontSize->24], "\n", StyleBox[\(Lambda =. \), FontSize->24], "\n", StyleBox[\(\(Gd =. \)\(\ \)\( (*\ input\ conductance\ of\ dendrite\ *) \)\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(X1 =. \), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Dia =. \), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Rm =. \), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Ri =. \), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Bjo =. \), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Gn =. \), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Gs =. \), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Rn =. \), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Ginf =. \), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Rend =. \), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Bo =. \), FontSize->24], "\[IndentingNewLine]", StyleBox[\(B1 =. \), FontSize->24], "\[IndentingNewLine]", StyleBox[\(\(B1o =. \)\(\ \)\( (*\ this\ is\ the\ B1\ value\ for\ calculating\ at\ the\ end\ of\ \ dendrites\ *) \)\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(\(Bjs =. \)\(\ \ \)\(\[IndentingNewLine]\) \( (*\ this\ is\ the\ Bjo\ value\ for\ the\ soma\ end\ of\ the\ dendrite\ \ when\ moving\ out\ *) \)\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Gend =. \), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Ssa =. \), FontSize->24], "\[IndentingNewLine]", StyleBox[\(DiaSoma =. \), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Ginf = 1\/2\ N[\[Pi]]\ 1\/\@\(Rm\ Ri\)\ Dia\^\(3/2\)\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(\(Ssa = \[Pi]\ DiaSoma\^2\)\(\ \ \)\( (*\ definition\ of\ soma\ surface\ area\ *) \)\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(\(Gs = Ssa\/Rm\)\(\ \)\( (*\ input\ conductance\ of\ soma\ as\ a\ function\ of\ Rm\ and\ Ssa\ *) \ \)\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Gd = B1\ Ginf\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Bjs = Gs\/Ginf\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(\(Gend = B1o\ Ginf\)\(\ \)\( (*\ this\ is\ for\ the\ end\ of\ the\ process\ when\ going\ out\ *) \)\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(\(B1 = \(Bjo + Tanh[X1]\)\/\(1 + Bjo\ Tanh[X1]\)\)\(\ \)\( (*\ for\ going\ in\ *) \)\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(\(B1o = \(Bjs + Tanh[X1]\)\/\(1 + Bjs\ Tanh[X1]\)\)\(\ \)\( \ (*\ for\ going\ out\ *) \)\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Lambda = \@\(\(Rm\ Dia\)\/\(Ri\ 4\)\)\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Gn = Gd + Gs\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Rn = 1\/Gn\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Rend = 1/Gend\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(X1 = Lngth\/Lambda\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Bjs = Gs/Ginf\), FontSize->24]}], "Input"], Cell[BoxData[ \(\(1.5707963267948966`\ Dia\^\(3/2\)\)\/\@\(Ri\ Rm\)\)], "Output"], Cell[BoxData[ \(DiaSoma\^2\ \[Pi]\)], "Output"], Cell[BoxData[ \(\(DiaSoma\^2\ \[Pi]\)\/Rm\)], "Output"], Cell[BoxData[ \(\(1.5707963267948966`\ B1\ Dia\^\(3/2\)\)\/\@\(Ri\ Rm\)\)], "Output"], Cell[BoxData[ \(\(2.`\ DiaSoma\^2\ \@\(Ri\ Rm\)\)\/\(Dia\^\(3/2\)\ Rm\)\)], "Output"], Cell[BoxData[ \(\(1.5707963267948966`\ B1o\ Dia\^\(3/2\)\)\/\@\(Ri\ Rm\)\)], "Output"], Cell[BoxData[ \(\(Bjo + Tanh[X1]\)\/\(1 + Bjo\ Tanh[X1]\)\)], "Output"], Cell[BoxData[ \(\(\(2.`\ DiaSoma\^2\ \@\(Ri\ Rm\)\)\/\(Dia\^\(3/2\)\ Rm\) + \ Tanh[X1]\)\/\(1 + \(2.`\ DiaSoma\^2\ \@\(Ri\ Rm\)\ Tanh[X1]\)\/\(Dia\^\(3/2\)\ \ Rm\)\)\)], "Output"], Cell[BoxData[ \(1\/2\ \@\(\(Dia\ Rm\)\/Ri\)\)], "Output"], Cell[BoxData[ \(\(DiaSoma\^2\ \[Pi]\)\/Rm + \(1.5707963267948966`\ Dia\^\(3/2\)\ \((Bjo \ + Tanh[X1])\)\)\/\(\@\(Ri\ Rm\)\ \((1 + Bjo\ Tanh[X1])\)\)\)], "Output"], Cell[BoxData[ \(1\/\(\(DiaSoma\^2\ \[Pi]\)\/Rm + \(1.5707963267948966`\ Dia\^\(3/2\)\ \ \((Bjo + Tanh[X1])\)\)\/\(\@\(Ri\ Rm\)\ \((1 + Bjo\ Tanh[X1])\)\)\)\)], \ "Output"], Cell[BoxData[ \(\(0.6366197723675814`\ \@\(Ri\ Rm\)\ \((1 + \(2.`\ DiaSoma\^2\ \@\(Ri\ \ Rm\)\ Tanh[X1]\)\/\(Dia\^\(3/2\)\ Rm\))\)\)\/\(Dia\^\(3/2\)\ \((\(2.`\ \ DiaSoma\^2\ \@\(Ri\ Rm\)\)\/\(Dia\^\(3/2\)\ Rm\) + Tanh[X1])\)\)\)], "Output"], Cell[BoxData[ \(\(2\ Lngth\)\/\@\(\(Dia\ Rm\)\/Ri\)\)], "Output"], Cell[BoxData[ \(\(2.`\ DiaSoma\^2\ \@\(Ri\ Rm\)\)\/\(Dia\^\(3/2\)\ Rm\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(Bjo = 0.0\), "\[IndentingNewLine]", \(Dia = 1\ 10^\(-4\)\ cm\), "\[IndentingNewLine]", \(Rm = 100000\ Ohm\ cm\^2\), "\[IndentingNewLine]", \(Ri = 110\ Ohm\ cm\), "\[IndentingNewLine]", \(DiaSoma = 25\ 10^\(-4\)\ cm\), "\[IndentingNewLine]", \(N[Lambda]\)}], "Input", PageWidth->Infinity, FontSize->24], Cell[BoxData[ \(0.`\)], "Output"], Cell[BoxData[ TagBox[\(1\/10000\ Centimeter\), UnitsWrapper]], "Output"], Cell[BoxData[ TagBox[\(100000\ \(Centimeter\^2\ Ohm\)\), UnitsWrapper]], "Output"], Cell[BoxData[ TagBox[\(110\ \(Centimeter\ Ohm\)\), UnitsWrapper]], "Output"], Cell[BoxData[ TagBox[\(1\/400\ Centimeter\), UnitsWrapper]], "Output"], Cell[BoxData[ TagBox[\(0.15075567228888181`\ Centimeter\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Convert[Lambda, \ Micron] // N\)], "Input"], Cell[BoxData[ TagBox[\(1507.556722888818`\ Micron\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(N[Lngth = Lambda]\)], "Input", PageWidth->Infinity, FontSize->24], Cell[BoxData[ TagBox[\(0.15075567228888181`\ Centimeter\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(X1 = Lngth\/Lambda\)], "Input", PageWidth->Infinity, FontSize->24], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(B1\)], "Input", PageWidth->Infinity, FontSize->24], Cell[BoxData[ \(0.7615941559557649`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Ginf = Convert[Ginf, \ Mho]\)], "Input", PageWidth->Infinity, FontSize->24], Cell[BoxData[ TagBox[\(4.736129125497415`*^-10\ Mho\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Gs", "=", RowBox[{ RowBox[{"Convert", "[", StyleBox[\(Gs, Mho\), FontSize->24], StyleBox["]", FontSize->24]}], StyleBox["//", FontSize->24], StyleBox["N", FontSize->24]}]}]], "Input"], Cell[BoxData[ TagBox[\(1.9634954084936207`*^-10\ Mho\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Gd = Convert[Gd, \ Mho]\)], "Input", PageWidth->Infinity, FontSize->24], Cell[BoxData[ TagBox[\(3.6070082638307184`*^-10\ Mho\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Gn = Convert[Gn, \ Mho]\)], "Input", PageWidth->Infinity, FontSize->24], Cell[BoxData[ TagBox[\(5.570503672324339`*^-10\ Mho\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ StyleBox["Rn", FontSize->24]], "Input"], Cell[BoxData[ TagBox[\(1.7951698065800602`*^9\ Ohm\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Convert", "[", StyleBox[\(Rn, \ Mega\ Ohm\), FontSize->24], StyleBox["]", FontSize->24]}], StyleBox[" ", FontSize->24], StyleBox[\( (*\ resistance\ in\ megohms\ *) \), FontSize->24]}]], "Input"], Cell[BoxData[ TagBox[\(1795.1698065800601`\ \(Mega\ Ohm\)\), UnitsWrapper]], "Output"] }, Open ]], Cell[TextData[{ "\tThis is the value of the input resistance at the soma in M", StyleBox["W", FontFamily->"Symbol"], "\n\tNow let's calculate the resistance at the end of the 1 ", StyleBox["l", FontFamily->"Symbol"], " process" }], "Text", FontSize->36], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[\(Bjs = Gs\/Ginf\), FontSize->24]], "Input", PageWidth->Infinity, FontSize->18], Cell[BoxData[ \(0.414578098794425`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[\(\(N[Ginf]\)\(\ \)\( (*\ Stays\ the\ same\ *) \)\), FontSize->24]], "Input"], Cell[BoxData[ TagBox[\(4.736129125497415`*^-10\ Mho\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[\(N[B1o]\), FontSize->24]], "Input", PageWidth->Infinity, FontSize->18], Cell[BoxData[ \(0.8939243504087786`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Gend", "=", RowBox[{"Convert", StyleBox["[", FontSize->24], StyleBox[\(Gend, \ Mho\), FontSize->24], StyleBox["]", FontSize->24]}]}]], "Input", PageWidth->Infinity, FontSize->24], Cell[BoxData[ TagBox[\(4.2337411519623726`*^-10\ Mho\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[\(N[Rend]\), FontSize->24]], "Input", PageWidth->Infinity, FontSize->18], Cell[BoxData[ TagBox[\(2.361977183079537`*^9\ Ohm\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[\(Convert[Rend, \ Mega\ Ohm]\), FontSize->24]], "Input", PageWidth->Infinity, FontSize->18], Cell[BoxData[ TagBox[\(2361.9771830795366`\ \(Mega\ Ohm\)\), UnitsWrapper]], "Output"] }, Open ]], Cell["\tNote that the input resistance is larger at the end", "SmallText", Evaluatable->False, FontSize->36], Cell["Now increase the Lambda value to 3.0", "Subsubsection", Evaluatable->False, FontSize->36], Cell[CellGroupData[{ Cell[BoxData[{ StyleBox[\(Lngth =. \), FontSize->24], "\n", StyleBox[\(Lambda =. \), FontSize->24], "\n", StyleBox[\(\(Gd =. \)\(\ \)\( (*\ input\ conductance\ of\ dendrite\ *) \)\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(X1 =. \), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Dia =. \), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Rm =. \), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Ri =. \), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Bjo =. \), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Gn =. \), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Gs =. \), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Rn =. \), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Ginf =. \), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Rend =. \), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Bo =. \), FontSize->24], "\[IndentingNewLine]", StyleBox[\(B1 =. \), FontSize->24], "\[IndentingNewLine]", StyleBox[\(\(B1o =. \)\(\ \)\( (*\ this\ is\ the\ B1\ value\ for\ calculating\ at\ the\ end\ of\ \ dendrites\ *) \)\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(\(Bjs =. \)\(\ \ \)\( (*\ this\ is\ the\ Bjo\ value\ for\ the\ soma\ end\ of\ the\ dendrite\ \ when\ moving\ out\ *) \)\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Gend =. \), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Ssa =. \), FontSize->24], "\[IndentingNewLine]", StyleBox[\(DiaSoma =. \), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Bjs = Gs\/Ginf\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Ginf = 1\/2\ N[\[Pi]]\ 1\/\@\(Rm\ Ri\)\ Dia\^\(3/2\)\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(\(Ssa = \[Pi]\ DiaSoma\^2\)\(\ \ \)\( (*\ definition\ of\ soma\ surface\ area\ *) \)\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(\(Gs = Ssa\/Rm\)\(\ \)\( (*\ input\ conductance\ of\ soma\ as\ a\ function\ of\ Rm\ and\ Ssa\ *) \ \)\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Gd = B1\ Ginf\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(\(Gend = B1o\ Ginf\)\(\ \)\( (*\ this\ is\ for\ the\ end\ of\ the\ process\ when\ going\ out\ *) \)\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(\(B1 = \(Bjo + Tanh[X1]\)\/\(1 + Bjo\ Tanh[X1]\)\)\(\ \)\( (*\ for\ going\ in\ *) \)\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(\(B1o = \(Bjs + Tanh[X1]\)\/\(1 + Bjs\ Tanh[X1]\)\)\(\ \)\( \ (*\ for\ going\ out\ *) \)\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Lambda = \@\(\(Rm\ Dia\)\/\(Ri\ 4\)\)\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Gn = Gd + Gs\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Rn = 1\/Gn\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Rend = 1/Gend\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(X1 = Lngth\/Lambda\), FontSize->24]}], "Input"], Cell[BoxData[ \(Gs\/Ginf\)], "Output"], Cell[BoxData[ \(\(1.5707963267948966`\ Dia\^\(3/2\)\)\/\@\(Ri\ Rm\)\)], "Output"], Cell[BoxData[ \(DiaSoma\^2\ \[Pi]\)], "Output"], Cell[BoxData[ \(\(DiaSoma\^2\ \[Pi]\)\/Rm\)], "Output"], Cell[BoxData[ \(\(1.5707963267948966`\ B1\ Dia\^\(3/2\)\)\/\@\(Ri\ Rm\)\)], "Output"], Cell[BoxData[ \(\(1.5707963267948966`\ B1o\ Dia\^\(3/2\)\)\/\@\(Ri\ Rm\)\)], "Output"], Cell[BoxData[ \(\(Bjo + Tanh[X1]\)\/\(1 + Bjo\ Tanh[X1]\)\)], "Output"], Cell[BoxData[ \(\(\(2.`\ DiaSoma\^2\ \@\(Ri\ Rm\)\)\/\(Dia\^\(3/2\)\ Rm\) + \ Tanh[X1]\)\/\(1 + \(2.`\ DiaSoma\^2\ \@\(Ri\ Rm\)\ Tanh[X1]\)\/\(Dia\^\(3/2\)\ \ Rm\)\)\)], "Output"], Cell[BoxData[ \(1\/2\ \@\(\(Dia\ Rm\)\/Ri\)\)], "Output"], Cell[BoxData[ \(\(DiaSoma\^2\ \[Pi]\)\/Rm + \(1.5707963267948966`\ Dia\^\(3/2\)\ \((Bjo \ + Tanh[X1])\)\)\/\(\@\(Ri\ Rm\)\ \((1 + Bjo\ Tanh[X1])\)\)\)], "Output"], Cell[BoxData[ \(1\/\(\(DiaSoma\^2\ \[Pi]\)\/Rm + \(1.5707963267948966`\ Dia\^\(3/2\)\ \ \((Bjo + Tanh[X1])\)\)\/\(\@\(Ri\ Rm\)\ \((1 + Bjo\ Tanh[X1])\)\)\)\)], \ "Output"], Cell[BoxData[ \(\(0.6366197723675814`\ \@\(Ri\ Rm\)\ \((1 + \(2.`\ DiaSoma\^2\ \@\(Ri\ \ Rm\)\ Tanh[X1]\)\/\(Dia\^\(3/2\)\ Rm\))\)\)\/\(Dia\^\(3/2\)\ \((\(2.`\ \ DiaSoma\^2\ \@\(Ri\ Rm\)\)\/\(Dia\^\(3/2\)\ Rm\) + Tanh[X1])\)\)\)], "Output"], Cell[BoxData[ \(\(2\ Lngth\)\/\@\(\(Dia\ Rm\)\/Ri\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ StyleBox[\(Bjo = 0.0\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(Dia = 1\ 10^\(-4\)\ cm\), FontSize->24], "\[IndentingNewLine]", RowBox[{ StyleBox["Rm", FontSize->24], StyleBox["=", FontSize->24], RowBox[{ StyleBox["100000", FontSize->24], StyleBox[" ", FontSize->24], StyleBox["Ohm", FontSize->24], StyleBox[" ", FontSize->24], StyleBox[\(cm\^2\), FontSize->24]}]}], "\[IndentingNewLine]", StyleBox[\(Ri = 110\ Ohm\ cm\ // N\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(DiaSoma = 25\ 10^\(-4\)\ cm\ // N\), FontSize->24], "\[IndentingNewLine]", StyleBox[\(N[Lambda]\), FontSize->24]}], "Input"], Cell[BoxData[ \(0.`\)], "Output"], Cell[BoxData[ TagBox[\(1\/10000\ Centimeter\), UnitsWrapper]], "Output"], Cell[BoxData[ TagBox[\(100000\ \(Centimeter\^2\ Ohm\)\), UnitsWrapper]], "Output"], Cell[BoxData[ TagBox[\(110.`\ \(Centimeter\ Ohm\)\), UnitsWrapper]], "Output"], Cell[BoxData[ TagBox[\(0.0025`\ Centimeter\), UnitsWrapper]], "Output"], Cell[BoxData[ TagBox[\(0.15075567228888181`\ Centimeter\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ StyleBox["Lngth", FontSize->24], StyleBox["=", FontSize->24], RowBox[{ StyleBox[\(Lambda\ 3\), FontSize->24], StyleBox[" ", FontSize->24], StyleBox["//", FontSize->24], StyleBox["N", FontSize->24], StyleBox[" ", FontSize->24], RowBox[{ StyleBox["(*", FontSize->24], StyleBox[" ", FontSize->24], RowBox[{ StyleBox["3", FontSize->24], StyleBox[" ", FontSize->24], StyleBox["l", FontFamily->"Symbol", FontSize->24], StyleBox[" ", FontFamily->"Symbol", FontSize->24], StyleBox["equivalent", FontFamily->"Courier New", FontSize->24], StyleBox[" ", FontFamily->"Courier New", FontSize->24], StyleBox["cylinder", FontFamily->"Courier New", FontSize->24]}], StyleBox[" ", FontFamily->"Courier New", FontSize->24], StyleBox["*)", FontFamily->"Courier New", FontSize->24]}]}]}]], "Input"], Cell[BoxData[ TagBox[\(0.45226701686664544`\ Centimeter\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Convert[Lngth, \ Micron]\)], "Input", FontSize->24], Cell[BoxData[ TagBox[\(4522.670168666455`\ Micron\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ StyleBox["X1", FontSize->24]], "Input"], Cell[BoxData[ \(3.0000000000000004`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ StyleBox["B1", FontSize->24]], "Input"], Cell[BoxData[ \(0.9950547536867305`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[\(Ginf = Convert[Ginf, \ Mho]\), FontSize->24]], "Input"], Cell[BoxData[ TagBox[\(4.736129125497415`*^-10\ Mho\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[\(Gd = Convert[Gd, \ Mho]\), FontSize->24]], "Input"], Cell[BoxData[ TagBox[\(4.712707800400379`*^-10\ Mho\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Gs", "=", RowBox[{"Convert", StyleBox["[", FontSize->24], StyleBox[\(Gs, \ Mho\), FontSize->24], StyleBox["]", FontSize->24]}]}]], "Input"], Cell[BoxData[ TagBox[\(1.9634954084936207`*^-10\ Mho\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Gn", "=", RowBox[{"Convert", StyleBox["[", FontSize->24], StyleBox[\(Gn, \ Mho\), FontSize->24], StyleBox["]", FontSize->24]}]}]], "Input"], Cell[BoxData[ TagBox[\(6.676203208894`*^-10\ Mho\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ StyleBox["Rn", FontSize->24]], "Input"], Cell[BoxData[ TagBox[\(1.4978573430296514`*^9\ Ohm\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Convert", "[", StyleBox[\(Rn, Mega\ Ohm\), FontSize->24], StyleBox["]", FontSize->24]}]], "Input", FontSize->24], Cell[BoxData[ TagBox[\(1497.8573430296512`\ \(Mega\ Ohm\)\), UnitsWrapper]], "Output"] }, Open ]], Cell[TextData[{ "\t\tCompare this relatively low value to that for the 0.5 and 1.0 ", StyleBox["l", FontFamily->"Symbol"], StyleBox[" ", FontFamily->"Courier New"], StyleBox["cylinders\n\t\tNow go to the end of the process", FontFamily->"Times New Roman"] }], "Text", FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ StyleBox["Bjs", FontSize->24]], "Input"], Cell[BoxData[ \(0.414578098794425`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ StyleBox["Ginf", FontSize->24]], "Input"], Cell[BoxData[ TagBox[\(4.736129125497415`*^-10\ Mho\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ StyleBox["B1o", FontSize->24]], "Input"], Cell[BoxData[ \(0.9979504436816511`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Gend", "=", RowBox[{"Convert", "[", StyleBox[\(Gend, \ Mho\), FontSize->24], StyleBox["]", FontSize->24]}]}]], "Input"], Cell[BoxData[ TagBox[\(4.726422162123735`*^-10\ Mho\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ StyleBox["Rend", FontSize->24]], "Input"], Cell[BoxData[ TagBox[\(2.115765299201855`*^9\ Ohm\), UnitsWrapper]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Convert", "[", StyleBox[\(Rend, \ Mega\ Ohm\), FontSize->24], StyleBox["]", FontSize->24]}]], "Input"], Cell[BoxData[ TagBox[\(2115.7652992018548`\ \(Mega\ Ohm\)\), UnitsWrapper]], "Output"] }, Open ]], Cell[TextData[{ StyleBox["\t\t", FontSize->18], StyleBox["Note that the end is higher in resistance", "Text", FontSize->24] }], "SmallText", Evaluatable->False, FontSize->24], Cell[BoxData[ \(<< Graphics`Graphics`\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(BarChart[{10.94, \ 2408, \ 1795\ , 1498\ \ }, {14.93, 2870, 2362, 2116}, \[IndentingNewLine]BarSpacing\ \[Rule] \ \(- .05\), \ BarGroupSpacing\ \[Rule] \ .1, \[IndentingNewLine]PlotLabel\ \[Rule] \ \ "\", \[IndentingNewLine]BarLabels\ \[Rule] \ \ {"\", \ "\<0.5 L\>", "\<1.0 L\>", "\<3.0 L\>"}]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations -0.164634 0.243902 0.0147151 0.000205088 [ [.13415 .00222 -8 -12.875 ] [.13415 .00222 8 0 ] [.37805 .00222 -17 -12.875 ] [.37805 .00222 17 0 ] [.62195 .00222 -17 -12.875 ] [.62195 .00222 17 0 ] [.86585 .00222 -17 -12.875 ] [.86585 .00222 17 0 ] [-0.0125 .11726 -18 -4.5 ] [-0.0125 .11726 0 4.5 ] [-0.0125 .2198 -24 -4.5 ] [-0.0125 .2198 0 4.5 ] [-0.0125 .32235 -24 -4.5 ] [-0.0125 .32235 0 4.5 ] [-0.0125 .42489 -24 -4.5 ] [-0.0125 .42489 0 4.5 ] [-0.0125 .52744 -24 -4.5 ] [-0.0125 .52744 0 4.5 ] [.5 .63053 -83 0 ] [.5 .63053 83 12.875 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .13415 .01472 m .13415 .02097 L s gsave .13415 .00222 -69 -16.875 Mabsadd m 1 1 Mabs scale currentpoint translate 0 20.875 translate 1 -1 scale /g { setgray} bind def /k { setcmykcolor} bind def /p { gsave} bind def /r { setrgbcolor} bind def /w { setlinewidth} bind def /C { curveto} bind def /F { fill} bind def /L { lineto} bind def /rL { rlineto} bind def /P { grestore} bind def /s { stroke} bind def /S { show} bind def /N {currentpoint 3 -1 roll show moveto} bind def /Msf { findfont exch scalefont [1 0 0 -1 0 0 ] makefont setfont} bind def /m { moveto} bind def /Mr { rmoveto} bind def /Mx {currentpoint exch pop moveto} bind def /My {currentpoint pop exch moveto} bind def /X {0 rmoveto} bind def /Y {0 exch rmoveto} bind def 63.000 13.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 0.000 0.000 rmoveto 63.000 13.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (MN) show 75.000 13.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 0.000 0.000 rmoveto 1.000 setlinewidth grestore .37805 .01472 m .37805 .02097 L s gsave .37805 .00222 -78 -16.875 Mabsadd m 1 1 Mabs scale currentpoint translate 0 20.875 translate 1 -1 scale /g { setgray} bind def /k { setcmykcolor} bind def /p { gsave} bind def /r { setrgbcolor} bind def /w { setlinewidth} bind def /C { curveto} bind def /F { fill} bind def /L { lineto} bind def /rL { rlineto} bind def /P { grestore} bind def /s { stroke} bind def /S { show} bind def /N {currentpoint 3 -1 roll show moveto} bind def /Msf { findfont exch scalefont [1 0 0 -1 0 0 ] makefont setfont} bind def /m { moveto} bind def /Mr { rmoveto} bind def /Mx {currentpoint exch pop moveto} bind def /My {currentpoint pop exch moveto} bind def /X {0 rmoveto} bind def /Y {0 exch rmoveto} bind def 63.000 13.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 0.000 0.000 rmoveto 63.000 13.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (0.5) show 87.000 13.000 moveto (L) show 93.000 13.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 0.000 0.000 rmoveto 1.000 setlinewidth grestore .62195 .01472 m .62195 .02097 L s gsave .62195 .00222 -78 -16.875 Mabsadd m 1 1 Mabs scale currentpoint translate 0 20.875 translate 1 -1 scale /g { setgray} bind def /k { setcmykcolor} bind def /p { gsave} bind def /r { setrgbcolor} bind def /w { setlinewidth} bind def /C { curveto} bind def /F { fill} bind def /L { lineto} bind def /rL { rlineto} bind def /P { grestore} bind def /s { stroke} bind def /S { show} bind def /N {currentpoint 3 -1 roll show moveto} bind def /Msf { findfont exch scalefont [1 0 0 -1 0 0 ] makefont setfont} bind def /m { moveto} bind def /Mr { rmoveto} bind def /Mx {currentpoint exch pop moveto} bind def /My {currentpoint pop exch moveto} bind def /X {0 rmoveto} bind def /Y {0 exch rmoveto} bind def 63.000 13.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 0.000 0.000 rmoveto 63.000 13.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (1.0) show 87.000 13.000 moveto (L) show 93.000 13.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 0.000 0.000 rmoveto 1.000 setlinewidth grestore .86585 .01472 m .86585 .02097 L s gsave .86585 .00222 -78 -16.875 Mabsadd m 1 1 Mabs scale currentpoint translate 0 20.875 translate 1 -1 scale /g { setgray} bind def /k { setcmykcolor} bind def /p { gsave} bind def /r { setrgbcolor} bind def /w { setlinewidth} bind def /C { curveto} bind def /F { fill} bind def /L { lineto} bind def /rL { rlineto} bind def /P { grestore} bind def /s { stroke} bind def /S { show} bind def /N {currentpoint 3 -1 roll show moveto} bind def /Msf { findfont exch scalefont [1 0 0 -1 0 0 ] makefont setfont} bind def /m { moveto} bind def /Mr { rmoveto} bind def /Mx {currentpoint exch pop moveto} bind def /My {currentpoint pop exch moveto} bind def /X {0 rmoveto} bind def /Y {0 exch rmoveto} bind def 63.000 13.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 0.000 0.000 rmoveto 63.000 13.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (3.0) show 87.000 13.000 moveto (L) show 93.000 13.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 0.000 0.000 rmoveto 1.000 setlinewidth grestore 0 .01472 m 1 .01472 L s 0 .11726 m .00625 .11726 L s [(500)] -0.0125 .11726 1 0 Mshowa 0 .2198 m .00625 .2198 L s [(1000)] -0.0125 .2198 1 0 Mshowa 0 .32235 m .00625 .32235 L s [(1500)] -0.0125 .32235 1 0 Mshowa 0 .42489 m .00625 .42489 L s [(2000)] -0.0125 .42489 1 0 Mshowa 0 .52744 m .00625 .52744 L s [(2500)] -0.0125 .52744 1 0 Mshowa .125 Mabswid 0 .03522 m .00375 .03522 L s 0 .05573 m .00375 .05573 L s 0 .07624 m .00375 .07624 L s 0 .09675 m .00375 .09675 L s 0 .13777 m .00375 .13777 L s 0 .15828 m .00375 .15828 L s 0 .17879 m .00375 .17879 L s 0 .19929 m .00375 .19929 L s 0 .24031 m .00375 .24031 L s 0 .26082 m .00375 .26082 L s 0 .28133 m .00375 .28133 L s 0 .30184 m .00375 .30184 L s 0 .34286 m .00375 .34286 L s 0 .36337 m .00375 .36337 L s 0 .38387 m .00375 .38387 L s 0 .40438 m .00375 .40438 L s 0 .4454 m .00375 .4454 L s 0 .46591 m .00375 .46591 L s 0 .48642 m .00375 .48642 L s 0 .50693 m .00375 .50693 L s 0 .54795 m .00375 .54795 L s 0 .56845 m .00375 .56845 L s 0 .58896 m .00375 .58896 L s 0 .60947 m .00375 .60947 L s .25 Mabswid 0 0 m 0 .61803 L s gsave .5 .63053 -144 -4 Mabsadd m 1 1 Mabs scale currentpoint translate 0 20.875 translate 1 -1 scale /g { setgray} bind def /k { setcmykcolor} bind def /p { gsave} bind def /r { setrgbcolor} bind def /w { setlinewidth} bind def /C { curveto} bind def /F { fill} bind def /L { lineto} bind def /rL { rlineto} bind def /P { grestore} bind def /s { stroke} bind def /S { show} bind def /N {currentpoint 3 -1 roll show moveto} bind def /Msf { findfont exch scalefont [1 0 0 -1 0 0 ] makefont setfont} bind def /m { moveto} bind def /Mr { rmoveto} bind def /Mx {currentpoint exch pop moveto} bind def /My {currentpoint pop exch moveto} bind def /X {0 rmoveto} bind def /Y {0 exch rmoveto} bind def 63.000 13.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 0.000 0.000 rmoveto 63.000 13.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor (Rn) show 81.000 13.000 moveto (at) show 99.000 13.000 moveto (Soma) show 129.000 13.000 moveto (and) show 153.000 13.000 moveto (cylinder) show 207.000 13.000 moveto (end) show 225.000 13.000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000 0.000 0.000 setrgbcolor 0.000 0.000 rmoveto 1.000 setlinewidth grestore 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath 1 0 0 r .01829 .01472 m .01829 .01696 L .14024 .01696 L .14024 .01472 L F 0 g .5 Mabswid .01829 .01472 m .01829 .01696 L .14024 .01696 L .14024 .01472 L .01829 .01472 L s 1 0 0 r .2622 .01472 m .2622 .50857 L .38415 .50857 L .38415 .01472 L F 0 g .2622 .01472 m .2622 .50857 L .38415 .50857 L .38415 .01472 L .2622 .01472 L s 1 0 0 r .5061 .01472 m .5061 .38285 L .62805 .38285 L .62805 .01472 L F 0 g .5061 .01472 m .5061 .38285 L .62805 .38285 L .62805 .01472 L .5061 .01472 L s 1 0 0 r .75 .01472 m .75 .32194 L .87195 .32194 L .87195 .01472 L F 0 g .75 .01472 m .75 .32194 L .87195 .32194 L .87195 .01472 L .75 .01472 L s 0 .4 1 r .12805 .01472 m .12805 .01778 L .25 .01778 L .25 .01472 L F 0 g .12805 .01472 m .12805 .01778 L .25 .01778 L .25 .01472 L .12805 .01472 L s 0 .4 1 r .37195 .01472 m .37195 .60332 L .4939 .60332 L .4939 .01472 L F 0 g .37195 .01472 m .37195 .60332 L .4939 .60332 L .4939 .01472 L .37195 .01472 L s 0 .4 1 r .61585 .01472 m .61585 .49913 L .7378 .49913 L .7378 .01472 L F 0 g .61585 .01472 m .61585 .49913 L .7378 .49913 L .7378 .01472 L .61585 .01472 L s 0 .4 1 r .85976 .01472 m .85976 .44868 L .98171 .44868 L .98171 .01472 L F 0 g .85976 .01472 m .85976 .44868 L .98171 .44868 L .98171 .01472 L .85976 .01472 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHg7oo0P0000=oo`0000000P0000=oo`00Ool01Goo0`000Woo0`001Goo0`000Woo0`0000Moo`00Ool0 07oo001oo`030005Ool300001Goo000007oo00000Woo100017oo0`000goo00@007ooOomoo`<0009o o`<00005Ool00000Ool00002Ool4000017oo000000000Woo0`000Woo0`0000Eoo`00001oo`00009o o`@002Eoo`00>Goo00<007oo00000Woo00<007oo00001Woo0`000Woo00<007ooOol01goo00d007oo Ool007oo001oo`00Ool007oo001oo`030005Ool30002Ool01@00Ool007oo00000Woo00<007ooOol0 0goo00<007ooOol00Woo00<007oo00000goo00<007ooOol00Woo00T007ooOomoo`00Ool007oo0000 0Woo00<007oo00000P000goo00@007ooOomoo`<000=oo`05001oo`00Ool00002Ool00`00Oomoo`0T Ool003Uoo`<0009oo`03001oo`0000Qoo`04001oogoo0007Ool20003Ool02@00Ool007oo001oo`00 Ool00003Ool00`00Oomoo`05Ool02000Oomoo`00Ool007oo0002Ool00`00Oomoo`03Ool00`00Ool0 0002Ool01000Ool000000Woo00<007ooOol00Woo00T007ooOomoo`00Ool007oo00000Woo00<007oo 00000Woo00<007ooOol00`0000Eoo`00Oomoo`00009oo`05001oo`00Ool00002Ool00`00Oomoo`0T Ool003Uoo`03001oo`0000=oo`03001oogoo00Eoo`8000=oo`<000Eoo`03001oo`0000=oo`07001o ogooOol007oo00000Woo0P001Woo0P0017oo00@007ooOomoo`<000Ioo`03001oogoo00Moo`05001o ogooOol00005Ool01000Oomoogoo0`000Woo0P0027oo0P0017oo00@007ooOomoo`<002Ioo`00>7oo 0`004Goo00<007ooOol01Woo0P0087oo00<007ooOol03Goo0P0017oo00<007ooOol027oo00<007oo Ool05Woo00<007ooOol097oo003HOol00001\ \>"], ImageRangeCache->{{{0, 215}, {132.438, 0}} -> {0.0464218, -322.451, \ 0.0170791, 20.3114}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]] }, FrontEndVersion->"5.2 for Microsoft Windows", ScreenRectangle->{{0, 1600}, {0, 1119}}, ScreenStyleEnvironment->"SlideShow", PrintingStyleEnvironment->"Presentation", WindowToolbars->"RulerBar", CellGrouping->Manual, WindowSize->{1592, 1088}, WindowMargins->{{-14, Automatic}, {Automatic, -4}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, -1}}, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, Magnification->0.75, StyleDefinitions -> "Classroom.nb" ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. *******************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[1754, 51, 178, 7, 115, "Title", Evaluatable->False], Cell[1935, 60, 485, 9, 114, "Section", Evaluatable->False], Cell[2423, 71, 238, 5, 70, "Input"], Cell[2664, 78, 174, 4, 37, "Text"], Cell[CellGroupData[{ Cell[2863, 86, 136, 2, 70, "Input"], Cell[3002, 90, 488, 7, 49, "Message"] }, Open ]], Cell[3505, 100, 604, 12, 134, "Subtitle"], Cell[CellGroupData[{ Cell[4134, 116, 5543, 145, 977, "Input"], Cell[9680, 263, 85, 1, 67, "Output"], Cell[9768, 266, 51, 1, 50, "Output"], Cell[9822, 269, 59, 1, 61, "Output"], Cell[9884, 272, 89, 1, 67, "Output"], Cell[9976, 275, 90, 1, 67, "Output"], Cell[10069, 278, 75, 1, 62, "Output"], Cell[10147, 281, 75, 1, 62, "Output"], Cell[10225, 284, 61, 1, 75, "Output"], Cell[10289, 287, 166, 2, 67, "Output"], Cell[10458, 291, 175, 3, 73, "Output"], Cell[10636, 296, 138, 2, 66, "Output"], Cell[10777, 300, 69, 1, 72, "Output"], Cell[10849, 303, 89, 1, 64, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[10975, 309, 48, 1, 51, "Input"], Cell[11026, 312, 75, 2, 49, "Output"] }, Open ]], Cell[11116, 317, 350, 7, 92, "Section", Evaluatable->False], Cell[11469, 326, 394, 8, 75, "Subsection", Evaluatable->False], Cell[CellGroupData[{ Cell[11888, 338, 297, 7, 184, "Input"], Cell[12188, 347, 90, 2, 50, "Output"], Cell[12281, 351, 86, 2, 49, "Output"], Cell[12370, 355, 80, 2, 49, "Output"], Cell[12453, 359, 80, 2, 49, "Output"], Cell[12536, 363, 37, 1, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[12610, 369, 81, 3, 56, "Input"], Cell[12694, 374, 94, 2, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[12825, 381, 58, 1, 51, "Input"], Cell[12886, 384, 89, 2, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[13012, 391, 42, 1, 51, "Input"], Cell[13057, 394, 94, 2, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[13188, 401, 58, 1, 51, "Input"], Cell[13249, 404, 89, 2, 49, "Output"] }, Open ]], Cell[13353, 409, 195, 5, 37, "Text"], Cell[13551, 416, 236, 6, 54, "Subsection", Evaluatable->False], Cell[CellGroupData[{ Cell[13812, 426, 82, 2, 56, "Input"], Cell[13897, 430, 94, 2, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[14028, 437, 74, 3, 56, "Input"], Cell[14105, 442, 53, 1, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[14195, 448, 74, 3, 56, "Input"], Cell[14272, 453, 53, 1, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[14362, 459, 315, 11, 51, "Input"], Cell[14680, 472, 89, 2, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[14806, 479, 187, 6, 51, "Input"], Cell[14996, 487, 89, 2, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[15122, 494, 311, 11, 51, "Input"], Cell[15436, 507, 90, 2, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[15563, 514, 358, 13, 51, "Input"], Cell[15924, 529, 88, 2, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[16049, 536, 63, 2, 56, "Input"], Cell[16115, 540, 88, 2, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[16240, 547, 1028, 39, 52, "Input"], Cell[17271, 588, 95, 2, 49, "Output"] }, Open ]], Cell[17381, 593, 143, 5, 51, "Text"], Cell[17527, 600, 110, 3, 48, "Text"], Cell[CellGroupData[{ Cell[17662, 607, 64, 2, 56, "Input"], Cell[17729, 611, 53, 1, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[17819, 617, 65, 2, 56, "Input"], Cell[17887, 621, 89, 2, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[18013, 628, 64, 2, 56, "Input"], Cell[18080, 632, 53, 1, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[18170, 638, 191, 6, 51, "Input"], Cell[18364, 646, 89, 2, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[18490, 653, 65, 2, 56, "Input"], Cell[18558, 657, 89, 2, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[18684, 664, 159, 5, 51, "Input"], Cell[18846, 671, 95, 2, 49, "Output"] }, Open ]], Cell[18956, 676, 203, 5, 37, "SmallText", Evaluatable->False], Cell[19162, 683, 190, 4, 69, "Section", Evaluatable->False], Cell[CellGroupData[{ Cell[19377, 691, 2145, 49, 1054, "Input"], Cell[21525, 742, 85, 1, 67, "Output"], Cell[21613, 745, 51, 1, 50, "Output"], Cell[21667, 748, 59, 1, 61, "Output"], Cell[21729, 751, 89, 1, 67, "Output"], Cell[21821, 754, 90, 1, 67, "Output"], Cell[21914, 757, 75, 1, 62, "Output"], Cell[21992, 760, 75, 1, 62, "Output"], Cell[22070, 763, 61, 1, 75, "Output"], Cell[22134, 766, 166, 2, 67, "Output"], Cell[22303, 770, 175, 3, 73, "Output"], Cell[22481, 775, 138, 2, 66, "Output"], Cell[22622, 779, 69, 1, 72, "Output"], Cell[22694, 782, 89, 1, 64, "Output"] }, Open ]], Cell[22798, 786, 101, 2, 55, "Subsubsection", Evaluatable->False], Cell[CellGroupData[{ Cell[22924, 792, 310, 8, 236, "Input"], Cell[23237, 802, 92, 2, 50, "Output"], Cell[23332, 806, 86, 2, 49, "Output"], Cell[23421, 810, 35, 1, 49, "Output"], Cell[23459, 813, 81, 2, 49, "Output"], Cell[23543, 817, 80, 2, 59, "Output"], Cell[23626, 821, 94, 2, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[23757, 828, 58, 1, 51, "Input"], Cell[23818, 831, 88, 2, 49, "Output"] }, Open ]], Cell[23921, 836, 115, 3, 32, "Text"], Cell[CellGroupData[{ Cell[24061, 843, 58, 1, 51, "Input"], Cell[24122, 846, 88, 2, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[24247, 853, 89, 3, 74, "Input"], Cell[24339, 858, 94, 2, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[24470, 865, 74, 3, 56, "Input"], Cell[24547, 870, 53, 1, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[24637, 876, 74, 3, 56, "Input"], Cell[24714, 881, 54, 1, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[24805, 887, 90, 2, 56, "Input"], Cell[24898, 891, 90, 2, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[25025, 898, 95, 3, 56, "Input"], Cell[25123, 903, 91, 2, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[25251, 910, 100, 3, 56, "Input"], Cell[25354, 915, 91, 2, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[25482, 922, 100, 3, 56, "Input"], Cell[25585, 927, 90, 2, 50, "Output"] }, Open ]], Cell[25690, 932, 136, 4, 48, "Text"], Cell[CellGroupData[{ Cell[25851, 940, 74, 3, 56, "Input"], Cell[25928, 945, 88, 2, 50, "Output"], Cell[CellGroupData[{ Cell[26041, 951, 257, 8, 51, "Input"], Cell[26301, 961, 95, 2, 49, "Output"] }, Open ]] }, Open ]], Cell[26423, 967, 253, 6, 94, "Text"], Cell[CellGroupData[{ Cell[26701, 977, 280, 7, 71, "Input"], Cell[26984, 986, 54, 1, 49, "Output"], Cell[27041, 989, 127, 4, 37, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[27205, 998, 115, 2, 56, "Input"], Cell[27323, 1002, 90, 2, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[27450, 1009, 190, 4, 56, "Input"], Cell[27643, 1015, 53, 1, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[27733, 1021, 199, 4, 56, "Input"], Cell[27935, 1027, 90, 2, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[28062, 1034, 65, 2, 56, "Input"], Cell[28130, 1038, 88, 2, 50, "Output"], Cell[CellGroupData[{ Cell[28243, 1044, 250, 8, 70, "Input"], Cell[28496, 1054, 96, 2, 49, "Output"] }, Open ]] }, Open ]], Cell[28619, 1060, 227, 8, 51, "Text"], Cell[28849, 1070, 153, 5, 54, "Subsection"], Cell[29005, 1077, 116, 2, 65, "Subsubsection", Evaluatable->False], Cell[CellGroupData[{ Cell[29146, 1083, 3282, 80, 982, "Input"], Cell[32431, 1165, 85, 1, 67, "Output"], Cell[32519, 1168, 51, 1, 50, "Output"], Cell[32573, 1171, 59, 1, 61, "Output"], Cell[32635, 1174, 89, 1, 67, "Output"], Cell[32727, 1177, 89, 1, 64, "Output"], Cell[32819, 1180, 90, 1, 67, "Output"], Cell[32912, 1183, 75, 1, 62, "Output"], Cell[32990, 1186, 183, 3, 80, "Output"], Cell[33176, 1191, 61, 1, 75, "Output"], Cell[33240, 1194, 166, 2, 67, "Output"], Cell[33409, 1198, 175, 3, 73, "Output"], Cell[33587, 1203, 244, 3, 80, "Output"], Cell[33834, 1208, 69, 1, 72, "Output"], Cell[33906, 1211, 89, 1, 64, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[34032, 1217, 351, 8, 185, "Input"], Cell[34386, 1227, 37, 1, 49, "Output"], Cell[34426, 1230, 82, 2, 59, "Output"], Cell[34511, 1234, 92, 2, 50, "Output"], Cell[34606, 1238, 86, 2, 49, "Output"], Cell[34695, 1242, 80, 2, 59, "Output"], Cell[34778, 1246, 94, 2, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[34909, 1253, 63, 1, 51, "Input"], Cell[34975, 1256, 88, 2, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[35100, 1263, 89, 3, 56, "Input"], Cell[35192, 1268, 94, 2, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[35323, 1275, 90, 3, 75, "Input"], Cell[35416, 1280, 35, 1, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[35488, 1286, 74, 3, 56, "Input"], Cell[35565, 1291, 53, 1, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[35655, 1297, 99, 3, 56, "Input"], Cell[35757, 1302, 90, 2, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[35884, 1309, 309, 11, 51, "Input"], Cell[36196, 1322, 91, 2, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[36324, 1329, 95, 3, 56, "Input"], Cell[36422, 1334, 91, 2, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[36550, 1341, 95, 3, 56, "Input"], Cell[36648, 1346, 90, 2, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[36775, 1353, 63, 2, 56, "Input"], Cell[36841, 1357, 89, 2, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[36967, 1364, 304, 10, 51, "Input"], Cell[37274, 1376, 96, 2, 49, "Output"] }, Open ]], Cell[37385, 1381, 273, 9, 98, "Text"], Cell[CellGroupData[{ Cell[37683, 1394, 116, 4, 74, "Input"], Cell[37802, 1400, 52, 1, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[37891, 1406, 110, 2, 56, "Input"], Cell[38004, 1410, 90, 2, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[38131, 1417, 108, 4, 56, "Input"], Cell[38242, 1423, 53, 1, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[38332, 1429, 273, 10, 56, "Input"], Cell[38608, 1441, 91, 2, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[38736, 1448, 109, 4, 56, "Input"], Cell[38848, 1454, 88, 2, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[38973, 1461, 128, 4, 56, "Input"], Cell[39104, 1467, 96, 2, 49, "Output"] }, Open ]], Cell[39215, 1472, 112, 2, 47, "SmallText", Evaluatable->False], Cell[39330, 1476, 99, 2, 65, "Subsubsection", Evaluatable->False], Cell[CellGroupData[{ Cell[39454, 1482, 3175, 77, 940, "Input"], Cell[42632, 1561, 42, 1, 59, "Output"], Cell[42677, 1564, 85, 1, 67, "Output"], Cell[42765, 1567, 51, 1, 50, "Output"], Cell[42819, 1570, 59, 1, 61, "Output"], Cell[42881, 1573, 89, 1, 67, "Output"], Cell[42973, 1576, 90, 1, 67, "Output"], Cell[43066, 1579, 75, 1, 62, "Output"], Cell[43144, 1582, 183, 3, 80, "Output"], Cell[43330, 1587, 61, 1, 75, "Output"], Cell[43394, 1590, 166, 2, 67, "Output"], Cell[43563, 1594, 175, 3, 73, "Output"], Cell[43741, 1599, 244, 3, 80, "Output"], Cell[43988, 1604, 69, 1, 72, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[44094, 1610, 816, 26, 154, "Input"], Cell[44913, 1638, 37, 1, 49, "Output"], Cell[44953, 1641, 82, 2, 59, "Output"], Cell[45038, 1645, 92, 2, 50, "Output"], Cell[45133, 1649, 88, 2, 49, "Output"], Cell[45224, 1653, 81, 2, 49, "Output"], Cell[45308, 1657, 94, 2, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[45439, 1664, 1328, 47, 52, "Input"], Cell[46770, 1713, 94, 2, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[46901, 1720, 73, 2, 56, "Input"], Cell[46977, 1724, 88, 2, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[47102, 1731, 63, 2, 56, "Input"], Cell[47168, 1735, 53, 1, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[47258, 1741, 63, 2, 56, "Input"], Cell[47324, 1745, 53, 1, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[47414, 1751, 90, 2, 56, "Input"], Cell[47507, 1755, 90, 2, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[47634, 1762, 86, 2, 56, "Input"], Cell[47723, 1766, 90, 2, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[47850, 1773, 230, 8, 51, "Input"], Cell[48083, 1783, 91, 2, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[48211, 1790, 230, 8, 51, "Input"], Cell[48444, 1800, 87, 2, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[48568, 1807, 63, 2, 56, "Input"], Cell[48634, 1811, 89, 2, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[48760, 1818, 171, 6, 56, "Input"], Cell[48934, 1826, 96, 2, 49, "Output"] }, Open ]], Cell[49045, 1831, 303, 9, 76, "Text"], Cell[CellGroupData[{ Cell[49373, 1844, 64, 2, 56, "Input"], Cell[49440, 1848, 52, 1, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[49529, 1854, 65, 2, 56, "Input"], Cell[49597, 1858, 90, 2, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[49724, 1865, 64, 2, 56, "Input"], Cell[49791, 1869, 53, 1, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[49881, 1875, 191, 6, 51, "Input"], Cell[50075, 1883, 90, 2, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[50202, 1890, 65, 2, 56, "Input"], Cell[50270, 1894, 88, 2, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[50395, 1901, 159, 5, 51, "Input"], Cell[50557, 1908, 96, 2, 49, "Output"] }, Open ]], Cell[50668, 1913, 190, 7, 37, "SmallText", Evaluatable->False], Cell[50861, 1922, 54, 1, 51, "Input"], Cell[CellGroupData[{ Cell[50940, 1927, 383, 5, 108, "Input"], Cell[51326, 1934, 26346, 734, 145, 10356, 532, "GraphicsData", "PostScript", \ "Graphics"], Cell[77675, 2670, 130, 3, 49, "Output"] }, Open ]] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)