(************** 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). 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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[ RowBox[{\(Off[General::spell1]\), " ", \( (*\ to\ turn\ off\ spell\ error\ messages\ *) \)}], FontSize->24], "\[IndentingNewLine]", StyleBox[\(Off[General::spell]\), FontSize->24]}], "Input"], 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[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["\<\ 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[BoxData[{ \(Rm = 5000\), "\n", \(Ri = 110\), "\n", \(Dia = 10\ 10^\(-4\)\), "\[IndentingNewLine]", \(DiaSoma = 70\/10\^4\), "\[IndentingNewLine]", \(Bjo = 0.0\)}], "Input", PageWidth->Infinity, FontSize->24], Cell[BoxData[ \(N[Lambda]\)], "Input", PageWidth->Infinity, FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ \(N[Lambda\ 10^4]\)], "Input"], Cell[BoxData[ \(\(\(N[Lambda]\)\(\[IndentingNewLine]\) \)\)], "Input"] }, 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[BoxData[ StyleBox[\(Lngth = Lambda\ 1.5\), FontSize->24]], "Input"], Cell[BoxData[ \(X1\)], "Input", PageWidth->Infinity, FontSize->24], Cell[BoxData[ \(B1\)], "Input", PageWidth->Infinity, FontSize->24], Cell[BoxData[ StyleBox["Ginf", FontSize->24]], "Input"], Cell[BoxData[ StyleBox["Gd", FontSize->24]], "Input"], Cell[BoxData[ StyleBox[\(N[Gs]\), FontSize->24]], "Input"], Cell[BoxData[ StyleBox[\(N[Gn]\), FontSize->24]], "Input"], Cell[BoxData[ StyleBox["Rn", FontSize->24]], "Input"], Cell[BoxData[ RowBox[{ StyleBox["Rn", FontSize->24], StyleBox["/", FontSize->24], RowBox[{ StyleBox["10", FontSize->24], StyleBox["^", FontSize->24], StyleBox["6", 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[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[BoxData[ StyleBox["Bjs", FontSize->24]], "Input"], Cell[BoxData[ StyleBox["Ginf", FontSize->24]], "Input"], Cell[BoxData[ StyleBox["B1o", FontSize->24]], "Input"], Cell[BoxData[ StyleBox["Gend", FontSize->24]], "Input"], Cell[BoxData[ StyleBox["Rend", FontSize->24]], "Input"], Cell[BoxData[ StyleBox[\(Rend/10^6\), FontSize->24]], "Input"], 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[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["Now calculate Lambda and the Bo factor", "Subsubsection", Evaluatable->False, FontSize->24], Cell[BoxData[{ \(Rm = 100000\), "\n", \(Ri = 110\), "\n", \(Bjo = 0\), "\[IndentingNewLine]", \(Dia = 1.0\/10\^4\), "\[IndentingNewLine]", \(DiaSoma = 25\/10\^4\), "\[IndentingNewLine]", \(Lambda\)}], "Input", PageWidth->Infinity, FontSize->24], Cell["\<\ \tThis is the length constant (Lambda) in cm, so multiply by 10^4 to get um \ or 1507.56 um\ \>", "Text"], Cell[BoxData[ \(Lngth = Lambda\/2\)], "Input", PageWidth->Infinity, FontSize->24], Cell[BoxData[ \(X1\)], "Input", PageWidth->Infinity, FontSize->24], Cell[BoxData[ \(B1\)], "Input", PageWidth->Infinity, FontSize->24], Cell[BoxData[ StyleBox["Ginf", FontSize->24]], "Input"], Cell[BoxData[ \(Gd\)], "Input", PageWidth->Infinity, FontSize->24], Cell[BoxData[ \(N[Gs]\)], "Input", PageWidth->Infinity, FontSize->24], Cell[BoxData[ \(N[Gn]\)], "Input", PageWidth->Infinity, FontSize->24], Cell["\<\ \t\tSo Gn is the sum of Gs (soma) and Gd (real value of the dendritic \ conductance at the soma)\ \>", "Text", FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ \(Rn\)], "Input", PageWidth->Infinity, FontSize->24], Cell[BoxData[ StyleBox[\(Rn = Rn/\(\(10^6\)\(\ \)\( (*\ convert\ to\ megohms\ *) \)\)\), FontSize->24]], "Input"] }, 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->24], 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["\<\ \t\tIn this case, we use Bjs to equal the Bjo when going in, but now we are \ going out\ \>", "Text"] }, Open ]], Cell[BoxData[ StyleBox[\(\(Ginf\)\(\ \)\( (*\ this\ remains\ the\ same\ *) \)\), FontSize->24]], "Input"], Cell[BoxData[ StyleBox[\(\(B1o\)\(\ \)\( (*\ this\ will\ be\ the\ B1\ factor\ for\ calculating\ the\ conductance\ \ at\ the\ distal\ end\ *) \)\), FontSize->24]], "Input"], Cell[BoxData[ StyleBox[\(\(Gend\)\(\ \)\( (*\ gives\ us\ the\ conductance\ at\ the\ end\ of\ the\ 0.5\ Lambda\ \ Process\ *) \)\), FontSize->24]], "Input"], Cell[CellGroupData[{ Cell[BoxData[ StyleBox["Rend", FontSize->24]], "Input"], Cell[BoxData[ RowBox[{ StyleBox[\(Rend/10^6\), FontSize->24], "\[IndentingNewLine]"}]], "Input"] }, Open ]], Cell[TextData[{ "\t\tHow does this compare to the resistance in the soma? 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