(*^ ::[paletteColors = 128; showRuler; currentKernel; fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e8, 24, "Times"; ; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e6, 18, "Times"; ; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, L1, e6, 14, "Times"; ; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, L1, a20, 18, "Times"; ; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, L1, a15, 14, "Times"; ; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, L1, a12, 12, "Times"; ; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12, "Times"; ; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L-5, 12, "Courier"; ; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; ; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R65535, L-5, 12, "Courier"; ; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; ; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B65535, L-5, 12, "Courier"; ; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, L1, 12, "Courier"; ; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, L1, 10, "Geneva"; ; fontset = header, inactive, noKeepOnOnePage, preserveAspect, M7, L1, 12, "Times"; ; fontset = Left Header, inactive, 12, "Times"; ; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7, L1, 12, "Times"; ; fontset = Left Footer, inactive, 12, "Times"; ; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12, "Times"; ; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12, "Times"; ; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12, "Times"; ; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12, "Times"; ; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12, "Times"; ; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12, "Times"; ; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12, "Times"; ;] :[font = input; inactive; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; ] from Exploring Neuron Models and Neural Networks with Mathematicaª © Jacques J. Vidal, 1990 ;[s] 5:0,1;4,2;54,1;65,2;92,0;93,-1; 3:1,10,8,Courier,1,10,0,0,0;2,12,10,New York,3,10,0,0,0;2,12,10,New York,1,10,0,0,0; :[font = subsection; inactive; noPageBreakInGroup; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; startGroup; ] OUTER LAYER OF MAMMALIAN RETINA Interaction between Photoreceptors and Horizontal Cells :[font = subsubsection; inactive; noPageBreakInGroup; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; startGroup; ] References for this Notebook :[font = text; inactive; noPageBreakInGroup; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; ] J. E. Dowling, The Retina: An Approachable part of the Brain, The Belknap Press, Cambridge, MA, 1987. :[font = text; inactive; noPageBreakInGroup; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; endGroup; ] T. Yagi, Y. Funahashi, and F. Ariki, "Dynamic Model of a Dual Layer Neural Network for Vertebrate Retina," in IJCNN89 Proceedings, Washington DC, June 1989. :[font = subsubsection; inactive; noPageBreak; noPageBreakInGroup; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; startGroup; ] Anatomy and Physiology :[font = input; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; ] :[font = input; output; dontPreserveAspect; rightWrapOffset = 414; pictureLeft = 82; pictureWidth = 263; pictureHeight = 106; fontSize = 10; fontName = "New York"; startGroup; ] Show[Graphics[{ Thickness[0.003802], Text[FontForm["Receptors", {"", 12}], {0.821293, 0.262357}, {-1, -1}], Text[FontForm["Synapses", {"", 12}], {0.821293, 0.102662}, {-1, -1}], Text[FontForm["Horizontal ", {"", 12}], {0.821293, 0.041825}, {-1, -1}], Text[FontForm["Cells", {"", 12}], {0.821293, 0.003802}, {-1, -1}], Polygon[{ {0.393536, 0.173004}, {0.355513, 0.161597}, {0.355513, 0.169202}, {0.355513, 0.180608}, {0.393536, 0.173004}}], Line[{{0.180608, 0.150190}, {0.355513, 0.169202}}], Text[FontForm["Gap ", {"", 12}], {0.007605, 0.148289}, {-1, -1}], Text[FontForm["Junctions", {"", 12}], {0.007605, 0.110266}, {-1, -1}], Line[{{0.591255, 0.173004}, {0.617871, 0.370722}}], Line[{{0.777567, 0.173004}, {0.750951, 0.370722}}], Line[{{0.617871, 0.150190}, {0.750951, 0.150190}}], Line[{{0.644487, 0.397338}, {0.724335, 0.397338}}], Circle[{0.752852, 0.176806}, {0.026616, 0.024715}, {270.0 Degree, 0.0 Degree}], Circle[{0.619772, 0.176806}, {0.026616, 0.024715}, {180.0 Degree, 270.0 Degree}], Circle[{0.726236, 0.374525}, {0.026616, 0.024715}, {0.0 Degree, 90.0 Degree}], Circle[{0.646388, 0.374525}, {0.026616, 0.024715}, {90.0 Degree, 180.0 Degree}], GrayLevel[1.00000], Polygon[{ {0.591255, 0.001901}, {0.777567, 0.001901}, {0.777567, 0.100761}, {0.591255, 0.100761}, {0.591255, 0.001901}}], GrayLevel[0.00000], Line[{ {0.591255, 0.001901}, {0.777567, 0.001901}, {0.777567, 0.100761}, {0.591255, 0.100761}, {0.591255, 0.001901}}], Line[{{0.724335, 0.138783}, {0.724335, 0.100761}}], GrayLevel[1.00000], Disk[{0.722434, 0.144487}, {0.005703, 0.005703}], GrayLevel[0.00000], Circle[{0.722434, 0.144487}, {0.005703, 0.005703}], Line[{{0.644487, 0.150190}, {0.644487, 0.112167}}], Disk[{0.642586, 0.106464}, {0.005703, 0.005703}], Circle[{0.642586, 0.106464}, {0.005703, 0.005703}], Line[{{0.404943, 0.173004}, {0.431559, 0.370722}}], Line[{{0.591255, 0.173004}, {0.564639, 0.370722}}], Line[{{0.431559, 0.150190}, {0.564639, 0.150190}}], Line[{{0.458175, 0.397338}, {0.538023, 0.397338}}], Circle[{0.566540, 0.176806}, {0.026616, 0.024715}, {270.0 Degree, 0.0 Degree}], Circle[{0.433460, 0.176806}, {0.026616, 0.024715}, {180.0 Degree, 270.0 Degree}], Circle[{0.539924, 0.374525}, {0.026616, 0.024715}, {0.0 Degree, 90.0 Degree}], Circle[{0.460076, 0.374525}, {0.026616, 0.024715}, {90.0 Degree, 180.0 Degree}], GrayLevel[1.00000], Polygon[{ {0.404943, 0.001901}, {0.591255, 0.001901}, {0.591255, 0.100761}, {0.404943, 0.100761}, {0.404943, 0.001901}}], GrayLevel[0.00000], Line[{ {0.404943, 0.001901}, {0.591255, 0.001901}, {0.591255, 0.100761}, {0.404943, 0.100761}, {0.404943, 0.001901}}], Line[{{0.538023, 0.138783}, {0.538023, 0.100761}}], GrayLevel[1.00000], Disk[{0.536122, 0.144487}, {0.005703, 0.005703}], GrayLevel[0.00000], Circle[{0.536122, 0.144487}, {0.005703, 0.005703}], Line[{{0.458175, 0.150190}, {0.458175, 0.112167}}], Disk[{0.456274, 0.106464}, {0.005703, 0.005703}], Circle[{0.456274, 0.106464}, {0.005703, 0.005703}], Line[{{0.218631, 0.173004}, {0.245247, 0.370722}}], Line[{{0.404943, 0.173004}, {0.378327, 0.370722}}], Line[{{0.245247, 0.150190}, {0.378327, 0.150190}}], Line[{{0.271863, 0.397338}, {0.351711, 0.397338}}], Circle[{0.380228, 0.176806}, {0.026616, 0.024715}, {270.0 Degree, 0.0 Degree}], Circle[{0.247148, 0.176806}, {0.026616, 0.024715}, {180.0 Degree, 270.0 Degree}], Circle[{0.353612, 0.374525}, {0.026616, 0.024715}, {0.0 Degree, 90.0 Degree}], Circle[{0.273764, 0.374525}, {0.026616, 0.024715}, {90.0 Degree, 180.0 Degree}], GrayLevel[1.00000], Polygon[{ {0.218631, 0.001901}, {0.404943, 0.001901}, {0.404943, 0.100761}, {0.218631, 0.100761}, {0.218631, 0.001901}}], GrayLevel[0.00000], Line[{ {0.218631, 0.001901}, {0.404943, 0.001901}, {0.404943, 0.100761}, {0.218631, 0.100761}, {0.218631, 0.001901}}], Line[{{0.351711, 0.138783}, {0.351711, 0.100761}}], GrayLevel[1.00000], Disk[{0.349810, 0.144487}, {0.005703, 0.005703}], GrayLevel[0.00000], Circle[{0.349810, 0.144487}, {0.005703, 0.005703}], Line[{{0.271863, 0.150190}, {0.271863, 0.112167}}], Disk[{0.269962, 0.106464}, {0.005703, 0.005703}], Circle[{0.269962, 0.106464}, {0.005703, 0.005703}], Polygon[{ {0.393536, 0.051331}, {0.355513, 0.055133}, {0.355513, 0.062738}, {0.359316, 0.070342}, {0.393536, 0.051331}}], Line[{{0.180608, 0.123574}, {0.355513, 0.062738}}] }], AspectRatio->0.40304, PlotRange->{{0.00000, 1.00000}, {0.00000, 0.40304}}] ;[o] -Graphics- :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 113; ] %! %%Creator: Mathematica %%AspectRatio: 0.40304 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0 1 0 1 [ [ 0 0 0 0 ] [ 1 0.40304 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave grestore 0 0 moveto 1 0 lineto 1 0.40304 lineto 0 0.40304 lineto closepath clip newpath gsave gsave / findfont 12 scalefont setfont [(Receptors)] 0.82129 0.26236 -1 -1 Mshowa grestore gsave / findfont 12 scalefont setfont [(Synapses)] 0.82129 0.10266 -1 -1 Mshowa grestore gsave / findfont 12 scalefont setfont [(Horizontal )] 0.82129 0.04183 -1 -1 Mshowa grestore gsave / findfont 12 scalefont setfont [(Cells)] 0.82129 0.0038 -1 -1 Mshowa grestore 0.39354 0.173 moveto 0.35551 0.1616 lineto 0.35551 0.1692 lineto 0.35551 0.18061 lineto 0.39354 0.173 lineto fill 0.0038 setlinewidth 0.18061 0.15019 moveto 0.35551 0.1692 lineto stroke gsave / findfont 12 scalefont setfont [(Gap )] 0.00761 0.14829 -1 -1 Mshowa grestore gsave / findfont 12 scalefont setfont [(Junctions)] 0.00761 0.11027 -1 -1 Mshowa grestore 0.59126 0.173 moveto 0.61787 0.37072 lineto stroke 0.77757 0.173 moveto 0.75095 0.37072 lineto stroke 0.61787 0.15019 moveto 0.75095 0.15019 lineto stroke 0.64449 0.39734 moveto 0.72434 0.39734 lineto stroke newpath matrix currentmatrix 0.026616 0.024715 scale 23.2856 7.15396 1 180 270 arc setmatrix stroke newpath matrix currentmatrix 0.026616 0.024715 scale 27.2858 15.154 1 0 90 arc setmatrix stroke newpath matrix currentmatrix 0.026616 0.024715 scale 24.2858 15.154 1 90 180 arc setmatrix stroke 1 setgray 0.59126 0.0019 moveto 0.77757 0.0019 lineto 0.77757 0.10076 lineto 0.59126 0.10076 lineto 0.59126 0.0019 lineto fill 0 setgray 0.59126 0.0019 moveto 0.77757 0.0019 lineto 0.77757 0.10076 lineto 0.59126 0.10076 lineto 0.59126 0.0019 lineto stroke 0.72434 0.13878 moveto 0.72434 0.10076 lineto stroke 1 setgray 0.72243 0.14449 moveto 0.72243 0.14449 0.0057 0 365.73 arc fill 0 setgray newpath 0.72243 0.14449 0.0057 0 365.73 arc stroke 0.64449 0.15019 moveto 0.64449 0.11217 lineto stroke 0.64259 0.10646 moveto 0.64259 0.10646 0.0057 0 365.73 arc fill newpath 0.64259 0.10646 0.0057 0 365.73 arc stroke 0.40494 0.173 moveto 0.43156 0.37072 lineto stroke 0.59126 0.173 moveto 0.56464 0.37072 lineto stroke 0.43156 0.15019 moveto 0.56464 0.15019 lineto stroke 0.45818 0.39734 moveto 0.53802 0.39734 lineto stroke newpath matrix currentmatrix 0.026616 0.024715 scale 16.2857 7.15396 1 180 270 arc setmatrix stroke newpath matrix currentmatrix 0.026616 0.024715 scale 20.2855 15.154 1 0 90 arc setmatrix stroke newpath matrix currentmatrix 0.026616 0.024715 scale 17.2858 15.154 1 90 180 arc setmatrix stroke 1 setgray 0.40494 0.0019 moveto 0.59126 0.0019 lineto 0.59126 0.10076 lineto 0.40494 0.10076 lineto 0.40494 0.0019 lineto fill 0 setgray 0.40494 0.0019 moveto 0.59126 0.0019 lineto 0.59126 0.10076 lineto 0.40494 0.10076 lineto 0.40494 0.0019 lineto stroke 0.53802 0.13878 moveto 0.53802 0.10076 lineto stroke 1 setgray 0.53612 0.14449 moveto 0.53612 0.14449 0.0057 0 365.73 arc fill 0 setgray newpath 0.53612 0.14449 0.0057 0 365.73 arc stroke 0.45818 0.15019 moveto 0.45818 0.11217 lineto stroke 0.45627 0.10646 moveto 0.45627 0.10646 0.0057 0 365.73 arc fill newpath 0.45627 0.10646 0.0057 0 365.73 arc stroke 0.21863 0.173 moveto 0.24525 0.37072 lineto stroke 0.40494 0.173 moveto 0.37833 0.37072 lineto stroke 0.24525 0.15019 moveto 0.37833 0.15019 lineto stroke 0.27186 0.39734 moveto 0.35171 0.39734 lineto stroke newpath matrix currentmatrix 0.026616 0.024715 scale 9.28577 7.15396 1 180 270 arc setmatrix stroke newpath matrix currentmatrix 0.026616 0.024715 scale 13.2856 15.154 1 0 90 arc setmatrix stroke newpath matrix currentmatrix 0.026616 0.024715 scale 10.2855 15.154 1 90 180 arc setmatrix stroke 1 setgray 0.21863 0.0019 moveto 0.40494 0.0019 lineto 0.40494 0.10076 lineto 0.21863 0.10076 lineto 0.21863 0.0019 lineto fill 0 setgray 0.21863 0.0019 moveto 0.40494 0.0019 lineto 0.40494 0.10076 lineto 0.21863 0.10076 lineto 0.21863 0.0019 lineto stroke 0.35171 0.13878 moveto 0.35171 0.10076 lineto stroke 1 setgray 0.34981 0.14449 moveto 0.34981 0.14449 0.0057 0 365.73 arc fill 0 setgray newpath 0.34981 0.14449 0.0057 0 365.73 arc stroke 0.27186 0.15019 moveto 0.27186 0.11217 lineto stroke 0.26996 0.10646 moveto 0.26996 0.10646 0.0057 0 365.73 arc fill newpath 0.26996 0.10646 0.0057 0 365.73 arc stroke 0.39354 0.05133 moveto 0.35551 0.05513 lineto 0.35551 0.06274 lineto 0.35932 0.07034 lineto 0.39354 0.05133 lineto fill 0.18061 0.12357 moveto 0.35551 0.06274 lineto stroke grestore % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; ] The Unformatted text for this cell was not generated. Use options in the Actions Preferences dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = text; inactive; noPageBreakInGroup; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; endGroup; ] The figure illustrates the interactions between cells in the outer plexiform layer of the retina. Retinal photoreceptors transduce incident light into transmembrane currents. The receptors are electrically coupled to their neighbours by simple electrical contact or gap junctions. The receptors form a plexus within which the induced current flows diffuse. The horizontal cells make up the next layer of the retina. They also form a syncytium, allowing electrical signals to diffuse from cell to cell. A sustained light input will inject steady currents into the receptor array at the location of and in proportion to the amount of light impinging on each receptor cell. The two layers interact "vertically" and through electrochemical synapses (rather than gap junctions). The receptors supply excitatory inputs to the horizontal cells. The latter provide negative feedback. They inhibit the receptors. This notebook only discusses the two outer layers of the five layered retina. The output of the horizontal cells connects to the next stage which consists of bipolar cells. Receptors, horizontal and bipolar cells then interact through the so-called triad synapses and provide input to the succeeding layers of amacrine and ganglion cells. Finally, the axons of the ganglion cells form the optic nerve. The present model is confined to the interaction of the receptor and horizontal cells. It illustrates the signal enhancement produced by the lateral inhibition resulting from the feedback between the two layers combined with the diffusion spread through the gap junctions. ;[s] 16:0,1;99,2;121,1;267,2;279,1;365,2;381,1;1073,2;1080,1;1164,2;1179,1;1225,2;1233,1;1238,2;1246,1;1592,0;1593,-1; 3:1,13,9,Times,0,12,0,0,0;8,13,10,New York,0,10,0,0,0;7,13,10,New York,2,10,0,0,0; :[font = subsubsection; inactive; noPageBreakInGroup; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; startGroup; ] The Electrical Network Model :[font = text; inactive; noPageBreakInGroup; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; ] The diffusion plexi are modelled as discrete networks of conductances, representing gap junctions and local transmembrane conductance. The synaptic interactions form mutual current injections, of the appropriate polarity, between corresponding nodes in the two resistance networks. There are only conductances in this steady-state model, proposed by T. Yagi et al.which assumes a time invariant, (but spatially distributed) input to the photoreceptors. A more complete model would involve capacitances and dynamic membrane mechanisms to accomodate time-varying inputs. Two interacting networks, S1 and S2, respectively represent the receptor and horizontal cell syncytia. The vectors V and W represent membrane voltages in the syncytia S1 and S2, respectively. The circuits for S1 and S2, are shown above: the conductances g1 and h1 model the gap junctions, while g2 and h2 represent the membrane conductance. A voltage dependent conductance (zWk) represent the coupling from S1 to S2, and yWk similarly models the back-coupling from S2 to S1. The photocurrents are applied at S1 and are represented by the vector Ip. :[font = input; PICT; formatAsPICT; output; inactive; wordwrap; thinLines; dontPreserveAspect; rightWrapOffset = 414; pictureLeft = 87; pictureWidth = 272; pictureHeight = 228; fontSize = 10; fontName = "New York"; endGroup; pictureID = 12150; ] :[font = subsubsection; inactive; noPageBreak; noPageBreakInGroup; preserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; startGroup; ] Prolog :[font = input; initialization; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; endGroup; ] *) Off[General::spell]; Off[General::spell1]; (* :[font = subsubsection; inactive; pageBreak; noPageBreakInGroup; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; startGroup; ] Establish Finite Difference Equations for a One Dimensional "Retina" :[font = text; inactive; noPageBreakInGroup; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; ] We first choose the number of cell pairs in the simulation: :[font = input; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; ] pMax=60; :[font = text; inactive; noPageBreakInGroup; preserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; ] and create named arrays to store the cell membrane potentials :[font = input; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; ] receptors=Array[v ,pMax]; horizontals = Array[w,pMax]; :[font = input; wordwrap; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; ] Clear[ g1 ,g2 , h1, h2, y, z,in]; :[font = text; inactive; noPageBreakInGroup; preserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; ] To avoid dealing with special end conditions in forming the finite differences, we assume that the cells are arranged in a loop (e.g., a circle) so that the first cells in the array are neighbors of the lasts. Then simply "rotating" the arrays will give us access to the neighbors. :[font = input; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; ] receptorOutFlows := -g1 ( RotateLeft[receptors] -2 receptors + RotateRight[receptors] ) +g2 receptors; :[font = input; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; ] receptorInputs := imageInputs - y horizontals; :[font = text; inactive; noPageBreakInGroup; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; ] Similarly, for the horizontalontal cell layers: ;[s] 2:0,1;47,0;48,-1; 2:1,16,12,Chicago,0,12,0,0,0;1,13,10,New York,0,10,0,0,0; :[font = input; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; ] horizontalOutFlows := -h1 ( RotateLeft[horizontals] -2 horizontals +RotateRight[horizontals] ) +h2 horizontals; :[font = input; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; ] horizontalInputs := - z receptors; :[font = text; inactive; noPageBreakInGroup; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; ] We apply Kirchoff's law to each array to produce two sets of simultaneous equations for the potentials; OutFlows and inputs must cancel out in each array: :[font = input; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; ] receptorEquations:=Map[Equal[#,0]&, Apply[Plus,{receptorOutFlows,receptorInputs}]] :[font = input; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; endGroup; ] horizontalEquations:=Map[Equal[#,0]&, Apply[Plus,{horizontalOutFlows,horizontalInputs}]] :[font = subsubsection; inactive; noPageBreakInGroup; preserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; startGroup; ] Define "getSolution" (solve the network equations) ;[s] 3:0,1;6,2;51,0;52,-1; 3:1,16,12,New York,1,12,0,0,0;1,13,10,New York,3,10,0,0,0;1,13,10,New York,1,10,0,0,0; :[font = text; inactive; noPageBreakInGroup; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; ] To get numerical solutions we need to assign boundary conditons, specify the receptor input and allocate numerical values to the parameters; We set the potentials to zero at both ends of the array. To avoid dealing with the boundary, we may let the cells wrap around (as if in a circle) and confining non-zero inputs to the center (retinal) region. :[font = text; inactive; noPageBreakInGroup; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; ] We calculate the solutions rPotentials and hPotentials for the voltages :[font = input; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; endGroup; ] getSolution:=(solution=Flatten[Solve[ Join[ receptorEquations, horizontalEquations], Join[receptors,horizontals]]]; rPotentials = ReplaceAll[receptors,solution]; hPotentials = ReplaceAll[horizontals,solution];) ;[s] 3:0,1;51,2;227,0;228,-1; 3:1,12,10,Courier,1,12,0,0,0;1,12,10,New York,1,10,0,0,0;1,11,9,Courier,1,10,0,0,0; :[font = subsubsection; inactive; noPageBreakInGroup; preserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; startGroup; ] Define showPlots ;[s] 3:0,1;6,2;16,0;17,-1; 3:1,16,12,New York,1,12,0,0,0;1,13,10,New York,3,10,0,0,0;1,13,10,New York,1,10,0,0,0; :[font = text; inactive; noPageBreakInGroup; preserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; ] We want to plot the input and the receptor potentials induced by the network action. :[font = input; preserveAspect; fontSize = 10; ] SetOptions[ ListPlot, PlotJoined -> True, Ticks -> {Automatic, Automatic}, AxesOrigin->{0,0},PlotRange->{-4,4}]; ;[s] 2:0,2;113,1;114,-1; 3:0,12,10,Courier,1,12,0,0,0;1,15,12,New York,1,12,0,0,0;1,12,10,New York,1,10,0,0,0; :[font = input; preserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; endGroup; ] showPlots:=Show[ {ListPlot[rPotentials,PlotStyle->{Thickness[0.01]}], ListPlot[imageInputs]}]; :[font = subsubsection; inactive; noPageBreakInGroup; preserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; startGroup; ] Choose Numerical Values for the Model Parameters :[font = text; inactive; noPageBreakInGroup; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; ] We pick approximate parameters consistent with the physiological evidence. :[font = input; wordwrap; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; endGroup; ] g1 = 1.0; g2 = 0.01; h1 = 60.0; h2 = 0.01; y = -1.0; z=1.0; :[font = subsubsection; inactive; noPageBreakInGroup; preserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; startGroup; ] Case 1: One Point of Light (Spatial Impulse response) :[font = text; inactive; noPageBreakInGroup; preserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; startGroup; ] Define the input as a list of pMax elements with value zero with the exception of the center element which gets a finite value. ;[s] 3:0,1;16,2;127,0;128,-1; 3:1,16,12,New York,0,12,0,0,0;1,13,10,New York,1,10,0,0,0;1,13,10,New York,0,10,0,0,0; :[font = input; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; endGroup; ] imageInputs=Table[0,{pMax}]; imageInputs[[ Floor[pMax/2] ]]=1; :[font = text; inactive; noPageBreakInGroup; preserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; startGroup; ] Compute and Plot the Solution :[font = input; preserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; ] getSolution; :[font = text; inactive; noPageBreakInGroup; preserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; ] Display Plots showing the inputted "image" and the resulting membrane potentials: :[font = input; preserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; startGroup; ] showPlots; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; rightWrapOffset = 414; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; fontSize = 10; fontName = "New York"; endGroup; endGroup; ] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.015873 0.309017 0.0772542 [ [(10)] 0.18254 0.30902 0 2 Msboxa [(20)] 0.34127 0.30902 0 2 Msboxa [(30)] 0.5 0.30902 0 2 Msboxa [(40)] 0.65873 0.30902 0 2 Msboxa [(50)] 0.81746 0.30902 0 2 Msboxa [(60)] 0.97619 0.30902 0 2 Msboxa [(-4)] 0.01131 0 1 0 Msboxa [(-3)] 0.01131 0.07725 1 0 Msboxa [(-2)] 0.01131 0.15451 1 0 Msboxa [(-1)] 0.01131 0.23176 1 0 Msboxa [(1)] 0.01131 0.38627 1 0 Msboxa [(2)] 0.01131 0.46353 1 0 Msboxa [(3)] 0.01131 0.54078 1 0 Msboxa [(4)] 0.01131 0.61803 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 0.61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave gsave 0.002 setlinewidth 0.18254 0.30902 moveto 0.18254 0.31527 lineto stroke grestore [(10)] 0.18254 0.30902 0 2 Mshowa gsave 0.002 setlinewidth 0.34127 0.30902 moveto 0.34127 0.31527 lineto stroke grestore [(20)] 0.34127 0.30902 0 2 Mshowa gsave 0.002 setlinewidth 0.5 0.30902 moveto 0.5 0.31527 lineto stroke grestore [(30)] 0.5 0.30902 0 2 Mshowa gsave 0.002 setlinewidth 0.65873 0.30902 moveto 0.65873 0.31527 lineto stroke grestore [(40)] 0.65873 0.30902 0 2 Mshowa gsave 0.002 setlinewidth 0.81746 0.30902 moveto 0.81746 0.31527 lineto stroke grestore [(50)] 0.81746 0.30902 0 2 Mshowa gsave 0.002 setlinewidth 0.97619 0.30902 moveto 0.97619 0.31527 lineto stroke grestore [(60)] 0.97619 0.30902 0 2 Mshowa gsave 0.001 setlinewidth 0.05556 0.30902 moveto 0.05556 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.0873 0.30902 moveto 0.0873 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.11905 0.30902 moveto 0.11905 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.15079 0.30902 moveto 0.15079 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.21429 0.30902 moveto 0.21429 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.24603 0.30902 moveto 0.24603 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.27778 0.30902 moveto 0.27778 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.30952 0.30902 moveto 0.30952 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.37302 0.30902 moveto 0.37302 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.40476 0.30902 moveto 0.40476 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.43651 0.30902 moveto 0.43651 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.46825 0.30902 moveto 0.46825 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.53175 0.30902 moveto 0.53175 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.56349 0.30902 moveto 0.56349 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.59524 0.30902 moveto 0.59524 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.62698 0.30902 moveto 0.62698 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.69048 0.30902 moveto 0.69048 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.72222 0.30902 moveto 0.72222 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.75397 0.30902 moveto 0.75397 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.78571 0.30902 moveto 0.78571 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.84921 0.30902 moveto 0.84921 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.88095 0.30902 moveto 0.88095 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.9127 0.30902 moveto 0.9127 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.94444 0.30902 moveto 0.94444 0.31277 lineto stroke grestore gsave 0.002 setlinewidth 0 0.30902 moveto 1 0.30902 lineto stroke grestore gsave 0.002 setlinewidth 0.02381 0 moveto 0.03006 0 lineto stroke grestore [(-4)] 0.01131 0 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.07725 moveto 0.03006 0.07725 lineto stroke grestore [(-3)] 0.01131 0.07725 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.15451 moveto 0.03006 0.15451 lineto stroke grestore [(-2)] 0.01131 0.15451 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.23176 moveto 0.03006 0.23176 lineto stroke grestore [(-1)] 0.01131 0.23176 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.38627 moveto 0.03006 0.38627 lineto stroke grestore [(1)] 0.01131 0.38627 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.46353 moveto 0.03006 0.46353 lineto stroke grestore [(2)] 0.01131 0.46353 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.54078 moveto 0.03006 0.54078 lineto stroke grestore [(3)] 0.01131 0.54078 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.61803 moveto 0.03006 0.61803 lineto stroke grestore [(4)] 0.01131 0.61803 1 0 Mshowa gsave 0.001 setlinewidth 0.02381 0.01545 moveto 0.02756 0.01545 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.0309 moveto 0.02756 0.0309 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.04635 moveto 0.02756 0.04635 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.0618 moveto 0.02756 0.0618 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.09271 moveto 0.02756 0.09271 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.10816 moveto 0.02756 0.10816 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.12361 moveto 0.02756 0.12361 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.13906 moveto 0.02756 0.13906 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.16996 moveto 0.02756 0.16996 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.18541 moveto 0.02756 0.18541 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.20086 moveto 0.02756 0.20086 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.21631 moveto 0.02756 0.21631 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.24721 moveto 0.02756 0.24721 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.26266 moveto 0.02756 0.26266 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.27812 moveto 0.02756 0.27812 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.29357 moveto 0.02756 0.29357 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.32447 moveto 0.02756 0.32447 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.33992 moveto 0.02756 0.33992 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.35537 moveto 0.02756 0.35537 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.37082 moveto 0.02756 0.37082 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.40172 moveto 0.02756 0.40172 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.41717 moveto 0.02756 0.41717 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.43262 moveto 0.02756 0.43262 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.44807 moveto 0.02756 0.44807 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.47898 moveto 0.02756 0.47898 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.49443 moveto 0.02756 0.49443 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.50988 moveto 0.02756 0.50988 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.52533 moveto 0.02756 0.52533 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.55623 moveto 0.02756 0.55623 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.57168 moveto 0.02756 0.57168 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.58713 moveto 0.02756 0.58713 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.60258 moveto 0.02756 0.60258 lineto stroke grestore gsave 0.002 setlinewidth 0.02381 0 moveto 0.02381 0.61803 lineto stroke grestore grestore 0 0 moveto 1 0 lineto 1 0.61803 lineto 0 0.61803 lineto closepath clip newpath gsave gsave gsave 0.01 setlinewidth 0.03968 0.30906 moveto 0.05556 0.30904 lineto 0.07143 0.30901 lineto 0.0873 0.30897 lineto 0.10317 0.30891 lineto 0.11905 0.30883 lineto 0.13492 0.30874 lineto 0.15079 0.30863 lineto 0.16667 0.30852 lineto 0.18254 0.30843 lineto 0.19841 0.30836 lineto 0.21429 0.30837 lineto 0.23016 0.30849 lineto 0.24603 0.30878 lineto 0.2619 0.3093 lineto 0.27778 0.31011 lineto 0.29365 0.31129 lineto 0.30952 0.31288 lineto 0.3254 0.31488 lineto 0.34127 0.31723 lineto 0.35714 0.31978 lineto 0.37302 0.32224 lineto 0.38889 0.32414 lineto 0.40476 0.32479 lineto 0.42063 0.32323 lineto 0.43651 0.31827 lineto 0.45238 0.30846 lineto 0.46825 0.2922 lineto 0.48413 0.2679 lineto 0.5 0.23425 lineto 0.51587 0.2679 lineto 0.53175 0.2922 lineto 0.54762 0.30846 lineto 0.56349 0.31827 lineto 0.57937 0.32323 lineto 0.59524 0.32479 lineto 0.61111 0.32414 lineto 0.62698 0.32224 lineto 0.64286 0.31978 lineto 0.65873 0.31723 lineto 0.6746 0.31488 lineto 0.69048 0.31288 lineto 0.70635 0.31129 lineto 0.72222 0.31011 lineto 0.7381 0.3093 lineto 0.75397 0.30878 lineto 0.76984 0.30849 lineto 0.78571 0.30837 lineto 0.80159 0.30836 lineto 0.81746 0.30843 lineto Mistroke 0.83333 0.30852 lineto 0.84921 0.30863 lineto 0.86508 0.30874 lineto 0.88095 0.30883 lineto 0.89683 0.30891 lineto 0.9127 0.30897 lineto 0.92857 0.30901 lineto 0.94444 0.30904 lineto 0.96032 0.30906 lineto 0.97619 0.30907 lineto Mfstroke grestore grestore gsave 0.004 setlinewidth 0.03968 0.30902 moveto 0.05556 0.30902 lineto 0.07143 0.30902 lineto 0.0873 0.30902 lineto 0.10317 0.30902 lineto 0.11905 0.30902 lineto 0.13492 0.30902 lineto 0.15079 0.30902 lineto 0.16667 0.30902 lineto 0.18254 0.30902 lineto 0.19841 0.30902 lineto 0.21429 0.30902 lineto 0.23016 0.30902 lineto 0.24603 0.30902 lineto 0.2619 0.30902 lineto 0.27778 0.30902 lineto 0.29365 0.30902 lineto 0.30952 0.30902 lineto 0.3254 0.30902 lineto 0.34127 0.30902 lineto 0.35714 0.30902 lineto 0.37302 0.30902 lineto 0.38889 0.30902 lineto 0.40476 0.30902 lineto 0.42063 0.30902 lineto 0.43651 0.30902 lineto 0.45238 0.30902 lineto 0.46825 0.30902 lineto 0.48413 0.30902 lineto 0.5 0.38627 lineto 0.51587 0.30902 lineto 0.53175 0.30902 lineto 0.54762 0.30902 lineto 0.56349 0.30902 lineto 0.57937 0.30902 lineto 0.59524 0.30902 lineto 0.61111 0.30902 lineto 0.62698 0.30902 lineto 0.64286 0.30902 lineto 0.65873 0.30902 lineto 0.6746 0.30902 lineto 0.69048 0.30902 lineto 0.70635 0.30902 lineto 0.72222 0.30902 lineto 0.7381 0.30902 lineto 0.75397 0.30902 lineto 0.76984 0.30902 lineto 0.78571 0.30902 lineto 0.80159 0.30902 lineto 0.81746 0.30902 lineto Mistroke 0.83333 0.30902 lineto 0.84921 0.30902 lineto 0.86508 0.30902 lineto 0.88095 0.30902 lineto 0.89683 0.30902 lineto 0.9127 0.30902 lineto 0.92857 0.30902 lineto 0.94444 0.30902 lineto 0.96032 0.30902 lineto 0.97619 0.30902 lineto Mfstroke grestore grestore % End of Graphics MathPictureEnd :[font = text; inactive; noPageBreakInGroup; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; endGroup; ] The response shows the center-on/surround-off response to a punctual light source and reveals the contrast enhancement already present at this early stage of retinal processing. :[font = subsubsection; inactive; pageBreak; dontNoPageBreakBelow; noPageBreakInGroup; noKeepOnOnePage; preserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; startGroup; ] Case 2: Response to a Matched Illumination :[font = text; inactive; noPageBreakInGroup; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; ] Next we will give the inputted image the same profile as the the response just obtained: :[font = input; noKeepOnOnePage; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; ] imageInputs=-rPotentials; :[font = input; preserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; ] getSolution; :[font = input; preserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; startGroup; ] showPlots; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; fontSize = 10; endGroup; endGroup; ] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.015873 0.309017 0.0772542 [ [(10)] 0.18254 0.30902 0 2 Msboxa [(20)] 0.34127 0.30902 0 2 Msboxa [(30)] 0.5 0.30902 0 2 Msboxa [(40)] 0.65873 0.30902 0 2 Msboxa [(50)] 0.81746 0.30902 0 2 Msboxa [(60)] 0.97619 0.30902 0 2 Msboxa [(-4)] 0.01131 0 1 0 Msboxa [(-3)] 0.01131 0.07725 1 0 Msboxa [(-2)] 0.01131 0.15451 1 0 Msboxa [(-1)] 0.01131 0.23176 1 0 Msboxa [(1)] 0.01131 0.38627 1 0 Msboxa [(2)] 0.01131 0.46353 1 0 Msboxa [(3)] 0.01131 0.54078 1 0 Msboxa [(4)] 0.01131 0.61803 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 0.61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave gsave 0.002 setlinewidth 0.18254 0.30902 moveto 0.18254 0.31527 lineto stroke grestore [(10)] 0.18254 0.30902 0 2 Mshowa gsave 0.002 setlinewidth 0.34127 0.30902 moveto 0.34127 0.31527 lineto stroke grestore [(20)] 0.34127 0.30902 0 2 Mshowa gsave 0.002 setlinewidth 0.5 0.30902 moveto 0.5 0.31527 lineto stroke grestore [(30)] 0.5 0.30902 0 2 Mshowa gsave 0.002 setlinewidth 0.65873 0.30902 moveto 0.65873 0.31527 lineto stroke grestore [(40)] 0.65873 0.30902 0 2 Mshowa gsave 0.002 setlinewidth 0.81746 0.30902 moveto 0.81746 0.31527 lineto stroke grestore [(50)] 0.81746 0.30902 0 2 Mshowa gsave 0.002 setlinewidth 0.97619 0.30902 moveto 0.97619 0.31527 lineto stroke grestore [(60)] 0.97619 0.30902 0 2 Mshowa gsave 0.001 setlinewidth 0.05556 0.30902 moveto 0.05556 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.0873 0.30902 moveto 0.0873 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.11905 0.30902 moveto 0.11905 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.15079 0.30902 moveto 0.15079 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.21429 0.30902 moveto 0.21429 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.24603 0.30902 moveto 0.24603 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.27778 0.30902 moveto 0.27778 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.30952 0.30902 moveto 0.30952 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.37302 0.30902 moveto 0.37302 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.40476 0.30902 moveto 0.40476 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.43651 0.30902 moveto 0.43651 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.46825 0.30902 moveto 0.46825 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.53175 0.30902 moveto 0.53175 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.56349 0.30902 moveto 0.56349 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.59524 0.30902 moveto 0.59524 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.62698 0.30902 moveto 0.62698 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.69048 0.30902 moveto 0.69048 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.72222 0.30902 moveto 0.72222 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.75397 0.30902 moveto 0.75397 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.78571 0.30902 moveto 0.78571 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.84921 0.30902 moveto 0.84921 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.88095 0.30902 moveto 0.88095 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.9127 0.30902 moveto 0.9127 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.94444 0.30902 moveto 0.94444 0.31277 lineto stroke grestore gsave 0.002 setlinewidth 0 0.30902 moveto 1 0.30902 lineto stroke grestore gsave 0.002 setlinewidth 0.02381 0 moveto 0.03006 0 lineto stroke grestore [(-4)] 0.01131 0 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.07725 moveto 0.03006 0.07725 lineto stroke grestore [(-3)] 0.01131 0.07725 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.15451 moveto 0.03006 0.15451 lineto stroke grestore [(-2)] 0.01131 0.15451 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.23176 moveto 0.03006 0.23176 lineto stroke grestore [(-1)] 0.01131 0.23176 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.38627 moveto 0.03006 0.38627 lineto stroke grestore [(1)] 0.01131 0.38627 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.46353 moveto 0.03006 0.46353 lineto stroke grestore [(2)] 0.01131 0.46353 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.54078 moveto 0.03006 0.54078 lineto stroke grestore [(3)] 0.01131 0.54078 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.61803 moveto 0.03006 0.61803 lineto stroke grestore [(4)] 0.01131 0.61803 1 0 Mshowa gsave 0.001 setlinewidth 0.02381 0.01545 moveto 0.02756 0.01545 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.0309 moveto 0.02756 0.0309 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.04635 moveto 0.02756 0.04635 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.0618 moveto 0.02756 0.0618 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.09271 moveto 0.02756 0.09271 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.10816 moveto 0.02756 0.10816 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.12361 moveto 0.02756 0.12361 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.13906 moveto 0.02756 0.13906 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.16996 moveto 0.02756 0.16996 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.18541 moveto 0.02756 0.18541 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.20086 moveto 0.02756 0.20086 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.21631 moveto 0.02756 0.21631 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.24721 moveto 0.02756 0.24721 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.26266 moveto 0.02756 0.26266 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.27812 moveto 0.02756 0.27812 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.29357 moveto 0.02756 0.29357 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.32447 moveto 0.02756 0.32447 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.33992 moveto 0.02756 0.33992 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.35537 moveto 0.02756 0.35537 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.37082 moveto 0.02756 0.37082 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.40172 moveto 0.02756 0.40172 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.41717 moveto 0.02756 0.41717 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.43262 moveto 0.02756 0.43262 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.44807 moveto 0.02756 0.44807 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.47898 moveto 0.02756 0.47898 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.49443 moveto 0.02756 0.49443 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.50988 moveto 0.02756 0.50988 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.52533 moveto 0.02756 0.52533 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.55623 moveto 0.02756 0.55623 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.57168 moveto 0.02756 0.57168 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.58713 moveto 0.02756 0.58713 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.60258 moveto 0.02756 0.60258 lineto stroke grestore gsave 0.002 setlinewidth 0.02381 0 moveto 0.02381 0.61803 lineto stroke grestore grestore 0 0 moveto 1 0 lineto 1 0.61803 lineto 0 0.61803 lineto closepath clip newpath gsave gsave gsave 0.01 setlinewidth 0.03968 0.31085 moveto 0.05556 0.31068 lineto 0.07143 0.31037 lineto 0.0873 0.30988 lineto 0.10317 0.30914 lineto 0.11905 0.30812 lineto 0.13492 0.30678 lineto 0.15079 0.3051 lineto 0.16667 0.30315 lineto 0.18254 0.30102 lineto 0.19841 0.29892 lineto 0.21429 0.29716 lineto 0.23016 0.29611 lineto 0.24603 0.29627 lineto 0.2619 0.29817 lineto 0.27778 0.30232 lineto 0.29365 0.3091 lineto 0.30952 0.31867 lineto 0.3254 0.33075 lineto 0.34127 0.34451 lineto 0.35714 0.3584 lineto 0.37302 0.37005 lineto 0.38889 0.37641 lineto 0.40476 0.37392 lineto 0.42063 0.35918 lineto 0.43651 0.32993 lineto 0.45238 0.28643 lineto 0.46825 0.23348 lineto 0.48413 0.1827 lineto 0.5 0.15498 lineto 0.51587 0.1827 lineto 0.53175 0.23348 lineto 0.54762 0.28643 lineto 0.56349 0.32993 lineto 0.57937 0.35918 lineto 0.59524 0.37392 lineto 0.61111 0.37641 lineto 0.62698 0.37005 lineto 0.64286 0.3584 lineto 0.65873 0.34451 lineto 0.6746 0.33075 lineto 0.69048 0.31867 lineto 0.70635 0.3091 lineto 0.72222 0.30232 lineto 0.7381 0.29817 lineto 0.75397 0.29627 lineto 0.76984 0.29611 lineto 0.78571 0.29716 lineto 0.80159 0.29892 lineto 0.81746 0.30102 lineto Mistroke 0.83333 0.30315 lineto 0.84921 0.3051 lineto 0.86508 0.30678 lineto 0.88095 0.30812 lineto 0.89683 0.30914 lineto 0.9127 0.30988 lineto 0.92857 0.31037 lineto 0.94444 0.31068 lineto 0.96032 0.31085 lineto 0.97619 0.3109 lineto Mfstroke grestore grestore gsave 0.004 setlinewidth 0.03968 0.30897 moveto 0.05556 0.30899 lineto 0.07143 0.30902 lineto 0.0873 0.30906 lineto 0.10317 0.30912 lineto 0.11905 0.3092 lineto 0.13492 0.3093 lineto 0.15079 0.3094 lineto 0.16667 0.30951 lineto 0.18254 0.30961 lineto 0.19841 0.30967 lineto 0.21429 0.30966 lineto 0.23016 0.30954 lineto 0.24603 0.30926 lineto 0.2619 0.30874 lineto 0.27778 0.30792 lineto 0.29365 0.30674 lineto 0.30952 0.30516 lineto 0.3254 0.30316 lineto 0.34127 0.30081 lineto 0.35714 0.29825 lineto 0.37302 0.29579 lineto 0.38889 0.29389 lineto 0.40476 0.29325 lineto 0.42063 0.2948 lineto 0.43651 0.29977 lineto 0.45238 0.30958 lineto 0.46825 0.32584 lineto 0.48413 0.35014 lineto 0.5 0.38378 lineto 0.51587 0.35014 lineto 0.53175 0.32584 lineto 0.54762 0.30958 lineto 0.56349 0.29977 lineto 0.57937 0.2948 lineto 0.59524 0.29325 lineto 0.61111 0.29389 lineto 0.62698 0.29579 lineto 0.64286 0.29825 lineto 0.65873 0.30081 lineto 0.6746 0.30316 lineto 0.69048 0.30516 lineto 0.70635 0.30674 lineto 0.72222 0.30792 lineto 0.7381 0.30874 lineto 0.75397 0.30926 lineto 0.76984 0.30954 lineto 0.78571 0.30966 lineto 0.80159 0.30967 lineto 0.81746 0.30961 lineto Mistroke 0.83333 0.30951 lineto 0.84921 0.3094 lineto 0.86508 0.3093 lineto 0.88095 0.3092 lineto 0.89683 0.30912 lineto 0.9127 0.30906 lineto 0.92857 0.30902 lineto 0.94444 0.30899 lineto 0.96032 0.30897 lineto 0.97619 0.30897 lineto Mfstroke grestore grestore % End of Graphics MathPictureEnd :[font = subsubsection; inactive; dontNoPageBreakBelow; noPageBreakInGroup; noKeepOnOnePage; preserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; startGroup; ] Case 3: Response to an illumination edge (Step response) :[font = text; inactive; noPageBreakInGroup; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; ] We test the model to a step input: :[font = input; noKeepOnOnePage; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; ] imageInputs=Join[Table[0,{Floor[pMax/2]}], Table[1,{Ceiling[pMax/2]}]]; :[font = input; preserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; ] getSolution; :[font = input; preserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; startGroup; ] showPlots; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; rightWrapOffset = 414; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; fontSize = 10; fontName = "New York"; endGroup; endGroup; ] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.015873 0.309017 0.0772542 [ [(10)] 0.18254 0.30902 0 2 Msboxa [(20)] 0.34127 0.30902 0 2 Msboxa [(30)] 0.5 0.30902 0 2 Msboxa [(40)] 0.65873 0.30902 0 2 Msboxa [(50)] 0.81746 0.30902 0 2 Msboxa [(60)] 0.97619 0.30902 0 2 Msboxa [(-4)] 0.01131 0 1 0 Msboxa [(-3)] 0.01131 0.07725 1 0 Msboxa [(-2)] 0.01131 0.15451 1 0 Msboxa [(-1)] 0.01131 0.23176 1 0 Msboxa [(1)] 0.01131 0.38627 1 0 Msboxa [(2)] 0.01131 0.46353 1 0 Msboxa [(3)] 0.01131 0.54078 1 0 Msboxa [(4)] 0.01131 0.61803 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 0.61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave gsave 0.002 setlinewidth 0.18254 0.30902 moveto 0.18254 0.31527 lineto stroke grestore [(10)] 0.18254 0.30902 0 2 Mshowa gsave 0.002 setlinewidth 0.34127 0.30902 moveto 0.34127 0.31527 lineto stroke grestore [(20)] 0.34127 0.30902 0 2 Mshowa gsave 0.002 setlinewidth 0.5 0.30902 moveto 0.5 0.31527 lineto stroke grestore [(30)] 0.5 0.30902 0 2 Mshowa gsave 0.002 setlinewidth 0.65873 0.30902 moveto 0.65873 0.31527 lineto stroke grestore [(40)] 0.65873 0.30902 0 2 Mshowa gsave 0.002 setlinewidth 0.81746 0.30902 moveto 0.81746 0.31527 lineto stroke grestore [(50)] 0.81746 0.30902 0 2 Mshowa gsave 0.002 setlinewidth 0.97619 0.30902 moveto 0.97619 0.31527 lineto stroke grestore [(60)] 0.97619 0.30902 0 2 Mshowa gsave 0.001 setlinewidth 0.05556 0.30902 moveto 0.05556 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.0873 0.30902 moveto 0.0873 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.11905 0.30902 moveto 0.11905 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.15079 0.30902 moveto 0.15079 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.21429 0.30902 moveto 0.21429 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.24603 0.30902 moveto 0.24603 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.27778 0.30902 moveto 0.27778 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.30952 0.30902 moveto 0.30952 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.37302 0.30902 moveto 0.37302 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.40476 0.30902 moveto 0.40476 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.43651 0.30902 moveto 0.43651 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.46825 0.30902 moveto 0.46825 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.53175 0.30902 moveto 0.53175 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.56349 0.30902 moveto 0.56349 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.59524 0.30902 moveto 0.59524 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.62698 0.30902 moveto 0.62698 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.69048 0.30902 moveto 0.69048 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.72222 0.30902 moveto 0.72222 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.75397 0.30902 moveto 0.75397 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.78571 0.30902 moveto 0.78571 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.84921 0.30902 moveto 0.84921 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.88095 0.30902 moveto 0.88095 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.9127 0.30902 moveto 0.9127 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.94444 0.30902 moveto 0.94444 0.31277 lineto stroke grestore gsave 0.002 setlinewidth 0 0.30902 moveto 1 0.30902 lineto stroke grestore gsave 0.002 setlinewidth 0.02381 0 moveto 0.03006 0 lineto stroke grestore [(-4)] 0.01131 0 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.07725 moveto 0.03006 0.07725 lineto stroke grestore [(-3)] 0.01131 0.07725 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.15451 moveto 0.03006 0.15451 lineto stroke grestore [(-2)] 0.01131 0.15451 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.23176 moveto 0.03006 0.23176 lineto stroke grestore [(-1)] 0.01131 0.23176 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.38627 moveto 0.03006 0.38627 lineto stroke grestore [(1)] 0.01131 0.38627 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.46353 moveto 0.03006 0.46353 lineto stroke grestore [(2)] 0.01131 0.46353 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.54078 moveto 0.03006 0.54078 lineto stroke grestore [(3)] 0.01131 0.54078 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.61803 moveto 0.03006 0.61803 lineto stroke grestore [(4)] 0.01131 0.61803 1 0 Mshowa gsave 0.001 setlinewidth 0.02381 0.01545 moveto 0.02756 0.01545 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.0309 moveto 0.02756 0.0309 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.04635 moveto 0.02756 0.04635 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.0618 moveto 0.02756 0.0618 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.09271 moveto 0.02756 0.09271 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.10816 moveto 0.02756 0.10816 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.12361 moveto 0.02756 0.12361 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.13906 moveto 0.02756 0.13906 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.16996 moveto 0.02756 0.16996 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.18541 moveto 0.02756 0.18541 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.20086 moveto 0.02756 0.20086 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.21631 moveto 0.02756 0.21631 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.24721 moveto 0.02756 0.24721 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.26266 moveto 0.02756 0.26266 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.27812 moveto 0.02756 0.27812 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.29357 moveto 0.02756 0.29357 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.32447 moveto 0.02756 0.32447 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.33992 moveto 0.02756 0.33992 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.35537 moveto 0.02756 0.35537 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.37082 moveto 0.02756 0.37082 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.40172 moveto 0.02756 0.40172 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.41717 moveto 0.02756 0.41717 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.43262 moveto 0.02756 0.43262 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.44807 moveto 0.02756 0.44807 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.47898 moveto 0.02756 0.47898 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.49443 moveto 0.02756 0.49443 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.50988 moveto 0.02756 0.50988 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.52533 moveto 0.02756 0.52533 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.55623 moveto 0.02756 0.55623 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.57168 moveto 0.02756 0.57168 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.58713 moveto 0.02756 0.58713 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.60258 moveto 0.02756 0.60258 lineto stroke grestore gsave 0.002 setlinewidth 0.02381 0 moveto 0.02381 0.61803 lineto stroke grestore grestore 0 0 moveto 1 0 lineto 1 0.61803 lineto 0 0.61803 lineto closepath clip newpath gsave gsave gsave 0.01 setlinewidth 0.03968 0.34642 moveto 0.05556 0.38759 lineto 0.07143 0.40444 lineto 0.0873 0.405 lineto 0.10317 0.3957 lineto 0.11905 0.38137 lineto 0.13492 0.36542 lineto 0.15079 0.35001 lineto 0.16667 0.3364 lineto 0.18254 0.32514 lineto 0.19841 0.31634 lineto 0.21429 0.30983 lineto 0.23016 0.30532 lineto 0.24603 0.30252 lineto 0.2619 0.30118 lineto 0.27778 0.30118 lineto 0.29365 0.30252 lineto 0.30952 0.30532 lineto 0.3254 0.30983 lineto 0.34127 0.31634 lineto 0.35714 0.32514 lineto 0.37302 0.3364 lineto 0.38889 0.35001 lineto 0.40476 0.36542 lineto 0.42063 0.38137 lineto 0.43651 0.3957 lineto 0.45238 0.405 lineto 0.46825 0.40444 lineto 0.48413 0.38759 lineto 0.5 0.34642 lineto 0.51587 0.27161 lineto 0.53175 0.23045 lineto 0.54762 0.2136 lineto 0.56349 0.21304 lineto 0.57937 0.22234 lineto 0.59524 0.23666 lineto 0.61111 0.25261 lineto 0.62698 0.26802 lineto 0.64286 0.28163 lineto 0.65873 0.29289 lineto 0.6746 0.3017 lineto 0.69048 0.30821 lineto 0.70635 0.31271 lineto 0.72222 0.31551 lineto 0.7381 0.31685 lineto 0.75397 0.31685 lineto 0.76984 0.31551 lineto 0.78571 0.31271 lineto 0.80159 0.30821 lineto 0.81746 0.3017 lineto Mistroke 0.83333 0.29289 lineto 0.84921 0.28163 lineto 0.86508 0.26802 lineto 0.88095 0.25261 lineto 0.89683 0.23666 lineto 0.9127 0.22234 lineto 0.92857 0.21304 lineto 0.94444 0.2136 lineto 0.96032 0.23045 lineto 0.97619 0.27161 lineto Mfstroke grestore grestore gsave 0.004 setlinewidth 0.03968 0.30902 moveto 0.05556 0.30902 lineto 0.07143 0.30902 lineto 0.0873 0.30902 lineto 0.10317 0.30902 lineto 0.11905 0.30902 lineto 0.13492 0.30902 lineto 0.15079 0.30902 lineto 0.16667 0.30902 lineto 0.18254 0.30902 lineto 0.19841 0.30902 lineto 0.21429 0.30902 lineto 0.23016 0.30902 lineto 0.24603 0.30902 lineto 0.2619 0.30902 lineto 0.27778 0.30902 lineto 0.29365 0.30902 lineto 0.30952 0.30902 lineto 0.3254 0.30902 lineto 0.34127 0.30902 lineto 0.35714 0.30902 lineto 0.37302 0.30902 lineto 0.38889 0.30902 lineto 0.40476 0.30902 lineto 0.42063 0.30902 lineto 0.43651 0.30902 lineto 0.45238 0.30902 lineto 0.46825 0.30902 lineto 0.48413 0.30902 lineto 0.5 0.30902 lineto 0.51587 0.38627 lineto 0.53175 0.38627 lineto 0.54762 0.38627 lineto 0.56349 0.38627 lineto 0.57937 0.38627 lineto 0.59524 0.38627 lineto 0.61111 0.38627 lineto 0.62698 0.38627 lineto 0.64286 0.38627 lineto 0.65873 0.38627 lineto 0.6746 0.38627 lineto 0.69048 0.38627 lineto 0.70635 0.38627 lineto 0.72222 0.38627 lineto 0.7381 0.38627 lineto 0.75397 0.38627 lineto 0.76984 0.38627 lineto 0.78571 0.38627 lineto 0.80159 0.38627 lineto 0.81746 0.38627 lineto Mistroke 0.83333 0.38627 lineto 0.84921 0.38627 lineto 0.86508 0.38627 lineto 0.88095 0.38627 lineto 0.89683 0.38627 lineto 0.9127 0.38627 lineto 0.92857 0.38627 lineto 0.94444 0.38627 lineto 0.96032 0.38627 lineto 0.97619 0.38627 lineto Mfstroke grestore grestore % End of Graphics MathPictureEnd :[font = subsubsection; inactive; noPageBreakInGroup; preserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; startGroup; ] To Explore Further with this Notebook :[font = text; inactive; noPageBreakInGroup; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; ] Verify and comment oln the following propositions: :[font = text; inactive; noPageBreakInGroup; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; ] As the lateral conductance g1 in S1 is increased, the extent of both the excitatory central region and the inhibitory surround must increases. The larger conductance allows the photocurrents to spread further within S1 but bercause of the increased effective conductance, the voltages in S1 and S2 are reduced . There is an overall decrease of the response in both the excitatory and inhibitory regions of the receptive field. ;[s] 12:0,1;27,2;29,1;33,2;35,1;216,2;218,1;288,2;290,1;295,2;297,1;427,0;428,-1; 3:1,16,12,Chicago,0,12,0,0,0;6,13,10,New York,0,10,0,0,0;5,13,10,New York,1,10,0,0,0; :[font = text; inactive; noPageBreakInGroup; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; ] Increasing h1 enlarges the excitatory center as well as the inhibitory surround increases. This is acompanied by an enhanced response in both parts of the receptive field due to the higher conductance of S2, the inhibition is weakened. Nevertheless, injected into S2 now spread further out, increasing both the extent and the strength of the inhibitory surround. ;[s] 8:0,1;11,2;13,1;205,2;208,1;267,2;269,1;366,0;367,-1; 3:1,16,12,Chicago,0,12,0,0,0;4,13,10,New York,0,10,0,0,0;3,13,10,New York,1,10,0,0,0; :[font = text; inactive; noPageBreakInGroup; dontPreserveAspect; rightWrapOffset = 414; fontSize = 10; fontName = "New York"; endGroup; endGroup; ] Increasing y, the inhibitory feedback from S2, decreases the extent and strength of both parts of the receptive field. y opposes the effect of h1. Shows how the model behaves when g1, h1, y and z are varied; ;[s] 14:0,2;11,3;13,2;43,3;45,2;121,3;122,2;145,3;147,2;183,3;200,2;203,3;204,2;211,1;212,-1; 4:0,16,12,Chicago,0,12,0,0,0;1,13,9,Times,0,12,0,0,0;7,13,10,New York,0,10,0,0,0;6,13,10,New York,1,10,0,0,0; ^*)