(*^ ::[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, 12, "Courier"; ; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, 12, "Courier"; ; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R65535, 12, "Courier"; ; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, 12, "Courier"; ; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B65535, 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, L1, 12, "Times"; ; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7, L1, 12, "Times"; ; fontset = Left Footer, inactive, center, L1, 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 = title; inactive; preserveAspect; ] Oculo-Motor Control :[font = subsubtitle; inactive; preserveAspect; left; ] includes data from a class project on saccades by Amir Doron andYoav Epstein :[font = section; inactive; Cclosed; preserveAspect; startGroup; ] Introduction :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 79; pictureWidth = 307; pictureHeight = 456; pictureID = 4260; ] :[font = text; inactive; preserveAspect; endGroup; ] There are two basic types of eye movements: smooth and saccadic. Smooth eye movement is utilized when the eye is tracking a moving object, such as a car or a pendulum. Saccadic eye movement is used when the eye needs to rapidly foviate on a new object . An example of a saccadic eye movement is the eyes jumping to an explosion that happens at the corner of a movie screen. Saccades that jump toward the middle of the face (the nose) are called medial saccades, while saccades that jump toward the ears are called lateral saccades. When the right eye performes a medial saccade, the left eye performs a lateral saccade (and vice-versa). Lateral and medial saccades are performed by different sets of neurons and muscles that function as a push-pull system. The steps involved in a saccade are as follows: 1. An object (target) is presented. 2. The CNS identifies the object, decides to foviate on it, and initiates a saccade. These steps cause a delay of 200-300 msec between the time the object is presented to the time the eyes start to move. 3. The eyes accelerate to a velocity of 500 degrees per second, and decelerate to bring the fovea accurately on to the target. The accuracy of the eye is astonishing. Despite the rapid speed with which the eye moves, saccades are usually completed without overshoot or oscilations. Saccades are the fastest and best controlled movements of which our body musculature is capable of [1]. In this project we model saccadic eye movement of a single left eye. The eye starts out looking straight ahead, and then jumps to a random point within its 120 degree field of vision. ;[s] 5:0,0;451,1;457,0;520,1;527,0;1644,-1; 2:3,13,9,Times,0,12,0,0,0;2,13,9,Times,2,12,0,0,0; :[font = section; inactive; preserveAspect; startGroup; ] Neurons :[font = subsection; inactive; noPageBreak; preserveAspect; startGroup; ] Definitions :[font = text; inactive; noPageBreak; preserveAspect; ] We model two kinds of neurons: linear and step. The output of a linear neuron varies linearly with its inputs. The output of a step neuron toggles between two values (i.e, its output is zero when the excitation is below the threshold, and some maximum value when the excitation exceeds the threshold. There are no in-between values). The ERR neuron is an example of a linear neuron, and the EBN is an example of a step neuron (see corresponding graphs below). ;[s] 13:0,0;32,1;38,0;43,1;47,0;66,1;72,0;130,1;134,0;343,1;346,0;396,1;399,0;465,-1; 2:7,13,9,Times,0,12,0,0,0;6,13,9,Times,2,12,0,0,0; :[font = input; noPageBreak; preserveAspect; endGroup; ] linearNeuron [i_,threshold_,min_,max_] := If [i > max,max,If [i <= threshold,min,i]] stepNeuron [i_,threshold_, min_,max_] := If [i < threshold,min,max] :[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ] Retinal Error Neuron :[font = text; inactive; noPageBreak; preserveAspect; ] The output of the retinal error neuron (ERR) is the difference between the current eye position and the desired eye position. ;[s] 3:0,0;18,1;31,0;126,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0; :[font = input; noPageBreak; preserveAspect; startGroup; ] errNeuron [i_] := linearNeuron [i,4,0,400] Plot [errNeuron[i],{i,-100,500}, PlotStyle->{{Thickness[0.01]}}] :[font 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Use options in the Actions Preferences dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = subsection; inactive; preserveAspect; startGroup; ] Excitory Burst Neuron :[font = text; inactive; noPageBreak; preserveAspect; ] The output of the excitory burst neron (EBN) toggles between 0 and 1000 spikes per second. 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Use options in the Actions Preferences dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = subsection; inactive; preserveAspect; startGroup; ] Tonic Neuron :[font = text; inactive; noPageBreak; preserveAspect; ] The output of the tonic neuron (TN) is proportional to the current eye position. It has a range between 0 and 400 spikes per second. A value of 200 indicates that the eye is fixated straight ahead. During a saccade, the tonic neuron changes its firing rate gradually between the pre- and post-saccadic rates. ;[s] 3:0,0;18,1;23,0;313,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0; :[font = input; noPageBreak; preserveAspect; startGroup; ] tnNeuron [i_] := linearNeuron [i,0,0,400] Plot [tnNeuron[i],{i,-100,500}, PlotStyle->{{Thickness[0.01]}}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; noPageBreak; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; ] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.18254 0.0015873 0.0147151 0.00147151 [ [(-100)] 0.02381 0.01472 0 2 Msboxa [(100)] 0.34127 0.01472 0 2 Msboxa [(200)] 0.5 0.01472 0 2 Msboxa [(300)] 0.65873 0.01472 0 2 Msboxa [(400)] 0.81746 0.01472 0 2 Msboxa [(500)] 0.97619 0.01472 0 2 Msboxa [(100)] 0.17004 0.16187 1 0 Msboxa [(200)] 0.17004 0.30902 1 0 Msboxa [(300)] 0.17004 0.45617 1 0 Msboxa [(400)] 0.17004 0.60332 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 0.02381 0.01472 moveto 0.02381 0.02097 lineto stroke grestore [(-100)] 0.02381 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.34127 0.01472 moveto 0.34127 0.02097 lineto stroke grestore [(100)] 0.34127 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.5 0.01472 moveto 0.5 0.02097 lineto stroke grestore [(200)] 0.5 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.65873 0.01472 moveto 0.65873 0.02097 lineto stroke grestore [(300)] 0.65873 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.81746 0.01472 moveto 0.81746 0.02097 lineto stroke grestore [(400)] 0.81746 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.97619 0.01472 moveto 0.97619 0.02097 lineto stroke grestore [(500)] 0.97619 0.01472 0 2 Mshowa gsave 0.001 setlinewidth 0.05556 0.01472 moveto 0.05556 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.0873 0.01472 moveto 0.0873 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.11905 0.01472 moveto 0.11905 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.15079 0.01472 moveto 0.15079 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.21429 0.01472 moveto 0.21429 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.24603 0.01472 moveto 0.24603 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.27778 0.01472 moveto 0.27778 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.30952 0.01472 moveto 0.30952 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.37302 0.01472 moveto 0.37302 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.40476 0.01472 moveto 0.40476 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.43651 0.01472 moveto 0.43651 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.46825 0.01472 moveto 0.46825 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.53175 0.01472 moveto 0.53175 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.56349 0.01472 moveto 0.56349 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.59524 0.01472 moveto 0.59524 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.62698 0.01472 moveto 0.62698 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.69048 0.01472 moveto 0.69048 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.72222 0.01472 moveto 0.72222 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.75397 0.01472 moveto 0.75397 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.78571 0.01472 moveto 0.78571 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.84921 0.01472 moveto 0.84921 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.88095 0.01472 moveto 0.88095 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.9127 0.01472 moveto 0.9127 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.94444 0.01472 moveto 0.94444 0.01847 lineto stroke grestore gsave 0.002 setlinewidth 0 0.01472 moveto 1 0.01472 lineto stroke grestore gsave 0.002 setlinewidth 0.18254 0.16187 moveto 0.18879 0.16187 lineto stroke grestore [(100)] 0.17004 0.16187 1 0 Mshowa gsave 0.002 setlinewidth 0.18254 0.30902 moveto 0.18879 0.30902 lineto stroke grestore [(200)] 0.17004 0.30902 1 0 Mshowa gsave 0.002 setlinewidth 0.18254 0.45617 moveto 0.18879 0.45617 lineto stroke grestore [(300)] 0.17004 0.45617 1 0 Mshowa gsave 0.002 setlinewidth 0.18254 0.60332 moveto 0.18879 0.60332 lineto stroke grestore [(400)] 0.17004 0.60332 1 0 Mshowa gsave 0.001 setlinewidth 0.18254 0.04415 moveto 0.18629 0.04415 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.07358 moveto 0.18629 0.07358 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.10301 moveto 0.18629 0.10301 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.13244 moveto 0.18629 0.13244 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.1913 moveto 0.18629 0.1913 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.22073 moveto 0.18629 0.22073 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.25016 moveto 0.18629 0.25016 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.27959 moveto 0.18629 0.27959 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.33845 moveto 0.18629 0.33845 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.36788 moveto 0.18629 0.36788 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.39731 moveto 0.18629 0.39731 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.42674 moveto 0.18629 0.42674 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.4856 moveto 0.18629 0.4856 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.51503 moveto 0.18629 0.51503 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.54446 moveto 0.18629 0.54446 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.57389 moveto 0.18629 0.57389 lineto stroke grestore gsave 0.002 setlinewidth 0.18254 0 moveto 0.18254 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 0.01 setlinewidth 0.02381 0.01472 moveto 0.06349 0.01472 lineto 0.10317 0.01472 lineto 0.14286 0.01472 lineto 0.1627 0.01472 lineto 0.17262 0.01472 lineto 0.17758 0.01472 lineto 0.18006 0.01472 lineto 0.1813 0.01472 lineto 0.18254 0.01472 lineto 0.18502 0.01701 lineto 0.1875 0.01931 lineto 0.19246 0.02391 lineto 0.20238 0.03311 lineto 0.22222 0.0515 lineto 0.2619 0.08829 lineto 0.30159 0.12508 lineto 0.34127 0.16187 lineto 0.38095 0.19865 lineto 0.42063 0.23544 lineto 0.46032 0.27223 lineto 0.5 0.30902 lineto 0.53968 0.3458 lineto 0.57937 0.38259 lineto 0.61905 0.41938 lineto 0.65873 0.45617 lineto 0.69841 0.49296 lineto 0.7381 0.52974 lineto 0.77778 0.56653 lineto 0.79762 0.58493 lineto 0.80754 0.59412 lineto 0.8125 0.59872 lineto 0.81498 0.60102 lineto 0.81622 0.60217 lineto 0.81746 0.60332 lineto 0.81994 0.60332 lineto 0.82242 0.60332 lineto 0.82738 0.60332 lineto 0.8373 0.60332 lineto 0.85714 0.60332 lineto 0.89683 0.60332 lineto 0.93651 0.60332 lineto 0.97619 0.60332 lineto stroke grestore grestore % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; 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 = subsection; inactive; preserveAspect; startGroup; ] Omnipause Neuron :[font = text; inactive; noPageBreak; preserveAspect; ] The output of the omnipause neuron (OPN) is 0 spikes per second during a saccade and 100 spikes per second any other time. When the OPN's output is not 0, saccades are inhibited. ;[s] 3:0,0;18,1;27,0;180,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0; :[font = input; noPageBreak; preserveAspect; startGroup; ] opnNeuron [i_] := stepNeuron [i,0,0,100] Plot [opnNeuron[i],{i,-100,500}, PlotStyle->{{Thickness[0.01]}}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; noPageBreak; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; ] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.18254 0.0015873 0.0147151 0.00588604 [ [(-100)] 0.02381 0.01472 0 2 Msboxa [(100)] 0.34127 0.01472 0 2 Msboxa [(200)] 0.5 0.01472 0 2 Msboxa [(300)] 0.65873 0.01472 0 2 Msboxa [(400)] 0.81746 0.01472 0 2 Msboxa [(500)] 0.97619 0.01472 0 2 Msboxa [(20)] 0.17004 0.13244 1 0 Msboxa [(40)] 0.17004 0.25016 1 0 Msboxa [(60)] 0.17004 0.36788 1 0 Msboxa [(80)] 0.17004 0.4856 1 0 Msboxa [(100)] 0.17004 0.60332 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 0.02381 0.01472 moveto 0.02381 0.02097 lineto stroke grestore [(-100)] 0.02381 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.34127 0.01472 moveto 0.34127 0.02097 lineto stroke grestore [(100)] 0.34127 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.5 0.01472 moveto 0.5 0.02097 lineto stroke grestore [(200)] 0.5 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.65873 0.01472 moveto 0.65873 0.02097 lineto stroke grestore [(300)] 0.65873 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.81746 0.01472 moveto 0.81746 0.02097 lineto stroke grestore [(400)] 0.81746 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.97619 0.01472 moveto 0.97619 0.02097 lineto stroke grestore [(500)] 0.97619 0.01472 0 2 Mshowa gsave 0.001 setlinewidth 0.05556 0.01472 moveto 0.05556 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.0873 0.01472 moveto 0.0873 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.11905 0.01472 moveto 0.11905 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.15079 0.01472 moveto 0.15079 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.21429 0.01472 moveto 0.21429 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.24603 0.01472 moveto 0.24603 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.27778 0.01472 moveto 0.27778 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.30952 0.01472 moveto 0.30952 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.37302 0.01472 moveto 0.37302 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.40476 0.01472 moveto 0.40476 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.43651 0.01472 moveto 0.43651 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.46825 0.01472 moveto 0.46825 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.53175 0.01472 moveto 0.53175 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.56349 0.01472 moveto 0.56349 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.59524 0.01472 moveto 0.59524 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.62698 0.01472 moveto 0.62698 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.69048 0.01472 moveto 0.69048 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.72222 0.01472 moveto 0.72222 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.75397 0.01472 moveto 0.75397 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.78571 0.01472 moveto 0.78571 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.84921 0.01472 moveto 0.84921 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.88095 0.01472 moveto 0.88095 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.9127 0.01472 moveto 0.9127 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.94444 0.01472 moveto 0.94444 0.01847 lineto stroke grestore gsave 0.002 setlinewidth 0 0.01472 moveto 1 0.01472 lineto stroke grestore gsave 0.002 setlinewidth 0.18254 0.13244 moveto 0.18879 0.13244 lineto stroke grestore [(20)] 0.17004 0.13244 1 0 Mshowa gsave 0.002 setlinewidth 0.18254 0.25016 moveto 0.18879 0.25016 lineto stroke grestore [(40)] 0.17004 0.25016 1 0 Mshowa gsave 0.002 setlinewidth 0.18254 0.36788 moveto 0.18879 0.36788 lineto stroke grestore [(60)] 0.17004 0.36788 1 0 Mshowa gsave 0.002 setlinewidth 0.18254 0.4856 moveto 0.18879 0.4856 lineto stroke grestore [(80)] 0.17004 0.4856 1 0 Mshowa gsave 0.002 setlinewidth 0.18254 0.60332 moveto 0.18879 0.60332 lineto stroke grestore [(100)] 0.17004 0.60332 1 0 Mshowa gsave 0.001 setlinewidth 0.18254 0.03826 moveto 0.18629 0.03826 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.0618 moveto 0.18629 0.0618 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.08535 moveto 0.18629 0.08535 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.10889 moveto 0.18629 0.10889 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.15598 moveto 0.18629 0.15598 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.17952 moveto 0.18629 0.17952 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.20307 moveto 0.18629 0.20307 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.22661 moveto 0.18629 0.22661 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.2737 moveto 0.18629 0.2737 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.29724 moveto 0.18629 0.29724 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.32079 moveto 0.18629 0.32079 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.34433 moveto 0.18629 0.34433 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.39142 moveto 0.18629 0.39142 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.41497 moveto 0.18629 0.41497 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.43851 moveto 0.18629 0.43851 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.46205 moveto 0.18629 0.46205 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.50914 moveto 0.18629 0.50914 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.53269 moveto 0.18629 0.53269 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.55623 moveto 0.18629 0.55623 lineto stroke grestore gsave 0.001 setlinewidth 0.18254 0.57977 moveto 0.18629 0.57977 lineto stroke grestore gsave 0.002 setlinewidth 0.18254 0 moveto 0.18254 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 0.01 setlinewidth 0.02381 0.01472 moveto 0.06349 0.01472 lineto 0.10317 0.01472 lineto 0.12302 0.01472 lineto 0.14286 0.01472 lineto 0.15278 0.01472 lineto 0.1627 0.01472 lineto 0.16766 0.01472 lineto 0.17262 0.01472 lineto 0.1751 0.01472 lineto 0.17758 0.01472 lineto 0.17882 0.01472 lineto 0.18006 0.01472 lineto 0.1813 0.01472 lineto 0.18254 0.60332 lineto 0.18502 0.60332 lineto 0.1875 0.60332 lineto 0.19246 0.60332 lineto 0.20238 0.60332 lineto 0.22222 0.60332 lineto 0.2619 0.60332 lineto 0.30159 0.60332 lineto 0.34127 0.60332 lineto 0.38095 0.60332 lineto 0.42063 0.60332 lineto 0.46032 0.60332 lineto 0.5 0.60332 lineto 0.53968 0.60332 lineto 0.57937 0.60332 lineto 0.61905 0.60332 lineto 0.65873 0.60332 lineto 0.69841 0.60332 lineto 0.7381 0.60332 lineto 0.77778 0.60332 lineto 0.81746 0.60332 lineto 0.85714 0.60332 lineto 0.89683 0.60332 lineto 0.93651 0.60332 lineto 0.97619 0.60332 lineto stroke grestore grestore % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; 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 = subsection; inactive; preserveAspect; startGroup; ] Motor Neuron :[font = text; inactive; noPageBreak; preserveAspect; ] The output of the motor neuron (MN) is 400 spikes per second during a saccade. When the eye is fixated, the output is proportional to the eye position. ;[s] 3:0,0;18,1;23,0;153,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0; :[font = input; noPageBreak; preserveAspect; startGroup; ] mnNeuron [i_] := linearNeuron [i,0,0,400] Plot [mnNeuron[i],{i,-100,500}, PlotStyle->{{Thickness[0.01]}}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; noPageBreak; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; ] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.18254 0.0015873 0.0147151 0.00147151 [ [(-100)] 0.02381 0.01472 0 2 Msboxa [(100)] 0.34127 0.01472 0 2 Msboxa [(200)] 0.5 0.01472 0 2 Msboxa [(300)] 0.65873 0.01472 0 2 Msboxa [(400)] 0.81746 0.01472 0 2 Msboxa [(500)] 0.97619 0.01472 0 2 Msboxa [(100)] 0.17004 0.16187 1 0 Msboxa [(200)] 0.17004 0.30902 1 0 Msboxa [(300)] 0.17004 0.45617 1 0 Msboxa [(400)] 0.17004 0.60332 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 0.02381 0.01472 moveto 0.02381 0.02097 lineto stroke grestore [(-100)] 0.02381 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.34127 0.01472 moveto 0.34127 0.02097 lineto stroke grestore [(100)] 0.34127 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.5 0.01472 moveto 0.5 0.02097 lineto stroke grestore [(200)] 0.5 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.65873 0.01472 moveto 0.65873 0.02097 lineto stroke grestore [(300)] 0.65873 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.81746 0.01472 moveto 0.81746 0.02097 lineto stroke grestore [(400)] 0.81746 0.01472 0 2 Mshowa gsave 0.002 setlinewidth 0.97619 0.01472 moveto 0.97619 0.02097 lineto stroke grestore [(500)] 0.97619 0.01472 0 2 Mshowa gsave 0.001 setlinewidth 0.05556 0.01472 moveto 0.05556 0.01847 lineto stroke grestore gsave 0.001 setlinewidth 0.0873 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Use options in the Actions Preferences dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = section; inactive; preserveAspect; startGroup; ] Model :[font = subsection; inactive; noPageBreak; preserveAspect; startGroup; ] Diagram :[font = subsubsection; inactive; noPageBreak; dontPreserveAspect; startGroup; ] Diagram :[font = input; PICT; formatAsPICT; output; inactive; noPageBreak; wordwrap; nohscroll; dontPreserveAspect; pictureWidth = 460; pictureHeight = 366; endGroup; pictureID = 340; ] :[font = subsubsection; inactive; preserveAspect; startGroup; ] Explanation :[font = text; inactive; noPageBreak; preserveAspect; endGroup; endGroup; ] The above is the control diagram for an eye saccade. The top half of the diagram, generally, controls the lateral rectus; the bottom half controls the medial rectus. Arrows represents excitory (+) or inhibittory (-) synapses between neurons. A saccade is generated in the following way: 1. Target position Lateral (TL) and Target postion Medial (TM) are set to the desired absolute activities of the lateral and medial rectus, respectively. The Trigger signal, inhibits the action of the omnipause neuron. TL, TM, and Trigger are generated by the CNS in a yet undetermined way. 2. The Retinal Error neurons (ERR) continuously calculate the difference between the desired and the current eye positions. 3. Because the omnipause neuron (OPN) is inhibited by the trigger, and because there can be only one ERR neuron firing, one Excitory Burst neuron (EBN) is excited. It will stay excited as long as there is a retinal error becuase it inhibits the OPN from firing again. 4. While the EBN is firing the corresponding motorneuron (MN) fires at full blast causing the matching rectus to move the eye at full speed, while the other rectus is relaxed. 5. The Tonic neuron (TN) is also excited by its corresponding EBN, or inhibited by the opposing EBN. In this way, the TN is continuosly updated to store the current position of the eye. 6. When the eye reaches its desired location, ERR is zero so it turns off the EBN. Once the EBN stops firing, the OPN begins firing and inhibits any further saccadic eye movements. Also, because the EBN no longer excites the MNs, their only excitation is from the TN. This changes the output of the MN to be the desired/current eye position which, in turn, holds the eye muscles in place. :[font = subsection; inactive; preserveAspect; startGroup; ] Defintions :[font = subsubsection; inactive; noPageBreak; dontPreserveAspect; startGroup; ] Definitions :[font = text; inactive; noPageBreak; dontPreserveAspect; ] The following defintions implement the above neural diagram. The neurons controlling the lateral rectus are defined first. They are followed by the medial neurons and finally the omnipause neuron. :[font = input; noPageBreak; preserveAspect; endGroup; ] errLat := N[errNeuron [TRAGETLAT - TNLAT]] ebnLat := N[ebnNeuron [ERRLAT/4 - OPN]] tnLat := N[tnNeuron [TNLAT + EBNLAT/500 - EBNMED/500]] mnLat := N[mnNeuron [EBNLAT + TNLAT - EBNMED]] errMed := N[errNeuron [TRAGETMED - TNMED]] ebnMed := N[ebnNeuron [ERRMED/4 - OPN]] tnMed := N[tnNeuron [TNMED + EBNMED/500 - EBNLAT/500]] mnMed := N[mnNeuron [EBNMED + TNMED - EBNLAT]] opn := N[opnNeuron [-TRIGGER/10 - EBNLAT/100 - EBNMED/100]] :[font = subsubsection; inactive; preserveAspect; startGroup; ] Explanations :[font = text; inactive; noPageBreak; preserveAspect; endGroup; endGroup; ] ERR: The retinal error is the difference between the desired eye position and the current eye position. EBN: The EBN must be inhibited when the OPN is firing (at 100 spikes per second). Because the maximum value of the ERR is 400, a division of the ERR by 4 will result in the EBN being inhibted whenever the OPN is excited. TN: Because a full saccade lasts 200 msec, and because the TN must change 400 units, the value of the TN must be modifed by 2 units per msec. A change of 2 units is achived by dividing the 1000 value of the EBN by 500. MN: Becuase the maximum output of the MN is 400, during a saccde, the EBN will drive the neuron to its maximun excitation. Conversly, because the maximum value of the TN is 400, an inhibitory EBN will drive the MN to its minimum value. When the eye is fixed the value of both EBN's is 0, hence the only input is the TN which will hold the MN at the same value as the TN. OPN: The OPN is inhibited by any input. Thus when either the Trigger or either of the EBN's are firing, the OPN is inhibited. ;[s] 10:0,1;3,0;106,1;109,0;330,1;332,0;552,1;554,0;926,1;929,0;1054,-1; 2:5,13,9,Times,0,12,0,0,0;5,13,9,Times,1,12,0,0,0; :[font = subsection; inactive; preserveAspect; startGroup; ] Implementation :[font = subsubsection; inactive; noPageBreak; preserveAspect; startGroup; ] Initialization :[font = text; inactive; noPageBreak; preserveAspect; ] The routine initLists initializes the values of all the neurons. It also initializes the lists that keep track of the value of a neuron at each time step. ;[s] 3:0,0;12,1;21,0;156,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,9,Times,1,12,0,0,0; :[font = input; noPageBreak; preserveAspect; endGroup; ] initLists := ( errLatList = {ERRLAT = 0}; ebnLatList = {EBNLAT = 0}; tnLatList = {TNLAT = 200}; mnLatList = {MNLAT = 200}; errMedList = {ERRMED = 0}; ebnMedList = {EBNMED = 0}; tnMedList = {TNMED = 200}; mnMedList = {MNMED = 200}; opnList = {OPN = 100};) :[font = subsubsection; inactive; preserveAspect; startGroup; ] Calculations :[font = text; inactive; noPageBreak; preserveAspect; ] The routine calculateNewEyePos calculates the output of the neurons for the next time step. The results are stored in a temporary value. ;[s] 3:0,0;12,1;30,0;138,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,9,Times,1,12,0,0,0; :[font = input; noPageBreak; preserveAspect; endGroup; ] calculateNewEyePos := ( TEMPERRLAT = N[errLat]; TEMPEBNLAT = N[ebnLat]; TEMPTNLAT = N[tnLat]; TEMPMNLAT = N[mnLat]; TEMPERRMED = N[errMed]; TEMPEBNMED = N[ebnMed]; TEMPTNMED = N[tnMed]; TEMPMNMED = N[mnMed]; TEMPOPN = N[opn];) :[font = subsubsection; inactive; preserveAspect; startGroup; ] List updates :[font = text; inactive; noPageBreak; preserveAspect; ] The routine updateEyePos sets the the values of the neurons to the values calculated by the preceding routine. It also appends the new values of the neurons to their respective time lists. ;[s] 3:0,0;12,1;24,0;190,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,9,Times,1,12,0,0,0; :[font = input; noPageBreak; preserveAspect; endGroup; endGroup; endGroup; ] updateEyePos := ( errLatList = Append [errLatList, ERRLAT = TEMPERRLAT]; ebnLatList = Append [ebnLatList, EBNLAT = TEMPEBNLAT]; tnLatList = Append [tnLatList, TNLAT = TEMPTNLAT]; mnLatList = Append [mnLatList, MNLAT = TEMPMNLAT]; errMedList = Append [errMedList, ERRMED = TEMPERRMED]; ebnMedList = Append [ebnMedList, EBNMED = TEMPEBNMED]; tnMedList = Append [tnMedList, TNMED = TEMPTNMED]; mnMedList = Append [mnMedList, MNMED = TEMPMNMED]; opnList = Append [opnList, OPN = TEMPOPN];) :[font = section; inactive; preserveAspect; startGroup; ] Simulation :[font = subsection; inactive; noPageBreak; preserveAspect; startGroup; ] Eye Drawing :[font = text; inactive; noPageBreak; preserveAspect; ] The following routines draw the field of vision, the eye, and the object. The rotuine drawEye puts the three routines together. ;[s] 3:0,0;87,1;94,0;129,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,9,Times,1,12,0,0,0; :[font = input; noPageBreak; preserveAspect; endGroup; ] fx[t_,d_]:=(t-1/6*Pi)/(4/6*Pi)*Cos[d] fy[t_,d_]:=(t-1/6*Pi)/(4/6*Pi)*Sin[d] fieldOfVisionPlot = ParametricPlot [{{Cos[t],Sin[t]}, {fx[t,1/6*Pi],fy[t,1/6*Pi]}, {-fx[t,1/6*Pi],fy[t,1/6*Pi]}}, {t,1/6*Pi,5/6*Pi},Axes->False, PlotRange->{{-1,1},{0,1}}, DisplayFunction->Identity]; fe[d_] := d/400*4/6*Pi+1/6*Pi eyePlot [eyePos_] := ParametricPlot [{fx[t,fe[eyePos]],fy[t,fe[eyePos]]}, {t,1/6*Pi,5/6*Pi},Axes->False, PlotRange->{{-1,1},{0,1}}, PlotStyle->{{Thickness[0.01]}}, DisplayFunction->Identity] objPlot[objPos_] := Graphics[Disk[{Cos[fe[objPos]],Sin[fe[objPos]]},.03]] drawEye[eyePos_,objPos_] := Show [fieldOfVisionPlot,eyePlot[eyePos],objPlot[objPos], DisplayFunction->$DisplayFunction] :[font = subsection; inactive; preserveAspect; startGroup; ] Placement of Object :[font = text; inactive; noPageBreak; preserveAspect; ] We randomly choose a position for the object. The desired values for the lateral rectus and medial rectus are complementry in the range 0..400. :[font = input; noPageBreak; preserveAspect; startGroup; ] TRAGETLAT = Random[Integer,400] TRAGETMED = 400 - TRAGETLAT; :[font = output; output; inactive; noPageBreak; preserveAspect; endGroup; endGroup; ] 302 ;[o] 302 :[font = subsection; inactive; preserveAspect; startGroup; ] Saccade :[font = text; inactive; noPageBreak; preserveAspect; ] Initialize the time lists and calculate the values of the network for 150 msec. The trigger for the saccade is enabled for 2 msec starting at the 20th msec of the simulation. 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Dashed lines represent the lateral neurons while solid lines represent the medial neurons. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ] Error Neurons :[font = text; inactive; noPageBreak; preserveAspect; ] Only one error neuron changes during the simulation, the other is steady at 0 error . In the first 20 msec, before the saccade starts, the error is constant. During the saccade, the error linearly decreases. After the saccade, both error neurons show zero error. :[font = input; noPageBreak; preserveAspect; startGroup; ] plotErrLatList = ListPlot[errLatList,PlotJoined->True, PlotStyle->{Dashing[{0.02,0.03}],Thickness[0.01]}, PlotRange->{0,200},DisplayFunction->Identity]; plotErrMedList = ListPlot[errMedList,PlotJoined->True, PlotStyle->{Thickness[0.01]}, PlotRange->{0,200},DisplayFunction->Identity]; Show[plotErrLatList,plotErrMedList,DisplayFunction->$DisplayFunction] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; ] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.00630716 0 0.00309017 [ [(0)] 0.02381 0 0 2 Msboxa [(20)] 0.14995 0 0 2 Msboxa [(40)] 0.2761 0 0 2 Msboxa [(60)] 0.40224 0 0 2 Msboxa [(80)] 0.52838 0 0 2 Msboxa [(100)] 0.65453 0 0 2 Msboxa [(120)] 0.78067 0 0 2 Msboxa [(140)] 0.90681 0 0 2 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