diff --git a/announcements/correlationcartoon.jpeg b/announcements/correlationcartoon.jpeg new file mode 100644 index 0000000..a9b3823 Binary files /dev/null and b/announcements/correlationcartoon.jpeg differ diff --git a/announcements/evalutation.pdf b/announcements/evalutation.pdf new file mode 100644 index 0000000..ec41ac7 Binary files /dev/null and b/announcements/evalutation.pdf differ diff --git a/announcements/evalutation.tex b/announcements/evalutation.tex new file mode 100644 index 0000000..c528f4f --- /dev/null +++ b/announcements/evalutation.tex @@ -0,0 +1,60 @@ +\documentclass[pdftex]{beamer} +\usepackage{beamerthemedefault} +\usepackage{multimedia} +\usepackage{wasysym} +\useoutertheme[subsection=false]{smoothbars} +\useinnertheme[shadow=true]{rounded} + +\setbeamercolor{block title}{fg=black,bg=gray} +\setbeamercolor{block title alerted}{use=alerted text,fg=black,bg=alerted text.fg!75!bg} +\setbeamercolor{block title example}{use=example text,fg=black,bg=example text.fg!75!bg} +\setbeamercolor{block body}{parent=normal text,use=block title,bg=block title.bg!25!bg} +\setbeamercolor{block body alerted}{parent=normal text,use=block title alerted,bg=block title alerted.bg!25!bg} +\setbeamercolor{block body example}{parent=normal text,use=block title example,bg=block title example.bg!25!bg} + +\usepackage{hyperref} +\usepackage[english]{babel} % language set to new-german +\usepackage[utf8]{inputenc} % coding of german special characters +\usepackage{graphicx} % \includegraphics[options]{file.eps} +\usepackage{array} +\usepackage{setspace} +\usepackage{color} +\usepackage{eqlist} +\usepackage{wallpaper} + +\definecolor{tug}{HTML}{CC0000} +\newcommand{\rot}[1]{{\color{tug} #1}} +\definecolor{green}{HTML}{1C8C1C} +\newcommand{\gruen}[1]{{\color{green} #1}} +\definecolor{blue}{HTML}{2424AA} +\newcommand{\blau}[1]{{\color{blue} #1}} +\definecolor{gray}{rgb}{0.8,0.8,0.8} +\definecolor{gray2}{rgb}{0.5,0.5,0.5} +\definecolor{gray2}{rgb}{0.5,0.5,0.5} +\newcommand{\grau}[1]{{\color{gray2} #1}} + +\setbeamercolor{footline}{fg=black,bg=gray} + +%\setbeamertemplate{headline}[text line]{ +% \begin{beamercolorbox}[wd=\paperwidth,ht=8ex,dp=4ex]{}%\vskip-4pt +% \insertnavigation{0.85\paperwidth} +% \hskip-1pt\rule{\paperwidth}{0.3pt} +% \end{beamercolorbox} +%} + +\setlength{\unitlength}{0.01\paperwidth} + +\setbeamercolor{background canvas}{bg=} + + +\begin{document} + +\begin{frame} + \frametitle{Evalutation} + In Ilias:\\[2ex] + Go to {\em Evaluation} and fill out the forms!\\[6ex] + \includegraphics[width=0.8\textwidth]{correlationcartoon} +\end{frame} + + +\end{document} diff --git a/linearalgebra/code/Chorthippus_biguttulus.wav b/linearalgebra/code/Chorthippus_biguttulus.wav new file mode 100644 index 0000000..2c57068 Binary files /dev/null and b/linearalgebra/code/Chorthippus_biguttulus.wav differ diff --git a/linearalgebra/code/PeterUndDerWolf.wav b/linearalgebra/code/PeterUndDerWolf.wav new file mode 100644 index 0000000..80aab98 Binary files /dev/null and b/linearalgebra/code/PeterUndDerWolf.wav differ diff --git a/linearalgebra/code/Tielli-Kraulis.wav b/linearalgebra/code/Tielli-Kraulis.wav new file mode 100644 index 0000000..8ad071a Binary files /dev/null and b/linearalgebra/code/Tielli-Kraulis.wav differ diff --git a/linearalgebra/code/coordinaterafo2.m b/linearalgebra/code/coordinaterafo2.m new file mode 100644 index 0000000..054ac6d --- /dev/null +++ b/linearalgebra/code/coordinaterafo2.m @@ -0,0 +1,52 @@ +% some vectors: +x = [ -1:0.02:1 ]; +y = x*0.5 + 0.1*randn( size(x) ); +plot( x, y, '.b' ); +hold on +plot( x(10), y(10), '.r' ); + +% new coordinate system: +%e1 = [ 3/5 4/5 ]; +%e2 = [ -4/5 3/5 ]; +e1 = [ 3 4 ]; +e2 = [ -4 3 ]; +me1 = sqrt( e1*e1' ); +e1 = e1/me1; +me2 = sqrt( e2*e2' ); +e2 = e2/me2; +quiver( 0.0, 0.0 , e1(1), e1(2), 1.0, 'r' ) +quiver( 0.0, 0.0 , e2(1), e2(2), 1.0, 'g' ) +axis( 'equal' ) + +% project [x y] onto e1 and e2: + +% % the long way: +% nx = zeros( size( x ) ); % new x coordinate +% ny = zeros( size( y ) ); % new y coordinates +% for k=1:length(x) +% xvec = [ x(k) y(k) ]; +% nx(k) = xvec * e1'; +% ny(k) = xvec * e2'; +% end +% plot( nx, ny, '.g' ); + +% the short way: +%nx = [ x' y' ] * e1'; +%nx = [x; y]' * e1'; +% nx = e1 * [ x; y ]; +% ny = e2 * [ x; y ]; +% plot( nx, ny, '.g' ); + +% even shorter: +n = [e1; e2 ] * [ x; y ]; +plot( n(1,:), n(2,:), '.g' ); + + + + + + + + + +hold off diff --git a/linearalgebra/code/coordinatetrafo.m b/linearalgebra/code/coordinatetrafo.m new file mode 100644 index 0000000..7c8507c --- /dev/null +++ b/linearalgebra/code/coordinatetrafo.m @@ -0,0 +1,37 @@ +% some vectors: +x=[-1:0.02:1]'; % column vector! +y=4*x/3 + 0.1*randn( size( x ) ); +plot( x, y, '.b' ); +axis( 'equal' ); +hold on; + +% new coordinate system: +% e1 = [ 3 4 ] +% e2 = [ -4 3 ] +% and normalized to unit length (factor 1/5): +e = [ 3/5 -4/5; 4/5 3/5 ]; +quiver( [0 0], [0 0], e(1,:), e(2,:), 1.0, 'r' ); + +% project [x y] onto e1 and e2: + +% % the long way: +% nx = x; +% ny = y; +% for k=1:length(x) +% nx(k) = [x(k) y(k)]*e(:,1); +% ny(k) = [x(k) y(k)]*e(:,2); +% end +% plot( nx, ny, '.g' ); + +% the short way +% nx = [x y]*e(:,1); +% ny = [x y]*e(:,2); +% plot( nx, ny, '.g' ); + +% even shorter: +n = [x y]*e; +plot( n(:,1), n(:,2), '.g' ); +xlabel( 'x' ); +ylabel( 'y' ); + +hold off; diff --git a/linearalgebra/code/correlationcoefficient.m b/linearalgebra/code/correlationcoefficient.m new file mode 100644 index 0000000..55472ab --- /dev/null +++ b/linearalgebra/code/correlationcoefficient.m @@ -0,0 +1,29 @@ +set( 0, 'DefaultTextFontSize', 22.0 ); +set( 0, 'DefaultAxesFontSize', 22.0 ); + +n = 10000; +x = 1.0*randn( n, 1 ); +z = 1.0*randn( n, 1 ); +for r=-1:0.2:1 + y = r*x + sqrt(1.0-r^2.0)*z; + cv = cov( x, y ); + cr = corrcoef( x, y ); + scatter( x, y, 'filled', 'MarkerEdgeColor', 'white' ); + xlabel( 'x' ); + ylabel( 'y' ); + text( 0.05, 0.9, sprintf( 'r = %.2f', r ), 'Units', 'normalized' ) + text( 0.05, 0.8, sprintf( 'cor = %.2f %.2f', cr(1,1), cr(1,2) ), 'Units', 'normalized' ) + text( 0.05, 0.7, sprintf( 'cov = %.2f %.2f', cv(1,1), cv(1,2) ), 'Units', 'normalized' ) + waitforbuttonpress; +end + +y = x.^2.0 + 0.5*z; +cv = cov( x, y ); +cr = corrcoef( x, y ); +scatter( x, y, 'filled', 'MarkerEdgeColor', 'white' ); +xlabel( 'x' ); +ylabel( 'y' ); +text( 0.05, 0.9, sprintf( 'x^2', r ), 'Units', 'normalized' ) +text( 0.05, 0.8, sprintf( 'cor = %.2f %.2f', cr(1,1), cr(1,2) ), 'Units', 'normalized' ) +text( 0.05, 0.7, sprintf( 'cov = %.2f %.2f', cv(1,1), cv(1,2) ), 'Units', 'normalized' ) + diff --git a/linearalgebra/simulations/covareigen.m b/linearalgebra/code/covareigen.m similarity index 83% rename from linearalgebra/simulations/covareigen.m rename to linearalgebra/code/covareigen.m index 1acf330..cd6025f 100644 --- a/linearalgebra/simulations/covareigen.m +++ b/linearalgebra/code/covareigen.m @@ -44,11 +44,16 @@ function covareigen( x, y, pm, w, h ) % plot eigenvectors: quiver( ones( 1, 2 ).*mean( x ), ones( 1, 2 )*mean(y), v(1,:).*sqrt(diag(d))', v(2,:).*sqrt(diag(d))', 'r', 'LineWidth', 3, 'AutoScale', 'off', 'AutoScaleFactor', 1.0, 'MaxHeadSize', 0.7 ) + xlabel( 'x' ); + ylabel( 'y' ); axis( 'equal' ); hold off; + % histogram of x values: subplot( 2, 2, 3 ); hist( x, 50, 'b' ); + xlabel( 'x' ); + ylabel( 'n_x' ); % sort the eigenvalues: [d,inx] = sort( diag(d), 'descend' ); @@ -59,24 +64,28 @@ function covareigen( x, y, pm, w, h ) x = x - mean( x ); y = y - mean( y ); % project onto eigenvectors: - nx = [ x y ] * v(:,inx(1)); - ny = [ x y ] * v(:,inx(2)); - cv = cov( [nx, ny] ) + nc = [ x y ] * v(:,inx); + cv = cov( nc ) [ v , d] = eig( cv ) if (pm > 0) & (nargin == 5) if pm == 1 - [n,c] = hist3([nx ny], { xp, yp } ); + [n,c] = hist3( nc, { xp, yp } ); contourf( c{1}, c{2}, n' ); else gauss = reshape( exp(-0.5*diag(xy*inv(cv)*xy'))/sqrt((2.0*pi)^2.0*det(cv)), size( xg ) ); contourf( xp, yp, gauss ) end else - scatter( nx, ny, 'b', 'filled', 'MarkerEdgeColor', 'white' ); + scatter( nc(:,1), nc(:,2), 'b', 'filled', 'MarkerEdgeColor', 'white' ); end + xlabel( 'x' ); + ylabel( 'y' ); axis( 'equal' ); hold off; + % histogram of new x values: subplot( 2, 2, 4 ); - hist( nx, 50, 'b' ); + hist( nc(:,1), 50, 'b' ); + xlabel( 'new x' ); + ylabel( 'n_newx' ); end diff --git a/linearalgebra/simulations/covareigen3.m b/linearalgebra/code/covareigen3.m similarity index 64% rename from linearalgebra/simulations/covareigen3.m rename to linearalgebra/code/covareigen3.m index c11f2de..636d9cb 100644 --- a/linearalgebra/simulations/covareigen3.m +++ b/linearalgebra/code/covareigen3.m @@ -20,25 +20,40 @@ function covareigen3( x, y, z ) ylabel( 'y' ); zlabel( 'z' ); grid on; - + % plot eigenvectors: quiver3( ones( 1, 3 ).*mean( x ), ones( 1, 3 )*mean(y), ones( 1, 3 )*mean(z), v(1,:).*sqrt(diag(d))', v(2,:).*sqrt(diag(d))', v(3,:).*sqrt(diag(d))', 'r', 'LineWidth', 3, 'AutoScale', 'off', 'AutoScaleFactor', 1.0, 'MaxHeadSize', 0.7 ) %axis( 'equal' ); hold off; + % 2D scatter plots: + subplot( 2, 6, 4 ); + scatter( x, y, 'b', 'filled', 'MarkerEdgeColor', 'white' ); + xlabel( 'x' ) + ylabel( 'y' ) + subplot( 2, 6, 5 ); + scatter( x, z, 'b', 'filled', 'MarkerEdgeColor', 'white' ); + xlabel( 'x' ) + ylabel( 'z' ) + subplot( 2, 6, 6 ); + scatter( y, z, 'b', 'filled', 'MarkerEdgeColor', 'white' ); + xlabel( 'y' ) + ylabel( 'z' ) + % sort the eigenvalues: [d,inx] = sort( diag(d), 'descend' ); - subplot( 2, 2, 2 ); + subplot( 2, 2, 4 ); hold on; % subtract means: x = x - mean( x ); y = y - mean( y ); % project onto eigenvectors: - nx = [ x y z ] * v(:,inx(1)); - ny = [ x y z ] * v(:,inx(2)); - scatter( nx, ny, 'b', 'filled', 'MarkerEdgeColor', 'white' ); + nx = [ x y z ] * v(:,inx); + scatter( nx(:,1), nx(:,2), 'b', 'filled', 'MarkerEdgeColor', 'white' ); + xlabel( 'ex' ) + ylabel( 'ey' ) axis( 'equal' ); hold off; diff --git a/linearalgebra/code/covareigen3examples.m b/linearalgebra/code/covareigen3examples.m new file mode 100644 index 0000000..f27c76c --- /dev/null +++ b/linearalgebra/code/covareigen3examples.m @@ -0,0 +1,19 @@ +scrsz = get( 0, 'ScreenSize' ); +set( 0, 'DefaultFigurePosition', [ scrsz(3)/2 scrsz(4)/2 scrsz(3)/2 scrsz(4)/2 ] ); + +n = 10000; + +% three distributions: +x = randn( n, 1 ); +y = randn( n, 1 ); +z = randn( n, 1 ); +dx = [ 0 8 0 0 ]; +dy = [ 0 0 8 0 ]; +dz = [ 0 0 0 8 ]; +for k = 1:4 + x((k-1)*n/4+1:k*n/4) = x((k-1)*n/4+1:k*n/4) + dx(k); + y((k-1)*n/4+1:k*n/4) = y((k-1)*n/4+1:k*n/4) + dy(k); + z((k-1)*n/4+1:k*n/4) = z((k-1)*n/4+1:k*n/4) + dz(k); +end +f = figure( 1 ); +covareigen3( x, y, z ); diff --git a/linearalgebra/simulations/covareigenexamples.m b/linearalgebra/code/covareigenexamples.m similarity index 88% rename from linearalgebra/simulations/covareigenexamples.m rename to linearalgebra/code/covareigenexamples.m index f360e8c..2f7a480 100644 --- a/linearalgebra/simulations/covareigenexamples.m +++ b/linearalgebra/code/covareigenexamples.m @@ -5,11 +5,11 @@ set( 0, 'DefaultFigurePosition', [ scrsz(3)/2 scrsz(4)/2 scrsz(3)/2 scrsz(4)/2 ] n = 10000; x = randn( n, 1 ); f = figure( 1 ); -for r = 0.01:0.199:1 +for r = 0.01:0.19:1 fprintf( 'Correlation = %g\n', r ); clf( f ); y = r*x + sqrt(1-r^2)*randn( n, 1 ); - covareigen( x, y, 2, 5.0, 3.0 ); + covareigen( x, y, 0, 5.0, 3.0 ); key = waitforbuttonpress; end return @@ -29,7 +29,7 @@ for d = 0:1:5 x = [ x1; x2 ]; y = [ y1+d2; y2-d2 ]; scrsz = get(0,'ScreenSize'); - covareigen( x, y, 1, 10.0, 7.0 ); + covareigen( x, y, 0, 10.0, 7.0 ); %key = waitforbuttonpress; pause( 1.0 ); end diff --git a/linearalgebra/code/extdata.mat b/linearalgebra/code/extdata.mat new file mode 100644 index 0000000..619c0c1 Binary files /dev/null and b/linearalgebra/code/extdata.mat differ diff --git a/linearalgebra/code/extracellularrecording.m b/linearalgebra/code/extracellularrecording.m new file mode 100644 index 0000000..8488cd1 --- /dev/null +++ b/linearalgebra/code/extracellularrecording.m @@ -0,0 +1,54 @@ +% this script simulates extracellularly recorded +% spike waveforms and saves them in extdata.mat. + +% generate spikes: +n = 1000; +misi = 0.01; +isis = exprnd( misi, n, 1 ); +isis = isis + 0.01; +spikes = cumsum( isis ); +p = rand( size( spikes ) ); + +% spike waveforms: +dt = 0.0001; +waveformt = -0.01:dt:0.01; +waveform1 = 2.0*exp( -(waveformt-0.0003).^2/2.0/0.0005^2 ) - 1.4*exp( -(waveformt-0.0005).^2/2.0/0.0005^2 ); +waveform2 = exp( -waveformt.^2/2.0/0.002^2 ).*cos(2.0*pi*waveformt/0.005); +p12 = 0.5; + +% voltage trace: +noise = 0.2; +time = 0:dt:spikes(end)+3.0*waveformt(end); +voltage = noise*randn( 1, length( time ) ); +for k = 1:length( spikes ) + inx = ceil( spikes(k)/dt ); + if p(k) < p12 + voltage(inx:inx+length(waveformt)-1) = voltage(inx:inx+length(waveformt)-1) + waveform1; + else + voltage(inx:inx+length(waveformt)-1) = voltage(inx:inx+length(waveformt)-1) + waveform2; + end +end + +figure( 1 ); +clf; +plot( time(time<1.0), voltage(time<1.0) ); +hold on; + +% find peaks: +thresh = 0.7; +inx = find( voltage(2:end-1) > thresh & voltage(1:end-2) voltage(3:end) ) + 1; +spiketimes = time(inx); +spikevoltage = voltage(inx); +tinx = inx; +for k=1:2 + inx = find( ( spiketimes(2:end-1)-spiketimes(1:end-2)>0.005 | spikevoltage(2:end-1) > spikevoltage(1:end-2) ) & ( spiketimes(3:end)-spiketimes(2:end-1)>0.005 | spikevoltage(2:end-1) > spikevoltage(3:end) ) )+1; + spiketimes = spiketimes(inx); + spikevoltage = spikevoltage(inx); + tinx = tinx(inx); +end +scatter( spiketimes(spiketimes<1.0), spikevoltage(spiketimes<1.0), 100.0, 'r', 'filled' ); +%scatter( t(tinx), v(tinx), 100.0, 'r', 'filled' ); +hold off; + +% save data to file: +save( 'extdata', 'time', 'voltage', 'spiketimes', 'waveformt', 'waveform1', 'waveform2' ); diff --git a/linearalgebra/code/gaussian.m b/linearalgebra/code/gaussian.m new file mode 100644 index 0000000..cdac6d5 --- /dev/null +++ b/linearalgebra/code/gaussian.m @@ -0,0 +1,25 @@ +% Gaussian density from histogram: +x = randn( 1000000, 1 ); +[ n, c ] = hist( x, 100 ); +n = n/sum(n)/(c(2)-c(1)); +bar( c, n ); +hold on; +% equation p(x): +xx = -5:0.01:5; +p=exp(-xx.^2/2.0)/sqrt(2.0*pi); +plot( xx, p, 'r', 'LineWidth', 3 ) + +% with mean=2 and sigma=0.5: +mu = 2.0; +sig = 0.5; +x = sig*x + mu; +[ n, c ] = hist( x, 100 ); +n = n/sum(n)/(c(2)-c(1)); +bar( c, n ); +% equation: +p=exp(-(xx-mu).^2/2.0/sig^2)/sqrt(2.0*pi*sig^2); +plot( xx, p, 'r', 'LineWidth', 3 ) +hold off; +xlabel( 'x' ); +ylabel( 'p(x)' ); +title( 'Gaussian distribution' ); diff --git a/linearalgebra/code/matrix2dtrafos.m b/linearalgebra/code/matrix2dtrafos.m new file mode 100644 index 0000000..2bd7546 --- /dev/null +++ b/linearalgebra/code/matrix2dtrafos.m @@ -0,0 +1,21 @@ +% Visualize some common matrix transformations +% in a 2D coordinate system + +disp( 'Click into the plot to advance to the next matrix' ) +matrixbox( [ 1 0; 0 1], 'Identity' ); +matrixbox( [ 2 0; 0 1], 'Scale x' ); +matrixbox( [ 1 0; 0 2], 'Scale y' ); +matrixbox( [ 2 0; 0 2], 'Scale x and y' ); +matrixbox( [ 0.5 0; 0 0.5], 'Scale x and y' ); +matrixbox( [ -1 0; 0 1], 'Flip x' ); +matrixbox( [ 1 0; 0 -1], 'Flip y' ); +matrixbox( [ -1 0; 0 -1], 'Flip both' ); +matrixbox( [ 1 2; 0 1], 'Shear x' ); +matrixbox( [ 1 0; 2 1], 'Shear y' ); +matrixbox( [ 0.5 1; 1 0.5], 'Something A' ); +matrixbox( [ 0.5 1; 1.2 0.6], 'Something B' ); +% rotation matrices: +for deg = 0:10:360 + phi = deg*pi/180.0; + matrixbox( [ cos(phi) -sin(phi); sin(phi) cos(phi)], sprintf( 'Rotate %.0f', deg ) ); +end diff --git a/linearalgebra/code/matrixbox.m b/linearalgebra/code/matrixbox.m new file mode 100644 index 0000000..6cc0aa1 --- /dev/null +++ b/linearalgebra/code/matrixbox.m @@ -0,0 +1,45 @@ +function matrixbox( m, s ) +% visualizes the effect of a matrix m on a set of vectors forming a box +% m: a 2x2 matrix +% s: a string plotted as the title of the plot +% this function is called by matrix2dtrafos.m + + % the 5 vectors that point into the cornes of a unit square: + v = [ 0 1 1 0 0; + 0 0 1 1 0 ]; + % transform v by means of m into new vector w: + w = m*v; + clf; + hold on; + % axis: + plot( [-2 2], [0 0], 'k', 'LineWidth', 1 ); + plot( [0 0], [-2 2], 'k', 'LineWidth', 1 ); + % old vectors: + plot( v(1,:), v(2,:), 'k', 'LineWidth', 1.0 ) + quiver( v(1,1), v(2,1), v(1,3), v(2,3), 1.0, 'k', 'LineWidth', 2.0 ); + scatter( v(1,2), v(2,2), 60.0, 'filled', 'k' ); + % transformed vectors: + plot( w(1,:), w(2,:), 'b', 'LineWidth', 2.0 ) + quiver( w(1,1), w(2,1), w(1,3), w(2,3), 1.0, 'b', 'LineWidth', 3.0 ); + scatter( w(1,2), w(2,2), 100.0, 'filled', 'b' ); + % eigenvectors: + [ev ed] = eig(m); + n = ev*ed; + if isreal( n ) + quiver( [0 0], [0 0], n(1,:), n(2,:), 1.0, 'r', 'LineWidth', 2.0 ); + text( 0.1, 0.2, sprintf( '\\lambda_1 = %.3g', ed(1,1) ), 'Units', 'normalized' ) + text( 0.1, 0.1, sprintf( '\\lambda_2 = %.3g', ed(2,2) ), 'Units', 'normalized' ) + end + hold off; + xlim( [ -2 2 ] ); + ylim( [ -2 2 ] ); + axis( 'equal' ); + m( abs(m) < 1e-3 ) = 0.0; % make zeros really a zero + text( 0.7, 0.15, 'A =', 'Units', 'normalized' ) + text( 0.8, 0.2, sprintf( '%.3g', m(1,1) ), 'Units', 'normalized' ) + text( 0.9, 0.2, sprintf( '%.3g', m(1,2) ), 'Units', 'normalized' ) + text( 0.8, 0.1, sprintf( '%.3g', m(2,1) ), 'Units', 'normalized' ) + text( 0.9, 0.1, sprintf( '%.3g', m(2,2) ), 'Units', 'normalized' ) + title( s ); + waitforbuttonpress; +end diff --git a/linearalgebra/code/matrixmultiplicationlatex.m b/linearalgebra/code/matrixmultiplicationlatex.m new file mode 100644 index 0000000..dbb66de --- /dev/null +++ b/linearalgebra/code/matrixmultiplicationlatex.m @@ -0,0 +1,85 @@ +% open a file to write in the LaTeX commands: +f = fopen( 'matrices.tex','w'); +fprintf( f, '\\documentclass{exam}\n' ); +fprintf( f, '\\usepackage{amsmath}\n' ); +fprintf( f, '\\begin{document}\n' ); + +for k = 1:40 + + % compute row and column numbers: + pd = rand( 1, 1 ); + if pd < 0.05 + % row vector multiplication that might not be possible: + am = 1; + an = randi( [ 2 4 ]); + bm = randi( [ 2 4 ]); + bn = 1; + elseif pd < 0.1 + % column vector multiplication that might not be possible: + am = randi( [ 2 4 ]); + an = 1; + bm = 1; + bn = randi( [ 2 4 ]); + elseif pd < 0.4 + % row vector multiplication that is possible: + am = 1; + an = randi( [ 2 4 ]); + bm = an; + bn = 1; + elseif pd < 0.5 + % column vector multiplication that is possible: + am = randi( [ 2 4 ]); + an = 1; + bm = 1; + bn = am; + elseif pd < 0.6 + % matrix multiplication that might not be possible: + am = randi( [ 2 4 ]); + an = randi( [ 2 4 ]); + bm = randi( [ 2 4 ]); + bn = randi( [ 2 4 ]); + else + % matrix multiplication that is possible: + am = randi( [ 2 4 ]); + an = randi( [ 2 4 ]); + bm = an; + bn = randi( [ 2 4 ]); + end + % generate the matrices: + a = randi( [-4 4], am, an ); + b = randi( [-4 4], bm, bn ); + % write them out as LaTeX code: + % matrix a: + fprintf( f, ' \\[ \\begin{pmatrix}' ); + for r = 1:size( a, 1 ) + for c = 1:size( a, 2 ) + if c > 1 + fprintf( f, ' &' ); + end + fprintf( f, ' %d', a(r,c) ); + end + if r < size( a, 1 ) + fprintf( f, ' \\\\' ); + end + end + fprintf( f, ' \\end{pmatrix} \\cdot\n' ); + % matrix b: + fprintf( f, ' \\begin{pmatrix}' ); + for r = 1:size( b, 1 ) + for c = 1:size( b, 2 ) + if c > 1 + fprintf( f, ' &' ); + end + fprintf( f, ' %d', b(r,c) ); + end + if r < size( b, 1 ) + fprintf( f, ' \\\\' ); + end + end + fprintf( f, ' \\end{pmatrix} = \\]\n\n' ); + +end + +% close the document and the file: +fprintf( f, '\\end{document}\n' ); +fclose( f ); diff --git a/linearalgebra/code/p-unit_spike_times.mat b/linearalgebra/code/p-unit_spike_times.mat new file mode 100644 index 0000000..75f53e6 Binary files /dev/null and b/linearalgebra/code/p-unit_spike_times.mat differ diff --git a/linearalgebra/code/p-unit_stimulus.mat b/linearalgebra/code/p-unit_stimulus.mat new file mode 100644 index 0000000..b6d0b25 Binary files /dev/null and b/linearalgebra/code/p-unit_stimulus.mat differ diff --git a/linearalgebra/code/pca2d.m b/linearalgebra/code/pca2d.m new file mode 100644 index 0000000..6879f8a --- /dev/null +++ b/linearalgebra/code/pca2d.m @@ -0,0 +1,95 @@ +function pca2d( x, y, pm, w, h ) +% computes covariance matrix from the pairs x, y +% diagonalizes covariance matrix, i.e. performs a PCA on [ x, y ] +% x and y are column vectors +% pm: 0 - scatter of data, 1 - histogram, 2 - multivariate gauss from +% covariance matrix, 1 and 2 require w and h +% w and h are the width and the height (in data units) for plotting +% histograms instead of scatter + + % covariance matrix: + cv = cov( [ x y ] ); + + % eigen values: + [ v , d] = eig( cv ) + s = sign( v ); + s(1,:) = s(2,:); + v = v .* s; + + % plots: + subplot( 2, 2, 1 ); + hold on; + + if (pm > 0) & (nargin == 5) + if pm == 1 + % histogram of data: + xp = -w:0.5:w; + yp = -h:0.5:h; + [n,c] = hist3([x y], { xp, yp } ); + contourf( c{1}, c{2}, n' ); + else + % bivariate Gaussian: + xp = -w:0.1:w; + yp = -h:0.1:h; + [xg,yg] = meshgrid( xp, yp ); + xy = [ xg(:) yg(:) ]; + gauss = reshape( exp(-0.5*diag(xy*inv(cv)*xy'))/sqrt((2.0*pi)^2.0*det(cv)), size( xg ) ); + contourf( xp, yp, gauss ) + end + colormap( 'gray' ); + else + % scatter plot: + scatter( x, y, 'b', 'filled', 'MarkerEdgeColor', 'white' ); + end + + % plot eigenvectors: + quiver( ones( 1, 2 ).*mean( x ), ones( 1, 2 )*mean(y), v(1,:).*sqrt(diag(d))', v(2,:).*sqrt(diag(d))', 'r', 'LineWidth', 3, 'AutoScale', 'off', 'AutoScaleFactor', 1.0, 'MaxHeadSize', 0.7 ) + + xlabel( 'x' ); + ylabel( 'y' ); + axis( 'equal' ); + hold off; + + % histogram of x values: + subplot( 2, 2, 3 ); + hist( x, 50, 'b' ); + xlabel( 'x' ); + ylabel( 'count' ); + + % sort the eigenvalues: + [d,inx] = sort( diag(d), 'descend' ); + + subplot( 2, 2, 2 ); + hold on; + % subtract means: + x = x - mean( x ); + y = y - mean( y ); + % project onto eigenvectors: + nc = [ x y ] * v(:,inx); + cv = cov( nc ) + [ v , d] = eig( cv ) + if (pm > 0) & (nargin == 5) + if pm == 1 + % histogram of data: + [n,c] = hist3( nc, { xp, yp } ); + contourf( c{1}, c{2}, n' ); + else + % bivariate Gaussian: + gauss = reshape( exp(-0.5*diag(xy*inv(cv)*xy'))/sqrt((2.0*pi)^2.0*det(cv)), size( xg ) ); + contourf( xp, yp, gauss ) + end + else + % scatter plot: + scatter( nc(:,1), nc(:,2), 'b', 'filled', 'MarkerEdgeColor', 'white' ); + end + xlabel( 'n_x' ); + ylabel( 'n_y' ); + axis( 'equal' ); + hold off; + + % histogram of new x values: + subplot( 2, 2, 4 ); + hist( nc(:,1), 50, 'b' ); + xlabel( 'n_x' ); + ylabel( 'count' ); +end diff --git a/linearalgebra/code/pca2dexamples.m b/linearalgebra/code/pca2dexamples.m new file mode 100644 index 0000000..252ceda --- /dev/null +++ b/linearalgebra/code/pca2dexamples.m @@ -0,0 +1,33 @@ +scrsz = get( 0, 'ScreenSize' ); +set( 0, 'DefaultFigurePosition', [ scrsz(3)/2 scrsz(4)/2 scrsz(3)/2 scrsz(4)/2 ] ); + +% correlation coefficients: +n = 10000; +x = randn( n, 1 ); +f = figure( 1 ); +for r = 0.01:0.19:1 + fprintf( 'Correlation = %g\n', r ); + clf( f ); + y = r*x + sqrt(1-r^2)*randn( n, 1 ); + pca2d( x, y, 0, 5.0, 3.0 ); + waitforbuttonpress; +end + +% two distributions: +n = 10000; +x1 = randn( n/2, 1 ); +y1 = randn( n/2, 1 ); +x2 = randn( n/2, 1 ); +y2 = randn( n/2, 1 ); +f = figure( 1 ); +pause( 'on' ); +for d = 0:1:5 + fprintf( 'Distance = %g\n', d ); + clf( f ); + d2 = d / sqrt( 2.0 ); + x = [ x1; x2 ]; + y = [ y1+d2; y2-d2 ]; + scrsz = get(0,'ScreenSize'); + pca2d( x, y, 0, 10.0, 7.0 ); + waitforbuttonpress; +end diff --git a/linearalgebra/code/pca3d.m b/linearalgebra/code/pca3d.m new file mode 100644 index 0000000..6f2a6bd --- /dev/null +++ b/linearalgebra/code/pca3d.m @@ -0,0 +1,60 @@ +function pca3d( x, y, z ) +% computes covariance matrix from the triples x, y, z +% diagonalizes covariance matrix, i.e. performs pca on [ x, y, z ] +% x, y and z are column vectors + + % covariance matrix: + cv = cov( [ x y z ] ); + + % eigen values: + [ v , d] = eig( cv ) + + % plots: + subplot( 1, 2, 1 ); + hold on; + + % scatter plot: + view( 3 ); + scatter3( x, y, z, 0.1, 'b', 'filled', 'MarkerEdgeColor', 'blue' ); + xlabel( 'x' ); + ylabel( 'y' ); + zlabel( 'z' ); + grid on; + + % plot eigenvectors: + quiver3( ones( 1, 3 ).*mean( x ), ones( 1, 3 )*mean(y), ones( 1, 3 )*mean(z), v(1,:).*sqrt(diag(d))', v(2,:).*sqrt(diag(d))', v(3,:).*sqrt(diag(d))', 'r', 'LineWidth', 3, 'AutoScale', 'off', 'AutoScaleFactor', 1.0, 'MaxHeadSize', 0.7 ) + + %axis( 'equal' ); + hold off; + + % 2D scatter plots: + subplot( 2, 6, 4 ); + scatter( x, y, 'b', 'filled', 'MarkerEdgeColor', 'white' ); + xlabel( 'x' ) + ylabel( 'y' ) + subplot( 2, 6, 5 ); + scatter( x, z, 'b', 'filled', 'MarkerEdgeColor', 'white' ); + xlabel( 'x' ) + ylabel( 'z' ) + subplot( 2, 6, 6 ); + scatter( y, z, 'b', 'filled', 'MarkerEdgeColor', 'white' ); + xlabel( 'y' ) + ylabel( 'z' ) + + % sort the eigenvalues: + [d,inx] = sort( diag(d), 'descend' ); + + subplot( 2, 2, 4 ); + hold on; + % subtract means: + x = x - mean( x ); + y = y - mean( y ); + % project onto eigenvectors: + nx = [ x y z ] * v(:,inx); + scatter( nx(:,1), nx(:,2), 'b', 'filled', 'MarkerEdgeColor', 'white' ); + xlabel( 'ex' ) + ylabel( 'ey' ) + axis( 'equal' ); + hold off; + +end diff --git a/linearalgebra/code/pca3dexamples.m b/linearalgebra/code/pca3dexamples.m new file mode 100644 index 0000000..d2d790f --- /dev/null +++ b/linearalgebra/code/pca3dexamples.m @@ -0,0 +1,19 @@ +scrsz = get( 0, 'ScreenSize' ); +set( 0, 'DefaultFigurePosition', [ scrsz(3)/2 scrsz(4)/2 scrsz(3)/2 scrsz(4)/2 ] ); + +n = 10000; + +% three distributions: +x = randn( n, 1 ); +y = randn( n, 1 ); +z = randn( n, 1 ); +dx = [ 0 8 0 0 ]; +dy = [ 0 0 8 0 ]; +dz = [ 0 0 0 8 ]; +for k = 1:4 + x((k-1)*n/4+1:k*n/4) = x((k-1)*n/4+1:k*n/4) + dx(k); + y((k-1)*n/4+1:k*n/4) = y((k-1)*n/4+1:k*n/4) + dy(k); + z((k-1)*n/4+1:k*n/4) = z((k-1)*n/4+1:k*n/4) + dz(k); +end +f = figure( 1 ); +pca3d( x, y, z ); diff --git a/linearalgebra/simulations/plotvector.m b/linearalgebra/code/plotvector.m similarity index 100% rename from linearalgebra/simulations/plotvector.m rename to linearalgebra/code/plotvector.m diff --git a/linearalgebra/simulations/plotvector3.m b/linearalgebra/code/plotvector3.m similarity index 100% rename from linearalgebra/simulations/plotvector3.m rename to linearalgebra/code/plotvector3.m diff --git a/linearalgebra/code/simplematrix2dtrafos.m b/linearalgebra/code/simplematrix2dtrafos.m new file mode 100644 index 0000000..0b0cfe4 --- /dev/null +++ b/linearalgebra/code/simplematrix2dtrafos.m @@ -0,0 +1,21 @@ +% Visualize some common matrix transformations +% in a 2D coordinate system +% this uses the simplematrixbox() function for visualization +% use the matrix2dtrafos script for the the nicer matrixbox() function. + +disp( 'Click into the plot to advance to the next matrix' ) +simplematrixbox( [ 1 0; 0 1], 'Identity' ); +simplematrixbox( [ 2 0; 0 1], 'Scale x' ); +simplematrixbox( [ 1 0; 0 2], 'Scale y' ); +simplematrixbox( [ 2 0; 0 2], 'Scale x and y' ); +simplematrixbox( [ 0.5 0; 0 0.5], 'Scale x and y' ); +simplematrixbox( [ -1 0; 0 1], 'Flip x' ); +simplematrixbox( [ 1 0; 0 -1], 'Flip y' ); +simplematrixbox( [ -1 0; 0 -1], 'Flip both' ); +simplematrixbox( [ 1 1; 0 1], 'Shear x' ); +simplematrixbox( [ 1 0; 1 1], 'Shear y' ); +% rotation matrices: +for deg = 0:10:360 + phi = deg*pi/180.0; + simplematrixbox( [ cos(phi) -sin(phi); sin(phi) cos(phi)], sprintf( 'Rotate %.0f', deg ) ); +end diff --git a/linearalgebra/code/simplematrixbox.m b/linearalgebra/code/simplematrixbox.m new file mode 100644 index 0000000..bb94254 --- /dev/null +++ b/linearalgebra/code/simplematrixbox.m @@ -0,0 +1,21 @@ +function simplematrixbox( a, s ) +% visualizes the effect of a matrix a on a set of vectors forming a box +% a: a 2x2 matrix +% s: a string plotted as the title of the plot +% this function is called by simplematrix2dtrafos.m +% this is a simple version of the more elaborate matrixbox() function. + + x = [ 0 1 1 0 0 1; 0 0 1 1 0 1 ]; + plot( x(1,:), x(2,:), '-b' ); + hold on; + y = a*x; + plot( y(1,:), y(2,:), '-r' ); + [ ev ed ] = eig( a ); + ev = ev * ed; + quiver( [0 0], [0 0], ev(1,:), ev(2,:), 1.0, 'g' ) + hold off; + xlim( [-2 2 ] ) + ylim( [-2 2] ) + title( s ); + waitforbuttonpress; +end diff --git a/linearalgebra/code/soundmixture.m b/linearalgebra/code/soundmixture.m new file mode 100644 index 0000000..7cef0c8 --- /dev/null +++ b/linearalgebra/code/soundmixture.m @@ -0,0 +1,72 @@ +% read in wav files: +[s1, fs1 ] = audioread( 'PeterUndDerWolf.wav' ); +[s2, fs2 ] = audioread( 'Tielli-Kraulis.wav' ); +[s3, fs3 ] = audioread( 'Chorthippus_biguttulus.wav' ); + +% take out left channel and minimum number of samples: +n = min( [ size( s1, 1 ), size( s2, 1 ), size( s3, 1 ) ] ); +%n = 300000; +x1 = s1(1:n,1)'; +x2 = s2(1:n,1)'; +x3 = s3(1:n,1)'; +clear s1 s2 s3; + +% plot the sound waves: +% dt = 1/fs1; +% time = 0:dt:(length(x1)-1)*dt; +% plot( time, x1 ); +% plot( time, x2 ); +% plot( time, x3 ); + +% play original sounds: +% disp( 'Playing x1 (Peter und der Wolf)' ) +% a = audioplayer( x1, fs1 ); +% playblocking( a ); +% disp( 'Playing x2 (Martin Tielli)' ) +% a = audioplayer( x2, fs2 ); +% playblocking( a ); +% disp( 'Playing x3 (Grasshopper song)' ) +% a = audioplayer( x3, fs3 ); +% playblocking( a ); + +% mix them: +m = [ 1.0 0.3 0.4; 0.1 0.6 0.4; 0.7 0.1 0.7 ]; +y = m*[ x1; x2; x3 ]; + +% add noise: +%y = y + 0.03*randn(size(y)); + +% play mixtures: +% disp( 'Playing mixture y1' ) +% a = audioplayer( y(1,:), fs1 ); +% playblocking( a ); +% disp( 'Playing mixture y2' ) +% a = audioplayer( y(2,:), fs2 ); +% playblocking( a ); +% disp( 'Playing mixture y3' ) +% a = audioplayer( y(3,:), fs3 ); +% playblocking( a ); + +% demix them: +disp( 'Inverse mixing matrix:' ) +dm = inv(m) +dx = dm*y; + +% estimate demixing matrix: +size( y ) +cv = cov( y' ); +[ ev ed ] = eig( cv ); +disp( 'Estimated demixing matrix:' ) +ev + +%play mixtures: +% disp( 'Playing x1 (Peter und der Wolf) recovered from mixtures' ) +% a = audioplayer( dx(1,:), fs1 ); +% playblocking( a ); +% disp( 'Playing x3 (Martin Tielli) recovered from mixtures' ) +% a = audioplayer( dx(2,:), fs2 ); +% playblocking( a ); +% disp( 'Playing x3 (Grasshopper song) recovered from mixtures' ) +% a = audioplayer( dx(3,:), fs3 ); +% playblocking( a ); + diff --git a/linearalgebra/code/spikesortingwave.m b/linearalgebra/code/spikesortingwave.m new file mode 100644 index 0000000..da7c251 --- /dev/null +++ b/linearalgebra/code/spikesortingwave.m @@ -0,0 +1,118 @@ +% load data into time, voltage and spiketimes +load( 'extdata' ); + +% indices into voltage trace of spike times: +dt = time( 2) - time(1); +tinx = round(spiketimes/dt)+1; + +% plot voltage trace with dettected spikes: +figure( 1 ); +clf; +plot( time, voltage, '-b' ) +hold on +scatter( time(tinx), voltage(tinx), 'r', 'filled' ); +xlabel( 'time [ms]' ); +ylabel( 'voltage' ); +hold off + +% spike waveform snippets: +w = ceil( 0.005/dt ); +vs = []; +for k=1:length(tinx) + vs = [ vs; voltage(tinx(k)-w:tinx(k)+w-1) ]; +end +ts = time(1:size(vs,2)); +ts = ts - ts(floor(length(ts)/2)); + +% % plot snippets: +figure( 2 ); +clf; +hold on +for k=1:size(vs,1) + plot( ts, vs(k,:), '-b' ); +end +xlabel( 'time [ms]' ); +ylabel( 'voltage' ); +hold off + +% pca: +cv = cov( vs ); +[ ev , ed ] = eig( cv ); +[d,dinx] = sort( diag(ed), 'descend' ); + +figure( 3 ); +clf; +subplot( 4, 2, 1 ); +imagesc( cv ); +xlabel( 'time bin' ); +ylabel( 'time bin' ); +title( 'covariance matrix' ); +caxis([-0.1 0.1]) + +% spectrum of eigenvalues: +subplot( 4, 2, 2 ); +scatter( 1:length(d), d, 'b', 'filled' ); +xlabel( 'index' ); +ylabel( 'eigenvalue' ); + +% features: +subplot( 4, 2, 5 ); +plot( 1000.0*ts, ev(:,dinx(1)), 'r', 'LineWidth', 2 ); +xlabel( 'time [ms]' ); +ylabel( 'eigenvector 1' ); +subplot( 4, 2, 6 ); +plot( 1000.0*ts, ev(:,dinx(2)), 'g', 'LineWidth', 2 ); +xlabel( 'time [ms]' ); +ylabel( 'eigenvector 2' ); + +% project onto eigenvectors: +nx = vs * ev(:,dinx(1)); +ny = vs * ev(:,dinx(2)); +%scatter( nx, ny, 'b', 'filled', 'MarkerEdgeColor', 'white' ); + +% clustering (two clusters): +%kx = kmeans( [ nx, ny ], 2 ); +% nx smaller or greater a threshold: +kthresh = 1.6; +kx = ones( size( nx ) ); +kx(nx 0 ) & ( inx+wr <= size( stimulus, 1 ) ) + nspikes = nspikes + 1; + snip = stimulus( inx-wl:inx+wr, 2 ); + spikesnippets( nspikes, : ) = snip; % - mean(snip); + end + end + end + % delete not used snippets: + spikesnippets(nspikes+1:end,:) = []; + stavgtime = [-left:dt:right]; + meanrate = nspikes/length(spikes)/(stimulus(end,1)-stimulus(1,1)); + + % spike-triggered average: + stavg = mean( spikesnippets, 1 ); + + % loop over stimulus: + nstim = size( stimulus, 1 )-nw+1; + stimsnippets = zeros( nstim, nw ); + stimsnippetstime = zeros( nstim, 1 ); + for j=1:nstim + snip = stimulus( j:j+wr+wl, 2 ); + stimsnippets(j,:) = snip; % - mean(snip); + stimsnippetstime(j) = j*dt; + end + + % projection onto sta: + spikeonsta = spikesnippets * stavg'; + stimonsta = stimsnippets * stavg'; + + % projection onto orthogonal to sta: + orthos = null( stavg ); + mixcoef = rand(size( orthos, 2 ), 1); + mixcoef(length(mixcoef)/10:end) = 0.0; + staortho = orthos*mixcoef; + % stavg*staortho + staortho = staortho/(staortho'*staortho); + spikeonstaortho = spikesnippets * staortho; + stimonstaortho = stimsnippets * staortho; + + % psth binned: + ratetime = 0:dt:stimulus(end,1); +% rate = zeros( size( ratetime ) ); +% for k = 1:length( spikes ) +% [n, ~] = hist( spikes{k}, ratetime ); +% rate = rate + n/dt/length( spikes ); +% end + % kernel psth: + kernelsigma = 0.001; + windowtime = -5.0*kernelsigma:dt:5.0*kernelsigma; + window = normpdf( windowtime, 0.0, kernelsigma )/length(spikes); +% plot( 1000.0*windowtime, window ); + w2 = floor( length( windowtime )/2 ); + kernelrate = zeros( size( ratetime ) ); + for k = 1:length( spikes ) + times = spikes{k}; + for j = 1:length(times) + % index of spike in rate: + inx = round(times(j)/dt); + if ( inx - w2 > 0 ) & ( inx + w2 < length( kernelrate ) ) + kernelrate(inx-w2:inx+w2) = kernelrate(inx-w2:inx+w2) + window; + end + end + end + + if nargout == 0 + % sta plot: + figure( 1 ); + subplot( 3, 2, 1 ) + plot( 1000.0*stavgtime, staortho, '-r', 'LineWidth', 3 ) + hold on; + plot( 1000.0*stavgtime, stavg, '-b', 'LineWidth', 3 ) + hold off; + xlabel( 'time [ms]' ); + ylabel( 'stimulus' ); + title( 'Spike-triggered average' ); + legend( 'ortho', 'STA' ); + + % 2D scatter of projections: + subplot( 3, 2, 2 ) + scatter( stimonsta, stimonstaortho, 40.0, 'b', 'filled', 'MarkerEdgeColor', 'white' ) + hold on; + scatter( spikeonsta, spikeonstaortho, 20.0, 'r', 'filled', 'MarkerEdgeColor', 'white', 'LineWidth', 1.0 ) + hold off; + xlabel( 'projection sta' ); + ylabel( 'projection ortho' ); + title( 'Projections' ); + + % histogram of projections onto sta: + subplot( 3, 2, 3 ) + [ n, projections ] = hist( stimonsta, 50 ); + bw = (projections(2)-projections(1)); + pstim = n / sum(n)/bw; + bar( projections, pstim, 'b' ); + hold on; + [ n, ~ ] = hist( spikeonsta, projections ); + pstimspike = n / sum(n)/bw; + bar( projections, pstimspike, 'r' ); + xlabel( 'projection x' ); + title( 'Projection onto STA' ); + xlim( [projections(1) projections(end)] ); + hold off; + legend( 'p(x)', 'p(x|spikes)' ); + + % nonlinearity for orthogonal projections: + subplot( 3, 2, 5 ) + nonlinearity = meanrate*pstimspike./pstim; + plot( projections(pstim>0.01), nonlinearity(pstim>0.01), '-r', 'LineWidth', 3 ); + xlim( [projections(1) projections(end)] ); + ylim( [0 1000 ]) + xlabel( 'projection x' ); + ylabel( 'meanrate*p(x|spikes)/p(x) [Hz]' ); + title( 'Nonlinearity'); + + % histogram of projections onto orthogonal sta: + subplot( 3, 2, 4 ) + [ n, ~] = hist( stimonstaortho, projections ); + pstimortho = n / sum(n)/bw; + bar( projections, pstimortho, 'b' ); + hold on; + [ n, ~ ] = hist( spikeonstaortho, projections ); + pstimspikeortho = n / sum(n)/bw; + bar( projections, pstimspikeortho, 'r' ); + xlim( [projections(1) projections(end)] ); + xlabel( 'projection x' ); + title( 'Projection onto orthogonal to STA' ); + hold off; + legend( 'p(x)', 'p(x|spikes)' ); + + % nonlinearity for orthogonal projections: + subplot( 3, 2, 6 ) + nonlinearityortho = meanrate*pstimspikeortho./pstimortho; + plot( projections(pstimortho>0.01), nonlinearityortho(pstimortho>0.01), '-r', 'LineWidth', 3 ); + xlim( [projections(1) projections(end)] ); + ylim( [0 1000 ]) + xlabel( 'projection x' ); + ylabel( 'meanrate*p(x|spikes)/p(x) [Hz]' ); + title( 'Nonlinearity'); + end + + % stimulus reconstruction with STA: + stareconstruction = zeros( size( stimulus, 1 ), 1 ); + for k = 1:length( spikes ) + times = spikes{k}; + for j = 1:length(times) + % index of spike in stimulus: + inx = round(times(j)/dt); + if ( inx-wl > 0 ) & ( inx+wr <= size( stimulus, 1 ) ) + stareconstruction( inx-wl:inx+wr ) = stareconstruction( inx-wl:inx+wr ) + stavg'; + end + end + end + stareconstruction = stareconstruction/length(spikes); + + if nargout == 0 + % linear stimulus reconstruction: + figure( 2 ) + ax1 = subplot( 3, 1, 1 ); + plot( 1000.0*stimulus(:,1), stimulus(:,2), 'g', 'LineWidth', 2 ); + hold on; + plot( 1000.0*stimulus(:,1), stareconstruction, 'r', 'LineWidth', 2 ); + ylabel( 'stimulus' ); + hold off; + legend( 'stimulus', 'reconstruction' ); + + % stimulus projection onto sta: + ax2 = subplot( 3, 1, 2 ); + % plot( 1000.0*ratetime, rate, '-b', 'LineWidth', 2 ) + plot( 1000.0*ratetime, kernelrate-mean(kernelrate), '-b', 'LineWidth', 2 ) + hold on; + plot( 1000.0*(stimsnippetstime+left), 2.0*stimonsta*meanrate, 'r', 'LineWidth', 2 ); + ylabel( 'rate [Hz]' ); + hold off; + legend( 'rate', 'stimulus projection on STA' ); + + % sta with nonlinearity: + ax3 = subplot( 3, 1, 3 ); + plot( 1000.0*ratetime, kernelrate, '-b', 'LineWidth', 2 ) + hold on; + xmax = max(projections(pstim>0.01)); + stimonstax = stimonsta; + stimonstax( stimonstax > xmax ) = xmax; + sinx = round((stimonstax-projections(1))/bw); + sinx( sinx < 1 ) = 1; + stastimulus = nonlinearity( sinx ); + plot( 1000.0*(stimsnippetstime+left), stastimulus, 'r', 'LineWidth', 2 ); + legend( 'rate', 'LNP' ); + xlabel( 'time [ms]' ); + ylabel( 'rate [Hz]' ); + hold off; + linkaxes( [ax1 ax2 ax3], 'x' ); + end + +end diff --git a/linearalgebra/code/stastcexample.m b/linearalgebra/code/stastcexample.m new file mode 100644 index 0000000..32500fd --- /dev/null +++ b/linearalgebra/code/stastcexample.m @@ -0,0 +1,31 @@ +clear; +addpath( '../../pointprocesses/simulations/' ); + +set( 0, 'DefaultTextFontSize', 22.0 ); +set( 0, 'DefaultAxesFontSize', 22.0 ); + +% load data +load( 'p-unit_stimulus.mat' ); +load( 'p-unit_spike_times.mat' ); +%load( 'pyramidal_noise_2014-09-02-af_cutoff_50_contr_5.mat' ); +% load( 'pyramidal_noise_2014-09-02-ab_cutoff_300_contr_5.mat' ); +% for k = 1:length( spike_times ) +% spike_times{k} = spike_times{k} * 0.001; +% end +% stimulus = stimulus(1:20000*5,:); + +%dt = stimulus(2,1) - stimulus(1,1); +%spike_times = lifadaptspikes( 20, 50+100.0*stimulus(:,2), dt, 0.1, 0.04, 10.0 ); +%spike_times = lifspikes( 20, 12+30.0*stimulus(:,2), dt, 0.1 ); +%spike_times = lifspikes( 20, 20.0+30.0*stimulus(:,2), dt, 0.0001 ); + +%figure( 1 ); +%clf; +%spikeraster( spike_times ); +%isivec = isis( spike_times ); +%isihist( isivec ); +%isiserialcorr( isivec, 20 ); +%fano( spike_times ); + +sta( stimulus(1:5:length(stimulus),:), spike_times, 0.05, 0.01 ); +%stc( stimulus(1:5:length(stimulus),:), spike_times, 0.05, 0.01 ); diff --git a/linearalgebra/code/stc.m b/linearalgebra/code/stc.m new file mode 100644 index 0000000..c787b78 --- /dev/null +++ b/linearalgebra/code/stc.m @@ -0,0 +1,66 @@ +function stc( stimulus, spikes, left, right ) +% computes the spike-triggered covariance matrix +% stimulus: the stimulus as a nx2 matrix with the first column being time +% in seconds and the second column being the actual stimulus +% spikes: a cell array of vectors of spike times +% left: the time to the left of each spike +% right: the time to the right of each spike + + % time indices: + dt = stimulus(2,1) - stimulus(1,1); + wl = round( left/dt ); + wr = round( right/dt ); + nw = wl+wr+1; + + % spike-triggered average with snippets: + [ stavg, stavgtime, spikesnippets, stimsnippets, meanrate ] = sta( stimulus, spikes, left, right ); + + % spike-triggered covariance matrix: + figure( 3 ); + subplot( 2, 2, 1 ); + spikescv = cov( spikesnippets ); + imagesc( spikescv ); + caxis([-0.1 0.1]) + + % stimulus covariance matrix: + subplot( 2, 2, 2 ); + stimcv = cov( stimsnippets ); + imagesc( stimcv ); + caxis([-0.1 0.1]) + + subplot( 2, 1, 2 ); + imagesc( spikescv-stimcv ); + caxis([-0.01 0.01]) + + % eigenvalues: + %[ ev , ed ] = eig( spikescv-stimcv ); + [ ev , ed ] = eig( spikescv ); + [d,dinx] = sort( diag(ed), 'descend' ); + + % spectrum of eigenvalues: + figure( 4 ); + subplot( 3, 1, 1 ); + scatter( 1:length(d), d, 'b', 'filled' ); +% scatter( 1:length(d), d/sum(abs(d)), 'b', 'filled' ); + xlabel( 'index' ); + ylabel( 'eigenvalue [% variance]' ); + + % features: + subplot( 3, 2, 5 ); + plot( 1000.0*stavgtime, ev(:,dinx(1)), 'g', 'LineWidth', 2 ); + xlabel( 'time [ms]' ); + ylabel( 'eigenvector 1' ); + subplot( 3, 2, 6 ); + plot( 1000.0*stavgtime, ev(:,dinx(2)), 'r', 'LineWidth', 2 ); + xlabel( 'time [ms]' ); + ylabel( 'eigenvector 2' ); + + % project onto eigenvectors: + nx = spikesnippets * ev(:,dinx(1)); + ny = spikesnippets * ev(:,dinx(2)); + subplot( 3, 1, 2 ); + scatter( nx, ny, 'b', 'filled', 'MarkerEdgeColor', 'white' ); + xlabel( 'projection onto eigenvector 1' ); + ylabel( 'projection onto eigenvector 2' ); + +end diff --git a/linearalgebra/exercises/correlations.pdf b/linearalgebra/exercises/correlations.pdf new file mode 100644 index 0000000..4ca95f8 Binary files /dev/null and b/linearalgebra/exercises/correlations.pdf differ diff --git a/linearalgebra/exercises/correlations.tex b/linearalgebra/exercises/correlations.tex new file mode 100644 index 0000000..222e15d --- /dev/null +++ b/linearalgebra/exercises/correlations.tex @@ -0,0 +1,95 @@ +\documentclass[addpoints,10pt]{exam} +\usepackage{url} +\usepackage{color} +\usepackage{hyperref} +\usepackage{graphicx} +\usepackage{amsmath} + +\pagestyle{headandfoot} +\runningheadrule +\firstpageheadrule + +\firstpageheader{Scientific Computing}{Principal Component Analysis}{Oct 29, 2014} +%\runningheader{Homework 01}{Page \thepage\ of \numpages}{23. October 2014} +\firstpagefooter{}{}{} +\runningfooter{}{}{} +\pointsinmargin +\bracketedpoints + +%\printanswers +\shadedsolutions + +\usepackage[mediumspace,mediumqspace,Gray]{SIunits} % \ohm, \micro + +%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\usepackage{listings} +\lstset{ + basicstyle=\ttfamily, + numbers=left, + showstringspaces=false, + language=Matlab, + breaklines=true, + breakautoindent=true, + columns=flexible, + frame=single, + captionpos=t, + xleftmargin=2em, + xrightmargin=1em, + aboveskip=10pt, + %title=\lstname, + title={\protect\filename@parse{\lstname}\protect\filename@base.\protect\filename@ext} + } + + +\begin{document} + +\sffamily +%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{questions} + \question \textbf{Gaussian distribution} + \begin{parts} + \part Use \texttt{randn} to generate 1000000 normally (zero mean, unit variance) distributed random numbers. + \part Plot a properly normalized histogram of these random numbers. + \part Compare the histogram with the probability density of the Gaussian distribution + \[ p(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}} \] + where $\mu$ is the mean and $\sigma^2$ is the variance of the Gaussian distribution. + \part Generate Gaussian distributed random numbers with mean $\mu=2$ and + standard deviation $\sigma=\frac{1}{2}$. + \end{parts} + + \question \textbf{Covariance and correlation coefficient} + \begin{parts} + \part Generate two vectors $x$ and $z$ with Gausian distributed random numbers. + \part Compute $y$ as a linear combination of $x$ and $z$ according to + \[ y = r \cdot x + \sqrt{1-r^2}\cdot z \] + where $r$ is a parameter $-1 \le r \le 1$. + What does $r$ do? + \part Plot a scatter plot of $y$ versus $x$ for about 10 different values of $r$. + What do you observe? + \part Also compute the covariance matrix and the correlation + coefficient matrix between $x$ and $y$ (functions \texttt{cov} and + \texttt{corrcoef}). How do these matrices look like for different + values of $r$? How do the values of the matrices change if you generate + $x$ and $z$ with larger variances? + \part Do the same analysis (Scatter plot, covariance, and correlation coefficient) + for \[ y = x^2 + 0.5 \cdot z \] + Are $x$ and $y$ really independent? + \end{parts} + + \question \textbf{Principal component analysis} + \begin{parts} + \part Generate pairs $(x,y)$ of Gaussian distributed random numbers such + that all $x$ values have zero mean, half of the $y$ values have mean $+d$ + and the other half mean $-d$, with $d \ge0$. + \part Plot scatter plots of the pairs $(x,y)$ for $d=0$, 1, 2, 3, 4 and 5. + Also plot a histogram of the $x$ values. + \part Apply PCA on the data and plot a histogram of the data projected onto + the PCA axis with the largest eigenvalue. + What do you observe? + \end{parts} + +\end{questions} + + +\end{document} diff --git a/linearalgebra/exercises/matrices.pdf b/linearalgebra/exercises/matrices.pdf new file mode 100644 index 0000000..3d38055 Binary files /dev/null and b/linearalgebra/exercises/matrices.pdf differ diff --git a/linearalgebra/exercises/matrices.tex b/linearalgebra/exercises/matrices.tex new file mode 100644 index 0000000..4897644 --- /dev/null +++ b/linearalgebra/exercises/matrices.tex @@ -0,0 +1,205 @@ +\documentclass[addpoints,10pt]{exam} +\usepackage{url} +\usepackage{color} +\usepackage{hyperref} +\usepackage{graphicx} +\usepackage{amsmath} + +\pagestyle{headandfoot} +\runningheadrule +\firstpageheadrule + +\firstpageheader{Scientific Computing}{Matrix multiplication}{Oct 28, 2014} +%\runningheader{Homework 01}{Page \thepage\ of \numpages}{23. October 2014} +\firstpagefooter{}{}{} +\runningfooter{}{}{} +\pointsinmargin +\bracketedpoints + +%\printanswers +\shadedsolutions + +\usepackage[mediumspace,mediumqspace,Gray]{SIunits} % \ohm, \micro + +%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\usepackage{listings} +\lstset{ + basicstyle=\ttfamily, + numbers=left, + showstringspaces=false, + language=Matlab, + breaklines=true, + breakautoindent=true, + columns=flexible, + frame=single, + captionpos=t, + xleftmargin=2em, + xrightmargin=1em, + aboveskip=10pt, + %title=\lstname, + title={\protect\filename@parse{\lstname}\protect\filename@base.\protect\filename@ext} + } + + +\begin{document} + +\sffamily +%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{questions} + \question \textbf{Matrix multiplication} + Calculate the results of the following matrix multiplications and + confirm the result using matlab. + \[ \begin{pmatrix} 2 \\ -4 \\ -1 \end{pmatrix} \cdot + \begin{pmatrix} 3 & -4 & -4 \end{pmatrix} = \] + + \[ \begin{pmatrix} 3 & -3 & -1 \end{pmatrix} \cdot + \begin{pmatrix} 2 \\ 3 \\ 0 \end{pmatrix} = \] + + \[ \begin{pmatrix} 4 & -1 & 2 \\ -1 & 3 & 1 \\ 4 & -2 & 1 \\ 4 & -3 & -2 \end{pmatrix} \cdot + \begin{pmatrix} -2 & -2 & 0 & -3 \\ 3 & -2 & 1 & 0 \\ 1 & -2 & -4 & 0 \end{pmatrix} = \] + + \[ \begin{pmatrix} 3 & 1 \\ 1 & 4 \end{pmatrix} \cdot + \begin{pmatrix} 0 & -3 & 4 & 1 \\ -2 & -1 & -2 & -3 \\ -3 & 1 & -2 & -3 \end{pmatrix} = \] + + \[ \begin{pmatrix} 1 & 1 & -4 \end{pmatrix} \cdot + \begin{pmatrix} -1 \\ 2 \end{pmatrix} = \] + + \[ \begin{pmatrix} 3 & 1 & -2 \\ 2 & 1 & 3 \\ 1 & 1 & 2 \end{pmatrix} \cdot + \begin{pmatrix} 2 & 2 \\ -3 & 3 \\ -4 & 1 \end{pmatrix} = \] + + \[ \begin{pmatrix} 3 \\ 2 \end{pmatrix} \cdot + \begin{pmatrix} -3 & 2 & -4 & 1 \end{pmatrix} = \] + + \[ \begin{pmatrix} -1 \\ -4 \\ -1 \end{pmatrix} \cdot + \begin{pmatrix} 0 & -4 & 1 \end{pmatrix} = \] + + \[ \begin{pmatrix} 4 & -2 & -2 & -4 \end{pmatrix} \cdot + \begin{pmatrix} 2 \\ 2 \\ 1 \\ -1 \end{pmatrix} = \] + + \[ \begin{pmatrix} -2 & -3 & -4 \\ 1 & 3 & 2 \\ -4 & -2 & 1 \end{pmatrix} \cdot + \begin{pmatrix} 1 & 2 & -2 & 4 \\ 3 & -1 & 1 & -1 \\ -3 & 2 & -1 & 2 \end{pmatrix} = \] + + \[ \begin{pmatrix} 2 & -4 & 4 & 4 \\ -3 & 3 & 2 & 1 \end{pmatrix} \cdot + \begin{pmatrix} 0 & 3 & 4 & -2 \\ -4 & -2 & -1 & 0 \\ 1 & 2 & -4 & -4 \\ 3 & 2 & -2 & -4 \end{pmatrix} = \] + + \[ \begin{pmatrix} 3 & 1 & -2 & -2 \end{pmatrix} \cdot + \begin{pmatrix} -4 \\ 3 \\ -2 \\ 4 \end{pmatrix} = \] + + \[ \begin{pmatrix} -1 & 3 & 4 \end{pmatrix} \cdot + \begin{pmatrix} -1 \\ 4 \\ -3 \end{pmatrix} = \] + + \[ \begin{pmatrix} 1 & -4 & 3 & 3 \end{pmatrix} \cdot + \begin{pmatrix} 1 \\ 0 \\ -4 \\ -1 \end{pmatrix} = \] + + \[ \begin{pmatrix} -4 & -4 & -3 \\ -2 & -2 & 4 \\ -3 & 4 & -3 \end{pmatrix} \cdot + \begin{pmatrix} 0 & 3 & -4 & 4 \\ -1 & -2 & -3 & 1 \\ 1 & -2 & 2 & 0 \end{pmatrix} = \] + + \[ \begin{pmatrix} -3 & 0 & 4 & 1 \\ 0 & 1 & 1 & 4 \end{pmatrix} \cdot + \begin{pmatrix} -4 & 3 & 1 & 4 \\ 1 & -4 & 1 & -3 \\ -4 & 0 & -4 & -4 \\ 1 & -2 & 4 & 4 \end{pmatrix} = \] + + \[ \begin{pmatrix} 4 \\ 3 \\ 4 \\ -2 \end{pmatrix} \cdot + \begin{pmatrix} 2 & 4 & 3 & 3 \end{pmatrix} = \] + + \[ \begin{pmatrix} 1 & 2 & 0 & 3 \end{pmatrix} \cdot + \begin{pmatrix} -3 \\ 1 \\ 1 \end{pmatrix} = \] + + \[ \begin{pmatrix} -4 & 0 & -1 & 3 \\ 0 & -4 & 3 & -3 \end{pmatrix} \cdot + \begin{pmatrix} -1 & -4 & -1 \\ 3 & 2 & 0 \\ -2 & 3 & -2 \\ 1 & 2 & -2 \end{pmatrix} = \] + + \[ \begin{pmatrix} 2 & 0 & 3 \\ 1 & -4 & -1 \\ 3 & 0 & -2 \end{pmatrix} \cdot + \begin{pmatrix} 0 & 2 & -1 & -2 \\ -1 & -1 & -3 & 4 \\ 2 & 4 & -4 & 1 \end{pmatrix} = \] + \[ \begin{pmatrix} -1 & 4 \end{pmatrix} \cdot + \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \] + + \[ \begin{pmatrix} -4 & 3 \\ -4 & 0 \\ -2 & -2 \end{pmatrix} \cdot + \begin{pmatrix} 0 & 1 & -4 & 2 \\ 2 & 3 & -2 & -1 \end{pmatrix} = \] + + \[ \begin{pmatrix} -2 & -1 \end{pmatrix} \cdot + \begin{pmatrix} 1 \\ -2 \end{pmatrix} = \] + + \[ \begin{pmatrix} -2 & 2 & -2 & -3 \\ 2 & -4 & -2 & 2 \\ 0 & 2 & -2 & -2 \\ 1 & -2 & -2 & -2 \end{pmatrix} \cdot + \begin{pmatrix} 1 & -2 & 2 \\ -4 & -2 & -2 \\ 3 & 1 & 4 \\ -4 & 1 & -2 \end{pmatrix} = \] + + \[ \begin{pmatrix} -1 & -3 & 0 & -1 \\ 4 & -2 & 1 & 2 \end{pmatrix} \cdot + \begin{pmatrix} -3 & -4 \\ -4 & 0 \end{pmatrix} = \] + + \[ \begin{pmatrix} -1 & 1 & -2 \\ -2 & 2 & -4 \\ 1 & -2 & -2 \end{pmatrix} \cdot + \begin{pmatrix} -1 & 2 & -4 \\ 1 & 3 & 0 \\ 1 & 4 & -4 \end{pmatrix} = \] + + \[ \begin{pmatrix} -3 & 3 \\ -3 & 2 \end{pmatrix} \cdot + \begin{pmatrix} 2 & -3 \\ -2 & -4 \end{pmatrix} = \] + + \[ \begin{pmatrix} 1 & 1 & -3 \end{pmatrix} \cdot + \begin{pmatrix} -1 \\ -2 \\ 3 \end{pmatrix} = \] + + \[ \begin{pmatrix} -4 & 2 & 1 \\ 4 & 0 & -2 \\ 2 & 3 & -3 \\ -2 & -2 & -2 \end{pmatrix} \cdot + \begin{pmatrix} -1 & 2 & 0 & -2 \\ 2 & -2 & 0 & -1 \\ -4 & 3 & -3 & 4 \end{pmatrix} = \] + + \[ \begin{pmatrix} -2 & -4 & 2 & 4 \\ 3 & -3 & 2 & 1 \end{pmatrix} \cdot + \begin{pmatrix} 0 & 4 & -1 & -4 \\ 2 & 3 & -4 & -1 \\ 3 & 2 & -2 & 4 \end{pmatrix} = \] + + \[ \begin{pmatrix} -3 & -2 & -1 & -3 \end{pmatrix} \cdot + \begin{pmatrix} 2 \\ -2 \\ 3 \\ -2 \end{pmatrix} = \] + + \[ \begin{pmatrix} 4 & 4 & 2 & 3 \end{pmatrix} \cdot + \begin{pmatrix} 3 \\ 3 \\ -2 \\ 1 \end{pmatrix} = \] + + \[ \begin{pmatrix} 3 & 2 & -2 \end{pmatrix} \cdot + \begin{pmatrix} 2 \\ 4 \\ 3 \end{pmatrix} = \] + + \[ \begin{pmatrix} 2 & -1 & 0 & -2 \\ 0 & -4 & -3 & -1 \end{pmatrix} \cdot + \begin{pmatrix} 4 & -3 & 2 & 4 \\ -3 & -4 & 1 & 1 \\ 1 & 3 & -2 & 3 \\ -1 & -2 & 3 & 0 \end{pmatrix} = \] + + \[ \begin{pmatrix} -3 & -3 & 3 & 2 \\ 2 & 2 & -3 & 1 \end{pmatrix} \cdot + \begin{pmatrix} 0 & 1 \\ 4 & 2 \\ -3 & -1 \\ -3 & 4 \end{pmatrix} = \] + + \[ \begin{pmatrix} -4 & -3 \end{pmatrix} \cdot + \begin{pmatrix} -2 \\ 3 \\ 4 \end{pmatrix} = \] + + \[ \begin{pmatrix} 4 & 4 \end{pmatrix} \cdot + \begin{pmatrix} 1 \\ 4 \\ -1 \end{pmatrix} = \] + + \[ \begin{pmatrix} 1 & -2 & 3 \end{pmatrix} \cdot + \begin{pmatrix} 3 \\ 1 \\ 2 \end{pmatrix} = \] + + \[ \begin{pmatrix} -3 & 2 \end{pmatrix} \cdot + \begin{pmatrix} -1 \\ 1 \end{pmatrix} = \] + + \[ \begin{pmatrix} -2 & -4 & -4 & 0 \\ 0 & 3 & 4 & -4 \\ 4 & 2 & -2 & -4 \\ 0 & 0 & 4 & -1 \end{pmatrix} \cdot + \begin{pmatrix} 0 & -1 \\ -1 & 1 \\ -4 & -3 \\ 2 & 1 \end{pmatrix} = \] + + \[ \begin{pmatrix} -3 \\ 3 \\ -3 \\ -4 \end{pmatrix} \cdot + \begin{pmatrix} 2 & 4 & -2 & 1 \end{pmatrix} = \] + + \[ \begin{pmatrix} 2 \\ 0 \end{pmatrix} \cdot + \begin{pmatrix} -1 & -3 & -2 & 2 \end{pmatrix} = \] + + \[ \begin{pmatrix} 0 & -4 & -4 & 4 \end{pmatrix} \cdot + \begin{pmatrix} 1 \\ 4 \\ 0 \\ 4 \end{pmatrix} = \] + + \[ \begin{pmatrix} -3 & -1 \\ -3 & -1 \end{pmatrix} \cdot + \begin{pmatrix} 0 & -3 & 3 & -2 \\ -4 & 1 & -1 & 4 \end{pmatrix} = \] + + \[ \begin{pmatrix} 4 & 0 \\ -1 & 4 \\ 1 & -3 \end{pmatrix} \cdot + \begin{pmatrix} -4 & -4 \\ -4 & 2 \end{pmatrix} = \] + + \[ \begin{pmatrix} -1 \\ 3 \\ 2 \\ 4 \end{pmatrix} \cdot + \begin{pmatrix} 0 & -1 & 0 & 0 \end{pmatrix} = \] + + \[ \begin{pmatrix} 3 \\ -2 \\ 2 \\ 3 \end{pmatrix} \cdot + \begin{pmatrix} -2 & -3 & -4 & 2 \end{pmatrix} = \] + + \[ \begin{pmatrix} 2 & -2 & -4 & 4 \\ 0 & 1 & -3 & -2 \\ -1 & 3 & 0 & -2 \end{pmatrix} \cdot + \begin{pmatrix} -4 & 1 \\ -4 & 3 \end{pmatrix} = \] + + \[ \begin{pmatrix} -4 & -1 & 3 \end{pmatrix} \cdot + \begin{pmatrix} -4 \\ -3 \\ 3 \end{pmatrix} = \] + + \question \textbf{Automatic generation of exercises} + Write some matlab code that generates exercises like this one automatically! :-) + +\end{questions} + + +\end{document} diff --git a/linearalgebra/resources/Pillow2007-STC-LNPmodel.pdf b/linearalgebra/resources/Pillow2007-STC-LNPmodel.pdf new file mode 100644 index 0000000..da7133a Binary files /dev/null and b/linearalgebra/resources/Pillow2007-STC-LNPmodel.pdf differ diff --git a/linearalgebra/simulations/covareigen3examples.m b/linearalgebra/simulations/covareigen3examples.m deleted file mode 100644 index 3d16628..0000000 --- a/linearalgebra/simulations/covareigen3examples.m +++ /dev/null @@ -1,14 +0,0 @@ -n = 10000; - -% three distributions: -x = randn( n, 1 ); -y = randn( n, 1 ); -z = randn( n, 1 ); -f = figure( 'Position', [ scrsz(3)/2 scrsz(4)/2 scrsz(3)/2 scrsz(4)/2 ]); -d = [ 0 4 -4 8 ]; -for k = 1:4 - x((k-1)*n/4+1:k*n/4) = x((k-1)*n/4+1:k*n/4) + d(k); - y((k-1)*n/4+1:k*n/4) = y((k-1)*n/4+1:k*n/4) - d(k); - z((k-1)*n/4+1:k*n/4) = z((k-1)*n/4+1:k*n/4) + d(k); -end -covareigen3( x, y, z ); diff --git a/linearalgebra/simulations/matrix2dtrafos.m b/linearalgebra/simulations/matrix2dtrafos.m deleted file mode 100644 index da6840e..0000000 --- a/linearalgebra/simulations/matrix2dtrafos.m +++ /dev/null @@ -1,74 +0,0 @@ -m = [ 1 0; 0 1]; -matrixbox( m ); -title( 'Identity' ); -waitforbuttonpress; - -m = [ 2 0; 0 1]; -matrixbox( m ); -title( 'Scale x' ); -waitforbuttonpress; - -m = [ 1 0; 0 2]; -matrixbox( m ); -title( 'Scale y' ); -waitforbuttonpress; - -m = [ 2 0; 0 2]; -matrixbox( m ); -title( 'Scale both' ); -waitforbuttonpress; - -m = [ -1 0; 0 1]; -matrixbox( m ); -title( 'Flip x' ); -waitforbuttonpress; - -m = [ 1 0; 0 -1]; -matrixbox( m ); -title( 'Flip y' ); -waitforbuttonpress; - -m = [ -1 0; 0 -1]; -matrixbox( m ); -title( 'Flip both' ); -waitforbuttonpress; - -m = [ 1 0; 1 1]; -matrixbox( m ); -title( 'Shear x' ); -waitforbuttonpress; - -m = [ 1 1; 0 1]; -matrixbox( m ); -title( 'Shear y' ); -waitforbuttonpress; - -phi = 0.1667*pi; -m = [ cos(phi) -sin(phi); sin(phi) cos(phi)]; -matrixbox( m ); -title( 'Rotate 30' ); -waitforbuttonpress; - -phi = 0.333*pi; -m = [ cos(phi) -sin(phi); sin(phi) cos(phi)]; -matrixbox( m ); -title( 'Rotate 60' ); -waitforbuttonpress; - -phi = 0.5*pi; -m = [ cos(phi) -sin(phi); sin(phi) cos(phi)]; -matrixbox( m ); -title( 'Rotate 90' ); -waitforbuttonpress; - -phi = 0.75*pi; -m = [ cos(phi) -sin(phi); sin(phi) cos(phi)]; -matrixbox( m ); -title( 'Rotate 135' ); -waitforbuttonpress; - -phi = 1.0*pi; -m = [ cos(phi) -sin(phi); sin(phi) cos(phi)]; -matrixbox( m ); -title( 'Rotate 180' ); -waitforbuttonpress; diff --git a/linearalgebra/simulations/matrixbox.m b/linearalgebra/simulations/matrixbox.m deleted file mode 100644 index 5abbfa0..0000000 --- a/linearalgebra/simulations/matrixbox.m +++ /dev/null @@ -1,24 +0,0 @@ -function matrixbox( m ) -% visualizes the effect of a matrix m on a set of vectors forming a box -% m: a 2x2 matrix - - v = [ 0 1 1 0 0; 0 0 1 1 0]; - w = m*v; - clf; - hold on; - %set(gca, 'Xtick', [], 'Ytick', [], 'box', 'off') - % axis: - plot( [-2 2], [0 0], 'k', 'LineWidth', 1 ); - plot( [0 0], [-2 2], 'k', 'LineWidth', 1 ); - % old box: - plot( v(1,:), v(2,:), 'k', 'LineWidth', 1.0 ) - quiver( [v(1,1)], [v(2,1)], [v(1,3)], [v(2,3)], 1.0, 'k', 'LineWidth', 2.0 ); - scatter( [v(1,2)], [v(2,2)], 60.0, 'filled', 'k' ); - % transfomred box: - plot( w(1,:), w(2,:), 'b', 'LineWidth', 2.0 ) - quiver( [w(1,1)], [w(2,1)], [w(1,3)], [w(2,3)], 1.0, 'b', 'LineWidth', 3.0 ); - scatter( [w(1,2)], [w(2,2)], 100.0, 'filled', 'b' ); - hold off; - xlim( [ -2 2 ] ); - ylim( [ -2 2 ] ); -end diff --git a/linearalgebra/simulations/spikesortingwave.m b/linearalgebra/simulations/spikesortingwave.m deleted file mode 100644 index 369915e..0000000 --- a/linearalgebra/simulations/spikesortingwave.m +++ /dev/null @@ -1,127 +0,0 @@ -% generate spikes: -n = 1000; -misi = 0.01; -isis = exprnd( misi, n, 1 ); -isis = isis + 0.01; -spikes = cumsum( isis ); -p = rand( size( spikes ) ); - -% spike waveforms: -dt = 0.0001; -x = -0.01:dt:0.01; -y1 = 2.0*exp( -(x-0.0003).^2/2.0/0.0005^2 ) - 1.4*exp( -(x-0.0005).^2/2.0/0.0005^2 ); -y2 = exp( -x.^2/2.0/0.002^2 ).*cos(2.0*pi*x/0.005); -p12 = 0.5; - -% voltage trace: -noise = 0.2; -t = 0:dt:spikes(end)+3.0*x(end); -v = noise*randn( 1, length( t ) ); -for k = 1:length( spikes ) - inx = ceil( spikes(k)/dt ); - if p(k) < p12 - v(inx:inx+length(x)-1) = v(inx:inx+length(x)-1) + y1; - else - v(inx:inx+length(x)-1) = v(inx:inx+length(x)-1) + y2; - end -end - -figure( 1 ); -clf; -plot( t(t<1.0), v(t<1.0) ); -hold on; - -% find peaks: -thresh = 0.7; -inx = find( v(2:end-1) > thresh & v(1:end-2) v(3:end) ) + 1; -spiket = t(inx); -spikev = v(inx); -tinx = inx; -for k=1:2 - inx = find( ( spiket(2:end-1)-spiket(1:end-2)>0.005 | spikev(2:end-1) > spikev(1:end-2) ) & ( spiket(3:end)-spiket(2:end-1)>0.005 | spikev(2:end-1) > spikev(3:end) ) )+1; - spiket = spiket(inx); - spikev = spikev(inx); - tinx = tinx(inx); -end -scatter( spiket(spiket<1.0), spikev(spiket<1.0), 100.0, 'r', 'filled' ); -%scatter( t(tinx), v(tinx), 100.0, 'r', 'filled' ); -hold off; - -% spike waveform snippets: -w = ceil( 0.005/dt ); -vs = []; -for k=1:length(tinx) - vs = [ vs; v(tinx(k)-w:tinx(k)+w-1) ]; -end -ts = t(1:size(vs,2)); -ts = ts - ts(floor(length(ts)/2)); -figure( 2 ); -clf; -hold on -for k=1:size(vs,1) - plot( ts, vs(k,:), '-b' ); -end -hold off - -% pca: -cv = cov( vs ); -[ ev , ed ] = eig( cv ); -[d,dinx] = sort( diag(ed), 'descend' ); -% spectrum of eigenvalues: -figure( 3 ); -clf; -subplot( 4, 1, 1 ); -scatter( 1:length(d), d, 'b', 'filled' ); -% project onto eigenvectors: -nx = vs * ev(:,dinx(1)); -ny = vs * ev(:,dinx(2)); -%scatter( nx, ny, 'b', 'filled', 'MarkerEdgeColor', 'white' ); -%hold on; - -% features: -subplot( 4, 2, 5 ); -plot( 1000.0*ts, ev(:,dinx(1)), 'g', 'LineWidth', 2 ); -subplot( 4, 2, 6 ); -plot( 1000.0*ts, ev(:,dinx(2)), 'r', 'LineWidth', 2 ); - -% clustering: -%kx = kmeans( [ nx, ny ], 2 ); -% nx smaller or greater a threshold: -kthresh = 1.6; -kx = ones( size( vs, 1 ), 1 ); -kx(nx>kthresh) = 2; - -subplot( 4, 1, 2 ); -scatter( nx(kx==1), ny(kx==1), 'g', 'filled', 'MarkerEdgeColor', 'white' ); -hold on; -scatter( nx(kx==2), ny(kx==2), 'r', 'filled', 'MarkerEdgeColor', 'white' ); -hold off; - -% show sorted spike waveforms: -subplot( 4, 2, 7 ); -hold on -kinx1 = find(kx==1); -for k=1:length(kinx1) - plot( 1000.0*ts, vs(kinx1(k),:), '-g' ); -end -plot( 1000.0*x, y1, '-k', 'LineWidth', 2 ); -xlim( [ 1000.0*ts(1) 1000.0*ts(end) ] ) -hold off -subplot( 4, 2, 8 ); -hold on -kinx2 = find(kx==2); -for k=1:length(kinx2) - plot( 1000.0*ts, vs(kinx2(k),:), '-r' ); -end -plot( 1000.0*x, y2, '-k', 'LineWidth', 2 ); -xlim( [ 1000.0*ts(1) 1000.0*ts(end) ] ) -hold off - -% spike trains: -figure( 1 ); -hold on; -six1 = find( spiket(kinx1)<1.0 ); -scatter( spiket(kinx1(six1)), spikev(kinx1(six1)), 100.0, 'g', 'filled' ); -six2 = find( spiket(kinx2)<1.0 ); -scatter( spiket(kinx2(six2)), spikev(kinx2(six2)), 100.0, 'r', 'filled' ); -hold off; diff --git "a/moduledescription/Einf\303\274hrung_wissenschaftl_Datenverarbeitung.pdf" "b/moduledescription/Einf\303\274hrung_wissenschaftl_Datenverarbeitung.pdf" new file mode 100644 index 0000000..bcf33f1 Binary files /dev/null and "b/moduledescription/Einf\303\274hrung_wissenschaftl_Datenverarbeitung.pdf" differ diff --git "a/moduledescription/Einf\303\274hrung_wissenschaftl_Datenverarbeitung.pptx" "b/moduledescription/Einf\303\274hrung_wissenschaftl_Datenverarbeitung.pptx" new file mode 100644 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a/pointprocesses/code/colorednoisepdf.m b/pointprocesses/code/colorednoisepdf.m new file mode 100644 index 0000000..28f421b --- /dev/null +++ b/pointprocesses/code/colorednoisepdf.m @@ -0,0 +1,10 @@ +function pcn = colorednoisepdf( x, misi, epsilon, tau ) +% returns the ISI pdf for PIF with colored noise drive +% x: the input ISI +% misis: the mean isi +% epsilon: a parameter +% tau: the correlation time of the noise + gamma1 = x/tau+exp(-x/tau)-1.0; + gamma2 = 1.0-exp(-x/tau); + pcn=exp(-(x-misi).^2./(4.0*epsilon*tau.^2.*gamma1)).*(((misi-x).*gamma2+2*gamma1*tau).^2./(2*gamma1*tau^2)-epsilon*(gamma2.^2-2*gamma1.*exp(-x/tau))) ./ (2*tau*sqrt(4*pi*epsilon*gamma1.^3)); +end diff --git a/pointprocesses/code/colorednoisepdfisifit.m b/pointprocesses/code/colorednoisepdfisifit.m new file mode 100644 index 0000000..1c653c0 --- /dev/null +++ b/pointprocesses/code/colorednoisepdfisifit.m @@ -0,0 +1,24 @@ +% misi = 0.02; +% epsilon = 1.0; +% tau = 0.1; +x=0:0.002:0.1; +% pcn = colorednoisepdf( x, misi, epsilon, tau )+10.0*randn( size( x ) ); +% plot( x, pcn ); + +spikes = lifouspikes( 10, 15, 50.0, 1.0, 1.0 ); +isivec = isis( spikes ); +misi = mean( isivec ); +1.0/misi +isibins = 0:0.0005:0.1; +[ n, c ] = hist( isivec, isibins ); +n = n / sum(n)/(isibins(2)-isibins(1)); +bar( c, n ); + +beta0 = [ 1.0, 0.01 ]; +b = nlinfit(c(1:end-2), n(1:end-2), @(b,x)(colorednoisepdf(x, misi, b(1), b(2))), beta0) + +pcn = colorednoisepdf( x, misi, b(1), b(2) ); +hold on +plot( x, pcn, 'r', 'LineWidth', 3 ); +hold off + diff --git a/pointprocesses/code/counthist.m b/pointprocesses/code/counthist.m new file mode 100644 index 0000000..d64fb23 --- /dev/null +++ b/pointprocesses/code/counthist.m @@ -0,0 +1,38 @@ +function [ counts, bins ] = counthist( spikes, w ) +% computes count histogram and compare them with Poisson distribution +% spikes: a cell array of vectors of spike times +% w: observation window duration for computing the counts + + % collect spike counts: + tmax = spikes{1}(end); + n = []; + r = []; + for k = 1:length(spikes) + for tk = 0:w:tmax-w + nn = sum( ( spikes{k} >= tk ) & ( spikes{k} < tk+w ) ); + %nn = length( find( ( spikes{k} >= tk ) & ( spikes{k} < tk+w ) ) ); + n = [ n nn ]; + end + rate = (length(spikes{k})-1)/(spikes{k}(end) - spikes{k}(1)); + r = [ r rate ]; + end + % histogram of spike counts: + maxn = max( n ); + [counts, bins ] = hist( n, 0:1:maxn+1 ); + counts = counts / sum( counts ); + if nargout == 0 + bar( bins, counts ); + hold on; + % Poisson distribution: + rate = mean( r ); + x = 0:1:20; + l = rate*w; + y = l.^x.*exp(-l)./factorial(x); + plot( x, y, 'r', 'LineWidth', 3 ); + xlim( [ 0 20 ] ); + hold off; + xlabel( 'counts k' ); + ylabel( 'P(k)' ); + end +end + diff --git a/pointprocesses/code/fano.m b/pointprocesses/code/fano.m new file mode 100644 index 0000000..a73cada --- /dev/null +++ b/pointprocesses/code/fano.m @@ -0,0 +1,39 @@ +function fano( spikes ) +% computes fano factor as a function of window size +% spikes: a cell array of vectors of spike times + + tmax = spikes{1}(end); + windows = 0.01:0.05:0.01*tmax; + mc = windows; + vc = windows; + ff = windows; + fs = windows; + for j = 1:length(windows) + w = windows( j ); + % collect counts: + n = []; + for k = 1:length(spikes) + for tk = 0:w:tmax-w + nn = sum( ( spikes{k} >= tk ) & ( spikes{k} < tk+w ) ); + %nn = length( find( ( spikes{k} >= tk ) & ( spikes{k} < tk+w ) ) ); + n = [ n nn ]; + end + end + % statistics for current window: + mc(j) = mean( n ); + vc(j) = var( n ); + ff(j) = vc( j )/mc( j ); + fs(j) = sqrt(vc( j )/mc( j )); + end + + subplot( 1, 2, 1 ); + scatter( mc, vc, 'filled' ); + xlabel( 'Mean count' ); + ylabel( 'Count variance' ); + + subplot( 1, 2, 2 ); + scatter( 1000.0*windows, fs, 'filled' ); + xlabel( 'Window W [ms]' ); + ylabel( 'Fano factor' ); +end + diff --git a/pointprocesses/code/inversegauss.m b/pointprocesses/code/inversegauss.m new file mode 100644 index 0000000..58a9512 --- /dev/null +++ b/pointprocesses/code/inversegauss.m @@ -0,0 +1,5 @@ +function y = inversegauss( x, m, d ) +% returns the inverse Gauss density with mean isi m and diffusion +% coefficent d + y = exp(-(x-m).^2./(4.0*d.*x.*m.^2.0))./sqrt(4.0*pi*d.*x.^3.0); +end diff --git a/pointprocesses/code/inversegaussplot.m b/pointprocesses/code/inversegaussplot.m new file mode 100644 index 0000000..c43e6d2 --- /dev/null +++ b/pointprocesses/code/inversegaussplot.m @@ -0,0 +1,28 @@ +f = figure; +subplot( 1, 2, 1 ); +dx=0.0001; +x = dx:dx:0.5; +hold all +m = 0.02; +for d = [ 0.1 1 10 50 200 ] + plot( 1000.0*x, inversegauss( x, m, d ), 'LineWidth', 3, 'DisplayName', sprintf( 'D=%.1f', d ) ); +end +title( sprintf( 'm=%g ms', 1000.0*m ) ) +xlim( [ 0 50 ] ); +xlabel( 'ISI [ms]' ); +ylabel( 'p(ISI)' ); +legend( '-DynamicLegend' ); +hold off; + +subplot( 1, 2, 2 ); +hold all; +d = 5.0; +for m = [ 0.005 0.01 0.02 0.05 ] + plot( 1000.0*x, inversegauss( x, m, d ), 'LineWidth', 3, 'DisplayName', sprintf( 'm=%g ms', 1000.0*m ) ); +end +title( sprintf( 'D=%g Hz', d ) ) +xlim( [ 0 50 ] ) +xlabel( 'ISI [ms]' ); +ylabel( 'p(ISI)' ); +legend( '-DynamicLegend' ) +hold off diff --git a/pointprocesses/code/isihist.m b/pointprocesses/code/isihist.m new file mode 100644 index 0000000..a9722fd --- /dev/null +++ b/pointprocesses/code/isihist.m @@ -0,0 +1,32 @@ +function isihist( isis, binwidth ) +% plot histogram of isis +% isis: vector of interspike intervals +% binwidth: optional width to be used for the isi bins + + if nargin < 2 + nperbin = 200; % average number of data points per bin + bins = length( isis )/nperbin; % number of bins + binwidth = max( isis )/bins; + if binwidth < 5e-4 % half a millisecond + binwidth = 5e-4; + end + end + bins = 0.5*binwidth:binwidth:max(isis); + % histogram: + [ nelements, centers ] = hist( isis, bins ); + % normalization (integral = 1): + nelements = nelements / sum( nelements ) / binwidth; + % plot: + bar( 1000.0*centers, nelements ); + xlabel( 'ISI [ms]' ) + ylabel( 'p(ISI) [1/s]') + % annotation: + misi = mean( isis ); + sdisi = std( isis ); + disi = sdisi^2.0/2.0/misi^3; + text( 0.5, 0.6, sprintf( 'mean=%.1f ms', 1000.0*misi ), 'Units', 'normalized' ) + text( 0.5, 0.5, sprintf( 'std=%.1f ms', 1000.0*sdisi ), 'Units', 'normalized' ) + text( 0.5, 0.4, sprintf( 'CV=%.2f', sdisi/misi ), 'Units', 'normalized' ) + %text( 0.5, 0.3, sprintf( 'D=%.1f Hz', disi ), 'Units', 'normalized' ) +end + diff --git a/pointprocesses/code/isireturnmap.m b/pointprocesses/code/isireturnmap.m new file mode 100644 index 0000000..d790e13 --- /dev/null +++ b/pointprocesses/code/isireturnmap.m @@ -0,0 +1,24 @@ +function isireturnmap( isis, lag2 ) +% plot return maps for lag 1 and lag lag2 + + clf; + subplot( 1, 2, 1 ); + lag = 1; + scatter( 1000.0*isis(1:end-lag)', 1000.0*isis(1+lag:end)', 'b', 'filled', 'MarkerEdgeColor', 'white' ); + xlabel( 'ISI T_i [ms]' ); + ylabel( 'ISI T_{i+1} [ms]' ); + maxisi = max( isis ); + maxy = ceil(maxisi/10)*10.0; + xlim( [0 1.5*maxy ]) + ylim( [0 maxy ]) + + subplot( 1, 2, 2 ); + lag = lag2; + scatter( 1000.0*isis(1:end-lag)', 1000.0*isis(1+lag:end)', 'b', 'filled', 'MarkerEdgeColor', 'white' ); + xlabel( 'ISI T_i [ms]' ); + ylabel( 'ISI T_{i+2} [ms]' ); + xlim( [0 1.5*maxy ]) + ylim( [0 maxy ]) + +end + diff --git a/pointprocesses/code/isis.m b/pointprocesses/code/isis.m new file mode 100644 index 0000000..b0b95ff --- /dev/null +++ b/pointprocesses/code/isis.m @@ -0,0 +1,15 @@ +function isivec = isis( spikes ) +% returns a single list of isis computed from all trials in spikes +% spikes: a cell array of vectors of spike times + + isivec = []; + for k = 1:length(spikes) + difftimes = diff( spikes{k} ); + if ( size( difftimes, 1 ) == 1 ) + isivec = [ isivec difftimes ]; + elseif ( size( difftimes, 2 ) == 1 ) + isivec = [ isivec difftimes' ]; + end + end +end + diff --git a/pointprocesses/code/isiserialcorr.m b/pointprocesses/code/isiserialcorr.m new file mode 100644 index 0000000..d5b44c1 --- /dev/null +++ b/pointprocesses/code/isiserialcorr.m @@ -0,0 +1,26 @@ +function isicorr = isiserialcorr( isis, maxlag ) +% serial correlation of isis +% isis: vector of interspike intervals +% maxlag: the maximum lag + + lags = 0:maxlag; + isicorr = zeros( size( lags ) ); + for k = 1:length(lags) + lag = lags(k); + if length( isis ) > lag+10 + cc = corrcoef( [ isis(1:end-lag)', isis(1+lag:end)' ] ); + isicorr(k) = cc( 1, 2 ); + end + end + + if nargout == 0 + % plot: + plot( lags, isicorr, '-b' ); + hold on; + scatter( lags, isicorr, 100.0, 'b', 'filled' ); + hold off; + xlabel( 'Lag k' ) + ylabel( '\rho_k') + end +end + diff --git a/pointprocesses/code/lifadaptspikes.m b/pointprocesses/code/lifadaptspikes.m new file mode 100644 index 0000000..2ef1874 --- /dev/null +++ b/pointprocesses/code/lifadaptspikes.m @@ -0,0 +1,53 @@ +function spikes = lifadaptspikes( trials, input, tmaxdt, D, tauadapt, adaptincr ) +% Generate spike times of a leaky integrate-and-fire neuron +% with an adaptation current +% trials: the number of trials to be generated +% input: the stimulus either as a single value or as a vector +% tmaxdt: in case of a single value stimulus the duration of a trial +% in case of a vector as a stimulus the time step +% D: the strength of additive white noise +% tauadapt: adaptation time constant +% adaptincr: adaptation strength + + tau = 0.01; + if nargin < 4 + D = 1e0; + end + if nargin < 5 + tauadapt = 0.1; + end + if nargin < 6 + adaptincr = 1.0; + end + vreset = 0.0; + vthresh = 10.0; + dt = 1e-4; + + if max( size( input ) ) == 1 + input = input * ones( ceil( tmaxdt/dt ), 1 ); + else + dt = tmaxdt; + end + spikes = cell( trials, 1 ); + for k=1:trials + times = []; + j = 1; + v = vreset; + a = 0.0; + noise = sqrt(2.0*D)*randn( length( input ), 1 )/sqrt(dt); + for i=1:length( noise ) + v = v + ( - v - a + noise(i) + input(i))*dt/tau; + a = a + ( - a )*dt/tauadapt; + if v >= vthresh + v = vreset; + a = a + adaptincr/tauadapt; + spiketime = i*dt; + if spiketime > 4.0*tauadapt + times(j) = spiketime - 4.0*tauadapt; + j = j + 1; + end + end + end + spikes{k} = times; + end +end diff --git a/pointprocesses/code/lifboltzmanspikes.m b/pointprocesses/code/lifboltzmanspikes.m new file mode 100644 index 0000000..14640e4 --- /dev/null +++ b/pointprocesses/code/lifboltzmanspikes.m @@ -0,0 +1,51 @@ +function spikes = lifboltzmanspikes( trials, input, tmaxdt, D, imax, ithresh, slope ) +% Generate spike times of a leaky integrate-and-fire neuron +% trials: the number of trials to be generated +% input: the stimulus either as a single value or as a vector +% tmaxdt: in case of a single value stimulus the duration of a trial +% in case of a vector as a stimulus the time step +% D: the strength of additive white noise +% imax: maximum output of boltzman +% ithresh: threshold of boltzman input +% slope: slope factor of boltzman input + + tau = 0.01; + if nargin < 4 + D = 1e0; + end + if nargin < 5 + imax = 20; + end + if nargin < 6 + ithresh = 10; + end + if nargin < 7 + slope = 1; + end + vreset = 0.0; + vthresh = 10.0; + dt = 1e-4; + + if length( input ) == 1 + input = input * ones( ceil( tmaxdt/dt ), 1 ); + else + dt = tmaxdt; + end + inb = imax./(1.0+exp(-slope.*(input - ithresh))); + spikes = cell( trials, 1 ); + for k=1:trials + times = []; + j = 1; + v = vreset; + noise = sqrt(2.0*D)*randn( length( input ), 1 )/sqrt(dt); + for i=1:length( noise ) + v = v + ( - v + noise(i) + inb(i))*dt/tau; + if v >= vthresh + v = vreset; + times(j) = i*dt; + j = j + 1; + end + end + spikes{k} = times; + end +end diff --git a/pointprocesses/code/liffano.m b/pointprocesses/code/liffano.m new file mode 100644 index 0000000..b90403f --- /dev/null +++ b/pointprocesses/code/liffano.m @@ -0,0 +1,23 @@ +input = 65.0; % lifadapt 100Hz +%input = 8.0; % lifadapt 10Hz +%input = 15.7; % lif 100Hz +%input = 8.3; % lif 10Hz +trials = 10; +tmax = 100.0; +Dnoise = 0.1; +Dounoise = 5e1; +outau = 10.0; +adapttau = 0.1; +adaptincr = 5.0; + +%spikes = lifouadaptspikes( trials, input, 1.0, Dnoise, Dounoise, outau, adapttau, adaptincr ); +%spikeraster( spikes ); +%return; + +% generate spikes: +%spikes = lifspikes( trials, input, tmax, noise ); +spikes = lifouspikes( trials, input, tmax, Dounoise, outau ); +%spikes = lifadaptspikes( trials, input, tmax, Dnoise, adapttau, adaptincr ); +%spikes = lifouadaptspikes( trials, input, tmax, Dnoise, Dounoise, outau, adapttau, adaptincr ); + +fano( spikes ); diff --git a/pointprocesses/code/lifficurves.m b/pointprocesses/code/lifficurves.m new file mode 100644 index 0000000..5106c12 --- /dev/null +++ b/pointprocesses/code/lifficurves.m @@ -0,0 +1,36 @@ +% lif: +noises = [ 1e-5 1e-4 1e-3 1e-2 1e-1 ]; +inputs = 0:0.1:20; +duration = 50.0; +% pif: +% noises = [ 1e-1 1 1e1 1e2 1e3 ]; +% inputs = -5:0.1:10; +% duration = 100.0; + +f = figure; +hold all; +for noise = noises + fprintf( 'noise=%.0e\n', noise ); + rates = []; + for input = inputs + spikes = lifspikes( 10, input, duration, noise ); + % spikes = pifspikes( 50, input, duration, noise ); + nspikes = 0; + for k = 1:length( spikes ) + nspikes = nspikes + length( spikes{k} ); + end + rate = nspikes/duration/length( spikes ); + %fprintf( 'I=%g N=%d rate=%g\n', input, length( spikes ), rate ) + rates = [ rates rate ]; + end + plot( inputs, rates, 'LineWidth', 2, 'DisplayName', sprintf( 'D=%.0e', noise ) ); +end +xlabel( 'Input' ); +xlim( [ inputs(1) inputs(end) ] ) +ylabel( 'Firing rate [Hz]' ); +%title( 'Leaky integrate-and-fire' ) +title( 'Perfect integrate-and-fire' ) +legend( '-DynamicLegend', 'Location', 'NorthWest' ) +hold off + + diff --git a/pointprocesses/code/lifinputdiscriminationslope.m b/pointprocesses/code/lifinputdiscriminationslope.m new file mode 100644 index 0000000..3a52ef0 --- /dev/null +++ b/pointprocesses/code/lifinputdiscriminationslope.m @@ -0,0 +1,65 @@ +%input = 15.7; % lif 100Hz +%input = 8.3; % lif 10Hz +trials = 10; +tmax = 50.0; +Dnoise = 1.0; +imax = 25.0; +ithresh = 10.0; +slope=0.2; + +% inputs = 0:2:30; +% rates = zeros( size( inputs ) ); +% for j = 1:length( inputs ) +% input = inputs(j); +% spikes = lifboltzmanspikes( trials, input, tmax, Dnoise, imax, ithresh, slope ); +% nspikes = 0; +% for k = 1:length( spikes ) +% nspikes = nspikes + length( spikes{k} ); +% end +% rate = nspikes/tmax/length( spikes ); +% rates(j) = rate; +% end +% plot( inputs, rates ); +% grid on; +% return + +input = 10.0; % 80 Hz + +window = 0.2; +slopes = 0.1:0.1:2.0; +pmax = zeros( size( slopes) ); +for j = 1:length( slopes ) + slope = slopes( j ); + spikes = lifboltzmanspikes( trials, input, tmax, Dnoise, imax, ithresh, slope ); + [ n1, bins1 ] = counthist( spikes, w ); + + spikes = lifboltzmanspikes( trials, input+1.0, tmax, Dnoise, imax, ithresh, slope ); + [ n2, bins2 ] = counthist( spikes, w ); + + subplot( 2, 1, 1 ); + bar( bins1, n1, 'b' ); + hold on; + bar( bins2, n2, 'r' ); + hold off; + + subplot( 2, 1, 2 ); + bmax = max( [ length( bins1 ), length( bins2 ) ] ); + decision1 = zeros( bmax, 1 ); + decision2 = zeros( bmax, 1 ); + cs1 = ones( bmax, 1 ); + cs1(1:length(n1)) = cumsum( n1 ); + cs2 = ones( bmax, 1 ); + cs2(1:length(n2)) = cumsum( n2 ); + cbins = 0:1:bmax-1; + plot( cbins, cs1, 'b' ); + hold on; + plot( cbins, cs2, 'r' ); + plot( cbins, cs1-cs2, 'g' ); + hold off; + pause( 0.1 ); + pmax(j) = max( cs1-cs2 ); +end + +clf; +subplot( 1, 1, 1 ); +plot( slopes, pmax ); diff --git a/pointprocesses/code/lifinputdiscriminationtime.m b/pointprocesses/code/lifinputdiscriminationtime.m new file mode 100644 index 0000000..9cd4603 --- /dev/null +++ b/pointprocesses/code/lifinputdiscriminationtime.m @@ -0,0 +1,51 @@ +input = 65.0; % lifadapt 100Hz +%input = 8.0; % lifadapt 10Hz +%input = 15.7; % lif 100Hz +%input = 8.3; % lif 10Hz +trials = 10; +tmax = 50.0; +Dnoise = 0.1; +Dounoise = 5e1; +outau = 10.0; +adapttau = 0.2; +adaptincr = 0.5; + +windows = 0.05:0.05:1.0; +pmax = zeros( size( windows ) ); +for j = 1:length( windows ) + w = windows( j ); + spikes = lifadaptspikes( trials, input, tmax, Dnoise, adapttau, adaptincr ); + %spikes = lifouspikes( trials, input, tmax, Dounoise, outau); + [ n1, bins1 ] = counthist( spikes, w ); + + spikes = lifadaptspikes( trials, input+10.0, tmax, Dnoise, adapttau, adaptincr ); + %spikes = lifouspikes( trials, input+10.0, tmax, Dounoise, outau ); + [ n2, bins2 ] = counthist( spikes, w ); + + subplot( 2, 1, 1 ); + bar( bins1, n1, 'b' ); + hold on; + bar( bins2, n2, 'r' ); + hold off; + + subplot( 2, 1, 2 ); + bmax = max( [ length( bins1 ), length( bins2 ) ] ); + decision1 = zeros( bmax, 1 ); + decision2 = zeros( bmax, 1 ); + cs1 = ones( bmax, 1 ); + cs1(1:length(n1)) = cumsum( n1 ); + cs2 = ones( bmax, 1 ); + cs2(1:length(n2)) = cumsum( n2 ); + cbins = 0:1:bmax-1; + plot( cbins, cs1, 'b' ); + hold on; + plot( cbins, cs2, 'r' ); + plot( cbins, cs1-cs2, 'g' ); + hold off; + pause( 0.1 ); + pmax(j) = max( cs1-cs2 ); +end + +clf; +subplot( 1, 1, 1 ); +plot( windows, pmax ); diff --git a/pointprocesses/code/lifisih.m b/pointprocesses/code/lifisih.m new file mode 100644 index 0000000..f4e7a8e --- /dev/null +++ b/pointprocesses/code/lifisih.m @@ -0,0 +1,35 @@ +%input = 65.0; % lifadapt 100Hz +%input = 8.0; % lifadapt 10Hz +input = 15.7; % lif 100Hz +%input = 8.3; % lif 10Hz +trials = 10; +tmax = 100.0; +noise = 1e-1; +adapttau = 0.1; +adaptincr = 5.0; + +% generate spikes: +spikes = lifspikes( trials, input, tmax, noise ); +%spikes = lifadaptspikes( trials, input, tmax, noise, adapttau, adaptincr ); + +% interspike intervals: +isivec = isis( spikes ); +% histogram +f = figure( 1 ); +isihist( isivec, 10e-4 ); +hold on +% theoretical density: +misi = mean( isivec ); +disi = var( isivec )/2.0/misi^3; +xmax = 3.0*misi; +x = 0:0.0001:xmax; +plot( 1000.0*x, inversegauss( x, misi, disi ), 'r', 'LineWidth', 3 ); +% plot details: +title( sprintf( 'LIF, input=%g, nisi=%d', input, length( isivec ) ) ) +xlim( [ 0.0 1000.0*xmax ] ) +legend( 'data', 'inverse Gaussian' ) +hold off + +% serial correlations: +f = figure( 2 ); +isiserialcorr( isivec, 10 ); diff --git a/pointprocesses/code/lifisistats.m b/pointprocesses/code/lifisistats.m new file mode 100644 index 0000000..1dd1321 --- /dev/null +++ b/pointprocesses/code/lifisistats.m @@ -0,0 +1,63 @@ +inputs = 0:0.1:20; % lif +inputs = 0:0.1:10; % pif +avisi = []; +sdisi = []; +cvisi = []; +dcisi = []; + +for input = inputs + input + % spikes = lifspikes( 100, input, 100.0, 1e-2 ); + spikes = pifspikes( 100, input, 100.0, 1e-1 ); + isivec = isis( spikes ); + if length( isivec ) <= 1 + av = Inf; + sd = NaN; + cv = NaN; + dc = NaN; + else + av = mean( isivec ); + sd = std( isivec ); + if av > 0.0 + cv = sd/av; + dc = sd^2.0/2.0/av^3; + else + cv = NaN; + dc = NaN; + end + end + avisi = [ avisi av ]; + sdisi = [ sdisi sd ]; + cvisi = [ cvisi cv ]; + dcisi = [ dcisi dc ]; +end + +f = figure; +subplot( 2, 2, 1 ); +plot( inputs, 1.0./avisi, '-b', 'LineWidth', 3 ); +xlabel( 'Input' ); +xlim( [ inputs(1) inputs(end) ] ) +title( 'Mean rate [Hz]' ); + +subplot( 2, 2, 2 ); +plot( inputs, 1000.0*avisi, '-b', 'LineWidth', 3 ); +hold on; +plot( inputs, 1000.0*sdisi, '-c', 'LineWidth', 3 ); +hold off; +xlabel( 'Input' ); +xlim( [ inputs(1) inputs(end) ] ) +ylim( [ 0 1000 ] ) +title( 'ISI [ms]' ); +legend( 's.d. ISI', 'mean ISI' ); + +subplot( 2, 2, 3 ); +plot( inputs, cvisi, '-b', 'LineWidth', 3 ); +xlabel( 'Input' ); +xlim( [ inputs(1) inputs(end) ] ) +title( 'CV' ); + +subplot( 2, 2, 4 ); +plot( inputs, dcisi, '-b', 'LineWidth', 3 ); +xlabel( 'Input' ); +xlim( [ inputs(1) inputs(end) ] ) +title( 'D [Hz]' ); diff --git a/pointprocesses/code/lifouadaptspikes.m b/pointprocesses/code/lifouadaptspikes.m new file mode 100644 index 0000000..7dddb2d --- /dev/null +++ b/pointprocesses/code/lifouadaptspikes.m @@ -0,0 +1,64 @@ +function spikes = lifouadaptspikes( trials, input, tmaxdt, D, Dou, outau, tauadapt, adaptincr ) +% Generate spike times of a leaky integrate-and-fire neuron +% with colored noise and an adaptation current +% trials: the number of trials to be generated +% input: the stimulus either as a single value or as a vector +% tmaxdt: in case of a single value stimulus the duration of a trial +% in case of a vector as a stimulus the time step +% D: the strength of additive noise +% Dou: the strength of additive colored noise +% outau: time constant of the colored noise +% tauadapt: adaptation time constant +% adaptincr: adaptation strength + + tau = 0.01; + if nargin < 4 + D = 1e0; + end + if nargin < 5 + Dou = 1e0; + end + if nargin < 6 + outau = 1.0; + end + if nargin < 7 + tauadapt = 0.1; + end + if nargin < 8 + adaptincr = 1.0; + end + vreset = 0.0; + vthresh = 10.0; + dt = 1e-4; + + if max( size( input ) ) == 1 + input = input * ones( ceil( tmaxdt/dt ), 1 ); + else + dt = tmaxdt; + end + spikes = cell( trials, 1 ); + for k=1:trials + times = []; + j = 1; + v = vreset; + n = 0.0; + a = 0.0; + noise = sqrt(2.0*D)*randn( length( input ), 1 )/sqrt(dt); + noiseou = sqrt(2.0*Dou)*randn( length( input ), 1 )/sqrt(dt); + for i=1:length( noise ) + n = n + ( - n + noiseou(i))*dt/outau; + v = v + ( - v - a + noise(i) + n + input(i))*dt/tau; + a = a + ( - a )*dt/tauadapt; + if v >= vthresh + v = vreset; + a = a + adaptincr; + spiketime = i*dt; + if spiketime > 4.0*tauadapt + times(j) = spiketime - 4.0*tauadapt; + j = j + 1; + end + end + end + spikes{k} = times; + end +end diff --git a/pointprocesses/code/lifouspikes.m b/pointprocesses/code/lifouspikes.m new file mode 100644 index 0000000..fc1f5d5 --- /dev/null +++ b/pointprocesses/code/lifouspikes.m @@ -0,0 +1,44 @@ +function spikes = lifouspikes( trials, input, tmaxdt, D, outau ) +% Generate spike times of a leaky integrate-and-fire neuron +% trials: the number of trials to be generated +% input: the stimulus either as a single value or as a vector +% tmaxdt: in case of a single value stimulus the duration of a trial +% in case of a vector as a stimulus the time step +% D: the strength of additive white noise +% outau: time constant of the colored noise + + tau = 0.01; + if nargin < 4 + D = 1e0; + end + if nargin < 5 + outau = 1.0; + end + vreset = 0.0; + vthresh = 10.0; + dt = 1e-4; + + if length( input ) == 1 + input = input * ones( ceil( tmaxdt/dt ), 1 ); + else + dt = tmaxdt; + end + spikes = cell( trials, 1 ); + for k=1:trials + times = []; + j = 1; + n = 0.0; + v = vreset; + noise = sqrt(2.0*D)*randn( length( input ), 1 )/sqrt(dt); + for i=1:length( noise ) + n = n + ( - n + noise(i))*dt/outau; + v = v + ( - v + n + input(i))*dt/tau; + if v >= vthresh + v = vreset; + times(j) = i*dt; + j = j + 1; + end + end + spikes{k} = times; + end +end diff --git a/pointprocesses/code/lifrateisicorr.m b/pointprocesses/code/lifrateisicorr.m new file mode 100644 index 0000000..6276e79 --- /dev/null +++ b/pointprocesses/code/lifrateisicorr.m @@ -0,0 +1,34 @@ +% relation between firing rate and serieller correlation + +input = 65.0; % lifadapt 100Hz +%input = 8.0; % lifadapt 10Hz +trials = 10; +tmax = 50.0; +noise = 1e-5; +adapttau = 0.1; +adaptincr = 0.5; + +clf; +for adapttau = 0.01:0.02:0.2 + inputs = 1:5:120; + iscs = zeros( size( inputs ) ); + rates = zeros( size( inputs ) ); + for k = 1:length(inputs) + input = inputs(k); + % generate spikes: + spikes = lifadaptspikes( trials, input, tmax, noise, adapttau, adaptincr ); + isivec = isis( spikes ); + isc = isiserialcorr( isivec, 10 ); + iscs(k) = isc(2); + rates(k) = 1.0/mean( isivec ); + end + + subplot( 2, 1, 1 ); + hold on; + plot( inputs, rates ); + hold off; + subplot( 2, 1, 2 ); + hold on; + plot( rates, iscs ); + hold off; +end diff --git a/pointprocesses/code/lifspikes.m b/pointprocesses/code/lifspikes.m new file mode 100644 index 0000000..cfa0f55 --- /dev/null +++ b/pointprocesses/code/lifspikes.m @@ -0,0 +1,38 @@ +function spikes = lifspikes( trials, input, tmaxdt, D ) +% Generate spike times of a leaky integrate-and-fire neuron +% trials: the number of trials to be generated +% input: the stimulus either as a single value or as a vector +% tmaxdt: in case of a single value stimulus the duration of a trial +% in case of a vector as a stimulus the time step +% D: the strength of additive white noise + + tau = 0.01; + if nargin < 4 + D = 1e0; + end + vreset = 0.0; + vthresh = 10.0; + dt = 1e-4; + + if length( input ) == 1 + input = input * ones( ceil( tmaxdt/dt ), 1 ); + else + dt = tmaxdt; + end + spikes = cell( trials, 1 ); + for k=1:trials + times = []; + j = 1; + v = vreset; + noise = sqrt(2.0*D)*randn( length( input ), 1 )/sqrt(dt); + for i=1:length( noise ) + v = v + ( - v + noise(i) + input(i))*dt/tau; + if v >= vthresh + v = vreset; + times(j) = i*dt; + j = j + 1; + end + end + spikes{k} = times; + end +end diff --git a/pointprocesses/code/pifouspikes.m b/pointprocesses/code/pifouspikes.m new file mode 100644 index 0000000..b6516cc --- /dev/null +++ b/pointprocesses/code/pifouspikes.m @@ -0,0 +1,44 @@ +function spikes = pifouspikes( trials, input, tmaxdt, D, outau ) +% Generate spike times of a perfect integrate-and-fire neuron +% trials: the number of trials to be generated +% input: the stimulus either as a single value or as a vector +% tmaxdt: in case of a single value stimulus the duration of a trial +% in case of a vector as a stimulus the time step +% D: the strength of additive white noise +% outau: time constant of the colored noise + + tau = 0.01; + if nargin < 4 + D = 1e0; + end + if nargin < 5 + outau = 1.0; + end + vreset = 0.0; + vthresh = 10.0; + dt = 1e-4; + + if length( input ) == 1 + input = input * ones( ceil( tmaxdt/dt ), 1 ); + else + dt = tmaxdt; + end + spikes = cell( trials, 1 ); + for k=1:trials + times = []; + j = 1; + n = 0.0; + v = vreset; + noise = sqrt(2.0*D)*randn( length( input ), 1 )/sqrt(dt); + for i=1:length( noise ) + n = n + ( - n + noise(i))*dt/outau; + v = v + ( n + input(i))*dt/tau; + if v >= vthresh + v = vreset; + times(j) = i*dt; + j = j + 1; + end + end + spikes{k} = times; + end +end diff --git a/pointprocesses/code/pifspikes.m b/pointprocesses/code/pifspikes.m new file mode 100644 index 0000000..397658c --- /dev/null +++ b/pointprocesses/code/pifspikes.m @@ -0,0 +1,38 @@ +function spikes = pifspikes( trials, input, tmaxdt, D ) +% Generate spike times of a perfect integrate-and-fire neuron +% trials: the number of trials to be generated +% input: the stimulus either as a single value or as a vector +% tmaxdt: in case of a single value stimulus the duration of a trial +% in case of a vector as a stimulus the time step +% D: the strength of additive white noise + + tau = 0.01; + if nargin < 4 + D = 1e-1; + end + vreset = 0.0; + vthresh = 10.0; + dt = 1e-4; + + if length( input ) == 1 + input = input * ones( ceil( tmaxdt/dt ), 1 ); + else + dt = tmaxdt; + end + spikes = cell( trials, 1 ); + for k=1:trials + times = []; + j = 1; + v = vreset; + noise = sqrt(2.0*D)*randn( length( input ), 1 )/sqrt(dt); + for i=1:length( noise ) + v = v + ( noise(i) + input(i))*dt/tau; + if v >= vthresh + v = vreset; + times(j) = i*dt; + j = j + 1; + end + end + spikes{k} = times; + end +end diff --git a/pointprocesses/code/poissonisih.m b/pointprocesses/code/poissonisih.m new file mode 100644 index 0000000..f006e2b --- /dev/null +++ b/pointprocesses/code/poissonisih.m @@ -0,0 +1,27 @@ +rate = 100.0; +trials = 50; +tmax = 100.0; + +% generate spikes: +spikes = poissonspikes( trials, rate, tmax ); +% interspike intervals: +isivec = isis( spikes ); +% histogram +f = figure( 1 ); +isihist( isivec ); +hold on +% theoretical density: +xmax = 5.0/rate; +x = 0:0.0001:xmax; +y = rate*exp(-rate*x); +plot( 1000.0*x, y, 'r', 'LineWidth', 3 ); +% plot details: +title( sprintf( 'Poisson spike trains, rate=%g Hz, nisi=%d', rate, length( isivec ) ) ) +xlim( [ 0.0 1000.0*xmax ] ) +ylim( [ 0.0 1.1*rate ] ) +legend( 'data', 'poisson' ) +hold off + +% serial correlations: +f = figure( 2 ); +isiserialcorr( isivec, 10 ); diff --git a/pointprocesses/code/poissonisistats.m b/pointprocesses/code/poissonisistats.m new file mode 100644 index 0000000..0c97e3d --- /dev/null +++ b/pointprocesses/code/poissonisistats.m @@ -0,0 +1,46 @@ +rates = 1:1:100; +avisi = []; +sdisi = []; +cvisi = []; + +for rate = rates + spikes = poissonspikes( 10, rate, 100.0 ); + isivec = isis( spikes ); + av = mean( isivec ); + sd = std( isivec ); + cv = sd/av; + avisi = [ avisi av ]; + sdisi = [ sdisi sd ]; + cvisi = [ cvisi cv ]; +end + +f = figure; +subplot( 1, 3, 1 ); +scatter( rates, 1000.0*avisi, 'b', 'filled' ); +hold on; +plot( rates, 1000.0./rates, 'r' ); +hold off; +xlabel( 'Rate \lambda [Hz]' ); +ylim( [ 0 1000 ] ); +title( 'Mean ISI [ms]' ); +legend( 'simulation', 'theory 1/\lambda' ); + +subplot( 1, 3, 2 ); +scatter( rates, 1000.0*sdisi, 'b', 'filled' ); +hold on; +plot( rates, 1000.0./rates, 'r' ); +hold off; +xlabel( 'Rate \lambda [Hz]' ); +ylim( [ 0 1000 ] ) +title( 'Standard deviation ISI [ms]' ); +legend( 'simulation', 'theory 1/\lambda' ); + +subplot( 1, 3, 3 ); +scatter( rates, cvisi, 'b', 'filled' ); +hold on; +plot( rates, ones( size( rates ) ), 'r' ); +hold off; +xlabel( 'Rate \lambda [Hz]' ); +ylim( [ 0 2 ] ) +title( 'CV' ); +legend( 'simulation', 'theory' ); diff --git a/pointprocesses/code/poissonspikes.m b/pointprocesses/code/poissonspikes.m new file mode 100644 index 0000000..f00291c --- /dev/null +++ b/pointprocesses/code/poissonspikes.m @@ -0,0 +1,20 @@ +function spikes = poissonspikes( trials, rate, tmax ) +% Generate spike times of a homogeneous poisson process +% trials: number of trials that should be generated +% rate: the rate of the Poisson process in Hertz +% tmax: the duration of each trial in seconds +% returns a cell array of vectors of spike times + + dt = 3.33e-5; + p = rate*dt; % probability of event per bin of width dt + % make sure p is small enough: + if p > 0.1 + p = 0.1 + dt = p/rate; + end + spikes = cell( trials, 1 ); + for k=1:trials + x = rand( 1, round(tmax/dt) ); % uniform random numbers for each bin + spikes{k} = find( x < p ) * dt; + end +end diff --git a/pointprocesses/code/raster.pdf b/pointprocesses/code/raster.pdf new file mode 100644 index 0000000..a20fa0e Binary files /dev/null and b/pointprocesses/code/raster.pdf differ diff --git a/pointprocesses/code/savefigpdf.m b/pointprocesses/code/savefigpdf.m new file mode 100644 index 0000000..fbe2dc2 --- /dev/null +++ b/pointprocesses/code/savefigpdf.m @@ -0,0 +1,28 @@ +function savefigpdf( fig, name, width, height ) +% Saves figure fig in pdf file name.pdf with appropriately set page size +% and fonts + +% default width: +if nargin < 3 + width = 11.7; +end +% default height: +if nargin < 4 + height = 9.0; +end + +% paper: +set( fig, 'PaperUnits', 'centimeters' ); +set( fig, 'PaperSize', [width height] ); +set( fig, 'PaperPosition', [0.0 0.0 width height] ); +set( fig, 'Color', 'white') + +% font: +set( findall( fig, 'type', 'axes' ), 'FontSize', 12 ) +set( findall( fig, 'type', 'text' ), 'FontSize', 12 ) + +% save: +saveas( fig, name, 'pdf' ) + +end + diff --git a/pointprocesses/code/spikeraster.m b/pointprocesses/code/spikeraster.m new file mode 100644 index 0000000..f7ff5d4 --- /dev/null +++ b/pointprocesses/code/spikeraster.m @@ -0,0 +1,17 @@ +function spikeraster( spikes ) +% Display a spike raster of the spike times given in spikes. +% spikes: a cell array of vectors of spike times + +ntrials = length(spikes); +for k = 1:ntrials + times = 1000.0*spikes{k}; % conversion to ms + for i = 1:length( times ) + line([times(i) times(i)],[k-0.4 k+0.4], 'Color', 'k' ); + end +end +xlabel( 'Time [ms]' ); +ylabel( 'Trials'); +ylim( [ 0.3 ntrials+0.7 ] ) + +end + diff --git a/pointprocesses/exercises/hompoissonisih.m b/pointprocesses/exercises/hompoissonisih.m new file mode 100644 index 0000000..cd2bc06 --- /dev/null +++ b/pointprocesses/exercises/hompoissonisih.m @@ -0,0 +1,18 @@ +% generate spike times: +rate = 20.0; +spikes = hompoissonspikes( 10, rate, 50.0 ); +% isi histogram: +isivec = isis( spikes ); +isihist( isivec ); +hold on +% theoretical density: +xmax = 5.0/rate; +x = 0:0.0001:xmax; +y = rate*exp(-rate*x); +plot( 1000.0*x, y, 'r', 'LineWidth', 3 ); +% plot details: +title( sprintf( 'Poisson spike trains, rate=%g Hz, nisi=%d', rate, length( isivec ) ) ) +xlim( [ 0.0 1000.0*xmax ] ) +ylim( [ 0.0 1.1*rate ] ) +legend( 'data', 'poisson' ) +hold off diff --git a/pointprocesses/exercises/hompoissonisistats.m b/pointprocesses/exercises/hompoissonisistats.m new file mode 100644 index 0000000..7ca3cf3 --- /dev/null +++ b/pointprocesses/exercises/hompoissonisistats.m @@ -0,0 +1,46 @@ +rates = 1:1:100; +avisi = []; +sdisi = []; +cvisi = []; + +for rate = rates + spikes = hompoissonspikes( 10, rate, 100.0 ); + isivec = isis( spikes ); + av = mean( isivec ); + sd = std( isivec ); + cv = sd/av; + avisi = [ avisi av ]; + sdisi = [ sdisi sd ]; + cvisi = [ cvisi cv ]; +end + +f = figure; +subplot( 1, 3, 1 ); +scatter( rates, 1000.0*avisi, 'b', 'filled' ); +hold on; +plot( rates, 1000.0./rates, 'r' ); +hold off; +xlabel( 'Rate \lambda [Hz]' ); +ylim( [ 0 1000 ] ); +title( 'Mean ISI [ms]' ); +legend( 'simulation', 'theory 1/\lambda' ); + +subplot( 1, 3, 2 ); +scatter( rates, 1000.0*sdisi, 'b', 'filled' ); +hold on; +plot( rates, 1000.0./rates, 'r' ); +hold off; +xlabel( 'Rate \lambda [Hz]' ); +ylim( [ 0 1000 ] ) +title( 'Standard deviation ISI [ms]' ); +legend( 'simulation', 'theory 1/\lambda' ); + +subplot( 1, 3, 3 ); +scatter( rates, cvisi, 'b', 'filled' ); +hold on; +plot( rates, ones( size( rates ) ), 'r' ); +hold off; +xlabel( 'Rate \lambda [Hz]' ); +ylim( [ 0 2 ] ) +title( 'CV' ); +legend( 'simulation', 'theory' ); diff --git a/pointprocesses/exercises/hompoissonspikes.m b/pointprocesses/exercises/hompoissonspikes.m new file mode 100644 index 0000000..12af153 --- /dev/null +++ b/pointprocesses/exercises/hompoissonspikes.m @@ -0,0 +1,19 @@ +function spikes = hompoissonspikes( trials, rate, tmax ) +% Generate spike times of a homogeneous poisson process +% trials: number of trials that should be generated +% rate: the rate of the Poisson process in Hertz +% tmax: the duration of each trial in seconds +% returns a cell array of vectors of spike times + + dt = 3.33e-5; + p = rate*dt; + if p > 0.2 + p = 0.2 + dt = p/rate; + end + x = rand( trials, ceil(tmax/dt) ); + spikes = cell( trials, 1 ); + for k=1:trials + spikes{k} = find( x(k,:) >= 1.0-p ) * dt; + end +end diff --git a/pointprocesses/exercises/iafisistats-solutions.pdf b/pointprocesses/exercises/iafisistats-solutions.pdf new file mode 100644 index 0000000..d1b6c63 Binary files /dev/null and b/pointprocesses/exercises/iafisistats-solutions.pdf differ diff --git a/pointprocesses/exercises/iafisistats.pdf b/pointprocesses/exercises/iafisistats.pdf new file mode 100644 index 0000000..c19617b Binary files /dev/null and b/pointprocesses/exercises/iafisistats.pdf differ diff --git a/pointprocesses/exercises/iafisistats.tex b/pointprocesses/exercises/iafisistats.tex new file mode 100644 index 0000000..53ca479 --- /dev/null +++ b/pointprocesses/exercises/iafisistats.tex @@ -0,0 +1,142 @@ +\documentclass[addpoints,10pt]{exam} +\usepackage{url} +\usepackage{color} +\usepackage{hyperref} +\usepackage{graphicx} + +\pagestyle{headandfoot} +\runningheadrule +\firstpageheadrule + +\firstpageheader{Scientific Computing}{Integrate-and-fire models}{Oct 28, 2014} +%\runningheader{Homework 01}{Page \thepage\ of \numpages}{23. October 2014} +\firstpagefooter{}{}{} +\runningfooter{}{}{} +\pointsinmargin +\bracketedpoints + +%\printanswers +\shadedsolutions + +\usepackage[mediumspace,mediumqspace,Gray]{SIunits} % \ohm, \micro + +%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\usepackage{listings} +\lstset{ + basicstyle=\ttfamily, + numbers=left, + showstringspaces=false, + language=Matlab, + breaklines=true, + breakautoindent=true, + columns=flexible, + frame=single, + captionpos=t, + xleftmargin=2em, + xrightmargin=1em, + aboveskip=10pt, + %title=\lstname, + title={\protect\filename@parse{\lstname}\protect\filename@base.\protect\filename@ext} + } + + +\begin{document} + +\sffamily +%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{questions} + \question \textbf{Statistics of integrate-and-fire neurons} + For the following use different variants of the leaky integrate-and-fire models provided in \texttt{lifspikes.m}, + \texttt{lifouspikes.m}, and \texttt{lifadaptspikes.m} do generate some spike train data. + Use the functions you wrote for the Poisson process to analyze the statistics of the spike trains. + \begin{parts} + \part Generate a few trials of the two models for two different inputs + that result in qualitatively different spike trains and display + them in a raster plot. Decide for a noise strength (good values to try are 0.001, 0.01, 0.1, 1). + \begin{solution} + \begin{lstlisting} +spikes = pifspikes( 10, 1.0, 0.5, 0.01 ); +%spikes = pifspikes( 10, 10.0, 0.5, 0.01 ); +%spikes = lifspikes( 10, 11.0, 0.5, 0.001 ); +%spikes = lifspikes( 10, 15.0, 0.5, 0.001 ); +spikeraster( spikes ) + \end{lstlisting} + \mbox{}\\[-3ex] + \colorbox{white}{\includegraphics[width=0.48\textwidth]{pifraster02}} + \colorbox{white}{\includegraphics[width=0.48\textwidth]{pifraster10}}\\ + \colorbox{white}{\includegraphics[width=0.48\textwidth]{lifraster10}} + \colorbox{white}{\includegraphics[width=0.48\textwidth]{lifraster15}} + \end{solution} + + + \part The inverse Gaussian describes the interspike interval distribution of a PIF driven with white noise: + \[ p(T) = \frac{1}{\sqrt{4\pi D T^3}}\exp\left[-\frac{(T-\langle T \rangle)^2}{4DT\langle T \rangle^2}\right] \] + where $\langle T \rangle$ is the mean interspike interval and + \[ D = \frac{\langle(T - \langle T \rangle)^2\rangle}{2 \langle T \rangle^3} \] + is the diffusion coefficient (variance of the interspike intervals + $T$ divided by two times the mean cubed). Show in two plots how + this distribution depends on $\langle T \rangle$ and $D$. + \begin{solution} + \lstinputlisting{simulations/inversegauss.m} + \lstinputlisting{simulations/inversegaussplot.m} + \colorbox{white}{\includegraphics[width=0.98\textwidth]{inversegauss}} + \end{solution} + + \part Extent your function plotting an interspike interval histogram + to also report the diffusion coefficient $D$. + \begin{solution} + \begin{lstlisting} +... +% annotation: +misi = mean( isis ); +sdisi = std( isis ); +disi = sdisi^2.0/2.0/misi^3; +text( 0.6, 0.7, sprintf( 'mean=%.1f ms', 1000.0*misi ), 'Units', 'normalized' ) +text( 0.6, 0.6, sprintf( 'std=%.1f ms', 1000.0*sdisi ), 'Units', 'normalized' ) +text( 0.6, 0.5, sprintf( 'CV=%.2f', sdisi/misi ), 'Units', 'normalized' ) +text( 0.6, 0.4, sprintf( 'D=%.1f Hz', disi ), 'Units', 'normalized' ) +... + \end{lstlisting} + \end{solution} + + \part Compare intersike interval histograms obtained from the LIF and PIF models with the inverse Gaussian. + \begin{solution} + \lstinputlisting{simulations/lifisih.m} + \colorbox{white}{\includegraphics[width=0.48\textwidth]{pifisih01}} + \colorbox{white}{\includegraphics[width=0.48\textwidth]{pifisih10}}\\ + \colorbox{white}{\includegraphics[width=0.48\textwidth]{lifisih08}} + \colorbox{white}{\includegraphics[width=0.48\textwidth]{lifisih16}} + \end{solution} + + \part Plot the firing rate (inverse mean interspike interval), + mean interspike interval, the corresponding standard deviation, + CV, and diffusion coefficient as a function of the input to the LIF + and the PIF with noise strength set to 0.01. + \begin{solution} + \lstinputlisting{simulations/lifisistats.m} + Leaky integrate-and-fire:\\ + \colorbox{white}{\includegraphics[width=0.8\textwidth]{lifisistats}}\\ + Perfect integrate-and-fire:\\ + \colorbox{white}{\includegraphics[width=0.8\textwidth]{pifisistats}} + \end{solution} + + \part Plot the firing rate as a function of input of the LIF and the PIF for various values + of the noise strength. + \begin{solution} + \lstinputlisting{simulations/lifficurves.m} + Leaky integrate-and-fire:\\ + \colorbox{white}{\includegraphics[width=0.7\textwidth]{lifficurves}}\\ + Perfect integrate-and-fire:\\ + \colorbox{white}{\includegraphics[width=0.7\textwidth]{pifficurves}} + \end{solution} + + \part Use the functions for computing serial correlations, count statistics and fano factors + to further explore the statistics of the integrate-and-fire models! + + \end{parts} + +\end{questions} + + +\end{document} diff --git a/pointprocesses/exercises/inversegauss.pdf b/pointprocesses/exercises/inversegauss.pdf new file mode 100644 index 0000000..798a9ec Binary files /dev/null and b/pointprocesses/exercises/inversegauss.pdf differ diff --git a/pointprocesses/exercises/lifficurves.pdf b/pointprocesses/exercises/lifficurves.pdf new file mode 100644 index 0000000..fc228b2 Binary files /dev/null and b/pointprocesses/exercises/lifficurves.pdf differ diff --git a/pointprocesses/exercises/lifisih08.pdf b/pointprocesses/exercises/lifisih08.pdf new file mode 100644 index 0000000..4908f72 Binary files /dev/null and b/pointprocesses/exercises/lifisih08.pdf differ diff --git a/pointprocesses/exercises/lifisih16.pdf b/pointprocesses/exercises/lifisih16.pdf new file mode 100644 index 0000000..5c4fb3f Binary files /dev/null and b/pointprocesses/exercises/lifisih16.pdf differ diff --git a/pointprocesses/exercises/lifisistats.pdf b/pointprocesses/exercises/lifisistats.pdf new file mode 100644 index 0000000..f573bae Binary files /dev/null and b/pointprocesses/exercises/lifisistats.pdf differ diff --git a/pointprocesses/exercises/lifraster10.pdf b/pointprocesses/exercises/lifraster10.pdf new file mode 100644 index 0000000..c6e5f67 Binary files /dev/null and b/pointprocesses/exercises/lifraster10.pdf differ diff --git a/pointprocesses/exercises/lifraster15.pdf b/pointprocesses/exercises/lifraster15.pdf new file mode 100644 index 0000000..fac5b4d Binary files /dev/null and b/pointprocesses/exercises/lifraster15.pdf differ diff --git a/pointprocesses/exercises/pifficurves.pdf b/pointprocesses/exercises/pifficurves.pdf new file mode 100644 index 0000000..8ef6ffa Binary files /dev/null and b/pointprocesses/exercises/pifficurves.pdf differ diff --git a/pointprocesses/exercises/pifisih01.pdf b/pointprocesses/exercises/pifisih01.pdf new file mode 100644 index 0000000..d485e57 Binary files /dev/null and b/pointprocesses/exercises/pifisih01.pdf differ diff --git a/pointprocesses/exercises/pifisih10.pdf b/pointprocesses/exercises/pifisih10.pdf new file mode 100644 index 0000000..bb0f667 Binary files /dev/null and b/pointprocesses/exercises/pifisih10.pdf differ diff --git a/pointprocesses/exercises/pifisistats.pdf b/pointprocesses/exercises/pifisistats.pdf new file mode 100644 index 0000000..3c38580 Binary files /dev/null and b/pointprocesses/exercises/pifisistats.pdf differ diff --git a/pointprocesses/exercises/pifraster02.pdf b/pointprocesses/exercises/pifraster02.pdf new file mode 100644 index 0000000..e23d1b4 Binary files /dev/null and b/pointprocesses/exercises/pifraster02.pdf differ diff --git a/pointprocesses/exercises/pifraster10.pdf b/pointprocesses/exercises/pifraster10.pdf new file mode 100644 index 0000000..c6534b0 Binary files /dev/null and b/pointprocesses/exercises/pifraster10.pdf differ diff --git a/pointprocesses/exercises/poisson-solutions.pdf b/pointprocesses/exercises/poisson-solutions.pdf new file mode 100644 index 0000000..4546a8d Binary files /dev/null and b/pointprocesses/exercises/poisson-solutions.pdf differ diff --git a/pointprocesses/exercises/poisson.pdf b/pointprocesses/exercises/poisson.pdf new file mode 100644 index 0000000..aa1756e Binary files /dev/null and b/pointprocesses/exercises/poisson.pdf differ diff --git a/pointprocesses/exercises/poisson.tex b/pointprocesses/exercises/poisson.tex new file mode 100644 index 0000000..4b8f0f0 --- /dev/null +++ b/pointprocesses/exercises/poisson.tex @@ -0,0 +1,160 @@ +\documentclass[addpoints,10pt]{exam} +\usepackage{url} +\usepackage{color} +\usepackage{hyperref} +\usepackage{graphicx} + +\pagestyle{headandfoot} +\runningheadrule +\firstpageheadrule + +\firstpageheader{Scientific Computing}{Homogeneous Poisson process}{Oct 27, 2014} +%\runningheader{Homework 01}{Page \thepage\ of \numpages}{23. October 2014} +\firstpagefooter{}{}{} +\runningfooter{}{}{} +\pointsinmargin +\bracketedpoints + +%\printanswers +\shadedsolutions + +\usepackage[mediumspace,mediumqspace,Gray]{SIunits} % \ohm, \micro + +%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\usepackage{listings} +\lstset{ + basicstyle=\ttfamily, + numbers=left, + showstringspaces=false, + language=Matlab, + breaklines=true, + breakautoindent=true, + columns=flexible, + frame=single, + captionpos=t, + xleftmargin=2em, + xrightmargin=1em, + aboveskip=10pt, + %title=\lstname, + title={\protect\filename@parse{\lstname}\protect\filename@base.\protect\filename@ext} + } + + +\begin{document} + +\sffamily +%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{questions} + \question \textbf{Homogeneous Poisson process} + We use the Poisson process to generate spike trains on which we can test and imrpove some + standard analysis functions. + + A homogeneous Poisson process of rate $\lambda$ (measured in Hertz) is a point process + where the probability of an event is independent of time $t$ and independent of previous events. + The probability $P$ of an event within a bin of width $\Delta t$ is + \[ P = \lambda \cdot \Delta t \] + for sufficiently small $\Delta t$. + \begin{parts} + + \part Write a function that generates $n$ homogeneous Poisson spike trains of a given duration $T_{max}$ + with rate $\lambda$. + \begin{solution} + \lstinputlisting{hompoissonspikes.m} + \end{solution} + + \part Using this function, generate a few trials and display them in a raster plot. + \begin{solution} + \lstinputlisting{simulations/spikeraster.m} + \begin{lstlisting} +spikes = hompoissonspikes( 10, 100.0, 0.5 ); +spikeraster( spikes ) + \end{lstlisting} + \mbox{}\\[-3ex] + \colorbox{white}{\includegraphics[width=0.7\textwidth]{poissonraster100hz}} + \end{solution} + + \part Write a function that extracts a single vector of interspike intervals + from the spike times returned by the first function. + \begin{solution} + \lstinputlisting{simulations/isis.m} + \end{solution} + + \part Write a function that plots the interspike-interval histogram + from a vector of interspike intervals. The function should also + compute the mean, the standard deviation, and the CV of the intervals + and display the values in the plot. + \begin{solution} + \lstinputlisting{simulations/isihist.m} + \end{solution} + + \part Compute histograms for Poisson spike trains with rate + $\lambda=100$\,Hz. Play around with $T_{max}$ and $n$ and the bin width + (start with 1\,ms) of the histogram. + How many + interspike intervals do you approximately need to get a ``nice'' + histogram? How long do you need to record from the neuron? + \begin{solution} + About 5000 intervals for 25 bins. This corresponds to a $5000 / 100\,\hertz = 50\,\second$ recording + of a neuron firing with 100\,\hertz. + \end{solution} + + \part Compare the histogram with the true distribution of intervals $T$ of the Poisson process + \[ p(T) = \lambda e^{-\lambda T} \] + for various rates $\lambda$. + \begin{solution} + \lstinputlisting{hompoissonisih.m} + \colorbox{white}{\includegraphics[width=0.48\textwidth]{poissonisih100hz}} + \colorbox{white}{\includegraphics[width=0.48\textwidth]{poissonisih20hz}} + \end{solution} + + \part What happens if you make the bin width of the histogram smaller than $\Delta t$ + used for generating the Poisson spikes? + \begin{solution} + The bins between the discretization have zero entries. Therefore + the other ones become higher than they should be. + \end{solution} + + \part Plot the mean interspike interval, the corresponding standard deviation, and the CV + as a function of the rate $\lambda$ of the Poisson process. + Compare the simulations with the theoretical expectations for the dependence on $\lambda$. + \begin{solution} + \lstinputlisting{hompoissonisistats.m} + \colorbox{white}{\includegraphics[width=0.98\textwidth]{poissonisistats}} + \end{solution} + + \part Write a function that computes serial correlations for the interspike intervals + for a range of lags. + The serial correlations $\rho_k$ at lag $k$ are defined as + \[ \rho_k = \frac{\langle (T_{i+k} - \langle T \rangle)(T_i - \langle T \rangle) \rangle}{\langle (T_i - \langle T \rangle)^2\rangle} = \frac{{\rm cov}(T_{i+k}, T_i)}{{\rm var}(T_i)} \] + Use this function to show that interspike intervals of Poisson spikes are independent. + \begin{solution} + \lstinputlisting{simulations/isiserialcorr.m} + \colorbox{white}{\includegraphics[width=0.98\textwidth]{poissonserial100hz}} + \end{solution} + + \part Write a function that generates from spike times + a histogram of spike counts in a count window of given duration $W$. + The function should also plot the Poisson distribution + \[ P(k) = \frac{(\lambda W)^ke^{\lambda W}}{k!} \] + for the rate $\lambda$ determined from the spike trains. + \begin{solution} + \lstinputlisting{simulations/counthist.m} + \colorbox{white}{\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz10ms}} + \colorbox{white}{\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz100ms}} + \end{solution} + + \part Write a function that computes mean count, variance of count and the corresponding Fano factor + for a range of count window durations. The function should generate tow plots: one plotting + the count variance against the mean, the other one the Fano factor as a function of the window duration. + \begin{solution} + \lstinputlisting{simulations/fano.m} + \colorbox{white}{\includegraphics[width=0.98\textwidth]{poissonfano100hz}} + \end{solution} + + \end{parts} + +\end{questions} + + +\end{document} diff --git a/pointprocesses/exercises/poissoncounthistdist100hz100ms.pdf b/pointprocesses/exercises/poissoncounthistdist100hz100ms.pdf new file mode 100644 index 0000000..aa25500 Binary files /dev/null and b/pointprocesses/exercises/poissoncounthistdist100hz100ms.pdf differ diff --git a/pointprocesses/exercises/poissoncounthistdist100hz10ms.pdf b/pointprocesses/exercises/poissoncounthistdist100hz10ms.pdf new file mode 100644 index 0000000..746f30b Binary files /dev/null and b/pointprocesses/exercises/poissoncounthistdist100hz10ms.pdf differ diff --git a/pointprocesses/exercises/poissonfano100hz.pdf 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index 0000000..a20fa0e Binary files /dev/null and b/pointprocesses/exercises/poissonraster100hz.pdf differ diff --git a/pointprocesses/exercises/poissonserial100hz.pdf b/pointprocesses/exercises/poissonserial100hz.pdf new file mode 100644 index 0000000..ff667d9 Binary files /dev/null and b/pointprocesses/exercises/poissonserial100hz.pdf differ diff --git a/pointprocesses/lecture/Makefile b/pointprocesses/lecture/Makefile new file mode 100644 index 0000000..7a243a5 --- /dev/null +++ b/pointprocesses/lecture/Makefile @@ -0,0 +1,142 @@ +BASENAME=pointprocesses + +TEXFILE=$(BASENAME).tex +DVIFILE=$(BASENAME).dvi +PSFILE=$(BASENAME).ps +PDFFILE=$(BASENAME).pdf + +FOILSFILE=foils.pdf +THUMBNAILSFILE=thumbnails.pdf + +HTMLBASENAME=$(BASENAME)h +HTMLTEXFILE=$(BASENAME)h.tex +HTMLDIR=$(BASENAME)h + +GPTFILES=$(wildcard *.gpt) +GPTTEXFILES=$(GPTFILES:.gpt=.tex) + + +all: ps pdf talk again watchps watchpdf foils thumbs html html1 epsfigs clean cleanup cleanplots help +.PHONY: epsfigs + + +# thumbnails: +thumbs: $(THUMBNAILSFILE) +$(THUMBNAILSFILE): $(TEXFILE) $(GPTTEXFILES) + sed -e 's/setboolean{presentation}{true}/setboolean{presentation}{false}/; s/usepackage{crop}/usepackage[frame]{crop}/' $< > thumbsfoils.tex + pdflatex thumbsfoils | tee /dev/stderr | fgrep -q "Rerun to get cross-references right" && pdflatex thumbsfoils || true + pdfnup --nup 2x4 --no-landscape --paper a4paper --trim "-1cm -1cm -1cm -1cm" --outfile $@ thumbsfoils.pdf '1-19' + rm thumbsfoils.* + +# transparencies: +foils: $(FOILSFILE) +$(FOILSFILE): $(TEXFILE) $(GPTTEXFILES) + sed -e 's/setboolean{presentation}{true}/setboolean{presentation}{false}/' $< > tfoils.tex + pdflatex tfoils | tee /dev/stderr | fgrep -q "Rerun to get cross-references right" && pdflatex tfoils || true + pdfnup --nup 1x2 --orient portrait --trim "-1mm -1mm -1mm -1mm" --frame true --delta "1cm 1cm" --paper a4paper --outfile tfoils2.pdf tfoils.pdf + pdfnup --nup 1x1 --orient portrait --trim "-2cm -2cm -2cm -2cm" --paper a4paper --outfile $@ tfoils2.pdf + rm tfoils.* tfoils2.pdf + +# talk: +talk: $(PDFFILE) +pdf: $(PDFFILE) +$(PDFFILE): $(TEXFILE) $(GPTTEXFILES) + pdflatex -interaction=scrollmode $< | tee /dev/stderr | fgrep -q "Rerun to get cross-references right" && pdflatex -interaction=scrollmode $< || true +# batchmode (no output, no stop on error) +# nonstopmode / scrollmode (no stop on error) +# errorstopmode (stop on error) + + +again : + pdflatex $(TEXFILE) + +watchpdf : + while true; do ! make -q pdf && make pdf; sleep 0.5; done + +# html +html : $(HTMLTEXFILE) $(GPTTEXFILES) + rm -f $(HTMLDIR)/* + htlatex $< + mkdir -p $(HTMLDIR) + mv $(HTMLBASENAME).html $(HTMLDIR) + mv $(HTMLBASENAME)*.* $(HTMLDIR) + mv z*.gif $(HTMLDIR) + cd $(HTMLDIR); for i in *.gif; do convert -page +0+0 $$i tmp.gif; mv tmp.gif $$i; done; rmtex $(HTMLBASENAME) + +#$(HTMLTEXFILE) : $(TEXFILE) Makefile +# sed 's/setboolean{html}{false}/setboolean{html}{true}/; s/\\colorbox{white}{\(.*\)}/\1/g' $< > $@ + +html1 : $(HTMLTEXFILE) $(GPTTEXFILES) + latex2html -dir $(HTMLDIR) -mkdir -subdir -nonavigation -noinfo -image_type png -notransparent -white -split 0 $< + sed 's-Date:--' $(HTMLDIR)/$(HTMLDIR).html > tmp.html + cp tmp.html $(HTMLDIR)/index.html + mv tmp.html $(HTMLDIR)/$(HTMLDIR).html + +$(HTMLTEXFILE) : $(TEXFILE) + sed '/^%nohtml/,/^%endnohtml/d; s/\\colorbox{white}{\(.*\)}/\1/g' $< > $@ + + +# eps of all figures: +epsfigs: + mkdir -p epsfigs; \ + for i in $(GPTFILES); do \ + { sed -n -e '1,/\\begin{document}/p' $(TEXFILE); echo "\texpicture{$${i%%.*}}"; echo "\end{document}"; } > tmp.tex; \ + latex tmp.tex; \ + dvips tmp.dvi; \ + ps2eps tmp.ps; \ + mv tmp.eps epsfigs/$${i%%.*}.eps; \ + rm tmp.*; \ + done + + +# plots: +%.tex: %.gpt whitestyles.gp + gnuplot whitestyles.gp $< + epstopdf $*.eps + + +clean : + rm -f *~ + rmtex $(BASENAME) + rm -f $(GPTTEXFILES) + +cleanup : + rm -f *~ + rmtex $(BASENAME) + rm -f $(PSFILE) $(PDFFILE) $(FOILSFILE) $(THUMBNAILSFILE) + rm -f $(GPTTEXFILES) + rm -f -r $(HTMLDIR) + + +cleanplots : + sed -n -e '/\\begin{document}/,/\\end{document}/p' $(TEXFILE) | fgrep '\input{' | grep -v '^%' | sed 's/.*input{\(.*\).tex}.*/\1.gpt/' > plot.fls + mkdir -p unusedplots + for i in *.gp*; do \ + grep -q $$i plot.fls || { grep -q $$i $$(> plot.fls + for i in $$( dat.fls + mkdir -p unuseddata + for i in *.dat; do \ + grep -q $$i dat.fls || mv $$i unuseddata; \ + done + rm dat.fls plot.fls + + +help : + @echo -e \ + "make pdf: make the pdf file of the talk.\n"\ + "make foils: make black&white postscript foils of the talk.\n"\ + "make thumbs: make color thumbnails of the talk.\n"\ + "make again: run latex and make the pdf file of the talk,\n"\ + " no matter whether you changed the .tex file or not.\n\n"\ + "make watchpdf: make the pdf file of the talk\n"\ + " whenever the tex file is modified.\n"\ + "make html: make a html version of the paper (in $(HTMLDIR)).\n\n"\ + "make clean: remove all intermediate files,\n"\ + " just leave the source files and the final .ps and .pdf files.\n"\ + "make cleanup: remove all intermediate files as well as\n"\ + " the final .ps and .pdf files.\n"\ + "make cleanplots: move all unused .gpt and .dat files\n"\ + " into unusedplots/ and unuseddata/, respectively." diff --git a/pointprocesses/lecture/UT_WBMW_Rot_RGB.pdf b/pointprocesses/lecture/UT_WBMW_Rot_RGB.pdf new file mode 100644 index 0000000..f54eedc --- /dev/null +++ b/pointprocesses/lecture/UT_WBMW_Rot_RGB.pdf @@ -0,0 +1,529 @@ +%PDF-1.5 %âãÏÓ +1 0 obj <>/OCGs[15 0 R 18 0 R 21 0 R 25 0 R 35 0 R 36 0 R 37 0 R 38 0 R]>>/Pages 2 0 R/Type/Catalog>> endobj 34 0 obj <>stream + + + + + application/pdf + + + EKUT_WortBildMarke_W_RGB + + + + + Adobe Illustrator CS3 + 2010-07-08T15:07+02:00 + 2010-09-21T13:10:33+02:00 + 2010-09-21T13:10:33+02:00 + + + + 256 + 68 + JPEG + /9j/4AAQSkZJRgABAgEASABIAAD/7QAsUGhvdG9zaG9wIDMuMAA4QklNA+0AAAAAABAASAAAAAEA AQBIAAAAAQAB/+4ADkFkb2JlAGTAAAAAAf/bAIQABgQEBAUEBgUFBgkGBQYJCwgGBggLDAoKCwoK 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+\author[]{Jan Benda} +\institute[]{Neuroethology} +\date[]{WS 14/15} +\titlegraphic{\includegraphics[width=0.3\textwidth]{UT_WBMW_Rot_RGB}} + +%%%%% beamer %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\mode +{ + \usetheme{Singapore} + \setbeamercovered{opaque} + \usecolortheme{tuebingen} + \setbeamertemplate{navigation symbols}{} + \usefonttheme{default} + \useoutertheme{infolines} + % \useoutertheme{miniframes} +} + +%\AtBeginSection[] +%{ +% \begin{frame} +% \begin{center} +% \Huge \insertsectionhead +% \end{center} +% \end{frame} +%} + +\setbeamertemplate{blocks}[rounded][shadow=true] +\setcounter{tocdepth}{1} + +%%%%% packages %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\usepackage[english]{babel} +\usepackage{amsmath} +\usepackage{bm} +\usepackage{pslatex} % nice font for pdf file +%\usepackage{multimedia} + +\usepackage{dsfont} +\newcommand{\naZ}{\mathds{N}} +\newcommand{\gaZ}{\mathds{Z}} +\newcommand{\raZ}{\mathds{Q}} +\newcommand{\reZ}{\mathds{R}} +\newcommand{\reZp}{\mathds{R^+}} +\newcommand{\reZpN}{\mathds{R^+_0}} +\newcommand{\koZ}{\mathds{C}} + +%%%% graphics %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\usepackage{graphicx} +\newcommand{\texpicture}[1]{{\sffamily\small\input{#1.tex}}} + +%%%%% listings %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\usepackage{listings} +\lstset{ + basicstyle=\ttfamily, + numbers=left, + showstringspaces=false, + language=Matlab, + commentstyle=\itshape\color{darkgray}, + keywordstyle=\color{blue}, + stringstyle=\color{green}, + backgroundcolor=\color{blue!10}, + breaklines=true, + breakautoindent=true, + columns=flexible, + frame=single, + captionpos=b, + xleftmargin=1em, + xrightmargin=1em, + aboveskip=10pt + } + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{document} + +\begin{frame}[plain] + \frametitle{} + \vspace{-1cm} + \titlepage % erzeugt Titelseite +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame} + \frametitle{Content} + \tableofcontents +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Point processes} + +\begin{frame} + \frametitle{Point process} + \vspace{-3ex} + \texpicture{pointprocessscetchA} + + A point process is a stochastic (or random) process that generates a sequence of events + at times $\{t_i\}$, $t_i \in \reZ$. + + For each point process there is an underlying continuous-valued + process evolving in time. The associated point process occurs when + the underlying continuous process crosses a threshold. + Examples: + \begin{itemize} + \item Spikes/heartbeat: generated by the dynamics of the membrane potential of neurons/heart cells. + \item Earth quakes: generated by the pressure dynamics between the tectonic plates on either side of a geological fault line. + \item Onset of cricket/frogs/birds/... songs: generated by the dynamics of the state of a nervous system. + \end{itemize} +\end{frame} + +\begin{frame} + \frametitle{Point process} + \texpicture{pointprocessscetchB} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Homogeneous Poisson process} + +\begin{frame} + \frametitle{Homogeneous Poisson process} + The probability $p(t)\delta t$ of an event occuring at time $t$ + is independent of $t$ and independent of any previous event + (independent of event history). + + The probability $P$ for an event occuring within a time bin of width $\Delta t$ + is + \[ P=\lambda \cdot \Delta t \] + for a Poisson process with rate $\lambda$. + \includegraphics[width=1\textwidth]{poissonraster100hz} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Interval statistics} + +\begin{frame} + \frametitle{Rate} + Rate of events $r$ (``spikes per time'') measured in Hertz. + \begin{itemize} + \item Number of events $N$ per observation time $W$: $r = \frac{N}{W}$ + \item Without boundary effects: $r = \frac{N-1}{t_N-t_1}$ + \item Inverse interval: $r = \frac{1}{\mu_{ISI}}$ + \end{itemize} +\end{frame} + +\begin{frame} + \frametitle{(Interspike) interval statistics} + \begin{itemize} + \item Histogram $p(T)$ of intervals $T$. Normalized to $\int_0^{\infty} p(T) \; dT = 1$ + \item Mean interval $\mu_{ISI} = \langle T \rangle = \frac{1}{n}\sum\limits_{i=1}^n T_i$ + \item Variance of intervals $\sigma_{ISI}^2 = \langle (T - \langle T \rangle)^2 \rangle$\vspace{1ex} + \item Coefficient of variation $CV_{ISI} = \frac{\sigma_{ISI}}{\mu_{ISI}}$ + \item Diffusion coefficient $D_{ISI} = \frac{\sigma_{ISI}^2}{2\mu_{ISI}^3}$ + \vfill + \end{itemize} + \includegraphics[width=0.45\textwidth]{poissonisih100hz}\hfill + \includegraphics[width=0.45\textwidth]{lifisih16} +\end{frame} + +\begin{frame} + \frametitle{Interval statistics of homogeneous Poisson process} + \begin{itemize} + \item Exponential distribution of intervals $T$: $p(T) = \lambda e^{-\lambda T}$ + \item Mean interval $\mu_{ISI} = \frac{1}{\lambda}$ + \item Variance of intervals $\sigma_{ISI}^2 = \frac{1}{\lambda^2}$ + \item Coefficient of variation $CV_{ISI} = 1$ + \end{itemize} + \vfill + \includegraphics[width=0.45\textwidth]{poissonisihexp20hz}\hfill + \includegraphics[width=0.45\textwidth]{poissonisihexp100hz} +\end{frame} + +\begin{frame} + \frametitle{Interval return maps} + Scatter plot between succeeding intervals separated by lag $k$. + \vfill + Poisson process $\lambda=100$\,Hz: + \includegraphics[width=1\textwidth]{poissonreturnmap100hz}\hfill +\end{frame} + +\begin{frame} + \frametitle{Serial interval correlations} + Correlation coefficients between succeeding intervals separated by lag $k$: + \[ \rho_k = \frac{\langle (T_{i+k} - \langle T \rangle)(T_i - \langle T \rangle) \rangle}{\langle (T_i - \langle T \rangle)^2\rangle} = \frac{{\rm cov}(T_{i+k}, T_i)}{{\rm var}(T_i)} \] + \begin{itemize} + \item $\rho_0=1$ (correlation of each interval with itself). + \item Poisson process: $\rho_k =0$ for $k>0$ (renewal process!) + \end{itemize} + \vfill + \includegraphics[width=0.7\textwidth]{poissonserial100hz} +\end{frame} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Count statistics} + +\begin{frame} + \frametitle{Count statistics} + Histogram of number of events $N$ (counts) within observation window of duration $W$. + + \vfill + \includegraphics[width=0.48\textwidth]{poissoncounthist100hz10ms}\hfill + \includegraphics[width=0.48\textwidth]{poissoncounthist100hz100ms} +\end{frame} + +\begin{frame} + \frametitle{Count statistics of Poisson process} + Poisson distribution: + \[ P(k) = \frac{(\lambda W)^ke^{\lambda W}}{k!} \] + + \vfill + \includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz10ms}\hfill + \includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz100ms} +\end{frame} + +\begin{frame} + \frametitle{Count statistics --- Fano factor} + Statistics of number of events $N$ within observation window of duration $W$. + \begin{itemize} + \item Mean count: $\mu_N = \langle N \rangle$ + \item Count variance: $\sigma_N^2 = \langle (N - \langle N \rangle)^2 \rangle$ + \item Fano factor (variance divided by mean): $F = \frac{\sigma_N^2}{\mu_N}$ + \item Poisson process: $F=1$ + \end{itemize} + \vfill + Poisson process $\lambda=100$\,Hz: + \includegraphics[width=1\textwidth]{poissonfano100hz} +\end{frame} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Integrate-and-fire models} + +\begin{frame} + \frametitle{Integrate-and-fire models} + Leaky integrate-and-fire model (LIF): + \[ \tau \frac{dV}{dt} = -V + RI + D\xi \] + Whenever membrane potential $V(t)$ crosses the firing threshold $\theta$, a spike is emitted and + $V(t)$ is reset to $V_{reset}$. + \begin{itemize} + \item $\tau$: membrane time constant (typically 10\,ms) + \item $R$: input resistance (here 1\,mV (!)) + \item $D\xi$: additive Gaussian white noise of strength $D$ + \item $\theta$: firing threshold (here 10\,mV) + \item $V_{reset}$: reset potential (here 0\,mV) + \end{itemize} +\end{frame} + +\begin{frame} + \frametitle{Integrate-and-fire models} + Discretization with time step $\Delta t$: $V(t) \rightarrow V_i,\;t_i = i \Delta t$.\\ + Euler integration: + \begin{eqnarray*} + \frac{dV}{dt} & \approx & \frac{V_{i+1} - V_i}{\Delta t} \\ + \Rightarrow \quad V_{i+1} & = & V_i + \Delta t \frac{-V_i+RI_i+\sqrt{2D\Delta t}N_i}{\tau} + \end{eqnarray*} + $N_i$ are normally distributed random numbers (Gaussian with zero mean and unit variance) + --- the $\sqrt{\Delta t}$ is for white noise. + + \includegraphics[width=0.82\textwidth]{lifraster16} +\end{frame} + +\begin{frame} + \frametitle{Interval statistics of LIF} + Interval distribution approaches Inverse Gaussian for large $I$: + \[ p(T) = \frac{1}{\sqrt{4\pi D T^3}}\exp\left[-\frac{(T-\langle T \rangle)^2}{4DT\langle T \rangle^2}\right] \] + where $\langle T \rangle$ is the mean interspike interval and $D$ + is the diffusion coefficient. + \vfill + \includegraphics[width=0.45\textwidth]{lifisihdistr08}\hfill + \includegraphics[width=0.45\textwidth]{lifisihdistr16} +\end{frame} + +\begin{frame} + \frametitle{Interval statistics of PIF} + For the perfect integrate-and-fire (PIF) + \[ \tau \frac{dV}{dt} = RI + D\xi \] + (the canonical model or supra-threshold firing on a limit cycle)\\ + the Inverse Gaussian describes exactly the interspike interval distribution. + \vfill + \includegraphics[width=0.45\textwidth]{pifisihdistr01}\hfill + \includegraphics[width=0.45\textwidth]{pifisihdistr10} +\end{frame} + +\begin{frame} + \frametitle{Interval return map of LIF} + LIF $I=15.7$: + \includegraphics[width=1\textwidth]{lifreturnmap16} +\end{frame} + +\begin{frame} + \frametitle{Serial correlations of LIF} + LIF $I=15.7$: + \includegraphics[width=1\textwidth]{lifserial16}\\ + Integrate-and-fire driven with white noise are still renewal processes! +\end{frame} + +\begin{frame} + \frametitle{Count statistics of LIF} + LIF $I=15.7$: + \includegraphics[width=1\textwidth]{liffano16}\\ + Fano factor is not one! +\end{frame} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame} + \frametitle{Interval statistics of LIF with OU noise} + \begin{eqnarray*} + \tau \frac{dV}{dt} & = & -V + RI + U \\ + \tau_{OU} \frac{dU}{dt} & = & - U + D\xi + \end{eqnarray*} + Ohrnstein-Uhlenbeck noise is lowpass filtered white noise. + \includegraphics[width=0.45\textwidth]{lifouisihdistr08-100ms}\hfill + \includegraphics[width=0.45\textwidth]{lifouisihdistr16-100ms}\\ + More peaky than the inverse Gaussian! +\end{frame} + +\begin{frame} + \frametitle{Interval return map of LIF with OU noise} + LIF $I=15.7$, $\tau_{OU}=100$\,ms: + \includegraphics[width=1\textwidth]{lifoureturnmap16-100ms} +\end{frame} + +\begin{frame} + \frametitle{Serial correlations of LIF with OU noise} + LIF $I=15.7$, $\tau_{OU}=100$\,ms: + \includegraphics[width=1\textwidth]{lifouserial16-100ms}\\ + OU-noise introduces positive interval correlations! +\end{frame} + +\begin{frame} + \frametitle{Count statistics of LIF with OU noise} + LIF $I=15.7$, $\tau_{OU}=100$\,ms: + \includegraphics[width=1\textwidth]{lifoufano16-100ms}\\ + Fano factor increases with count window duration. +\end{frame} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame} + \frametitle{Interval statistics of LIF with adaptation} + \begin{eqnarray*} + \tau \frac{dV}{dt} & = & -V - A + RI + D\xi \\ + \tau_{adapt} \frac{dA}{dt} & = & - A + \end{eqnarray*} + Adaptation $A$ with time constant $\tau_{adapt}$ and increment $\Delta A$ at spike. + \includegraphics[width=0.45\textwidth]{lifadaptisihdistr08-100ms}\hfill + \includegraphics[width=0.45\textwidth]{lifadaptisihdistr65-100ms}\\ + Similar to LIF with white noise. +\end{frame} + +\begin{frame} + \frametitle{Interval return map of LIF with adaptation} + LIF $I=10$, $\tau_{adapt}=100$\,ms: + \includegraphics[width=1\textwidth]{lifadaptreturnmap10-100ms}\\ + Negative correlation at lag one. +\end{frame} + +\begin{frame} + \frametitle{Serial correlations of LIF with adaptation} + LIF $I=10$, $\tau_{adapt}=100$\,ms: + \includegraphics[width=1\textwidth]{lifadaptserial10-100ms}\\ + Adaptation with white noise introduces negative interval correlations! +\end{frame} + +\begin{frame} + \frametitle{Count statistics of LIF with adaptation} + LIF $I=10$, $\tau_{adapt}=100$\,ms: + \includegraphics[width=1\textwidth]{lifadaptfano10-100ms}\\ + Fano factor decreases with count window duration. +\end{frame} + + +\end{document} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Non stationary} +\subsection{Inhomogeneous Poisson process} +\subsection{Firing rate} +\subsection{Instantaneous rate} +\subsection{Autocorrelation} +\subsection{Crosscorrelation} +\subsection{Joint PSTH} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Renewal process} +\subsection{Superthreshold firing} +\subsection{Subthreshold firing} +\section{Non-renewal processes} +\subsection{Bursting} +\subsection{Resonator} + + +\subsection{Standard distributions} +\subsubsection{Gamma} +\subsubsection{How to read ISI histograms} +refractoriness, poisson tail, sub-, supra-threshold, missed spikes + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Correlation with stimulus} +\subsection{Tuning curve} +\subsection{Linear filter} +\subsection{Spatiotemporal receptive field} +\subsection{Generalized linear model} + +\begin{frame} +\end{frame} diff --git a/pointprocesses/lecture/pointprocessscetchA.eps b/pointprocesses/lecture/pointprocessscetchA.eps new file mode 100644 index 0000000..95a12ae --- /dev/null +++ b/pointprocesses/lecture/pointprocessscetchA.eps @@ -0,0 +1,500 @@ +%!PS-Adobe-2.0 EPSF-2.0 +%%Title: pointprocessscetchA.tex +%%Creator: gnuplot 4.6 patchlevel 4 +%%CreationDate: Sun Oct 26 14:09:12 2014 +%%DocumentFonts: +%%BoundingBox: 50 50 373 135 +%%EndComments +%%BeginProlog +/gnudict 256 dict def +gnudict begin +% +% The following true/false flags may be edited by hand if desired. +% The unit 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grestore} bind def +/D4 {gsave translate 45 rotate 0 0 S4 stroke grestore} bind def +/D5 {gsave translate 45 rotate 0 0 S5 stroke grestore} bind def +/D6 {gsave translate 45 rotate 0 0 S6 stroke grestore} bind def +/D7 {gsave translate 45 rotate 0 0 S7 stroke grestore} bind def +/D8 {gsave translate 45 rotate 0 0 S8 stroke grestore} bind def +/D9 {gsave translate 45 rotate 0 0 S9 stroke grestore} bind def +/D10 {gsave translate 45 rotate 0 0 S10 stroke grestore} bind def +/D11 {gsave translate 45 rotate 0 0 S11 stroke grestore} bind def +/D12 {gsave translate 45 rotate 0 0 S12 stroke grestore} bind def +/D13 {gsave translate 45 rotate 0 0 S13 stroke grestore} bind def +/D14 {gsave translate 45 rotate 0 0 S14 stroke grestore} bind def +/D15 {gsave translate 45 rotate 0 0 S15 stroke grestore} bind def +/DiaE {stroke [] 0 setdash vpt add M + hpt neg vpt neg V hpt vpt neg V + hpt vpt V hpt neg vpt V closepath stroke} def +/BoxE {stroke [] 0 setdash exch hpt sub exch vpt add M + 0 vpt2 neg V hpt2 0 V 0 vpt2 V + hpt2 neg 0 V closepath stroke} def +/TriUE {stroke [] 0 setdash vpt 1.12 mul add M + hpt neg vpt -1.62 mul V + hpt 2 mul 0 V + hpt neg vpt 1.62 mul V closepath stroke} def +/TriDE {stroke [] 0 setdash vpt 1.12 mul sub M + hpt neg vpt 1.62 mul V + hpt 2 mul 0 V + hpt neg vpt -1.62 mul V closepath stroke} def +/PentE {stroke [] 0 setdash gsave + translate 0 hpt M 4 {72 rotate 0 hpt L} repeat + closepath stroke grestore} def +/CircE {stroke [] 0 setdash + hpt 0 360 arc stroke} def +/Opaque {gsave closepath 1 setgray fill grestore 0 setgray closepath} def +/DiaW {stroke [] 0 setdash vpt add M + hpt neg vpt neg V hpt vpt neg V + hpt vpt V hpt neg vpt V Opaque stroke} def +/BoxW {stroke [] 0 setdash exch hpt sub exch vpt add M + 0 vpt2 neg V hpt2 0 V 0 vpt2 V + hpt2 neg 0 V Opaque stroke} def +/TriUW {stroke [] 0 setdash vpt 1.12 mul add M + hpt neg vpt -1.62 mul V + hpt 2 mul 0 V + hpt neg vpt 1.62 mul V Opaque stroke} def +/TriDW {stroke [] 0 setdash vpt 1.12 mul sub M + hpt neg vpt 1.62 mul V + hpt 2 mul 0 V + hpt neg vpt -1.62 mul V Opaque stroke} def +/PentW {stroke [] 0 setdash gsave + translate 0 hpt M 4 {72 rotate 0 hpt L} repeat + Opaque stroke grestore} def +/CircW {stroke [] 0 setdash + hpt 0 360 arc Opaque stroke} def +/BoxFill {gsave Rec 1 setgray fill grestore} def +/Density { + /Fillden exch def + currentrgbcolor + /ColB exch def /ColG exch def /ColR exch def + /ColR ColR Fillden mul Fillden sub 1 add def + /ColG ColG Fillden mul Fillden sub 1 add def + /ColB ColB Fillden mul Fillden sub 1 add def + ColR ColG ColB setrgbcolor} def +/BoxColFill {gsave Rec PolyFill} def +/PolyFill {gsave Density fill grestore grestore} def +/h {rlineto rlineto rlineto gsave closepath fill grestore} bind def +% +% PostScript Level 1 Pattern Fill routine for rectangles +% Usage: x y w h s a XX PatternFill +% x,y = lower left corner of box to be filled +% w,h = width and height of box +% a = angle in degrees between lines and x-axis +% XX = 0/1 for no/yes cross-hatch +% +/PatternFill {gsave /PFa [ 9 2 roll ] def + PFa 0 get PFa 2 get 2 div add PFa 1 get PFa 3 get 2 div add translate + PFa 2 get -2 div PFa 3 get -2 div PFa 2 get PFa 3 get Rec + TransparentPatterns {} {gsave 1 setgray fill grestore} ifelse + clip + currentlinewidth 0.5 mul setlinewidth + /PFs PFa 2 get dup mul PFa 3 get dup mul add sqrt def + 0 0 M PFa 5 get rotate PFs -2 div dup translate + 0 1 PFs PFa 4 get div 1 add floor cvi + {PFa 4 get mul 0 M 0 PFs V} for + 0 PFa 6 get ne { + 0 1 PFs PFa 4 get div 1 add floor cvi + {PFa 4 get mul 0 2 1 roll M PFs 0 V} for + } if + stroke grestore} def +% +/languagelevel where + {pop languagelevel} {1} ifelse + 2 lt + {/InterpretLevel1 true def} + {/InterpretLevel1 Level1 def} + ifelse +% +% PostScript level 2 pattern fill definitions +% +/Level2PatternFill { +/Tile8x8 {/PaintType 2 /PatternType 1 /TilingType 1 /BBox [0 0 8 8] /XStep 8 /YStep 8} + bind def +/KeepColor {currentrgbcolor [/Pattern /DeviceRGB] setcolorspace} bind def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop 0 0 M 8 8 L 0 8 M 8 0 L stroke} +>> matrix makepattern +/Pat1 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop 0 0 M 8 8 L 0 8 M 8 0 L stroke + 0 4 M 4 8 L 8 4 L 4 0 L 0 4 L stroke} +>> matrix makepattern +/Pat2 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop 0 0 M 0 8 L + 8 8 L 8 0 L 0 0 L fill} +>> matrix makepattern +/Pat3 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop -4 8 M 8 -4 L + 0 12 M 12 0 L stroke} +>> matrix makepattern +/Pat4 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop -4 0 M 8 12 L + 0 -4 M 12 8 L stroke} +>> matrix makepattern +/Pat5 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop -2 8 M 4 -4 L + 0 12 M 8 -4 L 4 12 M 10 0 L stroke} +>> matrix makepattern +/Pat6 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop -2 0 M 4 12 L + 0 -4 M 8 12 L 4 -4 M 10 8 L stroke} +>> matrix makepattern +/Pat7 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop 8 -2 M -4 4 L + 12 0 M -4 8 L 12 4 M 0 10 L stroke} +>> matrix makepattern +/Pat8 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop 0 -2 M 12 4 L + -4 0 M 12 8 L -4 4 M 8 10 L stroke} +>> matrix makepattern +/Pat9 exch def +/Pattern1 {PatternBgnd KeepColor Pat1 setpattern} bind def +/Pattern2 {PatternBgnd KeepColor Pat2 setpattern} bind def +/Pattern3 {PatternBgnd KeepColor Pat3 setpattern} bind def +/Pattern4 {PatternBgnd KeepColor Landscape {Pat5} {Pat4} ifelse setpattern} bind def +/Pattern5 {PatternBgnd KeepColor Landscape {Pat4} {Pat5} ifelse setpattern} bind def +/Pattern6 {PatternBgnd KeepColor Landscape {Pat9} {Pat6} ifelse setpattern} bind def +/Pattern7 {PatternBgnd KeepColor Landscape {Pat8} {Pat7} ifelse setpattern} bind def +} def +% +% +%End of PostScript Level 2 code +% +/PatternBgnd { + TransparentPatterns {} {gsave 1 setgray fill grestore} ifelse +} def +% +% Substitute for Level 2 pattern fill codes with +% grayscale if Level 2 support is not selected. +% +/Level1PatternFill { +/Pattern1 {0.250 Density} bind def +/Pattern2 {0.500 Density} bind def +/Pattern3 {0.750 Density} bind def +/Pattern4 {0.125 Density} bind def +/Pattern5 {0.375 Density} bind def +/Pattern6 {0.625 Density} bind def +/Pattern7 {0.875 Density} bind def +} def +% +% Now test for support of Level 2 code +% +Level1 {Level1PatternFill} {Level2PatternFill} ifelse +% +/Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont +dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall +currentdict end definefont pop +Level1 SuppressPDFMark or +{} { +/SDict 10 dict def +systemdict /pdfmark known not { + userdict /pdfmark systemdict /cleartomark get put +} if +SDict begin [ + /Title (pointprocessscetchB.tex) + /Subject (gnuplot plot) + /Creator (gnuplot 4.6 patchlevel 4) + /Author (jan) +% /Producer (gnuplot) +% /Keywords () + /CreationDate (Sun Oct 26 17:34:18 2014) + /DOCINFO pdfmark +end +} ifelse +end +%%EndProlog +%%Page: 1 1 +gnudict begin +gsave +doclip +50 50 translate +0.050 0.050 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+2123 460 CircleF +2413 556 CircleF +2717 652 CircleF +3167 748 CircleF +4033 844 CircleF +4650 940 CircleF +5685 1036 CircleF +1.000 UP +2.000 UL +LT0 +LC0 setrgbcolor +1.00 1.00 1.00 C 910 268 CircleF +1412 364 CircleF +2123 460 CircleF +2413 556 CircleF +2717 652 CircleF +3167 748 CircleF +4033 844 CircleF +4650 940 CircleF +5685 1036 CircleF +1.500 UP +2.000 UL +LT0 +LC0 setrgbcolor +0.00 0.00 0.00 C 910 364 CircleF +1412 460 CircleF +2123 556 CircleF +2413 652 CircleF +2717 748 CircleF +3167 844 CircleF +4033 940 CircleF +4650 1036 CircleF +5685 1132 CircleF +1.000 UP +2.000 UL +LTb +LCb setrgbcolor +0.00 0.00 0.00 C stroke +grestore +end +showpage +%%Trailer diff --git a/pointprocesses/lecture/pointprocessscetchB.gpt b/pointprocesses/lecture/pointprocessscetchB.gpt new file mode 100644 index 0000000..e1cda52 --- /dev/null +++ b/pointprocesses/lecture/pointprocessscetchB.gpt @@ -0,0 +1,204 @@ +set out 'pointprocessscetchB.tex' + +set border 0 +set lmarg 4 +set rmarg 1 +set tmarg 1 +set bmarg 1 +set xrange [0:11] +unset xtics +set yrange [-0.7:1.8] +unset ytics +set label 1 at graph -0.07, graph 1.1 + +set multiplot + +set origin 0, 0.7 +set size 1, 0.3 +set label 1 '\normalsize Event times $\{t_i\}$' +set arrow 1 from 0, 0.5 to 11, 0.5 head filled +set label 2 'Time' at 11, -0.3 center +set label 3 "$t_{1}$" at 0.724649, -0.6 center +set label 4 "$t_{2}$" at 1.67586, -0.6 center +set label 5 "$t_{3}$" at 3.02389, -0.6 center +set label 6 "$t_{4}$" at 3.57466, -0.6 center +set label 7 "$t_{5}$" at 4.15121, -0.6 center +set label 8 "$t_{6}$" at 5.00412, -0.6 center +set label 9 "$t_{7}$" at 6.64549, -0.6 center +set label 10 "$t_{8}$" at 7.81657, -0.6 center +set label 11 "$t_{9}$" at 9.77964, -0.6 center +plot '-' w l lt 1 lc rgb 'black' lw 10 +0.724649 0 +0.724649 1 + +1.67586 0 +1.67586 1 + +3.02389 0 +3.02389 1 + +3.57466 0 +3.57466 1 + +4.15121 0 +4.15121 1 + +5.00412 0 +5.00412 1 + +6.64549 0 +6.64549 1 + +7.81657 0 +7.81657 1 + +9.77964 0 +9.77964 1 +e +unset label 3 +unset label 4 +unset label 5 +unset label 6 +unset label 7 +unset label 8 +unset label 9 +unset label 10 +unset label 11 + +set origin 0, 0.4 +set label 1 '\normalsize Intervals $\{T_i\}, \; T_i = t_{i+1} - t_i$' +set label 3 "$T_{1}$" at 1.20025, -0.5 center +set arrow 3 from 0.724649, 0.2 to 1.67586, 0.2 heads +set label 4 "$T_{2}$" at 2.34987, -0.5 center +set arrow 4 from 1.67586, 0.2 to 3.02389, 0.2 heads +set label 5 "$T_{3}$" at 3.29927, -0.5 center +set arrow 5 from 3.02389, 0.2 to 3.57466, 0.2 heads +set label 6 "$T_{4}$" at 3.86293, -0.5 center +set arrow 6 from 3.57466, 0.2 to 4.15121, 0.2 heads +set label 7 "$T_{5}$" at 4.57767, -0.5 center +set arrow 7 from 4.15121, 0.2 to 5.00412, 0.2 heads +set label 8 "$T_{6}$" at 5.82481, -0.5 center +set arrow 8 from 5.00412, 0.2 to 6.64549, 0.2 heads +set label 9 "$T_{7}$" at 7.23103, -0.5 center +set arrow 9 from 6.64549, 0.2 to 7.81657, 0.2 heads +set label 10 "$T_{8}$" at 8.79811, -0.5 center +set arrow 10 from 7.81657, 0.2 to 9.77964, 0.2 heads +plot '-' w l lt 1 lc rgb 'black' lw 10 +0.724649 0 +0.724649 1 + +1.67586 0 +1.67586 1 + +3.02389 0 +3.02389 1 + +3.57466 0 +3.57466 1 + +4.15121 0 +4.15121 1 + +5.00412 0 +5.00412 1 + +6.64549 0 +6.64549 1 + +7.81657 0 +7.81657 1 + +9.77964 0 +9.77964 1 +e +unset label 3 +unset label 4 +unset label 5 +unset label 6 +unset label 7 +unset label 8 +unset label 9 +unset label 10 +unset arrow 3 +unset arrow 4 +unset arrow 5 +unset arrow 6 +unset arrow 7 +unset arrow 8 +unset arrow 9 +unset arrow 10 + +set origin 0, 0 +set size 1, 0.4 +set border 2 +set yrange [-0.5:10.5] +set ytics 2 nomirror out +set arrow 1 from 0, 0.0 to 11, 0.0 head filled +set label 2 'Time' at 11, -2.2 center +set label 1 '\normalsize Event counts $\{ n_i \}$' at graph -0.07, graph 1.2 +plot '-' w l lt 1 lc rgb 'black' lw 3, \ + '-' w p lt 1 lc rgb 'black' pt 7 ps 1.5 lw 2, \ + '-' w p lt 1 lc rgb 'white' pt 7 ps 1.0 lw 2, \ + '-' w p lt 1 lc rgb 'black' pt 7 ps 1.5 lw 2 +0 0 +0.724649 0 + +0.724649 1 +1.67586 1 + +1.67586 2 +3.02389 2 + +3.02389 3 +3.57466 3 + +3.57466 4 +4.15121 4 + +4.15121 5 +5.00412 5 + +5.00412 6 +6.64549 6 + +6.64549 7 +7.81657 7 + +7.81657 8 +9.77964 8 + +9.77964 9 +10.7899 9 +e +0.724649 0 +1.67586 1 +3.02389 2 +3.57466 3 +4.15121 4 +5.00412 5 +6.64549 6 +7.81657 7 +9.77964 8 +e +0.724649 0 +1.67586 1 +3.02389 2 +3.57466 3 +4.15121 4 +5.00412 5 +6.64549 6 +7.81657 7 +9.77964 8 +e +0.724649 1 +1.67586 2 +3.02389 3 +3.57466 4 +4.15121 5 +5.00412 6 +6.64549 7 +7.81657 8 +9.77964 9 +e + +unset multiplot diff --git a/pointprocesses/lecture/pointprocessscetchB.pdf b/pointprocesses/lecture/pointprocessscetchB.pdf new file mode 100644 index 0000000..a4c7e8c Binary files /dev/null and b/pointprocesses/lecture/pointprocessscetchB.pdf differ diff --git a/pointprocesses/lecture/pointprocessscetchB.tex b/pointprocesses/lecture/pointprocessscetchB.tex new file mode 100644 index 0000000..14803c3 --- /dev/null +++ b/pointprocesses/lecture/pointprocessscetchB.tex @@ -0,0 +1,130 @@ +% GNUPLOT: LaTeX picture with Postscript +\begingroup + \makeatletter + \providecommand\color[2][]{% + \GenericError{(gnuplot) \space\space\space\@spaces}{% + Package color not loaded in conjunction with + terminal option `colourtext'% + }{See the gnuplot documentation for explanation.% + }{Either use 'blacktext' in gnuplot or load the package + color.sty in LaTeX.}% + \renewcommand\color[2][]{}% + }% + \providecommand\includegraphics[2][]{% + \GenericError{(gnuplot) \space\space\space\@spaces}{% + Package graphicx or graphics not loaded% + }{See the gnuplot documentation for explanation.% + }{The gnuplot epslatex terminal needs graphicx.sty or graphics.sty.}% + \renewcommand\includegraphics[2][]{}% + }% + \providecommand\rotatebox[2]{#2}% + \@ifundefined{ifGPcolor}{% + \newif\ifGPcolor + \GPcolortrue + }{}% + \@ifundefined{ifGPblacktext}{% + \newif\ifGPblacktext + \GPblacktexttrue + }{}% + % define a \g@addto@macro without @ in the name: + \let\gplgaddtomacro\g@addto@macro + % define empty templates for all commands taking text: + \gdef\gplbacktext{}% + \gdef\gplfronttext{}% + \makeatother + \ifGPblacktext + % no textcolor at all + \def\colorrgb#1{}% + \def\colorgray#1{}% + \else + % gray or color? + \ifGPcolor + \def\colorrgb#1{\color[rgb]{#1}}% + \def\colorgray#1{\color[gray]{#1}}% + \expandafter\def\csname LTw\endcsname{\color{white}}% + \expandafter\def\csname LTb\endcsname{\color{black}}% + \expandafter\def\csname LTa\endcsname{\color{black}}% + \expandafter\def\csname LT0\endcsname{\color[rgb]{1,0,0}}% + \expandafter\def\csname LT1\endcsname{\color[rgb]{0,1,0}}% + \expandafter\def\csname LT2\endcsname{\color[rgb]{0,0,1}}% + \expandafter\def\csname LT3\endcsname{\color[rgb]{1,0,1}}% + \expandafter\def\csname LT4\endcsname{\color[rgb]{0,1,1}}% + \expandafter\def\csname LT5\endcsname{\color[rgb]{1,1,0}}% + \expandafter\def\csname LT6\endcsname{\color[rgb]{0,0,0}}% + \expandafter\def\csname LT7\endcsname{\color[rgb]{1,0.3,0}}% + \expandafter\def\csname LT8\endcsname{\color[rgb]{0.5,0.5,0.5}}% + \else + % gray + \def\colorrgb#1{\color{black}}% + \def\colorgray#1{\color[gray]{#1}}% + \expandafter\def\csname LTw\endcsname{\color{white}}% + \expandafter\def\csname LTb\endcsname{\color{black}}% + \expandafter\def\csname LTa\endcsname{\color{black}}% + \expandafter\def\csname LT0\endcsname{\color{black}}% + \expandafter\def\csname LT1\endcsname{\color{black}}% + \expandafter\def\csname LT2\endcsname{\color{black}}% + \expandafter\def\csname LT3\endcsname{\color{black}}% + \expandafter\def\csname LT4\endcsname{\color{black}}% + \expandafter\def\csname LT5\endcsname{\color{black}}% + \expandafter\def\csname LT6\endcsname{\color{black}}% + \expandafter\def\csname LT7\endcsname{\color{black}}% + \expandafter\def\csname LT8\endcsname{\color{black}}% + \fi + \fi + \setlength{\unitlength}{0.0500bp}% + \begin{picture}(6462.00,3740.00)% + \gplgaddtomacro\gplbacktext{% + \csname LTb\endcsname% + \put(122,3587){\makebox(0,0)[l]{\strut{}\normalsize Event times $\{t_i\}$}}% + \put(6329,2947){\makebox(0,0){\strut{}Time}}% + \put(910,2865){\makebox(0,0){\strut{}$t_{1}$}}% + \put(1412,2865){\makebox(0,0){\strut{}$t_{2}$}}% + \put(2123,2865){\makebox(0,0){\strut{}$t_{3}$}}% + \put(2413,2865){\makebox(0,0){\strut{}$t_{4}$}}% + \put(2717,2865){\makebox(0,0){\strut{}$t_{5}$}}% + \put(3167,2865){\makebox(0,0){\strut{}$t_{6}$}}% + \put(4033,2865){\makebox(0,0){\strut{}$t_{7}$}}% + \put(4650,2865){\makebox(0,0){\strut{}$t_{8}$}}% + \put(5685,2865){\makebox(0,0){\strut{}$t_{9}$}}% + }% + \gplgaddtomacro\gplfronttext{% + }% + \gplgaddtomacro\gplbacktext{% + \csname LTb\endcsname% + \put(122,2465){\makebox(0,0)[l]{\strut{}\normalsize Intervals $\{T_i\}, \; T_i = t_{i+1} - t_i$}}% + \put(6329,1825){\makebox(0,0){\strut{}Time}}% + \put(1161,1770){\makebox(0,0){\strut{}$T_{1}$}}% + \put(1767,1770){\makebox(0,0){\strut{}$T_{2}$}}% + \put(2268,1770){\makebox(0,0){\strut{}$T_{3}$}}% + \put(2565,1770){\makebox(0,0){\strut{}$T_{4}$}}% + \put(2942,1770){\makebox(0,0){\strut{}$T_{5}$}}% + \put(3600,1770){\makebox(0,0){\strut{}$T_{6}$}}% + \put(4341,1770){\makebox(0,0){\strut{}$T_{7}$}}% + \put(5168,1770){\makebox(0,0){\strut{}$T_{8}$}}% + }% + \gplgaddtomacro\gplfronttext{% + }% + \gplgaddtomacro\gplbacktext{% + \colorrgb{0.00,0.00,0.00}% + \put(333,268){\makebox(0,0)[r]{\strut{}$0$}}% + \colorrgb{0.00,0.00,0.00}% + \put(333,460){\makebox(0,0)[r]{\strut{}$2$}}% + \colorrgb{0.00,0.00,0.00}% + \put(333,652){\makebox(0,0)[r]{\strut{}$4$}}% + \colorrgb{0.00,0.00,0.00}% + \put(333,844){\makebox(0,0)[r]{\strut{}$6$}}% + \colorrgb{0.00,0.00,0.00}% + \put(333,1036){\makebox(0,0)[r]{\strut{}$8$}}% + \colorrgb{0.00,0.00,0.00}% + \put(333,1228){\makebox(0,0)[r]{\strut{}$10$}}% + \csname LTb\endcsname% + \put(122,1487){\makebox(0,0)[l]{\strut{}\normalsize Event counts $\{ n_i \}$}}% + \put(6329,57){\makebox(0,0){\strut{}Time}}% + }% + \gplgaddtomacro\gplfronttext{% + }% + \gplbacktext + \put(0,0){\includegraphics{pointprocessscetchB}}% + \gplfronttext + \end{picture}% +\endgroup diff --git a/pointprocesses/lecture/poissoncounthist100hz100ms.pdf b/pointprocesses/lecture/poissoncounthist100hz100ms.pdf new file mode 100644 index 0000000..a120ecf Binary files /dev/null and b/pointprocesses/lecture/poissoncounthist100hz100ms.pdf differ diff --git a/pointprocesses/lecture/poissoncounthist100hz10ms.pdf b/pointprocesses/lecture/poissoncounthist100hz10ms.pdf new file mode 100644 index 0000000..4a9615c Binary files /dev/null and b/pointprocesses/lecture/poissoncounthist100hz10ms.pdf differ diff --git a/pointprocesses/lecture/poissoncounthistdist100hz100ms.pdf b/pointprocesses/lecture/poissoncounthistdist100hz100ms.pdf new file mode 100644 index 0000000..aa25500 Binary files /dev/null and b/pointprocesses/lecture/poissoncounthistdist100hz100ms.pdf differ diff --git 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b/pointprocesses/lecture/raster.pdf new file mode 100644 index 0000000..a20fa0e Binary files /dev/null and b/pointprocesses/lecture/raster.pdf differ diff --git a/pointprocesses/lecture/thumbs.pdf b/pointprocesses/lecture/thumbs.pdf new file mode 100644 index 0000000..61cb4e6 Binary files /dev/null and b/pointprocesses/lecture/thumbs.pdf differ diff --git a/pointprocesses/lecture/whitestyles.gp b/pointprocesses/lecture/whitestyles.gp new file mode 100644 index 0000000..1eeb255 --- /dev/null +++ b/pointprocesses/lecture/whitestyles.gp @@ -0,0 +1,67 @@ +set tics scale 0.8 +set tics out +set format '$%g$' +unset grid +unset key + +# on white background: +set style line 1 lt -1 lc rgb "black" lw 2 # border +set border back ls 1 +set style line 2 lt 1 lc rgb "black" lw 4 pt 7 ps 2 # thick line +set style line 3 lt 1 lc rgb "black" lw 2 # thin line +set style line 4 lt 3 lc rgb "black" lw 1 # grid line + +set style line 10 lt 1 lc rgb "#00DD00" lw 6 # stimulus thick, AM +set style line 11 lt 1 lc rgb "#00DD00" lw 3 # stimulus thin, AM +set style line 15 lt 1 lc rgb "black" lw 15 # stimulus bar + +set style line 20 lt 1 lc rgb "yellow" lw 4 # thin spikes cell 1 +set style line 21 lt 1 lc rgb "orange" lw 4 # thin spikes cell 2 +set style line 22 lt 1 lc rgb "yellow" lw 6 # thick spikes cell 1 +set style line 23 lt 1 lc rgb "orange" lw 6 # thick spikes cell 2 +set style line 24 lt 1 lc rgb "blue" lw 22 # big sync spikes +set style line 25 lt 1 lc rgb "blue" lw 6 # thick sync spikes +set style line 26 lt 1 lc rgb "orange" lw 4 # all spikes + +set style line 30 lt 1 lc rgb "dark-red" lw 6 pt 7 ps 2 # firing rate 1 thick +set style line 31 lt 1 lc rgb "red" lw 6 pt 7 ps 2 # firing rate 2 thick +set style line 32 lt 1 lc rgb "orange" lw 6 pt 7 ps 2 # firing rate 3 thick +set style line 33 lt 1 lc rgb "blue" lw 6 pt 7 ps 2 # firing rate 4 thick +set style line 34 lt 1 lc rgb "gray" lw 6 # firing rate gray thick +set style line 35 lt 1 lc rgb "dark-red" lw 3 pt 7 ps 0.7 # firing rate 1 thin +set style line 36 lt 1 lc rgb "red" lw 3 # firing rate 2 thin +set style line 37 lt 1 lc rgb "orange" lw 3 # firing rate 3 thin +set style line 38 lt 1 lc rgb "blue" lw 3 # firing rate 4 thin +set style line 39 lt 1 lc rgb "gray" lw 3 # firing rate gray thin + +set style line 40 lt 1 lc rgb "green" lw 6 pt 5 ps 2 # onset f-I curve thick +set style line 41 lt 1 lc rgb "green" lw 3 pt 5 ps 2 # onset f-I curve thin +set style line 42 lt 1 lc rgb "red" lw 6 pt 7 ps 2 # steady-state f-I curve thick +set style line 43 lt 1 lc rgb "red" lw 3 pt 7 ps 2 # steady-state f-I curve thin +set style line 44 lt 1 lc rgb "blue" lw 6 pt 9 ps 2 # adapted f-I curve thick +set style line 45 lt 1 lc rgb "blue" lw 3 pt 9 ps 2 # adapted f-I curve thin +set style line 47 lt 1 lc rgb "cyan" lw 6 # adaptation level thick +set style line 48 lt 1 lc rgb "cyan" lw 3 # adaptation level thin +set style line 49 lt 1 lc rgb "yellow" lw 3 pt 7 ps 4 # current adaptation point + +set style line 50 lt 1 lc rgb "dark-red" lw 4 # membrane voltage 1 +set style line 51 lt 1 lc rgb "red" lw 4 # membrane voltage 2 +set style line 52 lt 1 lc rgb "orange" lw 4 # membrane voltage 3 +set style line 55 lt 1 lc rgb "orange" lw 4 # nerve potential + +set style line 60 lt 1 lc rgb "blue" lw 4 # gating variable +set style line 61 lt 1 lc rgb "cyan" lw 4 # gating variable average + +set style line 70 lt 1 lc rgb "red" lw 4 # ionic current +set style line 71 lt 1 lc rgb "orange" lw 4 # ionic current average + +set style line 80 lt 1 lc rgb "#77FFFF" lw 4 pt 13 ps 2.4# baseline +set style line 81 lt 1 lc rgb "#FF5533" lw 4 pt 9 ps 2.8 # beat +set style line 82 lt 1 lc rgb "yellow" lw 4 pt 7 ps 2.4 # chirp +set style line 83 lt 1 lc rgb "#FF7700" lw 4 pt 11 ps 2.8# beat shuffled +set style line 84 lt 1 lc rgb "#FFAA00" lw 4 pt 5 ps 2.2 # chirp shuffled + +set style line 90 lt 1 lc rgb "#77FFFF" lw 2 # EOD thin +set style line 91 lt 1 lc rgb "#77FFFF" lw 4 # EOD thick + +set term epslatex input color dashed 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files /dev/null and b/resources/Nieder2013-Matlab_I_Tutorial.pdf differ diff --git a/resources/beautifulplots.txt b/resources/beautifulplots.txt new file mode 100644 index 0000000..e39588c --- /dev/null +++ b/resources/beautifulplots.txt @@ -0,0 +1 @@ +http://www.randalolson.com/2014/06/28/how-to-make-beautiful-data-visualizations-in-python-with-matplotlib/ \ No newline at end of file