MATLAB Tutorial CSHA Suzhou Course 2011

Transcription

1 MATLAB Tutorial CSHA Suzhou Course 2011 Basics Malte J. Rasch National Key Laboratory of Cognitive Neuroscience and Learning Beijing Normal University China July 15, 2011

2 Overview Simply introduction to basic algebra and matrix indexing Simply introduction to plotting and programming with MATLAB Exercises

3 Why MATLAB?

4 Why MATLAB? MathWorks Homepage: Over one million people around the world use MATLAB for technical computing. [...] By combining a powerful numeric engine and technical programming environment with interactive exploration and visualization tools, MATLAB has become the language of technical computing. [...]

5 Why MATLAB? MathWorks Homepage: Over one million people around the world use MATLAB for technical computing. [...] By combining a powerful numeric engine and technical programming environment with interactive exploration and visualization tools, MATLAB has become the language of technical computing. [...] Of course, this is an advertisement...

6 Why MATLAB? Pro: Matrix-like numerical operations very fast and easy to use Good plotting capabilities Script language for programming small to medium sized problems in applied mathematics (rapid prototyping) Widely used in the neuroscience community for data analysis as well as computational projects

7 Why MATLAB? Pro: Matrix-like numerical operations very fast and easy to use Good plotting capabilities Script language for programming small to medium sized problems in applied mathematics (rapid prototyping) Widely used in the neuroscience community for data analysis as well as computational projects Contra: Support of symbolic/analytic expressions less advanced Mathematica, Maple Often not flexible/ fast enough for big projects e.g. Python much more versatile specialized software for detailed/large neural network simulations (NEURON, PCSIM, NEST, GENESIS,...) Very expensive, especially when using on a cluster free alternative: (scientific) python

8 Starting MATLAB Easy: Just click on MATLAB symbol...

9 Scalar expressions Binary operations: work as expected, use = + - * / ^. Example (compute y = a2 x 2+a + b) >> a = 2; >> b = 1; >> x = 0.5; >> y = a^2*x/(2+a) + b; >> y y = Unary operations: called as functions with (), eg. sin cos tan atan sqrt log gamma Example ( compute y = sin x ln x ) >> x = 0.5; >> y = sqrt(sin(x))/log(x);

10 Getting help Each MATLAB function has a header which describes its usage Just type: >> help command or >> doc command Alternatively:

11 MATLAB == MATrix LABoratory Not suprisingly, in MATLAB everything is about matrices.

12 MATLAB == MATrix LABoratory Not suprisingly, in MATLAB everything is about matrices. However, the matrix-like data structure in MATLAB is better called a n-dimensional array, because it can be manipulated in non-algebraic ways.

13 MATLAB == MATrix LABoratory Not suprisingly, in MATLAB everything is about matrices. However, the matrix-like data structure in MATLAB is better called a n-dimensional array, because it can be manipulated in non-algebraic ways. (Almost) all functions will work on arrays as well usually element-wise Many MATLAB functions will produce arrays as output Array operations much faster than for-looped element-wise operation

14 Arrays: Overview How to initialize arrays Indexing Calculating with arrays Array manipulation

15 Syntax for initializing arrays 1 Implicitly, using function returning an array

16 Syntax for initializing arrays 1 Implicitly, using function returning an array 2 By explicit concatenation Concatenation of columns of arrays [ arr1, arr2,..., arrn] Concatenation of rows of arrays [ arr1; arr2;... ; arrn]

17 Syntax for initializing arrays 1 Implicitly, using function returning an array 2 By explicit concatenation Concatenation of columns of arrays [ arr1, arr2,..., arrn] Concatenation of rows of arrays [ arr1; arr2;... ; arrn] Note: an scalar is also regarded as an array (of size [1,1]). Note 2: arrays must have matching sizes.

18 Syntax for initializing arrays 1 Implicitly, using function returning an array 2 By explicit concatenation Concatenation of columns of arrays [ arr1, arr2,..., arrn] Concatenation of rows of arrays [ arr1; arr2;... ; arrn] Note: an scalar is also regarded as an array (of size [1,1]). Note 2: arrays must have matching sizes. Example (Concatenation) >> A = [1,2;3,4] A = >> B = [A,A] B = >> C = [A;A] C =

19 Syntax for initializing arrays: implicitly Functions that pre-allocate memory and set each array element to an initial value: zeros all zero ND-arrays ones all one ND-arrays rand random ND-arrays (equally in [0, 1])

20 Syntax for initializing arrays: implicitly Functions that pre-allocate memory and set each array element to an initial value: zeros all zero ND-arrays ones all one ND-arrays rand random ND-arrays (equally in [0, 1]) colon (:) for linear sequences, syntax: istart:[step:]iend

21 Syntax for initializing arrays: implicitly Functions that pre-allocate memory and set each array element to an initial value: zeros all zero ND-arrays ones all one ND-arrays rand random ND-arrays (equally in [0, 1]) colon (:) for linear sequences, syntax: istart:[step:]iend many others, for details type: help command randn, linspace, logspace, ndgrid, repmat,...

22 Syntax for initializing arrays: implicitly Functions that pre-allocate memory and set each array element to an initial value: zeros all zero ND-arrays ones all one ND-arrays rand random ND-arrays (equally in [0, 1]) colon (:) for linear sequences, syntax: istart:[step:]iend many others, for details type: help command randn, linspace, logspace, ndgrid, repmat,... Example (Functions initializing arrays) >> A = zeros(3,3); >> A = ones(4,4,4); >> size(a) ans = >> x = 3:-0.5:1 x = >> A = ones(2)

23 Indexing arrays 1 Subscript of a matrix: access the (i,j)-th element of a 2D-matrix W of dimension (m,n) >> W(i,j) = 1 Note: The first index is always 1 (not 0 as in most other languages)

24 Indexing arrays 1 Subscript of a matrix: access the (i,j)-th element of a 2D-matrix W of dimension (m,n) >> W(i,j) = 1 Note: The first index is always 1 (not 0 as in most other languages) 2 Linear index of a matrix: access the (i,j)th element of the 2D-matrix W of dimension (m,n) >> linearidx = i + (j-1)*m; >> W(linearidx)

25 Indexing arrays 1 Subscript of a matrix: access the (i,j)-th element of a 2D-matrix W of dimension (m,n) >> W(i,j) = 1 Note: The first index is always 1 (not 0 as in most other languages) 2 Linear index of a matrix: access the (i,j)th element of the 2D-matrix W of dimension (m,n) >> linearidx = i + (j-1)*m; >> W(linearidx) 3 Slice indexing with : access the i-th row and jth column in W, respectively >> wi = W(i,:) >> wj = W(:,j) get all elements as a concatenated column vector >> W(:)

26 Indexing arrays (2) 4 Multiple indices vectors of linear indices can be used >> W([1, 4, 5,6]) access the 1st to 4th rows of the 2D-matrix W of dimension (m,n) >> W(1:4,:) access the 2nd (m,n)-slice of a 3D-matrix V of dimension (m,n,p) >> W(:,:,2)

27 Indexing arrays (2) 4 Multiple indices vectors of linear indices can be used >> W([1, 4, 5,6]) access the 1st to 4th rows of the 2D-matrix W of dimension (m,n) >> W(1:4,:) access the 2nd (m,n)-slice of a 3D-matrix V of dimension (m,n,p) >> W(:,:,2) 5 Logical indexing logical matrices of the same size as W can be used as index (very useful) >> W(W>0) >> W(find(W>0)) = 1

28 Calculating with arrays 1 Element-wise interpretation For instance, sin cos log etc. Reserved symbols,.*./.^ 2 true matrix interpretation (with dot product) Symbols * / ^ etc. 3 Operations on one specified dimensions of the matrix For instance, sum mean max etc. 4 Array manipulations eg. reshape repmat permute circshift tril Example (element-wise product and dot product) >> A = ones(2,2); >> A.*A ans = >> A*A ans =

29 Array manipulation Usage of array manipulation is important to achieve fast codes. circshift shifts array circular reshape reshapes arrays to new size permute permutes dimensions of array repmat repeats and concattenates arrays and many more, eg. shiftdim, squeeze

30 Array manipulation Usage of array manipulation is important to achieve fast codes. circshift shifts array circular reshape reshapes arrays to new size permute permutes dimensions of array repmat repeats and concattenates arrays Example (reshape) and many more, eg. shiftdim, squeeze >> r = rand(10,10); >> r = reshape(r,[50,2]); >> size(r) ans = 50 2

31 Calculating with arrays is straightforward however, carefully check the size of matrices if element-wise or matrix-like operations are intended which matrix dimension to operate on

32 Calculating with arrays is straightforward however, carefully check the size of matrices if element-wise or matrix-like operations are intended which matrix dimension to operate on Example (compute y i = Wx i + b with b,x i R 2 ) >> W = [1,0.2;0.4,1]; >> b = [1;2] + 0.1; >> x = 2*randn(2,1); >> y = W * x + b y = >> N = 5; % same as b = [b,b,b,b,b] >> bi = repmat(b,[1,n]); >> xi = 2*randn(2,N); >> xi(:,1) = x; >> yi = W * xi + bi yi = [..] [..]

33 Useful operations on specified dimensions Often used functions on array elements include size number of elements in a dimension sum sum of elements along an array dimension prod product of elements all, any logical AND or OR, respectively mean mean of elements max maximal element of an array dimension and many more, eg. min var std median diff Functions are usually called as res = func(arr,dim); Note: Some functions have the syntax res = func(arr,[],dim); If unsure type: help func

34 Plotting with MATLAB Very flexible plotting tools Many functions for a variety of plot types and graphics drawing ND improvement in respect to p st [%] <= >=100 minimal ND Firing rates Fano Factors (FF) Population FF Burst rates Burst sizes ISI 2 ISI CV(ISI) Synchronization X Corr (distance) Improvement [%] overall (MND) average percent (ND) Parameter # optimized Parameter #

35 Overview Useful plotting commands plot plots all kinds of lines, dots, markers in 2D imagesc plots an 2-D pseudo image errorbar plots dots/line with error bars bar bar plot hist histogram surface 3D surface plot

36 Overview Useful plotting commands plot plots all kinds of lines, dots, markers in 2D imagesc plots an 2-D pseudo image errorbar plots dots/line with error bars bar bar plot hist histogram surface 3D surface plot Note: For an overview of 2D plotting commands: help graph2d Note 2: For fancy graphs see: help specgraph

37 How to plot How to plot: 1 Open a figure (window) with >> figure; 2 Create an axes object with >> axes; 3 Type the plotting command of your choice >> plot(x,y, b--, LineWidth,2); Note: graphics commands draw into the current axes (gca) on the current figure (gcf). They automatically create a new figure and axes if necessary. 4 Make the plot nicer by adding labels and setting limits, eg: >> xlabel( Time [s] ); >> ylabel( Response [spks/sec] ); >> xlim([-1,1]); ylim([-2,2]); >> title( Simulation )

38 plot command Basic syntax: handle = plot(x,y,linespec,optname1,val1,...); X,Y x- and y-values to plot. If Y is a 2-D array, all columns are plotted as different lines linespec a string with a short hand for color and line style and marker type. See help plot for an overview. Eg, linespec = :ko plots dotted (:) black line (k) with a circle at each given coordinate (x i,y i ) optname1 a string specifing the option name, eg. LineWidth val1 a corresponding value. handle graphics handle for later reference, eg. with >> set(handle,optname1,val1) Tip: To get an overview over all possible options, try get(handle)

39 Plotting example: empirical PDF of Gaussian 1 figure ; 2 3 % plot Gaussian PDF 4 x = 4:0.01:4; 5 y = exp( 0.5 x.ˆ2)/ sqrt (2 pi ) ; 6 plot (x, y, k, Linewidth,2); 7 8 % e mpirical pdf 9 % histogram of 1000 random values 10 [N, z]= hist ( randn (1000,1)); 11 %normalization 12 p = N/sum(N)/( z(2) z ( 1 ) ) ; hold on ; %adds a l l f o l l o w i n g graphs 15 %to the current axes 16 plot (z, p, rx, MarkerSize,10); 17 hold o f f ; xlabel ( Random v a r i a b l e x ) 20 ylabel ( P r o b a b i l i t y density ) 21 legend ( true PDF, e m pirical PDF ) Probability density true PDF empirical PDF Random variable x

40 MATLAB is a script language Scripts are blocks of code which can be called within MATLAB or within another script. They are text-files with extensions.m. They should contain all commands associated with a scientific project. (at least to easily reproduce and the calculations)

41 MATLAB is a script language Scripts are blocks of code which can be called within MATLAB or within another script. They are text-files with extensions.m. They should contain all commands associated with a scientific project. (at least to easily reproduce and the calculations) There are basically two types of m-files 1 m-file script A squential list of MATLAB commands The variable scope is that of the caller. 2 m-file function m-file functions accept input parameters and deliver outputs. The variable scope is different from that of the caller.

42 m-file scripts How to write an m-file script?

43 m-file scripts How to write an m-file script? 1 open a text-editor of your choice or use the editor provided with MATLAB >> edit myscript

44 m-file scripts How to write an m-file script? 1 open a text-editor of your choice or use the editor provided with MATLAB >> edit myscript 2 Write all your calculations in a text-file with extension.m (here myscript.m)

45 m-file scripts How to write an m-file script? 1 open a text-editor of your choice or use the editor provided with MATLAB >> edit myscript 2 Write all your calculations in a text-file with extension.m (here myscript.m) 3 Save file in the current working directory (or addpath to search path )

46 m-file scripts How to write an m-file script? 1 open a text-editor of your choice or use the editor provided with MATLAB >> edit myscript 2 Write all your calculations in a text-file with extension.m (here myscript.m) 3 Save file in the current working directory (or addpath to search path ) 4 Call your script by calling it from the Command Window >> myscript; (no compilation needed)

47 m-file scripts How to write an m-file script? 1 open a text-editor of your choice or use the editor provided with MATLAB >> edit myscript 2 Write all your calculations in a text-file with extension.m (here myscript.m) 3 Save file in the current working directory (or addpath to search path ) 4 Call your script by calling it from the Command Window >> myscript; (no compilation needed) 5 Note: The variable scope is that of the caller. This means that the variables of the script are now present in the workspace

48 m-file functions How to write an m-file function?

49 m-file functions How to write an m-file function? Identical to writing an m-file script, except:

50 m-file functions How to write an m-file function? Identical to writing an m-file script, except: In the first line input and output arguments are defined: function outarg = myfun(inarg1,inarg2,...);

51 m-file functions How to write an m-file function? Identical to writing an m-file script, except: In the first line input and output arguments are defined: function outarg = myfun(inarg1,inarg2,...); Call the function with >> result = myfun(p1,p2,...);

52 m-file functions How to write an m-file function? Identical to writing an m-file script, except: In the first line input and output arguments are defined: function outarg = myfun(inarg1,inarg2,...); Call the function with >> result = myfun(p1,p2,...); Note: The variable scope is different from that of the caller. That means, that variables defined in the function body are NOT present in the workspace after calling the function

53 m-file functions How to write an m-file function? Identical to writing an m-file script, except: In the first line input and output arguments are defined: function outarg = myfun(inarg1,inarg2,...); Call the function with >> result = myfun(p1,p2,...); Note: The variable scope is different from that of the caller. That means, that variables defined in the function body are NOT present in the workspace after calling the function Note 2: Input arguments are always referenced by value (although internally MATLAB does not copy a variable as long it is not changed inside function)

54 m-file functions How to write an m-file function? Identical to writing an m-file script, except: In the first line input and output arguments are defined: function outarg = myfun(inarg1,inarg2,...); Call the function with >> result = myfun(p1,p2,...); Note: The variable scope is different from that of the caller. That means, that variables defined in the function body are NOT present in the workspace after calling the function Note 2: Input arguments are always referenced by value (although internally MATLAB does not copy a variable as long it is not changed inside function) Note 3: When called the m-file file name is used.

55 Basic syntax The basic syntax is similar to other script languages MATLAB has the usual flow control structures, namely if.. else for decisions for-loop while-loop switch.. case for multiple if-else decisions try.. catch for handling errors

56 Basic syntax: flow control (1) if-else block syntax: 1 i f s c a l a r c o n d i t i o n 2 expressions 3 else 4 expressions 5 end Relational operators, eg.: == (equals), (or), && (and), ~ (not) for details type: help relop

57 Basic syntax: flow control (1) if-else block syntax: 1 i f s c a l a r c o n d i t i o n 2 expressions 3 else 4 expressions 5 end Relational operators, eg.: == (equals), (or), && (and), ~ (not) for details type: help relop Example (if-else) a = rand ( 1 ) ; i f a == 0.5 fprintf ( you are very lucky! ) ; end

58 Programming: basic flow control (2) for-loop block syntax: 1 for i = array 2 % i==array ( j ) in the j th loop 3 expressions 4 end (one can also use break and continue keywords)

59 Programming: basic flow control (2) for-loop block syntax: 1 for i = array 2 % i==array ( j ) in the j th loop 3 expressions 4 end (one can also use break and continue keywords) Example (for loop) a=0; for i = 1:100 a = a+i ; end

60 MATLAB provides many more things Useful Toolboxes Statistical Toolbox (lots of statistical tests and distributions) Neural Network Toolbox (training networks and much more) Signal Processing Toolbox (a lot of filter designs and spectral methods) Image Processing Toolbox (reading, writing, filtering images) Symbolic maths (Mathematica is more powerful though) Distributed Computing Toolbox (expensive...) Interfaces to C code (MEX) CUDA-kernels (using super-parallel CUDA-GPU hardware) fancy 3D plots

61 Exercises

62 Exercises Exercise (How to write an m-file script) Write an m-file computing the Stirling approximation n! ( n ) n 2πn e Hint: π is defined in MATLAB as pi, and e x as exp(x). Solution Reminder: 1 Open a text-editor >> edit myscript 2 Write calculations in text-file with extension.m 3 Save file into current working directory 4 Call your script >> myscript;

63 Exercises Exercise (p-series) Calculate the p-series (generalization of the Harmonic Series) for a given p up to a given m Use array notation. Solution µ m = m i=1 1 n p Advise: Use array notation and avoid for-loops whereever you can!

64 Exercises Exercise (blob movie) 1 Generate a two arrays with, x and y, ranging from -2 to 2 (and about 100 elements) 2 Generate two 100 by 100 grid-matrices, X and Y using meshgrid with x and y as input (look at X and Y to understand what meshgrid is doing). 3 calculate a matrix Z of the same size as X and Y where each element is given by z i = e (x2 i +y2 i ). 4 write a loop with t ranging from 0 to 1000 where you 1 plot the matrix Z (using imagesc) 2 circular shift the matrix Z by 1 element (using circshift) 3 force the plot to be drawn (using drawnow) Solution

65 Exercises Exercise (Logical indexing and basic plotting) 1 Generate a 100 by 2 matrix M of Gaussian random values 2 Plot a point cloud (first vs. second column of M) using blue circles 3 Calculate how many values of M are positive 4 Erase all rows in M which have at least one negative value 5 Plot the points of the modified array M using red crosses in the same graph as above. 6 Set both axes limits to 3 to 3 and plot dotted gray lines on the coordinate axes (where y = 0 or x = 0). 7 Label the axes and write a legend Solution

66 Exercises Exercise (writing a m-file function) Write an m-file function computing the Stirling approximation n! ( n ) n 2πn e for a given n Solution Reminder: In the first line input and output arguments are defined: function outarg = myfun(inarg1,inarg2,...); Call the function with >> result = myfun(p1,p2,...);

67 Exercises Exercise (Poisson spike trains) Write a function that generates a homogeneous Poisson spike train of rate λ having exactly N spikes. Use array notations (cumsum). Hint: Poisson spike intervals t are exponentially distributed. They can be generated by inverse transform sampling: Given uniform random variables u in 0 to 1 valid inter-spike intervals can be calculated as t = log u λ Solution Advise: Use array notation and avoid for-loops whereever you can!

68 Exercises Exercise (More on spike trains) Generate a long Poisson spike train (with the function from the last exercise). Compute the mean and standard deviation of the inter-spike interval distribution. Further, write a function that counts the number of spike times falling into time-bins of length t. Hint: Use diff to get the intervals from spike times. Hint 2: Use histc to bin the spike-times Solution

69 Exercises Exercise (Plot 2-D Gaussian point cloud) Write a function which plots a point cloud of n random samples drawn from a 2D Normal distribution (with 0 mean) and variance Σ = (RD) T (RD), where the rotation matrix is defined as «cos θ sin θ R = sin θ cos θ and D is a diagonal matrix of the standard deviation in the principal directions «σ1 0 D = 0 σ 2 The function should accept parameters σ 1, σ 2, and θ Hint: use randn to produce independent Normal-distributed random vectors x i and transform them according to y i = RDx i. Solution

70 Exercises Exercise (Simulate LIF-neuron network) Simulate a network of n simple LIF-neurons τ m dv dt = v + (Ws + RI in + RI noise ) and v i = v reset if v i > v thres. The synaptic currents dynamics is simplified as Unit # Time [sec] τ ds i dt = s i + Θ spike where Θ spike = 1 at times where the i-th neuron spikes and 0 otherwise. Connections W are sparse (probability p 0.2) and weights are normal distributed (with standard deviation g). Plot the spikes raster when constantly stimulating for a brief period of time. Hint: use the Euler method to integrate the equations v i (t + 1) = v i (t) + t dv i Solution dt

71 Advanced exercise Exercise (Optimizing MATLAB code) Optimize bad MATLAB code for generating self-organizing maps. See provided code and description in som_exercise.zip.

72 Solution (Writing a m-file script) 1 %compute the S t i r l i n g Formula 2 3 n = 50; 4 5 s t i r f a c n = sqrt (2 pi n ) ( n/exp (1))ˆ n ; 6 fac n = f a c t o r i a l (n ) ; 7 8 %n i c e r output 9 f p r i n t f ( S t i r l i n g approximation for %d! : \ n,n) 10 s t i r f a c n 11 f p r i n t f ( Exact %d! : \ n,n) 12 fac n back to text

73 Solution (p-series) 1 function mu = p s e r i e s (n, p ) ; 2 % PSERIES(N,P) computes the P s e r i e s up to N 3 4 i a r r = (1: n ).ˆ p ; 5 mu = sum(1./ i arr, 2 ) ; back to text

74 Solution (blob movie) 1 x = linspace ( 2,2,100); 2 y = linspace ( 2,2,100); 3 4 [X,Y] = meshgrid (x, y ) ; 5 6 Z = exp( X.ˆ2 Y. ˆ 2 ) ; 7 8 for t = 1: imagesc (Z ) ; 10 Z = c i r c s h i f t (Z, [ 0, 1 ] ) ; 11 drawnow ; 12 end back to text

75 Solution (Logical indexing and basic plotting) 1 %1. 2 M = randn (100,2); 3 4 %2. 5 plot (M(:,1),M(:,2), bo ) ; 6 7 %3. 8 npos = sum(m(:) >0) 9 10 %4. 11 M( any (M<0,2),:) = [ ] ; %5. 14 hold on ; 15 plot (M(:,1),M(:,2), rx ) ; %6. 18 l i m i t s = [ 3,3]; 19 xlim ( l i m i t s ) ; ylim ( l i m i t s ) 20 plot ( l i m i t s, [ 0, 0 ], :, Color, [ 0. 5, 0. 5, 0. 5 ] ) 21 plot ( [ 0, 0 ], l i m i t s, :, Color, [ 0. 5, 0. 5, 0. 5 ] ) %7. 24 xlabel ( X ) 25 ylabel ( Y ) 26 legend ({ Gaussian random samples, Points with X>0 and Y>0 }) back to text

76 Solution (Writing a m-file function) 1 function s t i r f a c n = s t i r f a c (n ) ; 2 %compute the S t i r l i n g Formula 3 4 s t i r f a c n = sqrt (2 pi n ) ( n/exp (1))ˆ n ; back to text

77 Solution (Poisson spike trains) 1 function spkt = poissonspikes ( lambda,n) 2 % SPKT = POISSONSPIKES(LAMBDA,N) generates a 3 % Poisson spike t r a i n with rate LAMBDA 4 % and exactly N spikes. 5 6 i s i s = log ( rand (N,1))/ lambda ; 7 spkt = cumsum( i s i s ) ; back to text

78 Solution (more on spike trains) 1 N = 1000; % number of spikes 2 lambda = 10; %spike rate 3 4 %Poisson spikes 5 spkt = cumsum( log ( rand (N,1))/ lambda ) ; 6 7 %mean of the I S I 8 misi = mean( d i f f ( spkt ) ) ; 9 10 %stddev of the I S I 11 s i s i = std ( d i f f ( spkt ) ) ; function S = spikebinning ( spkt, twin, dt ) 15 % S = SPIKEBINNING(SPKT,TWIN,DT) counts the spike times 16 % occurring in each bin of width DT in the time window 17 % from TWIN(1) and TWIN(2) S = h i s t c ( spkt, twin ( 1 ) : dt : twin ( 2 ) ) ; back to text

79 Solution (plot 2-D Gaussian point cloud) 1 function plot2dgaussian ( sig1, sig2, theta ) ; 2 % PLOT2DGAUSSIAN( SIG1, SIG2,THETA) p l o t s a Gaussian point cloud with 3 % standard d e v i a t i o n s SIG1 and SIG2 and o r i e n t a t i o n THETA. 4 5 n = 250; 6 r = randn (2, n ) ; 7 D = [ sig1, 0;0, sig2 ] ; 8 R = [ cos ( theta ), sin ( theta ) ; sin ( theta ), cos ( theta ) ] ; 9 10 x = R D r ; %p l o t t i n g 13 figure ; subplot (2,2,1); 14 plot ( x (1,:), x (2,:),. b, MarkerSize,6); %r e s i z e the axes 17 mxlen = 1.1 max( abs ( x ( : ) ) ) ; 18 xlim ([ mxlen, mxlen ] ) 19 ylim ([ mxlen, mxlen ] ) %plo t the 0 coordinates 22 hold on ; 23 plot ([ mxlen, mxlen ], [ 0, 0 ], k ) 24 plot ([0,0],[ mxlen, mxlen ], k ) 25 hold o f f ; back to text

80 Solution (simulate LIF-neuron network) 1 %parameters 2 n = 50; % number of neurons 3 p = 0.25; % prob. of connection 4 vthres = 0. 2 ; 5 v reset = 0.05; 6 s i g I = 0. 5 ; % input noise strength 7 tau s = 1; % synaptic time const. 8 g = 1; % weight s c a l e 9 tau m = 1; % membrane time const. 10 R = 1; % r e s i s t a n c e 11 dt = 0.05; % time step 12 tmax = 10; 13 tinput = [ 2, 6 ] ;% input time i n t e r v a l 14 I i n p u t = 0. 3 ; % input strength %connections j u s t random 18 connif = rand (n, n)<=p ; 19 %weights normal d i s t r i b u t i o n 20 W = g randn (n, n ). connif ; %i n i t i a l i z e v a r i a b l e s 23 t = 0: dt : tmax ; 24 spks = zeros (n, size ( t, 2 ) ) ; 25 v = zeros (n, 1 ) ; % voltage 26 s = zeros (n, 1 ) ; % synaptic current %Euler i n t e g r a t i o n 30 for i = 1: size ( t,2) 31 %noise + input 32 I = s i g I randn (n, 1 ) ; 33 i f t ( i)>=tinput (1) && t ( i)<tinput (2) 34 I = I + I i n p u t ; 35 end 36 %voltage update 37 dv = v + W s + R I ; 38 v = v + dv dt/tau m ; %find spikes and set to v reset 41 spks ( :, i ) = v>vthres ; 42 v ( find ( spks ( :, i ))) = v reset ; %update s 45 ds = s + spks ( :, i ) ; 46 s = s + dt/ tau s ds ; 47 end %plot spikes 50 [X,Y] = meshgrid ( t, 1 : n ) ; 51 idx = find ( spks ( : ) ) ; 52 plot (X( idx ),Y( idx ), x ) ; 53 xlim ([0, tmax ] ) ; ylim ([0, n ] ) 54 xlabel ( Time [ sec ] ) 55 ylabel ( Unit # ) back to text