MATLAB Tutorial – CCN Course 2012 - · PDF fileIntroduction Arrays Coding Plotting...

Introduction Arrays Coding Plotting Network model: Step 1 Step 2 Step 3 Step 4 Exercises Solutions

MATLAB Tutorial – CCN Course 2012How to code a neural network simulation

Malte J. Rasch

National Key Laboratory of Cognitive Neuroscience and LearningBeijing Normal University

China

July 17, 2012


Overview

Basic introduction to Matlab

Learn to code a neural network simulation

Further exercises with solutions


Why MATLAB ?

Pro:

Matrix-like numerical operations very fast and easy to useGood plotting capabilitiesScript language for programming small to medium sizedproblems in applied mathematics (rapid prototyping)Widely used in the neuroscience community for data analysisas well as computational projects


Why MATLAB ?

Pro:

Matrix-like numerical operations very fast and easy to useGood plotting capabilitiesScript language for programming small to medium sizedproblems in applied mathematics (rapid prototyping)Widely used in the neuroscience community for data analysisas 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 versatilespecialized software for detailed/large neural networksimulations(NEURON, PCSIM, NEST, GENESIS, ...)

Very expensive, especially when using on a cluster

free alternative: (scientific) python


Starting MATLAB

Easy: Just click on MATLAB symbol...


Scalar expressions

Binary operations: work as expected, use = + - * / ^.

Example (compute y = a2x2+a

+ b)

>> a = 2;

>> b = 1;

>> x = 0.5;

>> y = a^2*x/(2+a) + b;

>> y

y =

1.500

Unary operations: called as functions with (), eg.sin cos tan atan sqrt log gamma

Example ( compute y =√

sin xln x )

>> x = 0.5;

>> y = sqrt(sin(x))/log(x);


MATLAB == MATrix LABoratory

Not suprisingly, in MATLAB everything is about matrices.




However, the matrix-like datastructure in MATLAB is better calleda n-dimensional array, because it can be manipulated innon-algebraic ways.




However, the matrix-like datastructure in MATLAB is better calleda n-dimensional array, because it can be manipulated innon-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-wiseoperation


Arrays: Overview

How to initialize arrays

Indexing

Calculating with arrays


Calculating with arrays is

straight-forward

however, carefully check

the size of matricesif element-wise or matrix-like operations are intendedwhich matrix dimension to operate on



straight-forward

however, carefully check

the size of matricesif element-wise or matrix-like operations are intendedwhich matrix dimension to operate on

Example (compute yi = Axi + b with b, xi ∈ R2)

>> A = [1,0.2;0.4,1];

>> b = [1;2] + 0.1;

>> x = 2*randn(2,1);

>> y = A * x + b

y =

4.5535

1.4856

>> N = 5;

% same as b = [b,b,b,b,b]

>> bi = repmat(b,[1,N]);

>> xi = 2*randn(2,N);

>> xi(:,1) = x;

>> yi = A * xi + bi

yi =

4.5535 2.3126 [..] -0.9021

1.4856 6.8091 [..] -0.1080


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]




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.



1 Implicitly, using function returning an array2 By explicit concatenation

Concatenation of columns of arrays[ arr1, arr2, ... , arrn]

Concatenation of rows of arrays[ arr1; arr2; ... ; arrn]


Example (Concatenation)

>> A = [1,2;3,4]

A =

1 2

3 4

>> B = [A,A]

B =

1 2 1 2

3 4 3 4

>> C = [A;A]

C =

1 2

3 4

1 2

3 4


Syntax for initializing arrays: implicitly

Functions that pre-allocate memory and set each array element toan 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








istart:[step:]iend

– many others, for details type: help command

randn, linspace, logspace, ndgrid, repmat,...




zeros – all zero ND-arraysones – all one ND-arraysrand – random ND-arrays (equally in [0, 1])


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 =

4 4 4

>> x = 3:-0.5:1

x =

3.0 2.5 2.0 1.5 1.0

>> A = ones(2)

1 1

1 1


Indexing arrays

1 Subscript of a matrix:

access the (i , j)-th element of a 2D-matrix A of dimension (m, n)

>> A(i,j)

Note: The first index is always 1 (not 0 as in most other languages)


Indexing arrays



>> A(i,j)


2 Linear index of a matrix:

access the (i , j)th element of the 2D-matrix A of dimension (m, n)

>> linearidx = i + (j-1)*m;

>> A(linearidx)


Indexing arrays



>> A(i,j)



access the (i , j)th element of the 2D-matrix A of dimension (m, n)


>> A(linearidx)

3 “Slice” indexing with “:”

access the i-th row and jth column in A, respectively

>> A(i,:)

>> A(:,j)

get all elements as a cocatenated column vector

>> A(:)


Indexing arrays (2)

4 Multiple indices

vectors of linear indices can be used

>> A([1, 4, 5 ,6])

access the 1st to 4th rows of the 2D-matrix A of dimension (m, n)

>> A(1:4,:)

access the 2nd (m,n)-slice of a 3D-matrix B of dimension (m, n, p)

>> B(:,:,2)


Indexing arrays (2)

4 Multiple indices


>> A([1, 4, 5 ,6])

access the 1st to 4th rows of the 2D-matrix A of dimension (m, n)

>> A(1:4,:)

access the 2nd (m,n)-slice of a 3D-matrix B of dimension (m, n, p)

>> B(:,:,2)

5 Logical indexing

logical matrices of the same size as A can be used as indices

>> A(A>0)

>> A(find(A>0))



1 Element-wise interpretationFor instance, sin cos log etc.Reserved symbols, .* ./ .^

2 “true” matrix interpretation (with dot product)Symbols * / ^ etc.

3 Operations on one specified dimensions of the matrixFor instance, sum mean max etc.

4 Array manipulationseg. reshape repmat permute circshift tril

Example (element-wise product and dot product)

>> A = ones(2,2);

>> A.*A

ans =

1 1

1 1

>> A*A

ans =

2 2

2 2


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: max and min are exceptions! Here: res = max(arr,[],dim);

For details type: help func


MATLAB is a script language

Scripts are blocks of code which can be called within MATLAB orwithin another script.

They are text-files with extensions “.m”.

They should contain all commands associated with a scientificproject.(at least to easily reproduce and the calculations)


MATLAB is a script language

Scripts are blocks of code which can be called within MATLAB orwithin another script.

They are text-files with extensions “.m”.

They should contain all commands associated with a scientificproject.(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 commandsThe 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.


m-file scripts

How to write an m-file script?


m-file scripts


1 open a text-editor of your choice or use the editor providedwith MATLAB

>> edit myscript


m-file scripts



>> edit myscript

2 Write all your calculations in a text-file with extension “.m”(here myscript.m)


m-file scripts



>> edit myscript


3 Save file in the current working directory(or addpath to search path )


m-file scripts



>> edit myscript



4 Call your script by calling it from the “Command Window”

>> myscript;

(no compilation needed)


m-file scripts



>> edit myscript




>> myscript;

(no compilation needed)

5 Note: The variable scope is that of the caller. This means thatthe variables of the script are now present in the workspace


m-file scripts

Excercise #1 (How to write an m-file script)

Write an m-file computing the Stirling approximation

n! ≈√2πn

(n

e

)n

Hint: π is defined in MATLAB as pi, and ex as exp(x).Solution #1

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;


m-file functions

How to write an m-file function?


m-file functions


Identical to writing an m-file script, except:


m-file functions



In the first line input and output arguments are defined:

function outarg = myfun(inarg1,inarg2,...);


m-file functions





Call the function with

>> result = myfun(p1,p2,...);


m-file functions







Note: The variable scope is different from that of the caller. Thatmeans, that variables defined in the function body are NOT

present in the workspace after calling the function


m-file functions








present in the workspace after calling the functionNote 2: Input arguments are always referenced by value


m-file functions








present in the workspace after calling the functionNote 2: Input arguments are always referenced by valueNote 3: When called the m-file file name is used.


m-file functions

Excercise #2 (writing a m-file function)

Write an m-file function computing the Stirling approximation

n! ≈√2πn

(n

e

)n

for a given nSolution #2

Reminder:






Basic syntax

The basic syntax is similar to other script languages, eg. “python”.

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


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 n2 e x p r e s s i o n s3 e l s e

4 e x p r e s s i o n s5 end

Relational operators, eg.: == (equals), || (or), && (and), ~ (not)

for details type: help relop


Basic syntax: flow control (1)






Example (if-else)

a = rand ( 1 ) ;i f a == 0.5

f p r i n t f ( ’ you a r e v e r y l u c k y ! ’ ) ;end


Programming: basic flow control (2)

for-loop block syntax:

1 f o r i = a r r a y2 % i==a r r a y ( j ) i n the j−th l oop3 e x p r e s s i o n s4 end

(one can also use break and continue keywords)


Programming: basic flow control (2)




Example (for loop)

a=0;f o r i = 1 :100

a = a+i ;end


Plotting with MATLAB

Very flexible plotting tools

Many functions for a variety of plot types and graphicsdrawing

A

B C D EStatic

Res

pons

e

0° 90° 180°

High speed

→ M

ovem

ent d

ir. →

0° 90° 180°

Low speed

→ M

ovem

ent d

ir. →

0° 90° 180°

−20

2

−202−10

−5

0

5

10

SF [cyc/deg]

High speed

SF [cyc/deg]

Low speed

TF

[Hz]

Horz. RF

Vert. RF

t

Stimulus

x

y

RF filter Kθ,ρ,ω

vary θ t

Simple cell response

x

y

⟨ |⋅|2 ⟩x,y,t

(+ phase invariance)

θ2

θ3

:

0

0.5

1

1.5

2Response profile

Avg

. res

pons

e

0° 90° 180°


Overview

Just two most useful plotting commands

plot – plots all kinds of lines, dots, markers in 2D

imagesc – plots an 2-D pseudo image


Overview

Just two most useful plotting commands

plot – plots all kinds of lines, dots, markers in 2D

imagesc – plots an 2-D pseudo image

Note: For an overview of 2D plotting commands: help graph2d

Note 2: For fancy graphs see: help specgraph


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’)


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 areplotted as different lines

linespec – a string with a short hand for color and line style andmarker type. See help plot for an overview. Eg,

linespec = ’:ko’

plots dotted (:) black line (k) with a circle at each givencoordinate (xi , yi )

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)


Plotting example: empirical PDF of Gaussian

1 f i g u r e ;2

3 % p l o t Gaus s i an PDF4 x = −4 : 0 . 01 : 4 ;5 y = exp (−0.5∗ x . ˆ2 ) / s q r t (2∗ p i ) ;6 p l o t ( x , y , ’−k ’ , ’ L i n ew id th ’ , 2 ) ;7

8 % emp i r i c a l pd f9 % hi s tog ram o f 1000 random v a l u e s

10 [N, z ]= h i s t ( randn ( 1 0 00 , 1 ) ) ;11 %no rma l i z a t i o n12 p = N/sum (N)/( z (2)− z ( 1 ) ) ;13

14 ho ld on ; %adds a l l f o l l o w i n g graphs15 %to the c u r r e n t axe s16 p l o t ( z , p , ’ r x ’ , ’ Marke rS i ze ’ , 1 0 ) ;17 ho ld o f f ;18

19 x l a b e l ( ’Random v a r i a b l e x ’ )20 y l a b e l ( ’ P r o b a b i l i t y d e n s i t y ’ )21 l egend ( ’ t r u e PDF ’ , ’ em p i r i c a l PDF ’ )

−4 −3 −2 −1 0 1 2 3 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Random variable x

Pro

babi

lity

dens

ity

true PDFempirical PDF


Goal of tutorial

We will program a neural network simulation together.


We will practice on the way:

Writing scripts

Usage of array notation

How to integrate ODEs

How to plot results

How to simulate neurons and synapses

How to program a quite realistic network simulation


Network model simulation

What has to be done in principle:

n neurons, excitatory and inhibitory, are inter-connected withsynapses.

The each neuron and synapse follows a particular dynamicover time.

The simulation calculates the dynamic and spiking activity ofeach neuron for each t.

The network gets some input and reacts in a certain and theresponse of the network is plotted.


We will proceed in 4 successive steps

1 Simulate a single neuron with current step input2 Simulate a single neuron with Poisson input3 Simulate 1000 neurons (no recurrent connections)4 Simulate complete network (conductance based)

Network activity


Which neuron model to use?

Biophysical model (i.e. Hodgkin-Huxley model)

Cm

dVm

dt= − 1

Rm

(Vm − VL)−∑

i

gi (t)(Vm − Ei ) + Isyn + Iapp

Including non-linear dynamics of many channels in gi (t)


Which neuron model to use?

Biophysical model (i.e. Hodgkin-Huxley model)

Cm

dVm

dt= − 1

Rm

(Vm − VL)−∑

i

gi (t)(Vm − Ei ) + Isyn + Iapp

Including non-linear dynamics of many channels in gi (t)

Mathematical simplification (Izhikevich, book chapter 8)

v = (0.04v + 5) v + 150− u −∑

j

wjgj(v − Ej) + Iapp

u = a (b v − u)

v ← c , u ← u + d , if v ≥ 35

with gj = −gj/τg and gj ← gj + 1 if vj ≥ 35. It is b = 0.2, c = −65, andd = 8, a = 0.02 for exc. neurons and d = 2, a = 0.1 for inh. neurons.


Neuron model

regular spiking (RS) intrinsically bursting (IB) chattering (CH) fast spiking (FS)

40 ms

20 mV

low-threshold spiking (LTS)

para

met

er b

parameter c

para

met

er d

thalamo-cortical (TC)

-87 mV

-63 mV

thalamo-cortical (TC)

peak 30 mV

reset c

reset ddecay with rate a

sensitivity b

v(t)

u(t)0 0.1

0.05

0.2

0.25

RS,IB,CH FS

LTS,TC

-65 -55 -50

2

4

8

IB

CH

RS

FS,LTS,RZ

TC

0.02parameter a

resonator (RZ)

RZ

v(t)

I(t)

v'= 0.04v 2+5v +140 - u + I u'= a(bv - u) if v = 30 mV, then v c, u u + d


Step 1: Simulate a single neuron with injected current

Exercise 1

Simulate the neuron model for 1000ms and plot the resultingvoltage trace. Apply a current step (Iapp = 7pA) between time200ms and 700ms.


Step 1: Simulate a single neuron with injected current

Exercise 1

Simulate the neuron model for 1000ms and plot the resultingvoltage trace. Apply a current step (Iapp = 7pA) between time200ms and 700ms.

Neuron model:

v = (0.04v + 5) v + 140− u + Iapp

u = a (b v − u)

v ← c , u ← u + d , if v ≥ 35

with d = 8, a = 0.02, b = 0.2,

c = −65 for an excitatory neuron.


Step 1 in detail:Open MATLAB HowTo and create a new file (script) that will simulate the neuron

HowTo . If you do not know how to use a command get some help HowTo

Proceed as follows:

1 Initialize parameter values (∆t = 0.5ms, a = 0.02, d = 8, · · · )HowTo




Proceed as follows:


2 Reserve memory for voltage trace v and u (of length T = 1000/∆t)HowTo and set first element to −70 and −14, respectively.




Proceed as follows:



3 Loop over T − 1 time steps and do for each step t HowTo




Proceed as follows:




1 set Iapp ← 7 if t∆t is between 200 and 700 (otherwise 0)HowTo




Proceed as follows:





2 if vt < 35: update element t + 1 HowTo of v and u

according to HowTo

vt+1 ← vt +∆t {(0.04 vt + 5) vt − ut + 140 + Iapp}ut+1 ← ut +∆t a (b vt − ut)




Proceed as follows:






according to HowTo


3 if vt ≥ 35: set vt+1 ← c and ut+1 ← ut + d (and optionalvt ← 35)




Proceed as follows:






according to HowTo


3 if vt ≥ 35: set vt+1 ← c and ut+1 ← ut + d (and optionalvt ← 35)

4 Plot the voltage trace v versus t HowTo


Solution to step 1

1 % 1) i n i t i a l i z e pa ramete r s2 dt = 0 . 5 ;3 d = 8 ;4 a = 0 . 0 2 ;5 c = −65;6 b = 0 . 2 ;7 T = c e i l (1000/ dt ) ;89 % 2) r e s e r v e memory

10 v = ze ro s (T , 1 ) ;11 u = ze ro s (T , 1 ) ;12 v (1 ) = −70; % r e s t i n g p o t e n t i a l13 u (1 ) = −14; % s t eady s t a t e1415 %3) fo r−l oop ove r t ime16 f o r t = 1 :T−1;17 %3 . 1 ) ge t i n pu t18 i f t∗dt>200 && t∗dt<70019 Iapp = 7 ;20 e l s e

21 Iapp = 0 ;

22 end

2324 i f v ( t )<3525 %3 . 2 ) update ODE26 dv = (0 .04∗ v ( t )+5)∗v ( t )+140−u ( t ) ;27 v ( t+1) = v ( t ) + ( dv+Iapp )∗ dt ;28 du = a∗(b∗v ( t)−u ( t ) ) ;29 u ( t+1) = u ( t ) + dt∗du ;30 e l s e

31 %3 . 3 ) s p i k e !32 v ( t ) = 35 ;33 v ( t+1) = c ;34 u ( t+1) = u ( t )+d ;35 end

36 end

3738 % 4) p l o t v o l t a g e t r a c e39 p l o t ( ( 0 :T−1)∗dt , v , ’ b ’ ) ;40 x l a b e l ( ’ Time [ms ] ’ ) ;41 y l a b e l ( ’Membrane v o l t a g e [mV] ’ ) ;


Step 2: Single neuron with synaptic input

Exercise 2

Simulate the neuron model for 1000ms and plot the resultingvoltage trace. Assume that 100 synapses are attached to theneuron, with each pre-synaptic neuron firing with a Poissonprocess of rate frate = 2 Hz between time 200ms and 700ms.


Step 2: Single neuron with synaptic input

Exercise 2

Simulate the neuron model for 1000ms and plot the resultingvoltage trace. Assume that 100 synapses are attached to theneuron, with each pre-synaptic neuron firing with a Poissonprocess of rate frate = 2 Hz between time 200ms and 700ms.

Synaptic input model:

Isyn =∑

j

winj g

inj (t)(E

inj − v(t))

ginj = g

inj /τg

ginj ← g

inj + 1, if rj(t) < frate∆t

with τg = 10ms, weights w inj = 0.07,

Ej = 0, rj(t) ∈ [0, 1] uniform

random numbers drawn for each

step t, and j = 1 . . . 100.


Step 2 in detail:Use the last script, save it under a new file name, and add the necessary lines.

Proceed as follows:

1 Initialize new parameter values (τg = 10, frate = 0.002ms−1)



Proceed as follows:


2 Reserve memory and initialize gin = (g inj ), win = (w in

j ), andE = (Ej) (vectors of length nin = 100) with constant elements



Proceed as follows:




3 Inside the for-loop change/add the following:



Proceed as follows:





1 set pj = 1 if r ≤ frate∆t (otherwise 0) in the case when i∆t isbetween 200 and 700 (otherwise 0). r = (rj) is a vector ofuniform random numbers of length nin



Proceed as follows:





1 set pj = 1 if r ≤ frate∆t (otherwise 0) in the case when i∆t isbetween 200 and 700 (otherwise 0). r = (rj) is a vector ofuniform random numbers of length nin

2 before the vt update: implement the conductance dynamics gand set Iapp (using array notation HowTo carefully HowTo ):

g inj ← g in

j + pj

Iapp ← win ·(

gin ⊙ Ein)

−(

win · gin)

⊙ vt

g inj ← (1−∆t/τg ) g

inj


Solution to step 2

1 %i n i t i a l i z e pa ramete r s2 dt = 0 . 5 ;3 d = 8 ;4 a = 0 . 0 2 ;5 c = −65;6 b = 0 .278 %r e s e r v e memory9 T = c e i l (1000/ dt ) ;

10 v = ze ro s (T , 1 ) ;11 u = ze ro s (T , 1 ) ;12 v (1 ) = −70;13 u (1 ) = −14;1415 n i n = 100 ; %NEW16 r a t e = 2∗1e−3;%to [ms ] %NEW17 tau g = 10 ; %NEW18 g i n = ze ro s ( n in , 1 ) ; %NEW19 E in = ze ro s ( n in , 1 ) ; %NEW20 w in = 0.07∗ ones (1 , n i n ) ; %NEW2122 %fo r−l oop ove r t ime23 f o r t = 1 :T−1;24 %Po i s son i npu t25 i f t∗dt>200 && t∗dt<70026 p = rand ( n in ,1)< r a t e∗dt ; %NEW27 e l s e

28 p = 0 ; %NEW29 end

30 %conductance update o f g i n31 g i n = g i n + p ;32 Iapp = w in ∗( g i n .∗ E in ) ;33 Iapp = Iapp −(w in∗ g i n ) .∗ v ( t ) ;34 g i n = (1 − dt / tau g )∗ g i n ;3536 i f v ( t )<3537 %update ODE38 dv = (0 .04∗ v ( t )+5)∗v ( t )+140−u ( t ) ;39 v ( t+1) = v ( t ) + ( dv+Iapp )∗ dt ;40 du = a∗(b∗v ( t)−u ( t ) ) ;41 u ( t+1) = u ( t ) + dt∗du ;42 e l s e

43 %s p i k e s44 v ( t ) = 35 ;45 v ( t+1) = c ;46 u ( t+1) = u ( t )+d ;47 end

4849 end

5051 %p l o t v o l t a g e t r a c e52 p l o t ( ( 0 :T−1)∗dt , v , ’ b ’ ) ;53 x l a b e l ( ’ Time [ms ] ’ ) ;54 y l a b e l ( ’Membrane v o l t a g e [mV] ’ ) ;


Step 3: Simulate 1000 neurons (not inter-connected)

Exercise 3

Simulate 1000 neurons for 1000 ms and plot the resulting spikes.Assume that each neuron receives (random) 10% of the 100Poisson spike trains of rate frate = 2 Hz between time 200 ms and700 ms. Note that the neurons are not yet inter-connected.


Step 3: Simulate 1000 neurons (not inter-connected)

Exercise 3

Simulate 1000 neurons for 1000 ms and plot the resulting spikes.Assume that each neuron receives (random) 10% of the 100Poisson spike trains of rate frate = 2 Hz between time 200 ms and700 ms. Note that the neurons are not yet inter-connected.

Excitatory and inhibitory neurons:

A neuron is, with probability pI = 0.2,

a (fast-spiking) inhibitory neuron

(a = 0.1, d = 2), others are (regular

spiking) excitatory neurons (a = 0.02

and d = 8). Weights of the input

synapse j to inhibitory neuron i is

w in = 0.07 if connected (otherwise 0).


Step 3 in detail:Modify the last script (after saving it under new name).

Proceed as follows:

1 Initialize new parameter values (n = 1000)



Proceed as follows:


2 Initialize 2 logical vectors (for indexing HowTo ) kinh and kexc oflength n, where kinh has a 1 in element i with probability p = 0.2(marking an inhibitory neuron) and 0 otherwise. kexc = ¬kinh.



Proceed as follows:



3 Reserve memory and initialize vij , uij (now being n × T matrices).Set parameters ai and di according to kexc and kinh.



Proceed as follows:




4 The weights w inij = 0.07 now form a n × nin matrix. Set 90 %

random elements to 0 to account for the connection probability.



Proceed as follows:







1 Same update equations (for vi,t+1 and ui,t+1) but use arraynotation to update all i neurons simultaneously.



Proceed as follows:







1 Same update equations (for vi,t+1 and ui,t+1) but use arraynotation to update all i neurons simultaneously.

6 Plot the spike raster. Plot black dots at {(t, i)|vit ≥ 35} forexcitatory neuron i . Use red dots for inhibitory neurons.


Solution to step 31 % i n i t i a l i z e pa ramete r s2 dt = 0 . 5 ;3 n = 1000 ; % no o f neurons4 i nh = rand (n ,1)<0 .2 ; % k inh5 exc = ˜ inh ; % o t h e r s exc6 d = 8∗ exc + 2∗ i nh ; % f o r exc / i nh7 a = 0.02∗ exc + 0.1∗ i nh ;89 % r e s e r v e memory

10 T = c e i l (1000/ dt ) ;11 v = ze ro s (n ,T) ; % now : n x T12 u = ze ro s (n ,T) ; % now : n x T13 v ( : , 1 ) = −70; % v e c t o r s14 u ( : , 1 ) = −14;1516 n i n = 100 ;17 r a t e = 2∗1e−3;18 tau g = 10 ;19 g i n = ze ro s ( n in , 1 ) ;20 E i n = ze ro s ( n in , 1 ) ;21 W in = 0.07∗ ones (n , n i n ) ; % n x n i n22 W in ( rand (n , n i n )>0.1) = 0 ;%conn . prob2324 % fo r−l oop ove r t ime25 f o r t = 1 :T−1;26 i f t∗dt>200 && t∗dt<70027 p = rand ( n in ,1)< r a t e∗dt ;28 e l s e

29 p = 0 ;30 end

31 g i n = g i n + p ;

32 Iapp = w in ∗( g i n .∗ E in ) ;33 Iapp = Iapp −(W in∗ g i n ) .∗ v ( : , t ) ; %NEW34 g i n = (1 − dt / tau g )∗ g i n ;3536 %update v e c t o r i z e d ODE37 dv =(0.04∗ v ( : , t )+5).∗ v ( : , t )+140−u ( : , t ) ;38 v ( : , t+1) = v ( : , t ) + ( dv+Iapp )∗ dt ;39 du = a .∗ (0 . 2∗ v ( : , t)−u ( : , t ) ) ;40 u ( : , t+1) = u ( : , t ) + dt∗du ;4142 %hand l e s p i k e s ( r e s e t v , u )43 f i r e d = v ( : , t )>=35; % neurons f i r e d ?44 v ( f i r e d , t ) = 35 ;45 v ( f i r e d , t+1) = −65;46 u ( f i r e d , t+1) = u ( f i r e d , t )+d ( f i r e d ) ;47 end

4849 %p l o t s p i k e r a s t e r50 spks = doub l e ( v==35);51 spks ( inh , : ) = 2∗ spks ( inh , : ) ;52 c l f , ho ld on ;53 [X,Y] = meshgr id ( ( 0 :T−1)∗dt , 1 : n ) ;54 c o l = ’ k r ’ ;55 f o r k = 1 :2 % exc / i nh56 i d x = f i n d ( spks==k ) ;57 p l o t (X( i d x ) ,Y( i d x ) , [ c o l ( k ) ’ . ’ ] ) ;58 end

59 x l im ( [ 0 ,T∗dt ] ) ; y l im ( [ 0 , n ] )60 x l a b e l ( ’ Time [ms ] ’ )61 y l a b e l ( ’ Un i t #’ )


Step 4: Simulate recurrent network

Exercise 4

Simulate 1000 neurons as before but with added recurrentconnections.


Step 4: Simulate recurrent network

Exercise 4

Simulate 1000 neurons as before but with added recurrentconnections.

Recurrent synaptic activations

A neuron i is sparsely (with probabilityprc = 0.1) connected to a neuron j .Thus neuron i receives an additionalcurrent I syni of the form:

Isyni =

n∑

j=1

wijgj(t) (Ej − vi (t))

Weights are Gamma distributed (scale

0.003, shape 2). Inh. to exc.

connections are twice as strong.



Proceed as follows:

1 Initialize and allocate memory for the new variables ( g = (gj), Ej).Set Ej = −85 if j is an inhibitory neuron (otherwise 0).



Proceed as follows:


2 Reserve memory and initialize weights W = (wij ) to zero.Randomly choose 10% of the matrix elements.



Proceed as follows:



3 Set the chosen weight matrix elements to values drawn from aGamma distribution of scale 0.003 and shape 2 HowTo



Proceed as follows:




4 Make the weight matrix “sparse” HowTo



Proceed as follows:





5 Scale weights from inh. to exc. neurons by the factor of 2



Proceed as follows:





5 Scale weights from inh. to exc. neurons by the factor of 2


1 add the equations for recurrent conductances gj

gj ← gj + 1, if vj(t − 1) ≥ 35

Isyn ← W · (g ⊙ E)− (W · g)⊙ v

gj ← (1−∆t/τg ) gj


Solution to step 41 % i n i t i a l i z e pa ramete r s2 dt = 0 . 5 ;3 n = 1000 ;4 i nh = rand (n ,1)<0 .2 ;5 exc = ˜ inh ;6 d = 8∗ exc + 2∗ i nh ;7 a = 0.02∗ exc + 0.1∗ i nh ;89 % r e s e r v e memory

10 T = c e i l (1000/ dt ) ;11 v = ze ro s (n ,T) ;12 u = ze ro s (n ,T) ;13 v ( : , 1 ) = −70;14 u ( : , 1 ) = −14;1516 n i n = 100 ;17 r a t e = 2∗1e−3;18 tau g = 10 ;19 g i n = ze ro s ( n in , 1 ) ;20 E i n = ze ro s ( n in , 1 ) ;21 W in = 0.07∗ ones (n , n i n ) ;22 W in ( rand (n , n i n )>0.1) = 0 ;2324 g = ze ro s (n , 1 ) ; % NEW: i n i t RC25 E = ze ro s (n , 1 ) ;26 E( i nh ) = −85;27 W = ze ro s (n , n ) ; % i n i t W with 028 i d x = f i n d ( rand (n , n )<0.1) ;%choose e l e .29 W( i d x ) = gamrnd ( 2 , . 0 0 3 , l eng th ( i d x ) , 1 ) ;30 W( exc , i nh ) = 2∗W( exc , i nh ) ;%s c a l e I−>E31 W = spa r s e (W) ; %f a s t e r . .32 f i r e d = 0 ;33

34 % fo r−l oop ove r t ime35 f o r t = 1 :T−1;36 %Po i s son i npu t37 i f t∗dt>200 && t∗dt<70038 p = rand ( n in ,1)< r a t e∗dt ;39 e l s e

40 p = 0 ;41 end

42 % inpu t conduc tance s43 g i n = g i n + p ;44 Iapp = W in∗( g i n .∗ E in ) ;45 Iapp = Iapp −(W in∗ g i n ) .∗ v ( : , t ) ;46 g i n = (1 − dt / tau g )∗ g i n ;4748 %NEW r e c u r r e n t conduc tance s49 g = g + f i r e d ; % s p i k e s have a r r i v e d50 I s y n = W∗(g .∗E) − (W∗g ) .∗ v ( : , t ) ;51 Iapp = Iapp + I s y n ;52 g = (1 − dt / tau g )∗g ;5354 %update v e c t o r i z e d ODE55 dv =(0.04∗ v ( : , t )+5).∗ v ( : , t )+140−u ( : , t ) ;56 v ( : , t+1) = v ( : , t ) + ( dv+Iapp )∗ dt ;57 du = a .∗ (0 . 2∗ v ( : , t)−u ( : , t ) ) ;58 u ( : , t+1) = u ( : , t ) + dt∗du ;5960 %hand l e s p i k e s ( r e s e t v , u )61 f i r e d = v ( : , t )>=35; % neurons f i r e d ?62 v ( f i r e d , t ) = 35 ;63 v ( f i r e d , t+1) = −65;64 u ( f i r e d , t+1) = u ( f i r e d , t )+d ( f i r e d ) ;65 end

66 %p l o t s p i k e r a s t e r : as b e f o r e


Congratulation !

You have just coded and simulated a

quite realistic network model !


Optional exercises


Exercises

Exercise (p-series)

Calculate the p-series (generalization of the Harmonic Series) for agiven p up to a given m

µm =

m∑

i=1

1

np

Use array notation. Solution

Advise: Use array notation and avoid for-loops whereever you can!


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 usingmeshgrid with x and y as input (look at X and Y tounderstand what meshgrid is doing).

3 calculate a matrix Z of the same size as X and Y where eachelement is given by zi = 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


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 bluecircles

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 inthe same graph as above.

6 Set both axes limits to −3 to 3 and plot dotted gray lines onthe coordinate axes (where y = 0 or x = 0).

7 Label the axes and write a legend

Solution


Exercises

Exercise (Poisson spike trains)

Write a function that generates a homogeneous Poisson spike trainof rate λ having exactly N spikes. Use array notations (cumsum).

Hint: Poisson spike intervals t are exponentially distributed. Theycan be generated by inverse transform sampling: Given uniformrandom variables u in 0 to 1 valid inter-spike intervals can becalculated as

t = − log u

λSolution

Advise: Use array notation and avoid for-loops whereever you can!


Exercises

Exercise (More on spike trains)

Generate a long Poisson spike train (with the function from thelast exercise). Compute the mean and standard deviation of theinter-spike interval distribution.

Further, write a function that counts the number of spike timesfalling into time-bins of length ∆t.

Hint: Use diff to get the intervals from spike times.

Hint 2: Use histc to bin the spike-timesSolution


Exercises

Exercise (Plot 2-D Gaussian point cloud)

Write a function which plots a point cloud of n randomsamples drawn from a 2D Normal distribution (with 0mean) and variance Σ = (RD)T (RD), where therotation matrix is defined as

R =

(

cos θ − sin θsin θ cos θ

)

and D is a diagonal matrix of the standard deviation inthe principal directions

D =

(

σ1 00 σ2

)

The function should accept parameters σ1, σ2, and θ.

-5 0 5

-5

0

5

Hint: use randn to produce independent Normal-distributed random vectors xiand transform them according to yi = RDxi .

Solution


Advanced exercise

Exercise (Optimizing Matlab code)

Optimize “bad” Matlab code for generating self-organizing maps.

See provided code and description in som_exercise.zip.


Solution: P-series

1 f unc t i on 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 N3

4 i a r r = ( 1 : n ) . ˆ p ;5 mu = sum ( 1 . / i a r r , 2 ) ;

back to text


Solution: Blob movie1 x = l i n s pa c e (−2 ,2 ,100) ;2 y = l i n s pa c e (−2 ,2 ,100) ;3

4 [X ,Y] = meshgrid ( x , y ) ;5

6 Z = exp(−X.ˆ2 − Y . ˆ 2 ) ;7

8 f o r t = 1:10009 imagesc (Z ) ;

10 Z = c i r c s h i f t (Z , [ 0 , 1 ] ) ;11 drawnow ;12 end

back to text


Solution: Logical indexing and basic plotting1 %1 .2 M = randn ( 1 0 0 , 2 ) ;34 %2 .5 p l o t (M( : , 1 ) ,M( : , 2 ) , ’ bo ’ ) ;67 %3 .8 npos = sum (M(:)>0)9

10 %4 .11 M( any (M<0 ,2) , : ) = [ ] ;1213 %5 .14 ho ld on ;15 p l o t (M( : , 1 ) ,M( : , 2 ) , ’ r x ’ ) ;1617 %6 .18 l i m i t s = [−3 ,3 ] ;19 x l im ( l i m i t s ) ; y l im ( l i m i t s )20 p l o t ( l i m i t s , [ 0 , 0 ] , ’ : ’ , ’ Co l o r ’ , [ 0 . 5 , 0 . 5 , 0 . 5 ] )21 p l o t ( [ 0 , 0 ] , l i m i t s , ’ : ’ , ’ Co l o r ’ , [ 0 . 5 , 0 . 5 , 0 . 5 ] )2223 %7 .24 x l a b e l ( ’X ’ )25 y l a b e l ( ’Y ’ )26 l egend ({ ’ Gaus s i an random samples ’ , ’ Po i n t s w i th X>0 and Y>0 ’})

back to text


Solution: Poisson spike trains

1 f unc t i on spk t = p o i s s o n s p i k e s ( lambda ,N)2 % SPKT = POISSONSPIKES(LAMBDA,N) g e n e r a t e s a3 % Po i s son s p i k e t r a i n w i th r a t e LAMBDA4 % and e x a c t l y N s p i k e s .5

6 i s i s = −l og ( rand (N, 1 ) ) / lambda ;7 spk t = cumsum( i s i s ) ;

back to text


Solution: More on spike trains1 N = 1000 ; % number o f s p i k e s2 lambda = 10 ; %s p i k e r a t e3

4 %Po i s son s p i k e s5 spk t = cumsum(− l og ( rand (N, 1 ) ) / lambda ) ;6

7 %mean o f the I S I8 mi s i = mean( d i f f ( spk t ) ) ;9

10 %stddev o f the I S I11 s i s i = std ( d i f f ( spk t ) ) ;12

13

14 f unc t i on S = s p i k e b i n n i n g ( spkt , twin , dt )15 % S = SPIKEBINNING(SPKT,TWIN,DT) count s the s p i k e t imes16 % oc c u r r i n g i n each b in o f width DT i n the t ime window17 % from TWIN(1) and TWIN(2)18

19 S = h i s t c ( spkt , tw in ( 1 ) : dt : tw in ( 2 ) ) ;

back to text


Solution: Plot 2-D Gaussian point cloud1 f u n c t i o n p lo t2DGaus s i an ( s i g1 , s i g2 , t h e t a ) ;2 % PLOT2DGAUSSIAN( SIG1 , SIG2 ,THETA) p l o t s a Gaus s i an po i n t c l oud wi th3 % s tanda rd 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.45 n = 250 ;6 r = randn (2 , n ) ;7 D = [ s i g1 , 0 ; 0 , s i g 2 ] ;8 R = [ cos ( t h e t a ),− s i n ( t h e t a ) ; s i n ( t h e t a ) , cos ( t h e t a ) ] ;9

10 x = R∗D∗ r ;1112 %p l o t t i n g13 f i g u r e ; subp l o t ( 2 , 2 , 1 ) ;14 p l o t ( x ( 1 , : ) , x ( 2 , : ) , ’ . b ’ , ’ Marke rS i ze ’ , 6 ) ;1516 %r e s i z e the axe s17 mxlen = 1.1∗max( abs ( x ( : ) ) ) ;18 x l im ([−mxlen , mxlen ] )19 y l im ([−mxlen , mxlen ] )2021 %p l o t the 0−c o o r d i n a t e s22 ho ld on ;23 p l o t ([−mxlen , mxlen ] , [ 0 , 0 ] , ’−−k ’ )24 p l o t ( [ 0 , 0 ] , [ −mxlen , mxlen ] , ’−−k ’ )25 ho ld o f f ;

back to text

Howtos

How to start Matlab

Easy: Just click on Matlab symbol...

back

Howtos

How to get help

Each Matlab function hasa header which describes itsusage

Just type:

>> help command

or

>> doc command

Alternatively:http://www.mathworks.com/access/helpdesk/help/techdoc/

back

http://www.mathworks.com/access/helpdesk/help/techdoc/

Howtos

How to plot in general

How to plot:

1 Open a figure (window) with

>> figure;

2 Issue plotting command of your choice

>> plot(x,y,’b--’,’LineWidth’,2);

HowTo Note: graphics commands draw into the current axes (gca) on the

current figure (gcf). They automatically create a new figure and axes if

necessary.

3 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’)

back

Howtos

How to use the 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 areplotted as different lines

linespec – a string with a short hand for color and line style andmarker type. See help plot for an overview. Eg,

linespec = ’:ko’

plots dotted (:) black line (k) with a circle at each givencoordinate (xi , yi )

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)back

Howtos

How to initialize parameters

Just use the syntax

>> parname = value;

Example

>> a = 2;

>> vreset = 0;

>> tau = 0.02

tau =

0.02

back

Howtos

How to use scalar expressions

Binary operations: work as expected, use = + - * / ^

Example (compute y = a2x2+a

+ b)

>> a = 2;

>> b = 1;

>> x = 0.5;

>> y = a^2*x/(2+a) + b;

>> y

y =

1.500

back

Howtos

How to initialize arrays

1 Implicitly, using function returning an array2 By explicit concatenation

Concatenation of columns of arrays[ arr1, arr2, ... , arrn]

Concatenation of rows of arrays[ arr1; arr2; ... ; arrn]


Example (Concatenation)

>> A = [1,2;3,4]

A =

1 2

3 4

>> B = [A,A]

B =

1 2 1 2

3 4 3 4

>> C = [A;A]

C =

1 2

3 4

1 2

3 4

Howtos

How to pre-allocate memory

Functions for pre-allocating memory include:

colon (:) – for linear sequences

zeros – all zero array of given size

ones – all one array of given size

rand – random array of given size (equally in [0, 1])

To improve performance arrays should always be pre-allocated!

Example (Functions initializing arrays)

>> A = zeros(3,3);

>> A = ones(4,4,4);

>> size(A)

ans =

4 4 4

>> x = 3:-0.5:1

x =

3.0 2.5 2.0 1.5 1.0

>> A = ones(2)

1 1

1 1

back

Howtos


1 open a text-editor of your choice or use the editor providedwith Matlab

>> edit myscript




>> myscript;

Example (myscript.m)

1 %t h i s i s my f i r s t s c r i p t . I t d i s p l a y s a random number2 random number = rand ( 1 ) ;3 f p r i n t f ( ’A random number : %1.4 f :\ n ’ , random number ) ;

back

Howtos

How to use basic syntax: if-clause






Example (if-else)

a = rand ( 1 ) ;i f a == 0.5

f p r i n t f ( ’ you a r e v e r y l u c k y ! ’ ) ;end

back

Howtos

How to use a for-loop




Example (for loop)

a=0;f o r i = 1 :100

a = a+i ;end

back

Howtos

How to index arrays


access the (i , j)-th element of a 2D-matrix W of dimension (m, n)

>> W(i,j) = 1



access the (i , j)th element of the 2D-matrix W of dimension (m, n)


>> 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(:)

back

Howtos

How to index arrays (2)

4 Multiple indices


>> 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 (veryuseful)

>> W(W>0)

>> W(find(W>0)) = 1

back

Howtos


1 Element-wise interpretationFor instance, sin cos log etc.Reserved symbols, .* ./ .^

2 “true” matrix interpretation (with dot product)Symbols * / ^ etc.

3 Operations on one specified dimensions of the matrixFor instance, sum mean max etc.

4 Array manipulationseg. reshape repmat permute circshift tril

Example (element-wise product and dot product)

>> A = ones(2,2);

>> A.*A

ans =

1 1

1 1

>> A*A

ans =

2 2

2 2

back

Howtos


straightforward

however, carefully checkthe size of matricesif element-wise or matrix-like operations are intendedwhich matrix dimension to operate on

Example (compute yi = W xi + b with b, xi ∈ R2)

>> W = [1,0.2;0.4,1];

>> b = [1;2] + 0.1;

>> x = 2*randn(2,1);

>> y = W * x + b

y =

4.5535

1.4856

>> 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 =

4.5535 2.3126 [..] -0.9021

1.4856 6.8091 [..] -0.1080

back

Howtos

Gamma distribution

The Gamma probability density function is defined as

p(x |k , θ) = 1

Γ(k)θkxk−1e−

xθ

with shape k and scale θ and x ∈ [0,∞).

Example (Gamma random numbers)

>> shape = 2;

>> scale = 0.003;

>> n = 1e7;

>> r = gamrnd(shape,scale,n,1);

>> size(r)

ans =

1000000 1

>> hist(r,1000)

back

Howtos

How to use sparse matrices

In Matlab it is often more efficient (faster) to use sparse matricesinstead of regular matrices if the majority of matrix elements are 0.

To generate a sparse matrix:

>> W = sparse(W);

In general, the syntax for using sparse matrices is the same as forregular matrices.

Example (Sparse matrix)

>> W = double(rand(100,100)>0.9);

>> Ws = sparse(W);

>> y = Ws*x; % the same as y=W*x but faster

back

Howtos

Solution to Excercise #1

1 %compute the S t i r l i n g Formula2

3 n = 50 ;4

5 n f a c t o r i a l = sqr t (2∗ p i ∗n )∗ ( n/exp ( 1 ) ) ˆ n

back to text

Howtos

Solution to Excercise #2

1 funct ion n f a c t o r i a l = s t i r f a c ( n ) ;2 %compute the S t i r l i n g Formula3

4 n f a c t o r i a l = sqr t (2∗ p i ∗n )∗ ( n/exp ( 1 ) ) ˆ n ;

back to text

MATLAB Tutorial – CCN Course 2012 - · PDF fileIntroduction Arrays Coding Plotting...

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Transcript of MATLAB Tutorial – CCN Course 2012 - · PDF fileIntroduction Arrays Coding Plotting...