2. Integrate-and-Fire Model
To estimate the parameters inthisproject,we know the statisticsof the systemmustbe
preserved. Thus,we aimtoestimate the parameterssuchthatthe numerical resultsusingthe specified
parameters outputssimilarstatisticstothatof the data set.
PARAMETER ESTIMATES:
El= -65; vTH=-50; vSpike=60; vRst=-60; gl=0.03148;
C=1; D=0.022
(1) Iapp = 0.375
DATA NUMERICAL ERROR (%)
No. of ISI’s 8 3 55.5556
Mean Frequency 0.0824 0.0995 20.7414
Mean time for ISI’s 1.2140e+04 1.0054e+04 17.1784
0 0.5 1 1.5 2 2.5 3 3.5 4
x 10
4
0
0.5
1
1.5
2
2.5
3
Occurences
Length of Each ISI
Iapp = 0.375 ; D = 0.022 ;gl = 0.03148; #ISIs:3
Mean:12139.825
Mean:10054.4
DATA
NUMERICAL
3. (2) Iapp = 0.450
DATA NUMERICAL ERROR (%)
No. of ISI’s 674 677 0.4444
Mean Frequency 6.7567 6.7870 0.4479
Mean time for ISI’s 148.0006 147.3406 0.4459
Notice howthe error significantlydecreaseswiththe same parametervalueswithanincrease inIapp
fromgraph (1) to (2). This can be explainedbythe factthat, firstly,ingraph(1) the sample size of the
ISI’sismuch smaller. There are only8 ISI’sin(1) as opposedto674 in (2).Thus,(2) providesuswitha
greatersample size of ISI’s,andhence,abettermeasure forthe parametervalues.
Furthermore,the randomfactor,D, playsa more significantrole asIapp isdecreased. Thisiswhat
accounts fora highererrorwitha smallerIapp. There ismore randomnessprevalentinasystemwith
lowerIapp as comparedto a systemwithgreaterIapp. Observe thatas we increase Iapp,we see that our
parametervaluesdoinfact fitthe data quite well asexhibitedbythe table andhistogramin(2).
Finally,we note the ISI’smeanisinverselyproportional toIapp. This isintuitive since ahigherIapp allows
for more spikingwithinthe same interval of time,namely t€[0,10000]. More spikeswithinthe same
interval of time implyahigherfrequency,whichinturnimpliesasmallerISI. Thisintuitionisexemplified
by (1) and (2) where the meanof the ISIsdecreasesalmost100-fold.
0 100 200 300 400 500 600 700 800
0
50
100
150
200
250
300Occurences
Length of Each ISI
Iapp = 0.45 ; D = 0.022 ;gl = 0.03148; #ISIs:677
Mean:148.0006
Mean:147.3406
DATA
NUMERICAL
4. NUMERICAL MATLAB CODE:
clc
clear all
close all
A=importdata('C:/IF4.txt');
BINS=20;
str1='Iapp = ';
str2='D = ';
str3='gl = ';
%Integrate-and-Fire Neuron Model
...Iapp=1.50 ; D=0.0 - no spking
...Iapp=1.51 ; D=0.0 - spiking
...Iapp=1.50 ; D=0.1 - spiking
...Iapp=0.375; D=0.0110 ; gl=0.02906 -IF1.dat
...Iapp=0.400; D=0.0120 ; gl=0.03000 -IF2.txt
...Iapp=0.425; D=0.0100 ; gl=0.03029 -IF3.txt
...Iapp=0.450; D=0.0220 ; gl=0.03148 -IF4.txt
...Iapp=0.525; D=0.0200 ; gl=0.03035 -IF7.txt
%I: INITIALIZATION:
Iapp=0.45; gl=0.03148;
C=1;
D=0.022 ; ...Noise
El= -65; vTH=-50; vSpike=60; vRst=-60; R=1/gl; tau=C/gl;
SpikeTime=0;
% II: NUMERICAL SCHEME: Runga-Kutta order 2
%Discretize Grid
T = 100000; %#grid pts = T/dt
dt = 0.05; %time-step
t = 0:dt:T; %discretizing Grid
V = zeros(1,length(t)); %initalizing vector
V(1) = -65; %inital condition
i=0; %index for Spike Time
SpikeSwitch=0; %indicates Spiking
for j=1:length(t)-1
%Iapp=1.4+sin(2*pi*dt*j/1000);
kv1 = ((-V(j)+El+Iapp*R)/tau)+(sqrt(dt*2*D)*randn);
av = V(j)+kv1*dt;
kv2 = ((-av+El+Iapp*R)/tau)+(sqrt(dt*2*D)*randn);
V(j+1) = V(j) + ((kv1+kv2)*dt/2)+ (sqrt(dt*2*D)*randn);
%Condtion for spiking
if V(j+1)>=vTH
SpikeSwitch=1;
5. i=i+1;
SpikeTime(i)=dt*(j-1); %Store spike time
x=[dt*(j-1),dt*(j-1)]; %draw vertical
y=[V(j), vSpike]; ...lines
hold on
plot(x,y,'LineWidth',1)
V(j+1)=vRst;
end
end
%Calculate ISI's ONLY IF Conditon Satisified:
if numel(SpikeTime)>1
for i=1:numel(SpikeTime)-1
ISI(i)=SpikeTime(i+1)-SpikeTime(i);
end
else ISI=0;
end
% III. STATISTICS:
%MEAN of ISI's
sum=0;
for i=1:numel(ISI)
sum=sum+ISI(i);
end
meanISI=sum/numel(ISI);
%VARIANCE
for i=1:numel(ISI)
var(i)=abs(ISI(i)-meanISI);
end
%MEAN of VARIANCE
sum=0;
for i=1:numel(ISI)
sum=sum+var(i);
end
meanVar=sum/numel(var);
%COEFFICENT OF VARIATION
for i=1:numel(ISI)
COV(i)=var(i)/meanISI;
end
%MEAN of COEFFICENT OF VARIATION
sum=0;
for i=1:numel(ISI)
sum=sum+COV(i);
6. end
meanCOV=sum/numel(COV);
%FREQUENCY
for i=1:numel(ISI)
freq(i)=1/ISI(i);
end
meanFreq=1000/meanISI; %avg Spikes/sec
title([str1 num2str(Iapp) ' ; ' str2 num2str(D) ' ;' str3
num2str(gl) '; #ISIs:' num2str(numel(ISI))])
plot(t,V,'b','linewidth',1); %Numerical Solution
%axis([0 T -80 80])
%set(gca,'fontsize',15)
xlabel('t');
ylabel('V');
%data
figure (4)
plot(A(:, 1), A(:,2));
dataSpike=0;
i=0;
for j=1:numel(A(:,2))
if A(j,2)==vSpike
i=i+1;
dataSpike=dataSpike+1;
dataspikeTime(i)=A(j,1);
end
end
dataSpike;
dataspikeTime;
for i=1:numel(dataspikeTime)-1
dataISI(i)=dataspikeTime(i+1)-dataspikeTime(i);
end
%MEAN of data ISI's
sum=0;
for i=1:numel(dataISI)
sum=sum+dataISI(i);
end
7. %FREQUENCY for DATA
meandataISI=sum/numel(dataISI);
meandataFreq=1000/meandataISI; %avg Spikes/sec
%Error
ErrorISI=100*(abs(meanISI-meandataISI)/meandataISI);
ErrorFreq=100*(abs(meanFreq-meandataFreq)/meandataFreq);
ErrorSpik=100*(abs(numel(SpikeTime)-dataSpike)/dataSpike);
%DISPLAY RESULTS
disp('======ERROR========= ')
disp('ERROR ISIs (%): ');disp(ErrorISI)
disp('ERROR FREQ (%): ');disp(ErrorFreq)
disp('ERROR # Spikes (%): ');disp(ErrorSpik)
disp('======DATA========== ')
disp('Number of ISIs: ');disp(numel(dataISI))
disp('Mean Frequency: ');disp(meandataFreq)
disp('Mean of ISI: ');disp(meandataISI)
disp('=====Numerical====== ')
if numel(SpikeTime)>1
disp('Number of ISIs:');disp(numel(ISI))
disp('Mean Frequency:');disp(meanFreq)
disp('Mean ISI:');disp(meanISI)
%disp('Mean Variance:');disp(meanVar)
%disp('Mean Coefficent of Variation:');disp(meanCOV)
else if numel(SpikeTime)==1 && SpikeTime(1)>0
disp('1 spike')
SpikeTime
else
disp('no spiking')
end
end
figure (2)
hist(dataISI,BINS)
h = findobj(gca, 'Type','patch');
set(h,'FaceColor','r','EdgeColor','w','facealpha',0.75)
hold on
if numel(ISI)>1
