% This program is the first simulation. It just uses one neuron and simulate the three solution 
% of Differiantial  Equation viz. Exact Solution, Runge Kutta and Foward Euler


% Here are few of the aprameters which can be set by the user. These parameters are 

I=1.001; % I as input current.
deltaT=0.1; % deltaT as the time interval .
alpha=1; % alpha as the rate.
tKnot=1; % tKnot as the initial time.
uExactSolution(tKnot)=0; % uExactSolution as the membrane potential for exact solution.
uRungeKutta(tKnot)=0;  % uExactSolution as the membrane potential for Runge Kutta solution.
uFowardEuler(tKnot)=0;  % uExactSolution as the membrane potential for Foward Euler solution.


j=2;
for i=2:300 % Here the main loop begins and the potential is calculated for 300 time steps.
   uExactSolution(i)=((I/alpha)+(uExactSolution(tKnot)-(I/alpha))*(exp(-1*alpha*j*deltaT)));
   
   k1=((-1)*alpha*uRungeKutta(i-1)+(I));
   k2=((-1)*alpha*(uRungeKutta(i-1)+(0.5*k1*deltaT))+(I));
   k3=((-1)*alpha*(uRungeKutta(i-1)+(0.5*k2*deltaT))+(I));
   k4=((-1)*alpha*(uRungeKutta(i-1)+(k3*deltaT))+(I));
   
   uRungeKutta(i)=(uRungeKutta(i-1)+((deltaT/6)*(k1+2*k2+2*k3+k4)));
   uFowardEuler(i)=((uFowardEuler(i-1))+(deltaT*(((-1)*(alpha)*(uFowardEuler(i-1)))+(I))));
   if(uExactSolution(i)>=1)
 
 		uExactSolution(i)=0;
      j=1;
   end
   if(uRungeKutta(i)>=1)
      uRungeKutta(i)=0;
   end
   if(uFowardEuler(i)>=1)
      uFowardEuler(i)=0;
   end
   j=j+1;
end

% After calculating the complete solution the graphs are plotte for all three types of differential solution methods
figure;
subplot(3,1,1);
plot(uExactSolution);
title('Exact Solution');
xlabel('Time');
ylabel('Membrane Potential');
%figure;
subplot(3,1,2);
plot(uRungeKutta);
title('Runge Kutta');
xlabel('Time');
ylabel('Membrane Potential');
%figure;
subplot(3,1,3);
plot(uFowardEuler);
title('Foward Euler');
xlabel('Time');
ylabel('Membrane Potential');