function [Y] = HHSto15D(t, Ifunc, Area, NaNoise1, KNoise1,Y0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% EM_HH.m
% Written by Shusen Pu and Peter Thomas
% Oct, 2018
%%% Inputs
% t is vector of time values (ms)
% Ifunc is a function @(t)f(t) that returns stimulus value as a function of time (in ms)
% Area: Membrane Area (mu m^2)
% Noise Model (String), possible values:
% None: 'ODE'
% EM: 'Euler Maruyama', must also have value for Area
%%% Outputs
% Y(:,1) : t
% Y(:,2) : V
% Y(:,3:10) : fractions in Na states m00-m13
% Y(:,11:15) : fraction in K states n0-n4
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Initialize quantities needed to run solver
% time step size
dt = t(2)-t(1);
% Number of time steps
nt = length(t); % total
nt1 = nt-1; % at which to solve
% Initial Values
% from XPP code on the limit cycle
t0 = Y0(1);
V0=Y0(2);
Na_gates=Y0(3:10);
K_gates=Y0(11:15);
% Initialize Output
Y = zeros(nt,15);
Y(1,1) = t0;
Y(1,2) = V0;
Na_gates =Na_gates(:);
K_gates=K_gates(:);
Y(1,3:10)=Na_gates';
Y(1,11:15)=K_gates';
t=t(:);
Y(:,1)=t;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Parameter Values
% Number of Channels
NNa = round(Area*60); % Na
NK = round(Area*18); % K
% Capacitance
C = 1; % muF /cm^2
% Na Current
gNa = 120; % mS/cm^2
ENa = 50; % mV
% K Current
gK = 36; % mS/cm^2
EK = -77; % mV
% Passive Leak
gL = 0.3; % mS / cm^2
EL = -54.4; % mV
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Determine Which Noise Model and Do Some Necessary Setup
% Conductance Noise Model using Euler Maruyama (Thomas and Pu)
% Drift Na
ANa = @(V) ...
[ -3*alpham(V)-alphah(V) , betam(V) , 0 , 0 , betah(V) , 0 , 0 , 0 ;
3*alpham(V) ,-2*alpham(V)-betam(V)-alphah(V), 2*betam(V) , 0 , 0 , betah(V) , 0 , 0 ;
0 , 2*alpham(V) , -alpham(V)-2*betam(V)-alphah(V), 3*betam(V) , 0 , 0 , betah(V) , 0 ;
0 , 0 , alpham(V) , -3*betam(V)-alphah(V) , 0 , 0 , 0 , betah(V) ;
alphah(V) , 0 , 0 , 0 , -3*alpham(V) - betah(V) , betam(V) , 0 , 0 ;
0 , alphah(V) , 0 , 0 , 3*alpham(V) , -2*alpham(V)-betam(V)-betah(V) , 2*betam(V) , 0 ;
0 , 0 , alphah(V) , 0 , 0 , 2*alpham(V) , -alpham(V)-2*betam(V)-betah(V) , 3*betam(V) ;
0 , 0 , 0 , alphah(V) , 0 , 0 , alpham(V) , -3*betam(V)-betah(V)];
% Drift K
AK = @(V) ...
[-4*alphan(V), betan(V) , 0 , 0 , 0
4*alphan(V), -3*alphan(V)-betan(V), 2*betan(V) , 0, 0;
0, 3*alphan(V), -2*alphan(V)-2*betan(V), 3*betan(V), 0;
0, 0, 2*alphan(V), -alphan(V)-3*betan(V), 4*betan(V);
0, 0, 0, alphan(V), -4*betan(V)];
%Noise Term: adding white noise to each transaction
%defined in two functions namely, DNafull and DKfull
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%% HERE IS THE SOLVER %%%%%%%%%%%%%%%%%%%%%%%
%%%%%% USING EULER FOR ODEs, %%%%%%%%%%%%%%%%%%%%%%%
%%%%%% Milstein FOR SDEs, %%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for i=2:nt
% Input Current
I = Ifunc(t(i-1));
% update the gates
Na_gates = Na_gates + dt*ANa(V0)*Na_gates+ sqrt(dt)*DNafull(V0,Na_gates,NNa,NaNoise1(:,i-1));
K_gates = K_gates + dt*AK(V0) *K_gates + sqrt(dt)*DKfull(V0,K_gates,NK,KNoise1(:,i-1)) ;
% Compute Fraction of open channels
NaFraction= Na_gates(8);
KFraction=K_gates(5);
% Update Voltage
Vrhs = (-gNa*(NaFraction)*(V0 - ENa)-gK*(KFraction)*(V0 - EK) - gL*(V0-EL) + I)/C;
V0 = V0 + dt*Vrhs ; % VNoise is non-zero for Current Noise Model
% Save Outputs
% Y(i,1) = t(i-1);
Y(i,2) = V0;
Y(i,3:10)=Na_gates';
Y(i,11:15)=K_gates';
end % End loop over time for SDE solver
end
% End Function Definition
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%% END OF SOLVER %%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Define functions used above
% subunit kinetics (Hodgkin and Huxley parameters, modified from XPP code by PJT OCt, 2018)
function out = alpham(V)
out = .1*(V+40)/(1-exp(-(V+40)/10));
%0.1 * (25-V)/ (exp((25-V)/10)-1);
end
function out = betam(V)
out = 4*exp(-(V+65)/18);
%4 * exp(-V/18);
end
function out = alphah(V)
out = .07*exp(-(V+65)/20);
%0.07 * exp(-V/20);
end
function out = betah(V)
out = 1/(1+exp(-(V+35)/10));
%1/ (exp((30-V)/10)+1);
end
function out = alphan(V)
out = .01*(V+55)/(1-exp(-(V+55)/10));
%0.01 * (10-V) / (exp((10-V)/10)-1);
end
function out = betan(V)
out = .125*exp(-(V+65)/80);
%0.125 * exp(-V/80);
end
% Full Model (Thomas and Pu)
% Diffusion matrix for Na (Thomas and Pu)
function D = DNafull(V,Y,N,M)
%V is voltage, Y stands for probablity at each state (we use Y-Ybar)
%N is the dimension or number of Na Channels, M is randn 20 by 1
D = zeros(8,1);
Y=Y.*(Y>0); %to exclude negative numbers added by SP&PJT 10/08/2018
y00 = Y(1);
y10 = Y(2);
y20 = Y(3);
y30 = Y(4);
y01 = Y(5);
y11 = Y(6);
y21 = Y(7);
y31 = Y(8);
% noise for these 20 transactions
N20=zeros(20,1);
%save all these alpha and beta values for computational efficiency
ah=alphah(V);
bh=betah(V);
am=alpham(V);
bm=betam(V);
N20(1)=sqrt(ah*y00)*M(1); %00--01
N20(2)=sqrt(bh*y01)*M(2); %01--00
N20(3)=sqrt(3*am*y00)*M(3); %00--10
N20(4)=sqrt(bm*y10)*M(4); %10--00
N20(5)=sqrt(ah*y10)*M(5); %10--11
N20(6)=sqrt(bh*y11)*M(6); %11--10
N20(7)=sqrt(2*am*y10)*M(7);%10--20
N20(8)=sqrt(2*bm*y20)*M(8); %20--10
N20(9)=sqrt(ah*y20)*M(9); %20--21
N20(10)=sqrt(bh*y21)*M(10); %21--20
N20(11)=sqrt(am*y20)*M(11); %20--30
N20(12)=sqrt(3*bm*y30)*M(12);%30--20
N20(13)=sqrt(ah*y30)*M(13); %30--31
N20(14)=sqrt(bh*y31)*M(14); %31--30
N20(15)=sqrt(3*am*y01)*M(15);%01--11
N20(16)=sqrt(bm*y11)*M(16); %11--01
N20(17)=sqrt(2*am*y11)*M(17);%11--21
N20(18)=sqrt(2*bm*y21)*M(18);%21--11
N20(19)=sqrt(am*y21)*M(19);%21--31
N20(20)=sqrt(3*bm*y31)*M(20);%31--21
D(1) = -N20(1)-N20(3)+N20(2)+N20(4) ;
D(2) = -N20(4)-N20(5)-N20(7)+N20(3)+N20(6)+N20(8);
D(3) = -N20(8)-N20(9)-N20(11)+N20(7)+N20(10)+N20(12);
D(4) = -N20(12)-N20(13)+N20(11)+N20(14);
D(5) = -N20(2)-N20(15)+N20(1)+N20(16);
D(6) = -N20(6)-N20(16)-N20(17)+N20(5)+N20(15)+N20(18);
D(7) = -N20(10)-N20(18)-N20(19)+N20(9)+N20(17)+N20(20);
D(8) = -N20(14)-N20(20)+N20(13)+N20(19);
D = D/sqrt(N);
end
% Diffusion matrix for K (Thomas and Pu)
function D = DKfull(V,Y,N,M)
%V is voltage, Y stands for probablity at each state (we use Y-Ybar)
%N is the dimension or number of K Channels, M is randn 8 by 1
D = zeros(5,1);
% noise for these 8 transactions
N8=zeros(8,1);
%save all these alpha and beta values for computational efficiency
an=alphan(V);
bn=betan(V);
Y=Y.*(Y>0); %to exclude negative numbers added by SP&PJT 10/08/2018
N8(1)=sqrt(4*an*Y(1))*M(1); %0--1
N8(2)=sqrt(bn*Y(2))*M(2); %1--0
N8(3)=sqrt(3*an*Y(2))*M(3); %1--2
N8(4)=sqrt(2*bn*Y(3))*M(4); %2--1
N8(5)=sqrt(2*an*Y(3))*M(5); %2--3
N8(6)=sqrt(3*bn*Y(4))*M(6); %3--2
N8(7)=sqrt(an*Y(4))*M(7); %3--4
N8(8)=sqrt(4*bn*Y(5))*M(8); %4--3
D(1) = -N8(1)+N8(2);
D(2) = -N8(2)-N8(3)+N8(1)+N8(4);
D(3) = -N8(4)-N8(5)+N8(3)+N8(6);
D(4) = -N8(6)-N8(7)+N8(5)+N8(8);
D(5) = -N8(8)+N8(7);
D = D/sqrt(N);
end