function [Y,T_run] = OrioSoudry(t, Ifunc, Area)
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% 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
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% Initialize quantities needed to run solver
t_ind=tic;
% time step size
dt = t(2)-t(1);
% Number of time steps
nt = length(t); % total
nt1 = nt-1; % at which to solve
NaNoise1=randn(20,nt);
KNoise1=randn(8,nt);
% Initial Values
% from XPP code on the limit cycle
t0 = t(1);
V0 = -61.897274;
m00=0.4329406;m01=0.10034765;m02=0.0077529117;m03=0.00019966466;
m10=0.36696321;m11=0.085055299;m12=0.0065714167;
% m13=0.00016923703;
m13=1-m00-m01-m02-m03-m10-m11-m12; %normalize to 1
n0=0.13761529;n1=0.35331318;n2=0.34016079;n3=0.14555468;
%n4=0.023356047;
n4=1-n0-n1-n2-n3; %normalize to 1
% Initialize Output
Y = zeros(nt,1);
Y(1) = V0;
Na_gates =[m00;m01;m02;m03;m10;m11;m12;m13];
K_gates=[n0,n1,n2,n3,n4]';
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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)*DNaOrio(V0,Na_gates,NNa,NaNoise1(:,i-1));
K_gates = K_gates + dt*AK(V0) *K_gates + sqrt(dt)*DKOrio(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) = V0;
end % End loop over time for SDE solver
T_run=toc(t_ind);
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) for Orio 2012 methods
function D = DNaOrio(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 10 paris of transactions
N10=zeros(10,1);
%save all these alpha and beta values for computational efficiency
ah=alphah(V);
bh=betah(V);
am=alpham(V);
bm=betam(V);
N10(1)=sqrt(abs(3*ah*y00+bh*y01))*M(1); %00--01
N10(2)=sqrt(abs(3*am*y00+bm*y10))*M(2); %00--10
N10(3)=sqrt(abs(ah*y10+bh*y11))*M(3); %10--11
N10(4)=sqrt(abs(2*am*y10+2*bm*y20))*M(4);%10--20
N10(5)=sqrt(abs(ah*y20+bh*y21))*M(5); %20--21
N10(6)=sqrt(abs(am*y20+3*bm*y30))*M(6); %20--30
N10(7)=sqrt(abs(ah*y30+bh*y31))*M(7); %30--31
N10(8)=sqrt(abs(3*am*y01+bm*y11))*M(8);%01--11
N10(9)=sqrt(abs(2*am*y11+2*bm*y21))*M(9);%11--21
N10(10)=sqrt(abs(am*y21+3*bm*y31))*M(10);%21--31
D(1) = N10(1)+N10(2);
D(2) = -N10(2)+N10(3)+N10(4);
D(3) = -N10(4)+N10(5)+N10(6);
D(4) = -N10(6)+N10(7);
D(5) = -N10(1)+N10(8);
D(6) = -N10(3)-N10(8)+N10(9);
D(7) = -N10(5)-N10(9)+N10(10);
D(8) = -N10(7)-N10(10);
D = D/sqrt(N);
end
% Diffusion matrix for K (Orio and Soudry)
function D = DKOrio(V,Y,N,M)
%V is voltage, Y stands for probablity at each state
%N is the dimension or number of K Channels, M is randn 4 by 1
D = zeros(5,1);
% noise for these 8 transactions
N4=zeros(4,1);
%save all these alpha and beta values for computational efficiency
an=alphan(V);
bn=betan(V);
N4(1)=sqrt(abs(4*an*Y(1)+bn*Y(2)))*M(1); %0--1
N4(2)=sqrt(abs(3*an*Y(2)+2*bn*Y(3)))*M(2); %1--2
N4(3)=sqrt(abs(2*an*Y(3)+3*bn*Y(4)))*M(3); %2--3
N4(4)=sqrt(abs(an*Y(4)+4*bn*Y(5)))*M(4); %3--4
D(1) = N4(1);
D(2) = -N4(1)+N4(2);
D(3) = -N4(2)+N4(3);
D(4) = -N4(3)+N4(4);
D(5) = -N4(4);
D = D/sqrt(N);
end