When using ODE45, can I specify a variable to assume two different values during the timespan?

I have tried in different ways to see what happens to voltage V and gating conductances m, n and h when, at time step x, current I switched from 0 to 0.1, and then at time step x + n it gets back to 0. However, it looks like regardless of what I do, ODE45 still assumes I is either 0 or 0.1 for the whole time span.

This is the function. In this case the current I is 0.1 for the whole time span. Is there a way to make it 0 everywhere except for 10 ms in the middle, for example?

function ODE_Hodgkin_Huxley (varargin)

    t=0:0.1:25; %Time Array ms    
    V=-60; % Initial Membrane voltage
    
    m=alpham(V)/(alpham(V)+betam(V)); % Initial m-value
    n=alphan(V)/(alphan(V)+betan(V)); % Initial n-value
    h=alphah(V)/(alphah(V)+betah(V)); % Initial h-value
    y0=[V;n;m;h];
    tspan = [0,max(t)];
    
    %Matlab's ode45 function
    [time,V] = ode45(@ODEMAT,tspan,y0);
    OD=V(:,1);
    ODn=V(:,2);
    ODm=V(:,3);
    ODh=V(:,4);
    [r,c] = size(time);
     
    I = ones (r,c) ./ 10; %Current
    
    figure
    subplot(3,1,1)
    plot(time,OD);
    legend('ODE45 solver');
    xlabel('Time (ms)');
    ylabel('Voltage (mV)');
    title('Voltage Change for Hodgkin-Huxley Model');
    subplot(3,1,2)
    plot(time,I);
    legend('Current injected')
    xlabel('Time (ms)')
    ylabel('Ampere')
    title('Current')
    subplot(3,1,3)
    plot(time,[ODn,ODm,ODh]);
    legend('ODn','ODm','ODh');
    xlabel('Time (ms)')
    ylabel('Value')
    title('Gating variables'
    
end

function [dydt] = ODEMAT(t,y)
    
    %Constants
    ENa=55; % mv Na reversal potential
    EK=-72; % mv K reversal potential
    El=-49; % mv Leakage reversal potential
    
    %Values of conductances
    
    gbarl=0.003; % mS/cm^2 Leakage conductance  
    gbarNa=1.2; % mS/cm^2 Na conductance
    gbarK=0.36; % mS/cm^2 K conductancence
    I = 0.1; %Applied constant
    
    Cm = 0.01; % Capacitance
    
    % Values set to equal input values
    V = y(1);
    n = y(2);
    m = y(3);
    h = y(4);
    
    gNa = gbarNa*m^3*h;
    gK = gbarK*n^4;
    gL = gbarl;
    
    INa=gNa*(V-ENa);
    IK=gK*(V-EK);
    Il=gL*(V-El);
    
    dydt = [((1/Cm)*(I-(INa+IK+Il))); % Normal case
        alphan(V)*(1-n)-betan(V)*n;
        alpham(V)*(1-m)-betam(V)*m;
        alphah(V)*(1-h)-betah(V)*h];
end

I attach the other functions here.

Thank you!

Answer 1

[Steven Lord]

Solve the system with V = 0 up until the time when it should change. If you're not sure of the exact time when it should change, use an events function to determine when it should change. See the ballode example for how to use events functions with the ODE solvers.

Use the final result from the first call to the ODE solver to generate initial conditions for the second call to the ODE solver, which solves the ODE with V = 0.1 over the interval when it has that value.

Use the final result from the second call to the ODE solver to generate initial conditions for the third call to the ODE solver, which solves the ODE with V = 0 over the remaining interval.

[Samuele Bolotta]

Thank you!

What are the advantages of using this approach rather than the "ODE with Time-Dependent Terms" example that is mentioned in the ode45 documentation?

Best,

Samuele

Answer 2

[Jan]

ODE45 is designe to integrate smooth functions only. To change a parameter you have to integrate in chunks:

tSwitch1 = 10.0
tSwtich2 = 10.1;
t0 = 0;

...
I = 0.1;
[time1, V1] = ode45(@(t,y), @ODEMAT(t, y, I), [t0, tSwitch1], y0);

I  = 0.0;
y0 = V1(end, :);
[time2, V2] = ode45(@(t,y), @ODEMAT(t, y, I), [tSwitch1, tSwitch2], y0);

I  = 0.1;
y0 = V2(end, :);
[time3, V3] = ode45(@(t,y), @ODEMAT(t, y, I), [tSwitch1, tEnd], y0);

time = cat(1, time1, time2, time3);
V    = cat(1, V1, V2, V3);
...
 

function [dydt] = ODEMAT(t, y, I)
% Omit the definition of I inside this function.
...
end