Nonlinear fit instead of setting up a non linear equation?

Hello everyone,

I'll attach my script here, in case you want to run the simulation yourself.

My task is to find two values x and y so that if I multiply them by G_max_chl2 and G_max_glu2 (two scalar values that I pre-defined in previous parts of the code) respectively, this:

CPSC_2 = ((G_max_chl2 * x) .* (1 - exp(-t / tau_rise_In2)) .* (exp(-t / tau_decay_In2)) * (Vm - EChl2)) + ((G_max_glu2 * y) .* (1 - exp(-t / tau_rise_Ex2)) .* exp(-t / tau_decay_Ex2) * (Vm - EGlu2))

becomes the same as this:

CPSC_1 = ((G_max_chl) .* (1 - exp(-t / tau_rise_In)) .* (exp(-t / tau_decay_In)) * (Vm - EChl)) + (G_max_glu .* (1 - exp(-t / tau_rise_Ex)) .* exp(-t / tau_decay_Ex) * (Vm - EGlu))

1 All the variables above are the same for the two equations, except for the four "G_max" values. The only two variables that change are G_max_chl2 and G_max_glu2, so that they become the same as G_max_chl and G_max_glu.

2 Some of these variables are matrices, but in this case they are not involved and should remain the same (only the two scalars G_max_chl2 and G_max_glu2 should change)

I solved the task setting up a nonlinear equation:

% This is the difference that has to be minimized
Diff_CPSC  = abs((((G_max_chl2) .* ((1 - exp(-t / tau_rise_In2)) .* ...
    (exp(-t / tau_decay_In2))) * (Vm - EChl2)) + ((G_max_glu2) .* ...
    ((1 - exp(-t / tau_rise_Ex2)) .* exp(-t / tau_decay_Ex2)) * (Vm - EGlu2))) - (((G_max_chl) .* ...
    ((1 - exp(-t / tau_rise_In)) .* exp(-t / tau_decay_In)) * ...
    (Vm - EChl)) + ((G_max_glu) .* ((1 - exp(-t / tau_rise_Ex)) .* exp(-t / tau_decay_Ex)) * (Vm - EGlu))));

% EQUATION
% Create a nonlinear equation
x = optimvar('x',1);
y = optimvar('y',1);

% Define function to minimize
eq1 = (((G_max_chl2 * x) .* ((1 - exp(-t / tau_rise_In2)) .* ...
    (exp(-t / tau_decay_In2))) * (Vm - EChl2)) + ((G_max_glu2 * y) .* ...
    ((1 - exp(-t / tau_rise_Ex2)) .* exp(-t / tau_decay_Ex2))* ...
    (Vm - EGlu2))) == ((G_max_chl) .* ((1 - exp(-t / tau_rise_In)) .* ...
    exp(-t / tau_decay_In)) * (Vm - EChl)) + ((G_max_glu) .* ...
    ((1 - exp(-t / tau_rise_Ex)) .* exp(-t / tau_decay_Ex)) * (Vm - EGlu));

% Create an equation problem, and place the equation in the problem
prob = eqnproblem;
prob.Equations.eq1 = eq1;

% Show the problem
show(prob);

% Specify the initial point as a structure
x0.x = G_max_chl / G_max_chl2;
x0.y = G_max_glu / G_max_glu2;
[sol,fval,exitflag] = solve(prob,x0);

% View the solution point and convert to double
disp(sol.x);
disp(sol.y);
x = sol.x;
y = sol.y;

However, I find this procedure a bit laborious because, if I wanted to optimize other variables, I'd have to hard code the unknown variables everytime. My question is:

• This procedure forces me to know all of the values of CPSC_2, which in this case is true because I generated the simulation. However, if it were coming from experimental recordings, it wouldn't be the case anymore. Is there a way to fit a CPSC_2 (whose variables are unknown) to CPSC_1 (whose variables are known)?

Answer 1

[Matt J]

Your problem appears to be a linear fit. I see no non-linear dependence on x or y. You should therefore use lsqlin, if you also have bounds or linear constraints.

https://www.mathworks.com/help/optim/ug/lsqlin.html

Otherwise, you could just use mldivide() or linsolve().

[Samuele Bolotta]

Hi Matt, thank you very much for your help. This is exactly what I was looking for!

[Samuele Bolotta]

Hi Matt, sorry to bother you again. By using lsqlin, I end up:

a) Having more flexibility, because I don't have to specify to which parameters the unknown variables x or y or z should be coupled.

b) However, I lose information because I simply make matrix CPSC2 equal to CPSC, without being able to specify a cost function that I want to minimize (the difference between CPSC2 and CPSC) and without being able to trace back which parameters changed.

My supervisor says I should fit the matrix CPSC2 (whose underlying distribution is unknown, when it will come from experimental recordings) to the function that created CPSC:

((G_max_chl) .* (1 - exp(-t / tau_rise_In)) .* (exp(-t / tau_decay_In)) * (Vm - EChl)) + (G_max_glu .* (1 - exp(-t / tau_rise_Ex)) .* exp(-t / tau_decay_Ex) * (Vm - EGlu))

Does this (https://it.mathworks.com/help/curvefit/fit.html) seem a good candidate for that?

Thanks!

[Matt J]

You could use fmincon if you want more control over the cost function,

https://www.mathworks.com/help/optim/ug/fmincon.html