Curve-fit with zero offset « Presented Without Proof

Date: February 8th, 2010
Cate: engineering, research
Tags: calcium, curve fitting, ionomycin, latex, pstricks

Curve-fit with zero offset

Say you are measuring some signal, which you expect to show an exponential decay, but you are unsure of the zero-point on your measuring device. If you want to measure the time constant of the decay you need to know what it’s decaying to. For a signal s,

$s=s_{0}+A_0\exp(-t/\tau)$

In order to do conventional least-squares fitting to $A_0\exp(-t/\tau)$ you need to determine $s_0$ .

Step response of NIH 3T3 fibroblasts to ionomycin

For instance, the figure above shows the fluorescent intensity of some 3T3 cells to ionomycin. The traces are normalized to the initial values. Notice that the cells don’t recover to the levels at which they began. I wanted to know the time constant of the decay on the tail of the trace.

I solved this problem by performing the fit inside an iterative minimum-finding function. First I identified the point of fastest decline and selected that as the $t=0$ point. Then I iterated to find the value for $y_0$ which minimized the norm of the error:

$|| (y - y_0) - A\exp(-t/\tau) ||$

for best-fit A and $\tau$ .

y0_fit = fminbnd( @(y0) ...
                 norm( y-y0 - exp( polyval( polyfit( x, log(y-y0), 1), x))) ,...
                 -10*y(1), y(1) );

This, along with other analysis:

For a population of cells (Showing mean and standard deviation for each test point. numbers in parentheses indicate number of cells in the sample.):

Presented Without Proof(an exercise for the reader)

Curve-fit with zero offset

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