Help With Non-Linear Regression on a Log Scale

Hi. Wondering if I could get some help with a non-linear regression I am doing. I am an engineer doing what's called an electrochemical experiment where I make a step in voltage and then record a spike and decay in Current. This fits to a rather complex circuit equation where I am fitting the current resonse to the voltage step with two parameters (a time constant and what's called a constant phase element). Anyway I can fit the data and get a regression coefficiant of like .95 or something.

Here's the deal: this spike in current happens really quickly so that I have only a few data points at the beginning of the spike and through the decay. I have many data points at the end of the decay when the system settles out. So if I plot the time on a log scale it looks like the regression is fitting all that data at the end and not the good data at the spike and decay I want at the beggining. So it's not fitting what I really want, even though the regression coeeficiant is pretty good.

So how can I fix this? Can I some how wieght the first part of the data to have more impact on the regression? Can I somehow filter or transform the data to get rid of some of the end points in the data while keeping the important first few time points?