[Neuroscience] Re: Decay constants. What does "weighted" mean?

Trevor Lewis via neur-sci%40net.bio.net (by t.lewis from unsw.edu.au)
Thu Jan 15 17:46:53 EST 2009

Dear Bill,

A little more information to add to what r norman posted:
As explained, for a response that can be fitted with the sum of two 
exponentials -
  f(t) = a exp(-t/t1) + b exp(-t/t2)

where t1 and t2 are the time constants and a and b are the initial 
amplitudes of each component of the response at time t=0.

In some instances, rather than just quote both of the time constants 
and their respective initial amplitudes, a single 'weighed' decay 
constant is reported. Some times this is because a comparison is 
being made between different synapses that may or may not have single 
exponential decays.

To calculate the 'weighted' mean time constant, the individual 
initial amplitudes for each time constant are expressed as a fraction 
of the total initial amplitude. Lets call these fractions A and B, 
such that A = a / (a+b)  and B = b / (a+b)
Lets then call the 'weighted' mean decay constant T, and this is given by:
T = A * t1 + B * t2

So you see, you get a indication of the exponential decay by 
'weighting' the individual time constants according to the initial 
amplitude, to get a mean time constant. This is reasonable, given 
that the initial amplitude and the time constant of each component 
will determine the proportional contribution to the exponential decay 
that is being described.

I hope this helps.


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