Rescaling an exponential growth to model a brain activation

[servanm]

I'd like to model some brain activity as an exponential growth. Specifically, I'd like to rescale the exponential grow to control (i) the maximal amplitude of the activation, (ii) the rate of growth and (iii) the time at which the activation starts. Any idea?

 

[stapel]

I'd like to model some brain activity as an exponential growth. Specifically, I'd like to rescale the exponential grow to control (i) the maximal amplitude of the activation, (ii) the rate of growth and (iii) the time at which the activation starts. Any idea?

Um... Use statistical software (such as on a graphing calculator or a spreadsheet, or else specialized packages) to find the regression model that suits your needs...?

What are you asking for, specifically? Thank you!

 

[servanm]

Um... Use statistical software (such as on a graphing calculator or a spreadsheet, or else specialized packages) to find the regression model that suits your needs...?

What are you asking for, specifically? Thank you!

I don't think a software is necessary to do that. Here is what I've done so far to model he exponential growth:
M(t) = A[1-exp(-w1(t-T1))]

here M(t) is the activation time at time t, A the maximum activation, w1 the exponential rate constant, and T1 the time at which the activation starts. However, there seems to be something wrong in the equation (the parameters do not correspond to what they are suppose to do). Any idea?

 

[stapel]

I don't think a software is necessary to do that. Here is what I've done so far to model he exponential growth:
M(t) = A[1-exp(-w1(t-T1))]

here M(t) is the activation time at time t, A the maximum activation, w1 the exponential rate constant, and T1 the time at which the activation starts. However, there seems to be something wrong in the equation (the parameters do not correspond to what they are suppose to do). Any idea?

Since we have no idea what the inputs are, what the (desired? expected? required?) outputs are, or how the equation is "failing", I'm afraid I see no way to advise. Sorry.

 

[Ishuda]

I don't think a software is necessary to do that. Here is what I've done so far to model he exponential growth:
M(t) = A[1-exp(-w1(t-T1))]

here M(t) is the activation time at time t, A the maximum activation, w1 the exponential rate constant, and T1 the time at which the activation starts. However, there seems to be something wrong in the equation (the parameters do not correspond to what they are suppose to do). Any idea?

M(t) = activity at time t reaching a maximum in exponential time, i.e. a convex curve
T1 = Start time
B = Starting activity
A = Maximum activity
w1 = growth factor > 0

M(t) = (A-B) [1 - e^{-w1 (t-T1)}] + B

What else do you want? Do you want the typical concave curve of exponential growth, i.e.
M(t) ~ e^{a t}
for a greater than zero. If so you might want a hybrid curve where you have the concave exponential starting at tie T1 until some time T2 where it turns into the convex type of curve. That is maybe something modeled more on a hyperbolic tangent type curve, see
http://mathworld.wolfram.com/HyperbolicTangent.html

 

[servanm]

M(t) = activity at time t reaching a maximum in exponential time, i.e. a convex curve
T1 = Start time
B = Starting activity
A = Maximum activity
w1 = growth factor > 0

M(t) = (A-B) [1 - e^{-w1 (t-T1)}] + B

What else do you want? Do you want the typical concave curve of exponential growth, i.e.
M(t) ~ e^{a t}
for a greater than zero. If so you might want a hybrid curve where you have the concave exponential starting at tie T1 until some time T2 where it turns into the convex type of curve. That is maybe something modeled more on a hyperbolic tangent type curve, see
http://mathworld.wolfram.com/HyperbolicTangent.html

Thanks Ishuda, this is perfect.
As I mentioned, I would like to model some brain activity that shows an initial activation followed by a decay. Ideally, the parameters of the chosen function should index:
(i)start time of the activation
(ii) rate of growth
(ii) max activation
(iii)start time of decay
(iv) rate of decay

My idea was thus to model the initial activation as an exponential growth, and the decay as an exponential decay. Feel free to propose a better or simpler function.

 

[servanm]

Thanks Ishuda, this is perfect.
As I mentioned, I would like to model some brain activity that shows an initial activation followed by a decay. Ideally, the parameters of the chosen function should index:
(i) start time of the activation
(ii) rate of growth
(ii) max activation
(iii)start time of decay
(iv) rate of decay

My idea was thus to model the initial activation as an exponential growth, and the decay as an exponential decay. Feel free to propose a better or simpler function.

 

[Ishuda]

Thanks Ishuda, this is perfect.
As I mentioned, I would like to model some brain activity that shows an initial activation followed by a decay. Ideally, the parameters of the chosen function should index:
(i) start time of the activation
(ii) rate of growth
(ii) max activation
(iii)start time of decay
(iv) rate of decay

My idea was thus to model the initial activation as an exponential growth, and the decay as an exponential decay. Feel free to propose a better or simpler function.

It looks like a function of the sort
M(t) = \(\displaystyle A\, +\, \dfrac{ B\, +\, C\, e^{-r_1\, (t-t_1)}}{D\, +\, E\, e^{-r_2\, (t-t_2)}}\)
might be what you are looking for.

The function could be written in a different form so that some of the parameters disappear, i.e. let C1 be such that
C1 = C e^{-r₁(t₂-t₁)}
start counting time at t = t₁
and in a similar fashion absorb the A and re-write
M(t) = \(\displaystyle \dfrac{ A\, +\, B\, e^{-r_1\, t}}{C\, +\, D\, e^{-r_2\, (t-t_2)}}\)
We would then have t₂ as some sort of transition time.

 

[servanm]

It looks like a function of the sort
M(t) = \(\displaystyle A\, +\, \dfrac{ B\, +\, C\, e^{-r_1\, (t-t_1)}}{D\, +\, E\, e^{-r_2\, (t-t_2)}}\)
might be what you are looking for.

The function could be written in a different form so that some of the parameters disappear, i.e. let C1 be such that
C1 = C e^{-r₁(t₂-t₁)}
start counting time at t = t₁
and in a similar fashion absorb the A and re-write
M(t) = \(\displaystyle \dfrac{ A\, +\, B\, e^{-r_1\, t}}{C\, +\, D\, e^{-r_2\, (t-t_2)}}\)
We would then have t₂ as some sort of transition time.

Thanks a lot for your help, I really appreciate. Could you precise what your parameters A,B, C, D represent? It appears that I don't need to model the starting activity (B in your parameterization), so we might save one more parameter.