I keep forgetting this and having to re-derive it, so I'm putting it up here. Exponential decay a very simple response function. To a first approximation, the response function of a synapse might be modeled as an exponential function. The exponential isn't perfect, however, since synapses don't react immediately. Sometimes we use a function of the form $t*exp(-t)$, which has a rising phase as well as a falling phase. This function is actually the result of convolving two exponential decay functions together. Generally, the family $t^n* xp(-t)$ captures the idea of having n feed-forward variables coupled by exponential decay $X_n'=X_{n-1}-X_n$. However, I keep forgetting the appropriate constants to convert between the system of differential equations and the family of response functions. I'm not going to derive that right now, but rather show a series of different parameterizations of the response functions so I can remember how the response changes as $n$ varies.

The integral of $t^n exp(-t)$ actually gets larger for larger $n$. To keep the integral 1, divide by $n!$.

The peak response of $t^n*exp(-t)$ arrives at time $t=n$. Rescale time with $t:=nt$ to get a peak response at $t=1$.

Rescaling time changes the integral again, but this is easily fixed by multiplying by $n$.

Notice how the curves are getting progressively more smooth for larger $n$. I wonder what this looks like on a log axis ?

So, this family of functions appears to tend toward a log-gaussian in the limit. I don't know if thats particularly meaningful.

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