Neuro6.

Ok, we have N vesicles, and each one has a probability p of releasing its ACh across the membrane in response to the reception of an impulse at the synapse. Let m = Np, which is the mean (average) number of vesicles that will react to each impulse. If N is large, and p is small, then the second probability on the previous page can be approximated by the formula (replacing the number 4 by a general number k):

	[m^k e^-m]
P(any k yes, the other N-k no) = p_k =	________
	k!

This is the Poisson probability, and it is easy to calculate it on a calculator, and what's more the value of m can be fairly accurately determined by experiment, and the value of k is the number of MEPPs we measure with each trial. Notice that N and p do not appear explicitly in the expression for the probability.
There are a couple of ways we can calculate m. Since it is the mean number of MEPPs, if we perform a large number n of experiments (impulses) and compute the average number of MEPPs that arise from each trial, the result average will be m, or a close approximation to m.

Or... suppose we've done n trials. Let n_k be the number of those trials that gave rise to k MEPPs (so the sum of the n_k values is n). We expect, if n is large enough, that the n_k will be well approximated by the formula

n_k ~ np_k.

(This formula is fairly intuitive. If you roll a single die 120 times, since each number has a probability of 1/6 of coming up, you expect to have about 120(1/6) = 20 fives (or any of the other numbers). That is,
(the number of trials)x(probability of a given result)
= expected number of times to get that result.)
Therefore, since m⁰ = 1, and 0! = 1 (weird, but true), the value

n₀ ~ np₀ = ne^-m.

Therefore, e^-m ~ n₀/n, so e^m ~ n/n₀, implying

m ~ ln[n/n₀],

where ln is the natural logarithm (you'll see ln on any decent calculator). This gives us another way to approximate m: perform n experiments, count the number of times there is no response at all to the impulse (this will be n₀), then apply the formula above.

Now it's time to simulate this situation (actually it could already be simulated with the animation on page 4, but it would take a while to perform a large number of experiments).

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