NetStim Fractional Randomness: Mathematical Notation?

[agmccrei]

Hello,

I don't have a strong formal background in mathematics here, but I want to describe some of the mechanisms used in NEURON. Specifically, I am not sure if I have the correct mathematical notation for describing how fractional randomness is implemented in NEURON's NetStim function. So far, I have deduced the following from a previous discussion on the forum (see viewtopic.php?f=8&t=1839):


Any corrections or relevant references to this would be most helpful.

-Alex GM

[ted]

Everything depends on the meaning of words and notation. In plain english, what does your notation mean?

[agmccrei]

Right, so my understanding of how fractional randomness is implemented is that for each interspike interval, the interval is dependent on the values set for s.interval and s.noise (assuming s = new NetStim(x)). Specifically, the magnitude of the value of s.noise (from 0 to 1) will control the proportion by which the interval is dependent on s.interval vs random values sampled from a negative exponential distribution. For example if s.noise = 0.2, the actual interval will be 0.8*s.interval + a random duration sampled from a negative exponential distribution with a mean duration of 0.2*s.interval. I am mostly unsure about how this negative exponential distribution (i.e. X) is represented mathematically. Honestly, from my limited understanding I would've written it as follows:

If given a PDF negative exponential distribution according to the following: lambda*exp(-lambda*x), and 1/lambda = the mean of the distribution, then lambda would be equal to (1/s.interval*s.noise). This would give the following:

X ~ lambda*exp(-lambda*x)
X ~ (1/s.interval*noise) * exp( -x / (s.interval*s.noise) )

In this case, x would represent the range of durations that the exponential distribution covers, which can incorporate the proportion of the interval (i.e. (1-s.noise)*s.interval) according to the following:

X ~ (1/s.interval*noise) * exp( -(t - (1-s.noise)*s.interval)) / (s.interval*s.noise) )

Which ensures that x (i.e. t - (1-s.noise)*s.interval) ) must be larger than 0 in order to get event probabilities greater than zero (i.e. negative duration probabilities would be impossible), such that the total time elapsed, t, before a new event occurs is greater than (1-s.noise)*s.interval). Actually, going through this, again, I am thinking that the notation I had posted previously would be incorrect, because the PDF and the equation preceding it sort of incorporate the (1-s.noise)*s.interval twice, which I think could force the actual interval to be up to twice its set value. My corrected notation would be one of the following:



... or this (which I think would mean the same thing (?)):


Thank you for your time,

Alex GM

[ted]

Simple summary that correctly expresses the effect of NetStim's noise parameter:
If noise == 0, the interspike interval (ISI) is constant and equals s.interval .
If 0<noise<=1 then
mean ISI is s.interval
minimum ISI is (1-noise)*s.interval
maximum ISI is infinite
and the value of any particular ISI is calculated as the sum of two terms
a + b
where a = (1-noise)*s.interval, i.e. the minimum ISI,
and b = a value drawn from the negative exponential distribution that has a mean of noise*s.interval .
This last "sum of two terms" bit ensures that the expected (i.e. mean) ISI equals
(1-noise)*s.interval + noise*s.interval = s.interval

WRT mathematical notation: f is generally used to signify a probability density function, and X is used to signify a random variable. Your notes use X as if it were the pdf; maybe that's where the confusion is coming from.