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i'am currently reading " theoretical neuroscience by dayan abbott" and it says

" If we ignore the brief duration of an action potential (about 1 ms), an action potential sequence can be characterized simply by a list of the times when spikes occurred. For n spikes, we denote these times by $ti$ with i = 1, 2,... , n. The trial during which the spikes are recorded is taken to start at time zero and end at time T, so 0 ≤ $ti$ ≤ T for all i. The spike sequence can also be represented as a sum of infinitesimally narrow, idealized spikes in the form of Dirac δ functions . "

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then it says we average ρ(t) s that we get during the trials .

" We use angle brackets, <> , to denote averages over trials that use the same stimulus, so that for any quantity z is the sum of the values of z obtained from many trials involving the same stimulus, divided by the number of trials. The trial-averaged neural response function is thus denoted by <ρ(t)> "

how can we average ρ(t) . it consists of many infinity long spikes . so summing them and then dividing them by number of trials . is just like summing them . and does not give us the sense of average spikes .

the book further explains :

" In any integral expression, the neural response function generates a contribution whenever a spike occurs. If instead, we use the trial-average response function this generates contributions proportional to the fraction of trials on which a spike occurred. "

the materials I'm referring to are from pages 7 and 8 .

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ρ(t) in Equation 1.1 cannot be summed and averaged because the idealized spikes are infinitesimally narrow and the chance of two spikes from different trials occurring at the same time t is zero.

In Equation 1.2, the idea of convolving each idealized spike with a "well behaved function" h(t). If you use a rectangular function around zero as h(t), with a reasonable width (say 1ms), then Equation 1.2 will give you a timeseries that can be averaged. This is because the chances of two nonzero values from different trials occurring at the same t will become a more reasonable (nonzero) value.

If you have enough trials, then your trial-averaged neural response function using this more reasonable representation of spikes will give you a good sense of the average spike density.

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