# what is the “trial-averaged neural response function”?

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 . "

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 .