Editing Event-related potential (section)

==Calculation==

ERPs can be [[Reliability (statistics)|reliably]] measured using [[electroencephalograph]]y (EEG), a procedure that measures [[electricity|electrical]] activity of the brain over time using [[electrode]]s placed on the [[scalp]]. The EEG reflects thousands of simultaneously [[Ongoing brain activity|ongoing brain processes]]. This means that the brain response to a single stimulus or event of interest is not usually visible in the EEG recording of a single trial. To see the brain's response to a stimulus, the experimenter must conduct many trials and average the results together, causing random brain activity to be averaged out and the relevant waveform to remain, called the ERP.<ref>{{cite book | vauthors = Coles MG, Rugg MD |year=1995 |chapter=Event-related brain potentials: An introduction | veditors = Rugg MD, Coles MG |series=Oxford psychology series, No. 25 |title=Electrophysiology of mind: Event-related brain potentials and cognition |pages=1–26 |location=New York |publisher=Oxford University Press }}</ref><ref>{{Cite journal |last1=Clayson |first1=Peter E. |last2=Baldwin |first2=Scott A. |last3=Larson |first3=Michael J. |date= 2013|title=How does noise affect amplitude and latency measurement of event-related potentials ( ERPs )? A methodological critique and simulation study |url=https://onlinelibrary.wiley.com/doi/10.1111/psyp.12001 |journal=Psychophysiology |language=en |volume=50 |issue=2 |pages=174–186 |doi=10.1111/psyp.12001 |pmid=23216521 |issn=0048-5772|url-access=subscription }}</ref>

The random ([[Neural oscillation#Ongoing activity|background]]) brain activity together with other bio-signals (e.g., [[Electrooculography|EOG]], [[Electromyography|EMG]], [[Electrocardiography|EKG]]) and electromagnetic interference (e.g., [[Noise (electronics)|line noise]], fluorescent lamps) constitute the noise contribution to the recorded ERP. This noise obscures the signal of interest, which is the sequence of underlying ERPs under study.
From an engineering point of view it is possible to define the [[signal-to-noise ratio]] (SNR) of the recorded ERPs. Averaging increases the SNR of the recorded ERPs making them discernible and allowing for their interpretation. This has a simple mathematical explanation provided that some simplifying assumptions are made. These assumptions are:
# The signal of interest is made of a sequence of event-locked ERPs with invariable latency and shape
# The noise can be approximated by a zero-mean [[Gaussian process|Gaussian random process]] of variance <math>\sigma^2</math> which is uncorrelated between trials and not time-locked to the event (this assumption can be easily violated, for example in the case of a subject doing little tongue movements while mentally counting the targets in an experiment).
Having defined <math>k</math>, the trial number, and <math>t</math>, the time elapsed after the <math>k</math><sup>th</sup> event, each recorded trial can be written as <math>x(t,k)=s(t)+n(t,k)</math> where <math>s(t)</math> is the signal and <math>n(t,k)</math> is the noise (Under the assumptions above, the signal does not depend on the specific trial while the noise does).

The average of <math>N</math> trials is
:<math>\bar x(t) = \frac{1}{N} \sum_{k=1}^N x(t,k) = s(t) + \frac{1}{N} \sum_{k=1}^N n(t,k)</math> .

The [[expected value]] of <math>\bar x(t)</math> is (as hoped) the signal itself, <math>\operatorname{E}[\bar x(t)] = s(t)</math>.

Its [[variance]] is
:<math>\operatorname{Var}[\bar x(t)] = \operatorname{E}\left[\left(\bar x(t) - \operatorname{E}[\bar x(t)]\right)^2\right] = \frac{1}{N^2} \operatorname{E}\left[\left(\sum_{k=1}^N n(t,k)\right)^2\right] = \frac{1}{N^2} \sum_{k=1}^N \operatorname{E}\left[n(t,k)^2\right] = \frac{\sigma^2}{N}</math>.
For this reason the noise amplitude of the average of <math>N</math> trials is expected to deviate from the mean (which is <math>s(t)</math>) by less or equal than <math>\sigma/{\sqrt{N}}</math> in 68% of the cases. In particular, the deviation wherein 68% of the noise amplitudes lie is <math>1/{\sqrt{N}}</math> times that of a single trial. A larger deviation of <math>2 \sigma/{\sqrt{N}}</math> can already be expected to encompass 95% of all noise amplitudes.

Wide amplitude noise (such as eye blinks or movement [[Artifact (error)|artifacts]]) are often several orders of magnitude larger than the underlying ERPs. Therefore, trials containing such artifacts should be removed before averaging. Artifact rejection can be performed manually by visual inspection or using an automated procedure based on predefined fixed thresholds (limiting the maximum EEG amplitude or slope) or on time-varying thresholds derived from the statistics of the set of trials.{{citation needed|date=February 2020}}