  $Objective$: A characteristic of neurological signal processing is high
levels of noise from sub-cellular ion channels up to whole-brain processes. In
this paper, we propose a new model of electroencephalogram (EEG) background
periodograms, based on a family of functions which we call generalized van der
Ziel--McWhorter (GVZM) power spectral densities (PSDs). To the best of our
knowledge, the GVZM PSD function is the only EEG noise model which has
relatively few parameters, matches recorded EEG PSD's with high accuracy from 0
Hz to over 30 Hz, and has approximately $1/f^\theta$ behavior in the
mid-frequencies without infinities. $Methods$: We validate this model using
three approaches. First, we show how GVZM PSDs can arise in population of ion
channels in maximum entropy equilibrium. Second, we present a class of mixed
autoregressive models, which simulate brain background noise and whose
periodograms are asymptotic to the GVZM PSD. Third, we present two real-time
estimation algorithms for steady-state visual evoked potential (SSVEP)
frequencies, and analyze their performance statistically. $Results$: In
pairwise comparisons, the GVZM-based algorithms showed statistically
significant accuracy improvement over two well-known and widely-used SSVEP
estimators. $Conclusion$: The GVZM noise model can be a useful and reliable
technique for EEG signal processing. $Significance$: Understanding EEG noise is
essential for EEG-based neurology and applications such as real-time
brain-computer interfaces (BCIs), which must make accurate control decisions
from very short data epochs. The GVZM approach represents a successful new
paradigm for understanding and managing this neurological noise.
