ABSTRACT
The analysis of harmonics and non-sinusoidal waveform shape in neurophysiological data is growing in importance. However, a precise definition of what constitutes a harmonic is lacking. In this paper, we propose a rigorous definition of when to consider signals to be in a harmonic relationship based on an integer frequency ratio, constant phase, and a well-defined joint instantaneous frequency. We show this definition is linked to extrema counting and Empirical Mode Decomposition (EMD). We explore the mathematics of our definition and link it to results from analytic number theory. This naturally leads to us to define two classes of harmonic structures, termed strong and weak, with different extrema behaviour. We validate our framework using both simulations and real data. Specifically, we look at the harmonics structure in the FitzHugh-Nagumo model and the non-sinusoidal hippocampal theta oscillation in rat local field potential data. We further discuss how our definition helps to address mode splitting in EMD. A clear understanding of when harmonics are present in signals will enable a deeper understanding of the functional and clinical roles of non-sinusoidal neural oscillations.
Competing Interest Statement
The authors have declared no competing interest.
Footnotes
katie.warnaby{at}ndcn.ox.ac.uk, mark.woolrich{at}ohba.ox.ac.uk, andrew.quinn{at}psych.ox.ac.uk
This research was funded in part by the Wellcome Trust [Grant numbers 203139/Z/16/Z, 106183/Z/14/Z, 215573/Z/19/Z]. For the purpose of open access, the authors have applied a CC-BY public copyright licence to any Author Accepted Manuscript version arising from this submission. Research was also supported by the NIHR Oxford Health Biomedical Research Centre, the New Therapeutics in Alzheimer’s Diseases (NTAD) study supported by UK MRC and the Dementia Platform UK (RG94383/RG89702), an EU European Training Network grant (euSSN; 860563), and the MRC Development Pathway Funding Scheme (award reference MR/R006423/1).