PT  - JOURNAL ARTICLE
AU  - Marco S. Fabus
AU  - Mark W. Woolrich
AU  - Catherine W. Warnaby
AU  - Andrew J. Quinn
TI  - Understanding Harmonic Structures Through Instantaneous Frequency
AID  - 10.1101/2021.12.21.473676
DP  - 2021 Jan 01
TA  - bioRxiv
PG  - 2021.12.21.473676
4099  - http://biorxiv.org/content/early/2021/12/23/2021.12.21.473676.short
4100  - http://biorxiv.org/content/early/2021/12/23/2021.12.21.473676.full
AB  - 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 StatementThe authors have declared no competing interest.