Physiological and harmonic components in neural and muscular coherence in Parkinsonian tremor Shouyan Wang, Tipu Z. Aziz, John F. Stein, Peter G. Bain, Xuguang Liu Clinical Neurophysiology 117 (2006) 1487–1498

Take-Home message 1- Coherence doesn’t tell us anything about the magnitude or power of the signal. 2- Cross correlation is affected by both coherence and the magnitude of the signal. 3- Coherence is very sensitive to noise. Comments: 1- Fundamentals of Statistics 2- Important piece of knowledge not known in clinical EEG studies (disadvantage of SCAN and BESA) 3- Although shown here using an example of harmonics, but also applicable to other situations

Introduction Harmonics, non-sinusoidal wave: concatenation of a non-sinusoidal pattern

Introduction Harmonics, sinusoidal wave: concatenation of a leaking wave

Introduction Source of harmonics in power spectrum of the field potentials recorded from a group of oscillating cells: 1- a single group of cells oscillating in a non-sinusoidal pattern 2- several cell groups oscillating in different harmonic frequencies

Introduction Timmerman et al 2003: Cerebro-cerebral coherence mainly at harmonic frequency, but cerebro-muscular coherence at fundamental frequency. This article: 1- A large harmonic in coherence specrtum (Magnitude Squared Coherence: MSC) doesn’t mean a large peak in power spectrum. 2- Decreased coherence in original tremor frequency can be the effect of noise: The higher amplitude of ongoing brain oscillations near tremor frequencies.

Methods Signal x = Local Field Potentials (LFP) recorded from subthalamic nucleus in brain Signal y = EMG recorded from a Parkinsonian tremulous muscle Both signals are divided into n segments Therefore for each single frequency bin we have n data point pertaining to the magnitude of that frequency Therefore the magnitude of each frequency bin can be represented as a vector in an n dimensional time space magnitude squared coherence: MSC magnitude cross-spectrum

Methods Technical point: They are using the complex value (a + ib) that is the output of DFT as the X and Y: Therefore not only the congruence of changes in the magnitude but also the congruency of changes in the phase of the two signals is taken into account, considering the fact that the result of multiplication in complex conjugate would lay at an angle equal to the phase difference of the two vectors. denotes complex conjugate

Results (simulation) In the case one of the signals have large harmonics but the other one very little harmonics: - still considerable coherence may be obtained that is trivial or even misleading for biological interpretation - not much of a large inner product (cross-spectrum) that is congruent with its biological implications Harmonic addedNoise added

Results (simulation) The size of the harmonic peak in magnitude cross spectrum is congruent with the size (and therefore biological importance) of the harmonic peak in power spectrum, but this is not the case in coherence spectrum. Not a surprise considering the nature of the coherence that is normalized to the sizes of vectors! Noise addedHarmonic added

Methods The effect of noise: m= noise added to x n= noise added to y x’= x + m y’= y + n P= inner prodect n and m are not correlated to each other (I don’t agree, they may have some random correlation) and are not correlated to x and y, (i.e., they are orthogonal to each other and their inner product is zero) therefore some of the Ps in this formula is equal to zero. Importance of having a high dimensional time-space (that is a lot of data segments along time) is revealed here. Because in a low-dimensional space you can’t have this many vectors all orthogonal to each other. Hint: the inner product of a vector in itself is its size squared. x’. y’ = (x + m) (y + n) = xy + xn + ym + mn

Methods Therefore, the more the noise, the less the coherence. M and N: noise to signal ratios in x’ and y’

Results (simulation) The more the noise, the less the coherence

Results (simulation) The more the noise, the less the coherence Noise added Noise addedNoise added

Discussion - Presence of large harmonics in coherence spectrum does not literally mean that such harmonics are also prominent in the original signals that the coherence is calculated between them. - the non-existence of coherence in a harmonic or in a fundamental component does not mean that there is not much of that component present in the original signals, but it may simply be because of high noise to signal ratio, an important example of which is: the 4-6 Hz tremor related cortical activity is accompanied by the high amplitude Theta rhythm of brain. Therefore in a work like that of Timmerman in which Cerebro-cerebral coherence is reported mainly at harmonic frequency and not in fundamental tremor frequency, the results should be corrected for noise to signal ratio.

Results (biological) Simulation results is confirmed.

Results (biological) Simulation results is confirmed.

Results (biological) Simulation results is confirmed.

Results (biological) Simulation results is confirmed.

Results (biological) Simulation results is confirmed.

Results (biological) Simulation results is confirmed.