Mathematical Relations Between Measures of Brain Connectivity Estimated From Electrophysiological Recordings for Gaussian Distributed Data

A large variety of methods exist to estimate brain coupling in the frequency domain from electrophysiological data measured, e.g., by EEG and MEG. Those data are to reasonable approximation, though certainly not perfectly, Gaussian distributed. This work is based on the well-known fact that for Gaussian distributed data, the cross-spectrum completely determines all statistical properties. In particular, for an infinite number of data, all normalized coupling measures at a given frequency are a function of complex coherency. However, it is largely unknown what the functional relations are. We here present those functional relations for six different measures: the weighted phase lag index, the phase lag index, the absolute value and imaginary part of the phase locking value (PLV), power envelope correlation, and power envelope correlation with correction for artifacts of volume conduction. With the exception of PLV, the final results are simple closed form formulas. In an excursion we also discuss differences between short time Fourier transformation and Hilbert transformation for estimations in the frequency domain. We tested in simulations of linear and non-linear dynamical systems and for empirical resting state EEG on sensor level to what extent a model, namely the respective function of coherency, can explain the observed couplings. For empirical data we found that for measures of phase-phase coupling deviations from the model are in general minor, while power envelope correlations systematically deviate from the model for all frequencies. For power envelope correlation with correction for artifacts of volume conduction the model cannot explain the observed couplings at all. We also analyzed power envelope correlation as a function of time and frequency in an event related experiment using a stroop reaction task and found significant event related deviations mostly in the alpha range.


General remarks
Here we derive analytic results for the relation between linear and nonlinear coupling measures assuming Gaussian distributed data. The order of the measures is determined by mathematical complexity and differs from the order in the main paper. All considered measures are bivariate, and, being normalized quantities, also independent of global scales. Without loss of generality we can therefore set the cross-spectrum to be where c is the complex coherency. The stochastic variable is the vector consisting of two complex numbers z = (z 1 , z 2 ) T and for the following analysis we assume a Gaussian distribution with probability density function: Coupling measures are in general constructed from expected values of functions g(z) < g(z) >= Dz 1 Dz 2 g(z)p(z) In generic form, Dz denotes the infinitesimal element for the integration over a complex plain. For all our analysis we use spherical coordinates, z = r exp(iΦ), and then Dz = rdrdΦ. The integrals to be evaluated below will have two different forms. First, integrals of phases will always be of the form where δ n,m denotes the Kronecker delta function. Second, integrals over amplitudes r will always be integrals of a polynomial multiplied with a Gaussian function. Those integrals are standard and read for integer n and k ∞ 0 drr n exp(−ar 2 ) = k! 2a k+1 for n = 2k + 1 and ∞ 0 drr n exp(−ar 2 ) = (2k − 1)!! 2 k+1 a k π a for n = 2k (6) where (2k − 1)!! denotes the product of all odd integers from 1 to 2k − 1.

PEC without suppression of mixing artifacts
Power envelope correlation (PEC) between two complex variables z 1 and z 2 is defined as the usual correlation calculated for the powers |z 1 | 2 and |z 1 | 2 All expected values are to be evaluated for low order polynomials of z 1 and z 2 . This can be solved in closed form using a coordinate transformation. Also, power-power correlation is independent of the phase of coherency and we can therefore assume without loss of generality that c is real valued. Let U is real-valued, symmetric and orthogonal, i.e. U = U T = U † = U −1 . This U diagonalizes S and S −1 , specifically We now define new coordinates as Since, det(U ) = 1 we have for the infinitesimal elements Dx 1 Dx 2 = Dz 1 Dz 2 . The exponent of the exponential function reads apart from the overall sign Also, the polynomials as functions of z 1 and z 2 occurring in Eq.7 can be expressed as polynomials of the new coordinates, e.g.
Note, that in the first two terms the dependence on the phases drops out, which is not the case for the second two terms. These second two terms do not contribute to the expected value because these contributions vanish after integration with respect to the phases according to Eq.4. The expected values of the first two terms can be directly evaluated: These terms can be combined to give All other terms do not depend on c and we just present the results Inserting all these results into Eq.7 leads for real valued c to and in general, since PEC does not depend on the phase of c, to PEC with suppression of mixing artifacts PEC with suppression of mixing artifacts depends on phase differences, and the simplification to assume without loss of generality real valued coherency is not possible. However, the strategy to solve this problem is the same as in the previous section. We will present here only the main principles. We want to calculate the correlation of |z 1 | 2 and |z 2 − c R z 1 | 2 , i.e.
for coherency c = c R + ic I = |c| exp(iΦ). Now, let U is unitary, i.e. U † = U −1 , and it diagonalizes S −1 Using new coordinates x = U † z we get for the exponent in the probability density apart from the overall sign As before, all functions of z are now expressed as functions of x according to z 1 = (x 1 + x 2 )/ √ 2 and z 2 = exp(−iΦ)(x 1 − x 2 )/ √ 2. All terms are then products of functions of either x 1 or x 2 . Integrals over phases vanish unless the respective term is not phase dependent, and the integrals over the radial variables reduce to one-dimensional integrals which can all be calculated with Eq.5. These calculations are still very tedious and we here report only the results for the individual terms. In addition to Eq.16 and Eq.17 we get Inserting these results into Eq.20 we arrive at the final solution

Phase Lag Index
We first recall that PLI is defined as The calculation for PLI is similar to the one for PEC, but the integrals are slightly more complicated. We recall that PLI is invariant to real valued linear transformations. This can be exploited to consider the case of purely imaginary coherence. If S and U are given as in Eq.1 and Eq.8, respectively, then with c = c R + ic I we get Scaling to unit diagonal elements with Since PLI is invariant to the above transformations, we observe that it must be a function of lagged coherencec I . To calculate the functional form we need another transform which actually diagonalizes S. This can be achieved with for which det(T ) = 1 holds. Denoting the original coordinates (corresponding to covariance matrixŜ) as z, and defining new coordinates as x = T † z we get with r i = |x i |, and Note that phase dependencies disappear in the new coordinates and in the following we only sketch the major steps of the rather tedious but straight forward integrations along radial coordinates:

Phase locking value
The complex phase locking value (PLV) is defined as In contrast to power-power correlation, this coupling measure cannot be expressed in terms of expected values of low order polynomials of z 1 and z 2 . As a consequence, a coordinate transformation, as was done for PEC, is not useful here. For PLV we will therefore follow a different strategy. Since we here derive relations for the full complex PLV we cannot restrict ourselves to real valued c. We rewrite the exponential function using the abbreviation b = 1 − |c| 2 exp(−z † S −1 z) The second exponential function is expanded in a Taylor series The double sum over n and k will reduce to a single sum over n after integration with respect to the phases. We recall that z k = r k exp(iΦ k ). Apart from factors, which do not depend on phases, and including the factor z 1 z * 2 from the measure itself, we get for the integrals All remaining integrals are products of two one-dimensional integrals and can be evaluated with Eq.6 to give P LV = cf (|c|) with a correction factor with coefficients We recall that (2k − 1)!! denotes the product of all odd integers from 1 to 2k − 1 . The series expansion of Eq.42 converges poorly if |c| is close to 1. We therefore recommend an equivalent formulation as We also recommend that for the calculation of (2k − 1)!!/(k!2 k ) numerator and denominator should not be evaluated separately, but the whole ratio should rather be calculated as