Cross-bispectrum decomposition

In this repository you can find code for the decomposition method (bsfit.m), the statistical test (bsfit_stats.m), simulations (simulations/) and data analysis. The method itself has been developed by Guido Nolte and Stefan Haufe and is yet unpublished.

You can find the pipeline that was used for the simulation experiments in simulations/main_sim_pac.m. The main function for the real data analysis is main.m. A minimal demo script that shows the decomposition on simulated data without any complex analyses can be found in main_demo.m. Running code in this repository requires the installation of EEGLAB.

💡 If you have any questions about the code or the project, please reach out to [email protected], I am happy to answer your questions!

Problem formulation

The aim of this method is to identify $N$ brain source interactions from given estimates of the cross-bispectrum between $M$ sensors, with $N < M$. Compared to the biPISA approach Chella et al., 2016, which decomposes the antisymmetrized sensor cross-bispectrum into a set of pairwise interacting subsystems, this method is not restricted to pairwise source interactions but can be used to evaluate interactions between $n$ sources.

In its most general form, the sensor cross-bispectrum is defined over three channels $i, j, k$ and frequencies $f_1$ and $f_2$ as

$$ B_{ijk}(f_1, f_2) = \frac{1}{N_e} \ \sum_{e=1}^{N_e} x_{i,e}(f_1) \ x_{j,e}(f_2) \ x_{k,e}^{*}(f_1+f_2), $$

where $x_{i,e}(f)$ denotes the Fourier transform of the $e^{\text{th}}$ data epoch at frequency $f$ in channel $i$ and $^*$ denotes the complex conjugation. We make the usual assumption that the observed sensor signals $x_i(f)$ result from a linear superposition of the underlying source signals $s_m(f)$, which reads

$$ x_i(f) = \sum_{m=1}^{N} a_{im} \ s_m(f). $$

The sensor-level cross-bispectrum can therefore expressed as

$$ B_{ijk}(f_1, f_2) = \sum_{l,m,n}^N \ a_{il} \ a_{jm} \ a_{kn} \ D_{lmn}(f_1, f_2), $$

with

$$ D_{lmn}(f_1, f_2) = \frac{1}{N_e} \ \sum_{e=1}^{N_e} s_{l,e}(f_1) \ s_{m,e}(f_2) \ s_{n,e}^*(f_1+f_2). $$

$D_{lmn}(f_1, f_2)$ can be interpreted as the source-level cross-bispectrum. The aim of the method is to identify subsystems of interacting brain sources from estimates of the sensor cross-bispectrum. This implies finding a set of coefficients $a_{im}$ and a source cross-bispectrum $D_{lmn}(f_1, f_2)$ that together approximate the \textit{sensor} cross-bispectrum. The optimzation problem can therefore be expressed as

$$ \{ \boldsymbol{\hat{A}}, \hat{D}_{lmn} \} = argmin_{\{ \boldsymbol{A}, D_{lmn} \}} \ \frac{1}{| B_{ijk} |} \left|B_{ijk} - \sum_{l,m,n}^{N} a_{il} \ a_{jm} \ a_{kn} \ D_{lmn} \right| $$

where $\boldsymbol{A}$ is a $M \times N$ matrix that pulls together the individual coefficients $a_{im}$ for channels $M$ and sources $N$, where $N$ is smaller than $M$. The frequency arguments are omitted for ease of reading.