# Motivation#

## What is the bispectrum?#

The bispectrum is a higher-order statistic, based on the Fourier transform of the third order moment [1]. Two forms of computing the bispectrum exist: the direct approach, in which the Fourier coefficients of the data are computed, which in turn are used to compute the bispectum; or the indirect approach, in which the third order moments of the data are computed first before the Fourier transform is taken. PyBispectra uses the direct method. The bispectrum, \(\textbf{B}\), has the form

\(\textbf{B}_{kmn}(f_1,f_2)=<\textbf{k}(f_1)\textbf{m}(f_2)\textbf{n}^*(f_2+f_1)>\) ,

where \(kmn\) is a combination of signals with Fourier coefficients \(\textbf{k}\), \(\textbf{m}\), and \(\textbf{n}\), respectively; \(f_1\) and \(f_2\) correspond to a lower and higher frequency, respectively; and \(<>\) represents the average value over epochs.

A normalised form of the bispectrum also exists, termed bicoherence. Several forms of normalisation exist, however a common form is the threenorm: a univariate normalisation whereby the values of the bicoherence will be bound in the range \([-1, 1]\) in a manner that is independent of the coupling properties within or between signals [2]. The threenorm, \(\textbf{N}\), has the form

\(\textbf{N}_{kmn}(f_1,f_2)=(<|\textbf{k}(f_1)|^3><|\textbf{m}(f_2)|^3><|\textbf{n}(f_2+f_1)|^3>)^{\frac{1}{3}}\) .

The bicoherence, \(\boldsymbol{\mathcal{B}}\), is then computed as

\(\boldsymbol{\mathcal{B}}_{kmn}(f_1,f_2)=\Large\frac{\textbf{B}_{kmn}(f_1,f_2)}{\textbf{N}_{kmn}(f_1,f_2)}\) .

There are several possible uses of the bispectrum and bicoherence for signal analyses, including for phase-amplitude coupling (a form of cross-frequency coupling), the analysis of non-sinusoidal waveform features, and time delay estimation.

## Why analyse cross-frequency coupling, waveshape, and time delays?#

Cross-frequency coupling, waveshape analysis, and time delay estimation are relevant in a range of disciplines.

Cross-frequency coupling methods allow us to analyse interactions within and across signals between a lower frequency, \(f_1\), and a higher frequency, \(f_2\). Different forms of coupling exist, such as phase-phase coupling, amplitude-amplitude coupling, and phase-amplitude coupling. In phase-amplitude coupling, we examine the relationship between the phase of a signal at \(f_1\) and the amplitude of a signal at \(f_2\). Cross-frequency interactions have been posited as fundamental aspects of neuronal communication in the brain [3], with alterations in these relationships implicated in diseases such as Parkinson’s [4] and Alzheimer’s [5].

Additionally, a signal’s shape can contain information of interest. For example, non-sinusoidal features of signals may reflect particular forms of interneuronal communication [6], and have been shown to be correlated with symptoms of neurological diseases and altered by their treatments [7].

Finally, time delays, \(\tau\), between signals can also provide useful insights into systems. Such estimates are crucial for radar and sonar technologies [8], but also in neuroscience, where time delays can be used to infer features of the physical relationships between interacting brain regions [9].

## Why use the bispectrum for these analyses?#

The bispectrum offers several advantages over other methods for analysing phase-amplitude coupling, waveform shape, and time delay estimates.

For phase-amplitude coupling, common methods such as the modulation index can be practically challenging, requiring a precise set of filters to be applied to the data to extract the true underlying interactions (which are not readily apparent) as well as being computationally expensive (due to the requirement of Hilbert transforming the data) [10]. Furthermore, when analysing coupling between separate signals, the modulation index performs poorly at distinguishing genuine across-site coupling from within-site coupling in the presence of source mixing [11]. The bispectrum overcomes these issues, being computationally cheaper, lacking the need to precisely filter the data [10], and being more robust to spurious across-site coupling estimates [12].

Additionally, analyses of waveshape often rely on searching through the time-series signal [7], a computationally expensive procedure when handling long periods of high sampling-rate data. Furthermore, if information at particular frequencies is desired, the time-series must be bandpass filtered, distorting the shape of non-sinusoidal aspects of the underlying signal [13]. With the bispectrum, non-sinudoisal waveshape information can be extracted in a computationally cheap, frequency-resolved manner, without the need to bandpass filter.

Finally, traditional forms of time delay estimation often rely on cross-correlation. This method is perfectly adequate in noiseless situations or those where the noise of the signals are uncorrelated with one another as well as with the sources of interest [14][15]. This, however, is often not the case in many real-world contexts, leading to spurious time delay estimates. In contrast, the bispectrum is able to suppress the contribution of Gaussian noise sources to time delay estimates [14], and additional steps can be taken to minimise the effects of non-Gaussian noise sources, such as those associated with volume conduction [15].

## What is available in PyBispectra?#

PyBispectra offers tools for computing phase-amplitude coupling, time delay estimation, and waveshape feature analysis using the bispectrum and bicoherence. Additional tools are included for computing phase-phase coupling, amplitude-amplitude coupling, Fourier coefficients, time-frequency representations of data, spatio-spectral filters, as well as tools plotting results.

You can find the installation instructions here, as well as examples of how the package can be used here.