# Giuseppe de Alteriis: EiDA - A lossless approach for dynamic functional connectivity; application to fMRI data of a model of ageing

Consequently, dynamic functional connectivity (dFC) has become of major interest to understand brain function in health and disease (Cohen, 2018; Hutchison et al., 2013). dFC is an extension of functional connectivity (FC) (Fingelkurts et al., 2005), which aims to quantify the connectivity between signals in a functional sense, that is, where the connections are not anatomically informed, but based on a measure of similarity of the signal across the entire acquisition period (Friston, 2011; Stephan & Friston, 2009). Where FC is “static,” in the sense that it computes connectivity measures representative of the entire recording (e.g., Pearson Correlation), dFC analyses the time evolution of connectivity patterns (Arbabyazd et al., 2020; Calhoun et al., 2014; Cohen, 2018; Hansen et al., 2015; Hutchison et al., 2013).

In pursuing this objective, any dFC approach faces at least two challenges. The first pertains to the determination of the appropriate size of the observation window that has to be sequentially applied to the time-series (Hindriks et al., 2016). The second problem revolves around dimensionality as we are confronted with the task of analysing the temporal evolution of an

matrix (where is the number of signals), which can present difficulties in terms of both analysis and lossless storage.

A commonly used solution for addressing the challenge of dimensionality is the so-called Leading Eigenvector Dynamic Analysis (LEiDA) (Cabral et al., 2017). In LEiDA, functional connectivity or phase coupling is calculated using the Hilbert transform. The Hilbert transform is a signal processing technique to convert a real signal into a complex valued analytical signal , with an instantaneous amplitude and an instantaneous phase . The phase of the analytical signal is then extracted (Glerean et al., 2012) to generate the instantaneous Phase Alignment Matrix () (also known as instantaneous Phase Locking) by computing the pairwise phase differences. This matrix is then decomposed into its orthogonal components, the eigenvectors, where the first or leading eigenvector is selected and the remaining discarded, thus reducing the dimensionality of the matrix from to 1 dimensions. Clustering is then performed on the time-series of the first eigenvector to identify distinct and reproducible spatio-temporal patterns, or “modes” (sometimes referred to as “brain states”) of phase-alignment that the brain consistently exhibits throughout the recording period. The use of LEiDA has made important contributions to various functional brain paradigms, yielding insights across diverse areas of research, for example the study of sleep-wake transitions (Deco et al., 2019), the action of psychedelic drugs (Lord et al., 2019; Olsen et al., 2022), neurodevelopment (França et al., 2022), schizophrenia (Hancock, Rosas, et al., 2022), and depression (Figueroa et al., 2019; Martínez et al., 2020).

Using LEiDA, a number of dynamic indices of connectivity have been proposed such as the average duration of a mode, or its fractional occurrence. In addition, within modes, it is possible to define measures capturing ideas drawn from complex systems theory such as metastability, a metric reflecting simultaneous tendencies for global integration and functional segregation (Cavanna et al., 2018; Deco et al., 2017; Friston, 1997; Hancock, Cabral, et al., 2022).

However, three aspects of the current methodology are worth investigating. Firstly, one needs to quantify the information loss resulting from the retention of the leading eigenvector only (Cabral et al., 2017; Olsen et al., 2022). Secondly, there is a need to base current methodologies onto a formal mathematical characterisation of the

matrix—also to improve speed end efficiency of the algorithms (Hutchison et al., 2013). Thirdly, depending on the data, the methodology should be able to model the matrix both as a granular set of dynamic modes as well as a smooth transition across FC configurations (Battaglia et al., 2020; Lavanga et al., 2023; Petkoski et al., 2023).