Brain connectivity estimators

Brain connectivity estimators represent patterns of links in the brain. Connectivity can be considered at different levels of the brain's organisation: from neurons, to neural assemblies and brain structures. Brain connectivity involves different concepts such as: neuroanatomical or structural connectivity (pattern of anatomical links), functional connectivity (usually understood as statistical dependencies) and effective connectivity (referring to causal interactions).

Neuroanatomical connectivity is inherently difficult to define given the fact that at the microscopic scale of neurons, new synaptic connections or elimination of existing ones are formed dynamically and are largely dependent on the function executed, but may be considered as pathways extending over regions of the brain, which are in accordance with general anatomical knowledge. Diffusion Weighted Imaging (DWI) can be used to provide such information. The distinction between functional and effective connectivity is not always sharp; sometimes causal or directed connectivity is called functional connectivity. Functional connectivity, may be defined as the temporal correlation (in terms of statistically significant dependence between distant brain regions) among the activity of different neural assemblies, whereas effective connectivity may be defined as the direct or indirect influence that one neural system exerts over another. Some brain connectivity estimators evaluate connectivity from brain activity time series such as Electroencephalography (EEG), Local field potential (LFP) or spike trains, with an effect on the directed connectivity. These estimators can be applied to fMRI data, if the required image sequences are available. Among estimators of connectivity, there are linear and non-linear, bivariate and multivariate measures. Certain estimators also indicate directionality. Different methods of connectivity estimation vary in their effectiveness. This article provides an overview of these measures, with an emphasis on the most effective methods.

Classical estimators of connectivity are correlation and coherence. The above measures provide information on the directionality of interactions in terms of delay (correlation) or coherence (phase), however the information does not imply causal interaction. Moreover, it may be ambiguous, since phase is determined modulo 2π. It is also not possible to identify by means of correlation or coherence.

The most frequently used nonlinear estimators of connectivity are mutual information, transfer entropy, generalised synchronisation, the continuity measure, synchronization likelihood, and phase synchronization. Transfer entropy has been applied in neuroimaging studies to infer effective connectivity, particularly in dynamic systems like resting-state fMRI. Vincent Calhoun and colleagues have employed TE to identify connectivity alterations in disorders like schizophrenia. Mutual information and transfer entropy rely on the construction of histograms for probability estimates. The continuity measure, generalized synchronisations, and synchronisation likelihood are very similar methods based on phase space reconstruction. Among these measures, only transfer entropy allows for the determination of directionality. Nonlinear measures require long stationary segments of signals, are prone to systematic errors, and above all are very sensitive to noise. The comparison of nonlinear methods with linear correlation in the presence of noise reveals the poorer performance of non-linear estimators. In the authors conclude that there must be good reason to think that there is non-linearity in the data to apply non-linear methods. In fact it was demonstrated by means of surrogate data test, and time series forecasting that nonlinearity in EEG and LFP is the exception rather than the norm. On the other hand, linear methods perform quite well for non-linear signals. Finally, non-linear methods are bivariate (calculated pair-wise), which has serious implication on their performance.

Convergent Cross Mapping (CCM) is a method rooted in dynamical systems theory. CCM evaluates causality in coupled systems by assessing whether the states of one variable ${\displaystyle X}$ can be reconstructed from another variable ${\displaystyle Y}$ using its shadow manifold neighbourhood.

Reservoir computing causality extends the convergent cross-mapping principle by using a fixed, high-dimensional recurrent network (the reservoir) to model complex temporal patterns and interactions. A high-dimensional reservoir ${\displaystyle \mathbf {R} (t)}$ is composed of recurrently connected units to process temporal patterns. Ciezobka et al. (2025) demonstrated that RC is effective in modeling non-linear interactions in large-scale brain networks, making it a robust tool for effective connectivity analysis.

Comparison of performance of bivariate and multivariate estimators of connectivity may be found in, where it was demonstrated that in case of interrelated system of channels, greater than two, bivariate methods supply misleading information, even reversal of true propagation may be found. Consider the very common situation that the activity from a given source is measured at electrodes positioned at different distances, hence different delays between the recorded signals.

When a bivariate measure is applied, propagation is always obtained when there is a delay between channels., which results in a lot of spurious flows. When we have two or three sources acting simultaneously, which is a common situation, we shall get dense and disorganized structure of connections, similar to random structure (at best some "small world" structure may be identified). This kind of pattern is usually obtained in case of application of bivariate measures. In fact, effective connectivity patterns yielded by EEG or LFP measurements are far from randomness, when proper multivariate measures are applied, as we shall demonstrate below.