Effective connectivity

Effective connectivity measures the influence that one neuronal group or ROI exerts over another (or even on a whole collection of other groups at the same time, when multivariate analyses are performed), addressing specifically the directionality and the causal nature (either direct or indirect) of the connections (Vicente et al., 2011, Barrett et al., 2012). As a general statement, inferring effective connectivity requires successive steps of model specification (comprising

hypothesis about network arrangement), model identification (selection of the most suitable hypothesis) and, finally, causal modelling (explicit results yielded by the most likely network architecture) (Barrett et al., 2012). Similarly to functional connectivity (where structural connectivity delineates the general repertoire of possible functional connections), effective connectivity is also influenced to some extent by structural connectivity, although this time, structural information is not sufficient to provide significant clues about effective connectivity and directionality (even having a perfect knowledge of both structural and functional connectivity) (Stam et al., 2016). Additionally, as with other connectivity forms, effective connectivity is subject to plastic changes occurring across the lifespan; e.g., it has been observed that network organization in the infant is different compared to the adult brain (the latter being arranged in a more centralized way when directing causal synchronization) (Goldenberg and Galvan, 2015).

Effective connectivity can be studied either with tools specifically built for assessing neuronal responses in task-evoked experimental procedures (e.g., responses in the auditory cortex to music and how it drives limbic structures for emotional significance) or, conversely, in a more generalized and autonomous manner (i.e., context-independent, however, more dismissive of environmental inputs, not explicitly modeled) (Barrett et al., 2012, Liu and Aviyente, 2012). For the first approach, which includes external inputs to the brain directly into the equations, the most commonly used and a well-established method is the Direct Causal Modelling (DCM), originally developed for fMRI but applicable for EEG data too (Friston, 1994, Penny et al., 2009). A common alternative to DCM involves the use of Structural Equation Modelling (SEM), which rests on modelling effects of controlled experimental manipulations to the system and also uses network graphs to discover active links and their causal properties (Sommerlade et al., 2011, Valdes-Sosa et al., 2011).

Briefly, DCM relies on Bayesian modelling to estimate how changes in neuronal activity in one node are caused by activity of another one, providing parameters for connection strength (which are sensitive to the directionality of transmission) and brain modulation by experimental inputs (Friston, 2011, Friston et al., 2014). The use of

Bayesian statistics incorporated into the DCM theory makes it possible to find the causal network graph with highest evidence (i.e., explanatory power for observed functional connectivity at the causal level, as assessed by the Bayes theorem), and in case that several different networks were a priori compatible with the data, selects by default the most simple network (Park and Friston, 2013). Finally, it has been shown that DCM outperforms most alternative methods when dealing with networks with cyclic feedback loops (e.g., closed circuits allowing bi-directionality) and, in addition, DCM can take into account the previous temporal history of the network to calculate “memory footprints” (i.e., long-term effects of the past which still exert a causal influence) (i.e., long-term past effects which still exert a causal influence) (Valdes-Sosa et al., 2011).

On the other hand, effective connectivity is also studied through a second general family of models which are not explicitly conditioned by experimental set-ups and external causal manipulations, but purely descriptive of internally driven variations of neuronal activity in brain networks (Schelter et al., 2009, Vicente et al., 2011, Barrett et al., 2012). Therefore, whereas the previous approach takes a view on causality more focused on the modelling of biophysical influences (i.e., by changing causes –e.g., external inputs-, the effects on the network will also change), this approach understands causality as temporal precedence (causes always precede effects) (Valdes-Sosa et al., 2011).

The most prominent example for this approach is Granger causality (GC), along with all the subsequent improvements (e.g., frequency representations, multivariate analyses, etc.) (Granger, 1969, Kaminski et al., 2001). It is worth pointing out that practically all “temporal precedence” methods aiming to be totally general can be historically traced to GC as a kind of “common ancestor”, including partial directed coherence (PDC), directed transfer function (DTF) or isolated effective coherence (iCOH) (Baccala and Sameshima, 2001). The gist concept in this family of methods is that there is “causality” (in a mathematical sense) between variables A and B (where A causes B) whenever uncertainty or prediction error in the modeling of variable B is reduced by inclusion of previous states of variable A (Faes et al., 2010, Faes et al., 2012).

Finally, even if effective connectivity is a very powerful method for connectivity studies, as it was essentially designed to overcome most limitations of functional connectivity techniques, it also exhibits several caveats (Kralemann et al., 2014). First, a majority of effective connectivity methods are constrained to operate just at the level of measured signals (with the important exception of DCM, aiming to provide statistical tools to overcome it), hence, positing a hard problem when trying to model the underlying behavior at the neuronal level, which cannot be directly observed or probed with non-invasive techniques (Penny et al., 2009). It is likely to suppose that this limitation will be gradually improved with the introduction of more refined models enabling an accurate description of the biophysical link between neuronal activity and the resulting BOLD or EEG signals (called forward models), probably by addition of non-linear terms in the mathematical equations (i.e., broadening the simplicity of linearity) (Valdes-Sosa et al., 2011, Friston et al., 2014). Secondly, effective connectivity results might be misleading due to superposition of electrical fields produced by different brain regions on nearby electrodes and volume conduction effects (whose amplitude of propagation decreases with the inverse of the distance) (Vicente et al., 2011, Huang et al., 2015). Hence, conclusions drawn from effective connectivity analyses are prone to error when limited to the scalp level (Pascual-Marqui et al., 2014a). In this way, even when effective connectivity methods might detect correctly the coupling between regions, directionality aspects might be wrong (especially for complex networks), although there are methods more adept to overcome these limitations, which outperform many others (e.g., iCOH as compared to GC) (Kaminski and Blinowska, 2014, Goldenberg and Galvan, 2015).

Finally, it is expected that effective connectivity models will be updated in two interacting ways: first, by incorporating new discoveries on structural connectivity (hence, refining our theoretical comprehension), and secondly, by using more realistic computational modeling for both resting state and task-induced activity (however, this limitation is continuously being improved) (Park and Friston, 2013, Park et al., 2015).

2.2. Source localization with focus on low resolution