In order to show the starting point of synergizing between the three subfields of temporal network theory, multilayer network theory, and control theory, I will first illustrate how I approach them separately.
Temporal brain networks
The temporal brain network is marked by variation in connectivity strength and network topology across the span of seconds to minutes, schematically depicted below (Van Geest, …, Douw (2018) Brain and Behavior):
I have investigated these dynamics in both resting-state functional magnetic resonance imaging (rsfMRI) and magnetoencephalography (MEG), showing that these dynamics indeed relate to and may even predict future changes in individual cognitive performance (Douw et al (2015) PLoS ONE; Douw et al (2016) Neuroscience; Carbo, …, Douw (2017) Scientific Reports; Van Geest, …, Douw (2018) Brain and Behavior).
Multilayer brain networks
To illustrate how powerful multilayer network theory is, let us consider the European flight network. Each airport is a node, while connections between airports represent the average number of daily direct flights between them. Although this average network may give useful information on European coverage and airport importance, understanding its dynamics is problematic: suppose that a flight from Schiphol Airport to Frankfurt is delayed. Modeling the spread of delay based on the average network may predict delays across the entire continent and even the rest of the world, because of Frankfurt’s hub properties. However, flights are not all equal: some are operated by the Royal Dutch Airlines (KLM), others by Iberia or Lufthansa (see figure on the right). The delay of a Frankfurt-headed KLM flight will not have direct consequences for all Lufthansa flights departing from Frankfurt, but will first impact other KLM flights. Then it may spread to other airlines through interlayer connectivity (indicated in grey dotted lines), e.g. delayed ground personnel or a queue of airplanes forming to use a landing strip. Multilayer network algorithms take both intralayer and interlayer connections into account to accurately explain these flight delays, predict its dynamics, and ultimately manipulate the network to prevent further delays. Importantly, dynamics of the multilayer network supersede properties of individual layers: reducing the network to its separate layers yields misleading results. It is therefore essential to unify layers and implement mathematical analyses able to capture intra- and interlayer processes, as has been increasingly recognized in the field of complex networks.
In order to illustrate the aimed implementation of multilayer networks in neuroscience, I explain a pilot study that we performed to merge different modalities ánd the temporal dynamics of the functional brain network in individual subjects in two movies:
Controllable brain networks
Many measures reflecting network behavior and hypothetically cognition and mood have been introduced (e.g. efficiency, clustering, modularity). However, these measures are largely descriptive, and have only limited power when trying to explain behavior of the network in terms of individual cognitive functioning and mood. Control theory, which stems from electrical engineering, regards not just an instance, but the entire range of connectivity states that can be reached by nodes in the multilayer brain network. Thus, controllability refers to the possibility of manipulating particular interactions to steer a global system along a particular trajectory, and has only recently been mathematically defined for networks. In the context of the brain, let us view ‘a particular trajectory’ as intact cognitive functioning (for instance cognitive profile y in highly schematic figure on the right, which occupies a particular state space trajectory according to three dimensions), and consider ‘interactions’ as all multilayer connections of a node (as indicated in figure). Control theory can now determine how this specific interaction enables the entire network dynamics to reach (and maintain) the network state associated with cognitive profile y. More importantly, control theory also allows us to determine which input should be delivered in order to manipulate the trajectory of the system to a desired state associated with this cognitive profile, thereby overcoming the current limitation of many studies merely associating network topologies to cognitive (dys)functioning. Recent studies have indeed shown that controllability of the unilayer brain network may accurately describe brain functionality in general, and cognitive functioning specifically, although time has not been incorporated into these studies. Mathematical definitions of multilayer network controllability, even across differently time-scaled layers, have only recently been formulated, and have not been investigated in multilayer brain networks.