Fig 1.
Information structures and their dynamics.
A. Left: Lotka-Volterra 7-node dynamical system. Relevant parameters are those affecting just one node (αi) and those representing interactions, given by the weighted links (γij) between nodes (i and j). Right: Associated to each system (for given αi and γij parameters) there is an Information Structure (IS), containing the attractor landscape, that is, stationary points of the system dynamics and the solutions linking them. Black nodes in the stationary points represent a value greater than 0 of the corresponding node of the dynamical system. White nodes stand for 0 value of the corresponding node. Different γij and αi values produce different IS in both the globally asymptotically stable solution (GASS, in the bottom) and the nodes in the intermediate levels. The structure of the IS can be studied through several measures (see Fig 2) depending not only on the number of levels and nodes (NoEL and Frondosity, respectively) but also on the interaction between nodes (Synchronicity, Criticality and Cooperation measures). B. For a dynamical system with fixed connectivity (γij parameters) and changing activity among the nodes (αi) it is possible to study the evolution of the IS measures. The top graphic represents the change in time of the αi parameters in a Lotka-Volterra 7-node system with fixed connectivity γij. Associated to each instant there is an IS with the structure of the attractor underlying the dynamics of the system. The evolution of the system dynamics can be characterised by looking at the mean and standard deviation across time of the IS measures.
Fig 2.
Measures of information structures.
An information structure (IS) is shown on the left. It is determined by the structural connectivity matrix (fixed) and the αi values (variable at each instant) for the 7 networks in Yeo’s parcellation. The nodes of the information structure correspond to stationary states of the system. Networks with a value greater than 0 are coloured. The IS can be studied through several measures. NoEL (number of energy levels) corresponds to the number of layers in the IS (levels 0 to 4). Frondosity is the ratio of nodes in the IS with respect to the maximum for its NoEL. Cooperation measures focus on cooperative nodes, which are those (except for level 1) where a network that does not appear on its own occurs in presence of other networks. The highest cooperation level is given by the highest level at which a cooperative node appears. Cooperation values consider cooperative nodes in different ways. Cooperation value A counts all cooperative nodes, adding in each case the level of the node from which it proceeds. Cooperation value B considers only one level (the lowest) for each new network, and cooperation value C operates in the same way but by adding powers of 2. Other used measures are criticality and synchronicity. Criticality measures how close is the globally stable node to a bifurcation point that modifies which are its active and inactive networks. Finally, synchronicity measures the ratio of IS of a subject (across time) in which all networks appear in the same state (active or inactive) in the globally stable solution.
Fig 3.
Measuring the changes in the IS underlying brain activity.
The brain activity registered through fMRI is mapped into seven brain networks using Yeo’s 7-network parcellation. For each subject, given the timeseries of the activity and the structural connectivity matrix, Lotka-Volterra Transform produces the αi values at each instant. This allows the study of the Information Structures evolution in time. Machine learning is used to train classifiers identifying the state of each subject.
Table 1.
Wilcoxon Signed-Rank test for different IS measures.
Table 2.
Performance of the classifiers.
Fig 4.
Distributions of NoEL, frondosity and criticality.
Each subject brings one value to the distributions (the mean or standard deviation of the corresponding measure across time). Standard deviation of NoEL can be considered a measure of metastability, since it reflects the tendency of the system to change the attractors. Frondosity is related to integration: IS with low frondosity exhibit greater cooperative interactions between nodes (see the cooperation measures on Fig 6). Mean frondosity values very close to 1 (like those of UWS patients) denote a lack of integration in brain activity, since maximum frondosity coincides with the absence of cooperative nodes. Criticality indicates how close is the system to a major change on the attractor shape (affecting the GASS). The lower the criticality, the closer the system is to a bifurcation point and the easier it is for the IS to undergo large changes. The median criticality distributions show that the IS of healthy subjects move closer to bifurcation points.
Fig 5.
Distributions of synchronicity.
This measure is not applied to individual IS. It is the ratio of IS having the globally stable point with all nodes in the same state (either 0 or greater than 0). That is, the extreme values of NoEL. Subjects with low Synchronicity have information structures with more differentiated behaviour of the nodes. UWS patients have synchronicity values very close to 1, denoting undifferentiated brain activity.
Fig 6.
Distributions of cooperation measures.
These measures consider in several ways the cooperative interactions between different networks (Fig 2). It can be observed that the higher the consciousness level, the more cooperation is in the IS (mean) and with a higher variability (standard deviation).
Fig 7.
Presence of brain networks in the IS and cooperative interactions.
The proportion in which each network appears in the global stable point of the IS (which is equal to appearing in any point of the IS) is shown above. Values range from 0.8 to 1. The highest values appear in UWS patients, followed by MCS. This is due to higher values of NoEL and frondosity. However, this does not imply complex behaviour in brain dynamics. The proportion of occurrences of each network in the IS due to cooperative interactions of other networks is shown below. The values are between 0 and 0.1. The highest values belong to HC (the leading group in terms of coperative measures, followed by MCS, see Fig 6). A greater differentiation between networks is observed than when looking at the global stable point alone. The highest values in all three groups correspond to DMN followed by frontoparietal.