Information flow in networks and the law of diminishing marginal returns: evidence in human EEG recordings across cognitive and physiological states

• ¹ University of Gent, Faculty of Psychology and Pedagogical Sciences - Department of Data Analysis, Belgium
• ² Key Laboratory for NeuroInformation of Ministry of Education, School of Life Science and Technology, University of Electronic Science and Technology of China, China
• ³ University of Bari, Physics, Italy
• ⁴ University of Bari - University Hospital Policlinico, Neurology, Italy

Most social, biological, and technological systems can be modeled as complex networks, and display substantial non-trivial topological features. Moreover, time series of simultaneously recorded variables are available in many fields of science; the inference of the underlying network structure, from these time series, is an important problem that received great attention in the last years.

The inference of dynamical networks is also related to the estimation, from data, of the flow of information between variables, as measured by the transfer entropy [1]. Wiener [2] and Granger [3] formalized the notion that, if the prediction of one time series could be improved by incorporating the knowledge of past values of a second one, then the latter is said to have a causal influence on the former.
Initially developed for econometric applications, Granger causality has gained popularity also among physicists and eventually
became one of the methods of choice to study brain connectivity in neuroscience[4]. Multivariate Granger causality may be used to infer the structure of dynamical networks from data. It has been recently shown that for Gaussian variables Granger causality and transfer entropy are equivalent [5], and this framework has also been generalized to other probability densities [6]. Hence a weighted network obtained by Granger causality analysis can be given an interpretation in terms of flow of information between different components of a system. This way to look at information flow is particularly relevant for neuroscience, where it is crucial to shed light on the communication among neuronal populations, which is the mechanism underlying the information processing in the brain [7]. Furthermore, recent studies have investigated the economics implications of several network types mapping brain function[8,9].

In many situations it can be expected that each node of the network may handle a limited amount of information. This structural constraint suggests that information flow networks should exhibit some topological evidences of the law of diminishing marginal returns [10], a fundamental principle of economics which states that when the amount of a variable resource is increased, while other resources are kept fixed, the resulting change in the output will eventually diminish [11]. We have introduced [12] a simple dynamical network model where the topology of connections, assumed to be undirected, gives rise to a peculiar pattern of the information flow between nodes: a fat tailed distribution of the outgoing information flows while the average incoming information flow does not depend on the connectivity of the node. In the proposed model the units, at the nodes the network, are characterized by a transfer function that allows them to process just a limited amount of the incoming information. We have shown that a similar behavior is observed in another network model, which describes in a different fashion the law of diminishing marginal returns. Moreover, we also proposed an exactly solvable Ising model on sparse networks, in the limit of an infinite number of nodes, whose behavior may be seen in the light of the law of diminishing marginal returns.
Finally we have shown that this relevant topological feature is found as well in real neural data. In this work we present a comparison of the emergence of this property in three physiological/cognitive states.
For resting state EEG with eyes closed from healthy subjects the law of diminishing marginal returns is more expressed in temporal and occipital regions.
On the other hand, EEG from patients in vegetative state displayed a prominent evidence of this law in the frontal area.
The pattern resulting from the EEG of healthy subjects looking at emotional pictures was similar to the one observed for vegetative state, but with a completely different distribution of the pattern of incoming information.