Dynamical informational structures characterize the different human brain states of wakefulness and deep sleep

Abstract

The dynamical activity of the human brain describes an extremely complex energy landscape changing over time and its characterisation is central unsolved problem in neuroscience. We propose a novel mathematical formalism for characterizing how the landscape of attractors sustained by a dynamical system evolves in time. This mathematical formalism is used to distinguish quantitatively and rigorously between the different human brain states of wakefulness and deep sleep. In particular, by using a whole-brain dynamical ansatz integrating the underlying anatomical structure with the local node dynamics based on a Lotka-Volterra description, we compute analytically the global attractors of this cooperative system and their associated directed graphs, here called the informational structures. The informational structure of the global attractor of a dynamical system describes precisely the past and future behaviour in terms of a directed graph composed of invariant sets (nodes) and their corresponding connections (links). We characterize a brain state by the time variability of these informational structures. This theoretical framework is potentially highly relevant for developing reliable biomarkers of patients with e.g. neuropsychiatric disorders or different levels of coma. of the phase space described by a set of selected invariant global solutions of the associated dynamical system, such as stationary points (equilibria), connecting orbits among them, periodic solutions and limit cycles. The informational structure of the GA is defined as a directed graph composed of nodes associated with those invariants and links establishing their connections (see Methods and Supplementary Information for a formal rigorous definition). Informational Structure has been used to show the dependence between the topology, the value of the parameters and the state with respect to its level of integration 14 , as such pointing for the small world configuration of the brain 16 (although see recent controversies 17 ).Previous research has applied Informational Structure to population dynamics in complex networks 18,19 . Equally used for modelling mutualistic systems in Theoretical Ecology and Economy 20, 21 , Informational Structure can relate the topology of complex networks and their dynamics. In mutualistic systems when achieving robustness and life abundance -the so-called architecture of biodiversity 22, 23 -the dynamics is determined not only by the topology 24 but also by modularity 25 and the strength of the parameters 26 .These kinds of relationships are also at the heart of many important open questions in neuroscience 27,28 . Informational Structures and their continuous time-dependence on the strength of connections can allow to understand the dynamics of the system as a coherent process, whose information is structured, and potentially providing new insights into sudden bifurcations 29 .Here, we were interested in characterising human sleep, which is traditionally subdivided into different stages that alternate in the course...