Integrated Information Theory (IIT) has become nowadays the most sensible general theory of consciousness. In addition to very important statements, it opens the door for an abstract (mathematical) formulation of the theory. Given a mechanism in a particular state, IIT identifies a conscious experience with a conceptual structure, an informational object which exists, is composed of identified parts, is informative, integrated and maximally irreducible. This paper introduces a space-time continuous version of the concept of integrated information. To this aim, a graph and a dynamical systems treatment is used to define, for a given mechanism in a state for which a dynamics is settled, an Informational Structure, which is associated to the global attractor at each time of the system. By definition, the informational structure determines all the past and future behavior of the system, possesses an informational nature and, moreover, enriches all the points of the phase space with cause-effect power by means of its associated Informational Field. A detailed description of its inner structure by invariants and connections between them allows to associate a transition probability matrix to each informational structure and to develop a measure for the level of integrated information of the system.
Dynamical Systems and Graph Theory are naturally coupled since any real phenomenon is usually described as a complex graph in which the evolution of time produces changes in specific measures on nodes or links among them [1,2]. In this work, the starting point is any structural network, including a parcelling of the brain, possessing an intrinsic dynamics. For brain dynamics, the collective behavior of a group of neurons can be represented as a node with a particular dynamics along time [3–6]. In general, the mathematical way to describe and characterize dynamics is by (ordinary or partial) differential equations (continuous time) [7] or difference equations (discrete time) [8]. Global models on brain dynamics are grounded on anatomical structural networks built under parcelling of the brain surface [9–11]. Indeed, they are based on systems of differential equations described on complex networks, which may include noise, delays, and time-dependent coefficients. Thus, the designed dynamical system models the activity of nodes connected to each other by a given adjacency matrix. A global dynamics emerges through simulated dynamics at each node, which is coupled to others as detailed in the anatomical structural network (see, for instance, [12] for the structural networks on primate connectivity). Then, an empirical functional network and a simulated functional network emerge by correlation or synchronization of data on the structural network [4,13,14], showing a similar behavior and topology after a proper fitting of the parameters in the differential equations associated to the dynamics. We take advantage of this approach to apply some of the main results on the modern theory of dynamical systems showing that, given a dynamics on a network, there exists an object, the global attractor [15–18], determining all the asymptotic behaviour of each state of the network. The attractor exists and its nature is essentially informational, as it possesses the power to produce a curvature of the phase space enriching every point with the information on its possible past and future dynamics. The structure of the global attractors (or attracting complex networks [19,20]), described as composed structures by invariants and connections, naturally shows that its information is structured, composed by different parts, and can be unreachable from the study of the information of its parts, so allowing for a definition of integrated information. Integrated Information Theory (IIT, [21]), created by G. Tononi [22–24] starts with a phenomenological approach to the theory of consciousness. It assumes that consciousness exists and tries to describe it by defining the axioms that it satisfies. Having the axioms on hand, they serve to introduce the postulates that every physical mechanism has to obey in order to produce a conscious experience. This fact opens the door to the possibility of the mathematization of the theory by defining and describing postulates on concrete networks where a dynamics can be settled. It is then possible to define the appropriate structured dynamics which is supposed to explain a conscious experience by preserving its axioms. The IIT approach allows to represent a conscious experience and even to measure it quantitatively and qualitatively by the so called integrated information Φmax, which, at the same time indicates that, at the base of consciousness, there are essentially phenomena of causal processes of integrated information nature [25]. This fact links IIT to Information Theory and the Theory of Causality [26]. On the mathematical level, IIT approach is based on graphs consisting of logic gates and transition probabilities describing causality of consecutive discrete states on those graphs [21]. In this paper we present a continuous-time version (see Section Results for a formal description) related to IIT based on the theory of dynamical systems. IIT bases any particular experience on a mechanism, defined in a particular state, which possesses a well defined cause-effect power. Our starting point is given by a graph describing a mechanism and, thus, a graph is first defined. However, we focus our study on the network patterns arising from dynamical phenomena. So, a dynamical system and the associated mathematical objects (global attractor, equilibrium points, unstable invariant sets) have to be also defined. As a novelty in dynamical systems theory, the global attractor (which, for the gradient case consists of the equilibria and the heteroclinic connections between them [15,17,27]) is redefined as an object of informational nature, an Informational Structure (IS). An IS is a flow-invariant object of the phase space described by a set of selected invariant global solutions of the associated dynamical system, such as stationary points (equilibria) and connecting orbits among them (Fig 1). This set of invariants inside the IS creates a new structure, a new complex network with the power to ascertain the dynamics (past and future scenarios) of natural phenomena. Every IS posseses an associated Informational Field (IF), globally described from the attraction and repulsion rates on the nodes of the IS. We are able to translate the energy landscape caused by the IS and the IF into a transition probability matrix (TPM) to pass from one state to another within the system (see Section Results). Thus, the level of information of a mechanism in a state is going to be given by the global amount of deformation of the phase space caused by the intrinsic power of the IS and IF. The geometrical characterization of ISs can provide both the quality of the related information and, in particular, the shape in which it is integrated in the whole system, allowing to measure the level of integrated information it contains. Thus, the quality of the information comes from the detailed study of informational structure, which now possesses an intrinsic dynamics and enjoys a continuous change. This structure depends on the parameters of the underlying equations and has the ability to possibly rapidly change in the response to the change of those parameters (see Section Materials and methods). From this continuous approach, we are able to introduce first definitions for postulates of existence, composition, information and integration for a mechanism in a state. There is still a gap to the more elaborated formal definitions from IIT 3. 0 [21] (see Section Discussion), including the composition and exclusion postulates. However, our framework naturally leads to a study on the continuous dependence between the topology of the network and the level of integrated information for a given mechanism (see Section Results).
Many real phenomena can be described by a set of key nodes and their associated connections, building a (generically) complex network. In this way, we can always construct an application between a real situation and an abstract graph describing its essential skeleton. An undirected graph is an ordered pair G = (V, E) comprising a non-empty set V of vertices (or nodes) together with a set E of edges joining 2-element subsets of V. The order of a graph is given by the number of nodes, and its size by the number of edges. A directed graph or digraph is a graph in which edges (named arcs) have orientations. We want to study the behaviour on networks of systems of evolutionary differential equations as d u d t = F (t, u), (1) where F is a nonlinear map from (t, u) ∈ R × R N to R N. For modeling purposes, we could also add, for instance, delays, stochastic terms, or to make solution u (t) also depend on a subset Ω of the three dimensional space, i. e. u (t, x), for x ∈ Ω ⊂ R 3. Given an initial condition, suppose existence and uniqueness of solutions. If not, a multivalued approach could also be adapted. The phase space X (in our case X = R N) represents the framework in which the dynamics described by a group of transformations S (t): X → X is developed. Given a phase space X we define a dynamical system on X as a family of non-linear operators { S (t) } t ∈ R +, S (t): X → X u ∈ X, S (t) u ∈ X which describes the dynamics of each element u ∈ X. In particular, S (t) u0 = u (t; u0) is the solution of the differential Eq (1) at time t with initial condition u0. The global attractor is the central concept in dynamical system theory, since it describes all the future scenarios of a dynamical system. It is defined as follows [15–18,28,29]: A set A ⊆ X is a global attractor for {S (t): t ≥ 0} if it is Observe that (ii) is showing a crucial property of an attractor, as supposes a set with a proper intrinsic dynamics. Moreover, (iii) points that this set is determining all the future dynamics on the phase space X. We say that u* ∈ X is an equilibrium point (or stationary solution) for the semigroup S (t) if S (t) u* = u*, for all t ≥ 0. A stationary point is a trivial case for a global solution associated to S (t), i. e. , ξ: R → X such that ξ (t + s) = S (t) ξ (s) for all s∈R, t∈R+. Stationary points are the minimal invariant objects inside a global attractor. Every invariant set is a subset of the global attractor [15]. Generically, connections among invariant sets in the attractor describe its structure [27,30]. To this aim we need the following definitions, which also allow us to define the behaviour towards the past in a global attractor. The unstable set of an invariant set Ξ is defined by W u (Ξ) = { z ∈ X: there is a global solution ξ: R → X for S (t) satisfying ξ (0) = z and such that lim t → − ∞ dist (ξ (t), Ξ) = 0 }. The stable set of an invariant set Ξ is defined by W s (Ξ) = { z ∈ X: such that lim t → + ∞ dist (S (t) z, Ξ) = 0 }. We have to think in a global attractor as a set which does not depend on initial conditions, with an intrinsic proper dynamics, composed by a set of special solutions (global solutions), which are connecting particular invariants, so generating a complex directed graph. Moreover, the global attractor has the following properties [17,18]: The Fundamental Theorem of Dynamical Systems [33] states that every dynamical system on a compact metric space X (the one defined on a global attractor, for instance) has a geometrical structure described by a (finite or countable) number (indexed by I) of sets {Ei}i∈I with an intrinsic recurrent dynamics and a gradient-like dynamics outside them. In other words, when we define a dynamical system on a graph, the attractor can be always described by a (finite or countable) number of invariants and connections between them. The starting point in our approach is a system of connected elements where a dynamics is defined. This system is called a mechanism. Composition and exclusion postulates in IIT 3. 0 allow to consider mechanisms given by any subset of the system. Here, we are only focusing in the mechanism given by the whole system. A global attracting network can be characterized by the amount of information it provides to the mechanism, since the nature of a global attractor is essentially informational. Indeed, the informational nature of the attractor is based on the following assertions: The state of a mechanism is given by the state of its nodes. If those nodes take real values, the state is given by a vector of real numbers. When a dynamical system is defined for a mechanism in a state, it has cause-effect power, meaning that it conditions the possible past and future states of the mechanism. We can find the cause-effect power of a mechanism in a state by looking at its (local) attracting invariant sets. Typically, these are stationary points or periodic orbits [32,39–41], but it could also contain invariant sets with chaotic dynamics [42–44]. Following IIT, we will measure the amount of information on the structure made by these invariants, in the sense of its power to restrict the past and the future states. In IIT language, this is intrinsic information: differences (state of a mechanism) that make a difference (restricted set of possible past and future states) within a system (see the IIT glossary in [21]). Now we can define an Informational Structure (IS). Suppose we have a complex graph G given by N nodes and links among them. We denote by Gi every subset of G. An informational structure is a complex graph I = { G 1, ⋯, G M } which nodes are subgraphs Gi of G and links among them, with the following properties An IS (see Fig 2) becomes a natural emerging object (of informational nature) from a given mechanism in a state with intrinsic cause-effect power to the past and the future. Moreover, some properties of ISs can be given:
In this section we define a preliminary version of IIT postulates: existence, composition, information, integration and exclusion. We do not try to mimic or generalize the concepts of IIT 3. 0 (see Section Discussion), where they appear in a more elaborated fashion, allowing finer computations and insights. For example, IIT considers the integrated information of each subset of a mechanism (candidate set), while we always focus on the whole mechanism. Although ISs and mechanisms possess quantitative and qualitative major differences on the structure and topology of both networks, it is clear that ISs possess an strongly dependence of their mechanisms’ topology. To show this dependence (but no determination) between mechanisms and associated ISs, we have tried to model the continuous evolution of integrated information for simplified mechanisms. This is probably one of the virtues of our continuous approach to integrated information. In particular, we consider the cases of totally disconnected mechanisms, lattice ones, the presence of a hub, and totally connected mechanisms, showing the key functions of the topology and strength of connections with respect to integrated information. To allow the comparison of the different topologies, the reference value for αi is 1. 6 and for γij is 0. 1. The behaviour of ϕ-cause and ϕ-effect when some of these parameters change is shown for the different mechanisms.
The concept of informational structure is not only related to the understanding of brain processes and their functionality but, more broadly, as an inescapable tool when analyzing dynamics in complex networks [19,20]. For instance, in Theoretical Ecology and Economy related to the modeling of mutualistic systems [61,62], a very important subject is the study of the dependence between the topology of complex networks (lattice systems, unconnected systems, totally connected ones, random ones, “small world” networks) and the observed dynamics on their sets of solutions. The architecture of biodiversity [63,64] is thus referred as the way in which cooperative systems of plants-animals structure their connections in order to get a mechanism (complex graph) achieving optimal levels in robustness and life abundance. The nested organization of this kind of complex networks seems to play a key role for higher biodiversity. However, it is clear that the topology of these networks is not determining all the future dynamics [65], which seems also to be coupled to other inputs as modularity [66] or the strength of parameters [67]. This is also a very important task in Neuroscience [54,68]. Informational structures associated to phenomena described by dynamics on complex networks contain all the future options for the evolution of the phenomena, as they possess all the information on the forwards global behaviour of every state of the system and the paths to arrive into them. Moreover, the continuous dependence on the parameters measuring the strength of connections and the characterization of the informational structure allow to understand the evolution of the dynamics of the system as a coherent process, whose information is structured, so giving a comprehension to the appearance of sudden bifurcations [69]. From a dynamical system point of view [44], informational structures introduce several conceptual novelties in the standard theory. In particular, for phenomena associated with high dimensional networks or models by partial differential equations [15,18,29]. On the one hand, an informational structure has to be understood as a global attractor at each time, i. e. , we do not reach this compact set as the result of the long time behaviour of solutions, but it exists as a complex set made by selected solutions which is causing a curvature of the whole phase space at each time. Because we allow parameters to move in time, the informational structure also depends on time, so showing a continuous deformation of the phase space by time evolution. It is a real space-time process explaining all the possible forwards and internal backwards dynamics. On the other hand, the understanding of attracting networks as informational objects is also new in this framework, allowing a migration of this subject into Information Theory. It is remarkable that dynamical systems cover from trivial dynamics, as global convergence to a stationary point, to the much more richer one usually referred as chaotic dynamics [43,44,57] and dynamics of catastrophes [70]. While the first one can be found with total generality, attractors with chaotic dynamics can be only described in detail in low dimensional systems. Note that some of the classical concepts of dynamical systems have been reinterpreted. In particular, the global attractor it is understood as a set of selected solutions (organized in invariant sets and their connections in the past and future) which creates the informational structure. This network, moreover, produces a global transformation of the phase space (the informational field) enriching every point in the phase space on the crucial information on possible past states and possible future states. It is the continuous change on time of informational structures and fields what allows to talk on an informational flow for which we can analyse the postulates of IIT. Our approach deals with dynamics on a graph for which a global description of the asymptotic behaviour is posible by means of the existence of a global attractor. That is, dissipative dynamical system for which a global attractor exists. It is important to remark the Fundamental Theorem of Dynamical Systems in [33], inspired in the work of Conley, because it gives a general characterisation of the phase space by gradient and recurrent points, so leading to a global description of the phase space as invariants and connections among them. This is the really crucial point which allows for a general framework on the kind of systems to be considered. The Lotka-Volterra cooperative model we consider is for the applications to IIT, because in this case we have a detailed description of the global attractor, which allows to make concrete computations for the level of integrated information on it. Thus, in application to L-V systems, we are considering heteroclinic channels inside the global attractor. But all the treatment in the paper can be done for every dissipative systems for which a detailed description of the global attractor is available, i. e. , this description has not be to be restricted to heteroclinic channels coming from L-V models. Gradient dynamics [15,27,29,31] describes the dynamics from heteroclinic connections and associated Lyapunov functionals [29,32], and it naturally suits into higher order systems including infinite dimensional phase spaces [27,34,35]. Thus, although our description of informational structures associated to Lotka-Volterra systems is general enough to describe real complex networks, we think the concept may be also well adapted both for different topologies in the networks and also different kind of non-linearities defining the differential equations [51,52]. Actually, for comparison with data and experiments associated for real brain dynamics we certainly need to allow much more complex networks as primary mechanisms as well as different kinds of dynamical rules and nonlinearities. But, in all of them, we expect to find the existence of dynamical informational structures providing the information on the global evolution of the system. The concise geometrical description of these structures and their continuous dependence on sets of parameters is a real open problem in the dynamical systems theory. We have presented a preliminary approach to the notion of integrated information from informational structures, leading to a continuous framework for a theory inspired by IIT, in particular in order to define integrated information of a mechanism in a state. The detailed characterization of informational structures comes from strong theorems in dynamical systems theory about the characterization of global attractors. The description of causation in an informational structure by a coherent and continuous flow of integrated information is new in the literature. Indeed, the definition of TPMs between two equilibria in the IS by the strength of eigenvalues and direction of eigenvectors associated to these and intermediate equilibria produces a global analysis of the level of information contained in a particular IS. The same procedure for partitions of the associated mechanism allows a preliminary measure for integration. Moreover, the continuous dependence of the structure of ISs on parameters also opens the door to the analysis to sudden bifurcations, from an intrinsic point of view, for critical values of the parameters. It seems also very clear the close dependence between the topology of a mechanism, the actual value of the parameters and the current state with respect to its level of integrated information, so pointing for optimality for a “small world” configuration of the brain [71]. With respect to global brain dynamics, we think the concept of informational structures and informational fields could serve as the abstract objects to describe functionality and global observed dynamics now described by other concepts and methodologies as multistability [6] or metastability [72,73]. In [53] the notion of ghost attractors is used, which would seem to suit into our informational fields and its bifurcations under the movement of the coupling parameter in the connectivity matrix. We think it could also serve as a valuable tool in line of the perspectives developed in [5]. In [54] a detailed study of different attractor repertoires is studied for a global brain network for which a (local, global and dynamic) Hopfield model is defined. Note that, depending on parameters, the authors observe the bifurcation of dozens or even hundreds of stationary hyperbolic fixed points, which basis of attraction and stability properties has to be analyzed in an heuristic way by simulation of thousands of realizations related to initial conditions. The comparison with empirical data occurs at the edge of multistability. All of these phenomena could naturally enter into a broader global approach, by the study of an abstract formulation of their associated informational fields, creating a network of attractors and their connections, and their dependence on parameters to better understand transitions and bifurcations of structures. Moreover, the continuous approach for ISs possibly leads, if we are able to fit them to real data of brain activity, to a useful tool to analyze the functional connectivity dynamics [74]. The ability to mathematically represent human consciousness can help to understand the nature of conscious processes in human mind, and the dynamical systems approach may indeed be also a correct tool to start this trip towards a complementary mathematical approach to consciousness. Indeed, informational structures allow to associate the processes underlying consciousness to a huge functional and changing set of continuous flow structures. However, we think we are still far to the aim of describing a conscious experience with the actual development, which will be continued in the near future. We have introduced the postulates for IIT associated to ISs, and, in particular, we have developed a first approach for definitions of information and integration, leading to the introduction of measures as cei (cause-effect information) or ϕ (integrated information of a mechanism). However, in order to make this approximation comparable to the latest published IIT version for discrete dynamical systems of this theory [21], a series of additional developments are required: In IIT 3. 0 the exclusion postulate is applied to the causes and effects of the individual mechanisms, so that only the purview that generates the maximum value of integrated information intrinsically exists as the core cause (or core effect) of the candidate mechanism that generates a concept. In fact, a concept is defined as a mechanism in a state that specifies a maximally irreducible cause and effect, the core cause and the core effect, and both specify what the concept is about. The core cause (or effect) can be any set of elements, but only the set that generates the maximum value of integrated information in the past (or future) exists intrinsically as such. We can see the present work as a simplified particular case in which the candidates to core cause and effect, the candidate set to be a concept and the complete system match. But in general they will be different sets. In addition, elements outside the candidate set should be treated as background conditions. On the other hand, the partitions in this work isolate one part of the system from another for any instant of time. However, in line with IIT 3. 0, partitions should be performed in this way: two consecutive time steps are considered (past and present or present and future) and the elements of the candidate mechanism in the present are joined to the elements of the candidate to core cause in the past (or core effect in the future) to form a set that will be separated in two, so that in each of the parts there may be elements of the two time steps or not. It is also necessary to distinguish between mechanisms for which the value of ϕ for each concept or quale sensu stricto is calculated, and systems of mechanisms for which Φ for each complex or quale sensu lato is calculated. For the later a conceptual structure or constellation of concepts is defined as the set of all the concepts generated by a system of mechanisms in a state. It is necessary to specify how the conceptual information (CI) and the integrated conceptual information Φ of that system and its corresponding constellation of concepts are calculated. To this aim an extended version of the earth mover’s distance (EMD) is used. When the exclusion postulate is applied to systems of mechanisms, the complexes are defined as maximum irreducible conceptual structures (MICS), that is, local maxima of integrated conceptual information. To do this we should use unidirectional partitions with which to verify that integrated conceptual structures are formed and see if apart from the major complex, and in a non-overlapping way, minor complexes are formed. We are now working in all of these items, and hope, with a much higher computational complexity, a refinement of the results, also in relation to the topology of mechanisms and the integrated information. Note that the aim to relate (informational) fields and matter by a mathematical treatment is not restricted to our approach or even IIT theory. It is remarkable in this line of research the description of central neuronal systems (and more generically, biological systems) as a gauge theory where a lagrangian approach to model motion is the crucial adopted formalism ([75,76]). Our techniques are different, since we move into a more classical approach given by dynamical systems related to differential equations. In this way we take advantage of the huge theory on the qualitative analysis of differential equations. In particular, the curvature in the phase space produced by the informational field, closely related to the local stability of stationary points given by their associated eigenvalues, is translated into dynamical properties in the physical space, as a gauge theory approach. But the informational field, in our case, is, for each time, a global transformation of the phase space. This characterisation of an energy landscape (the phase space) evolves continuously in time. It remains an interesting and outstanding challenge to relate gauge theoretic treatments based upon information geometry (for example, the free energy principle) ([76], S3 appendix, or [77]) to our dynamical formulation; however, their common focus on information geometry (and manifold curvature) may admit some convergence in the future. This framework may provide a promising and fruitful perspective. Thus, our approach provides many open questions and possibilities for future work. Three key problems arise at this stage, showing the necessity for a really interdisciplinary work in the area: the necessity to define the level of information of a global attractor (a totally new question in dynamical systems’ theory), which notion of information better suits to our purposes, and how this concept of information is related to causality [78] on past and future events. We need to continue the development of the dynamical systems version of IIT 3. 0, finding the intrinsic representation of cause-effect repertoires and a measure to assess the quantity and quality of integrated information encoded in the informational structure. We also need to represent the evolution in continuous time of this information, and to find a way to measure the strength of attraction and repulsion through measure and dimension of stable and unstable sets and their intersections. Finally, use this notion to define the measure of integrated information compliant with IIT’s Φmax and describe the dynamics of this measure, by also creating computer codes for the informational structures evolution. Informational structures are really grounded in the topology of their associated mechanisms. In a very natural way, we can combine the (continuous) relation between structural networks and informational structures. A huge further research is probably deserved on this subject, not only in neuroscience, but in the huge area of dynamics on complex networks. This approach also opens the door to the development of a theory on the dependence of the structure of informational structures on parameters and on the topology on the complex graphs supporting them. To this aim, we need a theory on bifurcations of invariants and a theory on bifurcation of attractors, both in the autonomous and non-autonomous cases. In order to the comparison of this approach with real data, we need to develop global models for brain dynamics where the functionality of informational structures and informational fields can be tested. We will need to use the global models of brain dynamics [4,5, 13,74,79], allowing to the use of a brain simulator [80] which correlates with functional networks, and functional connectivity dynamics as developed from real data on human brains.
In this paper we introduce a space-time continuous version for the level of integrated information of a network on which a dynamics is defined. The concept of integrated information comes from the IIT of consciousness. By a strict mathematical formulation, we complement the existing IIT theoretical framework from a dynamical systems perspective. In other words, we develop the bases for a continuous mathematical approach to IIT introducing a dynamical system as the driving rule of a given mechanism. We also introduce and define the concepts of Informational Structure and Informational Field as the complex network with the power to ascertain the dynamics (past and future scenarios) of the studied phenomena. The detailed description of an informational structure is showing the cause-effect power of a mechanism in a state and thus, a characterization of the quantity and quality of information, and the way this is integrated. We firstly introduce how network patterns arise from dynamic phenomena on networks, leading to the concept of informational structure. Then, we formally introduce the mathematical objects supporting the theory, from graphs to informational structures, throughout the integration of dynamics on graphs with a global model of differential equations. After this, we formally present some of the IIT’s postulates associated to a given mechanism. Finally, we provide the quantitative and qualitative characterization of the integrated information, and how it depends on the geometry of the mechanism.
neuroscience
cognitive neuroscience
systems science
mathematics
probability distribution
algebra
computer and information sciences
consciousness
dynamical systems
differential equations
probability theory
eigenvectors
linear algebra
information theory
biology and life sciences
physical sciences
theories of consciousness
cognitive science
eigenvalues
Informational structures: A dynamical system approach for integrated information