Abstract:We propose new measures of shared information, unique information and synergistic information that can be used to decompose the mutual information of a pair of random variables (Y, Z) with a third random variable X. Our measures are motivated by an operational idea of unique information, which suggests that shared information and unique information should depend only on the marginal distributions of the pairs (X, Y ) and (X, Z). Although this invariance property has not been studied before, it is satisfied by other proposed measures of shared information. The invariance property does not uniquely determine our new measures, but it implies that the functions that we define are bounds to any other measures satisfying the same invariance property. We study properties of our measures and compare them to other candidate measures.

How can the information that a set {X1, . . . , Xn} of random variables contains about another random variable S be decomposed? To what extent do different subgroups provide the same, i.e. shared or redundant, information, carry unique information or interact for the emergence of synergistic information? Recently Williams and Beer proposed such a decomposition based on natural properties for shared information. While these properties fix the structure of the decomposition, they do not uniquely specify the values of the different terms. Therefore, we investigate additional properties such as strong symmetry and left monotonicity. We find that strong symmetry is incompatible with the properties proposed by Williams and Beer. Although left monotonicity is a very natural property for an information measure it is not fulfilled by any of the proposed measures. We also study a geometric framework for information decompositions and ask whether it is possible to represent shared information by a family of posterior distributions. Finally, we draw connections to the notions of shared knowledge and common knowledge in game theory. While many people believe that independent variables cannot share information, we show that in game theory independent agents can have shared knowledge, but not common knowledge. We conclude that intuition and heuristic arguments do not suffice when arguing about information.

Sleep encompasses approximately a third of our lifetime, yet its purpose and biological function are not well understood. Without sleep optimal brain functioning such as responsiveness to stimuli, information processing, or learning may be impaired. Such observations suggest that sleep plays a crucial role in organizing or reorganizing neuronal networks of the brain toward states where information processing is optimized.Increasing evidence suggests that cortical neuronal networks operate near a critical state characterized by balanced activity patterns, which supports optimal information processing. However, it remains unknown whether critical dynamics is affected in the course of wake and sleep, which would also impact information processing. Here, we show that signatures of criticality are progressively disturbed during wake and restored by sleep. We demonstrate that the precise power-laws governing the cascading activity of neuronal avalanches and the distribution of phase-lock intervals in human electroencephalographic recordings are increasingly disarranged during sustained wakefulness. These changes are accompanied by a decrease in variability of synchronization. Interpreted in the context of a critical branching process, these seemingly different findings indicate a decline of balanced activity and progressive distance from criticality toward states characterized by an imbalance toward excitation where larger events prevail dynamics. Conversely, sleep restores the critical state resulting in recovered power-law characteristics in activity and variability of synchronization. These findings support the intriguing hypothesis that sleep may be important to reorganize cortical network dynamics to a critical state thereby assuring optimal computational capabilities for the following time awake.

Abstract.Measures of complexity are of immediate interest for the field of autonomous robots both as a means to classify the behavior and as an objective function for the autonomous development of robot behavior. In the present paper we consider predictive information in sensor space as a measure for the behavioral complexity of a two-wheel embodied robot moving in a rectangular arena with several obstacles. The mutual information (MI) between past and future sensor values is found empirically to have a maximum for a behavior which is both explorative and sensitive to the environment. This makes predictive information a prospective candidate as an objective function for the autonomous development of such behaviors. We derive theoretical expressions for the MI in order to obtain an explicit update rule for the gradient ascent dynamics. Interestingly, in the case of a linear or linearized model of the sensorimotor dynamics the structure of the learning rule derived depends only on the dynamical properties while the value of the MI influences only the learning rate. In this way the problem of the prohibitively large sampling times for information theoretic measures can be circumvented. This result can be generalized and may help to derive explicit learning rules from complexity theoretic measures. PACS

In experiments, the dynamical behavior of systems is reflected in time series. Due to the finiteness of the observational data set, it is not possible to reconstruct the invariant measure up to an arbitrarily fine resolution and an arbitrarily high embedding dimension. These restrictions limit our ability to distinguish between signals generated by different systems, such as regular, chaotic, or stochastic ones, when analyzed from a time series point of view. We propose to classify the signal behavior, without referring to any specific model, as stochastic or deterministic on a certain scale of the resolution epsilon, according to the dependence of the (epsilon,tau) entropy, h(epsilon, tau), and the finite size Lyapunov exponent lambda(epsilon) on epsilon.

We consider biological individuality in terms of information theoretic and graphical principles. Our purpose is to extract through an algorithmic decomposition system-environment boundaries supporting individuality. We infer or detect evolved individuals rather than assume that they exist. Given a set of consistent measurements over time, we discover a coarsegrained or quantized description on a system, inducing partitions (which can be nested).Legitimate individual partitions will propagate information from the past into the future, whereas spurious aggregations will not. Individuals are therefore de ned in terms of ongoing, bounded information processing units rather than lists of static features or conventional replication-based de nitions which tend to fail in the case of cultural change. One virtue of this approach is that it could expand the scope of what we consider adaptive or biological phenomena, particularly in the microscopic and macroscopic regimes of molecular and social phenomena.

We analyze the classical and quantum properties of the integrable dimer problem. The classical version exhibits exactly one bifurcation in phase space, which gives birth to permutational symmetry broken trajectories and a separatrix. The quantum analysis yields all tunneling rates (splittings) in leading order of perturbation. In the semiclassical regime the eigenvalue spectrum obtained by numerically exact diagonalization allows one to conclude about the presence of a separatrix and a bifurcation in the corresponding classical model. PACS numbers: 05.45.+b, 03.20.+i, 03.65.Sq The problem of correspondence between classical and quantum-mechanical properties of nonlinear systems is currently an object of wide interest [1]. One interesting topic concerns Hamiltonian systems with a given symmetry (e.g., some permutational symmetry), where classical trajectories exist which are not invariant under the corresponding symmetry operation. This topic appears in analyzing selective bond excitation in chemistry and in the quantization of discrete breathers [2].We consider an integrable system with two degrees of freedom (TDF), whose classical version exhibits exactly one bifurcation (of periodic orbits) and separatrix manifold. This manifold cuts the phase space into three parts-one with invariant trajectories, and two with noninvariant trajectories, where the corresponding symmetry is the permutational one. By varying a single parameter it is possible to "switch" between these phase space parts by crossing the separatrix. It appears natural to expect in the quantum case a drastic change in the splittings of energy levels (which should be zero in the classical limit for the noninvariant phase space parts). However, the splittings are nonzero for any given value of the control parameter. The only way to avoid contradiction between the classical and quantum cases is to assume that the quantum level splittings tend to a steplike function (of, e.g., the level pair number) in the classical limit. The step should occur at the position of the classical separatrix. This problem can be coined also dynamical tunneling through a separatrix. There exist studies of the influence of classical chaos on dynamical tunneling [3]. This paper is an extension of previous studies on classical and quantum properties of the dimer system [4][5][6].We are able to trace the splittings of the level pairs using quantum perturbation methods. We consider the quasiclassical regime and show that the step indeed occurs. Therefore we are able to extract information about the classical separatrix and bifurcation. Further, we show that the quantum density of states (the second integral of motion is fixed) exhibits a sharp maximum at the separatrix energy. By calculating the corresponding classical quantity (with the help of Weyl's formula) we find that this singularity appears due to the integration over a part of the separatrix manifold which includes a hyperbolic isolated orbit.Let us consider the integrable dimer model with Hamiltonian [4]

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