scholarly journals Computing Integrated Information (Φ) in Discrete Dynamical Systems with Multi-Valued Elements

Entropy ◽  
2020 ◽  
Vol 23 (1) ◽  
pp. 6
Author(s):  
Juan D. Gomez ◽  
William G. P. Mayner ◽  
Maggie Beheler-Amass ◽  
Giulio Tononi ◽  
Larissa Albantakis
Keyword(s):  
Dynamical Systems ◽  
Software Package ◽  
Causal Structure ◽  
Original System ◽  
Causal Analysis ◽  
Discrete Dynamical Systems ◽  
Random Networks ◽  
Mathematical Framework ◽  
Integrated Information Theory ◽  
Integrated Information

Integrated information theory (IIT) provides a mathematical framework to characterize the cause-effect structure of a physical system and its amount of integrated information (Φ). An accompanying Python software package (“PyPhi”) was recently introduced to implement this framework for the causal analysis of discrete dynamical systems of binary elements. Here, we present an update to PyPhi that extends its applicability to systems constituted of discrete, but multi-valued elements. This allows us to analyze and compare general causal properties of random networks made up of binary, ternary, quaternary, and mixed nodes. Moreover, we apply the developed tools for causal analysis to a simple non-binary regulatory network model (p53-Mdm2) and discuss commonly used binarization methods in light of their capacity to preserve the causal structure of the original system with multi-valued elements.

scholarly journals Integrated Information Theory and Isomorphic Feed-Forward Philosophical Zombies

Entropy ◽  
2019 ◽  
Vol 21 (11) ◽  
pp. 1073 ◽  
Author(s):  
Jake R. Hanson ◽  
Sara I. Walker
Keyword(s):  
Information Theory ◽  
Subjective Experience ◽  
Original System ◽  
Necessary Condition ◽  
Mathematical Framework ◽  
Hard Problem ◽  
The Hard Problem ◽  
Labeling Scheme ◽  
Integrated Information Theory ◽  
Integrated Information

Any theory amenable to scientific inquiry must have testable consequences. This minimal criterion is uniquely challenging for the study of consciousness, as we do not know if it is possible to confirm via observation from the outside whether or not a physical system knows what it feels like to have an inside—a challenge referred to as the “hard problem” of consciousness. To arrive at a theory of consciousness, the hard problem has motivated development of phenomenological approaches that adopt assumptions of what properties consciousness has based on first-hand experience and, from these, derive the physical processes that give rise to these properties. A leading theory adopting this approach is Integrated Information Theory (IIT), which assumes our subjective experience is a “unified whole”, subsequently yielding a requirement for physical feedback as a necessary condition for consciousness. Here, we develop a mathematical framework to assess the validity of this assumption by testing it in the context of isomorphic physical systems with and without feedback. The isomorphism allows us to isolate changes in Φ without affecting the size or functionality of the original system. Indeed, the only mathematical difference between a “conscious” system with Φ > 0 and an isomorphic “philosophical zombie” with Φ = 0 is a permutation of the binary labels used to internally represent functional states. This implies Φ is sensitive to functionally arbitrary aspects of a particular labeling scheme, with no clear justification in terms of phenomenological differences. In light of this, we argue any quantitative theory of consciousness, including IIT, should be invariant under isomorphisms if it is to avoid the existence of isomorphic philosophical zombies and the epistemological problems they pose.

CHAOTIFICATION OF DISCRETE DYNAMICAL SYSTEMS IN BANACH SPACES

2006 ◽  
Vol 16 (09) ◽  
pp. 2615-2636 ◽  
Author(s):  
YUMING SHI ◽  
PEI YU ◽  
GUANRONG CHEN
Keyword(s):  
Dynamical Systems ◽  
Feedback Control ◽  
Banach Spaces ◽  
Original System ◽  
Real Space ◽  
Discrete Dynamical Systems ◽  
State Feedback Control ◽  
Controlled System ◽  
Partial Difference ◽  
Finite Dimensional

This paper is concerned with chaotification of discrete dynamical systems in Banach spaces via feedback control techniques. A criterion of chaos in Banach spaces is first established. This criterion extends and improves the Marotto theorem. Discussions are carried out in general and some special Banach spaces. All the controlled systems are proved to be chaotic in the sense of both Devaney and Li–Yorke. As a consequence, a controlled system described in a finite-dimensional real space studied by Wang and Chen is shown chaotic not only in the sense of Li–Yorke but also in the sense of Devaney. The original system can be driven to be chaotic by using an arbitrarily small-amplitude state feedback control in a certain space. In addition, the Chen–Lai anti-control algorithm via feedback control with mod-operation in a finite-dimensional real space is extended to a certain infinite-dimensional Banach space, and the controlled system is shown chaotic in the sense of Devaney as well as in the sense of both Li–Yorke and Wiggins. Differing from many existing results, it is not here required that the map corresponding to the original system has a fixed point in some cases. An application of the theoretical results to a class of first-order partial difference equations is given with some numerical simulations.

APPLICATION OF ORDER PARAMETER EQUATIONS FOR THE ANALYSIS AND THE CONTROL OF NONLINEAR TIME DISCRETE DYNAMICAL SYSTEMS

1999 ◽  
Vol 09 (08) ◽  
pp. 1618-1634 ◽  
Author(s):  
P. LEVI ◽  
M. SCHANZ ◽  
S. KORNIENKO ◽  
O. KORNIENKO
Keyword(s):  
Dynamical Systems ◽  
Order Parameter ◽  
Evolution Equations ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Original System ◽  
Discrete Dynamical Systems ◽  
Order Parameters ◽  
Control Mechanisms ◽  
Manifold Theory

This work is based on the concept of order parameters of synergetics. The order parameter equations describe the behavior of a system in the vicinity of an instability and are used here not only for the analysis but also for the control of nonlinear time discrete dynamical systems. Usually, the dimensionality of the evolution equations of the order parameters is less than the dimensionality of the original evolution equations. It is, therefore, convenient to introduce control mechanisms, first in the order parameter equations, and then to use the obtained results for the control of the original system. The aim of the control in this case is to avoid chaotic behavior of the system. This is achieved by shifting appropriate bifurcation points of a period-doubling cascade. In this work we concentrate on the shifting of only the first bifurcation point. The used control mechanisms are delayed feedback schemes. As an example the well-known Hénon map is investigated. The order parameter equation is calculated using both the adiabatic elimination procedure and the center manifold theory. Using the order parameter concept two types of control mechanisms are constructed, analyzed and compared.

scholarly journals Informational Structures and Informational Fields as a Prototype for the Description of Postulates of the Integrated Information Theory

Entropy ◽  
2019 ◽  
Vol 21 (5) ◽  
pp. 493 ◽  
Author(s):  
Piotr Kalita ◽  
José A. Langa ◽  
Fernando Soler-Toscano
Keyword(s):  
Information Theory ◽  
Phase Space ◽  
Dynamical Systems ◽  
Topological Constraints ◽  
The Past ◽  
Integrated Information Theory ◽  
Abstract Formulation ◽  
Integrated Information

Informational Structures (IS) and Informational Fields (IF) have been recently introduced to deal with a continuous dynamical systems-based approach to Integrated Information Theory (IIT). IS and IF contain all the geometrical and topological constraints in the phase space. This allows one to characterize all the past and future dynamical scenarios for a system in any particular state. In this paper, we develop further steps in this direction, describing a proper continuous framework for an abstract formulation, which could serve as a prototype of the IIT postulates.

scholarly journals Hidden Attractors in Discrete Dynamical Systems

Entropy ◽  
2021 ◽  
Vol 23 (5) ◽  
pp. 616
Author(s):  
Marek Berezowski ◽  
Marcin Lawnik
Keyword(s):  
Dynamical Systems ◽  
Chaos Theory ◽  
Scientific Literature ◽  
Discrete Systems ◽  
Discrete Dynamical Systems ◽  
Continuous Systems ◽  
Hidden Attractors ◽  
Negative Effects ◽  
Chemical Reactors ◽  
Points Of View

Research using chaos theory allows for a better understanding of many phenomena modeled by means of dynamical systems. The appearance of chaos in a given process can lead to very negative effects, e.g., in the construction of bridges or in systems based on chemical reactors. This problem is important, especially when in a given dynamic process there are so-called hidden attractors. In the scientific literature, we can find many works that deal with this issue from both the theoretical and practical points of view. The vast majority of these works concern multidimensional continuous systems. Our work shows these attractors in discrete systems. They can occur in Newton’s recursion and in numerical integration.

scholarly journals Awareness and integrated information theory identify plant meristems as sites of conscious activity

PROTOPLASMA ◽  
2021 ◽  
Author(s):  
Anthony Trewavas
Keyword(s):  
Information Theory ◽  
Nervous System ◽  
Complex Networks ◽  
Environmental Changes ◽  
Biological Function ◽  
Human Consciousness ◽  
Extrinsic Information ◽  
Integrated Information Theory ◽  
Conscious Activity ◽  
Integrated Information

AbstractLacking an anatomical brain/nervous system, it is assumed plants are not conscious. The biological function of consciousness is an input to behaviour; it is adaptive (subject to selection) and based on information. Complex language makes human consciousness unique. Consciousness is equated to awareness. All organisms are aware of their surroundings, modifying their behaviour to improve survival. Awareness requires assessment too. The mechanisms of animal assessment are neural while molecular and electrical in plants. Awareness of plants being also consciousness may resolve controversy. The integrated information theory (IIT), a leading theory of consciousness, is also blind to brains, nerves and synapses. The integrated information theory indicates plant awareness involves information of two kinds: (1) communicative, extrinsic information as a result of the perception of environmental changes and (2) integrated intrinsic information located in the shoot and root meristems and possibly cambium. The combination of information constructs an information nexus in the meristems leading to assessment and behaviour. The interpretation of integrated information in meristems probably involves the complex networks built around [Ca2+]i that also enable plant learning, memory and intelligent activities. A mature plant contains a large number of conjoined, conscious or aware, meristems possibly unique in the living kingdom.

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