Non-additive entropies and statistical mechanics at the edge of chaos: a bridge between natural and social sciences

The Boltzmann-Gibbs (BG) statistical mechanics constitutes one of the pillars of contemporary theoretical physics. It is constructed upon the other pillars-classical, quantum, relativistic mechanics and Maxwell equations for electromagnetism-and its foundations are grounded on the optimization of the BG (additive) entropic functional [Formula: see text]. Its use in the realm of classical mechanics is legitimate for vast classes of nonlinear dynamical systems under the assumption that the maximal Lyapunov exponent is positive (currently referred to as strong chaos), and its validity has been experimentally verified in countless situations. It fails however when the maximal Lyapunov exponent vanishes (referred to as weak chaos), which is virtually always the case with complex natural, artificial and social systems. To overcome this type of weakness of the BG theory, a generalization was proposed in 1988 grounded on the non-additive entropic functional [Formula: see text]. The index [Formula: see text] and related ones are to be calculated, whenever mathematically tractable, from first principles and reflect the specific class of weak chaos. We review here the basics of this generalization and illustrate its validity with selected examples aiming to bridge natural and social sciences. This article is part of the theme issue 'Thermodynamics 2.0: Bridging the natural and social sciences (Part 2)'.

How critical is brain criticality?

Criticality is the singular state of complex systems poised at the brink of a phase transition between order and randomness. Such systems display remarkable information-processing capabilities, evoking the compelling hypothesis that the brain may itself be critical. This foundational idea is now drawing renewed interest thanks to high-density data and converging cross-disciplinary knowledge. Together, these lines of inquiry have shed light on the intimate link between criticality, computation, and cognition. Here, we review these emerging trends in criticality neuroscience, highlighting new data pertaining to the edge of chaos and near-criticality, and making a case for the distance to criticality as a useful metric for probing cognitive states and mental illness. This unfolding progress in the field contributes to establishing criticality theory as a powerful mechanistic framework for studying emergent function and its efficiency in both biological and artificial neural networks.

Generation of Multi-Lobe Chua Corsage Memristor and Its Neural Oscillation

The Chua corsage memristor (CCM) is considered as one of the candidates for the realization of biological neuron models due to its rich neuromorphic behaviors. In this paper, a universal model for m-lobe CCM memristor is proposed. Moreover, a novel small-signal equivalent circuit with one capacitor is derived based on the proposed model to determine the edge of chaos and obtain the zero-pole diagrams and analyze the frequency response and oscillation mechanism of the m-lobe CCM system, which are discussed in detail. In view of existence of the edge of chaos, the frequency response and the oscillation mechanism of the simplest oscillator is analysed using the proposed model. Finally, the proposed model has exhibited some essential neural oscillation, including the stable limit cycle, supercritical Hopf bifurcation, spiking and bursting oscillation. This study also reveals a previously undiscovered behavior of bursting oscillation in a CCM system.

Dynamics and Information Import in Recurrent Neural Networks

Recurrent neural networks (RNNs) are complex dynamical systems, capable of ongoing activity without any driving input. The long-term behavior of free-running RNNs, described by periodic, chaotic and fixed point attractors, is controlled by the statistics of the neural connection weights, such as the density d of non-zero connections, or the balance b between excitatory and inhibitory connections. However, for information processing purposes, RNNs need to receive external input signals, and it is not clear which of the dynamical regimes is optimal for this information import. We use both the average correlations C and the mutual information I between the momentary input vector and the next system state vector as quantitative measures of information import and analyze their dependence on the balance and density of the network. Remarkably, both resulting phase diagrams C(b, d) and I(b, d) are highly consistent, pointing to a link between the dynamical systems and the information-processing approach to complex systems. Information import is maximal not at the "edge of chaos," which is optimally suited for computation, but surprisingly in the low-density chaotic regime and at the border between the chaotic and fixed point regime. Moreover, we find a completely new type of resonance phenomenon, which we call "Import Resonance" (IR), where the information import shows a maximum, i.e., a peak-like dependence on the coupling strength between the RNN and its external input. IR complements previously found Recurrence Resonance (RR), where correlation and mutual information of successive system states peak for a certain amplitude of noise added to the system. Both IR and RR can be exploited to optimize information processing in artificial neural networks and might also play a crucial role in biological neural systems.

Computational Power of Asynchronously Tuned Automata Enhancing the Unfolded Edge of Chaos

Asynchronously tuned elementary cellular automata (AT-ECA) are described with respect to the relationship between active and passive updating, and that spells out the relationship between synchronous and asynchronous updating. Mutual tuning between synchronous and asynchronous updating can be interpreted as the model for dissipative structure, and that can reveal the critical property in the phase transition from order to chaos. Since asynchronous tuning easily makes behavior at the edge of chaos, the property of AT-ECA is called the unfolded edge of chaos. The computational power of AT-ECA is evaluated by the quantitative measure of computational universality and efficiency. It shows that the computational efficiency of AT-ECA is much higher than that of synchronous ECA and asynchronous ECA.

Optimal Input Representation in Neural Systems at the Edge of Chaos

Shedding light on how biological systems represent, process and store information in noisy environments is a key and challenging goal. A stimulating, though controversial, hypothesis poses that operating in dynamical regimes near the edge of a phase transition, i.e., at criticality or the "edge of chaos", can provide information-processing living systems with important operational advantages, creating, e.g., an optimal trade-off between robustness and flexibility. Here, we elaborate on a recent theoretical result, which establishes that the spectrum of covariance matrices of neural networks representing complex inputs in a robust way needs to decay as a power-law of the rank, with an exponent close to unity, a result that has been indeed experimentally verified in neurons of the mouse visual cortex. Aimed at understanding and mimicking these results, we construct an artificial neural network and train it to classify images. We find that the best performance in such a task is obtained when the network operates near the critical point, at which the eigenspectrum of the covariance matrix follows the very same statistics as actual neurons do. Thus, we conclude that operating near criticality can also have-besides the usually alleged virtues-the advantage of allowing for flexible, robust and efficient input representations.

The Role of Entropy in the Development of Economics

The aim of this paper is to examine the role of thermodynamics, and in particular, entropy, for the development of economics within the last 150 years. The use of entropy has not only led to a significant increase in economic knowledge, but also to the emergence of such scientific disciplines as econophysics, complexity economics and quantum economics. Nowadays, an interesting phenomenon can be observed; namely, that rapid progress in economics is being made outside the mainstream. The first significant achievement was the emergence of entropy economics in the early 1970s, which introduced the second law of thermodynamics to considerations regarding production processes. In this way, not only was ecological economics born but also an entropy-based econometric approach developed. This paper shows that non-extensive cross-entropy econometrics is a valuable complement to traditional econometrics as it explains phenomena based on power-law probability distribution and enables econometric model estimation for non-ergodic ill-behaved (troublesome) inverse problems. Furthermore, the entropy economics has accelerated the emergence of modern econophysics and complexity economics. These new directions of research have led to many interesting discoveries that usually contradict the claims of conventional economics. Econophysics has questioned the efficient market hypothesis, while complexity economics has shown that markets and economies function best near the edge of chaos. Quantum economics has already appeared on the horizon, which recognizes money as a fundamental measurement device in the economy. The development of these sciences may indicate the need to reformulate all mainstream economics from its foundations.

Evolving Always-Critical Networks

Living beings share several common features at the molecular level, but there are very few large-scale “operating principles” which hold for all (or almost all) organisms. However, biology is subject to a deluge of data, and as such, general concepts such as this would be extremely valuable. One interesting candidate is the “criticality” principle, which claims that biological evolution favors those dynamical regimes that are intermediaries between ordered and disordered states (i.e., “at the edge of chaos”). The reasons why this should be the case and experimental evidence are briefly discussed, observing that gene regulatory networks are indeed often found on, or close to, the critical boundaries. Therefore, assuming that criticality provides an edge, it is important to ascertain whether systems that are critical can further evolve while remaining critical. In order to explore the possibility of achieving such “always-critical” evolution, we resort to simulated evolution, by suitably modifying a genetic algorithm in such a way that the newly-generated individuals are constrained to be critical. It is then shown that these modified genetic algorithms can actually develop critical gene regulatory networks with two interesting (and quite different) features of biological significance, involving, in one case, the average gene activation values and, in the other case, the response to perturbations. These two cases suggest that it is often possible to evolve networks with interesting properties without losing the advantages of criticality. The evolved networks also show some interesting features which are discussed.

Stochasticity of Biological Soft Matter: Emerging Concepts in Intrinsically Disordered Proteins and Biological Phase Separation

At the turn of this century, cardinal changes took place in the perceptions of the structure and function of proteins, as well as in the organizational principles of membrane-less organelles. As a result, the model of the organization of living matter is changing to one described by highly dynamic biological soft matter positioned at the edge of chaos. Intrinsically disordered proteins (IDPs) and membrane-less organelles are key examples of this new outlook and may represent a critical foundation of life, defining its complexity and the evolution of living things.

The edge of chaos: A nonlinear view of psychoanalytic technique

The field of nonlinear dynamics (or chaos theory) provides ways to expand concepts of psychoanalytic process that have implications for the technique of psychoanalysis. This paper describes how concepts of "the edge of chaos," emergence, attractors, and coupled oscillators can help shape analytic technique resulting in an approach to doing analysis which is at the same time freer and more firmly based in an enlarged understanding of the ways in which psychoanalysis works than some current recommendation about technique. Illustrations from a lengthy analysis of an analysand with obsessive-compulsive disorder show this approach in action.

Liquid computing on and off the edge of chaos with a striatal microcircuit

In reinforcement learning theories of the basal ganglia, there is a need for the expected rewards corresponding to relevant environmental states to be maintained and modified during the learning process. However, the representation of these states that allows them to be associated with reward expectations remains unclear. Previous studies have tended to rely on pre-defined partitioning of states encoded by disjunct neuronal groups or sparse topological drives. A more likely scenario is that striatal neurons are involved in the encoding of multiple different states through their spike patterns, and that an appropriate partitioning of an environment is learned on the basis of task constraints, thus minimizing the number of states involved in solving a particular task. Here we show that striatal activity is sufficient to implement a liquid state, an important prerequisite for such a computation, whereby transient patterns of striatal activity are mapped onto the relevant states. We develop a simple small scale model of the striatum which can reproduce key features of the experimentally observed activity of the major cell types of the striatum. We then use the activity of this network as input for the supervised training of four simple linear readouts to learn three different functions on a plane, where the network is stimulated with the spike coded position of the agent. We discover that the network configuration that best reproduces striatal activity statistics lies on the edge of chaos and has good performance on all three tasks, but that in general, the edge of chaos is a poor predictor of network performance.