Semi-Adaptive Evolution with Spontaneous Modularity of Half-Chaotic Randomly Growing Autonomous and Open Networks

Up until now, studies of Kauffman network stability have focused on the conditions resulting from the structure of the network. Negative feedbacks have been modeled as ice (nodes that do not change their state) in an ordered phase but this blocks the possibility of breaking out of the range of correct operation. This first, very simplified approximation leads to some incorrect conclusions, e.g., that life is on the edge of chaos. We develop a second approximation, which discovers half-chaos and shows its properties. In previous works, half-chaos has been confirmed in autonomous networks, but only using node function disturbance, which does not change the network structure. Now we examine half-chaos during network growth by adding and removing nodes as a disturbance in autonomous and open networks. In such evolutions controlled by a ‘small change’ of functioning after disturbance, the half-chaos is kept but spontaneous modularity emerges and blurs the picture. Half-chaos is a state to be expected in most of the real systems studied, therefore the determinants of the variability that maintains the half-chaos are particularly important in the application of complex network knowledge.

Supercritical dynamics at the edge-of-chaos underlies optimal decision-making

Critical dynamics, characterized by scale-free neuronal avalanches, is thought to underlie optimal function in the sensory cortices by maximizing information transmission, capacity, and dynamic range. In contrast, deviations from criticality have not yet been considered to support any cognitive processes. Nonetheless, neocortical areas related to working memory and decision-making seem to rely on long-lasting periods of ignition-like persistent firing. Such firing patterns are reminiscent of supercritical states where runaway excitation dominates the circuit dynamics. In addition, a macroscopic gradient of the relative density of Somatostatin (SST+) and Parvalbumin (PV+) inhibitory interneurons throughout the cortical hierarchy has been suggested to determine the functional specialization of low- versus high-order cortex. These observations thus raise the question of whether persistent activity in high-order areas results from the intrinsic features of the neocortical circuitry. We used an attractor model of the canonical cortical circuit performing a perceptual decision-making task to address this question. Our model reproduces the known saddle-node bifurcation where persistent activity emerges, merely by increasing the SST+/PV+ ratio while keeping the input and recurrent excitation constant. The regime beyond such a phase transition renders the circuit increasingly sensitive to random fluctuations of the inputs -i.e., chaotic-, defining an optimal SST+/PV+ ratio around the edge-of-chaos. Further, we show that both the optimal SST+/PV+ ratio and the region of the phase transition decrease monotonically with increasing input noise. This suggests that cortical circuits regulate their intrinsic dynamics via inhibitory interneurons to attain optimal sensitivity in the face of varying uncertainty. Hence, on the one hand, we link the emergence of supercritical dynamics at the edge-of-chaos to the gradient of the SST+/PV+ ratio along the cortical hierarchy, and, on the other hand, explain the behavioral effects of the differential regulation of SST+ and PV+ interneurons by neuromodulators like acetylcholine in the presence of input uncertainty.

Chaos arising from the hydrological behaviour of a floodplain river during the last century

The hydrological regime is the main factor governing the functioning of floodplain rivers. A full comprehension of its dynamic leads to a better understanding of the system’s behaviour and of the proper methods that must be used. We analysed the daily water level of the Paraná River during the last century at three gauge stations using linear and non-linear tools to characterise the hydrological dynamic and to analyse to what extent chaotic behaviour prevails. The three water level time series were characterised as non-linear and non-stationary by power spectrum, autocorrelation function, and surrogate test analyses. A strange attractor was developed when the phase space was reconstructed, having a low dimensional chaos supported by correlation dimension, positive maximum Lyapunov exponents, and recurrence quantification analysis. In line with this, the system resulted unpredictable with a threshold by sample entropy, and with an intermediate hydrological complexity, while Hurst exponent characterised the system as persistent and with sensitive dependence on initial conditions. In a general overview, all the evidence obtained indicates that the Paraná River’s behaviour is at the edge of chaos. A latitudinal gradient of decreasing chaoticity was observed as the floodplain extent increased, whereas complexity was highest at the intermediate river station due to the inflow of tributaries with different hydrology. This paper attempts to offer some additional insights for understanding the hydrological behaviour of floodplain rivers and the proper methods to understand their complexity.

Pattern Formation in a RD-MCNN with Locally Active Memristors

This chapter presents the mathematical investigation of the emergence of static patterns in a Reaction–Diffusion Memristor Cellular Nonlinear Network (RD-MCNN) structure via the application of the theory of local activity. The proposed RD-MCNN has a planar grid structure, which consists of identical memristive cells, and the couplings are established in a purely resistive fashion. The single cell has a compact design being composed of a locally active memristor in parallel with a capacitor, besides the bias circuitry, namely a DC voltage source and its series resistor. We first introduce the mathematical model of the locally active memristor and then study the main characteristics of its AC equivalent circuit. Later on, we perform a stability analysis to obtain the stability criteria for the single cell. Consequently, we apply the theory of local activity to extract the parameter space associated with locally active, edge-of-chaos, and sharp-edge-of-chaos domains, performing all the necessary calculations parametrically. The corresponding parameter space domains are represented in terms of intrinsic cell characteristics such as the DC operating point, the capacitance, and the coupling resistance. Finally, we simulate the proposed RD-MCNN structure where we demonstrate the emergence of pattern formation for various values of the design parameters.

Simulation Research on the Complexity of Life Game

The relationship between complexity and various factors is explored through the simulation of the three neighbor ways of the game of life. It mainly discusses the state evolution process of cell populations under various evolutionary laws, various environmental scales and various initial states. Based on the discovery of a novel, long-lived and simple cell with an initial state, the periodic and stable cell morphology in Game of Life is introduced, thus reflecting the related complexity factors and changes. By simulating various environmental boundaries and comparing the steady-state graphs, it is concluded that a closed system will cause certain limitations in the final outcome. The limited environment will prevent the cell from expanding outward, but it can also create more periodic patterns. A limited environment is simultaneously an important factor in simplifying the system. In addition to the environment, the edge of chaos is also an important factor in the complexity of the system. An appropriate evolution rule can help the entire system find a balance in the chaos and present stable and interesting patterns. In addition, the correct neighbor method has a positive effect on the change of the cell. Finally, an infinite loop mode is set up to illustrate once again the wonder and complexity of Game of Life.

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.