Edge of Chaos and Local Activity Domain of the Gierer–Meinhardt CNN

1998 ◽  
Vol 08 (12) ◽  
pp. 2321-2340 ◽  
Author(s):  
Radu Dogaru ◽  
Leon O. Chua
Keyword(s):  
Activity Theory ◽  
Parameter Space ◽  
Cell Parameter ◽  
Reaction Diffusion ◽  
Specific Reaction ◽  
Local Activity ◽  
Nonlinear Network ◽  
Edge Of Chaos ◽  
Homogeneous Lattice ◽  
Activity Domain

This paper present an application of the local activity theory [Chua, 1998] to a specific reaction–diffusion cellular nonlinear network (CNN) with cells defined by the model of morphogenesis first proposed in [Gierer & Meinhardt, 1972]. Both the local activity domain and a subset called the "edge of chaos" are identified in the cell parameter space. Within these domains, various cell parameter points were selected to illustrate the effectiveness of the local activity theory in choosing the parameters for the emergence of complex (static and dynamic) patterns in a homogeneous lattice formed by coupled locally active cells.

Edge of Chaos and Local Activity Domain of the Brusselator CNN

1998 ◽  
Vol 08 (06) ◽  
pp. 1107-1130 ◽  
Author(s):  
Radu Dogaru ◽  
Leon O. Chua
Keyword(s):  
Activity Theory ◽  
Parameter Space ◽  
Cell Parameter ◽  
Reaction Diffusion ◽  
Specific Reaction ◽  
Local Activity ◽  
Nonlinear Network ◽  
Edge Of Chaos ◽  
Homogeneous Lattice ◽  
Activity Domain

This paper presents an application of the local activity theory [Chua, 1998] to a specific reaction–diffusion cellular nonlinear network (CNN) with cells defined by a trimolecular model, called the Brusselator. Both the local activity domain and a subset called the "edge of chaos" are identified in the cell parameter space. Within these domains, various cell parameter points were selected to illustrate the effectiveness of the local activity theory in choosing the parameters for the emergence of complex (static and dynamic) patterns in a homogeneous lattice formed by coupled locally active cells.

ANALYTICAL CRITERIA FOR LOCAL ACTIVITY AND APPLICATIONS TO THE OREGONATOR CNN

2000 ◽  
Vol 10 (01) ◽  
pp. 25-71 ◽  
Author(s):  
LEQUAN MIN ◽  
KENNETH R. CROUNSE ◽  
LEON O. CHUA
Keyword(s):  
Cell Parameter ◽  
High Energy ◽  
Reaction Model ◽  
Cell Parameters ◽  
Local Activity ◽  
Nonlinear Network ◽  
Edge Of Chaos ◽  
Homogeneous Lattice ◽  
Coupled Cells ◽  
Emergence Of Complexity

This paper presents analytic criteria for local activity in one-port Cellular Nonlinear Network (CNN) cells [Chua, 1997, 1999], and gives the applications to the Oregonator CNN defined by the kinetic chemical reaction model of morphogenesis first introduced in [Field & Noyes, 1974]. Locally active domains, locally passive domains, and the edge of chaos are identified in the cell parameter space. Computer simulations of the dynamics of several Oregonator CNN's with specific selected cell parameters in the above-mentioned domains show genesis and the emergence of complexity. Furthermore, a novel phenomena is observed that "extremely high energy" is concentrated only on a few cells in the dynamic patterns of some Oregonator CNN's whose cell parameters are located in active domains; furthermore, relaxation oscillations and "transient oscillations" can exist if the cell parameters of the Oregonator CNN are located nearby or on the edge of chaos. This research illustrates once again the effectiveness of the local activity theory in choosing the system parameters for the emergence of complex patterns (static and dynamic) in a homogeneous lattice formed by coupled cells.

LOCAL ACTIVITY OF THE VAN DER POL CNN

2004 ◽  
Vol 14 (07) ◽  
pp. 2211-2222 ◽  
Author(s):  
LEQUAN MIN ◽  
GUANRONG CHEN
Keyword(s):  
Neural Networks ◽  
Activity Theory ◽  
Bifurcation Diagram ◽  
Fourth Order ◽  
Van Der Pol ◽  
One Dimensional ◽  
Local Activity ◽  
Edge Of Chaos ◽  
Dynamical Behaviors ◽  
Activity Domain

This paper studies a class of coupled Van der Pol (CVDP) cellular neural networks (CNNs) that can be realized via a coupled fourth-order circuit with two synaptic currents. The local activity theory, developed by Chua in 1997, is applied to study the CVDP CNN, thereby revealing that the bifurcation diagram of the CVDP CNN has a local activity domain with an edge of chaos, as well as a one-dimensional locally passive domain. Although no chaotic phenomena have been identified in simulations, many complex dynamical behaviors have been observed, such as the co-existence of one-periodic, divergent, and convergent orbits, at the edge of chaos.

MUTATIONS OF THE "GAME OF LIFE": A GENERALIZED CELLULAR AUTOMATA PERSPECTIVE OF COMPLEX ADAPTIVE SYSTEMS

2000 ◽  
Vol 10 (08) ◽  
pp. 1821-1866 ◽  
Author(s):  
RADU DOGARU ◽  
LEON O. CHUA
Keyword(s):  
Cellular Automata ◽  
Boolean Function ◽  
Parameter Space ◽  
Cell Parameter ◽  
Piecewise Linear ◽  
Cellular Systems ◽  
Boolean Logic ◽  
Game Of Life ◽  
Local Activity ◽  
Edge Of Chaos

This paper presents a novel approach for studying the relationship between the properties of isolated cells and the emergent behavior that occurs in cellular systems formed by coupling such cells. The novelty of our approach consists of a method for precisely partitioning the cell parameter space into subdomains via the failure boundaries of the piecewise-linear CNN (cellular neural network) cells [Dogaru & Chua, 1999a] of a generalized cellular automata [Chua, 1998]. Instead of exploring the rule space via statistically defined parameters (such as λ in [Langton, 1990]), or by conducting an exhaustive search over the entire set of all possible local Boolean functions, our approach consists of exploring a deterministically structured parameter space built around parameter points corresponding to "interesting" local Boolean logic functions. The well-known "Game of Life" [Berlekamp et al., 1982] cellular automata is reconsidered here to exemplify our approach and its advantages. Starting from a piecewise-linear representation of the classic Conway logic function called the "Game of Life", and by introducing two new cell parameters that are allowed to vary continuously over a specified domain, we are able to draw a "map-like" picture consisting of planar regions which cover the cell parameter space. A total of 148 subdomains and their failure boundaries are precisely identified and represented by colored paving stones in this mosaic picture (see Fig. 1), where each stone corresponds to a specific local Boolean function in cellular automata parlance. Except for the central "paving stone" representing the "Game of Life" Boolean function, all others are mutations uncovered by exploring the entire set of 148 subdomains and determining their dynamic behaviors. Some of these mutations lead to interesting, "artificial life"-like behavior where colonies of identical miniaturized patterns emerge and evolve from random initial conditions. To classify these emergent behaviors, we have introduced a nonhomogeneity measure, called cellular disorder measure, which was inspired by the local activity theory from [Chua, 1998]. Based on its temporal evolution, we are able to partition the cell parameter space into a class U "unstable-like" region, a class E "edge of chaos"-like region, and a class P "passive-like" region. The similarity with the "unstable", "edge of chaos" and "passive" domains defined precisely and applied to various reaction–diffusion CNN systems [Dogaru & Chua, 1998b, 1998c] opens interesting perspectives for extending the theory of local activity [Chua, 1998] to discrete-time cellular systems with nonlinear couplings. To demonstrate the potential of emergent computation in generalized cellular automata with cells designed from mutations of the "Game of Life", we present a nontrivial application of pattern detection and reconstruction from very noisy environments. In particular, our example demonstrates that patterns can be identified and reconstructed with very good accuracy even from images where the noise level is ten times stronger than the uncorrupted image.

scholarly journals Pattern Formation in a RD-MCNN with Locally Active Memristors

2021 ◽  
Author(s):  
Ahmet Samil Demirkol ◽  
Alon Ascoli ◽  
Ioannis Messaris ◽  
Ronald Tetzlaff
Keyword(s):  
Pattern Formation ◽  
Single Cell ◽  
Parameter Space ◽  
Voltage Source ◽  
Local Activity ◽  
Nonlinear Network ◽  
Edge Of Chaos ◽  
The Stability ◽  
Series Resistor ◽  
Compact Design

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.

scholarly journals SOME ANALYTICAL CRITERIA FOR LOCAL ACTIVITY OF TWO-PORT CNN WITH THREE OR FOUR STATE VARIABLES: ANALYSIS AND APPLICATIONS

2002 ◽  
Vol 12 (05) ◽  
pp. 931-963 ◽  
Author(s):  
LEQUAN MIN ◽  
NA YU
Keyword(s):  
Cell Model ◽  
Reaction Diffusion ◽  
State Variables ◽  
Periodic Patterns ◽  
Local Activity ◽  
Nonlinear Network ◽  
Edge Of Chaos ◽  
Excitable Membranes ◽  
Coupled Cells ◽  
Complex Patterns

The local activity principle of the Cellular Nonlinear Network (CNN) introduced by Chua [1997] has provided a powerful tool for studying the emergence of complex patterns in a homogeneous lattice formed by coupled cells. This paper presents some analytical criteria for the local activity of two-port CNN cells with three or four state variables. As a first application, a coupled excitable cell model (ECM) CNN is introduced, which has cells defined by the Chay equations representing ionic events in excitable membranes in terms of a Hodgkin–Huxley type formalism. The bifurcation diagram of the ECM CNN supplies a possible explanation for the mechanism of arrhythmia (from normal to abnormal until stopping) of excitable cells: the cell parameter is changed from an active unstable domain to an edge of chaos. The member potentials along fibers are simulated numerically, where oscillatory patterns, chaotic patterns as well as convergent patterns are observed. As a second application, a smoothed Chua's circuit (SCC) CNN with two ports is presented, whose prototype has been introduced by Chua as a dual-layer two-dimensional reaction–diffusion CNN in order to obtain Turing patterns. The bifurcation diagrams of the SCC CNN are the same as those with one port, which have only active unstable domains and edges of chaos. Numerical simulations show that in the active unstable parameter domains, the evolutions of the patterns of the state variables of the SCC CNNs can exhibit divergence, periodicity and chaos, where, in the parameter domains located in the edge of chaos, periodic patterns and divergent patterns are observed. These results demonstrate once again the effectiveness of the local activity theory in choosing the parameters for the emergence of complex patterns of CNNs.

Edge of Chaos and Local Activity Domain of FitzHugh-Nagumo Equation

1998 ◽  
Vol 08 (02) ◽  
pp. 211-257 ◽  
Author(s):  
Radu Dogaru ◽  
Leon O. Chua
Keyword(s):  
Activity Theory ◽  
Wide Spectrum ◽  
Reaction Diffusion ◽  
Fundamental Result ◽  
Activity Parameter ◽  
Cell Parameters ◽  
Parameter Domain ◽  
Local Activity ◽  
Edge Of Chaos ◽  
Nagumo Equation

The local activity theory [Chua, 97] offers a constructive analytical tool for predicting whether a nonlinear system composed of coupled cells, such as reaction-diffusion and lattice dynamical systems, can exhibit complexity. The fundamental result of the local activity theory asserts that a system cannot exhibit emergence and complexity unless its cells are locally active. This paper gives the first in-depth application of this new theory to a specific Cellular Nonlinear Network (CNN) with cells described by the FitzHugh–Nagumo Equation. Explicit inequalities which define uniquely the local activity parameter domain for the FitzHugh–Nagumo Equation are presented. It is shown that when the cell parameters are chosen within a subset of the local activity parameter domain, where at least one of the equilibrium state of the decoupled cells is stable, the probability of the emergence of complex nonhomogenous static as well as dynamic patterns is greatly enhanced regardless of the coupling parameters. This precisely-defined parameter domain is called the "edge of chaos", a terminology previously used loosely in the literature to define a related but much more ambiguous concept. Numerical simulations of the CNN dynamics corresponding to a large variety of cell parameters chosen on, or nearby, the "edge of chaos" confirmed the existence of a wide spectrum of complex behaviors, many of them with computational potentials in image processing and other applications. Several examples are presented to demonstrate the potential of the local activity theory as a novel tool in nonlinear dynamics not only from the perspective of understanding the genesis and emergence of complexity, but also as an efficient tool for choosing cell parameters in such a way that the resulting CNN is endowed with a brain-like information processing capability.

scholarly journals Edge of chaos in reaction diffusion CNN model

2017 ◽  
Vol 15 (1) ◽  
pp. 21-29 ◽  
Author(s):  
Angela Slavova ◽  
Ronald Tetzlaff
Keyword(s):  
Numerical Simulations ◽  
Parameter Space ◽  
Function Method ◽  
Method Comparison ◽  
Reaction Diffusion ◽  
Describing Function ◽  
Reaction Term ◽  
Nonlinear Network ◽  
Edge Of Chaos ◽  
Describing Function Method

Abstract In this paper, we study the dynamics of a reaction-diffusion Cellular Nonlinear Network (RD-CNN) nodel in which the reaction term is represented by Brusselator cell. We investigate the RD-CNN dynamics by means of describing function method. Comparison with classical results for Brusselator equation is provided. Then we introduce a new RD-CNN model with memristor coupling, for which the edge of chaos regime in the parameter space is determined. Numerical simulations are presented for obtaining dynamic patterns in the RD-CNN model with memristor coupling.

scholarly journals Theorems and Application of Local Activity of CNN with Five State Variables and One Port

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
Gang Xiong ◽  
Xisong Dong ◽  
Li Xie ◽  
Thomas Yang
Keyword(s):  
Bifurcation Diagram ◽  
Complex Dynamics ◽  
Nonlinear Dynamical Systems ◽  
Reaction Diffusion ◽  
Cellular Neural Network ◽  
State Variables ◽  
Nonlinear Dynamical ◽  
Local Activity ◽  
Homogeneous Lattice ◽  
Coupled Cells

Coupled nonlinear dynamical systems have been widely studied recently. However, the dynamical properties of these systems are difficult to deal with. The local activity of cellular neural network (CNN) has provided a powerful tool for studying the emergence of complex patterns in a homogeneous lattice, which is composed of coupled cells. In this paper, the analytical criteria for the local activity in reaction-diffusion CNN with five state variables and one port are presented, which consists of four theorems, including a serial of inequalities involving CNN parameters. These theorems can be used for calculating the bifurcation diagram to determine or analyze the emergence of complex dynamic patterns, such as chaos. As a case study, a reaction-diffusion CNN of hepatitis B Virus (HBV) mutation-selection model is analyzed and simulated, the bifurcation diagram is calculated. Using the diagram, numerical simulations of this CNN model provide reasonable explanations of complex mutant phenomena during therapy. Therefore, it is demonstrated that the local activity of CNN provides a practical tool for the complex dynamics study of some coupled nonlinear systems.

APPLICATIONS OF LOCAL ACTIVITY THEORY OF CNN TO CONTROLLED COUPLED OREGONATOR SYSTEMS

2008 ◽  
Vol 18 (11) ◽  
pp. 3233-3297 ◽  
Author(s):  
LEQUAN MIN ◽  
YAN MENG ◽  
LEON O. CHUA
Keyword(s):  
Numerical Simulations ◽  
Equilibrium Point ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Reaction Diffusion ◽  
Bifurcation Diagrams ◽  
Local Activity ◽  
Edge Of Chaos ◽  
Complex Patterns ◽  
Oregonator Model

The study of chemical reactions with oscillating kinetics has drawn increasing interest over the last few decades because it also contributes towards a deeper understanding of the complex phenomena of temporal and spatial organizations in biological systems. The Cellular Nonlinear Network (CNN) local activity principle introduced by Chua [1997, 2005] has provided a powerful tool for studying the emergence of complex patterns in a homogeneous lattice formed by coupled cells. Recently, Yang and Epstein proposed a reaction–diffusion Oregonator model with five variables for mimicking the Belousov–Zhabotinskii reaction. The Yang–Epstein model can generate oscillatory Turing patterns, including the twinkling eye, localized spiral and concentric wave structures. In this paper, we first propose a modified Yang–Epstein's Oregonator model by introducing a controller, and then map the revised Oregonator reaction–diffusion system into a reaction–diffusion Oregonator CNN. The Oregonator CNN has two cell equilibrium points Q1 = (0, 0, 0, 0, 0) and Q2, representing the "original" equilibrium point and an additional equilibrium point, respectively. The bifurcation diagrams of the Oregonator CNN are calculated using the analytical criteria for local activity. The bifurcation diagrams of the Oregonator CNN at Q1 have only locally active and unstable regions; and the ones at Q2 have locally passive regions, locally active and unstable regions, and edge of chaos regions. The calculated results show that the parameter groups of the Oregonator CNN which generate complex patterns are located on the edge of chaos regions, or on locally active unstable regions near the edge of chaos boundary. Numerical simulations show also that the Oregonator CNNs can generate similar dynamics patterns if the parameter groups are selected the same as those of the Yang–Epstein model. In particular, the parameters of the Yang–Epstein model which exhibit twinkling-eye patterns, and pinwheel patterns are located on the edges of chaos regions near the boundaries of locally active unstable regions with respect to Q2. The parameters of the Yang–Epstein models which exhibit labyrinthine stripelike patterns are located on the locally active unstable regions near the boundaries of the edge of chaos regions with respect to Q2. However the parameter group of the Yang–Epstein model with isolated spiral patterns is in the locally passive region near the boundary with edge of chaos with respect to Q2, whose trajectories tend to the equilibrium point Q2. Choosing a kind of triggering initial conditions given in [Chua, 1997], and the parameters of the Oregonator equations with the twinkling-eye patterns, pinwheel patterns, labyrinthine stripelike patterns, and isolated spiral patterns, three kinds of new spiral waves generated by the Oregonator CNNs were observed by numerical simulations. They seem to be essentially different patterns to those generated by the Oregonator CNNs with initial conditions consisting of equilibrium points plus small random perturbations. Our results demonstrate once again Chua's assertion that a wide spectrum of complex behaviors may exist if the corresponding CNN cell parameters are chosen in or near the edge of chaos region.

Sign in / Sign up

Export Citation Format

Share Document