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.

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.

Some Analytical Criteria for Local Activity of Three-Port CNN with Four State Variables: Analysis and Applications

2003 ◽  
Vol 13 (08) ◽  
pp. 2189-2239 ◽  
Author(s):  
Lequan Min ◽  
Jingtao Wang ◽  
Xisong Dong ◽  
Guanrong Chen
Keyword(s):  
Numerical Simulations ◽  
Wide Spectrum ◽  
Immune Surveillance ◽  
Reaction Diffusion ◽  
Diffusion Control ◽  
State Variables ◽  
Bifurcation Diagrams ◽  
Cell Parameters ◽  
Local Activity ◽  
Nonlinear Network

This paper presents some analytical criteria for local activity principle in reaction–diffusion Cellular Nonlinear Network (CNN) cells [Chua, 1997, 1999] that have four local state variables with three ports. As a first application, a cellular nonlinear network model of tumor growth and immune surveillance against cancer (GISAC) is discussed, which has cells defined by the Lefever–Erneaux equations, representing the densities of alive and dead cancer cells, as well as the number of free and bound cytotoxic cells, per unit volume. Bifurcation diagrams of the GISAC CNN provide possible explanations for the mechanism of cancer diffusion, control, and elimination. Numerical simulations show that oscillatory patterns and convergent patterns (representing cancer diffusion and elimination, respectively) may emerge if selected cell parameters are located nearby or on the edge of the chaos domain. As a second application, a smoothed Chua's oscillator circuit (SCC) CNN with three ports is studied, for which the original prototype was introduced by Chua as a dual-layer two-dimensional reaction–diffusion CNN with three state variables and two ports. Bifurcation diagrams of the SCC CNN are computed, which only demonstrate active unstable domains and edges of chaos. Numerical simulations show that evolution of patterns of the state variables of the SCC CNN can exhibit divergence, periodicity, and chaos; and the second and the fourth state variables of the SCC CNNs may exhibit generalized synchronization. These 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 nearby the edge of chaos.

EXTENDING LOCAL PASSIVITY THEORY AND HOPF BIFURCATION AT THE EDGE OF CHAOS IN OREGONATOR CNN

2012 ◽  
Vol 22 (11) ◽  
pp. 1250285
Author(s):  
CHANGBING TANG ◽  
FANGYUE CHEN ◽  
JIANBO WANG ◽  
XIANG LI
Keyword(s):  
Diffusion Coefficient ◽  
Hopf Bifurcation ◽  
Wide Spectrum ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Diffusion Term ◽  
Cell Parameters ◽  
Edge Of Chaos ◽  
Homogeneous Lattice ◽  
Coupled Cells

The fundamental local passivity theory asserts that a wide spectrum of complex behaviors may exist if the cells in the reaction–diffusion are not locally passive. This local passivity principle has provided a powerful tool for studying the complexity in a homogeneous lattice formed by coupled cells. In this paper, the complexity matrix YQ(s), which is the tool for testing the local passivity theory, is modified based on the characteristic polynomial AQ(λ). Then, the local passivity theory is applied to the study of the Oregonator CNN to judge if the cell parameters of a CNN are chosen at the edge of chaos. Analysis of the bifurcation and the numerical simulations show that nonzero diffusion term in Oregonator CNN may cause a reaction–diffusion equation oscillating under the appropriate choice of diffusion coefficient if the local passivity theory is not satisfied. That is, if the cell parameters of a CNN are chosen at the edge of chaos, the system is potentially unstable.

ANALYTICAL CRITERIA FOR LOCAL ACTIVITY OF REACTION–DIFFUSION CNN WITH FOUR STATE VARIABLES AND APPLICATIONS TO THE HODGKIN–HUXLEY EQUATION

2000 ◽  
Vol 10 (06) ◽  
pp. 1295-1343 ◽  
Author(s):  
LEQUAN MIN ◽  
KENNETH R. CROUNSE ◽  
LEON O. CHUA
Keyword(s):  
Reaction Diffusion ◽  
Purkinje Fiber ◽  
State Variables ◽  
Bifurcation Diagrams ◽  
Normal Heart ◽  
Cell Parameters ◽  
Local Activity ◽  
Edge Of Chaos ◽  
Pace Maker ◽  
Fixed Cell

This paper presents analytical criteria for local activity in reaction–diffusion Cellular Nonlinear Network (CNN) cells [Chua, 1997, 1999] with four local state variables. As a first application, we apply the criteria to a Hodgkin–Huxley CNN, which has cells defined by the equations of the cardiac Purkinje fiber model of morphogenesis that was first introduced in [Noble, 1962] to describe the long-lasting action and pace-maker potentials of the Purkinje fiber of the heart. The bifurcation diagrams of the Hodgkin–Huxley CNN's supply a possible explanation for why a heart with a normal heart-rate may stop beating suddenly: The cell parameter of a normal heart is located in a locally active unstable domain and just nearby an edge of chaos. The membrane potential along a fiber is simulated in a Hodgkin–Huxley CNN by a computer. As a second application, we present a smoothed Chua's circuit (SCC) CNN. The bifurcation diagrams of the SCC CNN's show that there does not exist a locally passive domain, and the edges of chaos corresponding to different fixed-cell parameters are significantly different. Our computer simulations show that oscillatory patterns, chaotic patterns, or divergent patterns may emerge if the selected cell parameters are located in locally active domains but nearby the edge of chaos. This research demonstrates once again 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.

scholarly journals TESTING FOR LOCAL ACTIVITY AND EDGE OF CHAOS

2001 ◽  
Vol 11 (06) ◽  
pp. 1495-1591 ◽  
Author(s):  
TAO YANG ◽  
LEON O. CHUA
Keyword(s):  
Reaction Diffusion ◽  
Active Regions ◽  
Second Order ◽  
Third Order ◽  
Cell Parameters ◽  
Nonlinear Networks ◽  
Local Activity ◽  
Edge Of Chaos ◽  
Fundamental Limitations ◽  
Emergence Of Complexity

In this paper we study the local activity, local passivity and edge of chaos of continuous-time reaction–diffusion cellular nonlinear networks (CNN) with one-port first-order, one-port second-order, two-port second-order, two-port third-order and three-port third-order cells. We prove that the local passive regions determined by cell impedance ZQ(s) and cell admittance YQ(s) for first- and second-order cells are equivalent to each other. We also present an efficient procedure to determine the edge-of-chaos parameter region by combining the local active regions derived from YQ(s) and the pole locations of ZQ(s). In order to characterize the fundamental limitations of local passivity on the emergence of complexity we study the local active property from a parameter space spanned by both the cell parameters and the external excitations called the cell's port currents (in view of its interpretations from classical circuit theory). Analytical results of locally passive, restricted locally passive, edge-of-chaos and locally active parameter regions for CNN cells modeled by cubic nonlinearities are presented. We illustrate our results by analyzing CNN cells modeled by Chua's circuits with a cubic nonlinearity. We find that the morphology of the edge-of-chaos and the local active parameter regions have a close connection to the pattern formation behaviors of CNNs. Simulation results are presented to verify our theoretical results.

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.

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.

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.

THE LOCAL ACTIVITY CRITERIA FOR "DIFFERENCE-EQUATION" CNN

2001 ◽  
Vol 11 (02) ◽  
pp. 311-419 ◽  
Author(s):  
VALERY I. SBITNEV ◽  
TAO YANG ◽  
LEON O. CHUA
Keyword(s):  
Conformal Mapping ◽  
Difference Equation ◽  
Laplace Transform ◽  
Difference Equations ◽  
Continuous Time ◽  
Reaction Diffusion ◽  
Surprising Result ◽  
Passive Region ◽  
Local Activity ◽  
Edge Of Chaos

In this paper we use an exponential conformal mapping and a z-transform to "translate" the local activity criteria for continuous-time reaction–diffusion cellular nonlinear networks (CNN) to those for difference-equation CNNs. A difference-equation CNN is modeled by a set of difference equations with a constant sampling interval δt>0. Since a difference-equation CNN tends to a continuous-time CNN when δt→0, we can view the Laplace transform of a continuous-time CNN as the limit of the conformal-mapping z-transform of a corresponding difference-equation CNN. Based on the relation between Laplace transform and our conformal-mapping z-transform, we extend the local activity criteria from a continuous-time CNN to a difference-equation CNN. We have proved the rather surprising result that the class of all reaction–diffusion difference-equation CNNs with two state variables and one diffusion coefficient is locally active everywhere, i.e. its local passive region is empty. In particular, as δt→0, the local-passive region of a continuous-time CNN cell transforms into the "edge-of-chaos" region of a corresponding difference-equation CNN cell with δt>0. Remarkably, as δt→0 the locally active edge-of-chaos region degenerates into a locally passive region as the difference equation tends to a differential equation. These results highlight a fundamental difference between the qualitative properties of systems of nonlinear differential- and difference-equations.

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