Analytical Criteria for Local Activity and Applications  to the Oregonator CNN

Lee, Min; Crounse, K.R.; Chua, Leon O.

doi:10.1142/s0218127400000037

Cited by 26 publications

(20 citation statements)

References 8 publications

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“…[Dogaru & Chua, 1998;Min et al, 2000a]. (Edge of chaos with respect to the equilibrium point Q i ) A "Reaction-Diffusion" CNN with one "diffusion coefficient" D 1 (resp.…”

Section: A Main Theorem Of Local Activitymentioning

confidence: 99%

“…By arbitrarily choosing parameters of CNNs within or nearby the edge of chaos domain, complex dynamic behaviors may become abundant, even with only a very small perturbations of parameters [Dogaru & Chua, 1998a-1998cMin et al, 2000aMin et al, , 2000bMin & Yu, 2000].…”

Section: A Main Theorem Of Local Activitymentioning

confidence: 99%

“…Now, we introduce a set of theorems for testing the local activity of the CNNs with three state variables and two ports (see [Min & Yu, 2000] for complete proofs). …”

Section: Theoremsmentioning

confidence: 99%

“…In particular, some analytical criteria for local activity of CNNs have been successfully applied to the study of the dynamics of the CNNs [Dogaru & Chua, 1998a, 1998b, 1998cMin et al 2000aMin et al , 2000bMin & Yu, 2000] related to the FitzHugh-Nagumo equation, the Brusselator equation, the Gierer-Meinhart equation, the Oregonator equation, the Hodgkin-Huxley equation, and the enzyme reaction equation. Some local activity criteria for "difference-equation" CNN has recently been set up, in [Sbitnev, et al, 2001].…”

Section: Introductionmentioning

confidence: 99%

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

Lee

2002

Int. J. Bifurcation Chaos

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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.

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“…[Dogaru & Chua, 1998;Min et al, 2000a]. (Edge of chaos with respect to the equilibrium point Q i ) A "Reaction-Diffusion" CNN with one "diffusion coefficient" D 1 (resp.…”

Section: A Main Theorem Of Local Activitymentioning

confidence: 99%

Section: A Main Theorem Of Local Activitymentioning

confidence: 99%

“…Now, we introduce a set of theorems for testing the local activity of the CNNs with three state variables and two ports (see [Min & Yu, 2000] for complete proofs). …”

Section: Theoremsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Lee

2002

Int. J. Bifurcation Chaos

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“…The principles and applications of local activity and edge of chaos for different classes of continuous-time CNNs had been presented in [Min et al, 2000;Dogaru & Chua, 1998c, 1998b, 1998aChua, 1999]. In this paper, we present new analytical results of locally passive and locally active parameter regions for different classes of CNNs.…”

Section: Introductionmentioning

confidence: 94%

Testing for Local Activity and Edge of Chaos

Yang

Chua

2001

Int. J. Bifurcation Chaos

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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 secondorder, 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 Z Q (s) and cell admittance Y Q (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 Y Q (s) and the pole locations of Z Q (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.

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2012

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Analytical Criteria for Local Activity and Applications to the Oregonator CNN

Cited by 26 publications

References 8 publications

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

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

Testing for Local Activity and Edge of Chaos

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