Memristor Synapse-Based Morris–Lecar Model: Bifurcation Analyses and FPGA-Based Validations for Periodic and Chaotic Bursting/Spiking Firings

Electromagnetic induction current sensed by the membrane potential in biological neurons can be characterized with a memristor synapse, which can be employed to demonstrate the real oscillating voltage patterns of Barnacle muscle fibers. This paper presents a 3D autonomous memristor synapse-based Morris–Lecar (abbreviated as m-ML) model, which is implemented through introducing a memristor synapse-based induction current to substitute the externally applied current in an existing 2D nonautonomous Morris–Lecar model. Making use of one- and two-parameter bifurcation plots and time-domain representations, diverse period-adding bifurcations as well as abundant periodic and chaotic burst firings are demonstrated. Through constructing the fold and Hopf bifurcation sets of fast spiking subsystem, bifurcation analyses of these chaotic and periodic burst firings are carried out. Moreover, the periodic and chaotic spiking firings and coexisting firing behaviors are illustrated by using one- and two-parameter bifurcation plots and local attraction basins. Finally, based on a field programmable gate array (FPGA) board, a compact digital electronic neuron is fabricated for the 3D m-ML model, from which periodic and chaotic bursting/spiking firings are experimentally measured to verify the results of the numerical simulations.

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Chaotic Bursting Dynamics and Coexisting Multistable Firing Patterns in 3D Autonomous Morris–Lecar Model and Microcontroller-Based Validations

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127419501347 ◽

2019 ◽

Vol 29 (10) ◽

pp. 1950134 ◽

Cited By ~ 11

Author(s):

Bocheng Bao ◽

Qinfeng Yang ◽

Lei Zhu ◽

Han Bao ◽

Quan Xu ◽

...

Keyword(s):

Neuron Model ◽

Phase Plane ◽

Three Dimensional ◽

Model Parameters ◽

Fast Subsystem ◽

Firing Patterns ◽

Chaotic Bursting ◽

Attraction Basins ◽

Initial States ◽

Electronic Neuron

A three-dimensional (3D) autonomous Morris–Lecar (simplified as M–L) neuron model with fast and slow structures was proposed to generate periodic bursting behaviors. However, chaotic bursting dynamics and coexisting multistable firing patterns have been rarely discussed in such a 3D M–L neuron model. For some specified model parameters, MATLAB numerical plots are executed by bifurcation plots, time sequences, phase plane plots, and 0–1 tests, from which diverse forms of chaotic bursting, chaotic tonic-spiking, and periodic bursting behaviors are uncovered in the 3D M–L neuron model. Furthermore, based on the theoretically constructing fold/Hopf bifurcation sets of the fast subsystem, the bifurcation mechanism for the chaotic bursting behaviors is thereby expounded qualitatively. Particularly, through numerically plotting the attraction basins related to the initial states under two sets of specific parameters, coexisting multistable firing patterns are demonstrated in the 3D M–L neuron model also. Finally, a digitally circuit-implemented electronic neuron is generated based on a low-power microcontroller and its experimentally captured results faultlessly validate the numerical plots.

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Two-Parameter Bifurcation Analysis of Axial Flow Compressor Dynamics

1991 American Control Conference ◽

10.23919/acc.1991.4791944 ◽

1991 ◽

Cited By ~ 2

Author(s):

Der-Cherng Liaw ◽

Raymond A. Adomaitis ◽

Eyad H. Abed

Keyword(s):

Bifurcation Analysis ◽

Axial Flow ◽

Axial Flow Compressor ◽

Flow Compressor ◽

Two Parameter ◽

Parameter Bifurcation

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Two-parameter bifurcation analysis of longitudinal flight dynamics

IEEE Transactions on Aerospace and Electronic Systems ◽

10.1109/taes.2003.1238764 ◽

2003 ◽

Vol 39 (3) ◽

pp. 1103-1112 ◽

Cited By ~ 8

Author(s):

Der-Cherng Liaw ◽

Chau-Chung Song ◽

Yew-Wen Liang ◽

Wen-Ching Chung

Keyword(s):

Bifurcation Analysis ◽

Flight Dynamics ◽

Two Parameter ◽

Parameter Bifurcation

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Computational Analysis of Ca2+ Oscillatory Bio-Signals: Two-Parameter Bifurcation Diagrams

Entropy ◽

10.3390/e23070876 ◽

2021 ◽

Vol 23 (7) ◽

pp. 876

Author(s):

Wieslaw Marszalek ◽

Jan Sadecki ◽

Maciej Walczak

Keyword(s):

Equilibrium Points ◽

Autonomous Systems ◽

Cytosolic Calcium ◽

Calcium Oscillations ◽

Dynamical Model ◽

Bifurcation Diagrams ◽

Step Size ◽

Oscillatory Systems ◽

Two Parameter ◽

Parameter Bifurcation

Two types of bifurcation diagrams of cytosolic calcium nonlinear oscillatory systems are presented in rectangular areas determined by two slowly varying parameters. Verification of the periodic dynamics in the two-parameter areas requires solving the underlying model a few hundred thousand or a few million times, depending on the assumed resolution of the desired diagrams (color bifurcation figures). One type of diagram shows period-n oscillations, that is, periodic oscillations having n maximum values in one period. The second type of diagram shows frequency distributions in the rectangular areas. Each of those types of diagrams gives different information regarding the analyzed autonomous systems and they complement each other. In some parts of the considered rectangular areas, the analyzed systems may exhibit non-periodic steady-state solutions, i.e., constant (equilibrium points), oscillatory chaotic or unstable solutions. The identification process distinguishes the later types from the former one (periodic). Our bifurcation diagrams complement other possible two-parameter diagrams one may create for the same autonomous systems, for example, the diagrams of Lyapunov exponents, Ls diagrams for mixed-mode oscillations or the 0–1 test for chaos and sample entropy diagrams. Computing our two-parameter bifurcation diagrams in practice and determining the areas of periodicity is based on using an appropriate numerical solver of the underlying mathematical model (system of differential equations) with an adaptive (or constant) step-size of integration, using parallel computations. The case presented in this paper is illustrated by the diagrams for an autonomous dynamical model for cytosolic calcium oscillations, an interesting nonlinear model with three dynamical variables, sixteen parameters and various nonlinear terms of polynomial and rational types. The identified frequency of oscillations may increase or decrease a few hundred times within the assumed range of parameters, which is a rather unusual property. Such a dynamical model of cytosolic calcium oscillations, with mitochondria included, is an important model in which control of the basic functions of cells is achieved through the Ca2+ signal regulation.

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Asymmetric and chaotic responses of dry friction oscillators with different static and kinetic coefficients of friction

Meccanica ◽

10.1007/s11012-021-01382-8 ◽

2021 ◽

Author(s):

Gábor Csernák ◽

Gábor Licskó

Keyword(s):

Dry Friction ◽

Continuation Method ◽

Friction Model ◽

Lugre Friction Model ◽

Kinetic Coefficients ◽

Friction Oscillator ◽

Lugre Friction ◽

Friction Parameters ◽

Two Parameter ◽

Parameter Bifurcation

AbstractThe responses of a simple harmonically excited dry friction oscillator are analysed in the case when the coefficients of static and kinetic coefficients of friction are different. One- and two-parameter bifurcation curves are determined at suitable parameters by continuation method and the largest Lyapunov exponents of the obtained solutions are estimated. It is shown that chaotic solutions can occur in broad parameter domains—even at realistic friction parameters—that are tightly enclosed by well-defined two-parameter bifurcation curves. The performed analysis also reveals that chaotic trajectories are bifurcating from special asymmetric solutions. To check the robustness of the qualitative results, characteristic bifurcation branches of two slightly modified oscillators are also determined: one with a higher harmonic in the excitation, and another one where Coulomb friction is exchanged by a corresponding LuGre friction model. The qualitative agreement of the diagrams supports the validity of the results.

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Two-Parameter Bifurcation in a Predator–Prey System of Ivlev Type

Journal of Mathematical Analysis and Applications ◽

10.1006/jmaa.1997.5700 ◽

1998 ◽

Vol 217 (2) ◽

pp. 349-371 ◽

Cited By ~ 54

Author(s):

Jitsuro Sugie

Keyword(s):

Predator Prey ◽

Prey System ◽

Two Parameter ◽

Parameter Bifurcation

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The Yamada Model for a Self-Pulsing Laser: Bifurcation Structure for Nonidentical Decay Times of Gain and Absorber

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127420300396 ◽

2020 ◽

Vol 30 (14) ◽

pp. 2030039

Author(s):

Robert Otupiri ◽

Bernd Krauskopf ◽

Neil G. R. Broderick

Keyword(s):

Three Dimensional ◽

Equation Model ◽

Decay Rates ◽

Phase Portraits ◽

Stable Phase ◽

Bifurcation Structure ◽

Decay Times ◽

Comprehensive Picture ◽

Two Parameter ◽

Parameter Bifurcation

We consider self-pulsing in lasers with a gain section and an absorber section via a mechanism known as [Formula: see text]-switching, as described mathematically by the Yamada ordinary differential equation model for the gain, the absorber and the laser intensity. More specifically, we are interested in the case that gain and absorber decay on different time-scales. We present an overall bifurcation structure by showing how the two-parameter bifurcation diagram in the plane of pump strength versus decay rate of the gain changes with the ratio between the two decay rates. In total, there are ten cases BI to BX of qualitatively different two-parameter bifurcation diagrams, which we present with an explanation of the transitions between them. Moroever, we show for each of the associated eleven cases of structurally stable phase portraits (in open regions of the parameter space) a three-dimensional representation of the organization of phase space by the two-dimensional manifolds of saddle equilibria and saddle periodic orbits. The overall bifurcation structure provides a comprehensive picture of the observable dynamics, including multistability and excitability, which we expect to be of relevance for experimental work on [Formula: see text]-switching lasers with different kinds of saturable absorbers.

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Forced oscillations of a self-oscillating bimolecular surface reaction model

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1988.0066 ◽

1988 ◽

Vol 417 (1853) ◽

pp. 363-388 ◽

Cited By ~ 11

Keyword(s):

Surface Reaction ◽

Period Doubling ◽

Reaction Model ◽

Forced Oscillations ◽

Hopf Bifurcations ◽

Codimension Two ◽

Homoclinic Tangles ◽

Two Parameter ◽

Parameter Bifurcation ◽

Surface Reaction Model

The effects of forced oscillations in the partial pressure of a reactant is studied in a simple isothermal, bimolecular surface reaction model in which two vacant sites are required for reaction. The forced oscillations are conducted in a region of parameter space where an autonomous limit cycle is observed, and the response of the system is characterized with the aid of the stroboscopic map where a two-parameter bifurcation diagram for the map is constructed by using the amplitude and frequency of the forcing as bifurcation parameters. The various responses include subharmonic, quasi-periodic, and chaotic solutions. In addition, bistability between one or more of these responses has been observed. Bifurcation features of the stroboscopic map for this system include folds in the sides of some resonance horns, period doubling, Hopf bifurcations including hard resonances, homoclinic tangles, and several different codimension-two bifurcations.

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