Dynamics of period-doubling bifurcation to chaos in the spontaneous neural firing patterns

Two different bifurcation scenarios of spontaneous neural firing patterns with decreasing extracellular calcium concentrations were observed in the biological experiment on identical pacemakers when potassium concentrations were fixed at two different levels. Six typical experimental scenarios manifesting dynamics closely matching those previously simulated using the Hindmarsh–Rose model and Chay model are provided as representative examples. Bifurcation scenarios from period-1 bursting to period-1 spiking via a complex process and via a simple process, period-doubling bifurcation to chaos, period-adding bifurcation with chaos, and period-adding bifurcation with stochastic burstings were identified. The results not only reveal that an experimental neural pacemaker is capable of generating different bifurcation scenarios but also provide a basic framework for bifurcations in neural firing patterns in a two-dimensional parameter space.

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Bifurcation Scenarios of Neural Firing Patterns across Two Separated Chaotic Regions as Indicated by Theoretical and Biological Experimental Models

Abstract and Applied Analysis ◽

10.1155/2013/374674 ◽

2013 ◽

Vol 2013 ◽

pp. 1-12

Author(s):

Huaguang Gu ◽

Baobao Pan ◽

Jian Xu

Keyword(s):

Period Doubling ◽

Prediction Method ◽

Experimental Models ◽

Nonlinear Prediction ◽

Neural Firing ◽

Firing Patterns ◽

Chaotic Bursting ◽

Chaotic Region ◽

Period 2 ◽

Experimental Findings

Nonlinear dynamics can be used to identify relationships between different firing patterns, which play important roles in the information processing. The present study provides novel biological experimental findings regarding complex bifurcation scenarios from period-1 bursting to period-1 spiking with chaotic firing patterns. These bifurcations were found to be similar to those simulated using the Hindmarsh-Rose model across two separated chaotic regions. One chaotic region lay between period-1 and period-2 burstings. This region has not attracted much attention. The other region is a well-known comb-shaped chaotic region, and it appears after period-2 bursting. After period-2 bursting, the chaotic firings lay in a period-adding bifurcation scenario or in a period-doubling bifurcation cascade. The deterministic dynamics of the chaotic firing patterns were identified using a nonlinear prediction method. These results provided details regarding the processes and dynamics of bifurcation containing the chaotic bursting between period-1 and period-2 burstings and other chaotic firing patterns within the comb-shaped chaotic region. They also provided details regarding the relationships between different firing patterns in parameter space.

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A quasi-analytical approach to period-doubling bifurcation computation and prediction

1999 European Control Conference (ECC) ◽

10.23919/ecc.1999.7099331 ◽

1999 ◽

Cited By ~ 1

Author(s):

Daniel W. Berns ◽

Jorge L. Moiola ◽

Guanrong Chen

Keyword(s):

Analytical Approach ◽

Period Doubling ◽

Period Doubling Bifurcation

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Period-doubling bifurcation analysis and chaos control for load torque using FLC

Complex & Intelligent Systems ◽

10.1007/s40747-021-00276-2 ◽

2021 ◽

Author(s):

Eman Moustafa ◽

Abdel-Azem Sobaih ◽

Belal Abozalam ◽

Amged Sayed A. Mahmoud

Keyword(s):

Floquet Theory ◽

Chaos Control ◽

Fuzzy Logic Controller ◽

Chaotic Behavior ◽

Period Doubling ◽

Parameter Variation ◽

Period Doubling Bifurcation ◽

Load Torque ◽

Nonlinear Behaviors ◽

Scientific Fields

AbstractChaotic phenomena are observed in several practical and scientific fields; however, the chaos is harmful to systems as they can lead them to be unstable. Consequently, the purpose of this study is to analyze the bifurcation of permanent magnet direct current (PMDC) motor and develop a controller that can suppress chaotic behavior resulted from parameter variation such as the loading effect. The nonlinear behaviors of PMDC motors were investigated by time-domain waveform, phase portrait, and Floquet theory. By varying the load torque, a period-doubling bifurcation appeared which in turn led to chaotic behavior in the system. So, a fuzzy logic controller and developing the Floquet theory techniques are applied to eliminate the bifurcation and the chaos effects. The controller is used to enhance the performance of the system by getting a faster response without overshoot or oscillation, moreover, tends to reduce the steady-state error while maintaining its stability. The simulation results emphasize that fuzzy control provides better performance than that obtained from the other controller.

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Complex dynamics and coexistence of period-doubling and period-halving bifurcations in an integrated pest management model with nonlinear impulsive control

Advances in Difference Equations ◽

10.1186/s13662-020-02971-9 ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Changtong Li ◽

Sanyi Tang ◽

Robert A. Cheke

Keyword(s):

Periodic Solution ◽

Integrated Pest Management ◽

Pest Management ◽

Pest Control ◽

Impulsive Control ◽

Period Doubling ◽

Dynamical Behaviour ◽

Period Doubling Bifurcation ◽

Predator Prey ◽

Predator Prey Model

Abstract An expectation for optimal integrated pest management is that the instantaneous numbers of natural enemies released should depend on the densities of both pest and natural enemy in the field. For this, a generalised predator–prey model with nonlinear impulsive control tactics is proposed and its dynamics is investigated. The threshold conditions for the global stability of the pest-free periodic solution are obtained based on the Floquet theorem and analytic methods. Also, the sufficient conditions for permanence are given. Additionally, the problem of finding a nontrivial periodic solution is confirmed by showing the existence of a nontrivial fixed point of the model’s stroboscopic map determined by a time snapshot equal to the common impulsive period. In order to address the effects of nonlinear pulse control on the dynamics and success of pest control, a predator–prey model incorporating the Holling type II functional response function as an example is investigated. Finally, numerical simulations show that the proposed model has very complex dynamical behaviour, including period-doubling bifurcation, chaotic solutions, chaos crisis, period-halving bifurcations and periodic windows. Moreover, there exists an interesting phenomenon whereby period-doubling bifurcation and period-halving bifurcation always coexist when nonlinear impulsive controls are adopted, which makes the dynamical behaviour of the model more complicated, resulting in difficulties when designing successful pest control strategies.

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IDENTIFYING DISTINCT STOCHASTIC DYNAMICS FROM CHAOS: A STUDY ON MULTIMODAL NEURAL FIRING PATTERNS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127409023135 ◽

2009 ◽

Vol 19 (02) ◽

pp. 453-485 ◽

Cited By ~ 37

Author(s):

MINGHAO YANG ◽

ZHIQIANG LIU ◽

LI LI ◽

YULIN XU ◽

HONGJV LIU ◽

...

Keyword(s):

Time Series ◽

Limit Cycle ◽

Stochastic Dynamics ◽

Nonlinear Time Series ◽

Special Importance ◽

Neuronal System ◽

Neural Firing ◽

Firing Patterns ◽

Theoretical Simulation ◽

Definition Of

Some chaotic and a series of stochastic neural firings are multimodal. Stochastic multimodal firing patterns are of special importance because they indicate a possible utility of noise. A number of previous studies confused the dynamics of chaotic and stochastic multimodal firing patterns. The confusion resulted partly from inappropriate interpretations of estimations of nonlinear time series measures. With deliberately chosen examples the present paper introduces strategies and methods of identification of stochastic firing patterns from chaotic ones. Aided by theoretical simulation we show that the stochastic multimodal firing patterns result from the effects of noise on neuronal systems near to a bifurcation between two simpler attractors, such as a point attractor and a limit cycle attractor or two limit cycle attractors. In contrast, the multimodal chaotic firing trains are generated by the dynamics of a specific strange attractor. Three systems were carefully chosen to elucidate these two mechanisms. An experimental neural pacemaker model and the Chay mathematical model were used to show the stochastic dynamics, while the deterministic Wang model was used to show the deterministic dynamics. The usage and interpretation of nonlinear time series measures were systematically tested by applying them to firing trains generated by the three systems. We successfully identified the distinct differences between stochastic and chaotic multimodal firing patterns and showed the dynamics underlying two categories of stochastic firing patterns. The first category results from the effects of noise on the neuronal system near a Hopf bifurcation. The second category results from the effects of noise on the period-adding bifurcation between two limit cycles. Although direct application of nonlinear measures to interspike interval series of these firing trains misleadingly implies chaotic properties, definition of eigen events based on more appropriate judgments of the underlying dynamics leads to accurate identifications of the stochastic properties.

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