The chaotic dynamics and multistability of two coupled Fitzhugh–Nagumo model neurons

Shim, Yoonsik; Husbands, Phil

doi:10.1177/1059712318789393

Cited by 18 publications

(12 citation statements)

References 30 publications

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“…( 41 ) and ( 43 ). For example, Shim and Husbands [ 32 ] considered the asymmetric case for which such additive parameters differ across the two coupled neurons. While it is beyond the scope of the present study to address such more sophisticated features of the coupled Fitzhugh–Nagumo model, future studies may take advantage of the approach presented in Sect.…”

Section: Discussionmentioning

confidence: 99%

Unstable eigenvectors and reduced amplitude spaces specifying limit cycles of coupled oscillators with simultaneously diagonalizable matrices: with applications from electric circuits to gene regulation

Mongkolsakulvong

Frank

2022

Eur. Phys. J. B

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A fascinating phenomenon is the self-organization of coupled systems to a whole. This phenomenon is studied for a particular class of coupled oscillatory systems exhibiting so-called simultaneously diagonalizable matrices. For three exemplary systems, namely, an electric circuit, a coupled system of oscillatory neurons, and a system of coupled oscillatory gene regulatory pathways, eigenvectors and amplitude equations are derived. It is shown that for all three systems, only the unstable eigenvectors and their amplitudes matter for the dynamics of the systems on their respective limit cycle attractors. A general class of coupled second-order dynamical oscillators is presented in which stable limit cycles emerging via Hopf bifurcations are solely specified by appropriately defined unstable eigenvectors and their amplitudes. While the eigenvectors determine the orientation of limit cycles in state spaces, the amplitudes determine the evolution of states along those limit cycles. In doing so, it is shown that the unstable eigenvectors define reduced amplitude spaces in which the relevant long-term dynamics of the systems under consideration takes place. Several generalizations are discussed. First, if stable and unstable system parts exhibit a slow-fast dynamics, the fast variables may be eliminated and approximative descriptions of the emerging limit cycle dynamics in reduced amplitude spaces may be again obtained. Second, the principle of reduced amplitude spaces holds not only for coupled second-order oscillators, but can be applied to coupled third-order and higher order oscillators. Third, the possibility to apply the approach to multifrequency limit cycle attractors and other types of attractors is discussed. Graphic abstract

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Section: Discussionmentioning

confidence: 99%

Unstable eigenvectors and reduced amplitude spaces specifying limit cycles of coupled oscillators with simultaneously diagonalizable matrices: with applications from electric circuits to gene regulation

Mongkolsakulvong

Frank

2022

Eur. Phys. J. B

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“…While a single isolated FHN (with δ = ε = 0) exhibits subcritical Hopf bifurcation at z = z h ≈0.38247, the coupled system can generate autonomous oscillations in a narrow range below z h . An interesting characteristic of this coupled FHN system is that it can generate a rich variety of dynamics ranging from multiple synchronised and quasiperiodic oscillations to chaotic orbits, depending on the two control inputs z 1 and z 2 [5,6,107] (Figs 9, 10, 11 and 12). In particular, it has been shown that the system exhibits chaos in a certain region of the parameter space defined by the values of z 1 and z 2 and the degree of their asymmetry.…”

Section: Cpgs Adjustable Chaoticity and Homeostatic Sensory Adaptationmentioning

confidence: 99%

Recent advances in evolutionary and bio-inspired adaptive robotics: Exploiting embodied dynamics

et al. 2021

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This paper explores current developments in evolutionary and bio-inspired approaches to autonomous robotics, concentrating on research from our group at the University of Sussex. These developments are discussed in the context of advances in the wider fields of adaptive and evolutionary approaches to AI and robotics, focusing on the exploitation of embodied dynamics to create behaviour. Four case studies highlight various aspects of such exploitation. The first exploits the dynamical properties of a physical electronic substrate, demonstrating for the first time how component-level analog electronic circuits can be evolved directly in hardware to act as robot controllers. The second develops novel, effective and highly parsimonious navigation methods inspired by the way insects exploit the embodied dynamics of innate behaviours. Combining biological experiments with robotic modeling, it is shown how rapid route learning can be achieved with the aid of navigation-specific visual information that is provided and exploited by the innate behaviours. The third study focuses on the exploitation of neuromechanical chaos in the generation of robust motor behaviours. It is demonstrated how chaotic dynamics can be exploited to power a goal-driven search for desired motor behaviours in embodied systems using a particular control architecture based around neural oscillators. The dynamics are shown to be chaotic at all levels in the system, from the neural to the embodied mechanical. The final study explores the exploitation of the dynamics of brain-body-environment interactions for efficient, agile flapping winged flight. It is shown how a multi-objective evolutionary algorithm can be used to evolved dynamical neural controllers for a simulated flapping wing robot with feathered wings. Results demonstrate robust, stable, agile flight is achieved in the face of random wind gusts by exploiting complex asymmetric dynamics partly enabled by continually changing wing and tail morphologies.

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“…Thus the system models a weakly coupled hardwired spinal circuit under supraspinal descending inputs, which operates as a half-center oscillator or as coupled pacemakers depending on the descending command signals. An interesting characteristic of this coupled FHN system is that it can generate a rich variety of dynamics ranging from multiple synchronised and quasiperiodic oscillations to chaotic orbits, depending on the two control inputs z 1 and z 2 [25][26][27] . In particular, it has been shown that the system exhibits chaos in a certain region of the parameter space defined by the values of z 1 and z 2 and the degree of their asymmetry.…”

Section: A Chaos In Two Coupled Fhn Equationsmentioning

confidence: 99%

“…While the behavior of these performance-driven systems is impressive, analysis of their dynamics has been limited. Partially qualitative 25,26 , and very recently detailed quantitative 27 , analyses have shown that the neural dynamics are chaotic. But the question remains: are the physical, bodily dynamics chaotic?…”

Section: Introductionmentioning

confidence: 99%

Embodied neuromechanical chaos through homeostatic regulation

Shim

Husbands

2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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In this paper we present detailed analyses of the dynamics of a number of embodied neuromechanical systems of a class that has been shown to efficiently exploit chaos in the development and learning of motor behaviors for bodies of arbitrary morphology. This class of systems has been successfully used in robotics, as well as to model biological systems. At the heart of these systems are neural central pattern generating (CPG) units connected to actuators which return proprioceptive information via an adaptive homeostatic mechanism. Detailed dynamical analyses of example systems, using high resolution LLE maps, demonstrate the existence of chaotic regimes within a particular region of parameter space, as well as the striking similarity of the maps for systems of varying size. Thanks to the homeostatic sensory mechanisms, any single CPG 'views' the whole of the rest of the system as if it was another CPG in a two coupled system, allowing a scale invariant conceptualization of such embodied neuromechanical systems. The analysis reveals chaos at all levels of the systems; the entire brain-body-environment system exhibits chaotic dynamics which can be exploited to power an exploration of possible motor behaviors. The crucial influence of the adaptive homeostatic mechanisms on the system dynamics is examined in detail, revealing chaotic behavior characterized by mixed mode oscillations (MMO). An analysis of the mechanism of the MMO concludes that they stems from dynamic Hopf bifurcation, where a number of slow variables act as 'moving' bifurcation parameters for the remaining part of the system.

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The chaotic dynamics and multistability of two coupled Fitzhugh–Nagumo model neurons

Cited by 18 publications

References 30 publications

Unstable eigenvectors and reduced amplitude spaces specifying limit cycles of coupled oscillators with simultaneously diagonalizable matrices: with applications from electric circuits to gene regulation

Unstable eigenvectors and reduced amplitude spaces specifying limit cycles of coupled oscillators with simultaneously diagonalizable matrices: with applications from electric circuits to gene regulation

Recent advances in evolutionary and bio-inspired adaptive robotics: Exploiting embodied dynamics

Embodied neuromechanical chaos through homeostatic regulation

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