Symmetry breaking in solitary solutions to the Hodgkin–Huxley model

Telksnys, Tadas; Navickas, Zenonas; Timofejeva, Inga; Marcinkevicius, Romas; Ragulskis, Minvydas

doi:10.1007/s11071-019-04998-4

Cited by 7 publications

(4 citation statements)

References 23 publications

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“…It is well known that separatrices in the phase space play a pivotal role in understanding the evolution of solutions to nonlinear dynamical systems. As a rule of thumb, separatrices are usually represented by soliton-type solutions [ 25 , 36 ]. A small impulse can be used to control the evolution of the transient trajectories of different nonlinear systems – provided it is possible to derive the analytic structure of separatrices in the phase space.…”

Section: Discussionmentioning

confidence: 99%

“…Kink solitary solutions do represent the separatrix between the silent mode and the firing mode of a dendritic neuron represented by a system of nonlinear differential equations [ 36 ]. Dark solitary solutions do represent a system of separatrices in the paradigmatic Hodgkin–Huxley model [ 36 ]. A control technique based on small impulses for silencing a random network of such neurons is proposed in [ 11 , 36 ].…”

Section: Discussionmentioning

confidence: 99%

“…Dark solitary solutions do represent a system of separatrices in the paradigmatic Hodgkin–Huxley model [ 36 ]. A control technique based on small impulses for silencing a random network of such neurons is proposed in [ 11 , 36 ].…”

Section: Discussionmentioning

confidence: 99%

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Higher order solitary solutions to the meta-model of diffusively coupled Lotka–Volterra systems

et al. 2021

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A meta-model of diffusively coupled Lotka–Volterra systems used to model various biomedical phenomena is considered in this paper. Necessary and sufficient conditions for the existence of nth order solitary solutions are derived via a modified inverse balancing technique. It is shown that as the highest possible solitary solution order n is increased, the number of nonzero solution parameter values remains constant for solitary solutions of order $n>3$ n > 3 . Analytical and computational experiments are used to illustrate the obtained results.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

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Higher order solitary solutions to the meta-model of diffusively coupled Lotka–Volterra systems

et al. 2021

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“…They proposed a non-invasive control mechanism that allows them to choose any attractor among the coexisting ones, which is very interesting to observe. Telksnys et al [24] investigated the phenomena of broken symmetry in the single solutions of the HodgkinHuxley neuron model. They also showed the necessary and sufficient conditions for bright and dark solitary solutions in that model.…”

Section: Introductionmentioning

confidence: 99%

Complex dynamics of a heterogeneous network of Hindmarsh-Rose neurons

et al. 2023

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This contribution is devoted to the study of the collective behavior of two HR neurons followed by a network of HR neurons. The collective behavior of the two coupled neuron was obtained from the connection between the traditional 3D HR and a memristive 2D HR neuron via a gap junction. The dynamical properties of this rst topology revealed that it is dissipative therefore can support complex phenomena. From numerical simulations, it is found that the coupled neurons display a variety of behaviors just by varying the control parameter. Amongst these behaviors found, we have periodic bursting or spiking, quasi-periodic bursting or spiking, and chaotic bursting or spiking. Non-synchronized motion is observed when the electrical coupling strength is weak. However, synchronized cluster states are observed when the coupling strength is increased. Also varied of cross ring networks made of combination of N = 100 these different HR neurons in the network are also investigated. It is discovered that the spatiotemporal patterns are affected by the network topology. The cluster states are represented in the non- homogenous network's ring and star structures. The ring and ring-star structures contain single and double-well chimera states. Finally, in the PSIM simulation environment, a comparable electronic circuit for the two coupled heterogeneous neurons is designed and investigated. The results obtained from the designed analog circuit and the mathematical model of the two coupled neurons match perfectly.

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