Farey–Lorenz Permutations for Interval Maps

Geller, William; Misiurewicz, Michał

doi:10.1142/s0218127418500219

Cited by 3 publications

(2 citation statements)

References 3 publications

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“…In this and the next section we mainly recall essential definitions and results of Geller and Misiurewicz (2018), which we will use in our analysis of the CNV model. Let f be a Lorenz-like map.…”

Section: Rotation Number and Intervalmentioning

confidence: 99%

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Spike patterns and chaos in a map-based neuron model

Bartłomiejczyk,

Llovera Trujillo,

Signerska-Rynkowska

2023

International Journal of Applied Mathematics and Computer Scien

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The work studies the well-known map-based model of neuronal dynamics introduced in 2007 by Courbage, Nekorkin and Vdovin, important due to various medical applications. We also review and extend some of the existing results concerning β-transformations and (expanding) Lorenz mappings. Then we apply them for deducing important properties of spike-trains generated by the CNV model and explain their implications for neuron behaviour. In particular, using recent theorems of rotation theory for Lorenz-like maps, we provide a classification of periodic spiking patterns in this model.

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Section: Rotation Number and Intervalmentioning

confidence: 99%

“…, q) is L if f i (x) ∈ I L and R otherwise (the sequence is given up to cyclic permutation). Proofs of Proposition 3 as well as Theorems 2 and 3 can be found in the work of Geller and Misiurewicz (2018).…”

Section: Finite Unions Of Periodic Orbits and Farey-lorenz Permutationsmentioning

confidence: 99%

Spike patterns and chaos in a map-based neuron model

Bartłomiejczyk,

Llovera Trujillo,

Signerska-Rynkowska

2023

International Journal of Applied Mathematics and Computer Scien

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Gluing and grazing bifurcations in periodically forced 2-dimensional integrate-and-fire models

Granados

Huguet

2019

Communications in Nonlinear Science and Numerical Simulation

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In this work we consider a general class of 2-dimensional hybrid systems. Assuming that the system possesses an attracting equilibrium point, we show that, when periodically driven with a square-wave pulse, the system possesses a periodic orbit which may undergo smooth and nonsmooth grazing bifurcations. We perform a semi-rigorous study of the existence of periodic orbits for a particular model consisting of a leaky integrateand-fire model with a dynamic threshold. We use the stroboscopic map, which in this context is a 2-dimensional piecewise-smooth discontinuous map. For some parameter values we are able to show that the map is a quasi-contraction possessing a (locally) unique maximin periodic orbit. We complement our analysis using advanced numerical techniques to provide a complete portrait of the dynamics as parameters are varied. We find that for some regions of the parameter space the model undergoes a cascade of gluing bifurcations, while for others the model shows multistability between orbits of different periods.

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Analysis of dynamics of a map-based neuron model via Lorenz maps

Bartłomiejczyk,

Llovera Trujillo,

Signerska-Rynkowska

2024

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Modeling nerve cells can facilitate formulating hypotheses about their real behavior and improve understanding of their functioning. In this paper, we study a discrete neuron model introduced by Courbage et al. [Chaos 17, 043109 (2007)], where the originally piecewise linear function defining voltage dynamics is replaced by a cubic polynomial, with an additional parameter responsible for varying the slope. Showing that on a large subset of the multidimensional parameter space, the return map of the voltage dynamics is an expanding Lorenz map, we analyze both chaotic and periodic behavior of the system and describe the complexity of spiking patterns fired by a neuron. This is achieved by using and extending some results from the theory of Lorenz-like and expanding Lorenz mappings.

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Farey–Lorenz Permutations for Interval Maps

Cited by 3 publications

References 3 publications

Spike patterns and chaos in a map-based neuron model

Spike patterns and chaos in a map-based neuron model

Gluing and grazing bifurcations in periodically forced 2-dimensional integrate-and-fire models

Analysis of dynamics of a map-based neuron model via Lorenz maps

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