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Piotr Bartłomiejczyk

Publications and source records attributed to Piotr Bartłomiejczyk.

9 recordsLinked to original sources
arXiv2026

Subthreshold oscillations and spiking in hybrid neuron models with a dynamic threshold

We investigate a class of two-dimensional hybrid non-autonomous periodically forced neuron models known as the Meng-Huguet-Rinzel neuron model, with a dynamic threshold and reset mechanism. We focus on the interplay between the continuous subthreshold dynamics and discrete spike-induced resets. The model consists of a linear voltage equation coupled with a threshold variable governed by a nonlinear function of the membrane potential (voltage), and incorporates periodic external forcing in the form of either pulse or rectified sinusoidal currents. We analyze the existence and uniqueness of periodic solutions and prove that the non-hybrid version of the system possesses a unique globally attracting periodic orbit. For the hybrid system, we show that there exists at most one periodic orbit without resets, while numerical simulations indicate the possibility of coexistence of a reset-free periodic orbit and periodic spiking attractors. We also prove that some natural polygons contained in the phase space of the hybrid system form compact positively invariant globally attracting sets of this system. These results provide a rigorous mathematical framework for the analysis of dynamic threshold mechanisms in neuron models and contribute to the theoretical understanding of transitions between subthreshold oscillations and spiking behavior under a time-periodic forcing.

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arXiv2025

Numerical Transitivity and Numerical Leo Properties for Lorenz Maps with Applications to Courbage-Nekorkin-Vdovin Neuron Model

This research investigates the dynamic behavior of one dimensional discrete systems using two computational algorithms, the numerical transitivity and the numerical locally eventually onto (LEO) tests. Both algorithms are systematically applied to a variety of interval maps, including classical examples such as beta transformations and expanding Lorenz maps, in order to assess and characterize their chaotic dynamics. We perform a detailed comparison of the two methods in terms of accuracy, computational efficiency, and their sensitivity in detecting transitions between regular and chaotic regimes. Particular emphasis is placed on the Courbage Nekorkin Vdovin (CNV) model of a single neuron, known for its rich, spiking like dynamics and its mathematical reducibility to Lorenz type maps. By analyzing both the piecewise linear and nonlinear versions of the CNV model, we illustrate how the proposed numerical tests reliably capture qualitative changes in the system dynamics, focusing on the onset of chaos and chaotic regimes. The results highlight the practical potential of these numerical approaches as diagnostic tools for studying complex dynamical systems arising in nonlinear science and mathematical neuroscience.

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arXiv2023

Periodic and chaotic dynamics in a map-based neuron model

Map-based neuron models are an important tool in modelling neural dynamics and sometimes can be considered as an alternative to usually computationally costlier models based on continuous or hybrid dynamical systems. However, due to their discrete nature, rigorous mathematical analysis might be challenging. We study a discrete model of neuronal dynamics introduced by Chialvo [Chaos, Solitons \& Fractals~5, 1995, 461--479]. In particular, we show that its reduced one-dimensional version can be treated as an independent simple model of neural activity where the input and the fixed value of the recovery variable are parameters. This one-dimensional model still displays very rich and varied dynamics. Using the fact that the map whose iterates define voltage dynamics is S-unimodal, we describe in detail both the periodic behaviour and the occurrence of different notions of chaos, indicating corresponding regions in parameter space. Our study is also complemented by a bifurcation analysis of the mentioned dynamical model.

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arXiv2019

Gradient versus proper gradient homotopies

We compare the sets of homotopy classes of gradient and proper gradient vector fields in the plane. Namely, we show that gradient and proper gradient homotopy classifications are essentially different. We provide a complete description of the sets of homotopy classes of gradient maps from $\mathbb{R}^n$ to $\mathbb{R}^n$ and proper gradient maps from $\mathbb{R}^2$ to $\mathbb{R}^2$ with the Brouwer degree greater or equal to zero.

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arXiv2018

Degree product formula in the case of a finite group action

Let $V, W$ be finite-dimensional orthogonal representations of a finite group $G$. The equivariant degree with values in the Burnside ring of $G$ has been studied extensively by many authors. We present a short proof of the degree product formula for local equivariant maps on $V$ and $W$.

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arXiv2016

The Hopf type theorem for equivariant gradient local maps

We construct a degree-type otopy invariant for equivariant gradient local maps in the case of a real finite dimensional orthogonal representation of a compact Lie group. We prove that the invariant establishes a bijection between the set of equivariant gradient otopy classes and the direct sum of countably many copies of $\mathbb{Z}$.

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