arXiv subjects

Justyna Signerska-Rynkowska

Publications and source records attributed to Justyna Signerska-Rynkowska.

9 recordsLinked to original sources

arXiv2025

Dynamic threshold curves and response precision in forced excitable systems

We investigate here various properties of the responses of excitable systems subject to periodic forcing and noise. While the properties of intrinsic oscillators, subject to added periodic signals, are well understood, much less is known about the factors that determine the response precision of excitable units, intrinsically at rest, when activated by periodic forcing and stochastic noise. One motivation for considering this issue comes from the behavior of auditory neurons. These neurons reportedly have the ability to fire spikes in a precise range of phases in response to incoming sound waves, a behavior for which the mechanism is unknown. To account for such a response precision, we introduce the notion of dynamic threshold curve (DTC), which estimates at each time the effective likelihood that noise will subsequently generate a spike. The DTC effectively summarizes, in a single curve, a representation of the response precision of an excitable model, as we demonstrate by showing that the distribution of spike times produced in this setting is well captured by the first passage time of a simple, Gaussian stochastic process to the distance to the DTC. This result shows that peaks and troughs of the DTC, but also their slopes, convey fine information about spike timing in response to noise. In particular, it explains properties of Type 2 and Type 3 excitable cells studied previously and provides a framework to predict the DTC properties necessary to support the response precision of auditory neurons, as we illustrate in a well-established auditory neuron model.

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arXiv2023

Testing topological conjugacy of time series

This paper considers a problem of testing, from a finite sample, a topological conjugacy of two dynamical systems $(X,f)$ and $(Y,g)$. More precisely, given $x_1,\ldots, x_n \subset X$ and $y_1,\ldots,y_n \subset Y$ such that $x_{i+1} = f(x_i)$ and $y_{i+1} = g(y_i)$ as well as $h: X \rightarrow Y$, we deliver a number of tests to check if $f$ and $g$ are topologically conjugated via $h$. The values of the tests are close to zero for conjugated systems and large for systems that are not conjugated. Convergence of the test values, in case when sample size goes to infinity, is established. A number of numerical examples indicating scalability and robustness of the methods are given. In addition, we show how the presented method specialize to a test of sufficient embedding dimension in Takens' embedding theorem. Our methods also apply to the situation when we are given two observables of deterministic processes, of a form of one or higher dimensional time-series. In this case, their similarity can be accessed by comparing the dynamics of their Takens' reconstructions.

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arXiv2022

Topological-numerical analysis of a two-dimensional discrete neuron model

We conduct computer-assisted analysis of the two-dimensional model of a neuron introduced by Chialvo in 1995 (Chaos, Solitons & Fractals 5, 461-479). We apply the method for rigorous analysis of global dynamics based on a set-oriented topological approach, introduced by Arai et al. in 2009 (SIAM J. Appl. Dyn. Syst. 8, 757-789) and improved and expanded afterwards. Additionally, we introduce a new algorithm to analyze the return times inside a chain recurrent set. Based on this analysis, together with the information on the size of the chain recurrent set, we develop a new method that allows one to determine subsets of parameters for which chaotic dynamics may appear. This approach can be applied to a variety of dynamical systems, and we discuss some of its practical aspects. The data and the software described in the paper are available at http://www.pawelpilarczyk.com/neuron/.

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arXiv2021

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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arXiv2020

Type III Responses to Transient Inputs in Hybrid Nonlinear Neuron Models

Experimental characterization of neuronal dynamics involves recording both of spontaneous activity patterns and of responses to transient and sustained inputs. While much theoretical attention has been devoted to the spontaneous activity of neurons, less is known about the dynamic mechanisms shaping their responses to transient inputs, although these bear significant physiological relevance. Here, we study responses to transient inputs in a widely used class of neuron models (nonlinear adaptive hybrid models) well-known to reproduce a number of biologically realistic behaviors. We focus on responses to transient inputs that have been previously associated with Type III neurons, arguably the least studied category in Hodgkin's classification, which are those neurons that never exhibit continuous firing in response to sustained excitatory currents. The two phenomena that we study are post-inhibitory facilitation, in which an otherwise subthreshold excitatory input can induce a spike if it is applied with proper timing after an inhibitory pulse, and slope detection, in which a neuron spikes to a transient input only when the input's rate of change is in a specific, bounded range. We analyze the origin of these phenomena in nonlinear hybrid models and provide a geometric characterization of dynamical structures associated with PIF in the system and an analytical study of slope detection for tent inputs. While the necessary and sufficient conditions for these behaviors are easily satisfied in neurons with Type III excitability, our proofs are quite general and valid for neurons that do not exhibit Type III excitability as well. This study therefore provides a framework for the mathematical analysis of these responses to transient inputs associated with Type III neurons in other systems and for advancing our understanding of these systems' computational properties.

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arXiv2019

Curlicues generated by circle homeomorphisms

We investigate the curves in the complex plane which are generated by sequences of real numbers being the lifts of the points on the orbit of an orientation preserving circle homeomorphism. Geometrical properties of these curves such as boundedness, superficiality, local discrete radius of curvature are linked with dynamical properties of the circle homeomorphism which generates them: rotation number and its continued fraction expansion, existence of a continuous solution of the corresponding cohomological equation and displacement sequence along the orbit.

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arXiv2016

Wild oscillations in a nonlinear neuron model with resets: (I) Bursting, spike adding and chaos

In a series of two papers, we investigate the mechanisms by which complex oscillations are generated in a class of nonlinear dynamical systems with resets modeling the voltage and adaptation of neurons. This first paper presents mathematical analysis showing that the system can support bursts of any period as a function of model parameters. In continuous dynamical systems with resets, period-incrementing structures are complex to analyze. In the present context, we use the fact that bursting patterns correspond to periodic orbits of the adaptation map that governs the sequence of values of the adaptation variable at the resets. Using a slow-fast approach, we show that this map converges towards a piecewise linear discontinuous map whose orbits are exactly characterized. That map shows a period-incrementing structure with instantaneous transitions. We show that the period-incrementing structure persists for the full system with non-constant adaptation, but the transitions are more complex. We investigate the presence of chaos at the transitions.

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arXiv2016

Integrate-and-fire models with an almost periodic input function

We investigate leaky integrate-and-fire models (LIF models for short) driven by Stepanov and $\mu$-almost periodic functions. Special attention is paid to the properties of a firing map and its displacement, which give information about the spiking behaviour of the system under consideration. We provide conditions under which such maps are well-defined for every $t \in \mathbb R$ and are uniformly continuous. Moreover, we show that the LIF model with a Stepanov almost periodic input has a uniformly almost periodic displacement map. We also show that in the case of a $\mu$-almost periodic drive it may happen that the displacement map corresponding to the LIF model is uniformly continuous, but is not $\mu$-almost periodic (and thus cannot be Stepanov or uniformly almost periodic). By allowing discontinuous inputs, we generalize some results of previous papers, showing, for example, that the firing rate for the LIF model with a Stepanov almost periodic drive exists and is unique. This is a starting point for the investigation of the dynamics of almost-periodically driven integrate-and-fire systems. The work provides also some contributions to the theory of Stepanov- and $\mu$-almost periodic functions.

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arXiv2015

Wild oscillations in a nonlinear neuron model with resets: (II) Mixed-mode oscillations

This work continues the analysis of complex dynamics in a class of bidimensional nonlinear hybrid dynamical systems with resets modeling neuronal voltage dynamics with adaptation and spike emission. We show that these models can generically display a form of mixed-mode oscillations (MMOs), which are trajectories featuring an alternation of small oscillations with spikes or bursts (multiple consecutive spikes). The mechanism by which these are generated relies fundamentally on the hybrid structure of the flow: invariant manifolds of the continuous dynamics govern small oscillations, while discrete resets govern the emission of spikes or bursts, contrasting with classical MMO mechanisms in ordinary differential equations involving more than three dimensions and generally relying on a timescale separation. The decomposition of mechanisms reveals the geometrical origin of MMOs, allowing a relatively simple classification of points on the reset manifold associated to specific numbers of small oscillations. We show that the MMO pattern can be described through the study of orbits of a discrete adaptation map, which is singular as it features discrete discontinuities with unbounded left- and right-derivatives. We study orbits of the map via rotation theory for discontinuous circle maps and elucidate in detail complex behaviors arising in the case where MMOs display at most one small oscillation between each consecutive pair of spikes.

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