Circuits with Oscillatory Hierarchical Farey Sequences and Fractal Properties

Marszałek, Wiesław

doi:10.1007/s00034-012-9392-3

Cited by 21 publications

(9 citation statements)

References 26 publications

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“…3 is that the distribution of periodic and chaotic phases is symmetric with respect to the central period-2 domain, a symmetry obviously not present in the Farey tree. Furthermore, although generated here for discrete-time dynamics, the typical phase diagrams found for maps agree very well with those obtained for rather more complex models governed by ordinary differential equations [16,17]. Similarly to what happens for differential equations, the generic unfolding of periodicities in the sigmoidal maps starts with a 1 → 2 period-doubling.…”

supporting

confidence: 78%

Stern-Brocot trees in spiking and bursting of sigmoidal maps

Freire¹,

Pöschel²,

Gallas³

2012

EPL

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We study the global organization of oscillations in sigmoidal maps, a class of models which reproduces complex locking behaviors commonly observed in lasers, neurons, and other systems which display spiking, bursting, and chaotic sequences of spiking and bursting. We find periodic oscillations to emerge organized regularly according to the elusive Stern-Brocot tree, a symmetric and more general tree which contains the better-known asymmetric Farey tree as a sub-tree. The Stern-Brocot tree provides a natural and encompassing organization to classify nonlinear oscillations. The mathematical algorithm for generating both trees is exactly the same, differing only in the initial conditions. Such degeneracy suggests that the wrong tree might have been attributed to locking phenomena reported in some of the earlier works.

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supporting

confidence: 78%

Stern-Brocot trees in spiking and bursting of sigmoidal maps

Freire¹,

Pöschel²,

Gallas³

2012

EPL

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“…Using selected dynamic simulations, we conjecture that the periodic MMOs build Farey sequences. Such Farey sequences have also been observed for various other systems displaying MMOs (see [27,[29][30][31][32] for examples); however, they have not been widely studied for neuron models (see [33] for an example). Given the diversity and abundance of neuron models that can generate MMOs [20], it will be interesting to explore the existence of Farey sequences in other neuron models as well.…”

Section: Discussionmentioning

confidence: 77%

Computational analysis of a 9D model for a small DRG neuron

Verma,

Kienle,

Flockerzi

et al. 2020

Preprint

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Small dorsal root ganglion (DRG) neurons are primary nociceptors which are responsible for sensing pain. Elucidation of their dynamics is essential for understanding and controlling pain. To this end, we present a numerical bifurcation analysis of a small DRG neuron model in this paper. The model is of Hodgkin-Huxley type and has 9 state variables. It consists of a Na v 1.7 and a Na v 1.8 sodium channel, a leak channel, a delayed rectifier potassium and an A-type transient potassium channel. The dynamics of this model strongly depends on the maximal conductances of the voltage-gated ion channels and the external current, which can be adjusted experimentally. We show that the neuron dynamics are most sensitive to the Na v 1.8 channel maximal conductance (g 1.8 ). Numerical bifurcation analysis shows that depending on g 1.8 and the external current, different parameter regions can be identified with stable steady states, periodic firing of action potentials, mixed-mode oscillations (MMOs), and bistability between stable steady states and stable periodic firing of action potentials. We illustrate and discuss the transitions between these different regimes. We further analyze the behavior of MMOs. As the external current is decreased, we find that MMOs appear after a cyclic limit point. Within this region, bifurcation analysis shows a sequence of isolated periodic solution branches with one large action potential and a number of small amplitude peaks per period. For decreasing external current, the number of small amplitude peaks is increasing and the distance between the large amplitude action potentials is growing, finally tending to infinity and thereby leading to a stable steady state. A closer inspection reveals more complex concatenated MMOs in between these periodic MMOs branches, forming Farey sequences. Lastly, we also find small solution windows with aperiodic oscillations, which seem to be chaotic. The dynamical patterns found here as a function of different parameters contain information of translational importance as their relation to pain sensation and its intensity is a potential source of insight into controlling pain.

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“…In this paper, we illustrate our bifurcation diagrams (mostly with two slowly varying parameters) with the use of an interesting autonomous system described in Appendix A . The system ( A3 ) has been used in the literature for a qualitative analysis of the oscillatory cytosolic calcium responses [ 12 , 13 , 14 , 15 , 16 , 17 , 18 ].…”

Section: Introductionmentioning

confidence: 99%

“…In general,

, with

and

. See [ 14 ] for more details and the relation of the

type of oscillations to the Ford circles, Stern–Brockot tree, sequences of firing numbers and the Riemann zeta function. The

mixed-mode periodic oscillations have also been considered in the context of two-parameter bifurcation diagrams for the modified Chua’s circuits in [ 6 ].…”

Section: Introductionmentioning

confidence: 99%

“…The

mixed-mode periodic oscillations have also been considered in the context of two-parameter bifurcation diagrams for the modified Chua’s circuits in [ 6 ]. The intermingling intervals of periodic and chaotic solutions (as in Figure 1 and Figure 2 ) can also be used to determine fractal dimension of ( A3 ) [ 14 ]. We chose to vary the same parameters of (A3) that were used in [ 12 , 15 , 16 , 17 , 18 ] for their one-parameter bifurcation analysis.…”

Section: Introductionmentioning

confidence: 99%

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Computational Analysis of Ca2+ Oscillatory Bio-Signals: Two-Parameter Bifurcation Diagrams

Marszałek

Sadecki

Walczak

2021

Entropy

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Two types of bifurcation diagrams of cytosolic calcium nonlinear oscillatory systems are presented in rectangular areas determined by two slowly varying parameters. Verification of the periodic dynamics in the two-parameter areas requires solving the underlying model a few hundred thousand or a few million times, depending on the assumed resolution of the desired diagrams (color bifurcation figures). One type of diagram shows period-n oscillations, that is, periodic oscillations having n maximum values in one period. The second type of diagram shows frequency distributions in the rectangular areas. Each of those types of diagrams gives different information regarding the analyzed autonomous systems and they complement each other. In some parts of the considered rectangular areas, the analyzed systems may exhibit non-periodic steady-state solutions, i.e., constant (equilibrium points), oscillatory chaotic or unstable solutions. The identification process distinguishes the later types from the former one (periodic). Our bifurcation diagrams complement other possible two-parameter diagrams one may create for the same autonomous systems, for example, the diagrams of Lyapunov exponents, Ls diagrams for mixed-mode oscillations or the 0–1 test for chaos and sample entropy diagrams. Computing our two-parameter bifurcation diagrams in practice and determining the areas of periodicity is based on using an appropriate numerical solver of the underlying mathematical model (system of differential equations) with an adaptive (or constant) step-size of integration, using parallel computations. The case presented in this paper is illustrated by the diagrams for an autonomous dynamical model for cytosolic calcium oscillations, an interesting nonlinear model with three dynamical variables, sixteen parameters and various nonlinear terms of polynomial and rational types. The identified frequency of oscillations may increase or decrease a few hundred times within the assumed range of parameters, which is a rather unusual property. Such a dynamical model of cytosolic calcium oscillations, with mitochondria included, is an important model in which control of the basic functions of cells is achieved through the Ca2+ signal regulation.

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Circuits with Oscillatory Hierarchical Farey Sequences and Fractal Properties

Cited by 21 publications

References 26 publications

Stern-Brocot trees in spiking and bursting of sigmoidal maps

Stern-Brocot trees in spiking and bursting of sigmoidal maps

Computational analysis of a 9D model for a small DRG neuron

Computational Analysis of Ca2+ Oscillatory Bio-Signals: Two-Parameter Bifurcation Diagrams

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