scholarly journals M-current induced Bogdanov–Takens bifurcation and switching of neuron excitability class

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Isam Al-Darabsah ◽  
Sue Ann Campbell
Keyword(s):  
Parameter Space ◽  
Neuron Model ◽  
Neuronal Excitability ◽  
Three Dimensional ◽  
Leak Current ◽  
Applied Current ◽  
Maximal Conductance ◽  
Dimensional Parameter ◽  
M Current ◽  
Numerical Bifurcation

AbstractIn this work, we consider a general conductance-based neuron model with the inclusion of the acetycholine sensitive, M-current. We study bifurcations in the parameter space consisting of the applied current $I_{app}$ I a p p , the maximal conductance of the M-current $g_{M}$ g M and the conductance of the leak current $g_{L}$ g L . We give precise conditions for the model that ensure the existence of a Bogdanov–Takens (BT) point and show that such a point can occur by varying $I_{app}$ I a p p and $g_{M}$ g M . We discuss the case when the BT point becomes a Bogdanov–Takens–cusp (BTC) point and show that such a point can occur in the three-dimensional parameter space. The results of the bifurcation analysis are applied to different neuronal models and are verified and supplemented by numerical bifurcation diagrams generated using the package . We conclude that there is a transition in the neuronal excitability type organised by the BT point and the neuron switches from Class-I to Class-II as conductance of the M-current increases.

COMPLEX BIFURCATION STRUCTURES IN THE HINDMARSH–ROSE NEURON MODEL

2007 ◽  
Vol 17 (09) ◽  
pp. 3071-3083 ◽  
Author(s):  
J. M. GONZÀLEZ-MIRANDA
Keyword(s):  
Phase Diagram ◽  
Bifurcation Diagram ◽  
Parameter Space ◽  
Neuron Model ◽  
Two Dimensional ◽  
Dimensional Parameter ◽  
Rapid Responses ◽  
Bifurcation Structures

The results of a study of the bifurcation diagram of the Hindmarsh–Rose neuron model in a two-dimensional parameter space are reported. This diagram shows the existence and extent of complex bifurcation structures that might be useful to understand the mechanisms used by the neurons to encode information and give rapid responses to stimulus. Moreover, the information contained in this phase diagram provides a background to develop our understanding of the dynamics of interacting neurons.

Effects of DC electric fields on neuronal excitability: A bifurcation analysis

2014 ◽  
Vol 28 (18) ◽  
pp. 1450114 ◽  
Author(s):  
Yanqiu Che ◽  
Huiyan Li ◽  
Chunxiao Han ◽  
Xile Wei ◽  
Bin Deng ◽  
...  
Keyword(s):  
Electric Field ◽  
Electric Fields ◽  
Parameter Space ◽  
Bifurcation Analysis ◽  
Neural Control ◽  
Neuronal Excitability ◽  
Modulation Method ◽  
Two Dimensional ◽  
Dimensional Parameter ◽  
Dc Electric Fields

In this paper, the effects of external DC electric fields on the neuro-computational properties are investigated in the context of Morris–Lecar (ML) model with bifurcation analysis. We obtain the detailed bifurcation diagram in two-dimensional parameter space of externally applied DC current and trans-membrane potential induced by external DC electric field. The bifurcation sets partition the two-dimensional parameter space in terms of the qualitatively different behaviors of the ML model. Thus the neuron's information encodes the stimulus information, and vice versa, which is significant in neural control. Furthermore, we identify the electric field as a key parameter to control the transitions among four different excitability and spiking properties, which facilitates the design of electric fields based neuronal modulation method.

scholarly journals Effects of generalized pooling on binocular disparity selectivity of neurons in the early visual cortex

2016 ◽  
Vol 371 (1697) ◽  
pp. 20150266 ◽  
Author(s):  
Daisuke Kato ◽  
Mika Baba ◽  
Kota S. Sasaki ◽  
Izumi Ohzawa
Keyword(s):  
Parameter Space ◽  
Three Dimensional ◽  
Stereoscopic Vision ◽  
Dimensional Parameter ◽  
Space Domain ◽  
Spatial Pooling ◽  
Proposed Model ◽  
Early Visual Cortex ◽  
Object Features ◽  
Theoretical Analyses

The key problem of stereoscopic vision is traditionally defined as accurately finding the positional shifts of corresponding object features between left and right images. Here, we demonstrate that the problem must be considered in a four-dimensional parameter space; with respect not only to shifts in space ( X , Y ), but also spatial frequency (SF) and orientation (OR). The proposed model sums outputs of binocular energy units linearly over the multi-dimensional V1 parameter space ( X , Y , SF, OR). Theoretical analyses and physiological experiments show that many binocular neurons achieve sharp binocular tuning properties by pooling the output of multiple neurons with relatively broad tuning. Pooling in the space domain sharpens disparity-selective responses in the SF domain so that the responses to combinations of unmatched left–right SFs are attenuated. Conversely, pooling in the SF domain sharpens disparity selectivity in the space domain, reducing the possibility of false matches. Analogous effects are observed for the OR domain in that the spatial pooling sharpens the binocular tuning in the OR domain. Such neurons become selective to relative OR disparity. Therefore, pooling allows the visual system to refine binocular information into a form more desirable for stereopsis. This article is part of the themed issue ‘Vision in our three-dimensional world’.

scholarly journals CLOSED CURVES OF GLOBAL BIFURCATIONS IN CHUA'S EQUATION: A MECHANISM FOR THEIR FORMATION

2003 ◽  
Vol 13 (03) ◽  
pp. 609-616 ◽  
Author(s):  
ANTONIO ALGABA ◽  
MANUEL MERINO ◽  
FERNANDO FERNÁNDEZ-SÁNCHEZ ◽  
ALEJANDRO J. RODRÍGUEZ-LUIS
Keyword(s):  
Parameter Space ◽  
Three Dimensional ◽  
Global Bifurcations ◽  
Dimensional Parameter ◽  
Closed Curves

In this work, the presence of closed bifurcation curves of homoclinic and heteroclinic connections has been detected in Chua's equation. We have numerically found and qualitatively described the mechanism of the formation/destruction of such closed curves. We relate this phenomenon to a failure of transversality in a curve of T-points in a three-dimensional parameter space.

Imaginary Scators Bound Set Under the Iterated Quadratic Mapping in 1 + 2 Dimensional Parameter Space

2016 ◽  
Vol 26 (01) ◽  
pp. 1630002 ◽  
Author(s):  
M. Fernández-Guasti
Keyword(s):  
Parameter Space ◽  
Complex Structure ◽  
Three Dimensional ◽  
Periodic Points ◽  
Mandelbrot Set ◽  
Three Dimensions ◽  
Quadratic Mapping ◽  
Two Dimensional ◽  
Dimensional Parameter ◽  
Nilpotent Elements

The quadratic iteration is mapped within a nondistributive imaginary scator algebra in [Formula: see text] dimensions. The Mandelbrot set is identically reproduced at two perpendicular planes where only the scalar and one of the hypercomplex scator director components are present. However, the bound three-dimensional S set projections change dramatically even for very small departures from zero of the second hypercomplex plane. The S set exhibits a rich fractal-like boundary in three dimensions. Periodic points with period [Formula: see text], are shown to be necessarily surrounded by points that produce a divergent magnitude after [Formula: see text] iterations. The scator set comprises square nilpotent elements that ineluctably belong to the bound set. Points that are square nilpotent on the [Formula: see text]th iteration, have preperiod 1 and period [Formula: see text]. Two-dimensional plots are presented to show some of the main features of the set. A three-dimensional rendering reveals the highly complex structure of its boundary.

INTERACTIVE GRAPHICAL EXPLORATION OF MULTIDIMENSIONAL NONLINEAR DYNAMICAL SYSTEMS

1992 ◽  
Vol 02 (02) ◽  
pp. 251-261 ◽  
Author(s):  
JUDY CHALLINGER
Keyword(s):  
Dynamical System ◽  
Parameter Space ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Three Dimensional ◽  
Initial Point ◽  
Unit Cube ◽  
Dimensional Parameter ◽  
Data Set ◽  
Nonlinear Dynamical

This paper discusses the application of an inherently three-dimensional graphical representation tool, isosurfaces, as a means to interactively explore and visualize the attractors of a nonlinear dynamical system with a fifteen-dimensional parameter space. A program has been written which allows the scientist to interactively select and visualize three-dimensional sub-spaces of the fifteen-dimensional parameter space. The dynamical system used to illustrate these concepts is a discrete-time, nonlinear, three-nation Richardson model with economic constraints. This dynamical system, which models the shifting alliances of nations in an arms race, maps an initial point in the unit cube to another point in the unit cube after multiple iterations of the model functions. Using an isosurface function on the resulting volumetric data set, surfaces indicating the changing alliances of nations are generated and rendered.

scholarly journals A unified framework for 21 cm tomography sample generation and parameter inference with progressively growing GANs

2020 ◽  
Vol 493 (4) ◽  
pp. 5913-5927 ◽  
Author(s):  
Florian List ◽  
Geraint F Lewis
Keyword(s):  
Brightness Temperature ◽  
Parameter Space ◽  
Three Dimensional ◽  
Training Data ◽  
X Rays ◽  
Unified Framework ◽  
Generative Adversarial Network ◽  
Dimensional Parameter ◽  
Adversarial Network ◽  
Approximate Bayesian

ABSTRACT Creating a data base of 21 cm brightness temperature signals from the Epoch of Reionization (EoR) for an array of reionization histories is a complex and computationally expensive task, given the range of astrophysical processes involved and the possibly high-dimensional parameter space that is to be probed. We utilize a specific type of neural network, a progressively growing generative adversarial network (PGGAN), to produce realistic tomography images of the 21 cm brightness temperature during the EoR, covering a continuous three-dimensional parameter space that models varying X-ray emissivity, Lyman band emissivity, and ratio between hard and soft X-rays. The GPU-trained network generates new samples at a resolution of ∼3 arcmin in a second (on a laptop CPU), and the resulting global 21 cm signal, power spectrum, and pixel distribution function agree well with those of the training data, taken from the 21SSD catalogue (Semelin et al.). Finally, we showcase how a trained PGGAN can be leveraged for the converse task of inferring parameters from 21 cm tomography samples via Approximate Bayesian Computation.

scholarly journals Extraction of Potential Energy from Geostrophic Fronts by Inertial–Symmetric Instabilities

2018 ◽  
Vol 48 (5) ◽  
pp. 1033-1051 ◽  
Author(s):  
Nicolas Grisouard
Keyword(s):  
Potential Energy ◽  
Parameter Space ◽  
Richardson Number ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Dimensional Parameter ◽  
Flow Cells ◽  
Hydrostatic Balance ◽  
Symmetric Instability

AbstractSubmesoscale oceanic density fronts are structures in geostrophic and hydrostatic balance, which are prone to inertial and/or symmetric instabilities. We argue in this article that drainage of potential energy from the geostrophic flow is a significant source of their growth. We illustrate our point with two-dimensional Boussinesq numerical simulations of oceanic density fronts on the f plane. A set of two-dimensional initial conditions covers the submesoscale portion of a three-dimensional parameter space consisting of the Richardson and Rossby numbers and a measure of stratification or latitude. Because we let the lateral density gradient decay with depth, the parameter space map is nontrivial, excluding low-Rossby, low-Richardson combinations. Dissipation and the presence of boundaries select a growing mode of inertial–symmetric instability consisting of flow cells that disturb isopycnal contours. Systematically, these isopycnal displacements correspond to a drainage of potential energy from the geostrophic fronts to the ageostrophic perturbations. In the majority of our experiments, this energy drainage is at least as important as the drainage of kinetic energy from the front. Various constraints, some physical, some numerical, make the energetics in our experiments more related to inertial rather than symmetric instabilities. Our results depend very weakly on the Richardson number and more on the Rossby number and relative stratification.

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