scholarly journals Liouville Integrability in a Four-Dimensional Model of the Visual Cortex

2021 ◽  
Vol 7 (12) ◽  
pp. 277
Author(s):  
Ivan Galyaev ◽  
Alexey Mashtakov
Keyword(s):  
Visual Cortex ◽  
Dimensional Space ◽  
Natural Extension ◽  
First Integral ◽  
Extended Model ◽  
Optimal Controls ◽  
Liouville Integrability ◽  
Dimensional Model ◽  
Abnormal Extremals ◽  
The Pontryagin Maximum Principle

We consider a natural extension of the Petitot–Citti–Sarti model of the primary visual cortex. In the extended model, the curvature of contours is taken into account. The occluded contours are completed via sub-Riemannian geodesics in the four-dimensional space M of positions, orientations, and curvatures. Here, M=R2×SO(2)×R models the configuration space of neurons of the visual cortex. We study the problem of sub-Riemannian geodesics on M via methods of geometric control theory. We prove complete controllability of the system and the existence of optimal controls. By application of the Pontryagin maximum principle, we derive a Hamiltonian system that describes the geodesics. We obtain the explicit parametrization of abnormal extremals. In the normal case, we provide three functionally independent first integrals. Numerical simulations indicate the existence of one more first integral that results in Liouville integrability of the system.

scholarly journals Meromorphic integrability of the Hamiltonian systems with homogeneous potentials of degree -4

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jaume Llibre ◽  
Yuzhou Tian
Keyword(s):  
Hamiltonian Systems ◽  
Homogeneous Polynomial ◽  
First Integral ◽  
Liouville Integrability ◽  
Linear Change ◽  
Change Of Variables ◽  
Polynomial First Integral ◽  
Homogeneous Potentials

<p style='text-indent:20px;'>We characterize the meromorphic Liouville integrability of the Hamiltonian systems with Hamiltonian <inline-formula><tex-math id="M2">\begin{document}$ H = \left(p_1^2+p_2^2\right)/2+1/P(q_1, q_2) $\end{document}</tex-math></inline-formula>, being <inline-formula><tex-math id="M3">\begin{document}$ P(q_1, q_2) $\end{document}</tex-math></inline-formula> a homogeneous polynomial of degree <inline-formula><tex-math id="M4">\begin{document}$ 4 $\end{document}</tex-math></inline-formula> of one of the following forms <inline-formula><tex-math id="M5">\begin{document}$ \pm q_1^4 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ 4q_1^3q_2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ \pm 6q_1^2q_2^2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ \pm \left(q_1^2+q_2^2\right)^2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$ \pm q_2^2\left(6q_1^2-q_2^2\right) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ \pm q_2^2\left(6q_1^2+q_2^2\right) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M11">\begin{document}$ q_1^4+6\mu q_1^2q_2^2-q_2^4 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M12">\begin{document}$ -q_1^4+6\mu q_1^2q_2^2+q_2^4 $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M13">\begin{document}$ \mu&gt;-1/3 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M14">\begin{document}$ \mu\neq 1/3 $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M15">\begin{document}$ q_1^4+6\mu q_1^2q_2^2+q_2^4 $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M16">\begin{document}$ \mu \neq \pm 1/3 $\end{document}</tex-math></inline-formula>. We note that any homogeneous polynomial of degree <inline-formula><tex-math id="M17">\begin{document}$ 4 $\end{document}</tex-math></inline-formula> after a linear change of variables and a rescaling can be written as one of the previous polynomials. We remark that for the polynomial <inline-formula><tex-math id="M18">\begin{document}$ q_1^4+6\mu q_1^2q_2^2+q_2^4 $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M19">\begin{document}$ \mu\in\left\{-5/3, -2/3\right\} $\end{document}</tex-math></inline-formula> we only can prove that it has no a polynomial first integral.</p>

scholarly journals First Integral Technique for Finding Exact Solutions of Higher Dimensional Mathematical Physics Models

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 783 ◽  
Author(s):  
Shumaila Javeed ◽  
Sidra Riaz ◽  
Khurram Saleem Alimgeer ◽  
M. Atif ◽  
Atif Hanif ◽  
...  
Keyword(s):  
Exact Solutions ◽  
Dimensional Space ◽  
Nonlinear Partial Differential Equations ◽  
First Integral ◽  
Mathematical Physics ◽  
Space Time ◽  
Mathematical Tool ◽  
Long Wave ◽  
Higher Dimensional ◽  
Dimensional Space Time

In this work, we establish the exact solutions of some mathematical physics models. The first integral method (FIM) is extended to find the explicit exact solutions of high-dimensional nonlinear partial differential equations (PDEs). The considered models are: the space-time modified regularized long wave (mRLW) equation, the (1+2) dimensional space-time potential Kadomtsev Petviashvili (pKP) equation and the (1+2) dimensional space-time coupled dispersive long wave (DLW) system. FIM is a powerful mathematical tool that can be used to obtain the exact solutions of many non-linear PDEs.

Necessary and sufficient conditions for optimal controls in viscous flow problems

1994 ◽  
Vol 124 (2) ◽  
pp. 211-251 ◽  
Author(s):  
H. O. Fattorini ◽  
S. S. Sritharan
Keyword(s):  
Maximum Principle ◽  
Viscous Flow ◽  
Sufficient Conditions ◽  
Optimal Controls ◽  
The Maximum Principle ◽  
Flow Problems ◽  
Hamilton Jacobi Bellman Equation ◽  
Hamilton Jacobi Bellman ◽  
Necessary And Sufficient ◽  
The Pontryagin Maximum Principle

A class of optimal control problems in viscous flow is studied. Main results are the Pontryagin maximum principle and the verification theorem for the Hamilton–Jacobi–Bellman equation characterising the feedback problem. The maximum principle is established by two quite different methods.

scholarly journals The geometry of masking in neural populations

2019 ◽  
Vol 10 (1) ◽  
Author(s):  
Dario L. Ringach
Keyword(s):  
Visual Cortex ◽  
Cortical Neurons ◽  
Dimensional Space ◽  
Geometric Analysis ◽  
Geometric Approach ◽  
Population Level ◽  
Contextual Modulation ◽  
Neural Populations ◽  
Early Visual Cortex ◽  
Low Dimensional

Abstract The normalization model provides an elegant account of contextual modulation in individual neurons of primary visual cortex. Understanding the implications of normalization at the population level is hindered by the heterogeneity of cortical neurons, which differ in the composition of their normalization pools and semi-saturation constants. Here we introduce a geometric approach to investigate contextual modulation in neural populations and study how the representation of stimulus orientation is transformed by the presence of a mask. We find that population responses can be embedded in a low-dimensional space and that an affine transform can account for the effects of masking. The geometric analysis further reveals a link between changes in discriminability and bias induced by the mask. We propose the geometric approach can yield new insights into the image processing computations taking place in early visual cortex at the population level while coping with the heterogeneity of single cell behavior.

GENERALIZED GAS DYNAMIC EQUATIONS WITH MULTIPLE TRANSLATIONAL TEMPERATURES

2009 ◽  
Vol 23 (03) ◽  
pp. 237-240 ◽  
Author(s):  
KUN XU ◽  
ZHAOLI GUO
Keyword(s):  
Flow Regime ◽  
Stokes Equations ◽  
Dimensional Space ◽  
Natural Extension ◽  
Dynamic Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Gas Dynamic ◽  
Continuum Flow ◽  
Gas Dynamic Equations

Based on a multiple stage BGK-type collision model and the Chapman–Enskog expansion, the corresponding macroscopic gas dynamics equations in three-dimensional space will be derived. The new gas dynamic equations have the same structure as the Navier–Stokes equations, but the stress strain relationship in the Navier–Stokes equations is replaced by an algebraic equation with temperature differences. In the continuum flow regime, the new gas dynamic equations automatically recover the standard Navier–Stokes equations. The current gas dynamic equations are natural extension of the Navier–Stokes equations to the near continuum flow regime and can be used for near continuum flow study.

Nonmonocentric Urban Configurations in a Two-Dimensional Space

1989 ◽  
Vol 21 (3) ◽  
pp. 363-374 ◽  
Author(s):  
H Ogawa ◽  
M Fujita
Keyword(s):  
Land Use ◽  
Dimensional Space ◽  
Urban Land ◽  
Urban Land Use ◽  
Two Dimensional ◽  
Dimensional Model ◽  
Circular Symmetry ◽  
One Dimensional ◽  
Urban Configurations ◽  
The One

A one-dimensional model of nonmonocentric urban land use is extended into a two-dimensional space. Under the assumption of circular symmetry, it is shown that the equilibrium urban configurations in the two-dimensional space are essentially the same as those in the one-dimensional space except for the conditions on the parameters.

scholarly journals Optimal Strategies for Psoriasis Treatment

2018 ◽  
Vol 23 (3) ◽  
pp. 45 ◽  
Author(s):  
Ellina Grigorieva ◽  
Evgenii Khailov
Keyword(s):  
Dendritic Cells ◽  
T Lymphocytes ◽  
Optimal Strategies ◽  
Time Interval ◽  
Optimal Controls ◽  
Controlled System ◽  
Psoriasis Treatment ◽  
System A ◽  
Bounded Controls ◽  
The Pontryagin Maximum Principle

Within a given time interval we consider a nonlinear system of differential equations describing psoriasis treatment. Its phase variables define the concentrations of T-lymphocytes, keratinocytes and dendritic cells. Two scalar bounded controls are introduced into this system to reflect medication dosages aimed at suppressing interactions between T-lymphocytes and keratinocytes, and between T-lymphocytes and dendritic cells. For such a controlled system, a minimization problem of the concentration of keratinocytes at the terminal time is considered. For its analysis, the Pontryagin maximum principle is applied. As a result of this analysis, the properties of the optimal controls and their possible types are established. It is shown that each of these controls is either a bang-bang type on the entire time interval or (in addition to bang-bang type) contains a singular arc. The obtained analytical results are confirmed by numerical calculations using the software “BOCOP-2.0.5”. Their detailed analysis and the corresponding conclusions are presented.

scholarly journals Rearrangement of lattice particles

1993 ◽  
Vol 16 (3) ◽  
pp. 593-608 ◽  
Author(s):  
M. Muraskin
Keyword(s):  
Dimensional Space ◽  
Natural Extension ◽  
Integration Scheme ◽  
Integration Path ◽  
Lattice Systems ◽  
Contour Lines ◽  
Exact Symmetry ◽  
Aesthetic Field ◽  
Point Data ◽  
Origin Point

We have studied origin point data which lead to soliton loop lattice systems when we specify an integration path in no integrability Aesthetic Field Theory. When we applied the integration scheme developed in previous paper we found that the solitons get rearranged. Close to the origin we saw a system more disorderly than the lattice. However, farther from the origin in two dimensional maps the location of planar maxima (minima) for fixedybecame regular. In this paper, we investigate various approaches with the aim of enlarging the nonsymmetric regions. Integrating inzdid not lead to an enlarged nonsymmetric region. We were able to enlarge the region by altering the magnitudes appearing in the origin point data. It is not clear if we can continually enlarge the nonsymmetric region by this method. We studied what we call an “imperfect” lattice which in a coarse sense can be thought of as being comprised of soliton loops when we specify an integration path. Here the integration scheme did not lead to an exact symmetry, but there was a repeat of “type” structures (as indicated by observations of contour lines in the maps). We then extended the system to higher dimensions. In particular, we studied a complex six dimensional space which is a natural extension of Minkowski space as an example. The system studied gave rise to a loop lattice, but with magnitudes of maxima (minima) of the different loops varying in an oscillatory way. When we applied the integration scheme to this system found no sign of the previously discussed symmetry in the domain studied although the system is not free from other regularities (this is also the case when magnitudes are altered).

scholarly journals LINEAR RELATIONS AND INTEGRABILITY FOR CLUSTER ALGEBRAS FROM AFFINE QUIVERS

2020 ◽  
pp. 1-38
Author(s):  
JOE PALLISTER
Keyword(s):  
Dynamical System ◽  
Constant Coefficient ◽  
Dimensional Space ◽  
Discrete Dynamical System ◽  
Cluster Algebras ◽  
Liouville Integrability ◽  
Linear Recurrences ◽  
Symplectic Map ◽  
Lower Dimensional Space ◽  
Lower Dimensional

Abstract We consider frieze sequences corresponding to sequences of cluster mutations for affine D- and E-type quivers. We show that the cluster variables satisfy linear recurrences with periodic coefficients, which imply the constant coefficient relations found by Keller and Scherotzke. Viewing the frieze sequence as a discrete dynamical system, we reduce it to a symplectic map on a lower dimensional space and prove Liouville integrability of the latter.

scholarly journals D = (0|2) DIRAC–MAXWELL–EINSTEIN THEORY AS A WAY FOR DESCRIBING SUPERSYMMETRIC QUARTIONS

1994 ◽  
Vol 09 (09) ◽  
pp. 1555-1568 ◽  
Author(s):  
DMITRIJ P. SOROKIN ◽  
DMITRIJ V. VOLKOV
Keyword(s):  
Gravitational Field ◽  
Dimensional Space ◽  
Equations Of Motion ◽  
Two Dimensional ◽  
Dimensional Model ◽  
Dirac Theory ◽  
Einstein Theory ◽  
Abelian Gauge ◽  
Dimensional Space Time

Drawing an analogy with the Dirac theory of fermions interacting with electromagnetic and gravitational field we write down supersymmetric equations of motion and construct a superfield action for particles with spin [Formula: see text] and [Formula: see text] (quartions), where the role of quartion momentum in effective (2+1)-dimensional space–time is played by an Abelian gauge superfield propagating in a basic two-dimensional Grassmann-odd space with a cosmological constant showing itself as the quartion mass. So, the (0|2) (0 even and 2 odd) dimensional model of quartions interacting with the gauge and gravitational field manifests itself as an effective (2 + 1)-dimensional supersymmetric theory.

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