Circular stationary solutions in two-dimensional neural fields

2001 ◽  
Vol 85 (3) ◽  
pp. 211-217 ◽  
Author(s):  
Herrad Werner ◽  
Tim Richter
Keyword(s):  
Stationary Solutions ◽  
Two Dimensional ◽  
Neural Fields

Solitary Waves of Integrate-and-Fire Neural Fields

1997 ◽  
Vol 9 (8) ◽  
pp. 1677-1690 ◽  
Author(s):  
David Horn ◽  
Irit Opher
Keyword(s):  
Solitary Waves ◽  
Two Dimensional ◽  
Neural Fields ◽  
Firing Activity ◽  
Firing Patterns ◽  
Integrate And Fire ◽  
Strong Inhomogeneity ◽  
Continuum Formulation ◽  
Dimensional Surface

Arrays of interacting identical neurons can develop coherent firing patterns, such as moving stripes that have been suggested as possible explanations of hallucinatory phenomena. Other known formations include rotating spirals and expanding concentric rings. We obtain all of them using a novel two-variable description of integrate-and-fire neurons that allows for a continuum formulation of neural fields. One of these variables distinguishes between the two different states of refractoriness and depolarization and acquires topological meaning when it is turned into a field. Hence, it leads to a topologic characterization of the ensuing solitary waves, or excitons. They are limited to pointlike excitations on a line and linear excitations, including all the examples noted above, on a two dimensional surface. A moving patch of firing activity is not an allowed solitary wave on our neural surface. Only the presence of strong inhomogeneity that destroys the neural field continuity allows for the appearance of patchy incoherent firing patterns driven by excitatory interactions.

ON STABILITY OF PHYSICALLY REASONABLE SOLUTIONS TO THE TWO-DIMENSIONAL NAVIER–STOKES EQUATIONS

2019 ◽  
pp. 1-52
Author(s):  
Yasunori Maekawa
Keyword(s):  
Stokes Equations ◽  
Stationary Solutions ◽  
Navier Stokes ◽  
Domain Modeling ◽  
Asymptotically Stable ◽  
Two Dimensional ◽  
Spatial Infinity ◽  
Navier Stokes Equations ◽  
Unique Existence ◽  
Fundamental Object

The flow past an obstacle is a fundamental object in fluid mechanics. In 1967 Finn and Smith proved the unique existence of stationary solutions, called the physically reasonable solutions, to the Navier–Stokes equations in a two-dimensional exterior domain modeling this type of flows when the Reynolds number is sufficiently small. The asymptotic behavior of their solution at spatial infinity has been studied in detail and well understood by now, while its stability has remained open due to the difficulty specific to the two-dimensionality. In this paper, we prove that the physically reasonable solutions constructed by Finn and Smith are asymptotically stable with respect to small and well-localized initial perturbations.

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