scholarly journals Inertial Manifolds via Spatial Averaging Revisited

2022 ◽  
Vol 54 (1) ◽  
pp. 268-305
Author(s):  
Anna Kostianko ◽  
Xinhua Li ◽  
Chunyou Sun ◽  
Sergey Zelik
Keyword(s):  
Inertial Manifolds ◽  
Spatial Averaging

scholarly journals An extension of the principle of spatial averaging for inertial manifolds

1999 ◽  
Vol 66 (1) ◽  
pp. 125-142 ◽  
Author(s):  
Hyukjin Kwean
Keyword(s):  
Diffusion Equation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Inertial Manifolds ◽  
Inertial Manifold ◽  
Specific Class ◽  
Spatial Averaging ◽  
The Laplace Operator

AbstractIn this paper we extend a theorem of Mallet-Paret and Sell for the existence of an inertial manifold for a scalar-valued reaction diffusion equation to new physical domains ωn ⊂ Rn, n = 2,3. For their result the Principle of Spatial Averaging (PSA), which certain domains may possess, plays a key role for the existence of an inertial manifold. Instead of the PSA, we define a weaker PSA and prove that the domains φn with appropriate boundary conditions for the Laplace operator, δ, satisfy a weaker PSA. This weaker PSA is enough to ensure the existence of an inertial manifold for a specific class of scalar-valued reaction diffusion equations on each domain ωn under suitable conditions.

scholarly journals Spatial averaging and apparent acceleration in inhomogeneous spaces

2011 ◽  
Vol 44 (2) ◽  
pp. 353-365 ◽  
Author(s):  
Antonio Enea Romano ◽  
Misao Sasaki
Keyword(s):  
Spatial Averaging ◽  
Apparent Acceleration

scholarly journals Spatial and Temporal Integration of Visual Motion Signals for Smooth Pursuit Eye Movements in Monkeys

2009 ◽  
Vol 102 (4) ◽  
pp. 2013-2025 ◽  
Author(s):  
Leslie C. Osborne ◽  
Stephen G. Lisberger
Keyword(s):  
Random Walk ◽  
Eye Movements ◽  
Smooth Pursuit ◽  
Visual Motion ◽  
Target Motion ◽  
Linear Filter ◽  
Smooth Pursuit Eye Movements ◽  
Spatial Averaging ◽  
Spatial Integration ◽  
Pursuit Eye Movements

To probe how the brain integrates visual motion signals to guide behavior, we analyzed the smooth pursuit eye movements evoked by target motion with a stochastic component. When each dot of a texture executed an independent random walk such that speed or direction varied across the spatial extent of the target, pursuit variance increased as a function of the variance of visual pattern motion. Noise in either target direction or speed increased the variance of both eye speed and direction, implying a common neural noise source for estimating target speed and direction. Spatial averaging was inefficient for targets with >20 dots. Together these data suggest that pursuit performance is limited by the properties of spatial averaging across a noisy population of sensory neurons rather than across the physical stimulus. When targets executed a spatially uniform random walk in time around a central direction of motion, an optimized linear filter that describes the transformation of target motion into eye motion accounted for ∼50% of the variance in pursuit. Filters had widths of ∼25 ms, much longer than the impulse response of the eye, and filter shape depended on both the range and correlation time of motion signals, suggesting that filters were products of sensory processing. By quantifying the effects of different levels of stimulus noise on pursuit, we have provided rigorous constraints for understanding sensory population decoding. We have shown how temporal and spatial integration of sensory signals converts noisy population responses into precise motor responses.

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