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Energy Manifold Natural Gradient Descent: From Riemannian Optimization to Modern Neuroscience, NeuroAI and Quantum Physics

Dr. Rishabh Pathak
When Geometry, Energy, Artificial Intelligence and Neuroscience Converge Modern Artificial Intelligence is rapidly moving beyond the idea that learning simply means minimizing an error function. Increasingly, researchers are asking a deeper question: what is the structure of the space in which learning takes place? This question becomes particularly important when the system being modelled is constrained, nonlinear, dynamic, or governed by physical principles. A recent work titled “Energy Manifold Natural Gradient Descent: Riemannian Optimization for Neural PDE Solvers” , by Zhangyong Liang and Huanhuan Gao, introduces Energy Manifold Natural Gradient Descent (EMNGD) , a mathematical framework that extends energy-based natural-gradient optimization from unconstrained Euclidean parameter spaces to constrained Riemannian parameter manifolds . At its core, the framework proposes a simple but powerful principle: An optimization algorithm should not only determine how to reduce error; it sh...
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Sensitive of surface morphology with respect to Stiffness Ratio

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Dr. Rishabh Pathak

The sensitivity of surface morphology with respect to the stiffness ratio between the cortex and subcortex is a crucial factor in understanding the mechanisms of cortical folding and brain development. Here are some key points regarding the sensitivity of surface morphology to the stiffness ratio:


1.  Influence on Folding Patterns: The stiffness ratio between the cortex and subcortex plays a significant role in shaping the folding patterns of the cerebral cortex. Variations in the stiffness ratio can lead to changes in the depth, frequency, and complexity of cortical folds, impacting the overall surface morphology of the brain.


2.  Stress Distribution: Differences in stiffness between the cortex and subcortex affect the distribution of mechanical stresses within the brain tissue. A mismatch in stiffness can result in uneven stress distribution, leading to alterations in cortical folding patterns and surface morphology.


3.     Surface Deformations: Changes in the stiffness ratio can influence the extent of surface deformations and the formation of cortical folds. A higher stiffness ratio may promote smoother brain surfaces with shallower folds, while a lower stiffness ratio can lead to more pronounced folding patterns.


4.     Mechanical Stability: The stiffness ratio contributes to the mechanical stability of the brain tissue and its ability to resist deformations. An optimal balance in stiffness between the cortex and subcortex is essential for maintaining structural integrity and preventing excessive folding or stretching of the cortical surface.


5.     Computational Modeling: Computational models can simulate the sensitivity of surface morphology to variations in the stiffness ratio by adjusting this parameter and observing the resulting changes in cortical folding patterns. These models provide insights into how the stiffness ratio influences the mechanical behavior and morphological features of the brain.


6.     Clinical Relevance: Abnormalities in the stiffness ratio between cortical layers have been associated with neurodevelopmental disorders and brain pathologies. Understanding the impact of the stiffness ratio on surface morphology can provide valuable insights into the underlying mechanisms of these conditions.


7. Biomechanical Interactions: The stiffness ratio is part of the complex biomechanical interactions that govern cortical folding and brain development. It interacts with other factors such as cortical thickness, growth rates, and genetic influences to shape the structural and functional properties of the cerebral cortex.


By investigating the sensitivity of surface morphology to the stiffness ratio, researchers can gain a deeper understanding of the mechanical principles underlying cortical folding and brain morphogenesis. This knowledge is essential for elucidating the intricate processes that govern brain development and for exploring the implications of mechanical factors in neurodevelopmental disorders and brain health.

 

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