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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 Cortical Thickness

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Dr. Rishabh Pathak

The sensitivity of surface morphology with respect to cortical thickness is a critical aspect in understanding the development and folding of the cerebral cortex. Here are some key points regarding the sensitivity of surface morphology to cortical thickness:


1.  Effect on Folding Patterns: The cortical thickness plays a significant role in determining the folding patterns of the cerebral cortex. Changes in cortical thickness can lead to alterations in the depth and complexity of cortical folds, influencing the overall surface morphology of the brain.


2.  Gyral Wavelength: Cortical thickness directly influences the gyral wavelength, which refers to the distance between adjacent cortical folds. Thicker cortices tend to have longer gyral wavelengths, resulting in smoother brain surfaces, while thinner cortices lead to shorter gyral wavelengths and increased cortical folding.


3.  Primary Folding: The primary folding of the cortex, characterized by the formation of gyri and sulci, is highly sensitive to variations in cortical thickness. Thicker cortices are associated with shallower folds, whereas thinner cortices exhibit more pronounced folding patterns.


4. Neurological Disorders: Abnormalities in cortical thickness can impact brain function and are associated with various neurological disorders. For example, conditions like lissencephaly (thickened cortex) and polymicrogyria (regionally thinned cortex) are linked to disruptions in cortical thickness and folding patterns.


5.     Surface-to-Volume Ratio: Changes in cortical thickness can affect the surface-to-volume ratio of the brain. Thicker cortices result in a smaller surface area relative to volume, while thinner cortices increase the surface area-to-volume ratio. These variations have implications for brain function and connectivity.


6.     Mechanical Properties: The mechanical properties of the cortex, such as stiffness and elasticity, interact with cortical thickness to influence surface morphology. Thicker cortices with different mechanical properties may exhibit distinct folding patterns compared to thinner cortices.


7.     Computational Modeling: Computational models can simulate the sensitivity of surface morphology to cortical thickness by varying this parameter and observing the resulting changes in cortical folding patterns. These models provide insights into how cortical thickness influences brain structure and function.


Understanding the sensitivity of surface morphology to cortical thickness is essential for elucidating the mechanisms underlying cortical folding and brain development. By investigating the relationship between cortical thickness and folding patterns, researchers can gain valuable insights into the factors shaping the complex structure of the cerebral cortex and their implications for brain function and pathology.

 

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