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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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How do cortical thickness, stiffness, and growth play a role in the folding process of the brain?

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Dr. Rishabh Pathak

Cortical thickness, stiffness, and growth are key factors that play crucial roles in the folding process of the brain. Here is an explanation of how each of these factors influences cortical folding:


1.  Cortical Thickness: The thickness of the cortex, the outer layer of the brain, directly influences the folding patterns of the brain surface. Thicker cortices tend to have longer intersulcal distances and may even suppress the formation of folds entirely. On the other hand, thinner cortices are associated with increased gyrification and the formation of more convoluted brain surfaces with smaller folds. Variations in cortical thickness can lead to different folding patterns and impact the overall morphology of the brain.


2.   Stiffness: The stiffness of the cortical tissue compared to the subcortical tissue also plays a significant role in cortical folding. The stiffness ratio between the cortex and subcortex influences the surface morphology of the brain. While the cortex is expected to be denser and have a higher mechanical stiffness due to the presence of neuronal cell bodies and synapses, the actual stiffness difference between the cortex and subcortex is relatively small. This stiffness ratio can affect the folding patterns, but it is not the sole driving force behind cortical folding.


3. Growth: Growth-induced processes in the brain, such as differential growth between the cortex and subcortex, can lead to the development of cortical folds. Abnormal growth rates can result in different behaviors of the brain tissue. Slow cortical growth can lead to a more fluid-like behavior in the subcortex, potentially suppressing folding, while fast cortical growth can create elastic solid-like behavior, provoking the formation of secondary folds. The growth ratio between the cortex and subcortex is a critical parameter in controlling irregular surface morphologies and secondary folding in the brain.


In summary, cortical thickness, stiffness, and growth are interconnected factors that influence the folding process of the brain. Understanding how these parameters interact and affect brain development is essential for unraveling the mechanisms behind cortical folding and associated neurological conditions.

 

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