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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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Increasing the Cortical Stiffness Increases the Gyral Wavelength

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Dr. Rishabh Pathak

Increasing the cortical stiffness has been shown to impact the gyral wavelength during brain development. Here is an explanation of how changes in cortical stiffness can influence the gyral wavelength:


1.     Physics-Based Models: Physics-based models predict that the gyral wavelength increases with the third root of the stiffness contrast between the cortex and subcortex. This relationship highlights the importance of the mechanical properties of the brain tissue, particularly the stiffness of the gray matter layer relative to the white matter core, in determining the folding patterns observed in the cerebral cortex.


2.     Mechanical Instabilities: Growth-induced surface buckling, which is essential for cortical folding, requires that the stiffness of the gray matter layer is equal to or greater than the stiffness of the white matter core. Changes in cortical stiffness can lead to alterations in the mechanical forces acting on the cortical tissue, affecting the formation of gyri and sulci. By modulating the stiffness properties, researchers can observe variations in the gyral wavelength and surface morphology of the brain.


3.     Gray-White Matter Interaction: The interaction between the gray and white matter layers plays a critical role in cortical folding. An increase in cortical stiffness, particularly in the gray matter, can influence the distribution of mechanical stresses within the cortex, leading to changes in folding amplitudes and the spacing between gyri. Understanding how alterations in cortical stiffness impact the gyral wavelength provides insights into the mechanical basis of cortical morphogenesis.


4.     Analytical Perspectives: Analytical studies have demonstrated that growth-induced instabilities in the brain tissue are initiated at the mechanically weakest spots. By manipulating the stiffness properties of different brain regions, researchers can observe how variations in cortical stiffness affect the folding patterns and surface complexity of the cerebral cortex. These analytical approaches help elucidate the relationship between cortical stiffness and gyral wavelength.


5.     Developmental Significance: The relationship between cortical stiffness and the gyral wavelength has developmental implications for brain structure and function. Changes in cortical stiffness can influence the mechanical stability of the developing brain, impacting the formation of gyri and sulci. Variations in cortical stiffness may contribute to individual differences in brain morphology and folding patterns, highlighting the role of mechanical factors in shaping the structural organization of the cerebral cortex.


In summary, increasing the cortical stiffness can lead to changes in the gyral wavelength, reflecting the intricate interplay between mechanical properties and cortical folding during brain development. By investigating how alterations in cortical stiffness affect folding patterns, researchers can enhance their understanding of the biomechanical mechanisms underlying cortical morphogenesis and its implications for brain structure and function.

 

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