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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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Mechanical Modeling explain surface Morphology of mammalian brains

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Dr. Rishabh Pathak

Mechanical modeling plays a crucial role in explaining the surface morphology of mammalian brains, particularly in understanding the mechanisms of cortical folding and brain development. Here are some key points regarding how mechanical modeling elucidates the surface morphology of mammalian brains:


1. Biomechanical Principles: Mechanical modeling provides a framework for applying biomechanical principles to study the structural properties of the brain tissue, including the cortex and subcortex. By considering the mechanical behavior of these brain regions, researchers can simulate how forces and stresses influence cortical folding patterns and overall brain morphology.


2.     Finite Element Analysis: Finite element analysis is a common technique used in mechanical modeling to simulate the behavior of complex structures like the brain. By constructing computational models based on finite element methods, researchers can investigate how variations in parameters such as cortical thickness, stiffness, and growth rates impact cortical folding and surface morphology.


3.  Stress Distribution: Mechanical models help in analyzing the distribution of mechanical stresses within the brain tissue during growth and development. By quantifying stress patterns in different regions of the cortex, researchers can understand how these stresses contribute to the formation of cortical folds and the overall surface morphology of the brain.


4.  Predictive Capabilities: Mechanical models have predictive capabilities that allow researchers to forecast how changes in mechanical properties, such as stiffness ratios or growth rates, may alter cortical folding patterns. By running simulations based on these models, researchers can anticipate the effects of varying parameters on brain morphology and validate these predictions against experimental observations.


5.     Comparative Studies: Mechanical modeling enables comparative studies across different mammalian species to investigate how variations in brain size, cortical thickness, and gyral morphology are influenced by mechanical factors. By analyzing the mechanical properties of brains from various species, researchers can gain insights into the evolutionary and developmental aspects of cortical folding.


6.  Clinical Relevance: Mechanical modeling of brain morphology has clinical relevance in understanding neurodevelopmental disorders and brain pathologies associated with abnormal cortical folding. By simulating the mechanical aspects of these conditions, researchers can identify potential mechanisms underlying disease states and explore therapeutic interventions targeting mechanical factors.


7.  Integration with Biological Data: Mechanical models can be integrated with biological data on cellular processes, gene expression, and neuronal development to provide a comprehensive understanding of brain morphogenesis. By combining mechanical insights with biological knowledge, researchers can elucidate the intricate interplay between mechanical forces and biological mechanisms in shaping brain structure.


Overall, mechanical modeling serves as a valuable tool for explaining the surface morphology of mammalian brains by elucidating the mechanical principles that govern cortical folding, growth, and development. By incorporating biomechanical perspectives into the study of brain morphology, researchers can advance our understanding of the complex processes underlying brain structure and function.

 

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