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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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Continuum Model of Subcortical Growth

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Dr. Rishabh Pathak

In the context of brain development, a continuum model of subcortical growth focuses on understanding the evolution of the brain's subcortical regions, which lie beneath the cortical surface. Here are the key aspects of a continuum model of subcortical growth:


1.  Representation of Subcortical Regions: The continuum model represents the subcortical regions of the brain as a continuous and deformable medium, distinct from the cortical layers. This allows researchers to study the growth and deformation of subcortical structures over developmental stages.


2.   Distinct Mechanical Properties: The model considers the subcortical regions to have different mechanical properties compared to the cortex, such as varying stiffness, elasticity, and viscoelasticity. These properties influence how the subcortical regions respond to growth-induced stresses and strains, leading to changes in their shape and morphology.


3. Growth Dynamics: The model incorporates growth dynamics specific to subcortical regions, including cell proliferation, differentiation, and migration processes that drive changes in the structure of these regions. By modeling these growth dynamics, researchers can simulate how the subcortical regions evolve over time.


4.  Interaction with Cortex: The continuum model accounts for the interactions between the subcortical regions and the overlying cortex. This interaction influences the growth patterns and morphological changes observed in both the subcortical and cortical layers, highlighting the importance of considering the brain as a coordinated system.


5.  Continuum Mechanics Principles: Similar to the cortical growth model, the subcortical growth model is based on principles of continuum mechanics to describe the behavior of the subcortical tissue under external forces and deformations. This framework allows researchers to analyze how growth processes affect the mechanical response of subcortical regions.


6. Computational Simulation: Computational methods, such as finite element analysis, are used to implement the continuum model of subcortical growth. By conducting computational simulations, researchers can predict how the subcortical regions deform and evolve over time, providing insights into the underlying mechanisms of subcortical growth.


7. Parameter Studies: Researchers can conduct parameter studies using the continuum model to investigate the effects of various factors on subcortical growth, such as growth rates, mechanical properties, and interactions with the cortex. By varying these parameters, researchers can explore the factors that influence the development of subcortical regions.


8.   Biological Relevance: The continuum model of subcortical growth aims to capture the biological relevance of subcortical development processes, offering a framework for understanding how mechanical forces, growth dynamics, and interactions with the cortex shape the subcortical structures of the developing brain. This approach helps elucidate the complex processes involved in subcortical growth and its coordination with cortical development.


In summary, a continuum model of subcortical growth provides a valuable framework for studying the mechanical and morphological aspects of subcortical brain regions during development. By integrating growth dynamics, mechanical properties, and computational simulations, researchers can gain insights into the processes driving subcortical growth and its coordination with cortical development.

 

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