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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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Patterns of Change in Gray Matter

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Dr. Rishabh Pathak

Gray matter undergoes dynamic changes throughout development, reflecting the maturation and specialization of neural circuits in the brain. Here are some key patterns of change in gray matter:


1.   Early Growth and Pruning: In early childhood, there is a period of rapid growth in gray matter volume, driven by increases in neuronal cell bodies, dendrites, and synapses. This phase is followed by a process of pruning, where weaker or unused connections are eliminated to optimize neural efficiency.


2.   Regional Variability: Gray matter changes exhibit regional variability, with different brain regions showing distinct patterns of growth and decline over time. For example, cortical regions involved in sensory and motor functions may mature earlier than areas responsible for higher-order cognitive processes.


3.  Inverted-U Shaped Trajectory: Many cortical regions, especially dorsal areas, exhibit an inverted-U shaped trajectory of gray matter development. This pattern involves an initial increase in gray matter volume during infancy and early childhood, followed by a gradual decrease starting in late childhood and continuing into adulthood, eventually reaching a plateau.


4.  Synaptic Density Changes: Changes in gray matter volume are closely related to synaptic density and complexity. Synaptic pruning, which involves the elimination of weaker synapses and the strengthening of important connections, contributes to the observed patterns of gray matter changes across development.


5. Hierarchical Development: Gray matter development follows a hierarchical sequence, with lower-order sensory and motor regions maturing earlier than higher-order association areas. This sequence of development reflects the phylogenetic organization of the brain and the progressive specialization of cortical functions.


6.  Age-Related Declines: While gray matter volume generally increases in childhood and peaks in early adulthood, there is a gradual decline in gray matter volume in later adulthood. Age-related declines in gray matter are associated with factors such as synaptic loss, neuronal atrophy, and changes in cortical thickness.


Understanding the patterns of change in gray matter provides insights into the structural and functional development of the brain across the lifespan. The dynamic nature of gray matter development reflects the ongoing refinement and optimization of neural circuits to support cognitive abilities, sensory processing, and motor functions.

 

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