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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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Until gestational Week 18 the cortex forms its six layered structure

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Dr. Rishabh Pathak

Until gestational week 18, a significant period in brain development unfolds as the cortex forms its six-layered structure. Here is an explanation of the importance of gestational week 18 in the context of cortical layer formation:


1.  Six-Layered Cortex Development: Between weeks 13 and 15, the ventricular zone undergoes changes, leading to the arrival of neurons destined for the middle layers of the cortex. By gestational week 18, the radial organization of the neocortex becomes clearly distinguishable, with the six distinct layers taking shape. This process involves the radial expansion of the cortical plate and subplate, marking a critical milestone in the structural development of the cerebral cortex.


2.  Distinct Cortical Layers: The six-layered structure of the cortex consists of layers with unique cellular compositions and functions. Each layer contains specific types of neurons that contribute to information processing and neural circuitry within the brain. The formation of these layers is essential for establishing the functional organization of the cerebral cortex and enabling complex cognitive processes.


3. Neuronal Maturation: As the cortex forms its six-layered structure, cortical neurons undergo maturation processes that are crucial for their functional integration into neural circuits. Neurons in different layers exhibit varying degrees of maturity, with older neurons in deeper layers forming connections earlier than younger neurons in superficial layers. This maturation process is essential for the establishment of functional connectivity within the developing cortex.


4. Developmental Gradients: During cortical layer formation, developmental gradients are observed in terms of neuronal age and morphology. Young cortical neurons in deep layers exhibit elongated cell bodies and descending axons, while older neurons in superficial layers have rounded cell bodies and elongated dendrites perpendicular to the cortical surface. These gradients reflect the temporal sequence of neuronal generation and migration, contributing to the establishment of the laminar organization of the cortex.


5.  Absence of Horizontal Connections: By gestational week 18, while the six-layered cortex is taking shape, horizontal intracortical connections have not yet developed. The focus during this period is on the radial expansion of the cortical plate and the establishment of the vertical organization of the cortical layers. The absence of horizontal connections highlights the early stages of cortical development and the ongoing processes shaping the structural framework of the cortex.


In summary, gestational week 18 represents a critical juncture in brain development when the cortex completes the formation of its six-layered structure. The establishment of distinct cortical layers, the maturation of cortical neurons, and the presence of developmental gradients contribute to the functional specialization and connectivity of the developing cerebral cortex. Understanding the events that occur until gestational week 18 is essential for unraveling the complexities of cortical development and the emergence of the mature brain's structural and functional organization.

 

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