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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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Cortical Folding is a Mechanical Instability Driven by Differential Growth

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Dr. Rishabh Pathak

Cortical folding is a complex phenomenon in brain development that is driven by differential growth processes. This mechanical instability arises from the differential growth rates between the cortical layers, leading to the formation of the characteristic gyri and sulci on the surface of the cerebral cortex. Here is an overview of how cortical folding is a mechanical instability driven by differential growth:


1.     Differential Growth: The process of cortical folding is fundamentally linked to the concept of differential growth, where different regions of the developing brain expand at varying rates. This uneven growth results in mechanical stresses within the cortical tissue, as certain areas experience more growth than others. The differential growth between the outer cortical layers and the underlying structures, such as the white matter, plays a key role in initiating cortical folding.


2. Physics-Based Approach: A physics-based approach has been increasingly utilized to understand cortical folding as a mechanical instability phenomenon. This perspective considers the mechanical forces generated by differential growth and how they influence the morphological changes in the brain. By modeling the cortical tissue as a multi-layered system undergoing constrained growth, researchers can simulate the patterns of cortical folding observed in the developing brain.


3.     Constrained Differential Growth: The theory of cortical folding as a constrained differential growth process suggests that the early radial expansion of the cortical plate is relatively uniform across its thickness and does not lead to folding. However, the later tangential expansion, particularly in the superficial cortical layers, is constrained by the inner layers and the underlying structures, promoting the formation of gyri and sulci. This differential growth pattern creates mechanical instabilities that drive the folding of the cortex.


4.     Role of Neuronal Connectivity: While the differential growth is a primary driver of cortical folding, other factors such as neuronal connectivity also play a significant role in shaping the folding patterns. The establishment of neural circuits and synaptic connections influences the distribution of mechanical forces within the cortex, further contributing to the folding process. Changes in synaptic pruning, myelination, and neuronal migration also impact the mechanical properties of the developing brain and influence cortical folding during different stages of development.


5. Implications for Developmental Disorders: Disruptions in the mechanisms underlying cortical folding and differential growth can lead to cortical malformations and neurodevelopmental disorders. Conditions such as lissencephaly, characterized by a smooth brain surface due to disrupted neuronal migration, highlight the importance of proper mechanical interactions in cortical development. Understanding the interplay between differential growth, mechanical forces, and neuronal processes is crucial for elucidating the origins of cortical malformations and associated neurological conditions.


In summary, cortical folding represents a dynamic interplay between differential growth processes and mechanical instabilities in the developing brain. By considering the physical principles that govern cortical morphogenesis, researchers can gain insights into the mechanisms driving the formation of gyri and sulci, as well as the implications of disrupted cortical folding for brain structure and function.

 

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