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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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Brain Mapping

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Dr. Rishabh Pathak


 

Brain mapping refers to the process of creating detailed representations or maps of the structure, function, and connectivity of the human brain. Various techniques and approaches are used in brain mapping to study different aspects of brain organization and activity. Here are some key methods and concepts related to brain mapping:


1.     Structural Brain Mapping:

§  MRI-based Structural Imaging: Techniques like structural MRI (sMRI) provide high-resolution images of the brain's anatomy, allowing researchers to visualize and study brain structures such as gray matter, white matter, and cortical regions.

§  Diffusion Tensor Imaging (DTI): DTI is used to map the brain's white matter tracts and study the connectivity between different brain regions based on the diffusion of water molecules along axonal pathways.

2.     Functional Brain Mapping:

§  Functional MRI (fMRI): fMRI measures changes in blood flow and oxygenation levels in the brain, providing insights into brain activity during tasks or at rest. It is widely used to map functional brain networks and identify regions involved in specific cognitive processes.

§  Electroencephalography (EEG) and Magnetoencephalography (MEG): EEG and MEG measure electrical or magnetic activity in the brain, respectively, with high temporal resolution. They are used to study brain dynamics, event-related potentials, and neural oscillations.

3.     Connectivity Mapping:

§  Resting-State fMRI: Resting-state fMRI measures spontaneous brain activity in the absence of tasks, allowing researchers to map functional connectivity networks and identify synchronized brain regions.

§  Diffusion MRI Tractography: DTI-based tractography is used to map structural connections in the brain by tracing the pathways of white matter fibers.

4.     Brain Atlases and Parcellation:

§  Brain Atlases: Atlases provide standardized maps of the brain's anatomy and functional regions, facilitating the comparison and localization of brain structures across individuals and studies.

§  Brain Parcellation: Parcellation divides the brain into distinct regions based on structural or functional criteria, enabling researchers to study specific brain areas and their interactions.

5.     Network Analysis:

§  Graph Theory: Graph theory is used to analyze brain networks as complex systems, identifying network properties such as connectivity patterns, hubs, and efficiency.

§  Connectomics: Connectomics focuses on mapping the brain's structural and functional connections to understand the brain as a network of interconnected regions.

6.     Clinical Applications:

§  Brain mapping techniques are used in clinical settings to study brain disorders, plan surgeries, assess brain function, and monitor treatment outcomes.

§  Mapping techniques help neuroscientists and clinicians understand the neural basis of neurological and psychiatric conditions.


Overall, brain mapping encompasses a diverse set of techniques and approaches aimed at unraveling the complexities of the human brain's structure, function, and connectivity. By combining multiple mapping methods, researchers can create comprehensive models of the brain's organization and dynamics, advancing our understanding of brain function and behavior.

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