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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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Volume Conduction Model (VCM)

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Dr. Rishabh Pathak

A Volume Conduction Model (VCM) is a computational model used in the field of neurostimulation, particularly in techniques like Transcranial Magnetic Stimulation (TMS) and Transcranial Current Stimulation (TCS). Here is an overview of Volume Conduction Modeling:


1.      Purpose:

oVCMs are designed to simulate the flow of electrical currents through different tissues in the head, including the scalp, skull, cerebrospinal fluid, and brain. These models help researchers and clinicians understand how electrical fields generated by external stimulations propagate and interact with neural tissue.

2.     Construction:

oA VCM typically divides the head into different compartments representing various tissues with distinct electrical properties, such as conductivity and permittivity. Common compartments include skin, skull, cerebrospinal fluid, gray matter, and white matter.

oGeometrically accurate boundaries between tissue compartments are defined to accurately represent the anatomical structure of the head.

3.     Simulation:

oBy applying the principles of electromagnetism, VCMs can calculate the distribution of electric fields induced by external stimulations, such as TMS coils or TCS electrodes, throughout the head.

oThese simulations provide insights into how the electric fields interact with neural tissue, including the strength, direction, and spatial extent of the induced fields.

4.    Applications:

oVCMs are valuable tools for optimizing stimulation protocols in neurostimulation techniques. They can help researchers determine the optimal placement of stimulation electrodes or coils to target specific brain regions effectively.

oThese models are also used to study the effects of stimulation parameters, such as intensity, frequency, and waveform, on neural activation and modulation.

5.     Advantages:

oVCMs offer a non-invasive and cost-effective way to predict and visualize the distribution of electric fields in the brain without the need for invasive measurements.

oThey allow researchers to explore the effects of stimulation on a macroscopic level, providing insights into how different brain regions are influenced by external electrical currents.

6.    Research Impact:

oVCMs have been instrumental in advancing our understanding of the mechanisms of action of neurostimulation techniques and optimizing their therapeutic applications.

o By integrating VCMs with experimental data and clinical observations, researchers can refine stimulation protocols, personalize treatments, and enhance the efficacy of neuromodulation therapies.

In summary, Volume Conduction Models (VCMs) play a crucial role in simulating and analyzing the distribution of electric fields in the head during neurostimulation procedures, offering valuable insights into the effects of external electrical stimuli on neural tissue and guiding the development of optimized stimulation protocols.

 

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