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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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Fascicle Arrangement

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Dr. Rishabh Pathak

Fascicle arrangement refers to the organization of muscle fibers within a muscle, which plays a significant role in determining the muscle's function, strength, and range of motion. Here are some common fascicle arrangements found in skeletal muscles:


1.    Parallel: In parallel muscle fibers, the fascicles run parallel to the long axis of the muscle. This arrangement allows for a greater range of motion but may sacrifice some strength compared to other arrangements. Examples of muscles with parallel fascicle arrangement include the sartorius and rectus abdominis.


2.    Pennate:

o    Unipennate: In unipennate muscles, the fascicles are arranged on one side of a tendon, resembling a feather. This arrangement provides a greater cross-sectional area for force generation, making unipennate muscles well-suited for generating high forces. Examples include the extensor digitorum longus.

o    Bipennate: Bipennate muscles have fascicles arranged on both sides of a central tendon, increasing the muscle's strength without sacrificing too much range of motion. The rectus femoris is an example of a bipennate muscle.

o  Multipennate: Multipennate muscles have multiple sets of fascicles arranged at various angles to the tendon, maximizing force production while maintaining some degree of flexibility. The deltoid muscle is an example of a multipennate muscle.

3.    Circular: Circular muscles, also known as sphincters, have fascicles arranged in concentric rings around an opening or orifice. This arrangement allows for control over the diameter of the opening, such as in the orbicularis oculi around the eye or the orbicularis oris around the mouth.


4.    Fusiform: Fusiform muscles have a spindle-shaped appearance with a central belly that tapers at both ends. This arrangement combines elements of parallel and pennate muscles, providing a balance between range of motion and force production. The biceps brachii is an example of a fusiform muscle.


5.    Radiate: In radiate muscles, the fascicles converge from a broad origin to a single tendon, resembling a fan. This arrangement allows for a combination of strength and the ability to produce movements in multiple directions. The pectoralis major is an example of a radiate muscle.

Understanding the fascicle arrangement of a muscle is essential for biomechanical analyses, as it influences the muscle's function, force production capabilities, and movement patterns. Different fascicle arrangements are suited to different tasks and can be optimized through specific training regimens to enhance performance in various activities.

 

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