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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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Direct Waves (DW)

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Dr. Rishabh Pathak

It seems there might be a misunderstanding in the term "Direct Waves (DW)" as it is not a standard term in the context of Transcranial Magnetic Stimulation (TMS) or neurophysiology. However, based on the provided excerpts, it appears that "Direct Waves" could be referring to "D-waves" in the context of TMS. Here is an explanation of D-waves in TMS:


1.      D-Waves in Transcranial Magnetic Stimulation (TMS):

o Definition: D-waves are direct motor responses evoked by TMS, specifically reflecting the direct activation of corticospinal neurons in the primary motor cortex (M1). These waves are part of the neurophysiological responses observed during TMS experiments.

o   Generation: When a TMS pulse is applied to the motor cortex, it can directly activate the corticospinal tract, leading to the generation of D-waves. D-waves are typically observed in electromyography (EMG) recordings of muscles innervated by the stimulated cortical area.

o  Characteristics: D-waves are characterized by their short latency and monophasic waveform. They represent the most direct pathway of neural activation in response to TMS, involving the excitation of pyramidal neurons in layer V of the motor cortex.

o Physiological Significance: D-waves provide insights into the excitability and integrity of the corticospinal pathway. Changes in D-wave amplitude or latency can indicate alterations in motor cortex function, corticospinal conductivity, or synaptic transmission efficiency.

2.     Relationship with I-Waves:

o   In addition to D-waves, TMS can also evoke indirect waves (I-waves) that reflect more complex neural activation patterns involving interneuronal circuits within the cortex. I-waves are generated through indirect pathways and contribute to the overall motor response observed during TMS.

o The interplay between D-waves and I-waves provides a comprehensive understanding of how TMS influences neural circuits in the motor cortex and modulates motor output. Different types of waves (e.g., I1-wave, I2-wave) represent distinct neural pathways and mechanisms of cortical activation.

3.     Clinical and Research Applications:

o  D-wave analysis in TMS studies is crucial for assessing motor cortex excitability, mapping corticospinal projections, and investigating motor system function in health and disease.

o  Researchers and clinicians use D-wave measurements to study motor recovery after stroke, evaluate corticospinal integrity in neurological disorders, and optimize TMS protocols for therapeutic interventions targeting motor dysfunction.

In summary, D-waves in TMS represent direct motor responses elicited by cortical stimulation and play a significant role in understanding motor cortex excitability and corticospinal pathway function. By studying D-waves along with other TMS-evoked responses, researchers gain valuable insights into neural activation patterns, motor system connectivity, and the effects of TMS on brain physiology.

 

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