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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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Vertex Sharp Transients compared to K Complexes

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Dr. Rishabh Pathak

Vertex Sharp Transients (VSTs) and K Complexes are both EEG patterns observed during sleep, but they have distinct characteristics and clinical significance. 

1.      Morphology:

§  VSTs: Typically exhibit a triphasic waveform, consisting of two small positive waves surrounding a larger negative sharp wave. They may also appear as diphasic or monophasic but are most commonly recognized in their triphasic form.

§  K Complexes: Characterized by a large, biphasic waveform that consists of a sharp negative deflection followed by a slower positive component. K Complexes are often more pronounced and can have a more complex morphology compared to VSTs.

2.     Timing and Context:

§  VSTs: Primarily occur during drowsiness and non-REM sleep, often spontaneously or in response to sensory stimuli, particularly auditory stimuli. They are considered a marker of the transition into sleep.

§  K Complexes: Typically occur during stage 2 sleep and can be triggered by external stimuli, such as sounds. They are thought to play a role in sleep maintenance and are often associated with the brain's response to environmental stimuli.

3.     Clinical Significance:

§  VSTs: Generally considered a normal finding during sleep and are not associated with any pathological conditions. They are useful as a marker for the transition into sleep.

§  K Complexes: While K Complexes are also considered a normal finding in sleep, their presence can be more variable. They may be associated with sleep disturbances or disorders if they occur excessively or inappropriately.

4.    Amplitude and Background Activity:

§  VSTs: Can vary in amplitude but typically do not exceed the amplitude of the background activity. They maintain a consistent morphology during a train of transients.

§  K Complexes: Often have a higher amplitude compared to the background activity and can be quite prominent in the EEG. They can also occur in bursts and may be followed by sleep spindles.

5.     Response to Stimulation:

§  VSTs: May be evoked by sensory stimuli and can reflect a mechanism to maintain sleep after stimulation.

§  K Complexes: Often arise in response to external stimuli, serving as a protective mechanism to help maintain sleep despite disturbances.

In summary, Vertex Sharp Transients are characterized by their triphasic waveform and are primarily associated with the transition into sleep, while K Complexes are larger, biphasic waveforms that occur during stage 2 sleep and can be triggered by external stimuli. Both patterns are generally considered normal findings in sleep, but they serve different roles and have distinct morphological features.

 

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