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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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Co-occurring Patterns of K Complexes

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Dr. Rishabh Pathak

K complexes are specific EEG waveforms that occur during non-REM sleep, particularly in stages 2 and 3. They often appear alongside various other EEG patterns and features. Here is the key co-occurring patterns associated with K complexes:

1. Sleep Spindles:

    • K complexes are frequently followed by sleep spindles, which are bursts of oscillatory brain activity. This co-occurrence is significant as both K complexes and sleep spindles are indicators of stage 2 non-REM sleep 17, 20. The presence of sleep spindles often enhances the identification of K complexes in the EEG.

2. Theta and Delta Activity:

    • During the periods when K complexes occur, the background EEG activity typically shows theta (4-8 Hz) and delta (0.5-4 Hz) waves. These frequency bands are characteristic of non-REM sleep and help to contextualize the presence of K complexes within the overall sleep architecture.

3. Positive Occipital Sharp Transients of Sleep (POSTS):

    • K complexes may also co-occur with positive occipital sharp transients of sleep, which are another type of EEG transient seen during stage 1 non-REM sleep. While K complexes are more prominent in stages 2 and 3, the presence of POSTS can sometimes be noted in the same sleep epochs.

4. Background Activity:

    • The background EEG during the occurrence of K complexes often shows a mix of slower waves (theta and delta) and may include bursts of higher frequency activity. This background activity is essential for distinguishing K complexes from other transients like vertex sharp transients (VSTs).

5. Arousals:

    • K complexes can occur in the context of arousals from sleep, particularly in response to external stimuli. This relationship highlights their role in sleep maintenance and the brain's ability to respond to environmental changes while still preserving sleep.

6. Clinical Patterns:

    • In certain clinical contexts, K complexes may be observed alongside other abnormal EEG patterns, such as those seen in epilepsy. For instance, K complexes with specific waveforms can occur during arousals from NREM sleep in patients with generalized or focal epilepsies.

Conclusion

K complexes are integral components of the sleep EEG and are often accompanied by various other patterns, including sleep spindles, theta and delta activity, and occasionally, arousals. Understanding these co-occurring patterns is crucial for accurate sleep staging and for assessing the overall health of sleep architecture.

 

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