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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 Lambda Waves

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Dr. Rishabh Pathak

Lambda waves are often associated with specific co-occurring patterns in EEG recordings, particularly during wakefulness and visual exploration. Here are the key co-occurring patterns associated with lambda waves:

1. Saccades and Ocular Artifacts

    • Lambda waves are frequently linked to saccadic eye movements, which are rapid movements of the eye as it shifts focus from one point to another. These waves may be time-locked to saccades, typically with a delay of less than 100 milliseconds 28.
    • The presence of lambda waves is often accompanied by ocular artifacts, such as lateral rectus spikes, which are generated by blinking and lateral gaze movements 28.

2. Posterior Dominant Rhythm (PDR)

    • Lambda waves occur during attentive wakefulness and are associated with the posterior dominant rhythm, which is a prominent alpha rhythm seen in the occipital region. However, the PDR is usually intermittently present when lambda waves occur, as the PDR can be blocked by eye opening 28, 30.

3. Blink Artifacts

    • In children, there is a strong association between lambda waves and blink artifacts. The presence of lambda waves is often noted during periods of blinking, which can create a mixture of lambda activity and blink-related artifacts in the EEG 31.

4. Positive Occipital Sharp Transients of Sleep (POSTS)

    • Although lambda waves and POSTS do not occur in similar behavioral states, individuals who exhibit lambda waves are more likely to also have POSTS. POSTS occur during non-REM sleep and are characterized by positive sharp transients in the occipital region 28.

5. Generalized Delta Activity

    • While lambda waves are primarily observed during wakefulness, they may also be seen in the context of generalized delta frequency range activity, particularly in certain clinical scenarios or during transitions between states of consciousness 43.

Conclusion

In summary, lambda waves co-occur with several patterns, including saccadic eye movements, ocular artifacts, the posterior dominant rhythm, blink artifacts, and occasionally with positive occipital sharp transients of sleep. Understanding these co-occurring patterns is essential for accurate interpretation of EEG recordings and for distinguishing lambda waves from other EEG phenomena.

 

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