What analytical model is used to estimate critical conditions at the onset of folding in the brain?

The analytical model used to estimate critical conditions at the onset of folding in the brain is based on the Föppl–von Kármán theory. This theory is applied to approximate cortical folding as the instability problem of a confined, layered medium subjected to growth-induced compression. The model focuses on predicting the critical time, pressure, and wavelength at the onset of folding in the brain's surface morphology.

The analytical model adopts the classical fourth-order plate equation to model the cortical deflection. This equation considers parameters such as cortical thickness, stiffness, growth, and external loading to analyze the behavior of the brain tissue during the folding process. By utilizing the Föppl–von Kármán theory and the plate equation, researchers can derive analytical estimates for the critical conditions that lead to the initiation of folding in the brain.

Analytical modeling provides a quick initial insight into the critical conditions at the onset of folding, allowing researchers to understand the fundamental mechanisms driving cortical folding. However, it may not fully capture the evolution of complex instability patterns in the post-critical regime. Therefore, while the analytical model helps in estimating the critical parameters for folding initiation, a computational model based on the continuum theory of finite growth is often employed to predict more realistic surface morphologies and complex folding patterns beyond the onset of folding.

In conclusion, the analytical model based on the Föppl–von Kármán theory provides a foundational framework for understanding the critical conditions that trigger folding in the brain's surface morphology. It serves as a valuable tool for estimating key parameters at the onset of cortical folding and guiding further computational modeling efforts to explore the evolution of brain surface morphologies.

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