Computational models of cortical folding: A review of common approaches

Axonal tension contributes to consistent fold placement

Cortical folding is a critical process during brain development, resulting in morphologies that are both consistent and distinct between individuals and species. While earlier studies have highlighted important aspects of cortical folding, most existing computational models, based on the differential growth theory, fall short of explaining why folds tend to appear in particular locations. The axon tension hypothesis may provide insight into this conundrum; however, there has been significant controversy about a potential role of axonal tension during the gyrification. The common opinion in the field is that axonal tension is inadequate to drive gyrification, but we currently run the risk of discarding this hypothesis without comprehensively studying the role of axonal tension. Here we propose a novel bi-layered finite element model incorporating the two theories, including characteristic axonal tension in the subcortex and differential cortical growth. We show that axon tension can serve as a perturbation sufficient to trigger buckling in simulations; similarly to other types of perturbations, the natural stability behavior of the system tends to determine some characteristics of the folding morphology (e.g. the wavelength) while the perturbation determines the location of folds. Certain geometries, however, can interact or compete with the natural stability of the system to change the wavelength. When multiple perturbations are present, they similarly compete with each other. We found that an axon bundle of reasonable size will overpower up to a 5% thickness perturbation (typical in the literature) and determine fold placement. Finally, when multiple axon tracts are present, even a slight difference in axon stiffness, representing the heterogeneity of axonal connections, is enough to significantly change the folding pattern. While the simulations presented here are a very simple representation of white matter connectivity, our findings point to urgent future research on the role of axon connectivity in cortical folding.

Correlation between gyral size, brain size, and head impact risk across mammalian species

A study on primates has established that gyral size is largely independent of overall brain size. Building on this-and other research suggesting that brain gyrification may mitigate the effects of head impacts-our study aims to explore potential correlations between gyral size and the risk of head impact across a diverse range of mammalian species. Our findings corroborate the idea that gyral sizes are largely independent of brain sizes, especially among species with larger brains, thus extending this observation beyond primates. Preliminary evidence also suggests a correlation between an animal's gyral size and its lifestyle, particularly in terms of head-impact risk. For instance, goats, known for their headbutting behaviors, exhibit smaller gyral sizes. In contrast, species such as manatees and dugongs, which typically face lower risks of head impact, have lissencephalic brains. Additionally, we explore mechanisms that may explain how narrower gyral sizes could offer protective advantages against head impact. Finally, we discuss a possible trade-off associated with gyrencephaly.

Toward a better understanding of how a gyrified brain develops

The size and shape of the cerebral cortex have changed dramatically across evolution. For some species, the cortex remains smooth (lissencephalic) throughout their lifetime, while for other species, including humans and other primates, the cortex increases substantially in size and becomes folded (gyrencephalic). A folded cortex boasts substantially increased surface area, cortical thickness, and neuronal density, and it is therefore associated with higher-order cognitive abilities. The mechanisms that drive gyrification in some species, while others remain lissencephalic despite many shared neurodevelopmental features, have been a topic of investigation for many decades, giving rise to multiple perspectives of how the gyrified cerebral cortex acquires its unique shape. Recently, a structurally unique germinal layer, known as the outer subventricular zone, and the specialized cell type that populates it, called basal radial glial cells, were identified, and these have been shown to be indispensable for cortical expansion and folding. Transcriptional analyses and gene manipulation models have provided an invaluable insight into many of the key cellular and genetic drivers of gyrification. However, the degree to which certain biomechanical, genetic, and cellular processes drive gyrification remains under investigation. This review considers the key aspects of cerebral expansion and folding that have been identified to date and how theories of gyrification have evolved to incorporate this new knowledge.

Scaling patterns of cortical folding and thickness in early human brain development in comparison with primates

Across mammalia, brain morphology follows specific scaling patterns. Bigger bodies have bigger brains, with surface area outpacing volume growth, resulting in increased foldedness. We have recently studied scaling rules of cortical thickness, both local and global, finding that the cortical thickness difference between thick gyri and thin sulci also increases with brain size and foldedness. Here, we investigate early brain development in humans, using subjects from the Developing Human Connectome Project, scanned shortly after pre-term or full-term birth, yielding magnetic resonance images of the brain from 29 to 43 postmenstrual weeks. While the global cortical thickness does not change significantly during this development period, its distribution does, with sulci thinning, while gyri thickening. By comparing our results with our recent work on humans and 11 non-human primate species, we also compare the trajectories of primate evolution with human development, noticing that the 2 trends are distinct for volume, surface area, cortical thickness, and gyrification index. Finally, we introduce the global shape index as a proxy for gyrification index; while correlating very strongly with gyrification index, it offers the advantage of being calculated only from local quantities without generating a convex hull or alpha surface.

Multi-scale measurement of stiffness in the developing ferret brain

Cortical folding is an important process during brain development, and aberrant folding is linked to disorders such as autism and schizophrenia. Changes in cell numbers, size, and morphology have been proposed to exert forces that control the folding process, but these changes may also influence the mechanical properties of developing brain tissue. Currently, the changes in tissue stiffness during brain folding are unknown. Here, we report stiffness in the developing ferret brain across multiple length scales, emphasizing changes in folding cortical tissue. Using rheometry to measure the bulk properties of brain tissue, we found that overall brain stiffness increases with age over the period of cortical folding. Using atomic force microscopy to target the cortical plate, we found that the occipital cortex increases in stiffness as well as stiffness heterogeneity over the course of development and folding. These findings can help to elucidate the mechanics of the cortical folding process by clarifying the concurrent evolution of tissue properties.

An integrated finite element method and machine learning algorithm for brain morphology prediction

The human brain development experiences a complex evolving cortical folding from a smooth surface to a convoluted ensemble of folds. Computational modeling of brain development has played an essential role in better understanding the process of cortical folding, but still leaves many questions to be answered. A major challenge faced by computational models is how to create massive brain developmental simulations with affordable computational sources to complement neuroimaging data and provide reliable predictions for brain folding. In this study, we leveraged the power of machine learning in data augmentation and prediction to develop a machine-learning-based finite element surrogate model to expedite brain computational simulations, predict brain folding morphology, and explore the underlying folding mechanism. To do so, massive finite element method (FEM) mechanical models were run to simulate brain development using the predefined brain patch growth models with adjustable surface curvature. Then, a GAN-based machine learning model was trained and validated with these produced computational data to predict brain folding morphology given a predefined initial configuration. The results indicate that the machine learning models can predict the complex morphology of folding patterns, including 3-hinge gyral folds. The close agreement between the folding patterns observed in FEM results and those predicted by machine learning models validate the feasibility of the proposed approach, offering a promising avenue to predict the brain development with given fetal brain configurations.

Exploring the role of the outer subventricular zone during cortical folding through a physics-based model

The human brain has a highly complex structure both on the microscopic and on the macroscopic scales. Increasing evidence has suggested the role of mechanical forces for cortical folding – a classical hallmark of the human brain. However, the link between cellular processes at the microscale and mechanical forces at the macroscale remains insufficiently understood. Recent findings suggest that an additional proliferating zone, the outer subventricular zone (OSVZ), is decisive for the particular size and complexity of the human cortex. To better understand how the OSVZ affects cortical folding, we establish a multifield computational model that couples cell proliferation in different zones and migration at the cell scale with growth and cortical folding at the organ scale by combining an advection-diffusion model with the theory of finite growth. We validate our model based on data from histologically stained sections of the human fetal brain and predict 3D pattern formation. Finally, we address open questions regarding the role of the OSVZ for the formation of cortical folds. The presented framework not only improves our understanding of human brain development, but could eventually help diagnose and treat neuronal disorders arising from disruptions in cellular development and associated malformations of cortical development.

Orchestrated neuronal migration and cortical folding: A computational and experimental study

Brain development involves precisely orchestrated genetic, biochemical, and mechanical events. At the cellular level, neuronal proliferation in the innermost zone of the brain followed by migration towards the outermost layer results in a rapid increase in brain surface area, outpacing the volumetric growth of the brain, and forming the highly folded cortex. This work aims to provide mechanistic insights into the process of brain development and cortical folding using a biomechanical model that couples cell division and migration with volumetric growth. Unlike phenomenological growth models, our model tracks the spatio-temporal development of cohorts of neurons born at different times, with each cohort modeled separately as an advection-diffusion process and the total cell density determining the extent of volume growth. We numerically implement our model in Abaqus/Standard (2020) by writing user-defined element (UEL) subroutines. For model calibration, we apply in utero electroporation (IUE) to ferret brains to visualize and track cohorts of neurons born at different stages of embryonic development. Our calibrated simulations of cortical folding align qualitatively with the ferret experiments. We have made our experimental data and finite-element implementation available online to offer other researchers a modeling platform for future study of neurological disorders associated with atypical neurodevelopment and cortical malformations.

Brain development and cortical folding is a highly dynamic process that results from the interaction between gene expression, cellular mechanisms, and mechanical forces. Here, we expand on existing mathematical models of brain development and cortical folding to capture the behavior of multiple different subpopulations of neurons. By calibrating our biomechanical model to our novel experiments on ferrets, we can track the distribution of neurons over time and observe how the brain grows and develops its characteristic folds. Our calibrated model captures interactions between cell behavior and tissue deformation and offers more detailed information about the orchestrated migration of neuronal subpopulations. This work offers new mechanistic insights into brain development and opens the door to future investigations of atypical brain development caused by disrupted neuronal activities, particularly those alterations associated with injury, exposure, or treatment at a specific location or time during development. Finally, our experimental data and numerical implementations are provided as a resource online for the use of other researchers.