Fig 1.
Schematic of neurodevelopment as represented in the proposed model.
Intermediate progenitor cells (light green) proliferate in the subventricular zone and mature into neurons before migrating outwards along radial glial fibers (orange) to arrive at the cortical plate. There, they arrange themselves into layers, with younger neurons (dark green) bypassing existing layers to reach the pial surface (top). The radial and tangential expansion of the cortical plate results in cortical folding (right). In the proposed model, the total deformation throughout development is described by the deformation gradient F, which can be decomposed into components that describe the local growth as a result of cellular proliferation and migration (Fg) and the passive physical deformation that accompanies cortical folding (Fe). The orientation of radial glial cells in undeformed and deformed states are denoted as unit vectors N and n, respectively. For detailed explanations of these variables, please refer to Section 4.4.
Fig 2.
Experimental data used for calibration of the computational model.
Top: time alignment between experiments and simulations. Ferrets were in utero electroporated between E31 and E37, and brain sections were prepared and imaged between E39 and P16; these dates correspond to simulation times between 0 and 27 days (0 d to 27 d). Images of ferret brains are reproduced from [34]. Bottom: regions of interest (ROI) taken at consistent locations in N = 8 typically developing ferret brains, grouped based on their imaging timepoint (E39–40, P5–6, and P16). For each timepoint, we show neuron cohorts born at different embryonic times (E31, E33–34, and E36–37) labeled with EGFP (bright green). As expected, younger neurons occupy the outermost of the cortex’s six laminae. For consistency, the data marker used in the results is shown for each of the eight samples; color indicates the cohort (cohort 1 is blue, cohort 2 is purple, and cohort 3 is red) while color saturation indicates the imaging timepoint (with lighter and darker colors representing earlier and later imaging, respectively). Scale bar, 500 μm. For detailed methods on the experimental approach, please refer to Section 4.2.
Fig 3.
Workflow of cell density calculation.
The ferret brain sections could be divided into five anatomical zones: marginal zone (MZ), cortical plate (CP), intermediate zone (IZ), outer subventricular zone (OSVZ), and inner subventricular zone (ISVZ). First, we select an ROI aligned radially, such that cell migration along radial glial fibers consistently occurs along the length of the domain. Secondly, the high-resolution image is imported into ImageJ [44] and neurons are segmented based on the color threshold. Each neuron and its coordinates are then labeled and recorded automatically. Thirdly, we post-process the data in Matlab to generate a neuron density plot by counting neurons in each subcell. Finally, the cell density profile along the y-axis is generated by averaging the data in the x-axis direction. Here we note that the small peak seen in the OSVZ represents GFP-positive fibers, not neurons. Scale bar, 500 μm. For detailed methods on the image analysis, please refer to Section 4.3.
Table 1.
Summary of material parameters.
Table 2.
Comparison of parameters between the original three-cohort model calibrated to the full set of experimental data, the model validation where the third timepoint was omitted, and the single-cohort model.
Percent differences relative to the original three-cohort model for the latter two models are shown. *The single-cohort model only has a single destination parameter, δv; here we compare it individually to the three separate destination parameters from the three-cohort model.
Fig 4.
Convergence and sensitivity analysis of model calibration.
A) the objective function fobj as a function of generation. B) sensitivity study of objective function fobj with respect to baseline division rate constant Gc, diffusion coefficient D, baseline velocity constant vi, and subcortical growth parameter ks. The red dots in each 2-D parameter space denote the calibrated parameter set.
Fig 5.
Neuronal cell density as a function of distance from the subventricular zone, compared between experiments (circles) and simulations (lines).
Results are organized both by neuronal cohort (left column) and imaging timepoint (right column). Note that colors represent neuronal cohorts based on IUE dates, while different markers and line styles differentiate imaging timepoints.
Fig 6.
Model validation via leave-one-out approach.
Our model was first calibrated to time points of A) E39-40 and B) P5-6, then used to predict the third time point of C) P16. D) The original model calibration at time 3 for the sake of comparison.
Fig 7.
Comparison between A) the single-cohort model and B) the three-cohort model in capturing the experimental data.
In both cases, data points represent summed cell densities across all three neuronal cohorts, . For comparison, the total cell density of the three-cohort model is calculated similarly, representing the sum of the different colored lines in Fig 5, right.
Fig 8.
2-D and 3-D simulations of cortical folding from 0d to 27d (corresponding to E31 to P16 in ferret development), with contour plots showing cell density ci for three cohorts of neurons.
Note that the time between shown timepoints is not consistent, as timepoints were selected on the basis of biological relevance.
Fig 9.
Factors that influence buckling point and wavelength in 2-D simulations.
The cortical-subcortical stiffness ratio βμ and tangential-radial growth ratio βk influence the contour plots of A) true strain ln(λ), B) the onset of buckling, and C) the normalized wavelength. Note that each simulation shown in A) is taken at the buckling point. D) The three- and single-cohort models predict similar buckling points, with only small differences.
Fig 10.
Simulated 3-D radial glial fiber orientations at 0d and 27d with contour plots showing total cell density c.
Fig 11.
Distributions of model parameters.
A) Heaviside function as a function of normalized cell density (ci − c0)/c0, B) Heaviside function Gx as a function of normalized distance r/R0, C) delta function
as a function of normalized time t/τ, D) the normalized velocity profile
, E) the normalized coupling parameter k/ks profile, and F) the normalized shear modulus μ/μs profile.
Table 3.
Algorithm for the calibration of our model parameters using a genetic algorithm.