Brain folding as a Fourier series yields a developmental clock

The human fetal cortex begins as a smooth spheroid and progressively folds into a stereotypical sulcal and gyral pattern that defines the final brain geometry. Two centuries ago, Joseph Fourier showed that any function on a bounded domain can be built from oscillations at successive frequencies, and that truncating the series at some maximum frequency yields a smoothed version of the original. Here I show the natural closed surface generalization of Fouriers construction, the spherical harmonic expansion, reproduces brain fetal developmental sequence in reverse: truncation of the fully formed cortex at low harmonic degree (L_max) recovers a younger fetal brain; at higher L_max, the primary, secondary and tertiary folds appear. The maximum harmonic degree is, therefore, a clock. From this observation, I obtained a closed-form 23-dimensional descriptor. The descriptor predicts gestational age from a single fetal MRI at 0.13 and 0.38 weeks across the two leading public atlases after a linear cross-cohort alignment (1.07 and 1.51 weeks without alignment); it orders all 28 longitudinal pairs of a modality- mismatched cohort; and it discriminates pathological from neurotypical fetuses at AUC 0.80 while resolving three biometrically concordant pathology subtypes. The equation also accounts for large- and small-scale brain folding mechanisms under one descriptor that become distinguishable as two distinct spectral signatures on the same curve. The descriptor is a new analytical dimension of brain folding. SignificanceJoseph Fouriers 1822 discovery, a sum of oscillations at successive frequencies, unexpectedly reproduces the temporal sequence in which the fetal cortex folds. The closed-surface generalization of Fouriers series truncated at low harmonic degrees recovers a younger fetal brain; and increasing the degree of truncation yields the primary, secondary and tertiary sulci. The truncation degree functions as a developmental clock and can be used as a descriptor that locates any fetal brain on the developmental curve, predicts gestational age, and discriminates pathological from neurotypical fetuses. The descriptor is a new analytical dimension of brain folding, and an unexpected application of an old mathematical principle.