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Scientific description of cubic spline (interpolation)

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Dear all,
I am writing a scientific article (quantitative economics), and I need to describe the 'spline' method (cubic spline interpolation) as in griddedInterpolant. I can't find much, except the description:
"The interpolated value at a query point is based on a cubic interpolation of the values at neighboring grid points in each respective dimension. The interpolation is based on a cubic spline using not-a-knot end conditions."
Can anybody help me with a more scientific description? Is this some kind of standard algorithm? I found some formula for the one dimensional case, but I need a more general description for multiple dimensions (at least up to 4 dimensions).
Sargondjani on 12 Nov 2021
Another reference: Spline Functions: Basic Theory by Schumaker (3rd edition, 2007).
And by same author also "Spline Functions: Computational Methods" (SIAM, 2015)

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