  Interactions between neighboring cells are essential for generating or
refining patterns in a number of biological systems. We propose a discrete
filtering approach to predict how networks of cells modulate spatially varying
input signals to produce more complicated or precise output signals. The
interconnections between cells determine the set of spatial modes that are
amplified or suppressed based on the coupling and internal dynamics of each
cell, analogously to the way a traditional digital filter modifies the
frequency components of a discrete signal. We apply the framework to two
systems in developmental biology: the Notch-Delta interaction that shapes
\textit{Drosophila} wing veins and the Sox9/Bmp/Wnt network responsible for
digit formation in vertebrate limbs. The latter case study demonstrates that
Turing-like patterns may occur even in the absence of instabilities. Results
also indicate that developmental biological systems may be inherently robust to
both correlated and uncorrelated noise sources. Our work shows that a spatial
frequency-based interpretation simplifies the process of predicting patterning
in living organisms when both environmental influences and intercellular
interactions are present.
