# A Spatial Filtering Approach to Biological Patterning

## TL;DR

This paper introduces a discrete spatial filtering framework to model how cellular interactions influence pattern formation in biological systems, providing insights into robustness and patterning mechanisms.

## Contribution

It presents a novel filtering approach to predict biological patterning, linking intercellular interactions to spatial frequency modulation, with applications to developmental biology.

## Key findings

- Patterns can form without instabilities, similar to Turing patterns.
- Biological systems show robustness to environmental and cellular noise.
- Spatial filtering simplifies understanding of pattern formation processes.

## Abstract

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.