Figure 3.

Formal definition of the model. All variables and parameters are continuous in that they can in principle assume any value on the real number line. The parameters (red) are trained and thus fixed during simulations. All variables are scaled by the boundary marker level $b$, except itself, representing its dampening effect (akin to a time constant)—the greater the value of $b$ (the greater the cell tends to be a boundary cell), the more dampened its overall activity, including the patterning state, cell properties and internal controller states, is. Subscripted parameters and variables denote unique projections of the corresponding parameters and variables that can be inferred from the connectivity diagram shown in Figure 2. In cases where the value of a variable is determined by inputs from multiple controllers, such as the gap-junctions and the boundary-markers, the averaging operator $g()$ computes the means of those contributions. The case of gap-junction weight updates exemplifies these concepts. The weight of every gap-junction $j_{\left(n-1\right)n}$ is determined by the intrinsic controllers of cells $\left(i-1\right)$ and $i$, specifically by the posterior column in the controller of the anterior cell and the anterior column of the posterior cell. Accordingly, the dimension of $r_g$ would be $3\times 2\times \left(n-1\right)$, where the dimensions $3\times 2$ represents the three nodes each of the two controller columns. This $r_g$ would then be partially vectorized, with the two columns contributing to each gap-junction concatenated into one, yielding a matrix with dimensions $6\times \left(n-1\right)$. In the same way, the dimension of $u_g$ would be $2\times 6$, representing the two gap-junctions each contributed to by one column of the generic cell’s intrinsic controller totaling six nodes (three nodes per column). Thus, the multiplication of $u_g$ with $r_g$ yields a matrix of dimensions $2\times \left(n-1\right)$. Finally, the averaging operator $g()$ computes the column-wise mean of the previous matrix resulting in a $1\times \left(n-1\right)$ vector of updates to all the gap-junctions.