# 3.1. HomogeneousModel similar to [Oh2014]¶

The homogeneous model we implement is similar to [Oh2014], and is a linear connectivity model via constrained optimization and linear regression of the form:

$\underset{w_{x, y} \geq 0 }{\text{min}} \sum_{i=1}^{|S_E|} \sqrt{( \sum_{x \in S_x} w_{x, y} PV(x \cap E_i) - PV(y) )^2}$

that best fits the data given by the injections in the set $$S_E$$.

This is perhaps more clearly represented as a nonnegative least squares regression problem:

$\underset{x}{\text{argmin}} \| Ax - b \|_2^2, \quad \text{subject to} \quad x \geq 0$

This model seeks set of positive linear weight coefficients $$w_{x,y}$$ that minimize the L2 prediction error. Because many injections overlap several regions, the model attempts to assign credit to each of the source regions by relying on multiple non-overlapping injections.

## 3.1.1. Assumptions¶

• Homogeneity: two injections of identical volume into region X result in the same fluorescence in a target region, irrespective of the exact position of the injection within the source area
• Additivity: the fluorescence observed in a target region can be explained by a linear sum of appropriately weighted sources.

## 3.1.2. Region selection criteria¶

This model only fits a connectivity matrix over a subset of the 292 summary structures. First, a region is only included if for at least one injection experiment the injection infected at least 50 voxels in the region. Additionally, since the injection matrix $$x$$ is poorly conditioned using all of the remaining regions, regions were heuristically removed one-by-one to reduce the condition number $$\kappa$$ to a predefined threshold of 1000.

### 3.1.2.1. Conditioning¶

The conditioning algorithm is implemented in models.homogeneous.backward_subset_selection_conditioning, and utilizes a singular value decomposition based technique to remove a set of columns that heuristically decreases the condition number.

References

 [Oh2014] (1, 2) “A mesoscale connectome of the mouse brain”, Oh et al, Nature. 2014. https://www.nature.com/articles/nature13186