# 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:

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:

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 |