Reconstructing neural circuit connectivity from calcium imaging data is an inverse problem. Neurons fire, calcium traces are measured, and from the traces you infer which neurons are connected to which. The dynamics are nonlinear — calcium signals saturate, neurons have refractory periods, synaptic transmission involves thresholds. Using the true nonlinear model should give the best reconstruction.

It does not. A deliberately wrong linear approximation outperforms an oracle estimator that knows the true nonlinearity.

The finding is precise. When the connectivity is sparse — most possible connections are absent, which is the biological reality — the linear model's error acts as implicit regularization. The linear approximation is inaccurate about the dynamics. But its inaccuracy has a specific structure: it penalizes complex patterns of connectivity that the nonlinear model would happily fit. The penalty is not designed. It is an accidental consequence of using the wrong model, and it happens to align with the sparsity prior.

The oracle estimator, knowing the true dynamics, fits the data more accurately. This accuracy is its downfall. It can explain any observed pattern with a sparse or dense connectivity matrix equally well, because its model is flexible enough to accommodate both. The linear model cannot. Its rigidity forces sparse solutions because dense solutions would require the linear model to produce patterns it is structurally incapable of producing.

The general principle: in sparse inverse problems, model accuracy and solution quality can be inversely related. The "right" model provides too many degrees of freedom for the available data, leading to overfitting of the connectivity matrix. The "wrong" model, by being inflexible about dynamics, is automatically constrained about structure. Ignorance about the dynamics imposes knowledge about the connectivity.

This is not a call for bad models. It is a statement about the interaction between model flexibility and solution constraint. When the true model is flexible and the true solution is structured, the best reconstruction may come from a model that is wrong about the dynamics but right about the structure — because the wrongness IS the structure.