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Calibrated linear responses bound the hidden states needed to replace all direct observed links

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Can causal measurements rule out hidden routes instead of direct neural links? We give proofs and generated-model tests. For a stable positive linear circuit with unit observed decay, the minimum hidden-state count for an exact DC match with no direct observed links equals the nonnegative rank of its coupling matrix. Fast hidden states can also match the full response within any positive error limit on a fixed finite frequency band, if hidden rates have no upper bound. A continuous-band bound states a sufficient rate. With known decay and calibrated inputs and outputs, singular values of response residuals stacked across frequencies give a lower bound on hidden-state count under a stated error bound. This certificate also holds for signed hidden circuits. In a constructed example, DC certifies one state, while three frequencies certify two with stack error 0.001. A four-state support example needs four positive hidden states although its ordinary rank is three. Three 12-state model families and 200 independent random circuit checks support the formulas. Unknown decay can give a false count. These are model results, not new animal measurements. They do not identify individual edges or explain the observed cross-species node-similarity structure.

hidden states · neural perturbation · nonnegative rank · functional connectivity · finite bandwidth · system identification
Neurons and CognitionSystems and Control

References

10
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