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Four covariance conditions identify a generic three-node linear drift up to time scale under fixed correlated drive

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We test whether unknown node-noise changes identify directed coupling from stationary covariance. In a fully observed stable linear model with fixed drift and fixed off-diagonal diffusion, an explicit rank test identifies drift up to a common time scale. Three total conditions cannot identify an unrestricted drift for two or more nodes. Four conditions suffice generically for three nodes, as shown by an exact rational witness. For two nodes, any number of conditions remains insufficient. In 90 selected geometric model cases with 3, 6 or 12 nodes and at least four conditions, exact recovery error was below 1.22e-11 times the recurrent-weight norm. Sampling was less favorable. With one million independent samples, median errors in 12-node models were 14.67 to 39.05 times that norm. An exact construction showed that changing correlated drive can select a wrong stable drift with a one-dimensional fitted kernel and near-zero residual. Thus the rank test cannot validate the drive assumption. These are proofs and synthetic tests, not animal measurements. Code, all 720 sample fits, exact evidence and offline instructions are included.

structure-function coupling · identifiability · unknown noise · Lyapunov equation · brain state · covariance · null space
Neurons and Cognition

References

9
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  4. reviewsThree controlled decay conditions can identify a generic linear drift with unknown fixed noise
  5. reviewsFour binary noise patterns reduced drift error in 16-node linear model tests
  6. reviewsLocal noise additions identify weight ratios beyond node gains in fully observed linear circuits with scalar filters
  7. doi:10.1137/22M1520311
  8. https://arxiv.org/abs/1506.02494
  9. https://arxiv.org/abs/2603.17142

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