A braided river meeting the sea. Photograph by Maximilian Brand.

02v1.0 · October 2026 · Arcas DynamicsNeuroAIBrain models

HypothesisNeuroscience + AI architecture

Perturbational equivalence: a testable criterion for replicating cognition.

Brain models are judged by how closely they match the brain under ordinary conditions. Yet every trained network has a continuum of exact twins that match it on every observational metric and still respond differently when a part of them is silenced. We propose a necessary condition for replication: intervening on matched components must change the model’s behaviour the way it changes the brain’s, within the variability between individuals.

The grain of the perturbation family sets the grain of the equivalence class.

The argument.

Why matching the brain under ordinary conditions cannot settle whether a model replicates it.

01Observation is not mechanismDistinct mechanisms can produce the same unperturbed activity. Among millions of three-cell circuit models, many different parameter sets gave similar output.
02Every network has twinsRotating a model’s units leaves its representational geometry, its linear predictivity and its behaviour exactly unchanged.
03Perturbation separates themSilencing a local group of units does not survive the rotation, so local perturbations tell twins apart. A global change in gain does not.
04A testable criterionA model is perturbationally equivalent if intervening on matched components changes its behaviour as it changes the brain’s, within the variation between individuals.

Live model.

A toy version of the paper’s simulation: the same units in two layouts. Watch what each loses when a patch is silenced.

Fig. 2a Same units, same readout, different layout

Fig. 2b Perturbational distance D, median of 10 drawsTolerance τ = 2.04 · RSA 0.712 for all four

Topographic model1.07Passes

Local-rotation twin1.06Passes

Permuted-layout twin3.70Fails

Global-rotation twin3.69Fails

Under local silencing, 2 of 4 variants pass the tolerance τ = 2.04.

Toy simulation from the paper: 20 simulated individuals, 144 units each, 16 silencing sites. Model estimates under illustrative parameters, not data.

The equations.

01 Effect of a perturbation

\Delta_{X}(p) = b_{X}(p) - b_{X}(0)

The change in behaviour, across every task condition, when perturbation p is applied to system X.

02 Between-individual variance

\sigma^{2}(p) = \frac{1}{n} \sum_{i} \bigl\lVert \Delta_{i}(p) - m_{-i}(p) \bigr\rVert^{2}

m₋ᵢ is the mean effect over every individual except i.

03 Perturbational distance

D_{M} = \frac{1}{|P_{\mathrm{test}}|} \sum_{p} \frac{1}{n} \sum_{i} \frac{\bigl\lVert \Delta_{M}(p) - m_{-i}(p) \bigr\rVert^{2}}{\sigma^{2}(p)} \le \htmlClass{hi}{\tau}

A typical individual scores about 1. The model passes if its distance is at most τ, the 95th percentile of the individual scores.

04 The invariance

\begin{gathered} W'^{\mathrm{T}} r' = W^{\mathrm{T}} Q^{\mathrm{T}} Q\, r = W^{\mathrm{T}} r \\ Q^{\mathrm{T}} E_{U}\, Q = E_{U} \;\Leftrightarrow\; \htmlClass{hi}{\text{silencing } U \text{ is unchanged}} \end{gathered}

Rotating a model’s units leaves its geometry, predictivity and behaviour unchanged, but not the effect of silencing them.

What it builds on.

The scale of today’s brain models, and the limits of observing them, as the paper cites them.

139,255

neurons in the adult fly connectome, joined by 54.5 million synapsesDorkenwald et al., 2024, Nature

91%

of 164 testable predictions correct for a whole-brain model built on that connectomeShiu et al., 2024, Nature

20 million+

three-cell circuit models, in which many parameter sets gave similar network activityPrinz et al., 2004, Nature Neuroscience

3,510

transistors lesioned one by one in a microprocessor, without revealing how the chip workedJonas & Kording, 2017, PLoS Computational Biology

Predictions.

Each one names the result that would refute it.

P1Ties come apartRefuted if: observational score, at its noise ceiling, accounts for all between-model variance in D.
P2Commitment mattersRefuted if: permuted or rotated twins reach the same D on held-out sites.
P3Degeneracy surfacesRefuted if: the spread across matched models is no larger under perturbation than at baseline.
P4Families do not transferRefuted if: rankings under input perturbations predict rankings under internal ones up to the noise ceiling.
P5D predicts new interventionsRefuted if: selection by observational score predicts a held-out perturbation type equally well or better.

How to test it.

First testPrimate IT reanalysis

Published muscimol inactivation data from inferior temporal cortex, compared against ventral-stream models across a range of observational scores, with each model’s exact twins as negative controls. No new animals.

P1P2P4

Experiment 2Connectome-constrained fly models

Silence and activate each identified cell type in silico, in a 50-model ensemble of the fly visual system and in a whole-brain model, and ask whether the spread across models grows under perturbation.

Experiment 3A held-out perturbation in mouse

Optogenetic silencing at one set of sites to fit models and excitation at another to test them, comparing models chosen by perturbational distance with models chosen by observational score.

Where it could be wrong.

Observational fit may be enough in practice.

If high observational scores reliably come with a pass, the criterion adds cost without telling models apart. That is the refutation condition for P1, and we would accept it.

The mapping may be doing the work.

The criterion is only as strong as the rule that places model units on the brain, and different declared rules could give different verdicts.

Biological perturbations are imperfect.

Muscimol spreads by an uncertain amount, optogenetic effects can reach beyond their target, and circuits compensate.

Necessary, not sufficient.

Passing for one family of perturbations licenses no claim about another, and the criterion does not cover recovery after a lesion.

About this paper.

Authors: Arcas Dynamics, Toronto, Canada
Version: 1.0, October 2026
Predictions: 5, none tested yet
DOI: Not yet assigned
Preprint: Not yet posted
Licence: CC BY 4.0. Share and adapt with credit.

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Changelog

v1.0 · October 2026First published.

How to cite

@misc{arcas2026perturbational,
  author       = {Arcas Dynamics},
  title        = {Perturbational equivalence: a testable criterion for replicating cognition},
  year         = {2026},
  month        = oct,
  note         = {Hypothesis paper 02, version 1.0, CC BY 4.0},
  howpublished = {\url{https://arcasdynamics.com/research/perturbational-equivalence}}
}