2. Hypotheses and falsification

H*: Per-neuron biophysical parameters are not free parameters to be fitted. They are the steady state of a regulatory process that each cell runs, and that process can be simulated.

Specifically: θ for each neuron is the outcome of activity-dependent homeostatic regulation: an equilibrium of that regulation, selected by the neuron's initial state and developmental history (04-physics.md, L7). That regulation is encoded per cell type, acts on the set of channel and receptor genes the cell type expresses, and runs while the neuron is embedded in the real connectome, which is embedded in a real body interacting with an environment.

If H* holds, the unknowns shrink from per-neuron conductances (~10⁹ in the fly) to per-cell-type regulatory programs (~10⁴ cell types × ~10–50 parameters). The behaviorally relevant (stiff) directions of the system then become identifiable from data that can actually be collected: whole-brain imaging, a few hundred targeted perturbations, and behavior.

What H* claims, and its rivals

The scientific content of H* is locality and activity dependence: each neuron's parameters are set by a rule that uses only that cell's own signals (its voltage, Ca²⁺ and second messengers) and its type's genetic program. The specific functional form (e.g. calcium-sensor rules after Liu et al. 1998) is a modeling choice, not the hypothesis, so CAKE also fits a learned rule family restricted to local signals (04-physics.md, L7). This keeps H* from being rejected merely because one rule form was wrong.

H* is tested against two named rivals:

Acute data cannot separate H* from R1. In an animal reared under standard conditions, a regulation fixed point and a per-type parameter set can produce the same activity. Optogenetic pair responses (F1) therefore mainly test H* against R2. Separating H* from R1 needs experiments that change a neuron's activity history and look for the parameter changes H* predicts and R1 forbids (F3, F7 below). These experiments are essential; 09-roadmap.md lists the data to commission.

One family, not three separate models. H*, R1 and R2 are limits of one parameterized family, so the comparison estimates interpretable quantities instead of only picking a winner. For neuron i of type c, in the canonical form of 04-physics.md, L7:

dz_i/dt = λ_c G_c e_i − D_c (z_i − m_i),     m_i ~ N(m_c, τ_c²),     s*_i ~ N(s*_c, Σ_η,c)

Three quantities summarize where the data put a cell type in this family. Each is estimated with an interval:

• Compensation fraction κ (F3). Let y be the measured activity statistic closest to the regulated sensors (e.g. mean Ca²⁺ fluorescence in the sensor band), y_ctrl its value in unperturbed controls, and y_frozen the prediction of the fitted model with regulation frozen, driven by the measured time course of the perturbation (e.g. degron depletion measured by a fluorescent tag). Then

  κ(t) = 1 − (y(t) − y_ctrl(t)) / (y_frozen(t) − y_ctrl(t))
  

  κ_fast is evaluated after the fast path has acted (minutes) and κ_slow after hours. R1 predicts κ_slow = 0. Integral control predicts κ_slow → 1 in the sensor statistic, provided the perturbation leaves the remaining regulated quantities able to restore the set point. A leak predicts an intermediate value set by the ratio of leak to loop gain.

• Persistence ρ (F7b): the fraction of a shift produced during a sustained manipulation that remains after the manipulation ends and the declared recovery period has passed. This distinguishes rule structures within H* (04-physics.md, "When history matters"); R1 predicts no shift to persist.

• Idiosyncrasy ι (F1, F4): the fraction of within-type variance not accounted for by the covariance the regulation model predicts from p₀,c, set-point jitter and circuit position.

Why this is plausible

• Neurons of the same type show stable function but variable channel expression with correlated conductances. Simple calcium-sensor-driven homeostatic rules reproduce these correlations (O'Leary et al. 2013, 2014).
• Cell identity is genetically specified by terminal selector transcription factors. Transcriptomes exist per neuron class in C. elegans (CeNGEN, Taylor et al. 2021) and per cell type in the fly (Fly Cell Atlas). These tell us which channel and receptor genes exist in each cell type, which is a hard structural prior on θ.
• Neurons develop and maintain their properties inside their circuit and body. A model that sets parameters outside that loop discards the mechanism that actually sets them in the animal.
• Cell-autonomous homeostasis exists in C. elegans itself: the abundance of presynaptic calcium channels is adjusted according to how efficiently vesicles are released, so release is kept in range by the neuron on its own.
• Regulation acts on two timescales: fast shifts in channel voltage-dependence give immediate compensation, while slower changes in channel density make adaptation persistent (PNAS 2026). Activity-dependent neuromodulation and calcium homeostasis also cooperate rather than compete (PLoS Comput. Biol. 2026). Both shape the rule family in L7.

What is new here

Novelty check, 2026-10-07: a search of recent literature found homeostatic regulation studied in single neurons and small circuits, and connectome-constrained models fitted directly to imaging data, but no work using simulated regulation as the inference mechanism for whole-connectome parameters. Repeat this check before each major release.

Homeostatic conductance regulation is well studied in single neurons and small circuits (STG models). Connectome-constrained whole-brain models are well studied with simplified neurons. As far as we know, nobody has used simulated regulation as the inference mechanism for whole-connectome biophysical parameters. In this approach:

1. Fitting targets change. We fit set points and regulatory gains per cell type, not conductances per neuron.
2. Degeneracy becomes a prediction. Within-type variability of conductances emerges from different initial states that regulation carries to different points of the same equilibrium set. To first order its covariance is P Σ₀ Pᵀ + G (J G)⁻¹ Σ_r (J G)⁻ᵀ Gᵀ (04-physics.md, L7). The combinations of conductances that the sensors control vary only as much as set points and inputs do. This structure can be tested against Patch-seq and single-cell RNA data (F4).
3. The model predicts slow responses to perturbation. A fitted static model cannot predict how a circuit compensates hours or days after a channel knockout, cell ablation or drug. A regulatory model must. This is the strongest falsification test available.

Two companion hypotheses

H2: Electrotonic anchoring. Effective synaptic weight should be computed, not fitted. The physically measurable part of a synapse's influence is its dendritic location, which sets the cable transfer impedance from the synapse to the cell's integration and output sites. EM morphology gives this geometry. Replacing "weight ∝ synapse count" with EM-derived transfer functions removes a large block of free parameters and should improve zero-shot prediction of functional connectivity.

Headroom versus validation. The B1 analysis (09-roadmap.md, Track B) measures headroom: how much replacing counts by EM-derived transfer changes the effective weights. Headroom is necessary for H2 to matter, but it is not evidence that H2 predicts biological responses better. Only F5, on functional data, tests that.

H3: Residual cartography of the invisible connectome. Once the chemically wired, electrotonically anchored, regulation-closed model is fitted, its structured residuals against functional perturbation data are not noise. They mark missing physics: gap junctions, neuropeptide/GPCR links and monoamine volume transmission. These latent edges get priors from gene expression (innexin/connexin co-expression plus membrane apposition in EM; ligand–receptor co-expression for peptides) and are inferred by Bayesian model selection. The output is a predicted wireless and electrical connectome that wet-lab experiments can test directly.

Residual structure is not evidence of missing edges by itself. Errors in channel models, stimulation strength, the observation model and morphology also produce structured residuals. H3 therefore attributes residuals with a structured decomposition in which these sources compete with latent edges. Each source enters with the algebraic signature it must have in the linearized pair-response residual R(ω): rows are responders, columns are stimulated neurons, measured pairs form the set Ω. Observation and stimulation errors act directly on the model prediction. The other sources perturb the linearized network operator by ΔW(ω), which appears in the residual to first order as L(ω) ΔW(ω) R_p(ω), where L and R_p are the model's linear propagators to responders and from stimulated sources.

The components are fitted jointly by penalized regression: ridge on row, column, diagonal and synaptic terms, group sparsity on gap-junction edges, sparsity on peptide-pair coefficients. Before fitting, an identifiability diagnostic computes, for each pair of components, the largest principal-angle cosine between their design subspaces restricted to Ω (mutual coherence). Component pairs above a pre-registered coherence threshold are reported as confounded in that dataset, not attributed. The active-learning queue proposes experiments that reduce the coherence. All of this is in the linear-response regime, so the residuals used come from small-signal stimulation or are locally linearized.

Falsification criteria (decided before building)

Every test below is scored with the task specifications of 03-evaluation.md: a defined loss, a normalized score S with a hierarchical-bootstrap interval, and a pre-registered equivalence margin δ. Each test has three outcomes:

• Pass: the interval lies entirely in the region the hypothesis predicts.
• Fail: the interval lies entirely in the rival region, and the test met its power requirement.
• Inconclusive: anything else, including any test that did not meet its power requirement.

The margins δ, κ_min, δ_κ, δ₇ and δ_ρ, the recovery period, the cell types tested and the power requirements are fixed at M0 (F1–F4, F6, F7) or in the Track B pre-registration (F5). They are not changed after data are seen.

F4 statistics. In log-expression (or log-conductance) coordinates, with J the model's sensor sensitivity for the type at its operating point:

1. Controlled-combination variance: v = tr(J Σ̂ Jᵀ) / E_null[tr(J Σ̂_null Jᵀ)]. The null shuffles cells independently per gene within the type, which keeps every marginal variance and removes covariance. Regulation predicts v ≪ 1.
2. Subspace alignment: the principal angles between the row space of J and the empirical lowest-variance k-dimensional eigenspace of Σ̂, compared with their distribution under the same null and under random k-dimensional subspaces.

Technical noise (estimated from the count model or spike-ins) is modeled, not subtracted after the fact. Statistic 1 is invariant to per-gene scale factors in log coordinates. Statistic 2 depends on the channel models through J, so F4 also tests the channel library.

F5 criteria and data requirements. For pathways j → i in pre-registered test types, let W^H2_ij(ω) = Σ over synapses of q · Z_syn→target(ω; θ_i), with the target set by transmission mode (spike-initiation zone for spiking neurons, output sites for graded neurons), and W^count_ij = q · n_ij. Let Δ(ω) = S_H2(ω) − S_count(ω) be the score difference in predicting measured transfer, in frequency bands.

• (i) Δ in the highest tested band > δ, with its interval, in the high-headroom types.
• (ii) The slope of Δ against log ω is positive, with its interval.
• (iii) Specificity: across types, Δ correlates positively with B1 headroom, and Δ is equivalent to zero in pre-registered low-headroom control types.

Test and control types are chosen from the B1 headroom ranking before any functional data are examined. The dataset is named in the pre-registration and must provide:

• temporal resolution and recording bandwidth of at least the highest tested frequency (calcium imaging resolves only the slow band; fast bands need electrophysiology or voltage imaging);
• transmission-mode labels per cell type;
• an observation model (indicator kinetics, sampling, noise).

Until such a dataset is named, B1 results remain headroom estimates, and F5 is reported as not yet run. F5 is tested mainly in the fly; most worm neurons are compact.

F6 specificity check (precondition). On the real measurement design (the same pairs, noise and stimulation), plant known errors of every non-edge source in the H3 table at realistic magnitudes, with and without planted gap-junction and peptide edges. The decomposition must:

• call latent edges at a false-discovery rate ≤ q_max when only non-edge errors are planted;
• recover planted edges with recall ≥ r_min.

Component pairs that the coherence diagnostic flags as confounded are excluded from F6. Verified links used to score F6 must be independent of the priors (leakage audit, 03-evaluation.md).

F1 fair-baseline protocol. F1 compares our model with a rival, so the rival must be as strong as we can make it:

• Same inputs. The direct-fit baselines get the same physics layers, training data, transcriptome priors and compute budget as the regulation-closed model.
• Two baselines. (a) per-neuron direct fit with a hierarchical prior that pulls neurons toward their cell type, with the prior strength tuned by cross-validation; (b) per-type direct fit (one θ per type, no regulation). H* must beat or match both.
• Tuned independently. Baseline hyperparameters are chosen on a validation split that the regulation-closed model's developers don't touch, and they are frozen before held-out scoring.
• Equal optimization effort. Every model is fitted from the same number of restarts. The best score must have stabilized: the last restart improves it by less than δ/4.
• Pre-registered. Metrics, splits and margins are written down at M0. Results are reported whichever way they come out.

Power requirements

Each test pre-registers:

• a minimal effect of interest in its own units (δ, κ_min, δ_ρ, enrichment odds ratio);
• α = 0.05 and a target power of 0.8.

The required numbers of animals, neurons, pairs and time points are computed by simulation in the M3 pipeline: data are simulated from each generating model (H* with both rule families, R1, R2) at the real experimental design, and the full fitting and testing pipeline is run on them. The same simulations give a model-recovery matrix: how often the pipeline selects each model when each one generated the data. Data requests to partner labs (09-roadmap.md) quote these numbers. A test whose data fall short of its requirement can only be inconclusive.

Decision rule (pre-registered)

The verdict separates two questions that a single table previously merged:

• Predictive model selection: which model family CAKE should use from here on. This is an engineering decision.
• Mechanistic evidence: what the results say about biology.

A model losing a prediction contest shows that the tested formulation is insufficient. It does not by itself show that the biological mechanism is absent.

Step 0: Validity gates

No verdict is drawn until all gates pass. A failed gate makes the overall verdict inconclusive (machinery), with the failure named.

Step 1: Predictive model selection

Candidates: the regulation-closed model (each rule family), baseline (b) and baseline (a).

• Rank them by held-out score on the perturbation axis (E5) first, then on unseen pairs and stimulated neurons (E2) (03-evaluation.md).
• Among models whose scores are equivalent to the best (within δ), choose the one with the fewest effective parameters: baseline (b), then the regulation-closed model, then baseline (a).

This is F1 read as a selection rule.

Step 2: Mechanistic evidence

Each claim is assessed from its own tests:

Conflicting tests. F3 can pass while F7a fails, or the reverse. This is not a contradiction: it is evidence for activity-dependent regulation in adults but not during development, or the reverse. The claim is then stated only for the setting where it passed. A failed powered test and a passed test on the same setting (e.g. F3 in two types) are reported per type, not pooled.

Before any negative mechanistic statement, four alternative explanations must each be checked and reported:

1. Missing mechanisms: Necessity Map ablations and H3 residual structure show no unmodeled process that could carry the effect.
2. Weak intervention: the measured efficacy of the manipulation (depletion fraction, silencing strength) exceeds its pre-registered minimum.
3. Insufficient measurement: the test met its power requirement.
4. Optimization failure: G4 passed for the models compared.

Only when all four are excluded does CAKE state "evidence against activity-dependent regulation of these parameters, in these types, under these conditions". Otherwise the statement is "the tested formulation does not account for the data", with the open alternatives listed.

Step 3: Consequences

F5 (H2) and F6 (H3) are reported independently and don't change the H* verdict.