The error that moves through time

Imagine two vending machines. Both deliver the same drink. Machine A has worked on every visit. Machine B failed yesterday and has flickered all morning. When both finally deliver, the liquid has the same objective amount. Yet the informational event is different. Delivery from A confirms a strong prediction. Delivery from B repairs a weak one. A learning system should update more after B.

Thought experiment 1 · The identical drink

Keep reward magnitude fixed. Change only expectation. If a signal represents reward amount, both machines should evoke the same response. If it represents signed reward prediction error, the unreliable machine should evoke the larger positive response. The manipulated property is expectation, not pleasure, consumption or sensory intensity.

This distinction made the classic dopamine result unusually powerful. In primate conditioning, phasic responses were stronger for unpredicted than predicted rewards, transferred towards reward-predicting cues during learning, and fell below baseline when an expected reward failed to arrive Schultz, Dayan and Montague, 1997. Later work showed that the timing of a negative response could follow the expected time of reward Hollerman and Schultz, 1998, that firing could scale quantitatively with the difference between obtained and expected value Bayer and Glimcher, 2005, and that identified dopamine neurons could share a common monotonic response function within a defined task Eshel et al., 2016.

The smallest useful formalisation is a one-step temporal-difference error:

δt = rt + γV(st+1) − V(st) δt is the error at time t; rt is reward received now; V(st) is the predicted future return in the current state; V(st+1) is the next-state prediction; and γ discounts later outcomes. The equation is valid only relative to a specified state representation and reward function.

Suppose a tone predicts one drop exactly one step later, and let γ equal 1. Before learning, both state values are zero. Delivery gives δ = 1 + 0 − 0 = +1. The tone state is then updated. Once its value reaches 1, delivery gives δ = 1 − 1 = 0. If the expected drop is omitted, δ = 0 − 1 = −1. The reward did not become less sweet. Its capacity to revise a prediction changed.

Worked example 1 · One cue, one drop

With a learning rate α = 0.2, an initial value of 0 and error +1, the cue value becomes V ← 0 + 0.2 × 1 = 0.2. The next identical reward is now only 0.8 better than predicted, so the next update is 0.16. Repetition shrinks the outcome error while building value at the predictor. The load-bearing idea is not that dopamine detects good things. It is that a transient can teach by reporting how wrong a value prediction was.

Reward prediction error shifts from outcome to cue during learning Three horizontal trial timelines show early learning, learned prediction and reward omission. Green and coral waveforms move from the outcome to the cue and dip at the expected outcome time. Learning changes where the error occurs Early trials Learned Omission cueexpected outcometime positive outcome error error transferred to predictor negative omission error

Temporal transfer is the signature that made RPE compelling

The cue acquires the positive error as it becomes the earliest reliable predictor. A correctly timed dip appears when the predicted outcome is omitted. These relations concern learning and expectation, not reward presence alone. Conceptual reconstruction of canonical conditioning findings.
Annotated numerical temporal-difference example Three panels show values before learning, after learning and on omission, with errors plus one, zero and minus one. The material reward stays fixed while informational value changes Before learning cue drop prediction V = 0reward r = 1 δ = 1 − 0 = +1Large update at delivery. After learning cue drop prediction V = 1reward r = 1 δ = 1 − 1 = 0No update at delivery. Omission cue none prediction V = 1reward r = 0 δ = 0 − 1 = −1Prediction must decrease.

Reward amount stays fixed while informational value changes

The numerical example exposes the causal variable. The outcome response is determined by the gap between observation and prediction, not by the physical reward alone. Illustrative; γ = 1 and a one-step task are simplifying assumptions.

Three signatures that should travel together

A reward prediction error claim becomes persuasive when three signatures converge. The first is sign. Better-than-expected and worse-than-expected outcomes should push the signal in opposite directions when sensory intensity is matched. The second is calibration through learning. As the prediction changes, the response should change with it, including transfer towards earlier predictors and a timed response to omission. The third is teaching consequence. Perturbing the transient should alter the next prediction or action in the direction implied by the error.

Any one signature can mislead. An unexpected stimulus may produce a burst without carrying value sign. A cue response may shift because action timing shifts. A perturbation may reduce behaviour by changing arousal rather than learning. Convergence matters because shape, computation and causal role are different claims.

Even sign needs care. Dopamine neurons often express negative errors as pauses from a limited baseline, so positive and negative responses need not be mirror images. Sensors can also saturate differently for increases and decreases. The right test is not visual symmetry. It is whether response magnitude changes monotonically with signed prediction error within the instrument’s usable range.

Worked example 2 · The probabilistic dispenser

A cue predicts one drop with probability 0.5. Its expected immediate value is 0.5 drops. On rewarded trials, the simple outcome error is 1 − 0.5 = +0.5. On omission trials, it is 0 − 0.5 = −0.5. A second cue predicts reward with probability 0.9. Receiving the same drop produces only +0.1, while omission produces −0.9. The physical event is identical within each outcome class, yet the learning signal differs because expectation differs.

This also separates RPE from uncertainty. The 0.5 cue is more uncertain before the outcome, but the sign of the outcome response depends on what happens. A signal that rises for both reward and omission under the uncertain cue may encode unsigned surprise or uncertainty rather than signed value error.

Reward probability is not the only source of expectation. Amount, delay, effort, internal need and inferred task state can all alter value. These additions make a model more realistic but also more vulnerable to post hoc rescue. The value function should be stated before the decisive neural result is inspected, or several pre-specified value models should compete on held-out trials.

Why the cue response is more than a correlation

A cue-locked burst might merely announce predicted reward. Temporal-difference theory makes a stronger claim: the transient is an error that trains earlier predictors. Causal tests can separate these roles. In second-order conditioning, suppressing dopamine at the start of an already valued cue impairs learning about a new predictor in a pattern consistent with removing a TD error Maes et al., 2020. Experimentally imposed dopamine transients can also produce learning effects expected from positive prediction errors Steinberg et al., 2013.

Evidence status

The bounded conclusion is strong: particular phasic dopamine signals can carry and causally implement TD-like teaching errors. It does not follow that every dopamine neuron, projection, release event or behavioural epoch computes the same scalar.

The hidden state inside the equation

The equation looks complete, but it quietly assumes that the system knows which state it is in. Consider a cue followed by reward at an uncertain time. As seconds pass without reward, has the animal entered a “late but still possible” state, or a “this trial is probably unrewarded” state? Those representations assign different values and therefore different errors to the same physical event.

Manipulations of temporal uncertainty and medial prefrontal cortex activity support the idea that dopamine errors depend on inferred hidden state, not only current sensory input Starkweather, Gershman and Uchida, 2018. Contingency-degradation experiments further show that temporal-difference models can account for behaviour and ventral-striatal dopamine when they use an appropriate prospective state representation Qian et al., 2025.

Mechanism

An RPE is always conditional on a predictive state model. A poor fit may mean that dopamine is not carrying an RPE. It may instead mean that the analyst supplied the wrong states, timing basis, outcome features or reward function. A decisive design must distinguish failure of the signal hypothesis from failure of the modeller’s representation.

The argument in one pass
Chart · flow
flowchart LR
  A["Opening case"] --> B["The error that moves through time"]
  B --> C["How the waveform is counterfeited"]
  C --> D["The value–salience–movement assay"]
  D -. "re-test" .-> A
  classDef first fill:#0a1424,stroke:#007D68,stroke-width:2px,color:#e6f3ff
  classDef step fill:#0a1424,stroke:#3f7da0,stroke-width:1px,color:#e6f3ff
  class A first
  class B,C,D step
A structural route through Reward prediction error and dopamine: from Opening case to Make rival dopamine mechanisms disagree. Reading use: follow the relation, then return to the surrounding argument for evidence and limits.

How the waveform is counterfeited

The classical pattern becomes overinterpreted when three steps collapse into one: a trace resembles an error; therefore the recorded signal encodes that error; therefore dopamine as a system has that function. The first step is descriptive. The second is causal. The third generalises across cells, projections and measurements. Each requires new evidence.

Counterfeit one: movement walks into the cue

During learning, animals do not remain mechanically constant. They orient, lick, accelerate, approach and initiate actions at new times. Distinct dopaminergic axons can carry rapid locomotion-related and reward-related signals in dorsal striatum Howe and Dombeck, 2016. Identified dopamine neurons can also show movement-related excitation or inhibition, and these components can depend on reward context. In one learning dataset, cue-related and movement-initiation-related components summed as action timing became more precise, producing apparent RPE correlates Coddington and Dudman, 2018.

Thought experiment 2 · The yoked movement

Two animals hear the same cue and receive the same reward. Animal A must initiate an approach. Animal B is carried towards the reward on a motorised platform while its limbs remain still. Now delay the movement in A without changing reward probability or delivery time. A scalar RPE should follow changed value prediction, not kinematics alone. A movement-linked signal should follow action onset. A mixed signal should split. The single changed property is self-initiated movement.

Counterfactual intervention separating value error from movement timing Two causal paths lead to an observed dopamine transient. One path runs from cue to expected value to reward prediction error. Another runs from cue to movement initiation. A movement-delay intervention changes only the second path. Hold reward statistics fixed; move action onset cue expected valueV(state) signed RPEoutcome − expectation movementinitiation time observed tracerecorded signal value pathmovement path intervention: delay movement without changing cue or reward

A useful intervention changes one parent of the trace

Delaying action while preserving cue and reward statistics creates a prediction difference between the RPE and movement accounts. Without this separation, timing correlations are underidentified. Illustrative causal design informed by movement-related dopamine findings.

Movement correlation does not imply that dopamine is a motor command. A transient may reinforce a state–action relation, alter future vigour, or reflect a prediction about action without directly initiating the movement being recorded. Dorsolateral striatal dopamine transients can shape the later statistical structure of spontaneous behaviour Markowitz et al., 2023. Conversely, disrupting rapid dopamine release can leave basic movement initiation intact while impairing reward-oriented conditioned behaviour Cai et al., 2024. Motor execution, action selection, action reinforcement, movement prediction error and motivational invigoration are distinct causal claims.

Counterfeit two: unsigned salience wears a signed mask

Unexpected events capture attention, trigger orienting and alter arousal. Those reactions often co-occur with positive reward errors because an unpredicted reward is also surprising. If a study contains only neutral baseline trials and surprising rewards, an unsigned surprise signal and a positive RPE make the same prediction.

Thought experiment 3 · The white flash

After a reward-predicting tone, insert a bright but value-neutral flash on rare trials. Match its novelty and sensory intensity to the unexpected reward as closely as the task permits. An RPE account predicts no positive value-signed response to the flash. A salience account predicts a transient to both. Now omit a strongly expected reward. A signed error should dip; unsigned salience should rise or remain positive. The changed property is value sign while surprise remains high.

Local dopamine release in nucleus accumbens can track perceived salience under conditions where value and salience are experimentally separated Kutlu et al., 2021. Yet “salience” is not a single mechanism. Novelty, sensory intensity, uncertainty, arousal and behavioural relevance can diverge. Salience is a rival only when it is operationalised strongly enough to make a different prediction. Calling every unexplained transient salient simply replaces one catch-all label with another.

Counterfeit three: the population is not one wire

Midbrain dopamine neurons differ by genetic subtype, projection target, input pattern and task response. Large-scale recordings have found specialised mixtures of sensory, motor and cognitive variables within ventral tegmental dopamine populations Engelhard et al., 2019. Genetic subtypes can map onto distinct functional response profiles Azcorra et al., 2023. Population averages are therefore useful descriptions of samples, not proof of a transmitter-wide scalar.

Dopamine subcircuits occupy different functional territories A stylised midbrain-to-striatum map shows dopamine subpopulations projecting to ventral, medial and lateral territories, with different balances of reward error, salience and movement signals. “Dopamine” names a distributed family of pathways midbrain ABC ventral targetmedial targetlateral / tail target reward error + motivationmixed sensory / cognitivemovement and action prediction

“Dopamine” is a family name

Different genetic and projection-defined populations can weight reward, movement, salience and other variables differently. A claim must name the sampled population and target territory. Conceptual synthesis from Howe and Dombeck, Engelhard et al., Azcorra et al. and related projection-specific work.

Recent work extends this diversity beyond a simple “reward versus movement” opposition. Dopamine input to the tail of striatum can carry an action prediction error, defined by the discrepancy between an action taken and the action predicted in that state, and can causally support value-free action learning Greenstreet et al., 2025. Striatal dopamine has also been reported to signal errors in predictions of value-neutral sensory relationships as well as valued outcomes Costa et al., 2025. The sharper distinction is reward prediction error versus broader predictive teaching, not prediction error versus every non-reward signal.

Counterfeit four: the measurement layer changes the message

A spike recorded at a dopamine cell body is not the same object as axonal calcium, transmitter release, extracellular concentration, receptor occupancy or downstream plasticity. Propagation can fail. Local cholinergic and presynaptic mechanisms can gate release. Uptake reshapes concentration. Receptors integrate over different timescales. A behavioural read-out adds another circuit transformation.

This matters empirically. Dopamine-cell spiking and nucleus-accumbens dopamine release can dissociate across learning and motivational epochs Mohebi et al., 2019. A statement such as “dopamine increased” therefore omits the very boundary needed to interpret the claim.

Thought experiment 4 · The projection swap

Record the same identified cell bodies while measuring release in two projection targets. Keep task, reward and movement fixed. Suppose soma spiking follows signed RPE, release in target X follows it closely, and release in target Y rises with movement regardless of error sign. Has the neuron stopped carrying RPE? No. Has “dopamine release” as a whole carried RPE? Also no. The changed property is the sampled measurement layer and target.

Measurement layers from dopamine cell firing to behaviour A horizontal cutaway shows soma spikes, axonal calcium, terminal release, extracellular concentration, receptor effects and behaviour. Transformations occur between every layer. Each instrument samples a different stage of the signal somaspikes axoncalcium terminalrelease outside cellconcentration receptorplasticity behaviourlearning/action propagationlocal gatingdiffusion + uptakeaffinity + timingcircuit state Typical instrumentelectrophysiology Typical instrumentcalcium photometry Typical instrumentsensor / voltammetry Typical instrumentpharmacology / imaging Read-outchoice, force

“Dopamine activity” is not one observable

Soma spikes, axonal calcium, extracellular concentration and receptor effects can diverge because local transformations intervene. Cross-layer agreement is evidence; it should not be assumed. Conceptual measurement cutaway; sensor and voltammetry measurements have distinct kinetics.
Worked example 3 · The pooled photometry trap

Suppose half the sampled axons respond +1 to positive value error and 0 to movement. The other half respond 0 to value error and +1 to acceleration. In a task where positive errors always trigger acceleration, the pooled signal is +1 on every relevant trial. Regressing it against RPE produces an excellent fit. Regressing it against movement produces the same fit. More trials reduce uncertainty around the wrong causal conclusion because the task never separates the predictors.

The value–salience–movement assay

The practical contribution of this paper is a compact causal protocol for deciding what a dopamine trace may be called. The value–salience–movement assay, or VSM assay, treats RPE attribution as an identifiability problem. It does not ask which model produces the prettiest average waveform. It asks which model keeps predicting when its usual correlates are pulled apart.

The assay has five commitments. Define the biological and measurement boundary. Manipulate signed value error independently of unsigned surprise. Manipulate movement timing independently of both. Compare explicit rival generators on held-out trials. Add a causal perturbation at the moment when the models disagree.

The value salience movement counterfactual cube A three-dimensional cube has axes signed value error, event salience and movement demand. Selected trial types occupy different corners, separating models that are usually correlated. Orthogonalise the three variables that usually travel together unexpected reward, no action novel neutral cue, action predicted reward, action withheld omission, no action action-only probe signed value error → salience → movement demand → A full factorial design is not always feasible. The minimum is a set of cells that breaks each major correlation at least once.

The VSM cube turns a naming dispute into an experiment

Trial types are selected so value error, salience and movement no longer rise and fall together. The decisive cells are counterfactual: same value error with different movement, or same salience with opposite value sign. Original decision instrument; illustrative trial placements.

1. State the candidate boundary before fitting anything

A useful claim names the cell class or projection, target region, recording method, task epoch and timescale. “VTA dopamine soma spikes during cue onset in this conditioning task” is testable. “Dopamine encodes reward prediction error” is too broad to fail cleanly. The boundary also determines which transformations have already occurred. A terminal-release claim cannot be inferred directly from a cell-body recording.

2. Build trials that separate the generators

The minimum design is not a large sample. It is a set of contrasts that changes one candidate cause while holding the others as steady as possible. The table below converts the three rival families into trial requirements.

ContrastHold steadyChangeRPE predictionRival prediction
Expectation contrastReward magnitude, movement, sensory eventPredicted probability or amountSigned response scales with actual minus expected valueReward detector stays similar
Salience-only probeExpected reward and movementNovelty or unsigned sensory surpriseNo value-signed responseSalience or sensory-error response appears
Movement-time shiftCue, outcome probability and timingAction onset or kinematicsTrace follows value errorMovement-linked component follows action
Signed surprise pairAbsolute surprise and sensory intensityBetter versus worse than expectedOpposite-signed responsesUnsigned salience responds similarly

3. Pre-register rival generators, not just extra regressors

A regression coefficient is not a mechanism. Write each candidate as a generator with an input, temporal rule and failure condition. The model should predict trials not used to choose its features. It should also specify when it is allowed to abstain because the relevant state or movement variable was not measured.

Candidate generatorMinimal formStrengthening observationWeakening observation
Signed value errorβvδvalue(t)Opposite sign for matched positive and negative errors; generalises across expectation levelsSame response to equally salient opposite-sign events
Unsigned salienceβs|surprise(t)|Novel neutral and valued surprises evoke comparable transientsReliable below-baseline response to negative value error
Movement-linked signalβmM(t)Response shifts when action timing changes but reward statistics do notTrace stays fixed under large movement-time shifts
Multiplexed signalβvδ + βsS + βmMComponents separate under orthogonal contrasts and transfer across tasksA single simpler generator wins held-out interventions

4. Use a perturbation where the models disagree

Observation tells us what covaries. A teaching-signal claim requires downstream consequence. Perturb the transient in a narrow window and test the update predicted by each model. Suppressing a candidate positive RPE should reduce subsequent value updating. Suppressing a movement-linked teaching signal may instead alter action repetition, stimulus–action association or future vigour. The critical outcome is more than less behaviour. It is the pattern of what fails to be learned.

5. Fit on one set of contrasts and decide on another

Train models on familiar conditions, then decide on counterfactual trials that break the usual covariance. Report held-out predictive performance, coefficient stability and uncertainty. If multiple models remain tied, the conclusion is underidentified. Do not resolve the tie by choosing the most familiar theory. Add a trial type that separates them.

Why prediction and intervention answer different questions

Imagine that the full VSM model predicts forty held-out trials better than the RPE-only model. The gain is concentrated in trials where movement begins before the cue. That result earns a descriptive claim: the measured trace contains movement-related information that improves prediction. It does not yet tell us whether that component teaches an action, energises performance, reflects an efference copy or merely covaries with a hidden state. Prediction ranks generators by compression of the observations. Intervention asks which variable the circuit actually changes.

Now suppress a cue-locked transient for 150 milliseconds only after unexpectedly good outcomes. Three patterns matter. If later choices reveal a smaller update in expected value while action execution remains intact, the result strengthens an RPE teaching interpretation. If value estimates remain stable but the animal is less likely to repeat the just-completed action, an action-learning account gains ground. If latency, speed and choice all deteriorate immediately, the manipulation may have disrupted performance rather than learning. The causal signature is the selective downstream update, not simply a behavioural decrement.

Mechanism distinction · Update, expression and state

A perturbation can change at least three things: the stored estimate learned from the event, the expression of an already learned policy, or the inferred state used to compute both. Separate these with a delayed choice test, a probe that changes motor demands, and a transfer trial in which the same inferred value guides a different action. Without those readouts, “learning was reduced” may be only a new label for slower movement.

Temporally precise stimulation and inhibition studies show why intervention can strengthen a teaching-signal claim, while newer work separates reward-guided updating from movement initiation and identifies value-free action prediction errors Steinberg et al., 2013; Maes et al., 2020; Cai et al., 2024; Greenstreet et al., 2025. The practical consequence is a crossed experiment: specify which transient is changed, when it is changed, and which learned quantity should fail. A model that predicts the waveform but not the selective consequence has described a correlate, not yet the teaching mechanism.

Worked example 4 · A configured VSM experiment

Train mice on two auditory cues. Cue H predicts a water drop with probability 0.8; cue L predicts the same drop with probability 0.2. On a minority of trials, a novel neutral light appears. Independently, a gate either permits an approach movement immediately or delays it by 600 milliseconds while reward timing remains fixed. Record optogenetically identified VTA soma spikes and dopamine release in one specified striatal target.

A reward after cue L creates a larger positive value error than the same reward after cue H. An omission after H creates a larger negative error than omission after L. The neutral light changes salience without reward value. The gate shifts movement without changing cue contingency. A time-locked perturbation on only one contrast then tests whether the transient teaches value, action or both. The design is small, but every included trial earns its place by forcing a disagreement.

Executable lab: generate a counterfeit dopamine trace

Choose a hidden generator, vary predictor collinearity, and compare held-out model performance. All data are synthetic and generated in your browser.

The lab demonstrates statistical underidentification. It does not replace biological intervention.

ModelHeld-out R²Key coefficients

The interactive lab intentionally lets movement correlate with RPE. When collinearity is high, a movement-generated trace can look well explained by an RPE-only model, especially under noise. Lowering the correlation creates the counterfactual trials that reveal the hidden generator. Increasing sample size in a confounded task is therefore not the same as improving identification.

Illustrative held-out model comparison A grouped bar chart compares RPE, salience, movement and full models under confounded and orthogonal trial designs. Orthogonal trials reveal the true multiplexed generator. Better identification can lower the wrong models’ scores 0.00.20.40.60.8 confounded trialsorthogonal trials RPE onlysalience onlymovement onlyfull VSM held-out R²

Better identification can lower the wrong models’ scores

Under confounding, several models inherit predictive power from the same trial structure. Counterfactual trials reduce those borrowed fits and allow the true mixed generator to separate. Illustrative values, not biological measurements.

The failure case: a perfect fit to the wrong task

Suppose every surprising reward causes approach, pupil dilation and an RPE at the same instant. A large dataset can estimate their common waveform with great precision. It still cannot identify which variable caused the recorded signal. The problem sits in the covariance structure of the task, not in the optimiser.

The collinearity failure path A diagonal path shows reward surprise causing movement, pupil dilation and a dopamine transient. Several models fit the same data. A red break point calls for counterfactual trials. Correlated predictors can share one apparent success rewardsurprise approachmovement pupilarousal dopaminetransient Observed dataset: RPE, movement and salience rise on the same trials RPE-only fit: high · movement-only fit: high · salience-only fit: high ! Needed: reward without movement, movement without reward, surprise without value change.

The adversary is the task’s covariance structure

When reward surprise, movement and arousal co-occur, every model can inherit predictive power from the same event. The repair is a new trial type, not a more elaborate regression on the old trials. Illustrative failure analysis.
Python reference implementation and assumptions

The browser lab uses ordinary least squares on a 70:30 chronological split. The code below reproduces the core comparison. It assumes linear mixing, observed predictors and independent Gaussian measurement noise. A positive result permits a candidate attribution within those assumptions. A negative result may indicate the wrong generator, the wrong state representation, non-linearity or insufficient separation. It cannot establish a biological teaching role without intervention.

Python · synthetic VSM comparison
from __future__ import annotations

from typing import Literal

import numpy as np

Mechanism = Literal["rpe", "salience", "movement", "multiplex"]


def make_trials(
    n: int = 480,
    confound: float = 0.70,
    noise: float = 0.45,
    mechanism: Mechanism = "multiplex",
    seed: int = 6801,
) -> dict[str, np.ndarray]:
    """Generate synthetic trials with a known hidden dopamine generator."""
    if not 0 <= confound < 1:
        raise ValueError("confound must be in [0, 1)")
    rng = np.random.default_rng(seed)
    expected = rng.choice([0.2, 0.5, 0.8], size=n)
    outcome = (rng.random(n) < expected).astype(float)
    rpe = outcome - expected
    novelty = (rng.random(n) < 0.22).astype(float)
    salience = np.abs(rpe) + 0.8 * novelty

    rpe_standard = (rpe - rpe.mean()) / max(rpe.std(), 1e-9)
    independent_move = rng.normal(size=n)
    movement = confound * rpe_standard + np.sqrt(1 - confound**2) * independent_move

    weights = {
        "rpe": (1.0, 0.0, 0.0),
        "salience": (0.0, 1.0, 0.0),
        "movement": (0.0, 0.0, 1.0),
        "multiplex": (0.8, 0.45, 0.6),
    }[mechanism]
    dopamine = (
        weights[0] * rpe
        + weights[1] * salience
        + weights[2] * movement
        + rng.normal(scale=noise, size=n)
    )
    return {"rpe": rpe, "salience": salience, "movement": movement, "dopamine": dopamine}


def fit_and_score(data: dict[str, np.ndarray], names: list[str]) -> tuple[float, np.ndarray]:
    """Fit ordinary least squares on 70% of trials and score the held-out 30%."""
    n = len(data["dopamine"])
    split = int(n * 0.7)
    X = np.column_stack([np.ones(n)] + [data[name] for name in names])
    y = data["dopamine"]
    beta, *_ = np.linalg.lstsq(X[:split], y[:split], rcond=None)
    pred = X[split:] @ beta
    truth = y[split:]
    denominator = np.sum((truth - truth.mean()) ** 2)
    r2 = 1 - np.sum((truth - pred) ** 2) / denominator
    return float(r2), beta


if __name__ == "__main__":
    trials = make_trials()
    for label, predictors in {
        "RPE": ["rpe"],
        "Salience": ["salience"],
        "Movement": ["movement"],
        "Full": ["rpe", "salience", "movement"],
    }.items():
        r2, beta = fit_and_score(trials, predictors)
        print(f"{label:9s} held-out R²={r2: .3f}  coefficients={beta}")
How the claim changes under pressure
Chart · state
stateDiagram-v2
  [*] --> Claim
  state "Three signatures that should<br/>travel together" as Claim
  state "Why the cue response is more<br/>than a correlation" as Test
  state "The hidden state inside the<br/>equation" as Revision
  Claim --> Test: expose mechanism
  Test --> Decision: evidence survives
  Test --> Revision: rival survives
  Revision --> Claim: narrow or rebuild
  Decision --> [*]
The revision loop begins with Three signatures that should travel… and keeps counterexamples inside the reasoning path. Reading use: follow the relation, then return to the surrounding argument for evidence and limits.

What survives the alternatives

The VSM assay is deliberately strict, but it can fail in two directions. It can over-credit RPE because movement and salience remain confounded. It can also under-credit RPE because the analyst uses the wrong value model, state representation or temporal basis. The remedy is not to weaken the identification standard. It is to make the alternatives predict something new.

The RPE may be richer than one scalar

Individual dopamine neurons can differ in how strongly positive and negative errors change firing. One influential interpretation is distributional reinforcement learning: heterogeneous neurons learn different expectiles of the future reward distribution, allowing a population to represent more than the mean Dabney et al., 2020. A scalar model can then fit poorly even when the population remains value-error-based.

Other extensions use belief states, outcome features, action-specific errors or several predictive channels. These are legitimate alternatives when they make new predictions. They become immunising stories when every deviation is absorbed after the result. A richer state model should be penalised for extra flexibility, fitted before the decisive contrast and tested on a different task structure. Perturbing a circuit proposed to carry the hidden-state inference should change the error in the direction that model predicts.

Open research hypothesis

A projection-defined dopamine population may multiplex a signed value error with a domain-specific error, while local release mechanisms reweight the mixture according to target state. This proposal is strengthened if the components separate under VSM contrasts, generalise across tasks and produce dissociable learning effects when perturbed. It is weakened if one simpler generator predicts all held-out interventions. A rival explanation is an unmeasured state representation. The smallest useful implementation is a two-target, two-measurement VSM task.

Do not infer subjective reward from a teaching signal

RPE evidence does not show that a transient is pleasure, wanting or conscious value. It identifies a computational relation between expectation, outcome and learning. Motivation, hedonic impact, effort allocation and felt experience require their own evidence. Likewise, a movement-linked dopamine signal does not establish that dopamine issues motor commands. The allowed conclusion stays at the level tested.

This boundary matters beyond neuroscience. Reinforcement-learning systems often name any scalar feedback “reward”, then slide from an optimisation variable to language about preference or desire. The biological case demonstrates why that move is unsafe. A teaching signal may alter future policy without representing what the organism likes, what it consciously feels, or what it would endorse under reflection.

Evidence matrix for dopamine reward prediction error attribution A matrix places waveform resemblance, held-out prediction, counterfactual trials, causal perturbation and cross-layer replication against the conclusions they permit. The conclusion should never outrun the strongest completed test EvidenceDescriptive fitRPE attributionTeaching roleGeneral claim Canonical cue / outcome waveformHeld-out value predictionVSM counterfactual contrastTime-locked causal perturbationProjection and measurement replication permitted not yet stronger task-specific causal role bounded only No row licences “dopamine as a whole equals reward prediction error”.

An attribution ladder, not a verdict stamp

Waveform resemblance supports a descriptive claim. Counterfactual separation supports pathway- and task-specific RPE attribution. A teaching-role claim needs causal perturbation. Replication broadens scope but does not erase biological heterogeneity. Original evidence matrix.
Decision rule

Call a trace a reward prediction error only when it shows value-signed responses, predicts held-out counterfactual trials, resists salience and movement alternatives, and causes the learning update expected from that error. Otherwise name the observation directly: cue transient, movement-linked activity, salience-correlated release or mixed predictive signal.

The evidence conversation
Chart · sequence
sequenceDiagram
  participant Q as Question
  participant M as Mechanism
  participant E as Evidence
  participant D as Decision
  Q->>M: Opening case
  M->>E: Three signatures that should travel together
  E-->>M: compare rival explanations
  E->>D: Make rival dopamine mechanisms disagree
  D-->>Q: revise scope or act
Question, mechanism, evidence and decision exchange the burdens developed in Reward prediction error and dopamine. Reading use: follow the relation, then return to the surrounding argument for evidence and limits.

Make rival dopamine mechanisms disagree

Stop asking whether dopamine “is” reward prediction error at the level of an entire transmitter system. Decide at the level where an intervention can be specified: this cell class, this projection, this measurement, this task state, this temporal window and this downstream update.

For experimental neuroscience, the changed decision is methodological. Do not spend the next cohort increasing power on a task in which reward surprise, movement and salience remain locked together. Add the counterfactual trial first. Pair observation with a perturbation whose learning consequence differs across models. Report the narrowest conclusion that survives.

For reinforcement-learning and agent designers, the changed decision is architectural. Do not build a single global variable called “dopamine” that mixes reward error, novelty, action surprise and response vigour. Keep those quantities typed and separately inspectable. Combine them only where the update rule states how and why. Biological multiplicity is not permission for an unstructured scalar.

The classical RPE account remains one of neuroscience’s most productive theory–experiment correspondences. Its strength comes from precise predictions about expectation, timing and learning. Preserving that strength now requires the same discipline that created it: make rival mechanisms disagree, then let the intervention decide.

Glossary

Reward prediction error
The signed difference between obtained and expected reward value, often including discounted next-state value.
Temporal-difference learning
A family of learning rules that updates predictions from differences between successive value estimates and observed reward.
Phasic dopamine
Brief changes in dopamine-neuron firing or dopamine concentration over subsecond to short timescales.
Salience
Behavioural importance or attention capture. It must be operationalised as novelty, surprise, arousal or another measurable construct.
Action prediction error
The discrepancy between an action that occurred and the action predicted in that state, independent of reward value in the strict formulation.
Hidden state
A task condition not directly given by current sensory input and therefore inferred from history or context.
Candidate boundary
The exact biological population, projection, measurement layer, task and time window to which a mechanism claim applies.
Multiplexing
Carrying more than one task variable in the activity of a population or pathway.

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