Fig 2. Logarithmic activation of dopamine dynamics explains scale invariant responses.

(A) Minimal circuit for dopamine responses. Dopamine (d) is activated by the logarithm of expected reward (log R) and is inhibited by negative feedback from GABAergic input (g). See Methods for equations and parameters. (B) The response of VTA dopaminergic neurons in mice (n = 5) to a water reward of variable volume (black squares, mean ± SEM, taken from Fig 1C in [38]) is well-described by the logarithmic relation Δd = μ log(au+b) (red line r² = 0.999, best-fit parameters a = 0.5±0.1, b = 1.5±0.05, μ = 4.9±0.45, N = 7 reward magnitudes). When the reward is preceded by an odor cue (gray squares) the response is well-described by subtracting a constant from the uncued response (pink line, r² = 0.99, best-fit subtraction constant is −3.2±0.1). (C) Simulation of dopamine and GABA responses to a step increase in expected reward input which corresponds to the presentation of a reward-predicting cue. The step is given by R(t) = R₀+λθ(t−t₀) where θ(t−t₀) is a unit step function, R₀ = 1 and λ = 7 for the black line (large reward) and λ = 1.8 for the gray line (small reward). Insets. Average change in firing rates from dopaminergic (type I) and GABAergic (type II) VTA neurons, in response to reward-predicting cues for a small reward (gray) or a large reward (black). Data from Fig 2D of [49]. (DE) Population responses of dopaminergic neurons of two Macaque monkeys to variable size liquid reward, either without a preceding cue (left panels, n = 55 neurons), or with a preceding visual cue that predicts reward delivery with 50% probability (right panels, n = 57 neurons). (D) The expected-reward input following the reward-predicting cue is R = 0.5(b+λu) (where λ is proportional to reward magnitude), which is doubled following reward delivery, R = b+λu, where u is the reward magnitude (we use b = 2 and λ = 10ml⁻¹). Dashed lines correspond to reward omission. (E) Experimentally measured average dopaminergic responses, using data from Fig 2A and Fig 4B of [26]. When the reward is delivered without a cue, dopaminergic responses increase with reward magnitude (left panel). When it is given after a cue that predicts reward delivery with 50% probability, dopaminergic responses to reward delivery are identical (right panel), as predicted by the FCD property. This is despite the 10-fold difference in reward magnitude. (F) Simulations of the dopamine model capture the experimentally observed dynamics. All simulation parameters are provided in Table 1.