Fig 1.
Correlational and Bayesian decoding approaches to uncertainty.
(A) In the Bayesian decoding approach, the posterior distribution over the stimulus values (right panels) is decoded from the full pattern of neuronal activity in the population (left panels: each circle denotes the spike count of a neuron with preferred stimulus indicated on the abscissa). This yields both an estimate of the stimulus and a measure of the uncertainty of this estimate. Uncertainty is defined operationally as the variance of such posterior (red bars in the right panels). (B-D) In the correlational approach, it is assumed that a specific feature of neuronal activity, such as tuning curve width (B) or gain (C), or noise correlations (D) encode sensory uncertainty. It is then possible to identify such a code by observing which features of neuronal activity correlate with the information content of the stimulus.
Fig 2.
Ideal observer reproduces owl localization behavior.
(A) In the auditory localization task, the azimuth of the sound source has to be estimated using the ITD (denoted by δ). (B) Barn owl head-turning responses at different levels of BC. Each curve corresponds to a different value of the true ITD. At low BC, head turns are on average biased towards the front (0 deg). (C) Variability of the head-turning responses. Different symbols correspond to different true ITDs. The continuous line is an exponential fit. At low BC, responses are more variable. (B,C) Replotted from Saberi et al (1998). Angles are measured with a precision of 4°. (D,E) Same as (B,C) but for the post-marginalization ideal observer. The true azimuth (values reported in the inset) is indicated by the dashed lines in (D).
Fig 3.
IC and OT models approximate the ideal observers.
(A) Schematic model of IC. Sounds reaching the two ears are first convolved with bandpass filters, then delayed, then cross-correlated at several delays, and lastly rectified to obtain tuning curves to ITD. (B) Average KL divergence between the posterior distribution computed by the ideal observer and the posterior decoded from the model population activity before (dashed line) and after (continuous) rectification. KL was normalized to the KL between the ideal observer posterior and the priors. Shaded areas represent s.e.m. Note that the pre-rectification KL divergence differs from 0 purely due to the fact that we cannot exactly decode from a finite amount of training examples. (C) Schematic model of OT. Outputs of the IC model are first combined across frequency bands and divided by the signal energy. Then, the outputs are passed through a static nonlinearity, filtered and rectified; these operations are equivalent to a non-linear filtering stage. The color of the units indicates their frequency preference (low: red, mid: blue, high: green). (D) Same as (B) but for the OT model activity, after rectification. (E) Example posterior distribution in one trial with true ITD = -145 and BC = 0.1, for the post-marginalization ideal observer (dashed pink line) and the reconstruction from the output of the OT model population (continuous black). (F) Same as (E), but with BC = 0.4. Note that the small secondary peak around ITD = 200 is present only in the reconstruction, not in the ideal observer’s posterior.
Fig 4.
Comparison of OT model and data.
(A) Tuning curve for ITD of a neuron recorded in owl’s OT. Each panel corresponds to a different BC. (B) Peak response as a function of BC. Thin lines represent individual neurons, thick line is the population average. Data replotted from Saberi et al.. (C,D) Same as (A,B) but for a model neuron.
Fig 5.
Uncertainty estimation quality from IC and OT neuronal activity.
(A) Tuning cuve width and inverse gain as a function of BC, expressed as a percent change from the value measured at the highest BC. Gray symbols: models. Blue: data from Cazettes et al. [14]. (B) Performance of different estimators as a function of BC. The estimators used IC neuronal activity to estimate the uncertainty of the pre-marginalization ideal observer. Quality was measured by the cross-validated R2 between the ideal observer’s posterior log-variance and estimated log-variance. Black: uncertainty of the reconstructed posterior. Blue: uncertainty estimated by a linear fit to single neuron activity. Red: uncertainty estimated by a linear fit to the total neuronal activity. Green: uncertainty estimated by a linear fit of the width of population activity. Line width represents 95% c.i. (C) Same as (B), but using OT neuronal activity to estimate the uncertainty of the post-marginalization ideal observer. In this case performance is still split by BC, but the linear fits were performed by combining trials across all BC (i.e. the estimators, as well as the ideal observer, had no knowledge of the true BC value).
Fig 6.
Modulation of OT activity by stimulus reliability.
(A,B) Average KL divergence (A) and performance of different estimators of uncertainty (B) from the population activity generated by the model of Saberi et al. [17]. Same conventions as in Fig 3B and Fig 5, respectively. Because this population is not a linear PPC, reconstruction of the uncertainty through linear decoding fails in this example, leading to negative R-square. The corresponding curve was thus removed. (C-F) ITD tuning curves of our OT model for different stimuli. (C) Original stimulus. Tuning curve replotted from Fig 4C-top. (D) Shorter stimulus duration, 5msec vs. 10msec for the original stimulus. (E) Low-passed filtered stimulus. (F) High-pass filtered stimulus. (G-J) Same as (C-F) but for the model of Saberi et al. [17].