Article Figures & Data
Figures
- Figure 1.
Perturbations shapes. We used the same 16 perturbation shapes for the two reference stimuli. The first 12 perturbation shapes were combinations of two Fourier components, and the last four ones were random combinations of them:
and
, with T the duration of the perturbation and t = 0 the beginning of the perturbation. The first perturbations j = 1…7 were
. For j = 8,…,10, they were the opposite of the three first ones:
. For j = 11, 12 we used
. Perturbations 13 and 14 were random combinations of perturbations 1, 2, 3, 11, and 12, constrained to be orthogonal. Perturbations 15 and 16 were random combinations of fj for j ∈ [1,8] and gk for k ∈ [1,7], allowing higher frequencies than perturbation directions 13 and 14. Perturbation direction 15 and 16 were also constrained to be orthogonal. The largest amplitude for each perturbation we presented was 115 µm. An exception was made for perturbations 15 and 16 applied to the second reference trajectory, as for this amplitude they had a discrimination probability below 70%. They were thus increased by a factor 1.5. The largest amplitude for each perturbation was repeated at least 93 times, with the exception of perturbation 15 (32 times) and 16 (40 times) on the second reference trajectory. - Figure 2.
Sensitivity of a neural population to visual stimuli. A, The retina is stimulated with repetitions of a reference stimulus (here the trajectory of a bar, in blue), and with perturbations of this reference stimulus of different shapes and amplitudes. Purple and red trajectories are perturbations with the same shape, of small and large amplitude. B, Mean response of three example cells to the reference stimulus (left column and light blue in middle and right columns) and to perturbations of small and large amplitudes (middle and right columns).
- Figure 3.
Closed-loop experiments to probe the range of stimulus sensitivity. A, Experimental setup: we stimulated a rat retina with a moving bar. Retinal ganglion cell (RGC) population responses were recorded extracellularly with a multielectrode array. Electrode signals were high-pass filtered and spikes were detected by threshold crossing. We computed the discrimination probability of the population response and adapted the amplitude of the next perturbation. B, left, The neural responses of 60 sorted RGCs are projected along the axis going through the mean response to reference stimulus and the mean response to a large perturbation. Small dots are individual responses, large dots are means. Middle, Mean and standard deviation (in gray) of response projections for different amplitudes of an example perturbation shape. Right, Distributions of the projected responses to the reference (blue), and to small (purple) and large (red) perturbations. Discrimination is high when the distribution of the perturbation is well separated from the distribution of the reference. C, Discrimination probability as a function of amplitude A. The discrimination increases as an error function,
, with
(gray line: fit). Ticks on the x-axis show the amplitudes that have been tested during the closed-loop experiment. - Figure 4.
Local model for responses to perturbations. A, The firing rates in response to a perturbation of a reference stimulus are modulated by filters applied to the perturbation. There is a different filter for each cell and each time bin. Because the model is conditionally independent across neurons we show the schema for one example neuron only. B, Raster plot of the responses of an example cell to the reference (blue) and perturbed (red) stimuli for several repetitions. C, PSTH of the same cell in response to the same reference (blue) and perturbation (red). Prediction of the local model for the perturbation is shown in green. D, Performance of the local model at predicting the change in PSTH induced by a perturbation, as measured by Pearson correlation coefficient between data and model, averaged over cells (green). The data PSTH were calculated by grouping perturbations of the same shape and of increasing amplitudes by groups of 20 and computing the mean firing rate at each time over the 20 perturbations of each group. The model PSTH was calculated by mimicking the same procedure. To control for noise from limited sampling, the same performance was calculated from synthetic data of the same size, where the model is known to be exact (black).
- Figure 5.
The Fisher information predicts the experimentally measured sensitivity. A, Sensitivity coefficients c for the two reference stimuli and 16 perturbation shapes, measured empirically and predicted by the Fisher information (Eq. 14) and the local model. The purple point corresponds to the perturbation shown in Figure 2. Dashed line stands for best linear fit. B, Same as A, but for responses simulated with the local model, with the same amount of data as in experiments. The discriminability of perturbations was measured in the same way than for recorded responses. Dots and error bars stand for mean and SEM over 10 simulations. Dashed line stands for identity.
- Figure 6.
Bayesian decoding of the local model outperforms the linear decoder. A, Responses to a perturbation of the reference stimulus (reference in blue, perturbation in red) are decoded using the local model (green) or a linear decoder (orange). For each decoder, the area shows one standard deviation from the mean. B, Decoding error as a function of amplitude, for an example perturbation shape. C, LSNR for perturbations with different frequencies (differing from the standard SNR definition to deal with locality in stimulus space and in time; Materials And Methods/Local signal to noise ratio in decoding). The performance of both decoders decreases for high frequency stimuli.
- Figure 7.
Signature of efficient coding in the sensitivity. A, Spectral density of the stimulus used in experiments, which is monotonically decreasing. B, Simple theory of retinal function: the stimulus is filtered by noisy photoreceptors, whose signal is then filtered by the noisy retinal network. The retinal network filter was optimized to maximize information transfer at constant output power. C, Sensitivity of the recorded retina to perturbations of different frequencies. Note the nonmonotonic behavior. D, Same as C, but for the theory of optimal processing. E, Information transmitted by the retina on the perturbations at different amplitudes. F, Same as E, but for the theory.
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