Retinal metric: a stimulus distance measure derived from population neural responses

The ability of the organism to distinguish between various stimuli is limited by the structure and noise in the population code of its sensory neurons. Here we infer a distance measure on the stimulus space directly from the recorded activity of 100 neurons in the salamander retina. In contrast to previously used measures of stimulus similarity, this "neural metric" tells us how distinguishable a pair of stimulus clips is to the retina, given the noise in the neural population response. We show that the retinal distance strongly deviates from Euclidean, or any static metric, yet has a simple structure: we identify the stimulus features that the neural population is jointly sensitive to, and show the SVM-like kernel function relating the stimulus and neural response spaces. We show that the non-Euclidean nature of the retinal distance has important consequences for neural decoding.Comment: 5 pages, 4 figures, to appear in Phys Rev Let

Stimulus-dependent maximum entropy models of neural population codes

Neural populations encode information about their stimulus in a collective fashion, by joint activity patterns of spiking and silence. A full account of this mapping from stimulus to neural activity is given by the conditional probability distribution over neural codewords given the sensory input. To be able to infer a model for this distribution from large-scale neural recordings, we introduce a stimulus-dependent maximum entropy (SDME) model---a minimal extension of the canonical linear-nonlinear model of a single neuron, to a pairwise-coupled neural population. The model is able to capture the single-cell response properties as well as the correlations in neural spiking due to shared stimulus and due to effective neuron-to-neuron connections. Here we show that in a population of 100 retinal ganglion cells in the salamander retina responding to temporal white-noise stimuli, dependencies between cells play an important encoding role. As a result, the SDME model gives a more accurate account of single cell responses and in particular outperforms uncoupled models in reproducing the distributions of codewords emitted in response to a stimulus. We show how the SDME model, in conjunction with static maximum entropy models of population vocabulary, can be used to estimate information-theoretic quantities like surprise and information transmission in a neural population.Comment: 11 pages, 7 figure

An overview of maximum entropy encoding models.

<p>The explicit dependence of single-neuron terms (, vertical axis, ‘T’ or ‘S’), and the absence or presence of pairwise terms (, horizontal axis, ‘1’ or ‘2’), together define the type of the maximum entropy model (e.g. pairwise SDME is ‘S2’). For completeness, the first row of the table includes static maximum entropy models of population vocabulary, , which have no explicit stimulus dependence. Full conditionally independent model (T1) reproduces exactly the instantaneous firing rate of every neuron, and thus fully captures the stimulus sensitivity, history effects, and adaptation on a single neuron level; for experimentally recorded rasters with stimulus repeats, simulated T1 rasters are often generated by taking the original data and, at each time point and for every neuron, randomly permuting the responses recorded on different stimulus repeats. “Total correlation” is the pairwise correlation matrix of activities, , averaged over all repetitions and all times in the experiment.</p

Response of a large population of ganglion cells to a 10 s long repeated visual stimulus.

<p>(<b>a</b>) White noise uncorrelated Gaussian stimulus presented at and the spiking patterns of 3 cells to repeated presentations of the stimulus. (<b>b</b>) Spike-trigerred averages of 110 simultaneously recorded cells; a subset of 100 cells was chosen for further analysis. (<b>c</b>) The histogram of pairwise correlation coefficients between cells for repeated Gaussian white noise stimulus (green). For comparison, the statistics of the response on repeated natural pixel movie (red), and non-repeated natural pixel movie (blue) is also shown, as documented in Ref. <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002922#pcbi.1002922-Ganmor2" target="_blank">[35]</a>. The significance cutoff for correlation coefficients is , 95% of correlations are above this cut (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002922#s4" target="_blank">Methods</a>). (<b>d</b>) Average pairwise correlation coefficient between cells as a function of the distance (mean and std are across pairs of cells at a given distance).</p

Surprise and information transmission estimated from the pairwise SDME (S2) model.

<p>(<b>a</b>) Surprise rate (blue) is estimated from the static ME and S2 models assuming independence of codewords across time bins. The instantaneous information rate (red) is the difference between the surprise and the noise entropy rate, estimated from the S2 model (see text). The information transmission rate is the average of the instantaneous information across time. (<b>b</b>) Population firing rate as a function of time shows that bursts of spiking strongly correlate with the bursts of surprise and information transmission in the population. (<b>c</b>) The stimulus (normalized to zero mean and unit variance) is shown for reference as a function of time.</p

Measured vs predicted noise correlations for the pairwise SDME (S2) model.

<p>Noise correlation (see text) is estimated from recorded data for every pair of neurons, and plotted against the noise correlation predicted by the S2 model (each pair of neurons = one dot; shown are dots for neurons; for significantly correlated pairs, the slope of the best fit line is , with ). Conditionally independent models predict zero noise correlation for all pairs.</p

Pairwise SDME (S2) model predicts the firing rate of single cells better than conditionally independent LN (S1) models.

<p>(<b>a</b>) Example of the PSTH segment for one cell (green), the best prediction of the S1 model (blue) and of the S2 model (red). (<b>b</b>) Correlation coefficient between the true PSTH and S2 model prediction (vertical axis) vs. the correlation between the true PSTH and the S1 model prediction (horizontal axis); each plot symbol is a separate cell, dotted line shows equality. S2 significantly outperforms S1 (, paired two-sided Wilcoxon test). The neuron chosen in panel (a) is shown in orange.</p

Pairwise SDME (S2) model predicts population activity patterns for neurons better than conditionally independent LN (S1) models.

<p>(<b>a</b>) The activity raster for 100 neurons across 626 repeats of the stimulus at a point in time where the retina is moderately active (). Dots represent individual spikes; training repeats denoted in black, test repeats in orange. (<b>b</b>) The diversity in retinal responses in a. Shown are all distinct patterns; their number is comparable to the number of repeats. Neurons are resorted by their instantaneous firing rate (high rate = top, low rate = bottom). (<b>c</b>) S2 model fit on the training repeats predicts the reliably estimated correlation coefficients between pairs of neurons at various time points where the retina is active. We identify all correlation coefficients whose value can be estimated from data with less than 25% relative error across many splits of the repeats into two halves. The value of these correlation coefficients is estimated on the test set (horizontal axis) and compared to the model prediction (vertical axis). (<b>d</b>) The log-likelihood ratio of the population firing patterns under the S2 model and under the S1 model, shown as a function of time (violet dots, scale at left) for an example (test) stimulus repeat. For reference, the average population firing rate is shown in grey (scale at right). The arrow denotes the time bin displayed in a, b.</p

Clustering of response patterns into basins of attraction centered on meta-stable patterns generalizes across repeats.

<p><b>a</b>) Every response pattern from data is assigned to its corresponding meta-stable pattern by descending on the energy landscape defined by the S2 model of <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002922#pcbi.1002922.e083" target="_blank">Eq (4)</a> until the local minimum is reached (see text). Across all test repeats and at each point in time (horizontal axis), we find the metastable states that are visited more than 30 times, plot their energy (vertical axis), and the number of repeats on which that metastable state is visited (shade of red). <b>b</b>) Inset: for (blue rectangle in a), we plot the frequency of visit to each metastable state (dots) in the training set (horizontal) against the frequency in the test set (vertical). Main panel: the same analysis across all time bins (different colors) superposed, dashed line is equality.</p