The effect of correlated neuronal firing and neuronal heterogeneity on population coding accuracy in guinea pig inferior colliculus

Abstract

It has been suggested that the considerable noise in single-cell responses to a stimulus can be overcome by pooling information from a large population. Theoretical studies indicated that correlations in trial-to-trial fluctuations in the responses of different neurons may limit the improvement due to pooling. Subsequent theoretical studies have suggested that inherent neuronal diversity, i.e., the heterogeneity of tuning curves and other response properties of neurons preferentially tuned to the same stimulus, can provide a means to overcome this limit. Here we study the effect of spike-count correlations and the inherent neuronal heterogeneity on the ability to extract information from large neural populations. We use electrophysiological data from the guinea pig Inferior-Colliculus to capture inherent neuronal heterogeneity and single cell statistics, and introduce response correlations artificially. To this end, we generate pseudo-population responses, based on single-cell recording of neurons responding to auditory stimuli with varying binaural correlations. Typically, when pseudo-populations are generated from single cell data, the responses within the population are statistically independent. As a result, the information content of the population will increase indefinitely with its size. In contrast, here we apply a simple algorithm that enables us to generate pseudo-population responses with variable spike-count correlations. This enables us to study the effect of neuronal correlations on the accuracy of conventional rate codes. We show that in a homogenous population, in the presence of even low-level correlations, information content is bounded. In contrast, utilizing a simple linear readout, that takes into account the natural heterogeneity, even of neurons preferentially tuned to the same stimulus, within the neural population, one can overcome the correlated noise and obtain a readout whose accuracy grows linearly with the size of the population.

Figure 1. Neuronal heterogeneity.

The conditional mean (over 200–500 trials for given stimulus BC) firing rate is plotted as a function of the binaural correlation level for the 30 different cells in the data set. The error bars depict the standard deviation of the firing rate. The firing rates were computed for the period of 100 ms following stimulus onset. Typically, the tuning curves are monotonic in the BC and approximately linear.

Figure 2. Signal correlations.

The signal correlations, i.e., the Pearson correlations between the tuning curves of the different neurons, are shown in color code.

Figure 3. Illustration of Step 2 of our algorithm: translating the continuous input variable to the discrete neural response.

The figure depicts the two cumulative distribution functions, [left, cumulative distribution of Gaussian input variable] and [right, empirical cumulative distribution of neuronal spike-count]. In the specific example shown in the figure, the random input variable is . The cumulative distribution of having input of equal or less than is in this case . To translate the input variable to spike count we choose the number of spikes that corresponds to the same cumulative distribution. In the specific example of the illustration, this corresponds to firing of four spikes, .

Figure 4. Mean spike-count correlation coefficient as function of input correlation for homogeneous pseudo-populations.

Pseudo-population responses of neurons were generated based on the response distribution of each of the 30 neurons in our data set, shown by the different panels. To compute the spike count correlations we simulated the response of the pseudo population for 1000 trials for every stimulus condition. The correlation coefficient matrix was averaged across all 21 different BC levels in our data set. The red line shows a linear fit that is forced through zero, for comparison, and the black line is the identity line.

Figure 5. Spike count correlation coefficient matrices for different levels of input correlation in a homogenous pseudo-population, using the response distribution of cell 9.

The correlation coefficients were averaged over all stimulus conditions, . For each stimulus condition the correlation coefficient matrix was estimated by generating 10,000 trials for the pseudo population response for the given stimulus.

Figure 6. Eigenvalue spectrum and eigenvectors of the spike count correlation coefficient matrix.

(A) The spectrum of the correlation matrix of a homogenous pseudo-population of 30 neurons, with input correlations of , and is shown,. The inset shows the largest eigenvalue (circle) and mean of all other eigenvalue (diamond) as function of population size. (B) The eigenvectors of the correlation matrices A are shown by the different colors. The thick blue lines show the eigenvector that corresponds to the largest eigenvalue.

Figure 7. The signal in a homogeneous population.

The signal, in terms of the covariance between the neural response and the stimulus is shown as a function of the neuron number in a homogeneous population based on the responses of cell 13. As the population is homogeneous the signal is distributed homogeneously in the responses of the different neurons in the population. The inset shows the distribution of the signal over the different eigenvectors of the neural spike-count correlation matrix (with uniform correlation coefficient of 0.2), as a function of the rank of their eigenvalue. The inset shows the projection of the signal vector on each eigenvalue in percent. Note that the first eigenvector corresponds to the uniform vector (cf Figure 7). The signal and the correlation matrix were estimated using 10,000 repetitions for every stimulus value .

Figure 8. OLE accuracy in a homogeneous pseudo population.

The OLE accuracy is shown in terms of: A one over the mean square estimation error, and its components: B the inverse of the bias and C the inverse of the variance, as a function of the number of cells for different levels of uniform correlations, by the different colors. The dots show the estimated OLE accuracy that was measured by first training the OLE weights using 10500 trials of psedo-population response (i.e., 500 trials per stimulus condition) and estimating the accuracy over 10500 trials of generalization, this procedure was repeated and averaged 100 times. Note that as the pseudo-populations are homogeneous and are uniquely determined by the marginal response distribution of a single neuron, there is no need to average over different realizations of the population. Specifically, here we have used the response distribution of cell 13 (Figure 1) to define the population response.

Figure 9. Spike count correlation coefficient matrices for a specific realization of a heterogeneous population of neurons for different levels of input correlation, from bottom to top.

The correlation coefficients matrix was computed for heterogeneous pseudo-population of neurons composed of all 30 neurons in the data set. The correlation coefficient matrix was computed by averaging over all stimulus conditions. For each stimulus (BC level), the conditional correlation coefficient matrix was estimated by generating 10,000 realizations of the pseudo population response.

Figure 10. Spike count correlation as function of the input correlation strength, , for a heterogeneous pseudo-population.

Neural responses for a population of 30 neurons containing all the cells in our data set were generated for all stimulus conditions (i.e., all 21 different BC levels in our data set) and the correlation coefficients for all the different cell pairs were averaged (blue circles), see Methods. The error-bars show the standard deviation of the correlation coefficients in the populations. The solid red line is a linear regression line with slope of 0.85, forced via the origin. For comparison the dashed black line shows the identity line.

Figure 11. Spectrum and collective modes of fluctuations in a heterogeneous population.

(A) Eigenvalue spectrum of the spike count correlation matrix of a heterogeneous pseudo-population containing all 30 neurons in our data set, and input correlations of and . The inset shows the largest eigenvalue (circle) and mean of all other eigenvalues (diamond) as function of population size. (B) The different eigenvectors of the spike-count correlation matrix ( and ) are shown by different colors. The thick blue line depicts the eigenvector with the largest eigenvalue.

Figure 12. The signal in a heterogeneous population.

The signal, in terms of the covariance between the neural response and the stimulus is shown as a function of the cell number in a heterogeneous population based on the responses of all cells in the data set. The inset shows the distribution of the signal over the different eigenvectors of the neural spike-count correlation matrix (with uniform correlation coefficient of 0.2), as a function of the rank of their eigenvalue, in percent. Note that the first eigenvector corresponds to the uniform vector (cf Figure 13). The signal and the correlation matrix were estimated using 10,000 repetitions for every stimulus value .

Figure 13. One over the Mean Square Error (A), bias (B) and variance (C) of OLE estimation, plotted as function of population size for heterogeneous pseudo-population with different input correlation level (different color).

All heterogeneous pseudo-populations created based on empirical response statistics of 30 cells. The OLE weights were learned from a training set of 500 trials per stimulus condition and the accuracy of the readout was estimated using a generalization set of the same size. For each value of population size the accuracy was averaged over 100 realizations of the neural population, where the different cells in each realization were drawn independently with equal probabilities (with repetitions) from the pool of all neurons.

Figure 14. Functionally dependent correlations.

(A) One over the Mean Square Error of OLE estimation, plotted as function of population size for heterogeneous pseudo-population with different functionally dependent input correlation level (different color). All heterogeneous pseudo-populations created based on empirical response statistics of 30 cells. The OLE weights were learned from a training set of 500 trials per stimulus condition and the accuracy of the readout was estimated using a generalization set of the same size. For each value of population size the accuracy was averaged over 100 realizations of the neural population, where the different cells in each realization were drawn independently with equal probabilities (with repetitions) from the pool of all neurons. (B–E) Four examples of the neuronal noise correlation matrix (in color code) for different values of functionally dependent input correlations, respectively. The matrices were estimated by averaging the conditional correlation coefficient matrices [estimated using 10,000 trials] over all stimulus conditions. (F) Eigenvalue spectrum of the functionally dependent correlation matrix of a heterogeneous pseudo-population containing all 30 neurons in our data set, and functionally dependent input correlations of . The inset shows the largest eigenvalue (circle) and mean of all other eigenvalues (diamond) as function of population size. (G) The eigenvectors plotted from output correlation matrix of the functionally dependent correlation matrix. Each vector plotted in different color. The thick blue line shows the eigenvector with the largest eigenvalues. (H) The signal distribution across the eigenvectors of the spike-count correlation matrix is shown as a function of the eigenvalue rank.

Figure 15. The overlap between the OLE weights and the uniform direction.

The cosine of the angle between the OLE weights vector and uniform vector is plotted as function of population size without correlations (blue) and with input correlation of 0.2 (green), in a heterogeneous population.

Figure 16. MSE of estimation made by an OLE based on pseudo-population of two neurons plotted as function of spike-count correlation Low heterogeneity pair (red) two cells generated using cells 28 & 30 response statistics, both tuning curves has slops that are monotonic and with the same sign.

Homogenous pair (blue) both cells response generated using cell 30 response statistics (cells tuning curves are identical). High heterogeneity pair (green) response generated based on the response of cells 30 & 8.