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A maximum entropy test for evaluating higher-order correlations in spike counts.

Onken A, Dragoi V, Obermayer K - PLoS Comput. Biol. (2012)

Bottom Line: Applying our test to artificial data shows that the effects of higher-order correlations on these divergence measures can be detected even when the number of samples is small.These results demonstrate that higher-order correlations can matter when estimating information theoretic quantities in V1.They also show that our test is able to detect their presence in typical in-vivo data sets, where the number of samples is too small to estimate higher-order correlations directly.

View Article: PubMed Central - PubMed

Affiliation: Technische Universit├Ąt Berlin, Berlin, Germany. arno.onken@unige.ch

ABSTRACT
Evaluating the importance of higher-order correlations of neural spike counts has been notoriously hard. A large number of samples are typically required in order to estimate higher-order correlations and resulting information theoretic quantities. In typical electrophysiology data sets with many experimental conditions, however, the number of samples in each condition is rather small. Here we describe a method that allows to quantify evidence for higher-order correlations in exactly these cases. We construct a family of reference distributions: maximum entropy distributions, which are constrained only by marginals and by linear correlations as quantified by the Pearson correlation coefficient. We devise a Monte Carlo goodness-of-fit test, which tests--for a given divergence measure of interest--whether the experimental data lead to the rejection of the hypothesis that it was generated by one of the reference distributions. Applying our test to artificial data shows that the effects of higher-order correlations on these divergence measures can be detected even when the number of samples is small. Subsequently, we apply our method to spike count data which were recorded with multielectrode arrays from the primary visual cortex of anesthetized cat during an adaptation experiment. Using mutual information as a divergence measure we find that there are spike count bin sizes at which the maximum entropy hypothesis can be rejected for a substantial number of neuronal pairs. These results demonstrate that higher-order correlations can matter when estimating information theoretic quantities in V1. They also show that our test is able to detect their presence in typical in-vivo data sets, where the number of samples is too small to estimate higher-order correlations directly.

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Subpopulation analysis of the data that are presented in Figure 6 C.(A) Overall firing rates of the 11 neurons in the data set from Figure 6 for the control and adaptation conditions. The rates were averaged over all stimuli. ( B) Fraction of neuronal pairs rejected by the maximum entropy test with the mutual information difference as the divergence measure () for the high firing rate (, cf. A) population of neurons. ( C) Same as in B but for the low firing rate population (). Rejection rates were averaged over all neuron pairs and all time bins. Simulated annealing [45] was applied to maximize the p-value (cf. Text S1). Number  of Monte Carlo samples was 1000. The false discovery rate of the rejections was corrected using the Benjamini-Hochberg procedure [35].
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pcbi-1002539-g007: Subpopulation analysis of the data that are presented in Figure 6 C.(A) Overall firing rates of the 11 neurons in the data set from Figure 6 for the control and adaptation conditions. The rates were averaged over all stimuli. ( B) Fraction of neuronal pairs rejected by the maximum entropy test with the mutual information difference as the divergence measure () for the high firing rate (, cf. A) population of neurons. ( C) Same as in B but for the low firing rate population (). Rejection rates were averaged over all neuron pairs and all time bins. Simulated annealing [45] was applied to maximize the p-value (cf. Text S1). Number of Monte Carlo samples was 1000. The false discovery rate of the rejections was corrected using the Benjamini-Hochberg procedure [35].

Mentions: Figure 7 A shows the overall firing rates of the individual neurons in the control and the adaptation conditions. The data suggest the existence of a high firing rate () and a low firing rate () population. Figures 7 B, C show the results of the maximum entropy test for the difference in the mutual information as a divergence measure separately for both populations. The figures show that the rejected pairs were significantly higher for the high firing rate population for bin sizes in both the control and the adaptation condition (paired t-test, ). Moreover, the rejection rates in this subpopulation were significantly higher in the adaptation condition than in the control condition for these bin sizes (paired t-test, ), which did not hold for the low firing rate population.


A maximum entropy test for evaluating higher-order correlations in spike counts.

Onken A, Dragoi V, Obermayer K - PLoS Comput. Biol. (2012)

Subpopulation analysis of the data that are presented in Figure 6 C.(A) Overall firing rates of the 11 neurons in the data set from Figure 6 for the control and adaptation conditions. The rates were averaged over all stimuli. ( B) Fraction of neuronal pairs rejected by the maximum entropy test with the mutual information difference as the divergence measure () for the high firing rate (, cf. A) population of neurons. ( C) Same as in B but for the low firing rate population (). Rejection rates were averaged over all neuron pairs and all time bins. Simulated annealing [45] was applied to maximize the p-value (cf. Text S1). Number  of Monte Carlo samples was 1000. The false discovery rate of the rejections was corrected using the Benjamini-Hochberg procedure [35].
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3369943&req=5

pcbi-1002539-g007: Subpopulation analysis of the data that are presented in Figure 6 C.(A) Overall firing rates of the 11 neurons in the data set from Figure 6 for the control and adaptation conditions. The rates were averaged over all stimuli. ( B) Fraction of neuronal pairs rejected by the maximum entropy test with the mutual information difference as the divergence measure () for the high firing rate (, cf. A) population of neurons. ( C) Same as in B but for the low firing rate population (). Rejection rates were averaged over all neuron pairs and all time bins. Simulated annealing [45] was applied to maximize the p-value (cf. Text S1). Number of Monte Carlo samples was 1000. The false discovery rate of the rejections was corrected using the Benjamini-Hochberg procedure [35].
Mentions: Figure 7 A shows the overall firing rates of the individual neurons in the control and the adaptation conditions. The data suggest the existence of a high firing rate () and a low firing rate () population. Figures 7 B, C show the results of the maximum entropy test for the difference in the mutual information as a divergence measure separately for both populations. The figures show that the rejected pairs were significantly higher for the high firing rate population for bin sizes in both the control and the adaptation condition (paired t-test, ). Moreover, the rejection rates in this subpopulation were significantly higher in the adaptation condition than in the control condition for these bin sizes (paired t-test, ), which did not hold for the low firing rate population.

Bottom Line: Applying our test to artificial data shows that the effects of higher-order correlations on these divergence measures can be detected even when the number of samples is small.These results demonstrate that higher-order correlations can matter when estimating information theoretic quantities in V1.They also show that our test is able to detect their presence in typical in-vivo data sets, where the number of samples is too small to estimate higher-order correlations directly.

View Article: PubMed Central - PubMed

Affiliation: Technische Universit├Ąt Berlin, Berlin, Germany. arno.onken@unige.ch

ABSTRACT
Evaluating the importance of higher-order correlations of neural spike counts has been notoriously hard. A large number of samples are typically required in order to estimate higher-order correlations and resulting information theoretic quantities. In typical electrophysiology data sets with many experimental conditions, however, the number of samples in each condition is rather small. Here we describe a method that allows to quantify evidence for higher-order correlations in exactly these cases. We construct a family of reference distributions: maximum entropy distributions, which are constrained only by marginals and by linear correlations as quantified by the Pearson correlation coefficient. We devise a Monte Carlo goodness-of-fit test, which tests--for a given divergence measure of interest--whether the experimental data lead to the rejection of the hypothesis that it was generated by one of the reference distributions. Applying our test to artificial data shows that the effects of higher-order correlations on these divergence measures can be detected even when the number of samples is small. Subsequently, we apply our method to spike count data which were recorded with multielectrode arrays from the primary visual cortex of anesthetized cat during an adaptation experiment. Using mutual information as a divergence measure we find that there are spike count bin sizes at which the maximum entropy hypothesis can be rejected for a substantial number of neuronal pairs. These results demonstrate that higher-order correlations can matter when estimating information theoretic quantities in V1. They also show that our test is able to detect their presence in typical in-vivo data sets, where the number of samples is too small to estimate higher-order correlations directly.

Show MeSH
Related in: MedlinePlus