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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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Effect of sample size on the Monte Carlo maximum entropy test results (solid black line) and on the maximum likelihood ratio test results (dashed blue line) with Poisson rate . The entropy difference was used as a divergence measure. Significance level was . Rates were estimated over 100 trials. ( A) Percent rejections of the maximum entropy hypothesis. Data were sampled from maximum entropy distributions with random correlation strengths. ( B) Percent rejections of the  hypothesis. Data were sampled from a copula-based mixture model with uniformly random mixture parameter (cf. , Equation 21).
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pcbi-1002539-g003: Effect of sample size on the Monte Carlo maximum entropy test results (solid black line) and on the maximum likelihood ratio test results (dashed blue line) with Poisson rate . The entropy difference was used as a divergence measure. Significance level was . Rates were estimated over 100 trials. ( A) Percent rejections of the maximum entropy hypothesis. Data were sampled from maximum entropy distributions with random correlation strengths. ( B) Percent rejections of the hypothesis. Data were sampled from a copula-based mixture model with uniformly random mixture parameter (cf. , Equation 21).

Mentions: We compared the likelihood ratio test to the proposed Monte Carlo test for varying sample sizes. Figure 3 A shows the percentages of rejections of the maximum entropy hypothesis for data that were sampled from a maximum entropy model (Type I error). For both tests the percentages do not depend on the number of samples. The percentages of rejections are generally smaller for the likelihood ratio test. Both tests, however, have a rejection rate below the significance level .


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

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

Effect of sample size on the Monte Carlo maximum entropy test results (solid black line) and on the maximum likelihood ratio test results (dashed blue line) with Poisson rate . The entropy difference was used as a divergence measure. Significance level was . Rates were estimated over 100 trials. ( A) Percent rejections of the maximum entropy hypothesis. Data were sampled from maximum entropy distributions with random correlation strengths. ( B) Percent rejections of the  hypothesis. Data were sampled from a copula-based mixture model with uniformly random mixture parameter (cf. , Equation 21).
© Copyright Policy
Related In: Results  -  Collection

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

pcbi-1002539-g003: Effect of sample size on the Monte Carlo maximum entropy test results (solid black line) and on the maximum likelihood ratio test results (dashed blue line) with Poisson rate . The entropy difference was used as a divergence measure. Significance level was . Rates were estimated over 100 trials. ( A) Percent rejections of the maximum entropy hypothesis. Data were sampled from maximum entropy distributions with random correlation strengths. ( B) Percent rejections of the hypothesis. Data were sampled from a copula-based mixture model with uniformly random mixture parameter (cf. , Equation 21).
Mentions: We compared the likelihood ratio test to the proposed Monte Carlo test for varying sample sizes. Figure 3 A shows the percentages of rejections of the maximum entropy hypothesis for data that were sampled from a maximum entropy model (Type I error). For both tests the percentages do not depend on the number of samples. The percentages of rejections are generally smaller for the likelihood ratio test. Both tests, however, have a rejection rate below the significance level .

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