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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 Universität 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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Flow diagram of the maximum entropy test.
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pcbi-1002539-g001: Flow diagram of the maximum entropy test.

Mentions: Figure 1 shows a flow diagram of the test. A step-by-step procedure is provided in Table 1. In step (1) the Poisson parameters and the correlation coefficient are initialized with their sample means that are obtained from the data set. The p-value is then maximized by applying an optimization algorithm like simulated annealing (step (2), cf. Section “Optimization of the Nuisance Parameters” in Text S1). To this end, we estimate the p-value based on the maximum entropy distribution subject to the optimization parameters. First, we calculate a divergence between the data and the maximum entropy distribution (step 2.2). We then draw many samples from the maximum entropy distribution and estimate an empirical distribution over divergences (step 2.3). By comparing the divergence to this empirical distribution we can assess how likely it is that the maximum entropy distribution generated the data. This gives us a p-value for a particular set of Poisson rates and a correlation coefficient (step 2.4). The maximization over these parameters then yields the most conservative p-value. In step (3) the second-order assumption is rejected if the p-value is 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)

Flow diagram of the maximum entropy test.
© Copyright Policy
Related In: Results  -  Collection

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

pcbi-1002539-g001: Flow diagram of the maximum entropy test.
Mentions: Figure 1 shows a flow diagram of the test. A step-by-step procedure is provided in Table 1. In step (1) the Poisson parameters and the correlation coefficient are initialized with their sample means that are obtained from the data set. The p-value is then maximized by applying an optimization algorithm like simulated annealing (step (2), cf. Section “Optimization of the Nuisance Parameters” in Text S1). To this end, we estimate the p-value based on the maximum entropy distribution subject to the optimization parameters. First, we calculate a divergence between the data and the maximum entropy distribution (step 2.2). We then draw many samples from the maximum entropy distribution and estimate an empirical distribution over divergences (step 2.3). By comparing the divergence to this empirical distribution we can assess how likely it is that the maximum entropy distribution generated the data. This gives us a p-value for a particular set of Poisson rates and a correlation coefficient (step 2.4). The maximization over these parameters then yields the most conservative p-value. In step (3) the second-order assumption is rejected if the p-value is 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 Universität 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