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Statistical analyses support power law distributions found in neuronal avalanches.

Klaus A, Yu S, Plenz D - PLoS ONE (2011)

Bottom Line: Consistent with critical state dynamics, avalanche size distributions exhibited robust scaling behavior in which the maximum avalanche size was limited only by the spatial extent of sampling ("finite size" effect).Using both the Kolmogorov-Smirnov statistic and a maximum likelihood approach, we found the slope to be close to -1.5, which is in line with previous reports.Finally, the power law model for neuronal avalanches was compared to the exponential and to various heavy-tail distributions based on the Kolmogorov-Smirnov distance and by using a log-likelihood ratio test.

View Article: PubMed Central - PubMed

Affiliation: Section on Critical Brain Dynamics, National Institute of Mental Health, Bethesda, MD 20892, USA. Andreas.Klaus@frontiersin.org

ABSTRACT
The size distribution of neuronal avalanches in cortical networks has been reported to follow a power law distribution with exponent close to -1.5, which is a reflection of long-range spatial correlations in spontaneous neuronal activity. However, identifying power law scaling in empirical data can be difficult and sometimes controversial. In the present study, we tested the power law hypothesis for neuronal avalanches by using more stringent statistical analyses. In particular, we performed the following steps: (i) analysis of finite-size scaling to identify scale-free dynamics in neuronal avalanches, (ii) model parameter estimation to determine the specific exponent of the power law, and (iii) comparison of the power law to alternative model distributions. Consistent with critical state dynamics, avalanche size distributions exhibited robust scaling behavior in which the maximum avalanche size was limited only by the spatial extent of sampling ("finite size" effect). This scale-free dynamics suggests the power law as a model for the distribution of avalanche sizes. Using both the Kolmogorov-Smirnov statistic and a maximum likelihood approach, we found the slope to be close to -1.5, which is in line with previous reports. Finally, the power law model for neuronal avalanches was compared to the exponential and to various heavy-tail distributions based on the Kolmogorov-Smirnov distance and by using a log-likelihood ratio test. Both the power law distribution without and with exponential cut-off provided significantly better fits to the cluster size distributions in neuronal avalanches than the exponential, the lognormal and the gamma distribution. In summary, our findings strongly support the power law scaling in neuronal avalanches, providing further evidence for critical state dynamics in superficial layers of cortex.

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Collapse of rescaled cluster size distributions in neuronal avalanches.A. Depiction of the rescaling approach for synthetic PMFs for maximum sizes N = 8, 16, 32, 64 (left). The system size, N, corresponds to the number of electrodes included in the analysis. Cluster sizes s were normalized by the system size N (ss/N) and the renormalized probability was obtained according to P(s)P(s)/A(N), resulting in a collapse of the cluster size distributions (right). Here, the definition of A(N) with upper bound N was used (Eq. 16). The vertical arrow indicates the system size (scaled to unity). B. Collapse of rescaled cluster size distributions for average in vitro distributions (n = 7), average in vivo distributions under anesthesia (rat, n = 7), and the two awake monkeys with low- and high-density array, respectively (from left to right). Note that the maximum cluster size for all data sets increases with N with the distribution showing a clear cut-off beyond the system size (s/N = 1). The exponent  for the empirical distributions was fitted individually for each system size N (see Materials and Methods).
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pone-0019779-g002: Collapse of rescaled cluster size distributions in neuronal avalanches.A. Depiction of the rescaling approach for synthetic PMFs for maximum sizes N = 8, 16, 32, 64 (left). The system size, N, corresponds to the number of electrodes included in the analysis. Cluster sizes s were normalized by the system size N (ss/N) and the renormalized probability was obtained according to P(s)P(s)/A(N), resulting in a collapse of the cluster size distributions (right). Here, the definition of A(N) with upper bound N was used (Eq. 16). The vertical arrow indicates the system size (scaled to unity). B. Collapse of rescaled cluster size distributions for average in vitro distributions (n = 7), average in vivo distributions under anesthesia (rat, n = 7), and the two awake monkeys with low- and high-density array, respectively (from left to right). Note that the maximum cluster size for all data sets increases with N with the distribution showing a clear cut-off beyond the system size (s/N = 1). The exponent for the empirical distributions was fitted individually for each system size N (see Materials and Methods).

Mentions: We used this property as an indicator of the power law scaling in neuronal avalanches. To study the invariance of cluster size distributions, we varied the number of electrodes, N, that were included for the detection of negative threshold crossings in the LFP (see Materials and Methods). Event sizes in the resulting size distributions were expressed in units of N by the basic rescaling approach ss/N. A proper renormalization of the probability mass functions (PMFs) resulted then in a collapse of power law distributions for different N, as shown in Figure 2A for theoretical power law distributions (see also Supporting Information, Text S1 and Figure S1). Figure 2B shows the collapse of cluster size distributions for the empirical data sets from Figure 1, indicating scale-free dynamics in neuronal avalanches independent of N. Importantly, the cluster size distributions for varying N showed a sharp cut-off at the system size (i.e., at s/N = 1) for the rescaled distributions (Figure 2B). Using a renormalization for time-shuffled cluster sizes based on either the power law assumption or the exponential model did not result in a collapse of the corresponding distributions (Figure S1). In addition, the maximum cluster size in the time-shuffled data decreased for increasing N (Figure S1), indicating that shuffling destroyed the scale-free behavior in the resulting distributions. These results, together with the finite-size scaling and the collapse of the original cluster size distributions in Figure 2 are consistent with critical state dynamics and the hypothesis of a power law distribution of neuronal avalanche sizes.


Statistical analyses support power law distributions found in neuronal avalanches.

Klaus A, Yu S, Plenz D - PLoS ONE (2011)

Collapse of rescaled cluster size distributions in neuronal avalanches.A. Depiction of the rescaling approach for synthetic PMFs for maximum sizes N = 8, 16, 32, 64 (left). The system size, N, corresponds to the number of electrodes included in the analysis. Cluster sizes s were normalized by the system size N (ss/N) and the renormalized probability was obtained according to P(s)P(s)/A(N), resulting in a collapse of the cluster size distributions (right). Here, the definition of A(N) with upper bound N was used (Eq. 16). The vertical arrow indicates the system size (scaled to unity). B. Collapse of rescaled cluster size distributions for average in vitro distributions (n = 7), average in vivo distributions under anesthesia (rat, n = 7), and the two awake monkeys with low- and high-density array, respectively (from left to right). Note that the maximum cluster size for all data sets increases with N with the distribution showing a clear cut-off beyond the system size (s/N = 1). The exponent  for the empirical distributions was fitted individually for each system size N (see Materials and Methods).
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Related In: Results  -  Collection

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getmorefigures.php?uid=PMC3102672&req=5

pone-0019779-g002: Collapse of rescaled cluster size distributions in neuronal avalanches.A. Depiction of the rescaling approach for synthetic PMFs for maximum sizes N = 8, 16, 32, 64 (left). The system size, N, corresponds to the number of electrodes included in the analysis. Cluster sizes s were normalized by the system size N (ss/N) and the renormalized probability was obtained according to P(s)P(s)/A(N), resulting in a collapse of the cluster size distributions (right). Here, the definition of A(N) with upper bound N was used (Eq. 16). The vertical arrow indicates the system size (scaled to unity). B. Collapse of rescaled cluster size distributions for average in vitro distributions (n = 7), average in vivo distributions under anesthesia (rat, n = 7), and the two awake monkeys with low- and high-density array, respectively (from left to right). Note that the maximum cluster size for all data sets increases with N with the distribution showing a clear cut-off beyond the system size (s/N = 1). The exponent for the empirical distributions was fitted individually for each system size N (see Materials and Methods).
Mentions: We used this property as an indicator of the power law scaling in neuronal avalanches. To study the invariance of cluster size distributions, we varied the number of electrodes, N, that were included for the detection of negative threshold crossings in the LFP (see Materials and Methods). Event sizes in the resulting size distributions were expressed in units of N by the basic rescaling approach ss/N. A proper renormalization of the probability mass functions (PMFs) resulted then in a collapse of power law distributions for different N, as shown in Figure 2A for theoretical power law distributions (see also Supporting Information, Text S1 and Figure S1). Figure 2B shows the collapse of cluster size distributions for the empirical data sets from Figure 1, indicating scale-free dynamics in neuronal avalanches independent of N. Importantly, the cluster size distributions for varying N showed a sharp cut-off at the system size (i.e., at s/N = 1) for the rescaled distributions (Figure 2B). Using a renormalization for time-shuffled cluster sizes based on either the power law assumption or the exponential model did not result in a collapse of the corresponding distributions (Figure S1). In addition, the maximum cluster size in the time-shuffled data decreased for increasing N (Figure S1), indicating that shuffling destroyed the scale-free behavior in the resulting distributions. These results, together with the finite-size scaling and the collapse of the original cluster size distributions in Figure 2 are consistent with critical state dynamics and the hypothesis of a power law distribution of neuronal avalanche sizes.

Bottom Line: Consistent with critical state dynamics, avalanche size distributions exhibited robust scaling behavior in which the maximum avalanche size was limited only by the spatial extent of sampling ("finite size" effect).Using both the Kolmogorov-Smirnov statistic and a maximum likelihood approach, we found the slope to be close to -1.5, which is in line with previous reports.Finally, the power law model for neuronal avalanches was compared to the exponential and to various heavy-tail distributions based on the Kolmogorov-Smirnov distance and by using a log-likelihood ratio test.

View Article: PubMed Central - PubMed

Affiliation: Section on Critical Brain Dynamics, National Institute of Mental Health, Bethesda, MD 20892, USA. Andreas.Klaus@frontiersin.org

ABSTRACT
The size distribution of neuronal avalanches in cortical networks has been reported to follow a power law distribution with exponent close to -1.5, which is a reflection of long-range spatial correlations in spontaneous neuronal activity. However, identifying power law scaling in empirical data can be difficult and sometimes controversial. In the present study, we tested the power law hypothesis for neuronal avalanches by using more stringent statistical analyses. In particular, we performed the following steps: (i) analysis of finite-size scaling to identify scale-free dynamics in neuronal avalanches, (ii) model parameter estimation to determine the specific exponent of the power law, and (iii) comparison of the power law to alternative model distributions. Consistent with critical state dynamics, avalanche size distributions exhibited robust scaling behavior in which the maximum avalanche size was limited only by the spatial extent of sampling ("finite size" effect). This scale-free dynamics suggests the power law as a model for the distribution of avalanche sizes. Using both the Kolmogorov-Smirnov statistic and a maximum likelihood approach, we found the slope to be close to -1.5, which is in line with previous reports. Finally, the power law model for neuronal avalanches was compared to the exponential and to various heavy-tail distributions based on the Kolmogorov-Smirnov distance and by using a log-likelihood ratio test. Both the power law distribution without and with exponential cut-off provided significantly better fits to the cluster size distributions in neuronal avalanches than the exponential, the lognormal and the gamma distribution. In summary, our findings strongly support the power law scaling in neuronal avalanches, providing further evidence for critical state dynamics in superficial layers of cortex.

Show MeSH
Related in: MedlinePlus