Marken JP et al. (2016), A Markovian Entropy Measure for the Analysis of...

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PLoS One 2016 Dec 15;1112:e0168342. doi: 10.1371/journal.pone.0168342.

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A Markovian Entropy Measure for the Analysis of Calcium Activity Time Series.

Marken JP , Halleran AD , Rahman A , Odorizzi L , LeFew MC , Golino CA , Kemper P , Saha MS .

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Methods to analyze the dynamics of calcium activity often rely on visually distinguishable features in time series data such as spikes, waves, or oscillations. However, systems such as the developing nervous system display a complex, irregular type of calcium activity which makes the use of such methods less appropriate. Instead, for such systems there exists a class of methods (including information theoretic, power spectral, and fractal analysis approaches) which use more fundamental properties of the time series to analyze the observed calcium dynamics. We present a new analysis method in this class, the Markovian Entropy measure, which is an easily implementable calcium time series analysis method which represents the observed calcium activity as a realization of a Markov Process and describes its dynamics in terms of the level of predictability underlying the transitions between the states of the process. We applied our and other commonly used calcium analysis methods on a dataset from Xenopus laevis neural progenitors which displays irregular calcium activity and a dataset from murine synaptic neurons which displays activity time series that are well-described by visually-distinguishable features. We find that the Markovian Entropy measure is able to distinguish between biologically distinct populations in both datasets, and that it can separate biologically distinct populations to a greater extent than other methods in the dataset exhibiting irregular calcium activity. These results support the benefit of using the Markovian Entropy measure to analyze calcium dynamics, particularly for studies using time series data which do not exhibit easily distinguishable features.

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	Fig 1. Measured calcium time series exhibit different levels of distinguishability for visual features.Representative single-cell calcium activity time series from (a) a progenitor neuron from embryonic Xenopus laevis or (b) a mature, synaptic neuron from embryonic mouse. Crosses represent individual time points from (a) our own dataset or (b) data received from [15]. While both time series exhibit calcium spikes, they are more easily identified in the cell in (b) than in the cell in (a). Furthermore, the calcium dynamics in (b) are governed almost entirely by spikes whereas in (a), one can see activity patterns that are more complex than the simple spiking behavior. The Xenopus neural progenitor time series consists of 900 data points measured at 0.25 Hz, while the murine synaptic neuron time series consists of 800 data points measured at 10 Hz.
	Fig 2. Illustrations of the Markov Processes which represent the observed calcium activity time series.Processes are defined with (a) n = 3 and k = 1, (b) n = 3 and k = 2, and (c) n = 4 and k = 1. Colored transitions in the observed schematic time series are correspondingly colored as state transitions in the Markov Process below the time series, and also designated in the state transition matrices. Note that the time series are identical between (a), (b), and (c). For clarity, each line between states in the central state-transition graph of (b) is condensed to represent both a forward and a reverse state transition.
	Fig 3. Parameter choices do not qualitatively change the biological interpretation of our information entropy measure.Cohen’s d statistic [29] comparing distributions of entropy values for cellular calcium activity of (a) stage 14 and stage 18 Xenopus laevis embryos, (b) stage 18 and stage 22 Xenopus laevis embryos, (c) stage 14 and stage 22 Xenopus laevis embryos, and (d) mature retrotrapezoid nucleus neurons from embryonic mice in pH 7.4 solution vs. pH 7.2 solution (data in (d) obtained from [15]). At large values of n and k, a sign change in d value occurs which is a technical artifact arising from there being more entries in the transition matrix than can be filled by data from our time series. The numerical values of d which generated this figure can be found in S2 Table.
	Fig 4. Distributions of Markovian Entropy and other analysis measures of calcium activity from Xenopus laevis neural progenitors.Univariate scatterplots represent the (a) Markovian Entropy, (b) Number of Spikes, (c) Average Power, and (d) Hurst Exponent of Xenopus laevis neural progenitor cells’ calcium activity at embryonic stages 14, 18, and 22. Lines represent mean ± SD of 2,176, 2,664, and 757 cells, respectively. All comparisons between distribution were statistically significant according to a Bonferroni-corrected two-sample Kolmogorov-Smirnov Test (p < 0.01). Hence stars are used to represent the effect size, rather than the significance of difference, between distributions via Cohen’s d statistic (: \|d\| ≥ 0.20, : \|d\| ≥ 0.50, : \|d\| ≥ 0.80, : \|d\| ≥ 1.00, ***: \|d\| ≥ 2.00) [29]. Markovian Entropy is calculated with n = 2 and k = 1.
	Fig 5. Separation between calcium activity distributions from two biologically distinct populations as a function of sample size.The p-value obtained from a two-sample Kolmogorov-Smirnov test between distributions of calcium activity traces processed by a given analysis method from stage 14 Xenopus neural progenitors and stage 22 Xenopus neural progenitors is used as a measure of separation between the two biologically distinct populations. A smaller p-value indicates a more confident separation between the distributions. Each point represents mean + SD of 5,000 comparisons between samples of a given size taken with replacement from the two distributions. Markovian Entropy is calculated with n = 2 and k = 1. A randomized control is included that compares two samples which both come from the stage 14 Xenopus population. The Cohen’s d values associated with this data can be found in S3 Fig.
	Fig 6. Distributions of Markovian Entropy and other analysis measures of calcium activity from synaptic neurons.Univariate scatterplots represent the (a) Markovian Entropy, (b) Number of Spikes, (c) Average Power, and (d) Hurst Exponent of murine retrotrapezoid nucleus neurons’ calcium activity in solution with pH 7.2 or 7.4. Data received from [15]. Lines represent mean ± SD of 397 and 244 cells, respectively. All comparisons between distributions were statistically significant according to a two-sample Kolmogorov-Smirnov Test (p < 0.01). Hence stars are used to represent the effect size, rather than the significance of difference, between distributions via Cohen’s d statistic (: \|d\| ≥ 0.20, : \|d\| ≥ 0.50, : \|d\| ≥ 0.80, : \|d\| ≥ 1.00, ***: \|d\| ≥ 2.00) [29]. Markovian Entropy is calculated with n = 2 and k = 2.
	Fig 7. Markovian entropy and spike counting detect two modes of calcium activity.Representative calcium traces from selected cells in the high-entropy / high-spiking cluster and the low-entropy / low-spiking cluster of retrotrapezoid nucleus neurons from embryonic mice in a pH 7.2 solution reveal that both methods detect two distinct modes of calcium activity dynamics in these cells. Lines represent mean ± SD of 397 cells. Markovian Entropy is calculated with n = 2 and k = 1.

References [+] :

Berridge, Calcium signalling: dynamics, homeostasis and remodelling. 2003, Pubmed