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 analysis:course-w16:week11 [2016/02/22 21:51]mvdm analysis:course-w16:week11 [2018/07/07 10:19] (current) Both sides previous revision Previous revision 2016/02/23 14:18 mvdm 2016/02/23 14:14 mvdm [Interactions between multiple signals: coherence, Granger causality, and phase-slope index] 2016/02/23 14:09 mvdm [Spectrally resolved Granger causality] 2016/02/23 11:33 mvdm [Phase-slope index] 2016/02/22 21:51 mvdm 2016/02/22 21:41 mvdm [Introduction] 2016/02/22 21:38 mvdm 2016/02/22 21:32 mvdm [Phase-slope index] 2016/02/22 21:19 mvdm [Phase-slope index] 2016/02/22 21:03 mvdm [Spectrally resolved Granger causality] 2016/02/22 21:01 mvdm [Spectrally resolved Granger causality] 2016/02/22 21:01 mvdm 2016/02/22 20:43 mvdm [Spectrally resolved Granger causality] 2016/02/22 20:21 mvdm [Generating artificial data] 2016/02/22 20:21 mvdm [Generating artificial data] 2016/02/22 20:20 mvdm 2016/02/22 19:15 mvdm [Generating artificial data] 2016/02/22 19:14 mvdm [Generating artificial data] 2016/02/22 15:49 mvdm [Generating artificial data] 2016/02/22 15:48 mvdm 2016/02/22 14:34 mvdm [Granger causality: introduction] 2016/02/22 14:33 mvdm [Granger causality: introduction] 2016/02/22 14:03 mvdm [Amplitude cross-correlation] 2016/02/21 18:25 mvdm 2016/02/21 18:20 mvdm 2016/02/21 17:51 mvdm [Beyond coherence] 2016/02/21 17:49 mvdm [Beyond coherence] 2016/02/21 17:45 mvdm [Time-frequency coherence analysis] 2016/02/21 17:42 mvdm [Comparison of vStr-HC coherence between experimental conditions] 2016/02/21 16:45 mvdm [Overall comparison of vStr-vStr and vStr-HC coherence] Next revision Previous revision 2016/02/23 14:18 mvdm 2016/02/23 14:14 mvdm [Interactions between multiple signals: coherence, Granger causality, and phase-slope index] 2016/02/23 14:09 mvdm [Spectrally resolved Granger causality] 2016/02/23 11:33 mvdm [Phase-slope index] 2016/02/22 21:51 mvdm 2016/02/22 21:41 mvdm [Introduction] 2016/02/22 21:38 mvdm 2016/02/22 21:32 mvdm [Phase-slope index] 2016/02/22 21:19 mvdm [Phase-slope index] 2016/02/22 21:03 mvdm [Spectrally resolved Granger causality] 2016/02/22 21:01 mvdm [Spectrally resolved Granger causality] 2016/02/22 21:01 mvdm 2016/02/22 20:43 mvdm [Spectrally resolved Granger causality] 2016/02/22 20:21 mvdm [Generating artificial data] 2016/02/22 20:21 mvdm [Generating artificial data] 2016/02/22 20:20 mvdm 2016/02/22 19:15 mvdm [Generating artificial data] 2016/02/22 19:14 mvdm [Generating artificial data] 2016/02/22 15:49 mvdm [Generating artificial data] 2016/02/22 15:48 mvdm 2016/02/22 14:34 mvdm [Granger causality: introduction] 2016/02/22 14:33 mvdm [Granger causality: introduction] 2016/02/22 14:03 mvdm [Amplitude cross-correlation] 2016/02/21 18:25 mvdm 2016/02/21 18:20 mvdm 2016/02/21 17:51 mvdm [Beyond coherence] 2016/02/21 17:49 mvdm [Beyond coherence] 2016/02/21 17:45 mvdm [Time-frequency coherence analysis] 2016/02/21 17:42 mvdm [Comparison of vStr-HC coherence between experimental conditions] 2016/02/21 16:45 mvdm [Overall comparison of vStr-vStr and vStr-HC coherence] 2016/02/21 16:01 mvdm [Example 1] 2016/02/21 15:45 mvdm [Definition and example] 2016/02/21 15:10 mvdm created Line 1: Line 1: ~~DISCUSSION~~ ~~DISCUSSION~~ - - :!: **Under construction,​ please do not use yet!** :!: ===== Interactions between multiple signals: coherence, Granger causality, and phase-slope index ===== ===== Interactions between multiple signals: coherence, Granger causality, and phase-slope index ===== Line 18: Line 16: * (background reading, a brief review) [[http://​www.ncbi.nlm.nih.gov/​pubmed/​16150631 | Fries (2005) ]] Communication through coherence paper * (background reading, a brief review) [[http://​www.ncbi.nlm.nih.gov/​pubmed/​16150631 | Fries (2005) ]] Communication through coherence paper * (optional, a nice example application) [[http://​www.ncbi.nlm.nih.gov/​pubmed/​17372196 | deCoteau et al. (2007)]] hippocampus-striatum coherence changes with learning * (optional, a nice example application) [[http://​www.ncbi.nlm.nih.gov/​pubmed/​17372196 | deCoteau et al. (2007)]] hippocampus-striatum coherence changes with learning + * (technical background) [[http://​arxiv.org/​pdf/​q-bio/​0608035v1.pdf | Ding et al. (2006)]] theory of Granger causality and applications to neuroscience ==== Introduction ==== ==== Introduction ==== Line 553: Line 551: === Spectrally resolved Granger causality === === Spectrally resolved Granger causality === - Given how ubiquitous oscillations are in neural data, it is often informative to not fit VAR models directly in the time domain (as we did in the previous section) but go to the frequency domain. Intuitively,​ //​spectrally resolved// Granger causality measures how much of the power in $X$, not accounted for by $X$ itself, can be attributed to $Y$. To explore this, we'll generate some more artificial data: + Given how ubiquitous oscillations are in neural data, it is often informative to not fit VAR models directly in the time domain (as we did in the previous section) but go to the frequency domain. Intuitively,​ //​spectrally resolved// Granger causality measures how much of the power in $X$, not accounted for by $X$ itself, can be attributed to $Y$ ([[http://​www.sciencedirect.com/​science/​article/​pii/​S1053811908001328 | technical paper]]). To explore this, we'll generate some more artificial data: Line 655: Line 653: ☛ Reverse the two signals and compute Granger cross-spectra,​ both for the zero-delay artifact case and for the true causal case above. Verify that this reverse-Granger test accurately distinguishes the two cases. [[http://​www.sciencedirect.com/​science/​article/​pii/​S105381191401009X | This paper]] discusses these issues in more detail and has thoughtful discussion. ☛ Reverse the two signals and compute Granger cross-spectra,​ both for the zero-delay artifact case and for the true causal case above. Verify that this reverse-Granger test accurately distinguishes the two cases. [[http://​www.sciencedirect.com/​science/​article/​pii/​S105381191401009X | This paper]] discusses these issues in more detail and has thoughtful discussion. - ==== Phase-slope index ==== ==== Phase-slope index ==== - If we have a situation such as the above, it is possible that a true lag or lead between two signals is obscured by different signal-to-noise ratios. If such a case is detected by the reverse-Granger analysis, how can we proceed? + If we have a situation such as the above, it is possible that a true lag or lead between two signals is obscured by different signal-to-noise ratios. If such a case is detected by the reverse-Granger analysis, how can we proceed ​with identifying the true delay? - A possible solution is offered by the analysis of **phase slopes**, an the idea that for a given lead or lag between two signals, the phase lag (or lead) should systematically depend on frequency ([[http://​arxiv.org/​pdf/​0712.2352.pdf | Nolte et al. (2008)]], see also precedents in the literature such as [[http://​science.sciencemag.org/​content/​308/​5718/​111.short | Schoffelen et al. (2005)]]).[[https://​github.com/​mvdm/​papers/​tree/​master/​Catanese_vanderMeer2016 | Catanese and van der Meer (2016)]] diagram the idea as follows: + A possible solution is offered by the analysis of **phase slopes**: the idea that for a given lead or lag between two signals, the phase lag (or lead) should systematically depend on frequency ([[http://​arxiv.org/​pdf/​0712.2352.pdf | Nolte et al. (2008)]], see also precedents in the literature such as [[http://​science.sciencemag.org/​content/​308/​5718/​111.short | Schoffelen et al. (2005)]]). + + [[https://​github.com/​mvdm/​papers/​tree/​master/​Catanese_vanderMeer2016 | Catanese and van der Meer (2016)]] diagram the idea as follows: {{ :​analysis:​course-w16:​psi_example.png?​nolink&​800 |}} {{ :​analysis:​course-w16:​psi_example.png?​nolink&​800 |}} - In the example in (**A**) above, the red signal always leads the blue signal by 5 ms, which results in a different phase lag across frequencies (20, 25 and 33.3 Hz in this example). ​The bottom panel shows the linear relationship between phase lag and frequency for the above examples, resulting a positive slope for the red-blue phase difference indicating a red lead (green phase slope). + In the example in (**A**) above, the red signal always leads the blue signal by 5 ms, which results in a different phase lag across frequencies (20, 25 and 33.3 Hz in this example). ​This is because 5ms is a much bigger slice of a full oscillation cycle at 33.3Hz than it is at 25Hz; the bottom panel shows the linear relationship between phase lag and frequency for the above examples, resulting ​in a positive slope for the red-blue phase difference indicating a red signal ​lead. + + (**B**) shows the raw phase differences for an example real data session in the top panel: note that the phase lag as a function of frequency contains approximately linear regions in the "​low-gamma"​ (45-65 Hz, green) and "​high-gamma"​ (70-90 Hz, red) frequency bands, with slopes in opposite directions. The phase slope (middle panel) is the derivative of the raw phase lag, and the reversal of the phase slope sign around 65-70 Hz indicates that high and low gamma are associated with opposite directionality in the vStr-mPFC system, with vStr leading for low gamma and mPFC leading for high gamma oscillations. The bottom panel shows the phase slope index (PSI) which normalizes the raw phase slope by its standard deviation. - (**B**) shows the raw phase differences in the top panel: note that the phase lag as a function of frequency ​contains approximately linear regions in the low-gamma ​(45-65 Hz, green) and high-gamma (70-90 Hz, red) frequency bands, with slopes in opposite directions. The phase slope (middle panel) ​is the derivative of the raw phase lag, and the reversal of the phase slope sign around 65-70 Hz indicates that high and low gamma are associated with opposite directionality in the vStr-mPFC system, with vStr leading for low gamma and mPFC leading for high gamma oscillations. The bottom panel shows the phase slope index (PSI) which normalizes ​the raw phase slope by its standard deviation. + Thus, to summarize, the phase slope index (PSI) is a normalized form of the phase slope -- obtained by dividing ​the raw phase slope at each frequency ​by its standard deviation ​(estimated using a bootstrap). The phase slope itself ​is obtained by taking ​the derivative ​(slope) ​of the raw phase differences across frequencies;​ as discussed above, these raw phase differences can be obtained by estimating ​the phase (angle) of the cross-spectrum. The time lag (or lead) between two signals given a phase slope is: The time lag (or lead) between two signals given a phase slope is: Line 675: Line 676: - where $t_{a-b}$ is the time lag (or lead) in seconds between signals $a$ and $b$, to be inferred from the phase differences $\phi_{a-b}$ + where $t_{a-b}$ is the time lag (or lead) in seconds between signals $a$ and $b$, to be inferred from the phase differences $\phi_{a-b}$ (in degrees) observed at frequencies $f$ and $f+df$. For instance, given a phase difference $\phi_{a-b} = 45^{\circ}$ between signals $a$ and $b$ at $f = 25$Hz, and $\phi_{a-b} = 36^{\circ}$ at $f = 20$Hz, $t_{a-b} = [(45-36)/​(25-20)]/​360 = 5$ms (the example in panel **A** above). As $df \to 0$, the fraction shown in square brackets above corresponds to the derivative $\phi_{a-b}'​(f)$,​ i.e. the phase slope. Positive time lags indicate that $a$ leads $b$. - (in degrees) observed at frequencies $f$ and $f+df$. For instance, given a phase difference $\phi_{a-b} = 45^{\circ}$ between signals $a$ + - and $b$ at $f = 25$Hz, and $\phi_{a-b} = 36^{\circ}$ at $f = 20$Hz, $t_{a-b} = [(45-36)/​(25-20)]/​360 = 5$ms. As $df \to 0$, the fraction shown in square brackets above corresponds to the derivative $\phi_{a-b}'​(f)$,​ i.e. the phase slope. Positive time lags indicate that $a$ leads $b$. + To test how this works, let's generate two signals with an ambiguous Granger-relationship:​ To test how this works, let's generate two signals with an ambiguous Granger-relationship:​ - %% SNR CASE nTrials = 1000; nTrials = 1000; Line 746: Line 744: ==== Challenges ==== ==== Challenges ==== - ★ If you have your own data with at least two signals that you suspect may be related, identify an appropriate functional connectivity analysis and apply it to the data. + ★ If you have your own data with at least two signals that you suspect may be related, identify an appropriate functional connectivity analysis and apply it to the data. Comment on why the chosen method was used. + + ★ The "​theta"​ rhythm, which is about 8 Hz in moving rodents, is important in coordinating the spike timing of hippocampal neurons. However, theta frequencies also appear in LFPs recorded from other brain areas, including the prefrontal cortex and the ventral striatum. One hypothesis is that those areas simply "​inherit"​ theta activity from their hippocampal inputs. Test this idea using data from R020, which has electrodes in hippocampus and ventral striatum, and your chosen connectivity analysis method. +