1. Multivariate time series analysis Bijan Pesaran Center for Neural Science New York University

2. Overview  Singular value decomposition  Application to space-time data  Application to space-frequency data  Periodic stacking method

3. Multivariate data,  Imaging data types MEG/EEG/LFP fMRI Optical imaging  Need to find spatial projection to reduce dimensionality  Combine spectral and multivariate tools

4. Singular value decomposition  Eigenvalue decomposition  Calculates directions/modes in data space that contain maximum variance Singular value spectrum Spatial mode Temporal mode

5. Application to space-time data

6. Spatial and temporal correlations  modes of spatial correlation matrix  modes of temporal correlation matrix

7. Truncation defines subspace Noise tail for a pxq matrix fMRI data set with 1877x500 data points, sampled at 5 Hz for 10 s. 2

8. Spectrum of temporal modes  Reveals physiological features across multiple modes

9. Application to space- frequency data

10. A geometric interpretation  Project time series into a subspace  Use an orthogonal basis set  Local-in-frequency projection operator

11. Advantages of local-in-frequency basis  Combine information across this basis Ensemble averaging  Choose properties of this basis Select time and frequency  Project onto multiple different subspaces centered on different frequencies

12. Space-frequency decomposition  Local-in-frequency projection  Dimensionality reduction

13. Multivariate Coherence,  Assess degree of low-dimensionality fMRI data set

14. Complex-valued spatial modes  Spatial segregation of physiological modes. 1 st order modes

15. fMRI example  Presence or absence of visual stimulus  Digitization rate: 5 Hz  Duration: 110 s  Visual stimulation with red LED patterns (8 Hz).

16. Visual response in fMRI signal No stimulus - dashed Visual stimulus - solid

17. Visual response in fMRI signal No stimulus Visual stimulus  Spatially restricted visual response  Coronal slice at the occipital pole

18. Optical imaging example  Isolated procerebral lobe of Limax  Presence of voltage sensitive dye  Digitization rate: 75 Hz  Duration: 23 s  600 um by 200 um

19. Optical imaging response  Limax procerebral lobe during olfactory stimulation

20. Principal spatial modes

21. Quantifying traveling waves  Express leading spatial mode 2.5 Hz 1.25 and 2.5 Hz x y

22. LFP example  Rubino et al. 2007  Electrode arrays in M1 and PMd of awake monkey  Digitization rate: 1 kHz  Duraction  Visual instructional cue response

23. Phase gradients in M1 PMd M1 PMd  Activity between 10 - 45 Hz

24. Waves reflect anatomical connections

25. Periodic stacking If you have repeated measurements of the response to multiple stimuli, you can order your data to take advantage of the multitaper harmonic analysis methods that we have been shown. 12 1212 Extraction of the average and differential dynamical response in stimulus-locked experimental data. J Neurosci Methods. 2005 Feb 15;141(2):223-9.

26. Odd harmonics - differences between responses Even harmonics - average dynamics Generalization to N stimuli: N’th harmonics are average dynamics, the rest are differences amongst the stimuli