1. The linear systems model of fMRI: Strengths and Weaknesses Stephen Engel UCLA Dept. of Psychology

2. Talk Outline Linear Systems –Definition –Properties Applications in fMRI (Strengths) Is fMRI Linear? (Weaknesses) Implications –Current practices –Future directions

3. Linear systems  System = input -> output Stimulus or Neural activity -> fMRI responses  System is linear if shows two properties Homogeneity & Superposition

5. Useful properties of linear systems Can add and subtract responses meaningfully Can characterize completely using impulse response Can use impulse response to predict output to arbitrary input via convolution Can characterize using MTF

6. Subtracting responses

7. Characterizing linear systems

9. Predicting block response

10. Characterizing linear systems

12. Talk Outline Linear Systems –Definition –Properties Applications in fMRI (Strengths) Is fMRI Linear? (Weaknesses) Implications –Current practices –Future directions

13. Uses of linear systems in fMRI If assume fMRI signal is generated by a linear system can: –Create model fMRI timecourses –Use GLM to estimate and test parameters –Interpret estimated parameters –Estimate temporal and spatial MTF

14. Simple GLM Example

15. Model fitting assumes homogeneity

16. Rapid designs assume superposition

17. Wagner et al. 1998, Results

18. Zarahn, ‘99; D’esposito et al.

19. D’Esposito et al.

20. More on GLM Many other analysis types possible –ANCOVA –Simultaneous estimate of HRF Interpretation of estimated parameters –If fMRI data are generated from linear system w/neural activity as input –Then estimated parameters will be proportional to neural activity Allows quantitative conclusions

21. MTF Boynton et al. (1996) estimated temporal MTF in V1 –Showed moving bars of checkerboard that drifted at various temporal frequencies –Generated periodic stimulation in retinotopic cortex –Plotted Fourier transform of MTF (which is impulse response)

23. Characterizing linear systems

25. MTF Engel et al. (1997) estimated spatial MTF in V1 –Showed moving bars of checkerboard that varied in spatial frequency but had constant temporal frequency –Calculated cortical frequency of stimulus –Plotted MTF –Some signal at 5 mm/cyc at 1.5 T in ‘97!

27. Talk Outline Linear Systems –Definition –Properties Applications in fMRI (Strengths) Is fMRI Linear? (Weaknesses) Implications –Current practices –Future directions

28. Is fMRI really based upon a linear system? Neural activity as input fMRI signal as output fMRI tests of temporal superposition Electrophysiological tests of homogeneity fMRI test of spatial superposition

29. Tests of temporal superposition Boynton et al. (1996) measured responses to 3, 6, 12, and 24 sec blocks of visual stimulation Tested if r(6) = r(3)+r(3) etc. Linearity fails mildly

31. Dale & Buckner ‘97 Tested superposition in rapid design  Full field stimuli  Groups of 1, 2, or 3 –Closely spaced in time –Responses overlap  Q1: 2-1 = 1?

32. Dale and Buckner, Design

34. fMRI fails temporal superposition Now many studies Initial response is larger than later response Looks OK w/3-5 second gap Possible sources –Attention –Neural adaptation –Hemodynamic non-linearity

35. Test of homogeneity Simultaneous measurements of neural activity and fMRI or optical signal Q: As neural activity increases does fMRI response increase by same amount?

36. Logothetis et al., ‘01

38. Optical imaging studies Measure electrophysiological response in rodents Various components of hemodynamic response inferred from reflectance changes at different wavelengths Devor ‘03 (whisker) and Sheth ‘04 (hindpaw)

41. Nonlinearities Optical imaging overestimates large neural responses relative to small ones –But Logo. found opposite fMRI overestimates brief responses relative to long ones –Amplified neural adaptation?

42. Spatial issue W/in a local region does signal depend upon sum or average activity? Or “is the whole garden watered for the sake of one thirsty flower?” (Grinvald)

43. Spatial Properties of HRF Thompson et al., 2003

44. Testing spatial superposition Need to measure responses of neurons from population a, population b, and both Where have intermingled populations that can activate separately? –LGN –Prediction twice as much fMRI response for two eye stimulation than for one eye Should be different in V1

45. Conclusions Linear model successful and useful but… Hemodynamic responses possibly not proportional to neural ones –Though could be pretty close for much of range –Take care interpreting differences in fMRI amplitude GLM results where neural responses overlap

46. Conclusions Temporal superposition of hemodynamic responses could still hold –Most applications of GLM may be OK w/proper interpretation and spacing to avoid neural adaptation –Run estimated fMRI amplitude through inverse of nonlinearity relating hemodynamics to neural activity (static nonlinearity)

47. Rapid designs assume superposition

48. Future Directions Better characterization of possible non- linearities Modeling of non-linearities Further tests of linearity –Hemodynamic superposition –Spatial superposition