The linear systems model of fMRI: Strengths and Weaknesses

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The linear systems model of fMRI Strengths and
Weaknesses

• Stephen Engel
• UCLA Dept. of Psychology

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Talk Outline

• Linear Systems
• Definition
• Properties
• Applications in fMRI (Strengths)
• Is fMRI Linear? (Weaknesses)
• Implications
• Current practices
• Future directions

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Linear systems

• System input -gt output
• Stimulus or Neural activity -gt fMRI responses
• System is linear if shows two properties
• Homogeneity Superposition

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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

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Subtracting responses
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Characterizing linear systems
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Predicting block response
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Characterizing linear systems
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Talk Outline

• Linear Systems
• Definition
• Properties
• Applications in fMRI (Strengths)
• Is fMRI Linear? (Weaknesses)
• Implications
• Current practices
• Future directions

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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

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Simple GLM Example
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Model fitting assumes homogeneity
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Rapid designs assume superposition
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Wagner et al. 1998, Results
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Zarahn, 99 Desposito et al.
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DEsposito et al.
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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

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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)

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Characterizing linear systems
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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!

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Talk Outline

• Linear Systems
• Definition
• Properties
• Applications in fMRI (Strengths)
• Is fMRI Linear? (Weaknesses)
• Implications
• Current practices
• Future directions

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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

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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

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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?

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Dale and Buckner, Design
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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

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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?

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Logothetis et al., 01
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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)

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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?

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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)

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Spatial Properties of HRF
Thompson et al., 2003
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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

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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

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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)

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Rapid designs assume superposition
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Future Directions

• Better characterization of possible
  non-linearities
• Modeling of non-linearities
• Further tests of linearity
• Hemodynamic superposition
• Spatial superposition