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Understanding Impulse Response Functions in Experimental Design

Explore arbitrary experimental designs with specific stimuli, response events, and convolution techniques for predicting responses. Investigate the shift-invariance and additivity principles in signal processing. Learn about Fourier Transforms and convolution theorems in functional MRI analysis.

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Understanding Impulse Response Functions in Experimental Design

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  1. Impulse response function h(t), or hi Stimulus: s = {1,0,0,0,0,….}

  2. Two events 2.5 2 1.5 1 0.5 0 0 5 10 15 20 time (s) Stimulus: s = {1,0,0,0,1,0….}

  3. Two events Stimulus: s = {1,0,0,0,.5,0….}

  4. 7 6 5 4 3 2 1 0 0 50 100 150 200 250 time (s) Arbitrary experimental design Stimulus: s = {0,0,1,0,1,1,0,0,1,……}

  5. 3.5 3 2.5 2 1.5 1 0.5 0 0 5 10 15 20 25 30 35 40 time (s) Response at time i=18 s4h14 + s6h12 + s10h8 Response at time i s1hi-1 + s2hi-2 + …+ si-1h1 Stimulus: s = {0,0,0,1,0,1,0,0,0,1,0,0,…}

  6. 2 2 1.5 1.5 1 1 0.5 0.5 0 0 0 20 40 60 80 0 20 40 60 80 time (s) time (s) 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 0 0 20 40 60 80 0 20 40 60 80 time (s) time (s) 6 4 2 0 0 10 20 30 40 50 60 70 80 time (s)

  7. Response R Design Matrix X Impulse response h x =

  8. if then

  9. Time-contrast separability Boynton, G.M., et al., J Neurosci, 1996. 16(13): p. 4207-21.

  10. Shift-invariance response to 6 sec pulse shift and duplicate 3 3 0 0 -3 -3 0 10 20 30 40 0 10 20 30 40 response to 12 sec pulse predict 12 sec from 6 sec pulses 3 3 0 0 -3 -3 0 10 20 30 40 0 10 20 30 40 Time (sec) Intensity Boynton, G.M., et al., J Neurosci, 1996. 16(13): p. 4207-21.

  11. 3 predicts 6 gmb 3 Response to pulsed stimulus Prediction from shift and sum 0 -3 0 10 20 30 40 3 predicts 12 6 predicts 12 3 3 0 0 -3 -3 0 10 20 30 40 0 10 20 30 40 3 predicts 24 6 predicts 24 12 predicts 24 3 3 3 0 0 0 -3 -3 -3 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 time (sec) Shift-invariance Intensity Boynton, G.M., et al., J Neurosci, 1996. 16(13): p. 4207-21.

  12. Additivity single stimulus paired same time SOA Stimuli: 0.5 c/deg, 8 Hz counterphase 30 deg 1sec duration 48 repetitions per condition 30 deg

  13. Nonlinearity after subtracting response to first pulse: single stimulus paired same

  14. single stimulus paired same paired orthogonal after subtracting response to first pulse:

  15. What does convolution have to do with sinusoids and Fourier Transforms? Convolution of sinusoid is a sinusoid:

  16. Convolution changes the amplitude and phase of a sinusoid – but NOT the frequency

  17. Fourier transform: Convolution of sinusoid is a sinusoid: Combined: Convolution of a function can be expressed via amplitude scaling and phase shifts of the function’s Fourier components.

  18. The amplitude scaling and phase shifts of the function’s Fourier components are determined by the Fourier transformation of the impulse response function. FFT of h(t)

  19. The convolution theorem: FFT of s(t) S(w) stimulus: s(t) FFT of h(t) H(w) HDR: h(t) convolution multiplication IFFT response: f(t)*h(t) S(w) H(w)

  20. fft of stimulus stimulus 3 0.12 2.5 0.1 0.08 2 amplitude 1.5 0.06 1 0.04 0.5 0.02 0 0 0 40 80 120 6 12 18 24 30 36 Time (sec) Cycles/scan response fft of response 0.1 1.2 1 0.08 0.8 0.06 amplitude 0.6 0.04 0.4 0.02 0.2 0 0 0 40 80 120 6 12 18 24 30 36 Time (sec) Cycles/scan

  21. Example: simple block design 10 9 8 7 6 5 4 3 2 1 0 0 50 100 150 200 250 time (s) Stimulus: s = {1,1,1,…,0,0,0,…,1,1,1,…,0,0,0,…}

  22. 3 0.8 2.5 0.6 2 amplitude 1.5 0.4 1 0.2 0.5 0 0 0 40 80 120 160 200 240 6 12 18 24 30 36 Time (sec) Cycles/scan 0.8 1.2 1 0.6 0.8 amplitude 0.4 0.6 0.4 0.2 0.2 0 0 0 40 80 120 160 200 240 6 12 18 24 30 36 Time (sec) Cycles/scan fft of stimulus stimulus response fft of response

  23. fMRI amplitude for different stimulus frequencies and contrasts

  24. fMRI amplitude at different frequencies for a 30 second period stimulus

  25. Time (s)

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