Linearity and Temporal Symmetry in Primary Auditory Cortex
Linearity and Temporal Symmetry in Primary Auditory Cortex Jonathan Z. Simon1,2 David J. Klein 2,3 Didier A. Depireux 4 Shihab A. Shamma 2,3 1Department of Biology 2Department of Electrical & Computer 3Institute for Systems Research
Engineering
University of Maryland, College Park 4Department of Anatomy & Neurobiology University of Maryland, Baltimore
Support from Office of Naval Research MURI # N00014-97-1-05001
Topics • The Spectro-Temporal Response Field (STRF) characterizes neuronal responses in Primary Auditory Cortex in ferret. • STRF can be measured independently by different stimulus types • STRF = Linear Statistic = Linearity in neuron? • Application of Singular Value Decomposition • Temporal Symmetry • Neural Connectivity vs. Temporal Symmetry Support from Office of Naval Research MURI # N00014-97-1-05001
Results • Three Different Stimuli give strongly similar STRFs: Linearity is strong across varied stimuli • Singular Value Decomposition optimally estimates STRFs with Low Rank approximation • Temporal Symmetry predominates • Simple models of neural connectivity inconsistent with temporal symmetry • Models of neural connectivity consistent with temporal symmetry if: Inputs are phase lagged Intracortical connections do not mix spectral response properties
Spectro-Temporal Response Field 4000 0
2000
50
100
15 0
200
S(t,x)= sin(2pwt + 2pWx + f)
25 0
x = log2(f / f0) w = ripple velocity, e.g. 4 Hz = 4 cycles/s W = ripple density, e.g. 0.4 cycles/octave = 2 cycles/5 octaves
1000 0
50
100
15 0
200
25 0
0
50
100
15 0
200
25 0
500 250 125
0
50
100
15
200
0 Time
4000
4000
2000
2000
1000
1000
500
500
250
250
125
25 0
Time
Impulse Responses, parametrized by spectral band
Fourier Space representation
W = 0.4 cyc/oct
Cross-section interpretations c.f. visual contrast gratings
w = 4 Hz
125
Rate
Rate
Spectral Response Fields, evolving in time Ripple (17.2)
Extent in Fourier Space W (cyc/oct)
Single Dynamic Ripple TORC (5.4)
STWN (1.0) -8
0 4
w (Hz)
24
Temporally Orthogonal Ripple Combination 0.2
Spectro-Temporal White Noise
Reverse Correlation Æ STRF Single Stimulus Single Dynamic Ripple
Reverse Correlation from Single Stimulus Fourier Domain
Spectro-Temporal
Reverse Correlation from Full Set Fourier Domain
Spectro-Temporal
SNR: 20.7 SNRcor: 30.4 0
125 t (ms)
250
0 8 w (Hz)
0
125 t (ms)
250
-24
-8 -4 0 4 8 w (Hz)
24
0
125 t (ms)
250
Temporally Orthogonal Ripple Combination
0
125 t (ms)
SNR: 3.4 SNRcor: 3.1
250
-24
-4 0 w (Hz)
0
125 t (ms)
250
-24
-4 0 4 w (Hz)
24
0
125 t (ms)
250
Spectro-Temporal White Noise
0
125 t (ms)
SNR: 1.3 SNRcor: 1.6
250
-24
-4 0 w (Hz)
24
0
125 t (ms)
250
-24
-4 0 4 w (Hz)
24
0
125 t (ms)
250
SNR: (Signal Power) / (Signal Variance) SNRcor: (Power from first 50% of STRF) / (Power in last 50% of STRF)
Singular Value Decomposition (SVD) Raw STRF
Singular Values
Cleaned STRF
Residual
b SVD : 4.8%
Estimated Noise Threshold: Max. Singular Value of STRF over last 50% of STRF
226/20a06 0
125 t (ms)
separable matrix #
250
0
125 t (ms)
250
0
125 t (ms)
250
bSVD: (unbiased) estimator of STRF power remaining in residual
bSVD: 6.7%
b SVD : 28.2%
235/22a05 0
125 t (ms)
separable matrix #
250
0
125 t (ms)
0
125 t (ms)
250
0
125 t (ms)
250
0
125 t (ms)
250
250
SVD used to estimate optimal approximation
b SVD : 6.6% Quad. 2
Quad. 1
235/22a05 0
125 t (ms)
separable matrix #
250 Quad. 2
-24
-24
-4 0 4 w (Hz)
24
0
125 t (ms)
250
0
125 t (ms)
250
SVD not used to estimate rank
Quad. 1
-4 0 4 w (Hz)
lower values = better approximations
24
-24
-4 0 4 w (Hz)
24
STRF Linearity/Robustness 1 DR
TC
WN
DR
TC
TC
rank-1
rank-2
rank-2
Correlations DR-TC:0.88 TC-WN:0.86 DR-WN:0.85
Correlation 0.82
Correlation 0.86
SNRcor DR:10.8 TC: 1.9
SNRcor
SNR cor DR:30.4 TC: 3.1 WN: 1.6
WN
0.9 0.8 0.7
TC:2.5 WN:1.7
rank-1
0.6
q-sep rank-2
0.5 226/20a 0
0.4
t (ms) 125
rank-2
rank-2
rank-1
Correlations DR-TC:0.68 TC-WN:0.69 DR-WN:0.72
Correlation 0.77
Correlation 0.91
SNRcor DR:12.0 TC: 2.7
SNRcor
SNRcor DR:12.4 TC: 1.7 WN: 1.3
raw
236/14a
226/24a
0.3 DR-TC TC-WN DR-WN localized to max.
0.2
TC:1.0 WN:1.7
0.1 SNRcor > 1
0 236/16b
228/08a
226/25a
rank-2
rank-1
rank-1
Correlations DR-TC:0.65 TC-WN:0.56 DR-WN:0.36
Correlation 0.83
Correlation 0.95
SNRcor DR:3.4 TC:0.9
SNRcor
SNR cor
1
g = 2.9 (rank-1) g = 1.9 (rank-2) g = 1.7 (q-sep)
0.8 0.6
TC:5.3 WN:1.3
DR:13.9 TC: 2.0 WN: 0.8
0.4 DR-TC TC-WN DR-WN
0.2 226/21a
229/04c
236/19a
0
N = 45: STRF measured with > 1 stimulus type, Anesthetized
0
1
2 3 SNRcor
4
5
SNRcor/(SNRcor+1): Prediction of Purely Linear System + Noise g SNRcor/(g SNRcor+1): Prediction of Purely Linear System + Noise + Noise Reduction (via SVD)
Temporal Symmetry Simulated STRF
Measured STRF
Temporal cross-sections +
8
Temporal cross-sections
8 400
4
4
2
2
1
1
0.5
0.5
0
0.25 0
Time (ms)
250
–
0
-400
250 0.25 0
Time (ms)
0
Time (ms)
250
same, overlaid Time (ms)
Temporal cross-sections, with Hilbert “rotations” and re-scalings
250
Temporal Symmetry Index: 0.99—complete overlap
Time (ms)
0
250
250
250
Time (ms)
Time (ms)
0
0
0
Time (ms)
Temporal cross-sections, with Hilbert “rotations” and re-scalings
250
same, overlaid Temporal Symmetry Index: 0.76—strong overlap
Temporal Symmetry Definition All temporal cross-sections equal, up to scaling and Hilbert “rotation” Temporal Symmetry Index Definition: (complex) correlation coefficient between 1st and 2nd (analytic) SVD temporal cross-sections Magnitude: between 0 (no temporal symmetry) and 1 (total temporal symmetry)
Temporal Symmetry Statistics Temporal Symmetry Index 25
SVD approximations by rank, across population Anesthetized: 49/73 Rank 1 (temporally symmetric, not shown) 22/73 Rank 2 (shown at left) 2/73 Rank 3 (not temporally symmetric)
Anesthetized Awake
20 15
Awake: 72/145 Rank 1 (temporally symmetric, not shown) 70/145 Rank 2 (shown at left) 3/145 Rank 3 (not temporally symmetric)
10 5 0
0
0.2
0.4
0.6
0.8
1
Spectral Symmetry Index 25
Compare:
20
Spectral Symmetry Definition All spectral cross-sections equal, up to scaling and Hilbert “rotation”
15
Spectral Symmetry Index Definition: (complex) correlation coefficient between 1st and 2nd (analytic) SVD spectral cross-sections Magnitude: between 0 (no spectral symmetry) and 1 (total spectral symmetry)
10 5 0
0
0.2
0.4
0.6
0.8
1
Caveats Sustained Portion of Response only Low Frequency (< 25 Hz) band of response only SNR > 2
Models with Temporal Symmetry q=0
q=0
FS
FS
TS q≠0 FS
TS
2 fully separable inputs from thalamus with identical temporal structure but with phase lag (spectral differences OK)
q≠0
q=0
.. . FS
q≠0
.. .
.. .
q=0
slow hA(t)
TS
Many fully separable inputs with different high-frequency temporal q≠0 structures but identical low-highfrequency temporal structure (and different phase lags), integrated (lowpassed) by a slower soma
Many fully separable inputs from thalamus with identical temporal structure and with possibly different phase lags
.. .
.. . FS
.. .
slow hA(t)
TS
1
h2(t)
TS
Same as on the left, but with feedforward to and feedback from another cortical neuron that conserves the spectral response properties
2
Models continued
q=0
h2(t)
Absurdly but plausibly complicated generalization—with additional feedforward and feedback among cortical cells that conserve the spectral response properties
2
.. . FS
q≠0
TS
.. .
slow hA(t)
TS
1
h3(t)
h4(t)
TS
3
TS
4
Other Models Inconsistent with Temporal Symmetry: • Inputs from thalamus with identical temporal structure but with time lag instead of phase lag • Feedforward to and feedback from another cortical neuron that changes the spectral response properties Caveats • Physiological, not Anatomical • Sustained Portion of Response only • Only for broadband dynamic stimuli • Describes linear response components only • Lag might arise from any of several mechanisms (e.g. inhibition, synaptic depression)
Suggested Reading Depireux DA, Simon JZ, Klein DJ, Shamma SA. 2001. Spectrotemporal response field characterization with dynamic ripples in ferret primary auditory cortex. J Neurophysiol 85: 1220-34 Eggermont JJ, Johannesma PM, Aertsen AM. 1983. Reversecorrelation methods in auditory research. Q Rev Biophys 16: 341-414 Klein DJ, Depireux DA, Simon JZ, Shamma SA. 2000. Robust spectrotemporal reverse correlation for the auditory system: optimizing stimulus design. J Comput Neurosci 9: 85-111 Stewart GW. 1993. Determining Rank in the Presence of Error. In Linear algebra for large scale and real-time applications, ed. MS Moonen, GH Golub, BLRd Moor. Dordrecht: Kluwer Academic Publishers
Engineering
University of Maryland, College Park 4Department of Anatomy & Neurobiology University of Maryland, Baltimore
Support from Office of Naval Research MURI # N00014-97-1-05001
Topics • The Spectro-Temporal Response Field (STRF) characterizes neuronal responses in Primary Auditory Cortex in ferret. • STRF can be measured independently by different stimulus types • STRF = Linear Statistic = Linearity in neuron? • Application of Singular Value Decomposition • Temporal Symmetry • Neural Connectivity vs. Temporal Symmetry Support from Office of Naval Research MURI # N00014-97-1-05001
Results • Three Different Stimuli give strongly similar STRFs: Linearity is strong across varied stimuli • Singular Value Decomposition optimally estimates STRFs with Low Rank approximation • Temporal Symmetry predominates • Simple models of neural connectivity inconsistent with temporal symmetry • Models of neural connectivity consistent with temporal symmetry if: Inputs are phase lagged Intracortical connections do not mix spectral response properties
Spectro-Temporal Response Field 4000 0
2000
50
100
15 0
200
S(t,x)= sin(2pwt + 2pWx + f)
25 0
x = log2(f / f0) w = ripple velocity, e.g. 4 Hz = 4 cycles/s W = ripple density, e.g. 0.4 cycles/octave = 2 cycles/5 octaves
1000 0
50
100
15 0
200
25 0
0
50
100
15 0
200
25 0
500 250 125
0
50
100
15
200
0 Time
4000
4000
2000
2000
1000
1000
500
500
250
250
125
25 0
Time
Impulse Responses, parametrized by spectral band
Fourier Space representation
W = 0.4 cyc/oct
Cross-section interpretations c.f. visual contrast gratings
w = 4 Hz
125
Rate
Rate
Spectral Response Fields, evolving in time Ripple (17.2)
Extent in Fourier Space W (cyc/oct)
Single Dynamic Ripple TORC (5.4)
STWN (1.0) -8
0 4
w (Hz)
24
Temporally Orthogonal Ripple Combination 0.2
Spectro-Temporal White Noise
Reverse Correlation Æ STRF Single Stimulus Single Dynamic Ripple
Reverse Correlation from Single Stimulus Fourier Domain
Spectro-Temporal
Reverse Correlation from Full Set Fourier Domain
Spectro-Temporal
SNR: 20.7 SNRcor: 30.4 0
125 t (ms)
250
0 8 w (Hz)
0
125 t (ms)
250
-24
-8 -4 0 4 8 w (Hz)
24
0
125 t (ms)
250
Temporally Orthogonal Ripple Combination
0
125 t (ms)
SNR: 3.4 SNRcor: 3.1
250
-24
-4 0 w (Hz)
0
125 t (ms)
250
-24
-4 0 4 w (Hz)
24
0
125 t (ms)
250
Spectro-Temporal White Noise
0
125 t (ms)
SNR: 1.3 SNRcor: 1.6
250
-24
-4 0 w (Hz)
24
0
125 t (ms)
250
-24
-4 0 4 w (Hz)
24
0
125 t (ms)
250
SNR: (Signal Power) / (Signal Variance) SNRcor: (Power from first 50% of STRF) / (Power in last 50% of STRF)
Singular Value Decomposition (SVD) Raw STRF
Singular Values
Cleaned STRF
Residual
b SVD : 4.8%
Estimated Noise Threshold: Max. Singular Value of STRF over last 50% of STRF
226/20a06 0
125 t (ms)
separable matrix #
250
0
125 t (ms)
250
0
125 t (ms)
250
bSVD: (unbiased) estimator of STRF power remaining in residual
bSVD: 6.7%
b SVD : 28.2%
235/22a05 0
125 t (ms)
separable matrix #
250
0
125 t (ms)
0
125 t (ms)
250
0
125 t (ms)
250
0
125 t (ms)
250
250
SVD used to estimate optimal approximation
b SVD : 6.6% Quad. 2
Quad. 1
235/22a05 0
125 t (ms)
separable matrix #
250 Quad. 2
-24
-24
-4 0 4 w (Hz)
24
0
125 t (ms)
250
0
125 t (ms)
250
SVD not used to estimate rank
Quad. 1
-4 0 4 w (Hz)
lower values = better approximations
24
-24
-4 0 4 w (Hz)
24
STRF Linearity/Robustness 1 DR
TC
WN
DR
TC
TC
rank-1
rank-2
rank-2
Correlations DR-TC:0.88 TC-WN:0.86 DR-WN:0.85
Correlation 0.82
Correlation 0.86
SNRcor DR:10.8 TC: 1.9
SNRcor
SNR cor DR:30.4 TC: 3.1 WN: 1.6
WN
0.9 0.8 0.7
TC:2.5 WN:1.7
rank-1
0.6
q-sep rank-2
0.5 226/20a 0
0.4
t (ms) 125
rank-2
rank-2
rank-1
Correlations DR-TC:0.68 TC-WN:0.69 DR-WN:0.72
Correlation 0.77
Correlation 0.91
SNRcor DR:12.0 TC: 2.7
SNRcor
SNRcor DR:12.4 TC: 1.7 WN: 1.3
raw
236/14a
226/24a
0.3 DR-TC TC-WN DR-WN localized to max.
0.2
TC:1.0 WN:1.7
0.1 SNRcor > 1
0 236/16b
228/08a
226/25a
rank-2
rank-1
rank-1
Correlations DR-TC:0.65 TC-WN:0.56 DR-WN:0.36
Correlation 0.83
Correlation 0.95
SNRcor DR:3.4 TC:0.9
SNRcor
SNR cor
1
g = 2.9 (rank-1) g = 1.9 (rank-2) g = 1.7 (q-sep)
0.8 0.6
TC:5.3 WN:1.3
DR:13.9 TC: 2.0 WN: 0.8
0.4 DR-TC TC-WN DR-WN
0.2 226/21a
229/04c
236/19a
0
N = 45: STRF measured with > 1 stimulus type, Anesthetized
0
1
2 3 SNRcor
4
5
SNRcor/(SNRcor+1): Prediction of Purely Linear System + Noise g SNRcor/(g SNRcor+1): Prediction of Purely Linear System + Noise + Noise Reduction (via SVD)
Temporal Symmetry Simulated STRF
Measured STRF
Temporal cross-sections +
8
Temporal cross-sections
8 400
4
4
2
2
1
1
0.5
0.5
0
0.25 0
Time (ms)
250
–
0
-400
250 0.25 0
Time (ms)
0
Time (ms)
250
same, overlaid Time (ms)
Temporal cross-sections, with Hilbert “rotations” and re-scalings
250
Temporal Symmetry Index: 0.99—complete overlap
Time (ms)
0
250
250
250
Time (ms)
Time (ms)
0
0
0
Time (ms)
Temporal cross-sections, with Hilbert “rotations” and re-scalings
250
same, overlaid Temporal Symmetry Index: 0.76—strong overlap
Temporal Symmetry Definition All temporal cross-sections equal, up to scaling and Hilbert “rotation” Temporal Symmetry Index Definition: (complex) correlation coefficient between 1st and 2nd (analytic) SVD temporal cross-sections Magnitude: between 0 (no temporal symmetry) and 1 (total temporal symmetry)
Temporal Symmetry Statistics Temporal Symmetry Index 25
SVD approximations by rank, across population Anesthetized: 49/73 Rank 1 (temporally symmetric, not shown) 22/73 Rank 2 (shown at left) 2/73 Rank 3 (not temporally symmetric)
Anesthetized Awake
20 15
Awake: 72/145 Rank 1 (temporally symmetric, not shown) 70/145 Rank 2 (shown at left) 3/145 Rank 3 (not temporally symmetric)
10 5 0
0
0.2
0.4
0.6
0.8
1
Spectral Symmetry Index 25
Compare:
20
Spectral Symmetry Definition All spectral cross-sections equal, up to scaling and Hilbert “rotation”
15
Spectral Symmetry Index Definition: (complex) correlation coefficient between 1st and 2nd (analytic) SVD spectral cross-sections Magnitude: between 0 (no spectral symmetry) and 1 (total spectral symmetry)
10 5 0
0
0.2
0.4
0.6
0.8
1
Caveats Sustained Portion of Response only Low Frequency (< 25 Hz) band of response only SNR > 2
Models with Temporal Symmetry q=0
q=0
FS
FS
TS q≠0 FS
TS
2 fully separable inputs from thalamus with identical temporal structure but with phase lag (spectral differences OK)
q≠0
q=0
.. . FS
q≠0
.. .
.. .
q=0
slow hA(t)
TS
Many fully separable inputs with different high-frequency temporal q≠0 structures but identical low-highfrequency temporal structure (and different phase lags), integrated (lowpassed) by a slower soma
Many fully separable inputs from thalamus with identical temporal structure and with possibly different phase lags
.. .
.. . FS
.. .
slow hA(t)
TS
1
h2(t)
TS
Same as on the left, but with feedforward to and feedback from another cortical neuron that conserves the spectral response properties
2
Models continued
q=0
h2(t)
Absurdly but plausibly complicated generalization—with additional feedforward and feedback among cortical cells that conserve the spectral response properties
2
.. . FS
q≠0
TS
.. .
slow hA(t)
TS
1
h3(t)
h4(t)
TS
3
TS
4
Other Models Inconsistent with Temporal Symmetry: • Inputs from thalamus with identical temporal structure but with time lag instead of phase lag • Feedforward to and feedback from another cortical neuron that changes the spectral response properties Caveats • Physiological, not Anatomical • Sustained Portion of Response only • Only for broadband dynamic stimuli • Describes linear response components only • Lag might arise from any of several mechanisms (e.g. inhibition, synaptic depression)
Suggested Reading Depireux DA, Simon JZ, Klein DJ, Shamma SA. 2001. Spectrotemporal response field characterization with dynamic ripples in ferret primary auditory cortex. J Neurophysiol 85: 1220-34 Eggermont JJ, Johannesma PM, Aertsen AM. 1983. Reversecorrelation methods in auditory research. Q Rev Biophys 16: 341-414 Klein DJ, Depireux DA, Simon JZ, Shamma SA. 2000. Robust spectrotemporal reverse correlation for the auditory system: optimizing stimulus design. J Comput Neurosci 9: 85-111 Stewart GW. 1993. Determining Rank in the Presence of Error. In Linear algebra for large scale and real-time applications, ed. MS Moonen, GH Golub, BLRd Moor. Dordrecht: Kluwer Academic Publishers
