Functional and Structural Implications of Non-Separability of Spectral and Temporal Responses in AI

Transcription

1 Functional and Structural Implications of Non-Separability of Spectral and Temporal Responses in AI Jonathan Z. Simon David J. Klein Didier A. Depireux Shihab A. Shamma and Dept. of Electrical & Computer Engineering Supported in part by MURI # N from the Office of Naval Research, # NIDCD T3 DC46- from the National Institute on Deafness and Other Communication Disorders, # NSFD CD883 from the National Science Foundation. This poster is available at <

2 Introduction We measure the response of cells in ferret Primary Auditory Cortex (AI) to dynamic, broadband sounds. The dynamic, broadband sounds are simple combinations of spectro-temporal basis functions, called moving ripples. By correlating the response ith the stimulus, e derive Spectro-Temporal Receptive Fields (STRFs), a linear, quantitative descriptor of ho a cell responds to dynamic sounds. The STRFs exhibit symmetries and patterns such as separability, and its generalization, quadrant separability. Quadrant separability does not arise from most neural netorks, and can be used to rule out some models of neural connectivity.

3 Summary The STRFs measured for all units in AI are quadrant separable or fully separable. Quadrant separability is incompatible ith simple summing of independent, fully separable sources. Summing to fully separable STRFs is quadrant separable if the temporal processing is in quadrature. There are three ays in hich a fully separable STRF can become quadrant separable: poer asymmetry, spectral asymmetry, & temporal asymmetry. Only poer and spectral asymmetry contribute to quadrant separability in AI, not temporal. Quadrant Separability is incompatible ith velocity selectivity. Quadrant Separability and persistence of temporal symmetry strongly constrain possible models of neural connectivity.

4 Cells are characterized their Spectro- Temporal Response Field (STRF)... STRFs in AI... or by the (Fourier domain) ripple Transfer Function (TF). STRF(t,x) X TF(,) (cycles/octave) x = log f/fo (octaves) t (ms) T 6/4a6.a - (cycles/second) Moving ripples form the basis for the Fourier domain description of dynamic spectra. At time t and frequency x, the amplitude S(t,x) is given by: x (octaves) 5 t (ms) 5.6. (cycles/octave) S(t,x)= sin(πt + πx + Φ) x = log [f / f ] = ripple velocity, modulation rate = ripple frequency, spectral density (Hz)

5 Temporally Orthogonal Ripple Combinations STRF measured by reverse-correlating ith dynamic spectrum of a broad-band stimulus. Temporally Orthogonal Ripple Combinations composed of ripples ith different modulation rates. Allo clean STRF estimates in relatively short time. stimulus S(t,x) 5 octaves response R(t) (spikes/ sec) - 5 ms 5 ms The stimuli shon contain ripples covering the same range of ripple velocities, but at different ripple frequencies...

6 Fully Separable STRF x = log [f / f ] STRF(t,x) TF(,) 4k Hz Q.8 cyc/oct Q g(x) Q Q 5 Hz f(t) 6/a.a 5 ms t -3 Hz 3 Hz The STRF and TF are a product of a single spectral response function ith a single temporal response function. Shon above are the impulse responses f(t) and receptive fields g(x) derived from quadrant (black) and quadrant (red) of the transfer function by inverse Fourier transformation. STRF(t,x) = f(t) g(x) f(t) g(x) F() G() TF(,) = F() G()

7 Quadrant Separable STRF x = log [f / f ] 8 khz STRF(t,x) TF(,).8 cyc/oct g i (x) Q Q 5 Hz f i (t) 5 ms This neuron responded tice as strong to rising frequencies than it did to falling frequencies. f i (t) g i (x) F i () G i () t T (, )= -3 Hz 3 Hz The STRF is not separable, but each quadrant of the transfer function is, i.e., there are different spectral and temporal responses for upards and donards frequency modulation. F( ) G( ) >, > F( ) G( ) <, > and for > : T(,) = T * (-,-)

8 Examples & Counterexamples Fully Separable Quadrant Separable - Velocity Selective is Inseparable - Quadrant - separability is incompatible ith velocity selectivity.

9 A Counterexample Fully Separable Fully Separable (displaced) - Sum of to Fully Separable is Inseparable - - Naive sum of to fully separable STRFs is inseparable.

10 An Example Fully Separable Same Fully Separable but Lagged (& shifted spectrally) Sum of Non-Lagged and Lagged is Separable Sum of to fully separable STRFs is separable if the temporal processing is in quadrature.

11 Measuring Separability ith SVD Singular Value Decomposition (SVD) can be used to estimate the separability of a Transfer Function (possibly corrupted by noise). It decomposes the Transfer Function into a sum of Quadrant Separable Transfer Functions, ordered by their poer. We apply SVD to each quadrant of the transfer function. Belo, an STRF and the three most significant quadrant-separable components, derived from SVD: Ra Estimate st Quadrant-separable Component nd = T, ( ) = ( ) ( ) + ( ) ( ) + ( ) ( ) + > > 3 3 F G F G F G K, ( ) ( ) + ( ) ( ) + ( ) ( ) + < > 3 3 F G F G F G K, x = t

12 Q SVD Example Q Fraction of total poer λ i Singular Value Number Bootstrap estimate of noise poer SVD naturally picks out high SNR components of a matrix. Large jumps in the singular values separate signal from noise. Jumps straddle bootstrap estimate of noise. Noise can be removed by discarding loer-magnitude components. All cells (3) measured in AI have a single dominant SVD component in each quadrant. All units measured in AI are quadrant separable (or fully separable).

13 Measure of Inseparability SVD supplies a natural measure of inseparability, α SVD α SVD = α SVD is fully separable α SVD >.3 is strongly inseparable λ λi Population Statistics α SVD =.6 i α SVD =.35 Frequency 5 Frequency Time 8/5b6.m 8/5b4.m.5.5 Time 5/33a6.m 5/33a5.m

14 Symmetry by Poer α d : Poer asymmetry breaks full separability, producing quadrant separability α d = (P - P )/(P + P ) α d is symmetric in poer P = (Poer in quadrant ) = (λ ) P = (Poer in quadrant ) = (λ ) Frequency α d >.3 is quite asymmetric in poer strongly inseparable Example STRF Population Statistics Contribution to α SVD α d =.79 α d.5 α SVD Time 9/a5.m 9/a4.m -.5 correlation α d.5

15 Spectral Symmetry α s : Asymmetry beteen spectral cross-sections G i (): α s is spectrally symmetric here the quantity inside the * G( ) G ( ) > α s = big absolute value bars is the G( ) G ( ) > (complex) correlation beteen G() and G() α s >.3 is spectrally asymmetric strongly inseparable Frequency Example STRF Population Statistics Contribution to α SVD α s =.65.5 α s α SVD Time 3/a6.m 3/a5.m.5.75 correlation α s.5

16 Temporal Symmetry α t : Asymmetry beteen temporal cross-sections F i (): α t = α t is temporally symmetric α t >.3 is temporally asymmetric strongly inseparable > > F( ) F( ) F F ( ) ( ) here the quantity inside the big absolute value bars is the (complex) correlation beteen F() and F * (-) Frequency Example STRF Population Statistics Contribution to α SVD α t =.3.5 α t α SVD Time 9/5b5.m 9/5b4.m.5.66 correlation.5 α t Distribution is strongly skeed toard temporal symmetry.

17 Symmetry Correlations Mean of 3 separate symmetry measures correlates ell ith full separability index. α SVD correlation.. Individual indices only partially correlated ith each other mean(ασ, ατ, α d ).8 correlation. correlation.7 correlation α t α d α d α s.5 α s.5 α t

18 Models fully separable MGB fully separable AI Not Quadrant Separable fully separable MGB fully separable same temporal function but lagged AI Quadrant Separable

19 Example STRF Recording Pairs Frequency (Hz) α SVD = % α d = 5% α s = 4% α t = 7% α SVD = 9% α d = % α s = 9% α t = 9% α SVD = 55% α d = 6% α s = 9% α t = 9% 5 34/7a3.ae. 5 34/3c.ae. 5 35/6a3.ae.3 Frequency (Hz) 4 5 α SVD = 5% α d = % α s = 3% α t = % 4 5 α SVD = 47% α d = 5% α s = 6% α t = 3% 8 4 α SVD = 77% α d = % α s = 6% α t = 3% 5 34/7a3.ae Time (ms) Very similar, Both fully separable [ αcorr 4% 6% ] s = 3% 4% [ ] αcorr t = % 7% 4% 5% 5 34/3c.a Time (ms) Different, Upper fully separable, Loer quadrant separable [ αcorr 6% % ] s = 3% % [ ] αcorr 6% 5% t = ε = ( %, 5%) ε = ( 4%, 5%) ε = ( 6%, 6%) % 7% /6a3.ae Time (ms) Very different, Both quadrant separable (somehat noisy though) [ αcorr 3% 55% ] s = 4% 5% [ ] αcorr t = % 3% 7% 59%

20 Selected References Spectro-Temporal Correlation Methods Klein DJ, Depireux DA, Simon JZ and Shamma SA, Robust spectro-temporal reverse correlation for the auditory system: Optimizing stimulus design, J. Computational Neurosci.. Eggermont JJ, Hearing Research 66 (993) 77-. Singular Value Decomposition Hansen PC, Rank-Deficient and Discrete Ill-Posed Problems, SIAM 998. Press WH, Flannery BP, Teukolsky SA, and Vetterling, WT, Numerical Recipes, Cambridge University Press 986. Separability Watson AB and Ahumada AJ, J. Opt. Soc. Am. A() (985) Saul AB and Humphrey AL, J. Neurophysiol. 64 (99) 6-4. Related techniques and models Koalski N, Depireux D and Shamma S, J. Neurophysiol. 76 (5) (996) , & Depireux DA, Simon JZ and Shamma SA, Comments in Theoretical Biology (998). Wang K and Shamma SA, IEEE Trans. on Speech and Audio (3) (994) 4-435, and 3() (995) Special thanks to Steve Bierer for his help in spike sorting.