{"title": "Perturbative M-Sequences for Auditory Systems Identification", "book": "Advances in Neural Information Processing Systems", "page_first": 180, "page_last": 186, "abstract": null, "full_text": "Perturbative M-Sequences for Auditory \n\nSystems Identification \n\nMark Kvale and Christoph E. Schreiner\u00b7 \n\nSloan Center for Theoretical Neurobiology, Box 0444 \n\nUniversity of California, San Francisco \n\n513 Parnassus Ave, San Francisco, CA 94143 \n\nAbstract \n\nIn this paper we present a new method for studying auditory sys(cid:173)\ntems based on m-sequences. The method allows us to perturba(cid:173)\ntively study the linear response of the system in the presence of \nvarious other stimuli, such as speech or sinusoidal modulations. \nThis allows one to construct linear kernels (receptive fields) at the \nsame time that other stimuli are being presented. Using the method \nwe calculate the modulation transfer function of single units in the \ninferior colli cui us of the cat at different operating points and discuss \nnonlinearities in the response. \n\n1 \n\nIntroduction \n\nA popular approach to systems identification, i.e., identifying an accurate analyt(cid:173)\nical model for the system behavior, is to use Volterra or Wiener expansions to \nmodel behavior via functional Taylor or orthogonal polynomial series, respectively \n[Marmarelis and Marmarelis1978]. Both approaches model the response r(t) as a \nlinear combination of small powers of the stimulus s(t). Although effective for mild \nnonlinearities, deriving the linear combinations becomes numerically unstable for \nhighly nonlinear systems. A more serious problem is that many biological systems \nare adaptive, i.e., the system behavior is dependent on the stimulus ensemble. For \ninstance, [Rieke et al.1995] found that in the auditory nerve of the bullfrog linearity \nand information rates depended sensitively on whether a white noise or naturalistic \nensemble is used. \n\nOne approach to handling these difficulties is to forgo the full expansion, and sim(cid:173)\nply compute the linear response to small (perturbative) stimuli in the presence \nof various different ensembles, or operating points. By collecting linear responses \n\n\u2022 Email: kvale@phy.ucsf.edu and chris@phy.ucsf.edu \n\n\fPerturbative M-Sequences for Auditory Systems Identification \n\n181 \n\nfrom different operating points, one may fit nonlinear responses as one fits a non(cid:173)\nlinear function with a piecewise linear approximation. For adaptive systems the \nsame procedure would be applied, with different operating points corresponding to \ndifferent points along the time axis. Perturbative stimuli have wide application in \ncondensed-matter physics, where they are used to characterize linear responses such \nas resistance, elasticity and viscosity, and in engineering, perturbative analyses are \nused in circuit analysis (small signal models) and structural diagnostics (vibration \nanalysis). In neurophysiology, however, perturbative stimuli are unknown. \n\nAn effective stimulus for calculating the perturbative linear response of a system \nis the m-sequence. M-sequences have a long history of use in engineering and the \nphysical sciences, with applications ranging from systems identification to cryp(cid:173)\ntography and cellular communication. In physiology, m-sequences have been used \nprimarily to compute system kernels [Marmarelis and Marmarelisl978], especially \nin the visual system [Pinter and Nabet1987]. In this work, we use perturbative m(cid:173)\nsequences to study the linear response of single units in the inferior colli cui us of a \ncat to amplitude-modulated (AM) stimuli. We add a small m-sequence signal to \nan AM carrier, which allows us to study the linear behavior of the system near a \nparticular operating point in a non-destructive manner, i.e., without changing the \noperating point. Perturbative m-sequences allow one to calculate linear responses \nnear the particular stimuli under study with only a little extra effort, and allow us \nto characterize the system over a wide range of stimuli, such as sinusoidal AM and \nnaturalistic stimuli. \n\nThe auditory system we selected to study was the response of single units in the \ncentral nucleus of the inferior colliculus (IC) of an anaesthetised cat. Single unit \nresponses were recorded extraceUularly. Action potentials were amplified and stored \non DAT tape, and were discriminated offline using a commercial computer-based \nspike sorter (Brainwave). 20 units were recorded, of which 10 yielded sufficiently \nstable responses to be analyzed. \n\n2 M-Sequences and Linear Systems \n\nA binary m-sequence is a two-level pseudo-random sequence of +1's and -1's. The \nsequence length is L = 2n - 1, where n is the order of the sequence. Typically, a \nbinary m-sequence can be generated by a shift register with n bits and feedback \nconnections derived from an irreducible polynomial over the multiplicative group Z2 \n[Golomb1982]. For linear systems identification, m-sequences have two important \nproperties. The first is that m-sequences have nearly zero mean: ~~':OI m[t] = -l. \nThe second is that the autocorrelation function takes on the impulse-like form \n\nSmm(T) = ~ m[t]m[t + T] = \n\nL-l \n\n{L \n\nif T = 0 \n\n-1 otherwise \n\n(1) \n\nImpulse stimuli also have a 8-function autocorrelation function. \nIn the context \nof perturbative stimuli, the advantage of an m-sequence stimulus over an impulse \nstimulus is that for a given signal to noise ratio, an m-sequence perturbation stays \nmuch closer to the original signal (in the least squares sense) than an impulse per(cid:173)\nturbation. Thus the perturbed signal does not stray as far from the operating point \nand measurement of linear response about that operating point is more accurate. \nWe model the IC response with a system F through which a scalar stimulus s(t) is \npassed to give a response r(t): \n\nr(t) = F[s(t)]. \n\n(2) \n\n\f182 \n\nM. Kvale and C. E. Schreiner \n\nFor the purposes of this section, the functional F is taken to be a linear functional \nplus a DC component. In real experiments, the input and output signal are sampled \ninto discrete sequences with t becoming an integer indexing the sequence. Then the \nsystem can be written as the discrete convolution \n\nL-1 \n\nr[t] = ho + L h[ttls[t - t1] \n\n(3) \n\nwith kernels ho and h[tt] to be determined. We assume that the system has a finite \nmemory of M time steps (with perhaps a delay) so that at most M of the h[t] \ncoefficients are nonzero. To determine the kernels perturbatively, we add a small \namount of m-sequence to a base stimulus so: \n\nCross-correlating the response with the original m-sequence yields \n\ns[t] = so[t] + am[t]. \n\nL-l \n\nt=o \n\nL-l L- l \n\nL-l \nL m[t]r[t + r] = L m[t]ho + L L h[ttlm[t]so[t + r - ttl \nL-1 L-l \n+ L L ah[tdm[t]m[t + r - tl]' \n\nt=o tl =0 \n\nt=o \n\n(4) \n\n(5) \n\nt=o tl=O \n\nUsing the sum formula for am -sequence above, the first sum in Eq. (5) can be \nsimplified to -ho. Using the autocorrelation Eq. (1), the third sum in Eq. (5) \nsimplifies, and we find \n\nRrm(r) = a(L + l)h[r] - ho - a L h[tl] + L L h[tt]m[t]so[t + r - ttl \n\nL-l \n\nL-l L-l \n\n(6) \n\ntl =0 \n\nt=o tl =0 \n\nAlthough the values for the kernels h(t) are set implicitly by this equation, the \nterms on the right hand side of Eq. (6) are widely different in size for large Land \nthe equation can be simplified. As is customary in auditory systems, we assume \nthe DC response ho is small. To estimate the size of the other terms, we compute \nstatistical estimates of their sizes and look at their scaling with the parameters. \nThe term a L:~-==~ h[tt] is a sum of M kernel elements; they may be correlated or \nuncorrelated, so a conservative estimate of their size is on the order of O( aM). \nThe last term in (6) is more subtle. We rewrite it as \n\nL-l L-l \nL L h[tdm[t]so[t + r - ttl = \n\nh=O t=o \n\ntl=O \n\nL-l L h[tt]p[r, ttl \nL-l L m[t]so[t + r - ttl \n\nt=o \n\n(7) \n\nThe time series of the ambient stimulus so[t] and m-sequence m[t] are assumed to \nbe uncorrelated. By the central limit theorem, the sum p[r, tl] will then have an \naverage of zero with a standard deviation of 0(L 1/ 2 ). If in turn, the terms p[r, ttl \nare un correlated with the kernels h[tl], we have that \n\nL-l L-l L L h[tt]m[t]so[t + r - ttl '\" 0(MI/2 Ll/2) \n\ntl=O t=o \n\n(8) \n\n\fPerturbative M-Sequences for Auditory Systems Identification \n\n183 \n\nIf N cycles of the m-sequence are performed, in which sort] is different for each \ncycle, all the terms in Eq. (6) scale with N as O(N), except for the double sum. \nBy the same central limits arguments above, the double sum scales as O(Nl/2). \n\nPutting all these results together into Eq. (6) and solving for the kernels yields \n\nh(r) \n\na(L 1+ 1) Rrm(r) - 0 ( ~) + 0 (aN~~~1/2 ) . \na(L + 1) Rrm(r) - C1 L + C2 aNI/2\u00a31/2' \n\nMl/2 \n\n1 \n\nM \n\n(9) \n\nwith the constants C1 , C2 '\" O(h[r]) depending neural firing rate, statistics, etc., \ndetermined from experiment. If we take the kernel element h(r) to be the first term \nin Eq. 9, then the last two terms in Eq. (9) contribute errors in determining the \nkernel and can be thought of as noise. Both error terms vanish as L -+ 00 and the \nprocedure is asymptotically exact for arbitrary uncorrelated stimuli sort]. In order \nfor the cross-correlation Ram (r) to yield a good estimate, the inequalities \n\n(10) \nmust hold. In practice, the kernel memory is much smaller than the sequence length, \nand the second inequality is the stricter bound. The second inequality represents a \ntradeoff among sequence length, number of trials and the size of the perturbation for \na given level of systematic noise in the kernel estimate. For instance, if L = 215 - 1, \nN = 10, M = 30, and noise floor at 10%, the perturbation should be larger than \na = 0.095. If no signal sort] is present, then the O(Ml/2a- 1(NL)-1/2) term drops \nout and the usual m-sequence cross-correlation result is recovered. \n\n3 M-Sequences for Modulation Response \n\nPrevious work, e.g., [M011er and Rees1986, Langner and Schreinerl988] has shown \nthat many of the cells in the inferior colliculus are tuned not only to a characteristic \nfrequency, but are also tuned to a best frequency of modulation of the carrier. A \nhighly simplified model of the IC unit response to sound stimuli is the Ll- N - L2 \ncascade filter, with L1 a linear tank circuit with a transfer function matching that \nof the frequency tuning curve, N a nonlinear rectifying unit, and L2 a linear cir(cid:173)\ncuit with a transfer function matching that of the modulation transfer function. \nDetecting this modulation is an inherently nonlinear operation and N is not well \napproximated by a linear kernel. Thus IC modulation responses will not be well \ncharacterized by ordinary m-sequence stimuli using the methods described in Sec(cid:173)\ntion 2. \nA better approach is to bypass the Ll - N demodulation step entirely and con(cid:173)\ncentrate on measuring L2. This can be accomplished by creating a modulation \nm-sequence: \n\n(11) \nwhere Iso[t]1 :::; 1 is the ambient signal, i.e., the operating point, m[t] E [-1,1] is an \nm-sequence added with amplitude b, and We is the carrier frequency. Demodulation \ngives the effective input stimulus \n\ns[t] = a (so[t] + bm[t]) sin[wet], \n\nsm[t] = a (so[t] + bm[t]) . \n\n(12) \n\nNote that there is little physiological evidence for a purely linear rectifier N. In \nfact, both the work of [M011er and Rees1986, Rees and M011er1987] and ours below \nshow that there is a nonlinear modulation response. Taking a modulation transfer \n\n\f184 \n\nM. Kvale and C. E. Schreiner \n\nfunction seriously, however, implies that one assumes that modulation response \nis linear, which implies that the static nonlinearity used is something like a half(cid:173)\nwave rectifier. Linearity is used here as a convenient assumption for organizing the \nstimulus and asking whether nonlinearities exist. \nFor full m-sequence modulation (so[t] = 1 and b = 1) the stimulus Sm and the \nneural response can be used to compute, via the Lee--Schetzen cross-correlation, \nthe modulation transfer function for the L2 system. Alternatively, for b \u00ab 1, the \nm-sequence is a perturbation on the underlying modulation envelope sort]. The \nderivation above shows that the linear modulation kernel can also be calculated \nusing a Lee--Schetzen cross-correlation. M-sequences at full modulation depth were \nfirst used by [M0ller and Rees1986, Rees and M011erI987] to calculate white-noise \nkernels. Here, we are using m-sequence in a different way-we are calculating the \nsmall-signal properties around the stimulus sort]. \nThe m-sequences used in this experiment were of length 215 -1 = 32,767. For each \nunit, 10 cycles of the m-sequence were presented back-to-back. After determining \nthe characteristic frequency of a unit, stimuli were presented which never differed \nfrom the characteristic frequency by more than 500 Hz. Figure 1 depicts the si(cid:173)\nnusoidal and m-sequence components and their combined result. The stimuli were \npresented in random order so as to mitigate adaptation effects. \n\nFigure 1: A depiction of stimuli used in the experiment. The top graph shows \na pure sine wave modulation at modulation depth 0.8. The middle graph shows \nan m-sequence modulation at depth 1.0. The bottom graph shows a perturbative \nm-sequence modulation at depth 0.2 added to a sinusoidal modulation at depth 0.8. \n\n4 Results \n\nFigure 2 shows the spike rates for both the pure sinusoid and the combined sinusoid \nand m-sequence stimuli. Note that the rates are nearly the same, indicating that \nthe perturbation did not have a large effect on the average response of the unit. \nThe unit shows an adaptation in firing rate over the 10 trials, but we did not find \n\n\fPerturbative M-Sequences for Auditory Systems Identification \n\n185 \n\na statistically significant change in the kernels of different trials in any of the units. \n\nG----e sinusoid \n~ sinusoid + m-sequence \n\n80.0 \n\n.-... o \nQ) \n~ \nC/) \nQ) \n~ \n'5. \n~ 60.0 \nQ) \n\u00abi \n\n\"-\n\n40.0 \n\n100.0 \n\n300.0 \n200.0 \nTime (sec) \n\n400.0 \n\n500.0 \n\nFigure 2: A plot of the unit firing rates for both the pure sinusoid and the sinusoid + \nm-sequence stimuli. The carrier frequency is 9 kHz and is close to the characteristic \nfrequency of the neuron. The sinusoidal modulation has a frequency of 20 Hz and \nthe m-sequence modulation has a frequency of 800 sec-I . \n\nFigure 3 shows modulation response kernels at several different values of the mod(cid:173)\nulation depth. Note that if the system was a linear, superposition would cause all \nthe kernels to be equivalent; in fact it is seen that the nonlinearities are of the same \nmagnitude as the linear response. In this particular unit, the triphasic behavior \nat small modulation depths gives way to monophasic behavior at high modulation \ndepths and an FFT of the kernel shows that the bandwidth of the modulation \ntransfer function also broadens with increasing depth. \n\n5 Discussion \n\nIn this paper, we have introduced a new type of stimulus, perturbative m-sequences, \nfor the study of auditory systems and derived their properties. We then applied \nperturbative m-sequences to the analysis of the modulation response of units in the \nIe, and found the linear response at a few different operation point. We demon(cid:173)\nstrated that the nonlinear response in the presence of sinusoidal modulations are \nnearly as large as the linear response and thus a description of unit response with \nonly an MTF is incomplete. We believe that perturbative stimuli can be an effective \ntool for the analysis of many systems whose units phase lock to a stimulus. \nThe main limiting factor is the systematic noise discussed in section 2, but it is \npossible to trade off duration of measurement and size of the perturbation to achieve \ngood results. The m-sequence stimuli also make it possible to derive higher order \ninformation [Sutter1987] and with a suitable noise floor, it may be possible to derive \nsecond-order kernels as well. \nThis work was supported by The Sloan foundation and ONR grant number N00014-\n94-1-0547. \n\n\f186 \n\nM. Kvale and C. E. Schreiner \n\nResponse vs. modulation depth \n\nsine wave @40Hz + pert. m-sequence \n\n-0.2 \n-0.4 \n- 0 . 6 \n-0 .8 \n-1 .0 \n\n5.0 \n\n10.0 \n\n15.0 \n\n20.0 \n\ntime from spike (milliseconds) \n\nCD \n\"C \n\n:::> \"\" a. \nE \nns \n\n90.0 \n\n70.0 \n\n50.0 \n\n30.0 \n\n10.0 \n\n-10.0 \n\n-30.0 \n\n-50.0 \n\n0.0 \n\nFigure 3: A plot of the temporal kernels derived from perturbative m-sequence \nstimuli in conjunction with sinusoidal modulations at various modulation depth. \nThe y-axis units are amplitude per spike and the x-axis is in milliseconds before the \nspike. \n\nReferences \n\n[Golomb1982] S. W. Golomb. Shift Register Sequences. Aegean Park Press, Laguna \n\nHills, CA, 1982. \n\n[Langner and Schreiner1988] G. Langner and C. E. Schreiner. Periodicity coding \n\nin the inferior colliculus of the cat: 1. neuronal mechanisms. Journal of Neuro(cid:173)\nphysiology, 60: 1799-1822, 1988. \n\n[Marmarelis and Marmarelis1978] Panos Z. Marmarelis and Vasilis Z. Marmarelis. \n\nAnalysis of Physiological Systems. Plenum Press, New York, NY, 10011, 1978. \n\n[M011er and Rees1986] Aage R. M011er and Adrian Rees. Dynamic properties of \nsingle neurons in the inferior colliculus of the rat. Hearing Research, 24:203-215, \n1986. \n\n[Pinter and Nabet1987] Robert B. Pinter and Bahram Nabet. Nonlinear Vision. \n\nCRC Press, Boca Raton, FL, 1987. \n\n[Rees and M011er1987] Adrian Rees and Aage R. M01ler. Stimulus properties in(cid:173)\n\nfluencing the responses of inferior colliculus neurons to amplitude-modulated \nsounds. Hearing Research, 27:129-143, 1987. \n\n[Rieke et al.1995] F. Rieke, D. A. Bodnar, and W. Bialek. Naturalistic stimuli \nincrease the rate and efficiency of information transmission by primary auditory \nafi\"erents. Proceedings of the Royal Society of London. Series B, 262:259-265, \n1995. \n\n[Sutter1987] E. E. Sutter. A practical non-stochastic approach to nonlinear time(cid:173)\ndomain analysis. In Vasilis Z. Marmarelis, editor, Advanced Methods of Physio(cid:173)\nlogical Modeling, Vol. 1, pages 303-315. Biomedical Simulations Resource, Uni(cid:173)\nversity of Southern California, Los Angeles, CA 90089-1451, 1987. \n\n\f", "award": [], "sourceid": 1430, "authors": [{"given_name": "Mark", "family_name": "Kvale", "institution": null}, {"given_name": "Christoph", "family_name": "Schreiner", "institution": null}]}