Perturbative M-Sequences for Auditory 
Systems Identification 
Mark Kvale and Christoph E. Schreiner* 
Sloan Center for Theoretical Neurobiology, Box 0444 
University of California, San Francisco 
513 Parnassus Ave, San Francisco, CA 94143 
Abstract 
In this paper we present a new method for studying auditory sys- 
tems based on m-sequences. The method allows us to perturba- 
tively study the linear response of the system in the presence of 
various other stimuli, such as speech or sinusoidal modulations. 
This allows one to construct linear kernels (receptive fields) at the 
same time that other stimuli are being presented. Using the method 
we calculate the modulation transfer function of single units in the 
inferior colliculus of the cat at different operating points and discuss 
nonlinearities in the response. 
I Introduction 
A popular approach to systems identification, i.e., identifying an accurate analyt- 
ical model for the system behavior, is to use Volterra or Wiener expansions to 
model behavior via functional Taylor or orthogonal polynomial series, respectively 
[Marmarelis and Marmarelis1978]. Both approaches model the response r(t) as a 
linear combination of small powers of the stimulus s(t). Although effective for mild 
nonlinearities, deriving the linear combinations becomes numerically unstable for 
highly nonlinear systems. A more serious problem is that many biological systems 
are adaptive, i.e., the system behavior is dependent on the stimulus ensemble. For 
instance, [Rieke et a/.1995] found that in the auditory nerve of the bullfrog linearity 
and information rates depended sensitively on whether a white noise or naturalistic 
ensemble is used. 
One approach to handling these difficulties is to forgo the full expansion, and sim- 
ply compute the linear response to small (perturbative) stimuli in the presence 
of various different ensembles, or operating points. By collecting linear responses 
*Email: kvale@phy. ucsf. edu and chris@phy. ucsf. edu 
Perturbative M-Sequences for Auditory Systems Identification 181 
from different operating points, one may fit nonlinear responses as one fits a non- 
linear function with a piecewise linear approximation. For adaptive systems the 
same procedure would be applied, with different operating points corresponding to 
different points along the time axis. Perturbative stimuli have wide application in 
condensed-matter physics, where they are used to characterize linear responses such 
as resistance, elasticity and viscosity, and in engineering, perturbative analyses are 
used in circuit analysis (small signal models) and structural diagnostics (vibration 
analysis). In neurophysiology, however, perturbative stimuli are unknown. 
An effective stimulus for calculating the perturbative linear response of a system 
is the m-sequence. M-sequences have a long history of use in engineering and the 
physical sciences, with applications ranging from systems identification to cryp- 
tography and cellular communication. In physiology, m-sequences have been used 
primarily to compute system kernels [Marmarelis and Marmarelis1978], especially 
in the visual system [Pinter and Nabetl987]. In this work, we use perturbative m- 
sequences to study the linear response of single units in the inferior colliculus of a 
cat to amplitude-modulated (AM) stimuli. We add a small m-sequence signal to 
an AM carrier, which allows us to study the linear behavior of the system near a 
particular operating point in a non-destructive manner, i.e., without changing the 
operating point. Perturbative m-sequences allow one to calculate linear responses 
near the particular stimuli under study with only a little extra effort, and allow us 
to characterize the system over a wide range of stimuli, such as sinusoidal AM and 
naturalistic stimuli. 
The auditory system we selected to study was the response of single units in the 
central nucleus of the inferior colliculus (IC) of an anaesthetised cat. Single unit 
responses were recorded extracellularly. Action potentials were amplified and stored 
on DAT tape, and were discriminated offline using a commercial computer-based 
spike sorter (Brainwave). 20 units were recorded, of which l0 yielded sufficiently 
stable responses to be analyzed. 
2 M-Sequences and Linear Systems 
A binary m-sequence is a two-level pseudo-random sequence of +l's and -l's. The 
sequence length is L = 2 n - 1, where n is the order of the sequence. Typically, a 
binary m-sequence can be generated by a shift register with n bits and feedback 
connections derived from an irreducible polynomial over the multiplicative group Z2 
[Golomb1982]. For linear systems identification, m-sequences have two important 
properties. The first is that m-sequences have nearly zero mean: tL__-01 re[t]: --1. 
The second is that the autocorrelation function takes on the impulse-like form 
L-1 
Smm(T) --- E m[t]m[t + ] = { L if r = 0 
- 1 otherwise (1) 
t=0 
Impulse stimuli also have a 5-function autocorrelation function. In the context 
of perturbative stimuli, the advantage of an m-sequence stimulus over an impulse 
stimulus is that for a given signal to noise ratio, an m-sequence perturbation stays 
much closer to the original signal (in the least squares sense) than an impulse per- 
turbation. Thus the perturbed signal does not stray as far from the operating point 
and measurement of linear response about that operating point is more accurate. 
We model the IC response with a system F through which a scalar stimulus s(t) is 
passed to give a response r(t): 
r(t) = F[8(t)]. (2) 
182 M. Kvale and C. E. Schreiner 
For the purposes of this section, the functional F is taken to be a linear functional 
plus a DC component. In real experiments, the input and output signal are sampled 
into discrete sequences with t becoming an integer indexing the sequence. Then the 
system can be written as the discrete convolution 
L-1 
r[t] = ho + E h[t]s[t- t] (3) 
tl=O 
with kernels ho and hit1] to be determined. We assume that the system has a finite 
memory of M time steps (with perhaps a delay) so that at most M of the hit] 
coefficients are nonzero. To determine the kernels perturbatively, we add a small 
amount of m-sequence to a base stimulus so: 
s[t] = so[t] + am[t]. (4) 
Cross-correlating the response with the original m-sequence yields 
L-1 
L-1 L-1 L-1 
lrm(7') = E /[t][t -Jr-T] = E /T[t]ho -Jr- E E h[tl][t]$[t -Jr- 
t=O t=O tl=O 
t----O 
L-1 L-1 
q- E E oh[tl]m[t]m[t q- r -- tl]. 
t----O t----O 
Using the sum formula for am -sequence above, the first sum in Eq. 
simplified to -ho. Using the autocorrelation Eq. 
simplifies, and we find 
(5) 
(5) can be 
(1), the third sum in Eq. (5) 
L-1 L-1 L-1 
Rrm(T) -- (L q- 1)h[r] - ho - c E h[tl] q- E E h[tl]m[t]so[t q- r - t] 
t!----O t--O tl----O 
(6) 
Although the values for the kernels h(t) are set implicitly by this equation, the 
terms on the right hand side of Eq. (6) are widely different in size for large L and 
the equation can be simplified. As is customary in auditory systems, we assume 
the DC response ho is small. To estimate the size of the other terms, we compute 
statistical estimates of their sizes and look at their scaling with the parameters. 
L-1 
The term a Et=o h[tl] is a sum of M kernel elements; they may be correlated or 
uncorrelated, so a conservative estimate of their size is on the order of O(aM). 
The last term in (6) is more subtle. We rewrite it as 
L-1 L-1 
E E h[tl]m[t]so[t + 
tl=O t=O 
L-1 
T -- tl] = E h[tl]p[T, tl] 
tl:0 
L-1 
p[%tl] -- E m[t]so[t q- -- t] 
t=0 
(7) 
The time series of the ambient stimulus so[t] and m-sequence re[t] are assumed to 
be uncorrelated. By the central limit theorem, the sum p[r, tl] will then have an 
average of zero with a standard deviation of O(LX/2). If in turn, the terms p[r, tl] 
are uncorrelated with the kernels h[tl], we have that 
L-1 L-1 
E E h[tl]m[t]so[t + 
tl =0 t=O 
T -- tl] " O(M1/2L 1/2) 
(s) 
Perturbative M-Sequences for Auditory Systems Identification 183 
If N cycles of the m-sequence are performed, in which so[t] is different for each 
cycle, all the terms in Eq. (6) scale with N as O(N), except for the double sum. 
By the same central limits arguments above, the double sum scales as 0(N1/2). 
Putting all these results together into Eq. (6) and solving for the kernels yields 
h(-) = a(L + 1) Rrm(T) - 0 + 0 otN1/2L1/2 . 
.. I Rrm(T) -- C1 M M 1/ 
 ct(L + 1) -- + C2ctN/2LX/2' (9) 
with the constants C, C2 - O(h[-]) depending neural firing rate, statistics, etc., 
determined from experiment. If we take the kernel element h(-) to be the first term 
in Eq. 9, then the last two terms in Eq. (9) contribute errors in determining the 
kernel and can be thought of as noise. Both error terms vanish as L - c and the 
procedure is asymptotically exact for arbitrary uncorrelated stimuli so[t]. In order 
for the cross-correlation Rsm(-) to yield a good estimate, the inequalities 
C1M  L and ct >> C2M1/2(NL) -1/2 (10) 
must hold. In practice, the kernel memory is much smaller than the sequence length, 
and the second inequality is the stricter bound. The second inequality represents a 
tradeoff among sequence length, number of trials and the size of the perturbation for 
a given level of systematic noise in the kernel estimate. For instance, if L = 225 - 1, 
N = 10, M = 30, and noise floor at 10%, the perturbation should be larger than 
ct = 0.095. If no signal so[t] is present, then the 0(M1/2o -I(NL) -/2) term drops 
out and the usual m-sequence cross-correlation result is recovered. 
3 M-Sequences for Modulation Response 
Previous work, e.g., [Mller and Rees1986, Langner and Schreiner1988] has shown 
that many of the cells in the inferior colliculus are tuned not only to a characteristic 
frequency, but are also tuned to a best frequency of modulation of the carrier. A 
highly simplified model of the IC unit response to sound stimuli is the L1 - N - L2 
cascade filter, with L1 a linear tank circuit with a transfer function matching that 
of the frequency tuning curve, N a nonlinear rectifying unit, and L2 a linear cir- 
cuit with a transfer function matching that of the modulation transfer function. 
Detecting this modulation is an inherently nonlinear operation and N is not well 
approximated by a linear kernel. Thus IC modulation responses will not be well 
characterized by ordinary m-sequence stimuli using the methods described in Sec- 
tion 2. 
A better approach is to bypass the L1 -N demodulation step entirely and con- 
centrate on measuring L2. This can be accomplished by creating a modulation 
m-sequence: 
s[t] = a(so[t] + bm[t])sin[wct], (11) 
where Is0[t]l _< 1 is the ambient signal, i.e., the operating point, re[t]  [-1, 1] is an 
m-sequence added with amplitude b, and wc is the carrier frequency. Demodulation 
gives the effective input stimulus 
sm[t] - a (so[t] + bin[t]). (12) 
Note that there is little physiological evidence for a purely linear rectifier N. In 
fact, both the work of [Moller and Rees1986, Rees and Moller1987] and ours below 
show that there is a nonlinear modulation response. Taking a modulation transfer 
184 M. Kvale and C. E. Schreiner 
function seriously, however, implies that one assumes that modulation response 
is linear, which implies that the static nonlinearity used is something like a half- 
wave rectifier. Linearity is used here as a convenient assumption for organizing the 
stimulus and asking whether nonlinearities exist. 
For full m-sequence modulation (so[t] - 1 and b - 1) the stimulus Sm and the 
neural response can be used to compute, via the Lee-Schetzen cross-correlation, 
the modulation transfer function for the L2 system. Alternatively, for b gg 1, the 
m-sequence is a perturbation on the underlying modulation envelope so[t]. The 
derivation above shows that the linear modulation kernel can also be calculated 
using a Lee-Schetzen cross-correlation. M-sequences at full modulation depth were 
first used by [Mller and Rees1986, Rees and M011er1987] to calculate white-noise 
kernels. Here, we are using m-sequence in a different way--we are calculating the 
small-signal properties around the stimulus s0[t]. 
The m-sequences used in this experiment were of length 215 - 1 = 32,767. For each 
unit, 10 cycles of the m-sequence were presented back-to-back. After determining 
the characteristic frequency of a unit, stimuli were presented which never differed 
from the characteristic frequency by more than 500 Hz. Figure 1 depicts the si- 
nusoidal and m-sequence components and their combined result. The stimuli were 
presented in random order so as to mitigate adaptation effects. 
Figure l: A depiction of stimuli used in the experiment. The top graph shows 
a pure sine wave modulation at modulation depth 0.8. The middle graph shows 
an m-sequence modulation at depth 1.0. The bottom graph shows a perturbative 
m-sequence modulation at depth 0.2 added to a sinusoidal modulation at depth 0.8. 
4 Results 
Figure 2 shows the spike rates for both the pure sinusoid and the combined sinusoid 
and m-sequence stimuli. Note that the rates are nearly the same, indicating that 
the perturbation did not have a large effect on the average response of the unit. 
The unit shows an adaptation in firing rate over the 10 trials, but we did not find 
Perturbative M-Sequences for Auditory Systems Identification 185 
a statistically significant change in the kernels of different trials in any of the units. 
100.0 
 80.0  ' 
 60.0 
 r- 40.0 
20.0 
0.0 100.0 200.0 300.0 400.0 500.0 
Time (sec) 
Figure 2: A plot of the unit firing rates for both the pure sinusoid and the sinusoid + 
m-sequence stimuli. The carrier frequency is 9 kHz and is close to the characteristic 
frequency of the neuron. The sinusoidal modulation has a frequency of 20 Hz and 
the m-sequence modulation has a frequency of 800 sec -]. 
Figure 3 shows modulation response kernels at several different values of the mod- 
ulation depth. Note that if the system was a linear, superposition would cause all 
the kernels to be equivalent; in fact it is seen that the nonlinearities are of the same 
magnitude as the linear response. In this particular unit, the triphasic behavior 
at small modulation depths gives way to monophasic behavior at high modulation 
depths and an FFT of the kernel shows that the bandwidth of the modulation 
transfer function also broadens with increasing depth. 
15 Discussion 
In this paper, we have introduced a new type of stimulus, perturbative m-sequences, 
for the study of auditory systems and derived their properties. We then applied 
perturbative m-sequences to the analysis of the modulation response of units in the 
IC, and found the linear response at a few different operation point. We demon- 
strated that the nonlinear response in the presence of sinusoidal modulations are 
nearly as large as the linear response and thus a description of unit response with 
only an MTF is incomplete. We believe that perturbative stimuli can be an effective 
tool for the analysis of many systems whose units phase lock to a stimulus. 
The main limiting factor is the systematic noise discussed in section 2, but it is 
possible to trade off duration of measurement and size of the perturbation to achieve 
good results. The m-sequence stimuli also make it possible to derive higher order 
information [Sutter1987] and with a suitable noise floor, it may be possible to derive 
second-order kernels as well. 
This work was supported by The Sloan foundation and ONR grant number N00014- 
94-1-0547. 
186 M. Kvale and C. E. Schreiner 
70.0 
50.0 
o.o 
10.0 
-10.0 
-30.0 
Response vs. modulation depth 
sine wave @40Hz + pert, m-sequence 
0.2 
0.4 
..... 0.6 
0.8 
-50.0 , I , I , I , 
0.0 5.0 10.0 15.0 20.0 
time from spike (milliseconds) 
Figure 3: A plot of the temporal kernels derived from perturbative m-sequence 
stimuli in conjunction with sinusoidal modulations at various modulation depth. 
The y-axis units are amplitude per spike and the x-axis is in milliseconds before the 
spike. 
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