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The Journal of Neuroscience, March 15, 2000, 20(6):2315-2331
Spectral-Temporal Receptive Fields of Nonlinear Auditory Neurons
Obtained Using Natural Sounds
Frédéric E.
Theunissen1,
Kamal
Sen4, and
Allison J.
Doupe2, 3, 4
1 Department of Psychology, University of California,
Berkeley, California 94720-1650 and Departments of
2 Psychiatry and 3 Physiology, and
4 Sloan Center for Theoretical Neuroscience, University of
California, San Francisco, California 94143-0444
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ABSTRACT |
The stimulus-response function of many visual and auditory neurons
has been described by a spatial-temporal receptive field (STRF), a
linear model that for mathematical reasons has until recently been
estimated with the reverse correlation method, using simple stimulus
ensembles such as white noise. Such stimuli, however, often do not
effectively activate high-level sensory neurons, which may be optimized
to analyze natural sounds and images. We show that it is possible to
overcome the simple-stimulus limitation and then use this approach to
calculate the STRFs of avian auditory forebrain neurons from an
ensemble of birdsongs. We find that in many cases the STRFs derived
using natural sounds are strikingly different from the STRFs that we
obtained using an ensemble of random tone pips. When we compare these
two models by assessing their predictions of neural response to the
actual data, we find that the STRFs obtained from natural sounds are
superior. Our results show that the STRF model is an incomplete
description of response properties of nonlinear auditory neurons, but
that linear receptive fields are still useful models for understanding higher level sensory processing, as long as the STRFs are estimated from the responses to relevant complex stimuli.
Key words:
Natural sounds; auditory cortex; spectro-temporal; receptive field; field L; reverse correlation
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INTRODUCTION |
Neuroscientists have successfully
used the concept of a receptive field (RF) (Hartline, 1940 ) to
characterize the stimulus features that are being encoded by sensory
neurons both at the periphery and at higher levels of processing. The
RF summarizes the encoding characteristics of a particular sensory
neuron by showing the feature that will elicit the maximal response.
This description has resulted in an understanding of the hierarchical computation underlying the feature extraction that occurs for example
in the visual system, where the simple center-surround RFs of visual
ganglion cells (Kuffler, 1953) become RFs for edge and bar detection in
complex cells of V1 (Hubel and Wiesel, 1962). A similar representation
is used in the auditory system, in frequency tuning curves, where the
spatial dimensions have been replaced by a spectral dimension. More
recently, visual and auditory neurophysiologists have also added the
dimension of time to static RFs, obtaining spatial-temporal RFs in the
visual system (DeAngelis et al., 1995; Cai et al., 1997; Ringach et
al., 1997a; De Valois and Cottaris, 1998), and spectral-temporal RFs in
the auditory system (Aertsen and Johannesma, 1981a,b; Aertsen et al.,
1981; Clopton and Backoff, 1991; Eggermont et al., 1983a,b; Kim
and Young, 1994; Kowalski et al., 1996a; Nelken et al., 1997;
deCharms et al., 1998) (in both visual and auditory systems referred to
as STRFs). The STRF shows which temporal succession of acoustical or
visual features would elicit the maximal neural response. Recent
research in the auditory cortex has suggested that auditory and visual
cortical STRFs have remarkably similar time varying shapes (deCharms et al., 1998).
The underlying assumption that allows one to reduce all aspects of
encoding by a neuron to an STRF is that the response to a novel
time-varying stimulus that was not used in the estimation of the STRF
can be predicted from the stimulus-response data used in the STRF
estimation by simple linear interpolation or extrapolation. The STRF of
a neuron can therefore be rigorously defined as the best linear model
that transforms any time-varying stimulus into a prediction of the
firing rate of a neuron. It is also this linear model of neural
encoding that allows one to easily obtain STRFs from experimental data.
If the visual or auditory spatial-temporal dimensions are sampled
uniformly and randomly, then one can estimate the STRF simply by
averaging the stimuli before each spike. This procedure is called the
reverse correlation method (Boer and Kuyper, 1968). However, the
uniform and random sampling requirement implies that one needs to use a
stimulus ensemble that is the equivalent of white noise in both the
spatial/spectral and temporal dimensions. If such an ensemble is not
used, then the spike-triggered average stimulus will, in general, not
be equal to the STRF (Aertsen et al., 1981; Eggermont et al., 1983a;
Ringach et al., 1997b).
Therefore, even though describing neural coding in terms of RF has been
a successful approach, the underlying linear assumptions and the use of
white noise to obtain the STRF raise important issues. First, white
noise has been shown to be a poor stimulus in higher sensory areas,
because it elicits very few spikes, making the calculation of the STRF
difficult for practical reasons (as the calculation requires large
amount of data) and the results of questionable validity for
methodological reasons (as the neurons might not be driven into their
full dynamical range). Second, because sensory neurons exhibit varying
degrees of linearity, it is essential to estimate how much of the
response of the neurons can be explained with the linear model implicit
in the STRF. For example, it is known that certain neurons in higher
auditory areas in primates (Rauschecker et al., 1995), bats (Ohlemiller
et al., 1996), and birds (Scheich et al., 1979; Margoliash, 1983, 1986) are not well driven by simple or white noise stimuli and seem to be
specifically tuned to the sounds of the animal's vocalizations. These
neurons show some degree of nonlinearity, because they do not respond
well to components of the vocalization presented alone but respond
strongly to the complete vocalization. Could we nonetheless explain
some of the encoding properties of such complex auditory neurons with
the linear STRF model for a subset of specialized sounds? And if so,
how would we calculate STRFs from such stimulus ensembles that in
general will show correlations in time and frequency? Finally, how does
one verify that the linear model is in fact capturing a significant
fraction of the encoding of such neurons?
In this work, we address these issues by demonstrating how STRFs can be
calculated from ensembles of natural sounds and by using simple methods
to quantify the accuracy of the linear assumption. We further examine
the degree of linearity by comparing the similarity and predictive
power of STRFs obtained from natural sounds to those obtained from an
ensemble of pure tone pips. The auditory neurons we examine lie in the
songbird forebrain auditory fields L1, L2a, L2b, and L3, which are
homologous both in their anatomical location in the chain of acoustical
processing and in their functional properties to primary and secondary
auditory cortical areas in mammals (Muller and Leppelsack, 1985;
Fortune and Margoliash, 1992; Vates et al., 1996). Understanding how
neurons in these high-level auditory areas process the sounds of the
conspecific's vocalizations, in particular conspecific song, is
behaviorally relevant, because songbirds must be able to recognize song
both to identify the singer and for young males to be able to copy a
tutor song (Marler, 1970; Konishi, 1985). In addition, auditory processing in field L (Lewicki and Arthur, 1996) must ultimately contribute to the extreme selectivity of song-selective auditory neurons found further along in the auditory hierarchy (in the song
system; Nottebohm et al., 1976). Such neurons respond highly nonlinearly only to the specific spectral and temporal context of the
bird's own song (Margoliash, 1983, 1986; Margoliash and Fortune, 1992;
Lewicki, 1996; Theunissen and Doupe, 1998).
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MATERIALS AND METHODS |
Electrophysiology. All physiological recordings were
done in urethane-anesthetized adult male zebra finches in acute
experiments. Extracellular waveforms were obtained using tungsten
electrodes that were inserted into the neostriatum of the bird at
locations that were previously marked with stereotaxic measurements.
The extracellular waveforms were transformed into spike trains by windowing the largest action potential. Single units or small multiunit
clusters were recorded in this manner. At the end of the experiment,
the bird was killed, and the location of the recordings was
verified histologically in Nissl-stained brain sections. A complete
description of similar experimental procedures can be found in
Theunissen and Doupe (1998). The location of the sites was classified
into anatomical subregions of Field L as described in Fortune and
Margoliash (1992). The data presented here were obtained from five
birds and 20 recording sites (five from area L1, two from L2a, five
from L2b, and eight from L3).
Stimuli. Two stimulus ensembles were used: a random tone pip
ensemble and a conspecific song ensemble. The random tone ensemble was
made of 20 different 2 sec trains of tone pips. The frequency, amplitude, length, and intertone interval were random. The distribution of frequencies was made to match the average power spectrum of zebra
finch song. The tone length and the intertone interval had Gaussian
distributions with the same mean and SD as the syllable length and
intersyllable interval of a large ensemble of zebra finch songs
recorded in our laboratory. The tone pip loudness was varied randomly
in logarithmic steps (0.125, 0.25, 0.5, 1.0). The loudest tones had an
intensity that was similar to the peak intensity of the songs (80 dB
SPL). The tones had onset and offset half-cosine ramps of 25 msec. The
conspecific song ensemble consisted of 21 songs from different adult
male zebra finches obtained in our colony including the bird's own
song. Ten spike train response trials were obtained for each of the 21 songs and each of the 20 2 sec train of tone pips. The intertrial
interval was 6 sec, and the trials from different stimuli were
interleaved in random order.
STRFs in the auditory domain. The STRF is defined as the
optimal linear filter that transforms a representation of a
time-varying stimulus into a prediction of the firing rate of the
neuron as estimated by the poststimulus time histogram (PSTH) of the
neuron. This optimal linear filter can be simply obtained from the
spike-triggered average stimulus (STA) if (1) the stimulus space used
in the experiment encompasses all stimuli that are capable of producing
significant neural responses, (2) if the sampling in the stimulus space
is done in a random and uniform manner, and (3) if the multiple spatial dimensions used to represent the stimulus are independent of each other. If these three conditions are satisfied, the STRF will be
proportional to the STA. This method for estimating the STRF is called
the reverse correlation (Boer and Kuyper, 1968). We explain below why
these three conditions can cause significant difficulties for the
estimation of STRFs for high-level auditory neurons and how one can
circumvent some of these constraints.
The first significant difficulty is the choice of a spatial
representation of sound. The definition of the spatial dimensions of
the stimulus in the visual domain is unambiguous because the natural
decomposition of an image into pixels yields a representation in which
the spatial dimensions are clearly independent of each other. In the
auditory domain, the spatial dimensions are usually taken to be the
amplitude values of each of the narrow-band signals obtained from a
decomposition of the sound into frequency bands (a spectrographic
representation of sound). This general form for the auditory spatial
dimensions makes sense physiologically, because sounds are decomposed
into narrow-band signals by frequency channels along the length of the
cochlea. However, as we will elaborate below, such spatial dimensions
will not be independent of each other when an invertible spectrographic
representation of sound is used. For this reason (violation of
condition 3), even when white noise sound is used, which would satisfy
both the complete (condition 1) and random (condition 2) sampling of the auditory space, the spike-triggered average spectrogram will only
be an approximation of the STRF. In addition, in both visual and
auditory modalities, one would like to be able to investigate neural
properties with stimuli other than white noise and therefore not be
constrained by the random and uniform sampling condition (condition 2).
STRFs in the auditory domain have been defined in two manners that can
be related to each other. Realizing that an STRF based on the
spike-triggered average spectrogram would depend not only on the
statistical properties of the stimulus (condition 2) but also on the
nature of the spectrographic representation (condition 3), Aertsen and
Johannesma (1981a) originally defined an invariant STRF as the
second-order Volterra kernel between the sound pressure waveform and
the PSTH. This particular definition of the STRF is therefore
independent of any choice of spectrographic representation and could
theoretically be estimated from stimulus ensembles with different
statistical properties. The second-order kernel that relates sound
pressure waveform to the PSTH can also be rewritten as a first-order
Volterra kernel that relates the Wigner function of the sound-pressure
waveform to the PSTH (Eggermont and Smith, 1990; Eggermont, 1993; Kim
and Young, 1994; Nelken et al., 1997). The Wigner function yields a
scale-free time-frequency representation of the sound (see also
Appendix). The second definition of the auditory STRF is the more
common one: it is the first-order Volterra kernel between a
spectrographic representation of sound and the PSTH (Eggermont et al.,
1983a; Escabi et al., 1998; Shamma et al., 1998). As shown in the
Appendix, the second definition of the STRF (called in this section the
spectrographic STRF) can be thought of as a filtered version of the
invariant STRF (see also Klein et al., 2000). This definition of the
STRF is favored because it illustrates the spectral-temporal patterns
that elicited neural responses in an explicit manner. However,
depending on the choice of the spectrographic representation, one can
obtain many spectrographic STRFs for a given neuron. Moreover, a
spectrographic STRF runs the risk of losing information as a result of
the filtering operation. To be able to relate the spectrographic STRF
to the invariant STRF, without any a priori assumptions, the
spectrographic representation of sound must be invertible (see
Appendix). That is, it should be possible to recover the original sound
from the spectrogram, except for a single absolute phase. Noninvertible
representations could be used if a priori knowledge of the neural
response properties tells us that any two different stimuli that are
not separately recoverable from the noninvertible representation of
sound yield the same neural responses. In such cases the invariant STRF
can still be recovered from the spectrographic STRF.
An invertible spectrographic representation of sound requires the use
of overlapping frequency bands, as explained in this paragraph and in
more mathematical detail in the Appendix. The fine temporal structure
of a sound is given by the relative phase of the narrow-band signals
obtained in the decomposition of the sound into frequency bands. The
phase of these narrow-band signals is thrown away in a spectrographic
representation, where only the amplitude envelopes of the narrow-band
signals are preserved. However, the relative phase of the narrow-band
signals can be recovered from the joint consideration of the amplitude
envelopes, as long as there is sufficient overlap among the frequency
bands (Cohen, 1995; Theunissen and Doupe, 1998). It is for this reason that, in a complete spectrographic representation, where frequency filters overlap, the spike-triggered average of even a white noise stimulus will only be an approximation of the spectrographic STRF. In a
spectrographic representation with nonoverlapping frequency bands, the
spike-triggered average spectrogram would be equal to the
spectrographic STRF but, in general, because such a spectrographic representation is noninvertible, one might not be able to obtain the
invariant STRF.
An invertible spectrographic representation of sound. In
this study, we used the spectrographic definition of the STRF (from here on called the STRF) and an invertible spectrographic
representation that is illustrated schematically in Figure
1. The sounds are represented by a set of
functions of time s{i}(t), where si(t) is taken to be the log of the amplitude
envelope of the signal in the frequency band i (the brackets
indicate that we refer to the entire set of time-varying functions).
The frequency bands were obtained with Gaussian filters of 250 Hz width
(SD). We used 31 frequency bands spanning center frequencies between 250 and 8000 Hz. In this manner, the center frequencies of neighboring bands are separated by exactly 250 Hz or the equivalent of 1 SD. It is
this large amount of overlap that allows this representation to be
invertible. We extracted the amplitude envelope of each frequency band using the analytical signal as explained in Theunissen and Doupe (1998) and characterized the sound by the difference between
the log value of the amplitude and the mean log amplitude for that
band. We have shown that the particular time-frequency scale used in
our spectrographic representation of sounds is the most efficient at
representing the spectral and temporal structure of songs that is
essential in eliciting the response of song-selective neurons in the
zebra finch (Theunissen and Doupe, 1998). The additional log
transformation was used because it improved our results significantly, as we will discuss in Results. Note that, in the auditory domain, the
STRF is a linear transformation between a nonlinear representation of
the stimulus and the PSTH. The calculation of the amplitude envelopes
that make up the spectrogram is a nonlinear operation (although in our
case it is invertible), and additional nonlinearities such as the log
transform are often used. This model, therefore, includes known or
determined static nonlinearities between the stimulus and the
response.
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Figure 1.
Schematic illustrating the spectrographic
decomposition and the calculation of the stimulus autocorrelation
matrix. The sound is decomposed into frequency bands by a bank of
Gaussian filters. The result is a set of narrowband signals with
time-varying amplitude and phase. Our representation of sound is based
on the time-varying amplitude envelopes. Although the time-varying
phase is discarded, the relative phase across frequency bands is
preserved because of the large overlap between adjoining filters. The
time-varying amplitude envelope or its log is what is usually
represented in a spectrogram. Our representation of sound and the
spectrograms shown in this paper are based on the log of the amplitude
envelopes. The stimulus autocorrelation function is then found by
cross-correlating the log-amplitude envelope of a particular band with
the log-amplitude envelope of all the other bands, including itself.
The autocorrelation for the entire ensemble is done by averaging the
correlation at each time point for all stimuli in a given ensemble.
Here we show the results of two of such pairwise correlations: the
correlation of band 4 (centered at 1250 Hz) with band 2 (centered at
500 Hz) and of band 4 with itself, for the song ensemble.
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Calculation of the STRF for any stimulus ensemble. In this
section we elaborate on the details of the analytical solution of the
STRF for any sound ensemble, which involves correcting for the
correlations in the stimulus ensemble. The STRF is defined as the
multidimensional linear Volterra filter
h{i}(t), such that:
where rpre(t) is the predicted firing
rate, s{i}(t) is the multidimensional
representation of the time-varying stimulus, and nf is the
total number of spatial dimensions or frequency bands in our case.
h{i}(t) is found by requiring that
rpre(t) be as close as possible to
rest(t), the estimated firing rate obtained from
a PSTH, in the mean square error sense. One finds that in the frequency
domain, the set of h{i} can be obtained by
solving a set of linear equations for each frequency w. This
set of equations is written in vector notation as:
Aw is the autocorrelation matrix of the
stimulus in the frequency domain:
Si(w) is the Fourier transform of
si(t), * denotes the complex conjugate, and
  indicates that we estimated the cross moments by averaging
over samples.  w is the Fourier transform
of the set of hi(t):
 w is the cross-correlation between
the spike trains and the stimulus amplitude envelopes in each band:
where R(w) is the Fourier transform of
rest(t). w is the Fourier
transform of the spike-triggered average spectrogram. To solve for
h{i}, the matrix Aw
is inverted for each frequency:
The inverse Fourier transform is then applied to
w to obtain the STRF in the time domain,
h{i}(t). As long as an invertible
representation of sound is used, and the stimulus autocorrelation
matrix, Aw, is also invertible, the
spectrographic STRF found with this procedure can be directly related
to the invariant STRF. As we will discuss below and show in Results, the autocorrelation matrices of the stimulus ensembles used in this
work were not invertible. This will happen when stimulus ensembles do
not sample all possible sounds. In such cases, the number of
independent dimensions needed to represent the stimulus ensemble is
less than the total number, nf, used in the initial representation of the stimuli. However, an estimated STRF can still be
obtained by performing a singular value decomposition (SVD) of the
autocorrelation matrix. The SVD method is used to determine numerically
the smallest number of independent spatial dimensions that are needed
to represent the stimulus ensemble (Press et al., 1992). The division
by the autocorrelation matrix is then performed only for the subset of
sounds spanned by these independent dimensions. The estimated STRF
obtained in this way can therefore only be equated to the invariant
STRF if the neural responses to the regions of sound that are not
sampled are insignificant.
Parameters used in our numerical calculations. The number of
frequency values, w, used to represent the stimulus
autocorrelation matrix and the stimulus-response cross-correlation in
the frequency domain depends on the number of time points used in the
time window in the estimation of the crossmoments. We used a sampling
rate of 1 kHz (the amplitude envelopes of our frequency bands are
effectively band limited to frequencies <500 Hz; Theunissen and Doupe,
1998) and looked at the cross moments in a 400 msec time window,
yielding 200 frequency values, w. Therefore, solving for
h{i} required inverting 200 31 × 31 complex matrices. [Note that there are two "frequencies": the
first one corresponds to the "spatial" dimension in our
spectrographic representation of sound: the 31 frequency bands (ranging
from 250 to 8000 Hz). The second is the 200 frequency values
w (ranging from 0 to 500 Hz) that characterize the temporal
modulations in and across each of the 31 bands in our representation of
sound.]
Figure 1 shows how the stimulus autocorrelation is calculated in the
time domain. For each frequency band, the autocorrelation of the log of
the amplitude envelope is obtained: these autocorrelations are the
diagonal terms in the stimulus autocorrelation matrix. The off-diagonal
terms are obtained by cross-correlating the amplitude envelope in one
frequency band with the envelope in another band. In this calculation
all the stimuli in one ensemble are used, and the results are averaged.
We therefore estimated our stimulus autocorrelation matrix from ~40
sec of sound (2 sec per stimulus times 20 stimuli). Figure 5 shows the
entire autocorrelation matrix for three stimulus ensembles: white
noise, tone pips, and zebra finch song. The particular correlation
properties of these stimulus ensembles will be explained in Results.
The estimation of the STRFs by this generalized cross-correlation
method required two additional numerical techniques. First, because of
limited data, the spike-triggered average is inherently noisy. The
noise corresponds to the expected deviations from the mean that are
obtained in any averaging procedure. Although this noise was relatively
small in our case, it is amplified in the calculation of the STRF in
the division by the autocorrelation matrix of the stimulus. This is
particularly true at high-amplitude modulation frequencies in which the
autocorrelation of the stimulus is very small. Therefore, to prevent
the amplification of high-frequency noise, the cross-correlation
between stimulus and spike trains was lowpass-filtered. The frequency
cut-off was found systematically by investigating which amplitude
modulation frequencies in the spike-triggered average had significant
power, using a jackknife resampling technique in the frequency domain
(Efron, 1982). In brief, we calculated the jackknife estimate of the SE
of the real and imaginary part of w. For
each element in the vector, we then estimated at what cut-off
frequency, w, the magnitude of each of the
S*wRw elements was
less than two times the SE. This cut-off frequency was used to lowpass
filter w.
The second numerical consideration deals with the inversion of the
stimulus autocorrelation matrix. Because each of the amplitude envelopes of our stimuli was band limited to low frequencies and because the temporal correlations were similar in all frequency bands,
the autocorrelation matrices are singular. In other words, for both our
song ensemble and our random tone ensemble, the 31 × 400 dimensions used to represent their second-order structure are
redundant. As mentioned above, in such situations one can find the
independent dimensions by using the SVD method. The number of
independent dimensions or eigenvectors of the autocorrelation matrix
was estimated by ignoring all the eigenvectors that had eigenvalues
that were smaller than the maximum eigenvalue of all the frequency
matrices for a given stimulus ensemble, multiplied by a tolerance
factor. The tolerance factor was found by testing a range of values and
choosing the one that gave the best predictive power to the STRF model.
The predictive power was estimated by calculating the coherence between
the actual response and the predicted response to novel stimuli from
the same ensemble, as explained below. The coherence is a function of
the frequency and is given by Marmeralis and Marmeralis (1978):
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The best Rpre is the one that gives the
largest coherence averaged over all frequencies. Although the number of
significant eigenvectors and therefore the tolerance value is a
property of the stimulus ensemble, we found that the best tolerance
value changed slightly for each neuronal site. We attribute this effect to variances in the noise in the cross-correlation between spike train
and stimulus (w). By slightly
varying the number of eigenvectors used in the estimation of the STRF,
this noise can be selectively filtered out (because the eigenvectors
with the smallest power represent the higher amplitude modulation
frequencies, which are the most corrupted by noise).
Goodness of fit. To be able to compare our results with
previous published work, we also calculated the goodness of the STRF model by calculating the cross-correlation coefficient between the
predicted firing rate and actual firing rates:
The predicted firing rate was obtained by convolving the STRF
with the stimulus, rectifying, and scaling the result to minimize the
square error between the predicted and estimated firing rates.
For each zebra finch song or 2 sec train of tone pips, we calculated a
STRF that was based on all the other stimuli in the same ensemble but
that specifically excluded the stimulus-response data being tested.
The CC calculated from such a stimulus ensemble-matched STRF was used
to estimate how well the STRF model obtained from a particular stimulus
ensemble fitted the stimulus-response function of any stimulus that
was not part of the ensemble but that had identical second-order
statistical properties. The CCs obtained from these calculations are
referred to as CC-matched.
For each song and train of tone pip stimuli, we also calculated the
predicted firing rate obtained using the STRF calculated from all the
stimuli in the other stimulus ensemble. For each neuronal site, the
STRF obtained from all the songs was used to predict the response to
tone pips, and vice versa. The CCs obtained from these calculations are
referred to as CC-switched.
The estimated firing rate was obtained by smoothing the PSTH with a
hanning window. The window that gave the maximal cross-correlation value for the matched ensemble was used. The width of the window was
similar to the width of the STRF and had values between 6 and 96 msec
(width at half maximum). This estimation of the cross-correlation coefficient has a negative bias because of the noise in the PSTH from
the limited number of samples (n = 10 trials for each
stimulus per neuronal site). We generated a bias-corrected
cross-correlation and calculated SEs by using the jackknife resampling
method after using the z-transform on our pseudosamples of
correlation coefficients (Miller, 1974; Hinkley, 1978). The
bias-corrected values were slightly higher, as expected (mean biased,
0.42; mean unbiased, 0.48). Note that this cross-correlation comparison
can still miss information in the spike trains that would be lost by
the averaging procedure and therefore still represents a lower bound of
the goodness of the fit.
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RESULTS |
To begin to understand the hierarchy of acoustical processing in
the avian auditory forebrain and in particular to determine the
stimulus-response function of complex field L auditory neurons when
stimulated with conspecific songs, we compared neural responses obtained using a random tone pip ensemble to those obtained with a
large ensemble of conspecific songs. The conspecific song ensemble was
chosen because we are interested in understanding the processing of
complex natural sounds. The random tone pip ensemble was designed to
have the same overall power density as the song ensemble and similar
temporal structure. Because the tone pips were independent of each
other, any zebra finch song could be generated by a linear combination
of a particular subset of such tone pips. By designing the tone
ensemble in this manner, we were able to compare the stimulus-response
function of neurons obtained from two quite different stimulus
ensembles that nonetheless sampled a similar spectral-temporal region
of auditory space.
Average properties of the stimulus ensembles and
mean responses
The random tone pip ensemble was designed so that the succession
of tone pips would have a similar temporal structure to the succession
of syllables in a zebra finch song. For the ensemble of 20 songs used
in this study, the mean syllable length was 95 msec with a SD of 37 msec. The mean intersyllable interval was 37 msec with a 21 msec SD. We
used these values and assumed a Gaussian distribution to generate tone
pips with similar temporal structure as the songs. The frequency of the
tone pips was also randomly chosen so that the song ensemble and the
tone-pip ensemble would have the same average power spectrum. The
average power density curve for the 20 songs used in this study is
shown in Figure 2.
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Figure 2.
Power spectrum of the ensemble of 20 zebra finch
songs used in this study. The curve shows the mean power density as a
function of frequency. The same power density was used to generate the
ensemble of tone pips.
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Figure 3 illustrates the responses of a
neuronal site in L2b to the two types of stimuli. This neuronal site
responds selectively to certain lower-frequency tone pips and seems to
respond to most of the syllables of the spectrally complex zebra finch
song shown. The mean firing rate of this neuron was much higher for the
song stimuli than for the tone-pip stimuli. This pattern was true for all of our neuronal sites, as shown in Figure
4 (p = 0.0001). The
maximal firing rate was more similar between ensembles than the average
rate, although it was still significantly higher for the song ensemble
(p = 0.03). Both ensembles elicited higher responses from most neurons than continuous white noise, although other broadband
stimuli [such as the spatially modulated stacks or ripples that have
been used by other investigators (Schreiner and Calhoun, 1994; Kowalski
et al., 1996a)] elicited rates that were as high as the song stimuli
(data not shown).
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Figure 3.
Samples of stimuli (top panels),
neuronal responses (middle panels), and mean spike rate
(bottom panels) for the two stimulus ensembles: a song-like
sequence of random tone pips (left column) and a zebra finch
song (right column). The top panels show the
spectrographic representation of the stimuli using the frequency
decomposition described in Materials and Methods. The
middle panels show 10 spike train recordings obtained in
response to 10 presentations of the stimuli for a neuronal site in area
L2b of one of the birds used in the experiment. The bottom
panel shows the estimated spike rate obtained by smoothing the
PSTH with a 6 msec time window.
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Figure 4.
Mean (A) and maximal (B)
firing rates of our Field L neurons relative to background, obtained
using the tone ensemble and the song ensemble.
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Spectral-temporal properties of the stimulus ensembles
As exhibited in Figure 3, the neuronal response in these high
auditory areas is reliably correlated with specific acoustical patterns
in the song or tone pip ensemble. A useful first-order description of
this neural encoding is the STRF, a linear model that relates the
spectral-temporal acoustical patterns of the stimulus to the
probability of spiking. To correlate such spectral-temporal patterns
with spike activity, one needs to transform the one-dimensional sound
pressure waveform into a two-dimensional time-frequency representation
of sound, of which many are possible. As described in detail in
Materials and Methods, Appendix, and Figure 1, we used an invertible
spectrographic representation of sound, that is, one from which the
original sound can be recovered.
If a white noise stimulus is used, the STRF can be estimated by the
reverse-correlation method by averaging the stimulus before each spike:
the spectral-temporal structure that is related to high probabilities
of spiking will add, and the other spectral-temporal patterns will
average to zero. This approach has two problems, however. First of all,
and most significantly, for most neurons in higher level auditory areas
white noise does not elicit many spikes. Secondly, and on a more
technical point, white noise that is completely free of correlations
cannot be achieved for an invertible spectral-temporal representation
such as the one used here (see Materials and Methods).
To overcome these shortcomings, we used an extended version of the
reverse-correlation method, in which the spike-triggered averaged
stimulus is normalized by the correlations present in any particular
stimulus ensemble. This normalization corrects for the fact that for
all stimuli, but much more significantly for natural stimuli such as
zebra finch songs, specific spectral and temporal patterns are
correlated with each other (so-called second-order structure). To
correct for these correlations, one has to effectively perform a
weighted average of the stimulus around each spike, in a mathematical
operation that involves a deconvolution in time and a decorrelation in
frequency of the spike-triggered average stimulus, as explained in
Materials and Methods.
This normalization procedure depends on the specific second-order
structure of the stimulus ensemble. The second-order structure is
described by the autocorrelation function of the stimulus. Because our
stimulus representation is based on the amplitude envelopes of the
decomposition of the sound into 31 frequency bands (our spectrographic
representation of sound as explained in Materials and Methods), the
autocorrelation consists of a matrix of 31 × 31 functions that
show the correlation of the amplitude envelopes in each band with the
amplitude envelopes of all the other bands (see Materials and Methods
and Fig. 1).
The entire autocorrelation matrix in the time domain is shown in the
top panels of Figure 5 for our two
stimulus ensembles and for white noise. To be able to simply equate the
spike-triggered spectrogram to the STRF, the stimulus autocorrelation
matrix in the time domain should consist of delta functions on the
diagonal and null functions everywhere else. In such a case, the
stimulus autocorrelation matrices in the frequency domain
Aw (see Materials and Methods) are all equal to
the unity matrix, and the STRF is therefore exactly equal to the
spike-triggered average spectrogram. As shown in Figure 5, white noise
approaches this ideal, but because our frequency bands overlap, the
correlations spread out slightly to nondiagonal terms. It is because of
this spread that, even when white noise is used, the STRF obtained by
just taking the spike-triggered spectrogram is an approximation.
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Figure 5.
Stimulus autocorrelation matrices (top
panels) and two-dimensional power spectra (bottom
panels) for a white noise ensemble and the random tone-pip and
song ensembles used in this paper. The diagonal corresponds to the
autocorrelation of the log of amplitude envelope of each frequency band
with itself. The off diagonal terms correspond to the cross-correlation
between the log of amplitude envelopes across frequency bands. The
bands are ordered in increasing frequency from left to
right and top to bottom. The top
left corner corresponds to the autocorrelation of the amplitude
envelope in the 250 Hz band. Only the top right side of the
matrix is shown because it is symmetric along the diagonal (with the
time axis inverted for the bottom left entries). The time
window of each autocorrelation function is from  200 to +200 msec as
shown in Figure 1. The ideal white noise signal would have null
functions in the off-diagonal terms and delta functions in the diagonal
terms. The white noise signal approaches this ideal. The random tone
pip ensemble is also closer to white noise than the song ensemble but
still has the spectral and temporal structure that is a result of our
design (see Materials and Methods and Results). The bottom
panels show the two-dimensional power spectra of the stimulus
ensemble. These two-dimensional power spectra are obtained by taking
the square of the two-dimensional Fourier transform of the stimulus in
their spectrographic representation. The two-dimensional power spectra
illustrate the temporal and spectral correlations of the stimulus in
the Fourier domain. The x-axis shows the frequency of
temporal modulations, and the y-axis the frequency of the
spectral modulations found in spectrograms of the different sounds. The
two-dimensional power spectra are symmetric around the origin and
therefore only the top quadrants are shown. The two-dimensional power
spectra can be obtained directly from the autocorrelation matrix of the
stimulus (top row), although the reverse is only true if the
correlations along the spatial (spectral) dimension are
stationary.
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The second-order statistical structure of the two stimulus ensembles
used in this paper are shown in the middle and right panels of Figure
5. There is more similarity between the random tone and white noise
autocorrelation matrices than between the song and white noise
autocorrelation matrices. As is the case for the white noise ensemble,
the off-diagonal terms of the autocorrelation of the random tone
ensemble quickly become small. The random tone autocorrelation is
different, however, in its temporal correlations, as exhibited by a
much wider central peak in the diagonal terms; in other words, the
amplitude envelopes in each frequency band vary slowly in time. In the
Fourier space, these temporal correlations are shown by the fact that
power in amplitude modulations is concentrated at the lower
frequencies, as shown in the middle bottom panel of Figure 5 and in
Figure 6. These temporal correlations are
similar to those found in the song ensemble. This is to be expected
because we designed our tones to have the same average duration and
gap-time as the syllables in our ensemble of zebra finch song. The
random tone pip ensemble also differs from white noise because tones in
each frequency band are presented in isolation. This leads to small
negative correlation values in the off-diagonal terms. Finally, the
song autocorrelation matrix shows both the temporal and spatial (across
frequency bands) correlations that are expected from the acoustical
properties of the zebra finch song. In zebra finch song, the temporal
correlations are similar in all frequency bands and also synchronized
across bands: the off diagonal terms have correlation functions similar
to those of the diagonal terms. The spatial and temporal correlations
can also be seen in the two-dimensional Fourier spectra of the stimulus
ensembles, shown in the bottom panels of Figure 5. Both the tone-pip
stimuli and the song ensemble have most of their power at low temporal
frequencies and low spatial frequencies. We will come back to this
point in the next section.
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Figure 6.
The nature of the second-order structure in the
stimulus ensembles can be assessed by estimating the number of
independent dimensions (or eigenvectors) of the autocorrelation matrix
as a function of the frequency of amplitude modulation, w.
The random-tone pip and the song ensemble have most of their
second-order structure in the low range of amplitude modulations (see
Results for details).
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The normalization procedure leading to the true STRF requires the
division of the spike-triggered average stimulus by the autocorrelation
matrix of the stimulus. This division effectively increases the
contribution of infrequent spectral-temporal patterns in the weighted
spike-triggered average. One problem arises when particular
spectral-temporal patterns are not sampled at all, because the
normalization procedure would require a division by 0. This is the
case, for example, with an ensemble of zebra finch songs: zebra finch
songs span only a restricted region of the stimulus space of all
possible sounds. To deal with this problem, we used an SVD of the
correlation matrix (see Materials and Methods). This method allows one
to assess the true dimensionality of the stimulus space by determining
the regions of the space where the probability of occurrence of
specific spectral-temporal patterns is significantly different from
zero. The subspace that is not sampled is then simply ignored in the
calculation. The level of significance was chosen by defining a noise
tolerance threshold that is expressed as a fraction of the power
density along the stimulus dimension where the probability of
occurrence of spectral-temporal patterns was maximal (see Materials and
Methods). The regions of space where the probability of occurrence was
below this level of significance were then ignored. We chose the
tolerance threshold that gave the STRF that best predicted neural
responses to novel stimuli. We found the mean optimal tolerance factor
to be 0.001 (range, 0.00005-0.005) for the song ensemble and 0.003 (range, 0.0001-0.005) for the tone ensemble.
The nature of the stimulus subspace being sampled significantly can be
examined by looking at the space covered by the independent dimensions
that are needed to describe that subspace. When the correlations have
stationary statistics, this subspace is shown in the two-dimensional
Fourier spectra of the stimulus, as illustrated in the bottom panels of
Figure 5. In our case, the temporal correlations were stationary but we
did not assume stationarity in our spatial correlations. Nonetheless,
the bottom panels of Figure 5 illustrate approximately the spectral and
temporal patterns that are sampled by our two stimulus ensembles, as
well as white noise. In addition, we show in Figure 6 the number of
dimensions needed to represent the stimulus subspace as a function of
the temporal frequencies (describing the power of the temporal
modulations in the amplitude envelopes in each of the frequency bands).
The number of dimensions depends both on the power at a particular
amplitude modulation frequency and on how correlated the amplitude
modulations in one frequency band are with those in a different
frequency band. At the tolerance values used here, the autocorrelation
matrix of a white noise ensemble obtained with the same number of data
points would have the maximum number of 31 dimensions for all
frequencies. The bottom panels of Figure 5 and Figure 6 show that both
of our stimulus ensembles sample mostly the low-frequency end of the temporal spectrum. In addition, in zebra finch song, the presence of
energy in one frequency band is correlated with the presence of energy
in many other frequency bands, yielding high spatial correlations. In
the tone-pip ensemble, the presence of energy is one frequency band is
positively correlated with energy in the neighboring bands and
negatively correlated with energy in frequency bands further away.
Because the spatial correlations are stronger in the song ensemble than
in the tone pip-ensemble, fewer dimensions are needed for the song
ensemble (Fig. 6). Also the large spatial extent of the correlations in
the song ensemble implies that only the very low spatial frequencies
are being sampled, as shown in the bottom right panel of Figure 5. The
tone pip ensemble also samples a restricted region of the lower spatial
frequencies, although a somewhat larger one than the song ensemble.
Therefore, our two stimulus ensembles sample a limited region of the
spectral-temporal space that could be occupied by any sound, as
illustrated by their spectra shown in the bottom panels of Figure 5. As
explained in Materials and Methods in mathematical detail, by sampling
such a limited stimulus, we are obtaining an estimation of the STRF
that would be valid only for stimuli that sample identical stimulus
space, unless the neural response to sounds outside the stimulus space
being sampled is nonsignificant. Although we did not investigate all
possible sounds, we found that for most neurons, the tone pip and the
song ensemble elicited relatively strong responses. Much smaller
responses were found, in many cases, with sustained white noise.
Therefore, by limiting the stimulus space as we did, we suggest that we
are effectively performing a better sampling of the acoustical patterns
that are relevant to the neurons. Stimuli that include higher spatial
and temporal frequencies (such as white noise) elicit much smaller neuronal responses because, in white noise, the specific sound patterns
that excite the neuron occur at much smaller power levels, as they are
embedded in other sounds. In addition, the neurons could exhibit
nonlinear stimulus-response properties that could further depress or
eliminate their responses to white noise.
Another major goal of this work was to compare for each neuron the STRF
obtained with songs to the one obtained with simpler synthetic stimuli,
similar to stimuli used in previous estimations of auditory STRFs using
reverse correlation methods (deCharms et al., 1998). For the comparison
to be valid, the space of sounds sampled by the synthetic stimulus
ensemble should at least encompass all sounds present in the song. In
theory, one might have chosen white noise, because it samples all
acoustical space. However, white noise turned out to be a poor stimulus
because it is unable to elicit sustained firing rates in most of the
neurons. We therefore used a synthetic ensemble of tone pips that
sampled a spectral-temporal space that was, by design, more restricted
than white noise and similar to the one of zebra finch songs. The tone
pips in our experiments had significant power in a similar (slightly
larger) region of temporal and spatial frequencies. In fact, we
designed our tone-pip ensemble so that any song could be well
approximated by a linear combination of such trains of tone pips,
although the reverse will not necessarily be true. If the neurons were linear, the STRF found with tone pips would then either be identical to
the one found with the song ensemble or would potentially include spectral-temporal patterns that are not found in the song ensemble. Moreover, in linear neurons, the tone pip-generated STRF should predict
the response to song as well as the song-generated STRF. We will show
that neither of these theoretical predictions were true in our data,
suggesting that our neurons are nonlinear and that higher order
properties of sound besides their average spectral-temporal spectra or
higher order characterizations of neural responses (such as adaptation)
need to be considered to fully understand the neural response.
Calculation of the STRF
To demonstrate that our numerical approach can generate STRFs that
have been properly corrected for the correlations present in the
stimulus ensemble, we generated artificial neuronal data based on an
STRF obtained from a simple neuron in L2a with a tone pip ensemble. The
time-varying firing rate of a model neuron was then obtained by
convolving the STRF with the spectrographic representation of each
stimulus in our two ensembles. The firing rate was scaled to obtain an
average of 10 spikes/sec which approximately matches the average firing
data of the actual neurons studied here. To generate trials of neuronal
data from our linear model neuron, we assumed Poisson firing statistics
with time-varying firing rate. We generated 10 trials of artificial
neuronal data for all the stimuli in our tone ensemble and in our song
ensemble. From these artificial data, we calculated the STRFs with our
numerical methods, in exactly the same way that we did with our real
neuronal data. Figure 7 shows how the
spike-triggered averages obtained from our two ensembles are radically
different but that, once normalized by the stimulus autocorrelation,
the estimated STRF obtained from the song and tone ensemble are in fact
very similar. Moreover, these STRF are almost identical to the one that
was used to generate the data. This simulation demonstrates that our method can compensate for the different correlations found in our two
ensembles. Note, however, that if we had used a STRF that exhibited
sensitivity to spectral temporal patterns that we had not sampled in
our stimulus ensemble (such as high-frequency amplitude modulations),
we would not have been able to recover the original STRF.
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Figure 7.
Validation of the STRF calculation
and illustration of the normalization procedure. The "Original
STRF" shown in the bottom left panel of the figure was
used to generate artificial neural response data for the song ensemble
and tone ensemble. The model neuron was linear and had Poisson firing
statistics with a similar average firing rate to the actual neurons in
our data set. These artificial data were then used to obtain estimates
of the STRF with our methodology. The top row shows the
spike-triggered average spectrograms that are obtained from these
artificial data by averaging the stimulus spectrogram around each spike
in a 400 msec window. These spike-triggered average spectrograms are
only equal to the STRF of the neuron if a white noise stimulus (in the
time/frequency representation of choice) is used. When correlations are
present in the stimulus, either in the time dimension or across the
frequency bands of the spectrogram, the spike-triggered average needs
to be normalized. This normalization involves a deconvolution in time
and decorrelation in frequency. The STRFs obtained from the
spike-triggered averages by the normalization procedure are shown on
the bottom right panels. As expected, a similar STRF
estimate is obtained for both ensembles and these estimates are very
close to the Original STRF that was used to generate the data.
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An example of the results of the STRF calculation on real data is
illustrated in Figure 8 for the neuron of
Figure 3. The spike-triggered spectrograms are shown on the left panels
of Figure 8A, and the normalized STRFs are shown on the
right panels for the two ensembles. As in the artificial data, the
spike-triggered average spectrograms for the two ensembles differ
significantly. In the case of actual data, however, it is unclear
whether the differences are attributable simply to the additional
spectral and temporal correlations that are found in the song ensemble or whether the differences reflect a different linear
stimulus-response function discovered by the use of a natural sound
ensemble. However, after we normalized the spike-triggered average
spectrograms by the autocorrelations of the stimulus ensembles, we
obtained the properly derived STRFs. The STRFs obtained for the two
ensembles are, in the case of this particular neuronal site, much more
similar than the spike-triggered averages. In both cases, the main
excitatory region is around 1.5 kHz ~15 msec after the onset of the
sound. Thus, most of the differences in the spike-triggered averages were attributable to stimulus properties and not to the neural responses. This example demonstrates that for presumably simpler, more
linear neurons, the linear stimulus-response function found with the
simple tone pip ensemble approximately matches the stimulus-response function found when complex natural sounds are used.
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Figure 8.
STRF calculation for a real neuron. The figure
illustrates the STRF calculation explained in Results and in
Figure 7 for the neuronal site of Figure 3. The calculation is based on
all the responses that were obtained for the song ensemble and tone
ensemble (10 trials for each of the 21 songs and 20 random tone
sequences). As shown in A, for this particular neuron, the
STRFs obtained from both ensembles are similar in that they exhibit
similar areas of excitation. On the other hand, the spike-triggered
average spectrograms were remarkably dissimilar. Most of the
differences in the spike-triggered average were therefore attributable
only to the statistical properties of the stimulus and not to the
stimulus-response properties of this neuronal site. B shows
examples of three song-based STRFs for this same neuron obtained with
different noise tolerance levels, as explained in Materials and
Methods. The time axis (x-axis) has been expanded
from A. The normalized coherence between predicted response
and actual response as a function of the tolerance level is plotted in
the rightmost panel. The best predictions were obtained with
a tolerance value of 0.0005.
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The tolerance level chosen for the normalization by the stimulus
autocorrelation can have a drastic effect on the STRF. Figure 8B illustrates the effect of the tolerance level on the
estimation of the STRF for the song ensemble for the example neuron of
Figures 8A and 3. The bottom right panel of Figure
8B shows the integrated coherence, a measure that quantifies
the quality of the STRF model by measuring its ability to predict
responses to novel stimuli (see Materials and Methods). The coherence
is maximal for a particular tolerance value, in this case 0.0005. This
tolerance value also yields the STRF that is by visual inspection the
closest to the one that is obtained with the tone-pip ensemble.
The STRF for song ensembles and the tone ensemble
We used the same procedure exemplified for the neuron of Figures 3
and 8 for all the neural sites that we recorded, obtaining a song-based
STRF and a tone pip-based STRF at each site. Figure 9A illustrates three examples
of STRFs in which the result obtained from the tone pip ensemble was
markedly different from the one obtained from the song ensemble. These
particular neuronal sites also had more complex STRFs than the one in
Figure 8 and are more characteristic of our data set.
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Figure 9.
STRFs derived using the random tone pip and song
ensemble for three neuronal sites (N1-N3) with complex stimulus
response properties (A) and decomposition of an STRF onto
its spectral axis and temporal axis (B). In A,
each row corresponds to the two STRFs that were obtained for a
particular site. The left column corresponds to the STRF
calculated from the random tone pip ensemble, and the right
column corresponds to the STRF calculated from the song ensemble.
These three examples were chosen to illustrate cases in which the STRF
obtained from the random tone pip ensemble was different from the one
obtained from the song ensemble. These particular sites were in L1 (N1
and N3) and L3 (N2). The STRFs differ both in amplitude and in shape.
The top neuronal site can be described as being sensitive to
a moving spectral edge. Similar STRFs have been found in some cortical
neurons (deCharms et al., 1998). The bottom neuronal site is
sensitive to a temporal combination of a low-frequency sound followed
by a high-frequency sound. Both of these complex spectral-temporal
responses became evident only when the song ensemble was used. In
B, the STRF of the neuron of Figure 8 is projected onto its
spectral axis (top right) and temporal axis (bottom
left). Such analyses can be used to extract the spectral and
amplitude modulation tuning of the neurons as explained in Results. The
solid line is the projection corresponding to the largest
excitatory point, and the dotted line is the projection
corresponding to the largest inhibitory point.
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From the STRF one can also extract the more traditional
characterizations of auditory processing such as spectral tuning and amplitude modulation rate tuning. The spectral tuning can be found by
projecting the STRF along the frequency axis for the time values that
show maximal and minimal firing probabilities (Fig. 9B,
solid and dashed lines, respectively). The
modulation rate tuning can be obtained by projecting the STRF onto the
time axis for the best excitatory frequency and measuring its power as
a function of temporal modulation frequencies. These procedures are
illustrated in Figure 9B for the neuronal site of Figures 3
and 8. For the entire data set, we found that the peak excitatory
spectral tuning ranged from 375 to 5125 Hz with tuning width factors (Q
values at half maximum) ranging from 0.4 to 4.1. These values are
similar to those found in avian forebrain by other groups (Zaretsky and Konishi, 1976; Muller and Leppelsack, 1985; Heil and Scheich, 1991).
The peaks of the amplitude modulation transfer function ranged from 10 to 40 Hz. Amplitude modulation transfer functions have not been
examined in detail in the avian forebrain (Heil and Scheich, 1991), but
these values are similar to those found in mammalian auditory cortex
(Schreiner and Urbas, 1988). In general, however, the STRF can give
more information than just the spectral and amplitude modulation
tuning, because it shows the specific spectral-temporal pattern that
optimally excites the neuron. Such patterns can only be described in
terms of their spectral and modulation rate tuning if the frequency and
time dimensions are independent. In contrast, the STRF of a neuron can
also show, for example, whether the neuron responds well to frequency
sweeps or to other more complex spectral-temporal patterns. The
neuronal sites shown on the top (N1) and bottom of Figure 9A
(N3) are examples of more complex STRFs. A detailed analysis of the
functionality that can be derived from the STRFs and the mapping of
these functions throughout the avian auditory forebrain will be
presented in a future publication.
Prediction of neural responses from the STRF
An additional advantage of describing the neural encoding with an
STRF, as opposed to a more classical description based on tuning
curves, is that the STRF can be used directly to predict the response
of the neuron to any dynamical stimuli. This is a very important
feature, because it allows validation of the description of the
stimulus-response properties of the neurons. In this work, we focused
our analysis on quantifying the goodness of fit of the predictions
obtained from the STRF model to the actual data. We also analyzed the
significance of differences that we found in these fits using two
different stimulus ensembles to derive the STRFs.
If a neuron encodes the stimulus parameters of our choice (which can
include a nonlinear transformation) in a linear fashion, then the STRF
model will be able to completely predict the deterministic part of the
response to any stimulus. Discrepancies between the predicted and the
actual response would then be attributed to random neuronal noise. A
problem arises immediately, however, when one finds very different
STRFs for the same neuron with two different stimulus ensembles, as was
the case for most of our data. In such cases, one finds two models to
characterize the neural encoding, and will obviously make two different
response predictions. Are the differences between these models
significant in the sense that the goodness of their prediction is
significantly different? Both to answer this question and to test the
validity of the underlying linear assumption, we used the STRFs to
obtain a prediction of the firing rate of each neuron to test stimuli taken from the same ensemble. This firing rate prediction was obtained
by convolving the STRF with the test stimulus in its spectrographic
representation. To prevent over-fitting, the particular stimulus-response data being tested were always excluded from the data
used for estimation of the STRF (see Materials and Methods). The
prediction could then be compared directly to the actual data, and the
validity of the linear model could be assessed.
We quantified the quality of the prediction by calculating an unbiased
correlation coefficient between predicted and actual spike rate (see
Materials and Methods). The correlation coefficients from our entire
data set are shown in Figure 10,
D and E. When the STRF model was used to predict
the response to a new stimulus that is nonetheless from the same type
of ensemble used to derive the STRF (i.e., songs or tones), we found
average correlations of 0.51 (range, 0.08-0.71) for the random tones
and 0.45 (range, 0.12-0.66) for the song ensemble (Fig.
10E, "CC-matched"). These two distributions of
correlation coefficients were very similar (p = 0.07,
Wilcoxon signed rank test). In both cases, there was a wide range of
fits showing that the linear model is a good approximation for some
neurons and a much poorer approximation for others. However, both the
response to tone pips and songs could be modeled with similar
effectiveness as long as the STRF used in the fitting was obtained with
the same type of stimulus ensemble.
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Figure 10.
Comparison of the predictions obtained from the
two STRFs calculated for each neuronal site. A shows the
actual firing rate of a particular neuronal site in response to random
tone-pip stimuli (left) and to a song (right).
B shows the firing rate predicted using the STRF calculated
with the corresponding stimulus ensemble. C shows the
predicted response calculated after switching the STRFs. This example
illustrates that the STRFs obtained from the different ensembles can
give radically different results. D, Scatter plot of the
correlation coefficients (CC) between the predicted and the
estimated firing rates, for switched STRFs versus matched STRFs. For
each neuronal site, the CCs are calculated for the songs and tone pips.
The solid dots show the CCs for the prediction to song
stimuli, obtained either with song-STRFs (CC matched on the
x-axis) or with the tone STRFs (CC switched on the
y-axis). The open dots show the predictions for
tone stimuli with the matched and switched filters. If the STRFs
generated with the two different stimulus ensembles were identical, the
points would lie on the x = y line shown in dots. Most
points are below the line, although some are close to the
line and some are closer to 0, showing the range of differences.
E, The data in D are replotted by projecting all
the points onto the x-axis (top plot) and onto
y-axis (bottom plot), thus showing the individual
distributions of CC-matched and CC-switched for the predictions to song
and tone-pip stimuli. The distributions are obtained by convolving the
raw data with a smoothing kernel of 0.05. The distribution of
CC-matched for the two types of stimuli are not significantly
different, but the CC-switched for the response to songs predicted
using the tone STRFs are significantly smaller than the CC-switched for
tone pips. In addition, all the CC-switched are smaller than or equal
to the CC-matched.
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The picture changed radically when the STRFs were switched, so that the
predicted response to a stimulus from one type of ensemble was
generated with the STRF obtained using the other ensemble. Figure
10A-C contrast the actual data with the predicted responses
obtained with the matched STRF and with the predicted responses
obtained with the switched STRF. When the STRFs were switched, the
prediction was a significantly poorer fit of the original stimulus
response. The decrease in the quality of the prediction for all our
data is shown in panels D and E, where the cross-correlation
coefficients for the switched case are compared to those found in the
matched case. The average switched correlation was 0.28 (range,
0.03-0.55) for the tone ensemble responses predicted using the song
STRF and 0.19 (range, 0.25-0.50) for the song ensemble responses
predicted using the tone STRF. These two distributions are not only
significantly different from the corresponding distributions in the
matched case (p = 0.0001), but also significantly different from
each other (p = 0.0003). The prediction of the response to the
song ensemble using the STRFs obtained with tone pips resulted in a
larger deterioration of the fit than did the prediction of tone
responses with the song-derived STRFs (Fig. 10E, "CC
switched"). Based on our results with these two ensembles, we
conclude that the STRFs obtained from the song ensemble are overall
better models of the stimulus-response function of these neurons.
The error in our estimate of the quality of the prediction (SE of CCs
for a particular neuron) is attributable in part to noise in the
estimation of the STRFs from a limited sample set. The noise in the
STRFs will also lead to a biased estimation of the CC (lower values
than actual). Although we corrected the bias in the CCs for the noise
in the PSTH, we did not correct for the bias caused by the noise in the
STRFs. An estimate of the magnitude of this bias can be obtained from
the SE in the CCs. The SE was obtained with the jackknife resampling
technique and is shown in the error bars of Figure 10D for
each data point. By comparing the error in the CCs obtained with song
STRFs to those obtained with tone STRFs, we can show that the
difference in the quality of the predictions of song responses by song-
and tone-derived STRFs is not attributable simply to differences in the
noise-generated bias in the CCs (as might have been true, for instance,
if all tone responses were less vigorous than song responses, leading to poorer signal-to-noise ratios of tone STRFs). The SE in the CCs was
similar for all predictions obtained with song and tone STRFs (mean CC
SE of song STRF on song = 0.041 vs mean CC SE of tone STRF on
song = 0.037; mean CC SE of tone STRF on tone = 0.06 vs mean
CC SE of song STRF on tone = 0.053). Therefore, we conclude that
the small differences of signal-to-noise ratio of the tone and
song-derived STRFs cannot explain the significant difference in song
and tone prediction seen in our results.
In all the data presented so far, we calculated the STRFs from the log
of the amplitude envelopes of our spectrographic decomposition of
sound. When the same calculations were done without using the log
transformation, the results were similar for the tone ensemble (mean
CC = 0.52) but much worse with the song ensemble (mean CC = 0.26). By including this additional preprocessing step of using the log
transform in our methodology, we illustrate how static nonlinearities
can easily be included in the calculation of STRFs. This also
demonstrates that nonlinearities in the neural encoding are different
for different stimulus ensembles. In this particular case we found that
the relationship between sound intensity and probability of firing was
more linear for the tone-pip stimuli than for the song stimuli.
Although we did not systematically investigate all possible nonlinear
transformations, we found that the log relationship worked well for the
natural ensemble.
| |
DISCUSSION |
Complex sensory cortical neurons are well known to exhibit many
nonlinearities, and these nonlinear effects become even more evident to
the experimenter when complex stimuli with natural statistics are used,
both in the auditory (Schwarz and Tomlinson, 1990; Calhoun and
Schreiner, 1998) and in the visual modalities (Gallant et al., 1998).
Because of this effect, the response to complex natural sounds
including vocalizations often cannot be explained from neural responses
to simpler sound stimuli (Rauschecker et al., 1995). Moreover, sensory
systems may be optimized for the encoding of natural stimuli (Rieke et
al., 1995; Dan et al., 1996). One would like to be able to
systematically study how such complex nonlinear neurons encode stimuli
and how they mediate natural behaviors. For example, for very nonlinear
neurons, referred to as combination-sensitive neurons, a systematic
decomposition of the ethologically relevant sound into small parts has
been revealed to be very useful (Suga, 1990; Margoliash and Fortune, 1992). However, for less specialized auditory neurons, such as the
majority of those found in auditory cortex of nonspecialized mammals or
in the avian forebrain homolog of auditory cortex, this approach will
fail. Such auditory neurons respond to a very large ensemble of sounds,
and their stimulus-response function cannot be explored systematically
by attempting to relate their firing rate to identified stretches of
sounds, such as the syllables in an animal's vocalizations. To be able
to extract the stimulus-response encoding function of these
nonspecialized complex auditory neurons, the concept of an STRF has
much to offer. First, the estimation of the STRF involves finding the
spectral-temporal sound patterns that excite the neuron in a systematic
manner. Second, the STRF is a model of the stimulus-response function
of the neuron that hypothetically could be used to model the response
to any stimulus. Finally, the STRF description encompasses classical
characterizations of higher auditory areas such as spectral tuning and
amplitude modulation tuning.
Calculating STRFs for complex neurons is a problem, however, because
the STRF is a linear model and the complex neurons of interest exhibit
significant nonlinearities. When the STRFs are calculated with broad
band stimuli, the spectral-temporal features that would elicit large
responses from complex neurons do not occur in isolation, but instead
occur with small power and are embedded in large amounts of what the
neurons evaluate as extraneous sound. For nonlinear neurons, such
broad-band stimuli yield no responses or responses that are much
smaller than expected from the neuronal encoding model based on STRFs
obtained with narrow band stimuli. Moreover, the simple reverse
correlation method that has been used so far to estimate STRFs relies
on the use of such random broadband stimuli. For these reasons and
because STRFs derived from such stimulus ensembles are poor predictors of neuronal responses to natural sounds, it has been suggested that the
concept of a linear STRF is of limited use in understanding the
auditory system (Aertsen and Johannesma, 1981b; Eggermont et al.,
1983a; Nelken et al., 1997). On the other hand, the approach has had
much greater success in explaining the responses of neurons in A1 when
the STRF is obtained from complex synthetic stimuli with limited
spectral and temporal modulation bandwidth and/or with methods that do
not rely on the reverse correlation (Kowalski et al., 1996b; Escabi et
al., 1998; Shamma et al., 1998; Versnel and Shamma, 1998). Our study
with natural sounds supports these studies and extends the use of the
STRF to the analysis of the neural response of complex auditory neurons.
In our analysis, we extended the reverse-correlation method so that we
could derive an STRF from any stimulus ensemble. This extension
involves a normalization of the spike-triggered average stimulus by the
correlations (or second order statistics) of the spectral-temporal
patterns of the stimulus ensemble. By doing so, we were able to
estimate STRFs for ensembles that had limited spectral and temporal
modulation bandwidth and specific spectral-temporal correlations. This
was particularly useful because we found that auditory neurons found in
the field L region of the zebra finch responded strongly to the sounds
of conspecific song. Our analysis involves describing the second-order
statistics of relevant stimulus ensembles and estimating for each a
"stimulus ensemble-dependent" STRF that is nonetheless correctly
normalized for the stimulus structure. We could then compare the STRFs
obtained with different stimulus ensembles to obtain a first-order
description of the stimulus-response function of our neurons.
We found that zebra finch songs have limited spectral and temporal
modulation bandwidths and very high correlations in the amplitude
envelopes across frequency bands. For a stimulus ensemble consisting of
these songs, we showed that the estimated STRFs yielded spectral and
amplitude modulation tuning that were in accord with previous work
using simple synthetic sounds. We also quantified the goodness of the
linear assumption underlying the STRF model by using the STRF to
predict the neural response to novel stimuli and comparing that to the
actual response. We found that the quality of the linear approximation
of the response characteristics of neurons in L1, L2, and L3 varied
over a wide range, working well for certain neurons but not for other,
presumably more nonlinear, neurons. In the few other experiments in
which a direct estimation of the validity of the classic STRF model has
been made, the quality of the model (measured, as we did, by the mean
and range of the correlation coefficients between predicted and actual
response) was similar to what we found here in the "matched" case,
where the STRFs used to predict the response were obtained from stimuli of the same type as the test stimuli (Eggermont et al., 1983b; Kowalski
et al., 1996b). In this respect our neurons are as linear (or
nonlinear) as other auditory neurons both in the periphery (Eggermont
et al., 1983b) and in primary auditory cortex (Kowalski et al.,
1996b).
We then tested the generality of the STRF model and further examined
the linearity of our neurons by also calculating a second STRF for each
neuron, using an ensemble of tone pips that had similar spectral and
temporal modulation bandwidth. The tone-pip stimulus ensemble was
designed so that it would include all sounds in the zebra finch song
but with clearly different statistical structure. This stimulus
ensemble also elicited relatively high firing rates from field L
neurons, although it yielded much smaller mean rates than the song
ensemble. By comparing the results obtained from the two STRFs, we
showed that many neurons in the avian auditory forebrain respond in a
nonlinear fashion to the particular spectral and temporal statistics of
species-specific vocalizations, because the STRF obtained from the song
ensemble was significantly different from the STRF obtained from
simpler tone pips. Moreover, when we tested how well the linear model
would generalize by using the STRF calculated from one ensemble to
predict the responses to stimuli from a different ensemble, we found
that the quality of the prediction decreased significantly. In
particular, the STRF obtained from tone pips is a poor model for the
responses found to natural sounds. This represents another way in which to quantify the non-linearity of the neurons. Although this effect had
been observed in a qualitative manner (Eggermont et al., 1983b; Nelken
et al., 1997), it had not been quantified, because STRFs from natural
sounds or images have not been estimated previously.
Despite this nonlinearity, we found that by calculating the STRFs from
natural sounds, we could predict the response to new songs as well as
the tone-derived STRFs could predict the response to simple stimuli. In
fact, the prediction to the natural sounds with the song STRFs for the
forebrain auditory neurons studied here is as good as the prediction
that has been found in visual thalamic neurons (which are presumably
more linear) to natural scenes, obtained with a classic STRF (Dan et
al., 1996). These results show that the limitations of the STRF model
are not only caused by its underlying linear assumption, but also by
the fact that STRFs have previously been estimated with nonoptimal
stimulus ensembles. Finally, the fact that song STRFs predicted tone
responses better than the tone-pip STRFs predicted song responses
raises the possibility that, in general, a better fit for the
stimulus-response functions of neurons will be obtained with a
stimulus ensemble with richer spectral-temporal correlations such as
those found in natural sounds. Figure
11 exemplifies how a particular
nonlinear stimulus response function could lead to all the effects that we have described in our discussion about the neurons in our data set.
View larger version (26K):
[in this window]
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|
Figure 11.
For illustrative purposes, we show the
distribution of stimulus response points for a hypothetical nonlinear
neuron that exhibits the properties of the neurons in our data set. The
multidimensional space (31 frequency bands × 400 points in time
or 1600 dimensions in our case) used to represent the stimulus is
collapsed onto the one-dimensional x-axis, and neural
response is shown on the y-axis. White noise stimuli sample
the entire space, whereas tone pips and songs only sample a region of
that space. The hypothetical neuron has poor responses to white noise
but significant responses to tone pips and songs, with songs being the
preferred stimuli. A linear fit for each stimulus subset shows that
relatively good stimulus-response predictions can be found, although
the fit obtained from one ensemble is not a good model for the
stimulus-response function found in a different ensemble. In
particular, the fit to the white noise data is a very poor predictor.
The fit to the song ensemble is better at predicting the response to
tones than vice versa. The data in our ensemble exhibit similar
patterns (albeit with on average much smaller correlations) in
multidimensional space where the line is replaced by a hyperplane. An
STRF is the set of coefficients that define such a hyperplane.
|
|
Our methodology and results also suggest ways in which the STRF
framework could be further extended. We have shown that by calculating
STRFs from multiple stimulus subspaces one can greatly improve the fit
of the actual stimulus-response function. One might therefore
model the nonlinear stimulus-response function by effectively doing a
piece-wise linear fit in multiple subspaces of stimuli. Such a
description of the stimulus-response function of a neuron would go
hand in hand with the description of the relevant statistics (in this
case second order correlations) of the different stimulus ensembles.
This is crucial, because the experimenter is now only sampling a
limited subspace of all possible stimuli. For completeness, one would
have to ensure that either the sampling of sound stimuli is based on
ethological distributions and/or on a broad investigation of all sounds
capable of eliciting significant responses. Finally, we have also shown
how to improve on this first-order model of the STRF by including
particular static nonlinearities in the calculation (see Results). The
nonlinearities can also be studied by examining the systematic
deviations of the data from the predicted response obtained with the
STRF model (Shamma et al., 1998). Our methodology combined with such
potential extensions could allow for a better characterization of
high-level sensory processing: nonlinear effects are clearly important,
but at the same time the linear model can explain a large fraction of
the response and will undoubtedly be part of a more complete model. Our
approach is general and could also be applied at various levels of the
visual system to validate the effectiveness of the classical STRF and
to further understand the response of complex visual cortical neurons
to natural images.
| |
FOOTNOTES |
Received July 8, 1999; revised Dec. 17, 1999; accepted Dec. 17, 1999.
This work was supported by research grants from the Alfred P. Sloan
Foundation (to F.E.T., K.S., and A.J.D.) and the National Institutes of
Health (NS 34385 to A.J.D. and MH 11209 and MH 59189 to F.E.T.). We
thank Steve Lisberger for critical reading of an earlier version of
this manuscript.
Correspondence should be addressed to Frédéric E. Theunissen, University of California, Berkeley, Department of
Psychology, 3210 Tolman, Berkeley, CA 94720-1650. E-mail:
fet{at}socrates.berkeley.edu.
| |
APPENDIX |
Invertible spectral-temporal representation signals
All of the possible time-frequency representations can be
derived from the Wigner distribution of a signal, which, in a
mathematical sense, can be thought of as the canonical distribution
(Cohen, 1995, their Chapter 9). The Wigner distribution satisfies the requirements of probability theory that ensure that it has the properties of a generalized joint-distribution function (Cohen, 1995,
their Chapter 8). Moreover the signal can be recovered from the Wigner
distribution up to a constant phase factor because the Wigner
distribution is unique and invertible (Cohen, 1995, pp 127, 128). The
Wigner distribution of a signal, s(t), is given by:
|
(1)
|
All other time-frequency representations of s(t),
G(w, t) can be written as a two-dimensional filtered version of
the W(t, w):
|
(2)
|
where g(t, w) is the two-dimensional filter. In
practice, Equation 2 is written in the Fourier domain, where the
convolution can be replaced by a product. In that formulation, the
two-dimensional Fourier transform of g(t, w) is called the
kernel of G(t, w) and is written as  ( ,  ):
|
(3)
|
Using Equations 1 and 3, Equation 2 can then be rewritten
as (Cohen, 1995, their Chapter 9):
|
(4)
|
The spectrographic class of time-frequency representations is
given by:
where k(t) is the time window used in the short-time
Fourier transform. It can be shown that the kernel of a spectrographic representation is given by the Wigner distribution of k(t)
(Cohen, 1995, p 141). A spectrographic representation is therefore
invertible (except for the fixed phase) if the Wigner distribution of
k(t) is not equal to zero anywhere where the Fourier
transform of the Wigner distribution of the signal is nonzero.
Therefore, to be generally invertible, a spectrographic representation
must have k(t) with a nonzero Wigner distribution everywhere
(for a mathematically equivalent proof, see also Cohen, 1995, p 108).
The Gaussian window used in our work satisfies this constraint: writing
k(t) as:
one finds that W(t, w)k is equal to
The remaining issue deals with the transformation from the
continuous theoretical framework to the discrete representation that is
used in practice. Samples in time and frequency must be taken at
intervals that are small enough to capture all significant spatial
frequencies, where space is the time-frequency grid. The amplitude
envelopes obtained in a spectrographic representation of sound are
effectively band-limited: the upper frequency is given by the bandwidth
of the time window used to calculate the spectrogram (Flanagan, 1980;
Theunissen and Doupe, 1998). A fine frequency grid must also be used to
ensure that all frequencies are represented. We used these properties
to define our sampling rate in time and the overlap of our frequency
bands in frequency. We have shown empirically in our previous work that
the entire signal (except for an absolute fixed phase) can effectively
be recovered from our specific decomposition (Theunissen and Doupe, 1998).
Generalized STRF
Because there are many time-frequency representations, there
will be correspondingly many possible definitions of the STRF. For
theoretical reasons, an STRF based on the Wigner distribution has many
appeals. First of all, it is equivalent to the second-order Volterra
term in the functional expansion between the sound pressure waveform
and the neural response, and in this sense it is the natural extension
of the first-order expansion that is used for phase-locked units.
Second, it is based on a unique and invertible time-frequency
representation. For these reasons, the STRF in auditory research was
originally defined based on this mathematically appealing definition
(Aertsen et al., 1981; Kim and Young, 1994). However, the STRF based on
the Wigner distribution, called here invariant STRF or
STRFI, has a distributed representation that is
often difficult to interpret without some sort of smoothing procedure
(Kim and Young, 1994). Recently researchers have, therefore, adopted a
more general definition of the STRF that is based on other
time-frequency representations, called here spectrographic STRF or
STRFS, effectively performing the desired smoothing
operation. Indeed, because other representations are in fact filtered
versions of the Wigner distribution, it can be shown that any
STRFS can be obtained from STRFI by a
two-dimensional filtering (or smoothing operation) (Klein et al.,
2000). The two-dimensional filter is the inverse of the two-dimensional
filter that is used to transform the Wigner distribution into the
general time-frequency representation. The proof goes as follows: The
response r(t) is found by convolving the STRF with the
time-frequency representation of the signal s(t):
|
(5)
|
in terms of G(t, w), or:
|
(6)
|
in terms of W(t, w).
Replacing Equation 2 in Equation 5,
|
(7)
|
Therefore,
or
|
(8)
|
Therefore, STRFS will only be theoretically equivalent
to STRFI if one is able to invert the filtering operation
shown in Equation 8. Just as for the transformation between
G and W, this operation will be possible only if
the kernel of G is nonzero at all Fourier values where
STRFI has significant power. Because STRFI is a
priori not known, a rigorous implementation would require a
time-frequency distribution with nonzero kernel. In this manner, one
would be able not only to find STRFI, but also to
compare the STRFs found by different research groups.
| |
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TOP
ABSTRACT
INTRODUCTION
