INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Only a few general organizational features are known in primary auditory cortex (AI). They include a spatially ordered tonotopic axis (Evans et al. 1965), bands of alternating binaural response properties (Imig and Adrian 1977; Middlebrooks et al. 1980), and a variety of other response features that change systematically along the isofrequency planes such as thresholds (Heil et al. 1994; Schreiner et al. 1992), bandwidths (Schreiner and Sutter 1992), FM selectivity (Heil et al. 1992; Mendelson et al. 1993; Shamma et al. 1993), and asymmetry of response areas (RAs; the span of frequencies that influence, both through excitation and inhibition, the response of a cell) (Shamma et al. 1993). To derive a functionally coherent picture of these maps, it is necessary to integrate these features within a comprehensive descriptor of the unit responses; one that can be quantitatively derived and employed to predict responses to novel stimuli.

Traditionally measured response areas are inadequate because they rarely include response dynamics and cannot be used to predict responses quantitatively. An alternative is the response field (RF) (Schreiner and Calhoun 1994; Shamma et al. 1995), a static, purely spectral function analogous to the RA except for the use of broadband sounds (but see Nelken et al. 1994; Sutter et al. 1996). A dynamic generalization of the RF is the spectro-temporal response field (STRF), a characteristic function of a neuron obtained using broadband sounds (Aertsen and Johannesma 1981; deCharms et al. 1998; Eggermont 1993 and references therein; Escabi and Schreiner 1999; Kowalski et al. 1996a; Kvale and Schreiner 1995; Theunissen et al. 2000). A schematic of an idealized STRF is illustrated in Fig. 1. Qualitatively, its spectral axis reflects the range of frequencies that influence the response or firing rate of the neuron being characterized, and its temporal axis reflects how this influence changes as a function of time. Positive-valued regions of the STRF describe excitatory influence, and negative regions describe inhibitory influence. The interplay between the spectral and temporal axes can give multiple interpretations to the STRF, e.g., as a time-evolving spectral response field or a family of impulse responses labeled by frequency band.

View larger version (71K): [in this window] [in a new window]   Fig. 1. An idealized spectro-temporal response field (STRF) with spectral and temporal 1-dimensional sections. The time axis is convolved with the time axis of the spectral envelope of a stimulus (as in Fig. 2) to predict the cell's response. For instance, a burst of energy between 1 and 2 kHz will produce a maximum firing rate after about 20 ms followed by inhibition.

Over the last few years, we have developed new methods to derive the STRFs and characterize the responses of both single and multiple units in the ferret AI (Kowalski et al. 1996a,b). These methods use "moving ripples": time-varying broadband sounds with sinusoidal spectral envelopes that drift a constant velocity along the logarithmic frequency axis. Figure 2 illustrates the spectrogram of such a stimulus. Neuronal responses are vigorous and well phase-locked to these spectral and temporal envelope modulations over a range of ripple velocities and densities. Measuring the amplitude and phase of the locked component of the response enables one to construct transfer functions. A transfer function can be inverse-Fourier transformed to obtain the STRF that characterizes a unit's dynamics and selectivity along the tonotopic axis.

View larger version (72K): [in this window] [in a new window]   Fig. 2. Envelope of a moving ripple, w = 2 Hz,  = 0.4 cycle/octave,  = 90°, with a 10 dB AM around a 60 dB base with spectral and temporal 1-dimensional sections. Ripple phase changes linearly with time and spectral position (in octaves).

In developing these measurement and analysis methods, we use two fundamental assumptions. The first is that the responses are substantially linear with respect to the time-varying spectral envelope of stimuli. In particular, this implies that the response to the spectro-temporally rich stimuluswhose envelope can always be described as the sum of multiple moving rippleswill be the sum of its responses to the individual ripple components. This assumption was confirmed by successfully predicting responses to the superposition of multiple ripples (Kowalski et al. 1996b).

The second important assumption deals with the separability of the temporal and spectral aspects of the responses. Specifically we have demonstrated in other reports that temporal and spectral transfer functions can be measured independently of each other and then combined with a simple product to compute the total transfer function (Kowalski et al. 1996a). The importance of this finding stems from its experimental implications for measuring the STRFs and theoretical consequences for the biophysical and functional models of the STRFs. On the experimental side, separability makes it possible to infer responses to all ripple velocities and peak densities based on only a pair of temporal and spectral transfer functions. Without this assumption, measuring the two-dimensional transfer function is difficult because of the extended times needed to collect adequate spike counts. On the theoretical side, separability suggests that certain features of the STRF (as we shall discuss in detail in the following text) are formed by independent (and likely sequential) spectral and temporal processing stages.

In our earlier study (Kowalski et al. 1996a), separability was validated for ripples moving only in one direction (spectral envelope moving downward in frequency), a notion also known as "quadrant separability." In this report, we compare the separable functions (spectral and temporal) across upward and downward quadrants. If the functions are the same across quadrants, the responses are "fully separable" (i.e., they are separable); otherwise they are quadrant separable, which is a (specialized) form of inseparability.

Like quadrant separability, full separability has experimental and theoretical implications. On the experimental side, fully separable STRFs can be measured with either upward or downward moving ripples. Theoretically, fully separable responses imply an STRF that is fully decomposable into the product of a purely temporal impulse response and a purely spectral response field. It also implies a unit that responds equally well to upward and downward moving ripples and hence has necessarily a symmetric transfer function magnitude with respect to direction (Watson and Ahumada 1985). By contrast, cells that are only quadrant separable necessarily respond in asymmetric fashion with respect to direction, i.e., are direction sensitive.

We restrict our presentation in this paper to measurements with singly presented moving ripples in contrast to simultaneously presented ripples discussed in Klein et al. (2000).

There are several goals of this paper. We present a method of measuring the complete descriptor of the linear spectro-temporal properties of an auditory cell, the STRF. We describe examples of STRFs measured in AI and summarize the distribution of the STRF and transfer function parameters encountered. We show that there is a directional sensitivity in the response to the upward versus downward moving components of a sound's spectral envelope. This breaks the symmetry of full spectro-temporal separability and produces quadrant separability. We propose measures to quantify quadrant and full separability. Finally, we discuss the significance of the results and their relationship to results from similar auditory and analogous visual experimental paradigms.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Surgery and animal preparation

Data were collected from a total of 11 domestic ferrets (Mustela putorius) supplied by Marshall Farms (Rochester, NY). The ferrets were anesthetized with pentobarbital sodium (40 mg/kg) and maintained under deep anesthesia during the surgery. Once the recording session started, a combination of ketamine (8 mg · kg¹ · h¹), xylazine (1.6 mg · kg¹ · h¹), atropine (10 µg · kg¹ · h¹), and dexamethasone (40 µg · kg¹ · h¹) was given throughout the experiment by continuous intravenous infusion, together with dextrose, 5% in Ringer solution, at a rate of 1 ml · kg¹ · h¹ to maintain metabolic stability. The ectosylvian gyrus, which includes the primary auditory cortex, was exposed by craniotomy and the dura was reflected. The contralateral ear canal was exposed and partly resected, and a cone-shaped speculum containing a miniature speaker (Sony MDR-E464) was sutured to the meatal stump. For more details on the surgery, see Shamma et al. (1993).

Recordings

Action potentials from single units were recorded using glass-insulated tungsten microelectrodes with 5-7 M tip impedance at 1 kHz. Neural signals were fed through a window discriminator, and the time of spike occurrence relative to stimulus delivery was stored using a computer. In each animal, electrode penetrations were made orthogonal to the cortical surface. In each penetration, cells were typically isolated at depths of 350-600 µm corresponding to cortical layers III and IV (Shamma et al. 1993). In many instances, it was difficult to isolate reliably a single unit for extended recordings, and hence several units were recorded instead. Such data were labeled "multiunit recordings" and are explicitly designated as such and separated from the single-unit records in all data presentations in the paper.

Acoustic stimuli

All stimuli are computer synthesized. For each unit isolated, initial tests are carried out using tonal stimuli to measure the basic frequency response at several intensities to determine the best frequency (BF) and response threshold. All other stimuli used in these experiments have broadband spectra with a sinusoidally modulated (or rippled) envelope. We used the knowledge of the cell's BF to adjust the frequency range of the broadband sound so that the cell's excitatory and inhibitory regions lay well within the frequency range of the sounds.

In practice, it is hard to generate noise and then shape it with filters to a desired dynamic spectral envelope, so we generate ripples over a range of five octaves by taking logarithmically spaced pure tones with random (temporal) phases. The amplitude S(t, x) of each tone is then

(1)

where x = log₂ (f/f₀) is the number of octaves above the base frequency f₀. The ripple envelope resembles a drifting one-dimensional grating as illustrated in Fig. 2. Five independent parameters characterize the ripple envelope: background level or loudness of the stimulus (L); AM of the ripple (A) in percentage or decibels; ripple velocity (w) in units of cycles/s (or Hz); ripple density () in units of cycles/octave; and the initial phase of the ripple . The spectra consist either of 20 or 100 tones per octave equally spaced along the logarithmic frequency axis or with a spacing of 1 tone/Hz with an amplitude decay producing equal power per octave. The spectra typically span five octaves (e.g., 0.25-8 kHz) with the range chosen such that the response area of the cell tested lay within the stimulus spectrum. The choice of a density of 20 or 100 tones per octave does not alter the cortical responses; hence we do not specify which density was used.

A single-ripple stimulus at overall level L dB SPL would typically be composed of N logarithmically spaced components, each at L 10 log₁₀ (N)  L 20 dB for N = 101. The overall stimulus level was chosen on the basis of threshold at BF; typically L was set 10-20 dB above threshold. High levels (L > 70 dB) were avoided to ensure the linearity of our stimulus delivery system. The amplitude of a single ripple was defined as the maximum percentage or logarithm change in the component amplitudes. Ripple amplitudes were either 90% (linear) or 10 dB (logarithmic) modulations.

The ripple velocities w and ripple densities  used were determined by the response properties of the neuron, but the typical range was |w| < 25 Hz (with some units requiring up to 100 Hz) and || < 1.6 cycles/octaves (with some units requiring up to 4 cycles/octaves). Single ripples were always presented with  = 0.

By the convention established in Eq. 1, a ripple whose spectral envelope is moving downward in frequency, as in Fig. 2, has positive w and positive ; equivalently, it can be described by a ripple with negative w and negative , and an added phase shift of , by Eq. 1 and the identity sin () = sin ( + ). A ripple whose spectral peaks are moving upward in frequency has negative w and positive , or by Eq. 1 and the same identity, positive w, negative , and an added phase shift of .

The stimulus bursts had an 8-ms rise/fall time and duration of 1.0 or 1.7 s, repeated every 3-4 s. All stimuli were gated and fed through an equalizer into the earphone. Calibration of the sound delivery system (to obtain a flat frequency response up to 20 kHz) was performed in situ with the use of a -in Brüel and Kjaer 4170 probe microphone. The microphone was inserted into the ear canal through the wall of the speculum to within 5 mm of the tympanic membrane. The speculum and microphone setup resembles closely that suggested by Evans (1979).

Theoretical considerations

DEFINING THE STRF. The fundamental tool to measure linearity and separability of primary cortical cell is to measure their STRF. The STRF is a spectro-temporal function STRF(t, x). The linear response rate y(t) of a cell is related to its STRF(t, x) and the spectro-temporal envelope of the stimulus S(t, x) by y(t) =  dt'dxS(t'  t, x) · STRF(t, x), i.e., convolution along the time dimension t and integration along the spectral dimension x.

w

w

(2)

w,

(3)

(4)

View larger version (17K): [in this window] [in a new window]   Fig. 3. To measure the complete ripple transfer function of an arbitrary STRF, we would need to measure the response of the cell to all the ripples represented by large circles. The small circles correspond to redundant ripples by complex conjugation. The value of the transfer function along the w = 0 axis is set to 0, because the modulation transfer function is not well defined there. Quadrant separability permits one to measure only the responses to ripples enclosed by the solid boxes. The transfer function in the dashed box is equal to the transfer function in the bottom half of the vertical box but with the opposite phase.

DEFINING AND ASSESSING SEPARABILITY. Separability is an important property of the transfer functions. A fully separable transfer function is one that factorizes into a function of w and a function of  over all quadrants: T(w, ) = F(w) · G(). This implies that STRF(t, x) is time-spectrum separable: STRF(t, x) = IR(t) · RF(x). In this case, one needs only measure the transfer function for all  at a convenient w and for all w at a convenient . F(w) and G() are each complex-conjugate symmetric [F(w) = F*(w), G() = G*()] because IR(t) and RF(x) are real, so one needs only consider the positive values of each. This dramatically decreases the number of measurements needed to characterize the STRF.

(5)

(6)

(7)

(8)

(9)

(10)

EFFECT OF FINITE SAMPLING. We measure the transfer function of cells by varying two parameters, ripple velocity and ripple density. For consistency's sake, we used the same range of parameters for a majority of cells. However, for some cells, the transfer function has not decreased significantly at the "edges" (for instance, in Fig. 9C, the temporal transfer function is clearly still strong at ±64 Hz and above). This is equivalent to multiplying the true transfer function by a rectangular function which is zero everywhere except between 64 and 64 Hz, over which range it is 1. In the dual Fourier space of the transfer function space, that is, in the STRF space with coordinates t and x, this corresponds to convolving along each dimension the STRF with the Fourier transform of a rectangular pulse, that is, with sin (x)/x. This leads to spurious oscillations in the display of the STRF as can be seen in Fig. 9C and others. These oscillations would disappear if we had measured the transfer functions all the way to their vanishing values.

DEVIATIONS FROM LINEARITY. Because the STRF is a measure of the linear part of the dynamics of a cell, we only consider effects that might modify the measurement of the first component of the Fourier transform of the period histograms. The most prominent nonlinearities are (approximate) half-wave rectification and compression. The half-wave rectification is primarily due to the positivity of spike rates (ordinarily the steady-state response to a flat spectrum is significantly less than half the peak firing rate of the unit); the distortion of a sinusoid due to half-wave rectification does not affect the phase of the response, and its effect on the amplitude of the first Fourier component is a constant factor, independent of w and . The distortion due to compression or saturation, similarly, does not affect the phase of the Fourier transform components of the response and similarly affects the amplitude only by an overall constant factor for stimuli of moderate level.

Data reduction

Many of the data analysis methods described here are similar or straightforward extensions of those developed earlier in Kowalski et al. (1996a), and those will be only briefly reviewed here. Figures 4 and 5 illustrate the nature of the responses to the ripple stimuli and the analysis to extract the spectral (Fig. 4) and temporal (Fig. 5) transfer functions. In Fig. 4A, the ripples are presented at 8 Hz for ripple densities from 1.6 to 1.6 cycle/octave in steps of 0.2 cycle/octave. Each stimulus is presented 15 times.

View larger version (57K): [in this window] [in a new window]   Fig. 4. Data analysis for ripples of fixed ripple velocity and varying ripple densities. A: raster plot of responses. Each point represents an action potential, and each ripple stimulus is presented 15 times. Note the position of the peaks changes linearly with ripple density. B: period histogram for 3 example ripple densities, with their sinusoidal fits. C: magnitude and phase of the period histogram fits. With the phase convention used for these stimuli, ripples with  < 0 (quadrant 4) are equivalent to ripples with w < 0 (quadrant 2), using the conversion (w, , )  (w, ,  + ).

View larger version (55K): [in this window] [in a new window]   Fig. 5. Data analysis from ripples of fixed ripple density and varying ripple velocities. A: raster plot of responses. Each point represents an action potential, and each ripple stimulus is presented 15 times. B: period histogram for 3 example ripple velocities. Note how the position of the peak of the best fit changes linearly with ripple velocity. C: magnitude and phase of the period histogram fits.

For each ripple density, we compute at 16-bin period histogram based on the responses starting at 120 ms (to exclude the onset response; Fig. 4B). A 16-point Fourier transform (FFT) is then performed on the period histogram, and the amplitude and phase of the first component is taken to be the amplitude and phase of the transfer function. If the modulation of the response was that of a purely linear system, the higher FFT coefficients would be negligible, but because of half-wave rectification and compression, they sometimes are significant. In general T_w() can be written as

(11)

where j = . Figure 4C illustrates the magnitude |T_w()| and the unwrapped phase _w() of the transfer function T_w(). The ripple density at which |T_w()| is a maximum is designated as _m (= 0.0 octave/cycle in Fig. 4C).

Analogous steps are followed in measuring the temporal transfer function as shown in Fig. 5 where ripples are presented at 0.2 cycle/octave for ripple velocities from 24 to 24 Hz in steps of 4 Hz.

Note that in the previous paper (Kowalski et al. 1996a), we weighted the measurement of the first component of the Fourier transforms of the period histograms by a weighted sum of the higher frequency components of the transform. This, however, is not compatible with the idea of a linear system so that the resultant STRF or equivalently the ripple transfer function T would not be expected to be the best possible predictor of the response to new sounds. Therefore in this paper, the values of T correspond directly to the first component of the Fourier transform.

Once the ripple transfer function has been measured, it can be inverse Fourier transformed to display the STRF. Since the transfer function is typically measured over fewer than 8 points along each dimension in each quadrant, the resulting STRF as computed would look very jagged even if the underlying STRF was smooth. We therefore interpolate to a smooth STRF for display purposes, padding the transfer function with zeros to a size of 64 × 64. All statistics and predictions use the measured unsmoothed STRF.

To construct the two-dimensional transfer function, we assume quadrant separability, measure the transfer function along the cross-sections shown in Fig. 3, to combine these spectral and temporal cross sections as illustrated in Fig. 6. For each quadrant, the transfer function is the outer product of the cross-section, divided by the (complex) value of the transfer function at the crossover (×) point. In Fig. 6, the point is (w_×1, _×1) = (8 Hz, 0.2 cycle/octave) in quadrant 1 and (w_×2, _×2) = (8 Hz, 0.2 cycles/octave) in quadrant 2. 

(12)

where q = 1 and q = 2 are the independent quadrants 1 and 2. In practice, the value of the transfer function along the two cross-sections was measured at two different times, giving two measurements of the transfer function at each crossover point T(w_×q, _×q). The results of the two measurements may differ, and so we use the (complex) geometric mean of the two measured values as the divisor in Eq. 12, T_eff(w_×q, _×q) = [T_1st(w_×q, _×q)T_2nd(w_×q, _×q)]^1/2.

View larger version (62K): [in this window] [in a new window]   Fig. 6. Deriving the spectro-temporal transfer function, STRF, and related parameters. A: magnitude of the temporal (left) and spectral (right) transfer function cross-sections, normalized by the values at the cross-over points (Eq. 12). The error bars are computed by the bootstrap method, explained below. B: the magnitude of the full transfer function, resulting from the outer product of the functions in A. C: the STRF of the cell computed by an inverse Fourier transform of the complex transfer functions. To the right is the error estimate of the STRF, using the same scale multiplied by a factor of 5 (for legibility), resulting in error parameters of  = 0.03 and  = 0.06. See Table 1 for details.

The ratio T_1st(w_×q, _×q)/T_2nd(w_×q, _×q), which should be unity, reflects noise in the system and is used to estimate reliability in the following text.

The value of the transfer function along the w = 0 axis is set to zero because the modulation transfer function is not well defined there, i.e., there is no modulation of firing rate around the DC (average) rate with a frequency of 0 Hz. The value of the transfer function along the  = 0 axis is not measured directly, so the value used is the mean of the value inferred from being the boundary of quadrant 1 and that inferred from being the boundary of quadrant 2.

Once the values of transfer functions for quadrants 1 and 2 and their boundaries are measured, the values for quadrants 3 and 4 are given by Eq. 4 (see also Fig. 3). The STRF is then computed by an inverse Fourier transform (as in Eq. 3) and is illustrated in Fig. 6B (left). This interpolated version of the STRF (used for display) is obtained by using Eq. 3 on the transfer function padded with zeros at high |w| and || (see Fig. 6A).

Deriving STRF parameters from the phase functions

Numerous parameters can be derived from the STRF (or equivalently the transfer function) that are analogous to traditional response measures such as BF, tuning curve bandwidth, and latency. Most of these parameters are best derived from analysis of the phase of the transfer functions (Fig. 7).

We model the phase of the transfer function within each quadrant ^q(w, ), q = 1, 2 (see Eq. 2) as a linear function of w and  

(13)

where is the mean or group delay of the STRF (a portion of which comes from the response latency), x = log (f/f₀) is the mean frequency (in octaves above the base frequency of the ripple, see Eq. 1) around which the STRF is centered (putting it near the BF), and ^q is a constant phase angle, for each quadrant q. The complex-conjugate symmetry of the transfer function means that these six independent parameters describe the phase everywhere in the w   plane. The convention of the minus sign before _d allows the time-dependent responses to be functions of (t  _d) as is appropriate for a delay.

The justification for assuming linear fits of the phase functions has been discussed in detail earlier in (Depireux et al. 1998) and is strongly motivated by the data (Kowalski et al. 1996a). Note, however, that the assumption of phase linearity is used only for parameter estimation and is not assumed in computing the STRF. The first linear term in Eq. 13 stems from the fact that auditory units differing in their mean neural delays will exhibit linear phase dependence on w with different slope depending on delay. Analogous arguments apply for units that are located at different places along the tonotopic axis: the response phase of different units (with otherwise identical STRFs) changes linearly with  at different rates, depending on the relative center frequency locations. In both cases, the slopes of the linear phase function indicate the absolute shift of the STRF relative to the origin, i.e., the mean time delay relative to the start of the stimulus, and the center frequency x relative to the low frequency edge of the ripple spectrum. The linear phase model does not assume that the linear phase shifts, and x, are equal across quadrants, but tonotopy suggests that f and f should be approximately equal and    since the temporal delays of the neural inputs are not segregated by quadrant. This is shown experimentally in the following text.

An interpretation of _d, for each quadrant, is that it is the sum of the pure response latency and (roughly) half the temporal width of the STRF. This is in contrast to the STRF's peak delay, _STRF, defined to be the delay for which the STRF achieves its maximum value, which may lead or lag _d, depending on the constant temporal phase shift, , defined in the following text. Similarly, f_m for each quadrant may or may not fall on the STRF's best frequency, BF_STRF, defined to be the frequency at which the STRF achieves its maximum value, depending on the constant spectral phase shift, , defined in the following text.

A convenient convention for interpreting the constant component of the phase is to break up the constant phase angle ^q into two parts

(14)

 and  are, respectively, the temporal polarity and spectral asymmetry of the STRF. Spectral asymmetry parameterizes the balance of the STRF along the spectral axis about its center. For example, a unit with  = 0 would have its BF_STRF in the center of the spectral envelope of the STRF, possibly surrounded by inhibitory regions. A unit with  > 0 would have its BF_STRF at a lower frequency than the center of the STRF with an inhibitory sideband above BF_STRF. A unit with  < 0 would have its BF_STRF at a higher frequency than the center of the STRF, with an inhibitory sideband below BF_STRF (see example in Fig. 4C of Shamma et al. 1995). Similarly the temporal polarity parametrizes the balance of the STRF along the temporal axis about its center: whether the peak response occurs before or after regions of inhibition, respectively,  < 0 ("onset response at BF") or  > 0 ("offset response at BF"). There is an ambiguity in fixing  and  that we remove by restricting  to lie between 90 and +90°, while  ranges the full 180 to +180°. See Fig. 7 as an illustration of the phase behavior in the different quadrants.

View larger version (19K): [in this window] [in a new window]   Fig. 7. A: the phase of the transfer function can be well described by a linear fit containing 6 parameters over most of the relevant regions of the w- plane. B: in this cartoon, the slope is constant for most of the curves, after (left) 2w has been subtracted in each quadrant, corresponding to a center frequency that is independent of the ripple density, and (right) after 2 has been subtracted, corresponding to a delay that is independent of ripple velocity. At very small ripple densities (long ripple periodicity), center frequency is less meaningful, and similarly for small ripple velocity and delay, respectively. At large ripple velocity the slope asymptotes to the signal front delay, but when this occurs, the small amplitude of the transfer function makes it difficult to measure the phase. See Dong and Atick (1995) and Papoulis (1962).

In past reports (Kowalski et al. 1996a),  and  could be measured without measuring the transfer function in the upward moving quadrant 2 by measuring the constant component of the phase in quadrant 1 (¹ =  + ) and along the w axis, where the constant component of the phase is expected to be the mean across the quadrants [(^1  2)/2 = ; note the change in convention of    between the present work and Kowalski et al. (1996a)].

Because of response variability, we only fit to those points of the transfer function that have more than half of the response power in the first component of the Fourier transform. Then the fit is done across the entire two-dimensional phase plane for each quadrant. Ultimately our unwrapping method is less than ideal, and estimates of  and  especially reflect that (Ghiglia and Pritt 1998).

Estimating response variability: the bootstrap method

Variability in our experiments originates from multiple sources, including internal neural mechanisms (e.g., Poisson-like distributions of spike times), extracellular recording/identifying methods, and equipment noise. Quantitative estimates of the reliability of our measurements is crucial to its analysis and subsequent interpretation. A method of variability estimation that is especially appropriate to these measurements is the bootstrap method (Efron and Tibshirani 1993; Politis 1998).

The essence of this method is to use "resamples," in which N samples of bootstrap data are drawn with replacement from the N original samples of data. Repeating this procedure a large number of times creates a population of bootstrap resamples whose probability distribution is a good estimator of the probability distribution from which the original data were drawn.

To illustrate this procedure, consider measuring the transfer function at a point (w, ). This is done by presenting the same (w, ) stimulus N times and constructing a period histogram based on all N sweeps. The amplitude and phase of the first Fourier component of the period histogram are assigned to the amplitude and phase of the transfer function. A single bootstrap resampling of the responses will have N sweeps, where, because they are drawn from the original responses with replacement, some will be duplicated and some will be unused. Nevertheless a period of histogram is constructed, and the bootstrap estimate of the transfer function is assigned to its first Fourier component. Performing a large number of bootstrap resamples results in a population of estimates for the transfer function. This population has a mean, variance, and higher-order moments. These moments are estimators of the moments of the original population (of all transfer functions of all allowable neuronal responses to the stimulus). For example, the standard deviation of all bootstrap estimates of the transfer function is an estimator of the standard deviation of measurements of the transfer function. This allows us to put error bars on our transfer functions and STRFs.

Effects of crossover point errors

Another significant source of error is the difference between the responses of repeated measurements at the transfer function crossover points. The ratio of these independent measurements, T_1st(w, )/T_2nd(w, ) should be unity. When not unity, it reflects the same variability measured by the bootstrap method but also additional systematic error from having measured the two transfer function cross-sections at different times. To account for this disparity, the total squared error of the STRF is set to the sum of the bootstrap STRF variance and the square of the crossover error _×

(15)

where _×(t, x) captures the systematic error from not having taken all data at the same time and is given by

(16)

Finally, we collapse the error over the entire (t, x) plane into two dimensionless terms  and  

(17)

(18)

where T and X are the length of time and number of octaves over which the STRF was measured.

is a measure of the average standard deviation in units of the maximum of the STRF.  is a measure of the variance in units of power. If noise is additive, then  = P/(P + P) = 1/(SNR + 1), with P = power, P = noise power, and SNR = signal-to-noise ratio.  should go down with the number of recordings, assuming the system can be described as the time-invariant random process.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Data presented here were collected from 22 single-unit and 54 multiunit recordings in 11 ferrets. In the summary histograms, both single units and multiunit are included but are distinguished from each other.

Most units encountered in AI respond well to moving ripples. Responses are typically phase-locked to the moving envelope of the ripple over a range of ripple velocities and densities. However, of a total of 172 recordings made, only 76 cases provided adequate quality and quantity of responses. The reasons for this low yield vary. For example, we have encountered responses from a few units that were either poorly phase-locked or were inconsistent from trial to trial; such units were abandoned since our analysis methods are unsuitable for their characterization. Also because of extended recording times, typically over an hour, units were sometimes lost before sufficient data could be collected to carry out a full analysis. In other cases, the unit or animal changed state during the recording session, rendering the data unreliable. The reason for the extended recording time is to present ripple sounds and other sounds consisting of combinations of ripples, so we can verify linearity by using the STRFs to predict the response of the cell to new sounds. We found empirically that about 10,000 spikes are typically needed to obtain an STRF with well-defined features in response to single ripples, which with our sound paradigm usually corresponds to a 20-min presentation per cross-section. To eliminate data corresponding to unreliable cells, as described in the preceding text, we use units only with values of   0.12 and   0.7 (see METHODS) as the threshold for rejecting the data. These reliability statistics takes into account most of the preceding sources of error. The values of 0.12 and 0.7 are somewhat arbitrary, though we found that cells tended to separate themselves into two populations above and below these thresholds, respectively, and that the mathematical criteria of reliable versus noisy cell corresponded well with our intuitive perception based on visual inspection.

Responses to moving ripples

On average, AI units synchronize their responses to upward and downward moving ripples equally effectively with ripple velocities ranging from 2 to over 100 Hz, and ripple densities up to 4 cycle/octave. Examples of several temporal and spectral transfer function magnitudes are shown in Figs. 8-10, each with its corresponding STRF. In all cases, units respond well only over a specific range of ripple velocities and ripple densities, but the detailed shape and extent of the transfer functions vary from one unit to another. For instance, the unit in Fig. 9A responds well only to ripple velocities of ±4 Hz, whereas the unit in Fig. 9C responds well at least up to ±64 Hz. The unit in Fig. 6 responds well to ripple densities within ±0.4 cycle/octave, whereas the unit in Fig. 10A responds over a wider range of densities but poorly at 0 cycle/octave.

View larger version (69K): [in this window] [in a new window]   Fig. 8. Three examples of spectro-temporal transfer function sections, corresponding STRFs. For each row A-C: magnitude of the temporal (left) and spectral (middle) transfer functions. All other details are as in Fig. 6; (right), STRFs.

View larger version (67K): [in this window] [in a new window]   Fig. 9. Further examples of spectro-temporal transfer function sections, corresponding STRFs. Conventions as in Fig. 8. A: with narrow ripple velocity bandwidth. B: wit