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The Journal of Neuroscience, December 1, 2002, 22(23):10434-10448
Energy Integration Describes Sound-Intensity Coding in an Insect Auditory System
Tim Gollisch, Hartmut Schütze, Jan Benda, and Andreas V. M. Herz Institute for Theoretical Biology, Department of Biology, Humboldt University, 10115 Berlin, Germany
ABSTRACT
TOP
ABSTRACT
INTRODUCTION
We investigate the transduction of sound stimuli into neural responses and focus on locust auditory receptor cells. As in other mechanosensory model systems, these neurons integrate acoustic inputs over a fairly broad frequency range. To test three alternative hypotheses about the nature of this spectral integration (amplitude, energy, pressure), we perform intracellular recordings while stimulating with superpositions of pure tones. On the basis of online data analysis and automatic feedback to the stimulus generator, we systematically explore regions in stimulus space that lead to the same level of neural activity. Focusing on such iso-firing-rate regions allows for a rigorous quantitative comparison of the electrophysiological data with predictions from the three hypotheses that is independent of nonlinearities induced by the spike dynamics. We find that the dependence of the firing rates of the receptors on the composition of the frequency spectrum can be well described by an energy-integrator model. This result holds at stimulus onset as well as for the steady-state response, including the case in which adaptation effects depend on the stimulus spectrum. Predictions of the model for the responses to bandpass-filtered noise stimuli are verified accurately. Together, our data suggest that the sound-intensity coding of the receptors can be understood as a three-step process, composed of a linear filter, a summation of the energy contributions in the frequency domain, and a firing-rate encoding of the resulting effective sound intensity. These findings set quantitative constraints for future biophysical models.
Key words: mechanosensory transduction; spectral integration; auditory receptor; hearing; sound intensity; energy; model; locust
INTRODUCTION
Auditory receptor cells are commonly characterized by their responses to pure tones. For example, threshold curves characterize the minimum intensity needed to evoke a response as a function of the frequency of a pure tone; rate-intensity functions describe how the response depends on the tone's intensity. Natural signals, however, are only rarely restricted to single frequencies, and receptor cells often show a broad frequency tuning. Our understanding of auditory coding is thus not satisfactory as long as we do not know how the relative intensities of different frequencies contained in a sound signal are integrated by auditory receptors. Investigating this spectral integration helps us also to scrutinize basic principles of the mechanosensory transduction process.
In general, the response of the receptor could be any complicated, nonlinear function of the frequency spectrum. One may hope, however, that the underlying mechanism is simple enough to allow for a straightforward phenomenological description. One such way of combining different spectral contents would be the extraction of a single physical stimulus property. Its nature is intensely debated with respect to the question of temporal integration, i.e., how stimulus intensities are combined over time. Psychoacoustic measurements of intensity-duration tradeoffs suggest that the stimulus energy is the crucial variable (Garner, 1947
; Plomp and Bouman, 1959
Spectral integration, on the other hand, depends at least in insects almost exclusively on the mechanosensory transduction process; any fluctuations on the several kilohertz scale of relevant sound frequencies that were still present after the transduction (i.e., in the cell-membrane conductance) would be highly attenuated by the low-pass filter properties of the cell membrane (Koch, 1999
For these reasons, we develop a descriptive model for the responses of auditory receptor neurons to stationary stimuli with arbitrary power spectrum. The model comprises three steps, which correspond to the coupling, the transduction, and the encoding of the primary signal (Eyzaguirre and Kuffler, 1955
MATERIALS AND METHODS
Electrophysiology. All experiments were performed on adult Locusta migratoria. The tympanal auditory organ of these animals is located in the first abdominal segment. After decapitation, removal of the legs, wings, intestines, and the dorsal part of the thorax, the animal was waxed to a holder, and the metathoracic ganglion and auditory nerve were exposed. Action potentials from auditory receptor cells were recorded intracellularly in the auditory nerve with standard glass microelectrodes (borosilicate, GC100F-10; Harvard Apparatus Ltd., Edenbridge, UK) filled with a 1 M KCl solution (50-110 M
resistance). The signals were amplified (BRAMP-01; NPI Electronic, Tamm, Germany) and recorded by a data acquisition board (PCI-MIO-16E-1; National Instruments, München, Germany) with a sampling rate of 10 kHz. Detection of action potentials and generation of acoustic signals were controlled on-line by the custom-made Online Electrophysiology Laboratory (OEL) software. Stimuli were transmitted by the above-mentioned data acquisition board with a conversion rate of 100 kHz to the loudspeakers [Esotec D-260, Dynaudio (Skanderborg, Denmark) on a DCA 450 amplifier (Denon Electronic GmbH, Ratingen, Germany)]. These were mounted at 30 cm distance on each side of the animal so that the incidence of sound-pressure waves was orthogonal to the body axis. Stimuli were played only by the loudspeaker ipsilateral to the recorded auditory nerve. The linearity of the loudspeakers for superpositions of multiple tones was verified by playing samples of the stimuli used in the experiments while recording the sound at the site of the animals with a high-precision microphone [40AC, G.R.A.S. Sound & Vibration (Vedbæk, Denmark) on a 2690 conditioning amplifier (Brüel & Kjær, Langen, Germany)]. During the experiments, animals were kept either at room temperature, which was ~20°C or at a constant temperature of 30°C. No systematic trends regarding a possible temperature dependence of the studied phenomena were observed. All experiments were performed in a Faraday cage lined with sound-attenuating foam to reduce echoes. Recordings from 45 receptor cells stemming from 18 animals (with at most 4 cells from the same animal) were used in this study.
The experimental protocol complied with German law governing animal care.
Measurement of rate-intensity functions. In general, each sound stimulus was presented for a duration of 100 msec, separated by pauses of at least 400 msec. To investigate adaptation effects, control experiments with longer stimuli and pauses (300/500 msec or 500/750 msec) were performed. All of these stimuli are decidedly longer than typical integration times of insect auditory receptors (1-3 msec as determined by reverse correlation for locust auditory receptors; data not shown) (see also Tougaard, 1998
Rate-intensity functions were determined in the following way. First, the stimulus was presented in steps of 5 dB between 20 and 100 dB sound pressure level (SPL) (for a definition see Eq. 21 in the Appendix) to obtain the general shape of the rate-intensity function. These data were used to identify the intensity range that gave rise to firing rates between 50 and 250 Hz. Within this dynamic range of ~10-15 dB, additional measurements in steps of 1 or 2 dB were performed, and these were repeated 4-10 times to yield average firing rates and their SDs.
Stimulus intensities corresponding to given firing rates were obtained by fitting a straight line through the four points closest to the desired firing rate as shown in Figure 1. Errors on these measurements follow from the errors of the fitted parameters according to the law of error propagation. Thresholds were determined by linear extrapolation to zero firing rate from data points with a low, but significant firing rate.
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Figure 1. Determination of sound intensities corresponding to given firing rates. A, Example of a spike train recorded intracellularly from an axon of a receptor cell. Calibration is given to the right. The thick bar below the voltage trace denotes the 500 msec pure-tone stimulus. The vertical bars below show the spike times as determined by the spike-detection algorithm. The firing rate is calculated by counting the spikes and averaging over several stimulus repetitions. B, Example of the rising part of a rate-intensity function ( ) measured in steps of 1 dB. Each stimulus was repeated multiple times. Vertical bars denote the SD of each measurement. Linear fits through the four points closest to the firing rates of interest, here 100 and 150 Hz, are depicted as dotted and dashed lines, respectively. The arrows indicate the readout of the corresponding intensities.
Superposition of pure tones. Measuring rate-intensity functions for pure tones allows one to understand how the firing rate r depends on the amplitude A of a single tone for a certain sound frequency, r = r(A). Investigating spectral integration amounts to asking whether this understanding can be extended to stimuli that contain multiple tones simultaneously. We therefore try to obtain a description of the firing rate r depending on the amplitudes A1, A2, ... of the different frequency components of such stimuli, r = r(A1, A2, ...).
In a first set of experiments, stimuli were sound-pressure waves S(t) consisting of two or three pure tones of amplitudes An, frequencies fn, and phase offsets
n, n = 1, 2, 3:
with A3 = 0 for the two-tone experiments. We used stimuli that were far longer than the periods of the sine waves and avoided combinations of frequencies that are related to each other by small integer factors. This makes the measurements insensitive to the relative phases of the individual sine tones, which we cannot control in our experiments because of putative phase shifts at the tympanal membrane (Michelsen, 1971b
(1) · 10 kHz
9.55 kHz, the three-tone experiments with f1 = 4 kHz, f2 = 3/
Within the present approach, we are concerned only with the encoding of sound intensity and not with temporal aspects. We thus restricted our attention to stationary stimuli with constant envelope as described above. This is justified because the responses of locust auditory receptors do not phase lock to sound frequencies in the kilohertz range (Suga, 1960
The experiments were designed to identify, for individual receptors, sets of amplitude combinations (A1, A2) or (A1, A2, A3), respectively, that result in the same firing rate. The recorded data were analyzed within a model framework, which includes explicit predictions about how these amplitude combinations should be related to each other. In Results, the model is systematically developed and discussed. Here, we only present the main aspects and cover technical issues and questions regarding the model's role within the data analysis.
In summary, we compute the average firing rate of a receptor cell in the following three-step process.
(1) The stimulus is a sound-pressure wave S(t), a superposition of pure tones with frequencies fn, amplitudes An, and phase offsets
An sin(2
This means that every tone receives a gain factor 1/Cn. In addition, the phase may change from
(2) n. The inverse of the filter constant Cn thus corresponds to the sensitivity for the frequency fn: the smaller the Cn, the more sensitive the receptor at the corresponding sound frequency. (2) An effective sound intensity J is computed according to one of the following three hypotheses:
where
(t) is the filtered signal from Equation 2, |x| denotes the absolute value of x, and y(t)
is the temporal average of y(t).
(3) The average firing rate r is determined according to a single nonlinear function r(J).
Note that the effective sound intensity J as defined above is distinct from the physical sound intensity, commonly measured in decibels SPL (compare Eq. 21 in the Appendix), which we denote by I throughout the text. Whereas I measures the stimulus itself, J is a derived quantity that incorporates the filter constants Cn and therefore also reflects the sensitivity of the specific receptor cell. Furthermore, I is defined as a logarithmic measure (relative to a predefined reference intensity); J is not, which facilitates the notation.
Within the model framework, the filter constants Cn are determined only up to a common factor, which can be absorbed in the function r(J). In other words, the model remains unchanged if all Cn are multiplied by the same constant and r(J) is at the same time adjusted appropriately. It follows that one way to determine the Cn is to choose a fixed firing rate, find for each frequency fn the amplitude Ân that leads to this firing rate, and set Cn = Ân.
In the following description of the experimental procedure, we will for simplicity focus on the case of superpositions of two tones. The generalization of concepts and formulas to the three-tone case is straightforward.
The three alternative hypotheses result in different predictions about which combinations of amplitudes (A1, A2) are expected to lead to the same firing rate. Because the model implies that equal firing rate follows from equal effective sound intensity J (step 3), curves of constant firing rate can easily be calculated for each hypothesis by setting J constant in the equations of the second step in the model. These "iso-firing-rate curves" are shown in Figure 2. From the amplitude hypothesis, pairs (A1, A2) yielding the same firing rate are expected to lie on a straight line. Likewise, from the energy hypothesis, they are expected to lie on an ellipse. For the pressure hypothesis, they should fall on an even more strongly bent curve. The corresponding shape has to be computed numerically by solving the equation:
for pairs:
(3)
The duration
has to be chosen large enough to cover many cycles of the sine waves in the signal, so that the phases
n can be neglected. Note that the shape of these three alternative iso-firing-rate curves is not influenced in any way by the form of r(J).
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Figure 2. Prediction of iso-firing-rate curves for the superposition of two pure tones. Depending on the model, the effective sound intensity J as well as the firing rate are expected to be constant along different curves in the two-dimensional space of amplitude combinations. A1 and A2 denote the amplitudes of the respective components. According to the amplitude hypothesis (AH), iso-firing-rate curves are straight lines (one example shown by the dashed line); according to the energy hypothesis (EH), they are ellipses (solid line); and according to the pressure hypothesis (PH), they are even more strongly bent curves (dash-dotted line), the exact shape of which has to be determined numerically. The scale of the axes is given by the filter constants C1 and C2. Note that when the hypotheses are fitted to the data, the obtained filter constants will in general be different for each model, and the intersection points with the axes will not coincide because C1 and C2 are free parameters for each model. The gray arrows indicate equally spaced directions along which the rate-intensity curves are measured. In each direction, the intensity increases with increasing amplitudes A1 and A2, whereas A1/A2 is kept fixed and determined by the angle . (One example for this angle is denoted in the figure.) The intersection points of the arrows with the iso-firing-rate curves denote the amplitude combinations that are expected to yield the specified firing rate according to each of the three alternative hypotheses. Because the three intersection points on each gray arrow clearly differ from each other, the measurements of the iso-firing-rate curves can be used to distinguish between the hypotheses.
To relate these predictions to experimental results, we determined a set of amplitude combinations leading to the same average firing rate in the following way. We start by measuring a first rate-intensity function for a single pure tone with frequency f1. From this rate-intensity function, we determine the amplitude A
that leads to a firing rate of, e.g., 150 Hz as shown in Figure 1. (In the notation A , the subscript i refers to the frequency fi at which the amplitude is measured, and the superscript n indicates the number of the measurement, 1 n
, 0), i.e., a point on the A1 axis in a graph such as that of Figure 2. The same procedure is performed for a pure tone with frequency f2, leading to a second data point (0, A ) on the A2 axis that also corresponds to a firing rate of 150 Hz. These two amplitudes A and A can already serve as estimates of the filter constants C1 and C2, respectively. We proceed by measuring rate-intensity functions for superpositions of the two tones where the ratio of the amplitudes A1 and A2 is held fixed. To do so, we set A1 = k·A2 and then jointly vary the intensity of A1 and A2. This corresponds to measuring the rate-intensity functions along straight lines in radial direction as pictured by the gray arrows in Figure 2. It is also evident from the figure that the radial direction is well suited for accurate measurements of the iso-firing-rate curves and for discriminating between the hypotheses. The resulting rate-intensity functions are similar in shape to the ones for the pure tones, and we can again determine the stimulus that leads to a firing rate of 150 Hz as in Figure 1. This yields a third data point (A , A ) with A /A = k. The procedure is continued for several different ratios k so that a set of amplitude pairs (A , A ) is obtained. A technical but important question is which ratios k should be used in the experiment. If the neuron is much more sensitive to one of the sound frequencies and if both amplitudes are comparable in size, i.e., A1
of a straight line:
In the off-line analysis, the parameters C1 and C2 were newly determined by
2 fits of each of the three curves in Figure 2 to the complete data set (A
, A ). These fitted values of C1 and C2 should be more reliable than the initial estimates, which were obtained on-line from the pure-tone rate-intensity functions only. A further technical detail concerns the choice of the fitting procedure. The procedure should treat A1 and A2 in a symmetric fashion, and it should not be affected by potentially large differences in the relative sensitivities for the two tones. This discards, e.g., the simplest choice of regarding A2 as a function of A1 or vice versa. Instead, we normalized the amplitudes by the filter constants and looked at the radial distance of the data points:
from the origin, which is given by:
as a function of the ratio:
This is a natural choice because the rate-intensity functions that led to the data points were measured in this radial direction. For the three hypotheses, we denote the predicted radial distance by d
, where m stands for the particular model hypothesis (m = AH, EH, or PH). d can be obtained from the model as a function of
for the energy hypothesis, d
= 1, and for the pressure hypothesis, d has to be determined numerically using the solutions of Equation 3. Estimating C1 and C2 then corresponds to minimizing the
with respect to C1 and C2. The contributions of the data points are weighted by the measurement errors
(4) n, which follow from the measurement errors
A
and for A and A , respectively, by the law of error propagation as:
The fitted curves and the
(5) For the control experiments with stimulus lengths of 300 or 500 msec, the onset response and the steady-state response were analyzed individually. For the onset, only spikes in the first 30 msec after stimulus onset were taken into account; for the steady state, the first 200 msec of the response were disregarded. The control experiments were aimed at investigating the effect of adaptation on our model description. We therefore performed the same analysis as explained above on the firing rates obtained for the onset and the steady state and fitted the filter constants C1 and C2 separately in each case. In addition, we compared C1 and C2 as well as their ratio R = C1/C2 for the onset with the respective values for the steady state. The relative change of R was computed from the onset value, RO, and the steady-state value, RS, as
RS|/RS. For the total response, the ratio of C1 and C2 is denoted by Rtotal. To estimate the significance of changes in C1, C2, and R between the onset and the steady state, error measures for these parameters were computed for each cell individually by taking several nonoverlapping stretches of 30 msec during the steady state for the analysis, determining C1, C2, and R in each case, and computing the respective SDs.
Experiments with superpositions of three pure tones were performed and analyzed in the same way as the two-tone experiments. We first measured rate-intensity functions for each pure tone and from these obtained initial estimates of the respective filter constants C1, C2, and C3. Subsequently, rate-intensity functions were measured along different directions in the three-dimensional stimulus space:
The ratios of:
were taken as 1:1:1, 2:1:1, 1:2:1, and 1:1:2. Final fits of the model parameters C1, C2, and C3 were obtained in an analogous way as for the superposition of two tones.
Statistical analysis. The
The Bayesian probability of a model given the data can be used as a measure for the preference of one hypothesis over another. It is calculated from Bayes' formula:
where p(data) =
(6) m p(data|model m)·p(model m). If there is no a priori evidence for any model, the prior probabilities for the models are to be set to p(model m) = 1/M, where M is the number of models investigated. The probabilities p(data|model m) were calculated from the difference between:
and the corresponding model predictions d
by assuming independent errors with a Gaussian distribution of SDs
(7)
An analogous formula was used in the case of superpositions of three pure tones.
Trends in the data were tested for statistical significance by a standard run test (Barlow, 1989
Comparison of pure-tone and noise stimuli. In another set of experiments, we tested whether our understanding of spectral integration allows accurate predictions of firing rates for more complex stimuli and focused on bandpass-filtered noise. To calibrate the model for a specific receptor, we measured the rate-intensity function for a pure tone as well as a set of filter constants in the relevant frequency band of the noise stimulus. According to our model, we can use these pure-tone results to calculate a prediction for the rate-intensity function of the noise stimulus (see below). The filter constants are needed for the calculation of the effective sound intensity J for the noise stimulus (model step 2), and the pure-tone rate-intensity function is needed because it implicitly contains the information about the shape of the response function r(J) of model step 3. To assess the reliability of the prediction, the rate-intensity function for the noise stimulus was also measured experimentally. The particular noise stimulus that we used was Gaussian white noise, cut off at ±3 SDs and bandpass filtered between 5 and 10 kHz, and the frequency of the pure-tone stimulus was 4 kHz. Note that when the amplitude of a noise stimulus is varied, all amplitudes in the signal are scaled by a common factor.
The prediction for the rate-intensity function of the noise signal is obtained in the following way. According to our model, the rate-intensity function of the pure tone, rpt(I), and the rate-intensity function of the noise stimulus, rnoise(I), should have the same shape and be related to each other by a shift
For notational simplicity, we always use the same symbol r to denote the firing rate regardless of whether we consider its dependence on the sound intensity I, r(I), or on the effective sound intensity J, r(J). Strictly speaking, r(I) and r(J) are different functions, but from the context, it will always be clear to which function we refer.
(8) Let us briefly describe the reason for the relation of Equation 8. For concreteness, we focus on the energy hypothesis; the amplitude and the pressure hypotheses can be dealt with in an analogous way. Consider an arbitrary sound signal S(t) composed of a set of pure tones with amplitudes An. From these, we can calculate the intensity, which is defined as:
as well as the effective sound intensity:
(9)
The essential observation is that multiplying every An by the same factor k amounts to adding a constant 20log10k to the intensity I (if k < 1, this constant is negative), whereas JEH is multiplied by a factor k2.
We now consider a noise stimulus with intensity Inoise and effective sound intensity J
. To compare the response with that of a pure tone, we find the intensity Ipt that yields the same firing rate as the noise stimulus by setting both effective sound intensities equal:
The parameter Cpt denotes the filter constant for the pure tone. From the preceding equation, we can calculate the pure-tone amplitude Apt and thus the intensity Ipt of the pure tone, for which the firing rate is the same as for the noise signal with given intensity Inoise. Let us denote the difference between Inoise and Ipt by
If we multiply all amplitudes by the same factor k, the amplitudes of the noise signal as well as Apt, the intensities Inoise and Ipt are changed by the same amount. Consequently, the difference between the new intensities is still given by
= J . Because the firing rate depends only on the value of J, this means that for the new intensities, the firing rates are also equal. It follows that whenever the intensities of the pure tone and the noise signal differ by The derivation shows that the predicted
for the energy hypothesis and by:
(10)
for the pressure hypothesis. The two predictions for
(11) Evaluating Equations 10 and 11 is possible if one knows the filter constants and the A
for the amplitudes in the noise signal. The latter are given by the power spectrum of the noise signal, which we calculated in discretized bins of width 0.05 kHz (using a triangular Bartlett window). Filter constants Cn were measured for pure tones between 5 and 10 kHz at every 0.2-1 kHz (depending on the length of the recording) by determining the amplitude that led to a firing rate of 260 Hz. Additional filter constants Cn, for all center frequencies of the power-spectrum bins, can be determined by linear interpolation from the measured filter constants. The prediction for the noise-stimulus rate-intensity function that results from shifting the pure-tone rate-intensity function rpt(I) by
of the pure-tone rate-intensity function a corresponding Î , and similarly for every intensity I of the noise rate-intensity function a corresponding Î . Because ideally, these should be related by Î = I + = I
Because the subthreshold part and the saturation are not important in the determination of the actual shift, only data points (I, r) with r between 20 and 80% of the maximum firing rate of the cell were taken into account.
(12)
RESULTS
The objective of this study is to develop a descriptive model for the responses of auditory receptor neurons to arbitrary stationary acoustic stimuli. This is done to identify the dominant physical stimulus property governing the encoding of sound intensity. We first develop a general mathematical framework for the transformation of the incoming sound into the neural response. Subsequently, we apply the model to locust auditory receptors and show that the experimental data are well described by only one of three rival hypotheses about the nature of the primary signal transduction.
Derivation of the mathematical model
In locusts, auditory signals are encoded by 60-80 receptor neurons at each ear with similar general properties but considerable variability in the parameter values describing the sensitivity of individual neurons to specific sound frequencies (Römer, 1985
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Figure 3. Firing-rate responses of a locust auditory receptor cell. A, Rate-intensity function for a 7 kHz pure tone. The observed sigmoidal shape of the rate-intensity function is typical for many receptor types. Below a threshold of ~45 dB SPL, the cell shows virtually no response. B, Rate-intensity functions of the same neuron for many different pure tones between 1.25 and 28 kHz. Connected points belong to the same sound frequency. Curves farther to the left correspond to frequencies at which the cell is more sensitive. Although there are large differences concerning the intensity range where the individual rate-intensity functions rise from threshold to saturation, their overall shape is very similar. For example, all measured rate-intensity functions have approximately the same slope in the rising part of the curves and saturate at around the same level. C, The same rate-intensity functions as in B, now shifted along the decibel axis such that they align at a firing rate of 250 Hz. This demonstrates the generic shape of the rate-intensity functions. D, Curves denoting equal firing rates at different sound intensities for the same cell. The threshold curve (solid line) and the intensities corresponding to constant firing rates of 150 Hz (dashed line) and 300 Hz (dotted line) are shown for pure tones between 1.25 and 28 kHz. The three curves are approximately parallel to each other, reflecting the similarity of the rate-intensity functions for different frequencies.
The frequency-resolved sensitivity of the receptors can be characterized by a threshold curve, i.e., the dependence of the threshold on the sound frequency (Fig. 3D). The receptors are fairly broadly tuned with characteristic frequencies in the range of 4 kHz (low-frequency receptors) to 15 kHz (high-frequency receptors), and the absolute sensitivities vary strongly between individual neurons (Römer, 1985
Measuring rate-intensity functions from a single receptor cell for many different sound frequencies reveals another property of the receptors; to good approximation, the rate-intensity functions are shifted versions of one another along the intensity axis, where intensity is measured in the logarithmic units of sound-pressure level, decibels SPL. This phenomenon has been reported previously by Suga (1960)
These key findings indicate that over the whole frequency range, the coupling of the physical stimulus is not substantially influenced by mechanical nonlinearities. In fact, a simple filtering mechanism captures the essence of the observed phenomenon. Let us assume that for all pure tones the firing rate is given by a single function r(An/Cn). An denotes the amplitude of a specific pure tone of frequency fn, and Cn is a frequency-dependent filter constant such that the firing rate depends only on the ratio An/Cn. This corresponds to a gain factor of 1/Cn for each sound frequency. For two different frequencies f1 and f2, the firing rates r(A1/C1) and r(A2/C2) are then the same when A1/C1 = A2/C2, i.e., when the amplitudes take on a constant ratio A1/A2 = C1/C2. Because the intensity I in decibels SPL is defined as a logarithmic measure of the amplitude, I = 20log10(A/(
·20 µPa)), this constant amplitude ratio corresponds to a constant intensity difference, Step 1: coupling to the stimulus
The sound pressure wave S(t), written as a Fourier series:
where the fn denote the frequencies,
(13) (t):
The amplitudes are multiplied by frequency-dependent gain factors 1/Cn. These describe the frequency-resolved sensitivity, i.e., the tuning of the receptor cell, and correspond directly to the values of the threshold curve at the frequencies fn. In addition, a putative phase shift turns
(14) n. Although the above reasoning for using a linear filter as the first model step is based on electrophysiological observations only, it corresponds well with biophysical findings regarding the tympanal membrane. Schiolten et al. (1981)
Step 2: mechanosensory transduction
Receptor cells are attached to the tympanal membrane with a cilium protruding from the dendrite and several auxiliary cells surrounding a receptor (Gray, 1960
Following ideas from the literature concerning temporal integration in auditory receptor cells (Tougaard, 1996
(t). Amplitude hypothesis (AH)
J corresponds to the maximum amplitude of(t). This is the common view of a threshold: a response occurs once the signal reaches a certain value. In the case of few frequency components, J is given by the sum of the scaled amplitudes:
(15) Energy hypothesis (EH)
J corresponds to the temporal mean of the squared signal [throughout what follows,
From Parseval's Theorem (Press et al., 1992
(16)
Because the square of the amplitude of a sinusoidal oscillation is proportional to the energy contained in the oscillation, this hypothesis reflects an energy-integration mechanism.
(17) Pressure hypothesis (PH)
J corresponds to the temporal mean of the absolute value of(t):
This hypothesis complies with a pressure-integration mechanism after half-wave rectification.
(18) Step 3: encoding by firing rates
The response of an auditory receptor to a signal of constant intensity can be characterized by a mean firing rate r. The rate is obtained from a one-dimensional, nonlinear transformation of the effective sound intensity J:
Note that the effective sound intensity J is a theoretical construct, which does not necessarily correspond to a biophysical property. It is used here to describe regions of constant firing rate in stimulus space because these correspond to regions of constant J. Therefore, instead of the specifically simple versions of J given above, we could also use any transformation
(19) = f(J) with some appropriate function f. This transformation does not affect the shape of the regions of constant J, but we can speculate that for the correct choice of f, has a direct biophysical interpretation, such as the change in membrane conductance caused by the stimulus. Measured spike-train responses have a strong transient attributable to adaptation. In a first approach, we average over this temporal structure in the response and consider only the total number of spikes elicited by the stimulus. In a second, more detailed analysis, we analyze individual parts of the response to explicitly test how this structure in the spike trains might affect our model description.
Electrophysiological experiments
Experimental strategy
To directly address the question of spectral integration and the hypotheses in step 2 of our model, we compare only stimuli that lead to the same firing rate of a given neuron. With this strategy, we circumvent complications attributable to the nonlinearity induced by the spike-generation mechanism. In terms of our model, </
