INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B REFERENCES

The sensitivity of the auditory system to amplitude transients is well documented, both physiologically and psychoacoustically. Psychoacoustical studies have demonstrated the importance of the temporal structure of amplitude envelope to auditory perception in general (e.g., Drullman 1995; Drullman et al. 1994a,b; Shannon et al. 1995; Turner et al. 1994), and to the segregation process of complex auditory scenes in particular (Bregman et al. 1994a,b). These studies demonstrate that both the magnitude and duration of amplitude transients affect auditory perception. However, it is still unclear which physical parameters of the amplitude transients most affect auditory perception of the transient.

Animal studies have shown that temporal changes in amplitude envelope in general, and amplitude onset in particular, generate strong neural responses throughout the auditory pathways (Eggermont 1993; Kitzes et al. 1978; Phillips 1988; Rees and Møller 1983; Schreiner and Langner 1988a; Suga 1971). Several studies of the dependence of neuronal responses on the shape of an onset ramp (Barth and Burkard 1993; Heil 1997a,b; Heil and Irvine 1996, 1997; Phillips 1988, 1998; Phillips and Burkard 1999; Phillips et al. 1995) have shown that neural response characteristics can neither be ascribed to a simple function of onset plateau level nor to onset duration per se. Rather, the dynamics of the onset, such as the rate or acceleration of peak pressure, shape the neural response. These phenomena are evident across multiple levels of the auditory pathways. Furthermore, they have been demonstrated using a variety of experimental procedures, such as single-cell recordings from the cat primary auditory cortex and posterior field (Heil 1997a,b; Heil and Irvine 1996, 1998b; Phillips 1988, 1998), inferior colliculus potential of the awake chinchilla (Phillips and Burkard 1999), and human brain stem-evoked response (Barth and Burkard 1993).

The dependence of neural responses on the dynamics of the amplitude envelope raises the possibility that these responses reflect the computation of temporal auditory edges. Following this assumption, we suggest a neural model for the detection of amplitude transients (auditory temporal edges), which is inspired by visual edge detector models. The model responses are compared to published physiological responses to amplitude transients, and its predictions regarding the responses to amplitude transients that have not been examined before are verified experimentally. In addition, we attempt to define the physical parameters of amplitude transient that affect human perception of amplitude discontinuity, in order to characterize the psychophysical properties of perceived auditory temporal edge.

Our results suggest that the same physical parameters may govern both physiological and psychophysical responses to amplitude transients. Moreover, we show that both physiological and psychoacoustical responses can be explained by our simple neural model for auditory temporal edge detection. These results suggest that the sensitivity of the auditory system to amplitude transients is a realization of auditory temporal edge calculation that may have a primary role in neural auditory processing.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B REFERENCES

Neural model principles

In line with the auditory-visual edge detection analogy, we adapted a model of visual edge detection to the auditory modality. The fundamental principle of the operation of visual edge detector is the calculation of a local brightness gradient. This is accomplished by differentiating the brightness function along some spatial direction or directions, using a combination of inhibitory and excitatory connections. The spatial organization of these connections in terms of the retinal image induces a receptive field that might be functionally described as an edge detector. Although there are recent and more elaborated visual receptive fields models, the simplest edge detecting receptive field model (Marr 1982; Rodieck 1965), which has an on-center off-surround (or vice versa) response pattern, suffices for our purpose. This receptive field describes the responses of edge detector neurons that can be found mostly in sub-cortical visual centers. The spatial properties of an idealized receptive field can be approximated by the second derivative of a gaussian or a difference of two gaussians (DOG), one wider than the other.

To adapt such a mechanism to auditory temporal edge detection, we hypothesize the existence of a temporal delay dimension, analogous to the visual spatial dimensions. The stimulus is progressively delayed along this delay dimension. Information related to the temporal dynamics of the amplitude envelope (e.g., its rate of change) can be made explicit by differentiating the stimulus along this dimension, as the visual brightness gradient is made explicit by differentiating the stimulus along a spatial dimension.

We construct the delay dimension by using the well-known temporal characteristics of a standard version of the integrate-and-fire model (I&F). Our I&F makes use of a kernel function in the form

(1)

The kernel function, when convolved with the neuron's presynaptic input, determines its postsynaptic potential (Gerstner 1999a). _m is the membrane time constant that may range from 3 to 25 ms (McCormick et al. 1985). Higher _m values induce greater delay in the neuron's response (Agmon-Snir and Segev 1993). Inducing a receptive field in the delay dimension can be done by connecting the neurons with increasing _ms to an edge detector neuron using inhibitory and excitatory connections with various efficacies that reflect the receptive field shape. Differentiation of the stimuli is obtained by using a receptive field shape of a first-order derivative of a gaussian.

Figure 1 presents a schematic diagram of the model and the flow of data along the different model components. Each model component is annotated with an approximate expression for its operation on its input. These formulations will be used in the analysis of the model. Exact implementation details are given in APPENDIX A. The inputs tested consisted of tone bursts shaped with ON and OFF ramps of various shapes. An example of a tone burst with linear ramps is displayed in Fig. 1A.

View larger version (29K): [in this window] [in a new window]   Fig. 1. Schematic diagram of the model. See detailed explanation in METHODS.

NEURAL REPRESENTATION. The neural representation (Fig. 1B) is roughly the expected peripheral representation of sound by the inner hair cell DC potential. This representation is generated using a simple preprocessing that includes demodulation to extract the temporal envelope, non-linear compression and low-pass filtering. In our analysis we formulate the demodulation and the non-linear compression using the amplitude envelope of the input converted to dB SPL scale (the constants in Fig. 1B are set to A = 20/ln (10) and P₀ = 2 · 10⁵ Pa). The form of the argument to the log transformation eases the analysis for near-zero values of t and has negligible effect for t  0. The low-pass filtering is formulated by convolving the log-envelope with an alpha kernel function (Eq. 1) with a time constant of ₁, which is in the millisecond range (Hewitt and Meddis 1990; Smith 1988). This preprocessing stage can be replaced by a more realistic inner hair cell model (which produces simulation of auditory nerve firing probabilities) (Hewitt and Meddis 1990; as implemented by Slaney 1998) without any qualitative change in the response characteristics of the model.

DELAY LAYER. The preprocessed input is fed to the delay layer of the model, which consists of standard integrate & fire (I&F) neurons with ascending membrane time decay constants. Each unit U(t, ₂) in the delay layer represents a population of neurons with identical characteristics. The population response is modeled as an analogue variable, by convolving the neuronal representation N(t), with a kernel (Gerstner 1999b) whose time constant is ₂. I&F kernel functions and membrane time-constant values are shown for several units (Fig. 1C). The membrane potential of each neuron in the delay layer is then saturated using a sigmoidal function

(2)

where F_max is the maximal instantaneous output firing rate (225 spikes/s) and C is a scaling factor, which determines the dynamic range of the transformation. In Fig. 3, the outputs of the delay layer neurons (including various amounts of saturation) are shown for stimuli, similar to the stimulus presented in Fig. 1.

RECEPTIVE FIELD. The delay layer neurons are connected to an edge detector neuron using inhibitory and excitatory connections with various efficacies (Fig. 1D) that reflect the receptive field shape, which is a first derivative of a gaussian. The output of the receptive field, R(t), is shown in Fig. 3 for stimuli similar to the stimulus presented in Fig. 1 and is approximately a smoothed first derivative of the outputs of the delay neurons along the ₂ dimension.

EDGE DETECTOR NEURON. The edge detector neuron (Fig. 1E) is a single I&F neuron with a membrane time constant ₃. The output of the edge detector neuron is also the output of the model. In the numerical implementation of the model, a noisy integration was used (Gerstner 1999a). For the analytical treatment presented here, the membrane potential of the edge detection neuron, M(t), is modeled as a low-pass filter operating on the output of the receptive field operator, R(t).

PARAMETERS OF THE MODEL. The responses of the model are adjusted to fit the response of a specific neuron by adjusting two parameters. The first parameter is C, the scaling factor of the delay layer saturation transformation (Eq. 2), and the second parameter is ₃, the membrane time constant of the edge detector neuron. In addition, the threshold of the edge detector neuron was varied. However, the threshold was not manipulated independently; instead, its value was always set to best approximate the threshold of the neuron that was fitted. There are six additional fixed parameters of the model; three of them are parameters of the I&F model. These parameters and their values are listed in full in APPENDIX A. Their specific values have only minor or redundant effect on the responses of the model. For example, changing the value of ₁ or the range of ₂ that are used in the delay layer can be in large extent be compensated by adjusting the value of ₃.

Physiological methods

ANIMALS AND PREPARATION. Neurons have been recorded in primary auditory cortex (AI) and medial geniculate body (MGB) of two halothane-anesthetized adult cats. The methods have been described in details elsewhere (Nelken et al. 1999). In short, the cats were premedicated with xylazine (0.1 ml im), and anesthesia was induced by ketamine (30 mg/kg im). The radial vein, the femoral artery, and the trachea were cannulated. Blood pressure and CO₂ levels in the trachea were continuously monitored. The cat was respirated with a mixture of O₂/N₂O (30%/70%) and halothane (0.2-1.5%, as needed). Halothane level was set so that arterial blood pressure was kept around 100 mmHg on the average. Under these conditions, the cat usually could be respirated without the use of muscle relaxants. In case muscle relaxants were required, the depth of anesthesia was evaluated by testing paw withdrawal reflexes before administering low levels (pancuronium bromide, 0.05-0.1 mg iv, typically once every 2-3 h). Lactated ringer was continuously given through the venous catheter (10 ml/h). Every 8-12 h a chemical analysis of arterial blood was performed. When the cat developed acidosis, bicarbonate was given (typically 5 ml iv, every 8 h).

DATA ACQUISITION. Glass-coated tungsten electrodes (locally made) were used for recording neuronal activity. The activity from the electrodes was amplified (MCP8 Plus, Alpha-Omega), and spikes were detected on-line by a spike sorter (MSD, Alpha-Omega). The times of the spikes were recorded (ET1, TDT) and written into a file for off-line analysis.

ACOUSTIC STIMULATION. Stimuli were generated digitally converted to analog waveforms and attenuated using TDT equipment. All stimuli were tone bursts, 230 ms long including the symmetrical onset and offset ramps. Six types of onset/offset window shapes were used, cos² (t), cos⁴ (t), t, t², t⁴, and squared exponential. By denoting the plateau peak pressure in Pascal units with P, and the onset rise time in milliseconds with D, the peak pressure (in Pa) during the onset is given by

(3)

for the t, t², and t⁴ windows, and is given by

(4)

for the cos² (t) and cos⁴ (t) windows. For the squared exponential window, the peak level (in dB instead of in Pa) is given by Eq. 3 with n = 2, except that P is given in dB. To accommodate the peak pressure close to 0 Pa (at the beginning of the onset and the end of the offset), where the dB scale is singular, a short linear ramp was used up to peak sound levels of about 0 dB SPL.

Psychoacoustical methods

The main goal of our psychoacoustical experiments was to test whether the perception of amplitude changes is determined by the gradient of the change, or by some other combination of its duration and magnitude. A secondary goal was to rule out the possibility that the sensitivity of the auditory system to amplitude changes is due to a spectral splatter that may be induced by the sudden amplitude change. In order to accomplish these goals we used a direct measure of the way in which the amplitude change is being perceived, rather than measuring amplitude change effect on higher perceptual tasks. This enabled us to isolate the perception of the amplitude transient from the context of more elaborate auditory phenomena such as auditory source segregation, in order to avoid high-level cognitive influences. Two sets of experiments were conducted; the first measured the discontinuity perception of ramped sinusoids (experiment 1), while the second measured the perception of ramped noise bursts (experiment 2).

PARTICIPANTS. All participants were normal hearing volunteer adults, who participated with full informed consent. Data for experiment 1 were obtained from 10 participants. All except for one, who is one of the authors (YY), had no previous listening experience in psychoacoustical experiments. Data for experiment 2 were obtained from five participants. None had participated in experiment 1, and none had previous listening experience in psychoacoustic experiments.

STIMULI. Experiment 1 stimuli are pure tones with an amplitude envelope as illustrated in Fig. 2 (solid line). Onset and offset times are 150 ms, and both plateau amplitude periods are 1 s. The first plateau level (A₁), the amplitude ramp size (A) and duration (T), and the frequency of the tone were manipulated. The values used appear in Table 1. The set of stimuli is a full combination of the variable's values, thus forming a set of 224 unique stimuli, each of which was presented once. The stimuli were generated digitally and played over a Silicon-Graphics Indigo workstation at sampling rate of 16,000 Hz at 16-bit resolution.

View larger version (12K): [in this window] [in a new window]   Fig. 2. Illustration of the amplitude envelopes of the stimuli that were used in experiments 1 (solid line) and 2 (dashed line). In both experiments A, T, and A1 were manipulated. Note that the time scales for the 2 experiments are different. The total duration of the tone stimuli used in experiment 1 is 2,300 + T ms, while the duration of the noise bursts used in experiment 2 is 700 ms.

PROCEDURE. An identical procedure was used in both experiments. The stimuli were presented binaurally through Yamaha HP-2 earphones to the participants who were seated in a soundproof room. The psychophysical task was to judge whether the transition between the two plateau amplitude levels was a continuous or discontinuous one. The participants were asked to indicate their choice for each of the stimuli using a two-alternative forced choice procedure. A random training subset of 40 trials was presented to the listeners, followed by the entire set presented in random order. The listeners were unaware of the fact that the first trials were training trials. Participants had unlimited time to respond after each trial and were presented with the next trial 2 s after their response.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B REFERENCES

Neural model: general observations

The model was capable of reproducing all the physiological characteristics of onset responses in AI neurons. In particular, the model was capable to produce the shortening of latencies with increase in tone level, and was capable of generating both monotonic and non-monotonic rate-level functions.

Figure 3 illustrates the way the model responds to amplitude transients and the effect of the delay layer's saturation on the timing and strength of the responses. The log-compressed envelopes of linearly shaped 30-dB SPL and 90-dB SPL tone bursts are shown in Fig. 3A. The response of the model components to these stimuli is considered in two different saturation conditions. A model with a highly saturated delay layer, which yields non-monotonic responses, is described in Fig. 3, B, D, and F, while a model with only weakly saturated delay layer is described in Fig. 3, C, E, and G. For clarity, we consider a simplified delay layer that consists of only two neurons with time constants of 3 and 6 ms. Figure 3, B and C, demonstrates the different delays of the stimulus envelope that are being induced by the two neurons. The outputs of the two neurons are subtracted by connecting them to the edge detector neuron with weights of equal magnitude and opposite signs (Fig. 3, D and E). The prominent effect of the amount of saturation on the model responses emerges at this stage. For example, the total current (the integrated presynaptic input) that is being injected to the edge detector neuron in the highly saturated model (Fig. 3D) is higher in response to the 30-dB tone than to the 90-dB tone (155 vs. 89.7 in arbitrary units, respectively). In the weakly saturated model (Fig. 3E), the integrated presynaptic input is lower in response to the 30-dB tone that the 90-dB tone (81.9 vs. 246.8, respectively). The non-monotonicity of the highly saturated model is enhanced by the low-pass properties of the membrane potential of the edge detector neuron (Fig. 3F). The effect of the delay layer's saturation on the non-monotonicity of the model is being mathematically analyzed in APPENDIX B. Another effect of the saturation is decreasing the first-spike latency and shortening the period of neural activity. For the purpose of mathematical treatment, it can be reasonably assumed that a neuron starts to fire when its membrane potential hits a fixed threshold and that its spike count is proportional to the area enclosed by this threshold and the neuron's membrane potential (Fig. 3G).

View larger version (22K): [in this window] [in a new window]   Fig. 3. The output of several of the model components as response to sound bursts of 2 amplitude levels (A). Two model settings are shown, the 1st includes highly saturated delay layer (B, D, and F), while the 2nd is only moderately saturated (C, E, and G). The figure demonstrates how the input is being progressively delayed along a simplified delay layer (B and C) and differentiated using the receptive field (D and E) that is formed by the connections from the delay layer neurons to the edge detector neuron (F and G). For mathematical analysis of the model we define the 1st-spike latency of the model as the time from stimulus onset to the 1st time the edge detector membrane potential hits a fixed threshold level (L30 and L90 in G). The spike count is assumed to be proportional to the area enclosed by the membrane potential and the threshold level (striped area in G).

Evaluation of the neural model: single-neuron data

We evaluated the adequacy of the model to match reported neural response to sound bursts by feeding the model with the amplitude envelope of the stimuli and comparing several aspects of the model output with those of the reported responses. The properties of the output examined were the first spike latency of the response, the response strength measured by the number of spikes that followed a stimulus, and the relationships between the two.

LATENCY. Heil and his co-workers (Heil 1997a; Heil and Irvine 1996) studied the latency of primary auditory cortex neurons (AI) as a function of the shape, amplitude, and duration of the rise time of a best frequency tone. Two kinds of onset envelope functions were used, linear and cosine-squared. The peak amplitude during a linear onset is described by a power function as described in Eq. 3 with n = 1. The peak amplitude during a cosine-squared onset is described by Eq. 4, with n = 2.

(5)

(6)

View larger version (36K): [in this window] [in a new window]   Fig. 4. Experimental (replotted from Heil 1997a) vs. model simulated data for 1st-spike latency of the onset response. A and B: the latency as a function of the amplitude level. C and D: the latency as a function of maximal acceleration of the cosine-squared onset and the curve fitted by Eq. 6. Our fit for the neuron in C yielded S = 4.53 and Lmin = 10.87 ms, and for the simulated data in D the best fit yielded S = 5.12 and Lmin = 8.1 ms. E and F: the latency as a function of the rate of linearly shaped onset and the curve fitted by Eq. 5. The fit for the neuron in E yielded S = 4.9 and Lmin = 11.55 ms and for the model (F) S = 5.09 and Lmin = 6.7 ms. Neuron identity, neuron characteristic frequency (CF), and the model parameters that were used are shown above each plot. The difference between experimental and simulated Lmin values reflects constant delays (acoustic, cochlear, and neural delays), which are not included in the model. Model S values are consistently somewhat higher than those estimated from the data, as explained in DISCUSSION.

View larger version (30K): [in this window] [in a new window]   Fig. 5. Latency data from the cat posterior field (replotted from Phillips 1998) vs. model simulated data. The latency is plotted as a function of maximum acceleration of peak pressure and is fitted using Heil's functional form. Our estimated S value for neuron 93K010.24 (A) is 3.99, and the estimated value for the corresponding simulated latency (B) is 4.62. Estimated S for neuron 93K013.12 (C) is 4.42 and for the corresponding model setting (D) the estimation is 4.57.

FIXED-THRESHOLD MODEL DOES NOT FIT THE DATA. A possible explanation for the latency phenomena is that the neuron first spike occurs when the input stimuli level hits a fixed threshold (Kitzes et al. 1978; Phillips 1988; Suga 1971). Indeed, it is easy to show that such a simple model predicts a reciprocal relation between the first-spike latency of a neuron and the rate (P/D) of linear onsets and maximum acceleration (²P/2D²) of cosine-squared onsets. While these predictions roughly approximate the experimental results, the later systematically deviate from the predictions. On these grounds, Heil and Irvine (1996) argue against the simple threshold model. Their claims can be summarized by two main points that are illustrated in Fig. 6.

View larger version (22K): [in this window] [in a new window]   Fig. 6. A: mean 1st-spike latency of a neuron from cat primary auditory cortex (solid lines) as a function of rise time of a CF tone of 22 kHz (replotted from Heil and Irvine 1996). The dashed lines plot the best fit of the data according to a fixed-threshold model. Note that the latency is not a linear function of rise time and that the slope ratio of any two quasi-linear iso-step-size curves does not match the inverse ratio of the step sizes. The model reproduces these phenomena (B). Note that the latency axis is translated with respect to A for greater clarity.

MATHEMATICAL ANALYSIS OF LATENCY PHENOMENA. In our analysis we use the formulations given in Fig. 1, and we assume that the amplitude envelope of the input stimulus, E(t), can be approximated during the onset (for t  D) by a power function such as described in Eq. 3 for any n > 0. For simplicity sake we will restrict our analysis to t  D; this assumption is equivalent to the statement that the first spike occurred during the onset ramp (after taking into account constant latency components that are independent of the sound level).

COMPARING MODEL PREDICTIONS WITH LATENCY RESULTS OF PHYSIOLOGICAL EXPERIMENTS. Figure 7 shows the latency data of one AI unit in response to three types of onset windows, linear (Fig. 7A), cosine squared (Fig. 7C), and squared exponential (Fig. 7E). The latency for each window is plotted as a function of the predicted invariant measure, and the alignment of the latency data along a single curve for each rise function validates our predictions. Numerical simulations reproduce these phenomena (Fig. 7, B, D, and F). Figure 8A shows the latency of a MGB neuron in response to four types of amplitude rise function, cos² (t), cos⁴ (t), t², and t⁴. The latency of each rise function is plotted as a function of the predicted invariant measure, which is P/Dⁿ for the tⁿ rise functions and ⁿP/2ⁿDⁿ for the cosⁿ (t) rise functions {Taylor's series approximation of cosⁿ [(/2)t + (/2)] is (ⁿ/2ⁿ)tⁿ + o(tⁿ⁺²) for even n}. This way the latency data collected with the t² and the cos² (t) rise function aligns along a single curve, and the latency data collected with the t⁴ and the cos⁴ (t) rise function aligns along another curve. The model predictions also hold for the responses of a neuron in primary auditory cortex (Fig. 8C) and are being reproduced by the numerical simulations of the model (Fig. 8, B and D).

View larger version (32K): [in this window] [in a new window]   Fig. 7. First-spike latency of a single unit of a cat [primary auditory cortex (AI)] as response to 3 rise functions, linear (A, experimental; B, simulated), cosine squared (C, experimental; D, simulated), and squared exponential (E, experimental; F, simulated). Latency is plotted as a function of the predicted invariant measure of each rise function.

View larger version (31K): [in this window] [in a new window]   Fig. 8. First-spike latency measured using cos2 (t), cos4 (t), t2, and t4 rise functions from a single unit of a cat medial geniculate body (MGB; A and B, simulated) and AI (C and D, simulated). Latency data are plotted as a function of the predicted invariant measure [P/Dn for the tn rise functions and nP/2nDn for the cosn (t) rise functions].

SPIKE COUNT. Neurons in AI of anesthetized cat show a low spontaneous rate of fire, and their typical response to sound bursts is a single spike or a short burst of a few spikes immediately following the onset of the stimulus (e.g., Heil 1997b). Examining the spike count as a function of plateau peak pressure alone reveals a non-monotonic pattern that is shared by many AI neurons to various degrees (e.g., Heil 1997b; Heil and Irvine 1998a; Phillips 1988; Schreiner and Mendelson 1990). Furthermore, the non-monotonicity is enhanced at the shorter rise times. Figure 9 demonstrates the typical response patterns of two types of neurons, as replotted from Heil's (1997b) data. Figure 9A shows a highly non-monotonic neuron, whereas Fig. 9C shows a more monotonic neuron. The spike-count data are plotted as a function of plateau peak pressure and are organized along iso-rise-time curves. The model reproduces these phenomena over a wide range of degrees of monotonicity. Figure 9, B and D, demonstrates a good correspondence between experimental and simulated results. The correspondence is apparent for curve shapes as well as for order of displacement of the iso-rise-time curves, although the displacement of the model curves along the abscissa are much larger than those of the neural curves.

View larger version (37K): [in this window] [in a new window]   Fig. 9. Experimental vs. simulated spike-count data. Iso-rise-time curves of spike counts are plotted as a function of amplitude level of a cosine-squared onset, for a non-monotonic neuron (A, experimental; B, simulated), and a monotonic neuron (C, experimental; D, simulated). Experimental data are replotted from Heil (1997b). Simulated data for both neurons were obtained using the same sets of parameters that were used to match their latency data (see Fig. 4).

View larger version (28K): [in this window] [in a new window]   Fig. 10. Experimental and simulated spike-count iso-rise-time curves are closely aligned when plotted as a function of stimulus peak pressure at 1st-spike generation. Experimental data of Heil (1997a,b) (A, C, and E) and of Phillips (1998) (G) are recorded from single units of the cat AI. Original data from a single unit of the cat MGB are shown in I. Note that the model (B, D, F, H, and J) reproduces this phenomenon over a broad range of parameters.

Evaluation of the neural model: evoked auditory brain stem responses

The ability of the model to match reported evoked auditory brain stem responses in humans (Barth and Burkard 1993) and inferior colliculus potential (ICP) in the awake chinchilla (Phillips and Burkard 1999) in response to sound bursts, was tested by feeding the model with the amplitude envelope of the stimuli and comparing the model output to the reported responses. The membrane potential of our modeled edge detector neuron (Fig. 1E) was used as an estimate of the combined activity of a large population of brain stem neurons (Gerstner 1999b). The model activity was then differentiated to mimic the analogue highpass filter (with a slope of 6 dB/oct) used in these experiments.

Figure 11A shows a typical measure of the inferior colliculus potential in response to a tone burst as replotted from Barth and Burkard (1993). Figure 11B shows the differentiated membrane potential of the edge detector neuron of the model. This figure also illustrates the definitions of the latency and amplitude of the response. The two adjustable parameters that shape the model's response to amplitude transients (C and ₃) were adjusted to fit the latency and amplitude of the experimental responses.

View larger version (14K): [in this window] [in a new window]   Fig. 11. A: a typical wave-V brain stem auditory evoked response (BAER) as response to a 60-dB nHL 1.25-ms rise-time noise burst replotted from Barth and Burkard (1993). B: a differentiated membrane potential of the model edge detector neuron as a response to the same stimulus without the addition of noise. The response latency is measured with respect to the peak of the BAER, and the response amplitude is measured from the peak to the following trough, as illustrated in A.

In contrast to the stimuli that were used in single-cell recordings, whose total durations were 50-100 ms (Phillips 1998) or 400 ms (Heil 1997a,b), the stimuli that were used by Barth and Burkard (1993) and by Phillips and Burkard (1999) were much shorter and included plateau-level durations of 2-5 ms. For the model to accurately reproduce the experimental responses to these very short bursts, we had to reduce the time constant of the delay layer units from a range of 3-5 ms to a range of 0.5-1 ms, since higher time constants oversmoothed the envelope. The problem of using very short time constants when modeling mammalian inferior colliculus neurons has also been encountered in other modeling studies (Hewitt and Meddis 1994).

LATENCY. Phillips and Burkard (1999) measured the latency of the ICP in the awake chinchilla in response to cosine-squared onsets of various rise times and amplitude levels. Although Phillips and Burkard reported that there were strong similarities between the latency behavior of the ICP and that of cortical single cells, they did not use Heil's functional expression (see Eq. 6) to match the latency data according to the maximum acceleration of the onset envelope. Figure 12A shows that replotting the ICP latency data as a function of maximum acceleration of the envelope brings the iso-rise-time curves to converge along a single curve that can be fitted using Eq. 6 and by using the same value of the constant parameter (A_c) that was used by Heil (1997a). Figure 12B shows that the model reproduces the ICP latency data.

View larger version (32K): [in this window] [in a new window]   Fig. 12. A: replotting inferior colliculus potential (ICP) response latencies (Phillips and Burkard 1999) as a function of maximum acceleration of cosine-squared onsets yields a good alignment of the latency data along a curve that can be fitted by Heil's (1997a) functional form (Eq. 6). Our fit for the experimental ICP data yields S = 5.65 and Lmin = 3.46. The model reproduces these results (B), with a fit of S = 5.97 and Lmin = 1.18. C: replotting wave-V latencies (Barth and Burkard 1993) as a function of the rate of linear onsets reveals good alignment along a curve that only moderately fits Heil's functional form (Eq. 5) with S = 5.65 and Lmin = 6.43. The model matches the experimental results (D) but is better fitted by the functional form (S = 6.17 and Lmin = 2.27). Note that both Barth and Burkard (1993) and Phillips and Burkard (1999) used 0-ms rise time onsets. To allow a valid calculation of the envelope maximum acceleration and rate of change for these stimuli, we replaced the zero rise time by a 0.185-ms value. This value was found to best match the fitted curves for both the ICP and the BAER latency data.

RESPONSE AMPLITUDE. The effect of onset rise time and amplitude level on the ICP and on wave V of BAER response amplitude are similar to their effect on the spike count of monotonic cortical single cells. The response amplitude increased with ascending amplitude levels and with descending onset rise times. Figure 13 replots Phillips and Burkard's (1999) ICP amplitude response (Fig. 13A) and Barth and Burkard's (1993) BAER wave V response amplitude (Fig. 13C); both are plotted as a function of the plateau peak level. The simulated response amplitudes are presented in Fig. 13, B and D, and are scaled in order to match the experimental measurements.

View larger version (41K): [in this window] [in a new window]   Fig. 13. Response amplitude of ICP (A) replotted from Phillips and Burkard (1999), and response amplitude of wave-V BAER (B) replotted from Barth and Burkard (1993) as a function of plateau peak pressure. Note the resemblance between the 2 experimental findings in response to stimuli of comparable parameters and between the experimental and simulated data (B and D).

Results of the psychoacoustic experiments

The results of the two experiments were analyzed using a stepwise logistic regression. The dependent variable was set to be the probability of eliciting a discontinuous response, and the independent variables included the stimuli parameters used in each experiment (as detailed in Tables 1 and 2, respectively). In addition, motivated by our model, we added to the two sets of independent variables: the logarithm of the normalized rate of change of the ramp peak pressure re the base peak pressure (this is the invariant measure for the stimuli used here, see next section).

EXPERIMENT 1. The regression results show that the variable that accounts for most of the variance is the normalized rate of the ramp peak pressure [F_(1,2238) = 1,948.6, P < 10¹⁵]. Other significant variables are the duration of the change, [F_(1,2237) = 60.8, P < 10¹²]; and the tone frequency [F_(1,2236) = 32.5, P < 10⁷].

EXPERIMENT 2. The results of the second experiment also found the normalized rate of peak pressure to be the variable that accounts for most of the variance [F_(1,898) = 509.6, P < 10¹⁵]. Other significant variables were the first plateau amplitude level, [F_(1,897) = 29.2, P < 10⁷]; the step amplitude [F_(1,896) = 11.5, P < 0.0007] and the step duration [F_(1,895) = 11.07, P < 0.0009].

View larger version (37K): [in this window] [in a new window]   Fig. 14. A: mean results across all participants for experiment 1 (solid lines) and experiment 2 (dashed lines). The probability for the amplitude ramp to be perceived as a discontinuous change is plotted as a function of the normalized rate of the ramp peak pressure re the pedestal (see text). This produces a good congruence of the data along a typical psychometric curve. A plot of the simulation results (B) shows a good fit with the psychoacoustic data. C: a replot of the discrimination score from Bregman et al. (1994b) as a function of the normalized rate of change of the incremented partials. The model matches the data only moderately (D).

Evaluation of the neural model: psychoacoustic data

In the following section we will compare the model responses with the results of three psychoacoustical experiments. These experiments include the experiment reported above that tested the perception of amplitude discontinuity; an experiment that tested the effect of amplitude transients on auditory segregation (Bregman et al. 1994b); and a forward masking experiment (Turner et al. 1994) that tested the effect of the probe rise time on the degree of masking. Although these experiments investigate different auditory phenomena, we demonstrate that by identifying the psychoacoustical measures with the responses of the neural model to the amplitude transients presented in the experiments, the model is able to reasonably reproduce the psychoacoustical results.

In two of these experiments (Bregman et al. 1994b and the experiment reported here) the stimuli contained an amplitude ramp rising above a pedestal. The invariant measure for these stimuli is not the rate of rise of the amplitude ramp per se, but rather the normalized rate of rise re the pedestal, P*/Dⁿ, where P* is the plateau peak pressure of the ramp normalized by the ratio between the pedestal peak pressure and P₀ (see Fig. 1B). Intuitively, this follows from the fact that the essential operation of the model is differentiating the log-compressed amplitude envelope. Therefore the output of the receptive field (Fig. 1D) is not changed by multiplying the input stimuli by a constant factor. In consequence, the response to a ramp rising above a pedestal is identical to the response to the onset of a sound with the same size in dB re P₀ and with the same shape. Note that we arbitrarily set the value of P₀ to 0 dB SPL for simplicity sake. Using different P₀ values can be compensated by adjusting the threshold value of the edge detector neuron. P₀ value is significant only when fitting the model responses with the responses of a specific neuron to both the onset of a sound and to a ramp rising above a pedestal. In these cases P₀ may be adjusted to best fit the neural responses to both types of stimuli.

PERCEPTION OF AMPLITUDE DISCONTINUITY. To compare our psychophysical results and the model predictions, a function of the neural response compatible with the dichotic nature of the psychophysical responses is required. As mentioned earlier, the modeled neurons have low spontaneous activity, and their responses to sound bursts consist of a short burst of 1-3 spikes. Therefore it seemed plausible to define a response to a stimulus as one or more spikes, and to identify the probability of response as the probability that a participant would report a discontinuous amplitude change in the psychophysical experiment. This measure did in fact yield a good match between the simulated (Fig. 14B) and the experimental results.

EFFECT OF AMPLITUDE TRANSIENTS ON AUDITORY SEGREGATION. One of the few studies that tested the effect of both the duration and magnitude of amplitude changes on auditory segregation tasks has been reported in Bregman et al. (1994b). They presented a 3.5-s long complex tone consisting of five harmonics of 500 Hz. The amplitudes of an adjacent pair of the three middle frequencies (1,000, 1,500, and 2,000 Hz) were incremented in succession in random order. A sufficiently large amplitude increment caused the partials to be segregated from the complex tone, and to be perceived as separate tones. To measure the degree of segregation, the participants had to judge whether the perceived pitch pattern, caused by the segregated partials, went up or down. Three levels of increments were used (1, 3, and 6 dB) and six increment durations (30, 90, 270, 730, 910, and 970), resulting in a total of 18 experimental conditions. The overall amplitude level of the complex tone in its steady state was 65 dB SPL. Bregman et al. reported that both the amplitude increment level and the increment duration had a significant effect on the participants' performance. Longer increment duration resulted in poorer discrimination performance, while larger increment levels led to better discrimination. These results suggest that the gradient of the increment had a dominant effect on discrimination performance. However, Bregman et al. did not include the gradient of the increment in their statistical analysis, and therefore it is impossible to determine the exact influence of the amplitude gradient of a tone on the ability to segregate it from a mixture of tones. When the results of Bregman et al. are replotted as a function of the normalized rate of peak pressure of the amplitude increment, the data fall along a single curve (Fig. 14C).

EFFECT OF AMPLITUDE TRANSIENTS ON RELEASE FROM FORWARD MASKING. In forward masking, the masker (which can be a tone or a noise burst) masks a target tone that appears just after the masker ends. The degree of masking depends on many factors such as masker level, bandwidth, duration, and the inter-stimulus interval. Turner et al. (1994) studied the effect of the target tone rise time and duration on forward masking levels. They used two types of target tones, one with a total duration of 25 ms including 2-ms cosine-squared rise/fall ramps, and the second with a total duration of 22 ms including 10-ms cosine-squared rise/fall ramps. Growth of masking (GOM) functions were measured using noise maskers at levels of 10-90 dB SPL. Their results show that targets with 10-ms rise time were masked more than targets of 2-ms rise time. In addition, Turner et al. showed that in contrast with the psychoacoustical results, there was no significant effect of the target rise time on the amount of masking that was measured in single auditory-nerve fibers of the chinchilla. This s