A phenomenological model of the synapse between the inner hair cell and auditory nerve: Long-term adaptation with power-law dynamics
Illustration of the dynamics of adaptation for exponential (dotted lines) and power-law (solid lines) models for four different time scales (0–0.1, 0–1, 0–10, and 0–100 s) in response to a unit step function. The parameters for the exponential adaptation are and . The parameters for the power-law adaptation are and . The solid curves retaining a similar shape across different time scales demonstrate the “scale-invariance” property of the power-law adaptation.
(A) Schematic diagram of the model for the auditory periphery. The input to the model is an instantaneous pressure waveform of the stimulus (in pascals) and the output is a series of AN spike times. The model includes a middle-ear filter, a feed-forward control-path, a signal-path (C1) filter and a parallel-path (C2) filter, the IHC section followed by the synapse model, and the discharge generator. Abbreviations: outer hair cell (OHC), low-pass (LP) filter, static nonlinearity (NL), characteristic frequency (CF), and inverting nonlinearity (INV). and are scaling constants that specify OHC and IHC status, respectively. From Zilany and Bruce (2006, with permission). (B) IHC-AN synapse model: exponential adaptation (three-store diffusion model by Westerman and Smith 1987, 1988) followed by parallel power-law adaptation models (slow and fast). Fractional Gaussian noise added at the input of the slow power-law adaptation model results in the desired distribution of spontaneous rates.
Histogram of actual (upper panel) and model (lower panel) SR estimates from 30-s recordings from 738 fibers in the auditory nerves of cats (binwidth of ). (A) Actual AN SR histograms from Liberman (1978, with permission). (B) Model histogram of SR estimates using the same paradigm as in Liberman (1978) for 738 independent simulations. Three parameter sets for the fGn were used, applied to the proportions of different SR fibers reported in Liberman (1978). fGn parameters are provided in Table I.
Illustration of the “scale-invariance” property of the PLA model. Here the output of the synapse model is shown before the discharge generator. The fGn was not included in the model to avoid fluctuation in the output. The dotted line indicates the spontaneous rate of the fiber. The stimulus was a 10-kHz tone at CF, 12 dB above threshold. The duration of the signal varied from 100 ms to 1 s, but the inter-stimulus interval was fixed at 200 ms in all cases. Responses to 50 repetitions of the stimulus were averaged. The dynamics of recovery from the stimulus offset to spontaneous rate scales according to the duration of the signal.
Effects of spontaneous rate on recovery in experimental [upper panels (A)] and model [lower panels (B)–(D): previous model, new PLA model with approximate and actual implementations, respectively] histograms of two AN fibers in response to 500-ms duration constant-amplitude stimuli. The stimuli were presented once a second. Each histogram represents 2 min of data collection. Left panels: CF = 1.82 kHz, HSR (unit 43 in data); right panels: CF = 10.34 kHz, LSR (unit 41 in data). (A) From Kiang (1965, with permission). (B) Model histograms of AN fibers at 25 dB SPL using the previous model (Zilany and Bruce, 2007) that has only exponential adaptation in the synapse model. (C) PLA model histograms at 25 dB SPL with approximate power-law implementation. (D) PLA model histograms at 25 dB SPL with actual power-law implementation.
Effect of exposure level on recovery for an AN fiber with CF = 2.15 kHz (HSR). The stimulus paradigm is illustrated at the top. Left panels [(A) and (C)] show the actual experimental responses, and the right panels [(B) and (D)] show the corresponding model responses. Duration of the exposure signal was 60 s, and the exposure levels were 29, 59, 74, and 89 dB SPLs (shown on the right/above of each curve). The test signal (100-ms long applied once per second) was also at CF with level 19 dB SPL (fixed). However, the test signal level in the model responses was at 9 dB SPL to match with the level of the experimental fiber with respect to its threshold. Total durations of the pre- and post-exposure test signals were 10 and 60 s, respectively. Recoveries of the post-exposure responses (fitted to an exponential) are shown with their corresponding time constant values. [(A) and (C)] From Young and Sachs (1973, with permission). [(B) and (D)] Model responses of recovery employing the same experimental condition as in the data. Responses to ten repetitions of the same stimulus were averaged.
Effect of inter-stimulus intervals on the responses of high and low SR fibers. (A) PST histograms of the PLA model in response to a 100-ms signal (tone at CF = 2 kHz), 40 dB above threshold for a HSR (upper panel) and a LSR (lower panel) fiber. The inter-stimulus intervals were 1.9, 0.303, and 0.103 s. With decreasing inter-stimulus interval, the peak of the onset was more reduced for the LSR fiber than for the corresponding HSR fiber. [(B) and (C)] Averaged value of normalized magnitude of the onset peak of PST histograms vs inter-stimulus intervals (0.103 and 0.303 s) for high (solid bar) and low (open bar) SR fibers is shown. The onset peak for each neuron was normalized by the onset peak of that neuron when the inter-stimulus interval was 1.9 s. The normalized values were averaged for all neurons within a group for different inter-stimulus intervals. (B) From Relkin and Doucet (1991, with permission). (C) Model responses from AN fibers with CFs ranging from 1 to 20 kHz for both high and low SRs.
AN fiber histograms and conservation of adaptation in both rapid and short-term components for the amplitude increment response paradigm (binwidth of 2 ms). [(A) and (C)] Physiological responses from gerbil (Westerman and Smith, 1987). [(B) and (D)] PLA model responses. The stimulus was at CF (5.99 kHz, HSR) with duration of 600 ms. The initial levels of the tone were 5, 10, 15 and 20 dB above threshold (background). At 300 ms, the intensity was increased to 43 dB above threshold (increment) in all cases. (A) Actual AN fiber histograms from Mongolian gerbil: from Westerman and Smith (1987, with permission). (B) Model histograms using the same paradigm as above, except that the highest level of the background tone was 25 dB above threshold (because the model fiber shows a wider dynamic range than the corresponding AN fiber of the physiological data). (C) Mean values of rapid and short-term components from six fibers; from Westerman and Smith (1987, with permission). (D) Model transient responses (for one AN fiber) from the corresponding model histograms (one fiber) shown in (B) using the same method as employed in the data.
Effects of prior adaptation on increment and decrement responses. (A) Model increment responses as a function of time for three different conditions of the synapse model (with only exponential adaptation, exponential followed by slow power-law adaptation, and exponential followed by both slow and fast power-law adaptations). (B) Increment responses from both actual and PLA model responses. (C) Decrement responses. The stimulus was a 60-ms CF tone 13 dB above threshold, and subsequently levels were either increased or decreased by 6 dB at different delays from onset. The resulting increment/decrement responses are obtained by subtracting the response to the constant-intensity tone (pedestal) from the response to the tone with an increment/decrement in level. Changes in rate responses for both the onset window (first 0.64 ms, circles) and a larger window (first 10.2 ms, downward triangles) are shown. Dotted lines with symbols show the actual data (data points from Figs. 5 and 7 of Smith et al., 1985), and the solid lines with symbols represent the corresponding PLA model responses. (B) Increment responses. CF at 4.16 kHz (HSR). (C) Decrement responses. CF at 3.58 kHz (HSR).
Actual [left panels (A)] and model [right panels (B)] post-stimulus recovery as a function of delay between masker offset and probe onset. Masker: 2.75-kHz tone (fiber’s CF, HSR), 30 dB above threshold (+30 dB), 100-ms duration. Probe stimulus: 2.75-kHz tone, +20 dB, 15-ms duration. Each data point represents the average number of spikes evoked by the probe as a percent of the control response (probe alone). The PST histograms on the right are the source for the data points on the left. (A) From Harris and Dallos (1979, with permission). (B) Model responses for the same paradigm as in the experiment. The solid line with filled circles represents the responses of the PLA model, and the dashed line with open circles indicates the responses of the previous model (Zilany and Bruce, 2007).
Forward-masking recovery functions for a population of fibers; masker level is the parameter. Masker stimuli were tones with frequency matched to CF, 100-ms duration. Probe stimuli were also tones at CF, +20 dB, 15-ms duration. (A) Actual median recovery functions from 37 fibers with CFs ranging from 0.5 to 16 kHz. From Harris and Dallos (1979, with permission). (B) Model average recovery functions from ten CFs (HSR and LSR) spaced logarithmically (range of 0.5–16 kHz).
Effect of increasing modulation depth on synchrony for a HSR fiber with a CF of 20.2 kHz in response to amplitude-modulated tones with a carrier frequency matched to CF, modulation frequency , and SPL = 49 dB (threshold = 32 dB SPL). Left panels [(A) and (C)] show the actual data and the right panels [(B) and (D)] represent the corresponding PLA model responses. Upper and lower panels show the period histograms, and their corresponding synchrony and gain, respectively, at different modulation depths. Modulation depths were varied from 0 to 0.99, and each histogram is flanked by a half-wave rectified version of the respective input AM stimulus to the right (two cycles of the responses are shown). [(A) and (C)] From Joris and Yin (1992, with permission). [(B) and (D)] Model period histograms and their corresponding synchrony and gain as a function of modulation depth using the same paradigm as employed in the experiment (the level of the stimulus is 17 dB above threshold). The dotted straight line represents 0-dB gain. Solid line with circles indicates the responses of the PLA model presented in this paper, and the dashed line with circles represents the responses of the previous model that had only exponential adaptation in the synapse model.
Effect of modulation depth and frequency on the actual [left panels: (A) and (C)] and PLA model [right panels: (B) and (D)] synchrony-level function. In the study of the effect of modulation depth (upper panels), CF is at 2 kHz (HSR fiber). For the effect of modulation frequency (lower panels) on synchrony-level function, CF is at 20 kHz (HSR fiber) with . [(A) and (C)] From Joris and Yin (1992, with permission). [(B) and (D)] Model responses using the same paradigm as in the experiment.
MTFs of high CF (>10 kHz) fibers. Upper panel shows the actual MTFs from cat, and the lower panel represents PLA model responses. (A) From Joris and Yin (1992, with permission). (B) Model MTFs for a population of fibers with CFs spaced logarithmically (range of 10–20 kHz) at level 10 dB above threshold for high, medium, and low SR fibers. Responses for 24 AN fibers (according to the proportions of the distribution of SRs) are simulated. Solid lines show the responses of the PLA model presented in this paper, and the two dashed lines (CF at 10 and 20 kHz) indicate the responses of the previous model.
MTF-3-dB cut-off frequencies vs CF and 10-dB bandwidth for high (plus) and low (down triangle) SR fibers. Left panels [(A) and (C)] show the actual responses from cat, and the right panels [(B) and (D)] represent the PLA model responses. (A) and (C) From Joris and Yin (1992, with permission). [(B) and (D)] Model responses for a population of fibers with CFs ranging from 250 Hz to 20 kHz (100 fibers spaced logarithmically) for high (61), medium (23), and low (16) SR fibers. Medium SR fibers are included in the low SR fibers, as treated in Joris and Yin (1992). Notice that the abscissae in the right panels (model responses) are different from those in the left panels (actual responses).
Effect of SR on maximum synchronization to . Upper panel shows the actual data from cat, and the lower panel shows the PLA model responses. (A) From Joris and Yin (1992, with permission). (B) Model responses for a population of fibers with CFs ranging from 250 Hz to 20 kHz for high, medium, and low SR fibers. Each fiber operates at 10 dB above threshold, and the maximum synchrony is chosen from responses to a wide range of (10 Hz–2 kHz).
Upper panels: central-peak height of normalized SAC to broadband noise (70-dB SPL) vs CF for a population of AN fibers. Lower panels: ratio of the value at delay 0 of XAC and SAC for a population of fibers. Left panels [(A) and (C)] show the actual responses from cat, and the right panels [(B) and (D)] represent the corresponding PLA model responses. Each point represents response from a single fiber. [(A) and (C)] From Louage et al. (2004, with permission). [(B) and (D)] Model responses for a population of fibers with CFs ranging from 250 Hz to 20 kHz (20 fibers logarithmically spaced) for high (plus), medium (circle), and low (downward triangle) SR fibers.
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