Fig 1.
Empirical response patterns of GBC and schematic drawings of the model structure and PSTH output.
A-B: PSTHs of cat GBCs: a PLN-type (A) and an OnL-type (B). Adapted from [24]. C-D: Low-frequency phase histograms of a cat auditory nerve (C) and GBC (D). Adapted from [25]. CF: characteristic frequency; VS: vector strength. E: Operation of the adaptive coincidence counting model (see Materials and Methods for the equations). The model neuron receives ME excitatory synaptic inputs (red) simulated by the AN model (ME = 3 in this example). Each presynaptic input spike (small vertical red bar) induces a postsynaptic response of a length WE and amplitude AE in the model neuron (black rectangular bump on left). The adaptive threshold θ(t) (green), parameterized by a time scale TA and strength SA, develops according to the summed input count v(t) (black). When the input count reaches or exceeds the threshold (arrowhead), an output spike is generated (blue). After each spike output, the model is in the refractory period of a duration TR (gray), in which no further spikes can be generated. Insufficient synchrony of input spikes (three inputs on right) leads to a failure of output spike generation due to the adaptive threshold. F: Components of a primary-like-with-notch (PLN) peristimulus time histogram (PSTH) used for judging the plausibility of each instance of the modeled bushy cell. Depending on the parameters used, a second notch may or may not exist while the other components almost always appear.
Table 1.
The total number of parameter combinations is 7×10×9×9×10×10 = 567,000. The excitatory input amplitude does not have a unit, as it is defined as the relative amplitude with respect to the static threshold of θS = 1. See "Adaptive coincidence counting model" and "Selection of model instances" in Materials and Methods for detailed explanations of the parameters and their ranges used. The parameter values used for Figs 2, 3, 4, 12A–12C, 13A–13C and 14 are shown in bold.
Fig 2.
Response patterns of the model input and output.
A: PSTHs of the AN model. B: PSTHs of a representative GBC model instance that receive the AN model inputs. C: Period histograms of the AN model. D: Period histograms of the GBC model. E: ISIHs of the AN model. F: ISIHs of the GBC model. Each column corresponds to either low- (350 Hz), medium- (1400 Hz), or high-frequency (7000 Hz) tonal stimulation. The sound intensity was fixed at 70 dB SPL. SR: average spontaneous rate (in spikes/sec); DR: average sound-driven rate (in spikes/sec); CV': modified coefficient of variation; VS: vector strength; EI: entrainment index; see Materials and Methods for their definitions.
Fig 3.
Level-dependent activity of model input and output.
A-C: Level-dependent spiking rates (top) and phase-locking (bottom) of AN (red) and GBC (blue) models driven by low (A: 350 Hz), medium (B: 1400 Hz), and high (C: 7000 Hz) frequency tonal stimulation. Filled circles in A (bottom) and C (top) indicate the responses at 70 dB SPL, the level at which the GBC model was further tested. D: Level-dependent PSTH shapes of the AN (red inset) and GBC (blue) models for 7000 Hz tonal stimulation.
Fig 4.
Frequency-dependent activity of model input and output.
A: Spiking rates for tonal stimulation below 1 kHz. Dotted line shows the spiking rate being equal to the stimulus frequency. B: Spiking rates for tonal stimulation between 0 and 12 kHz. C: Coefficients of variation of the interspike intervals. Vertical dotted lines show empirical lower and upper bounds for bushy cells [30]. D: Vector strength (VS) plotted along the scale of log(1-VS). Symbols show experimentally observed phase-lockers (PhL, diamond: units with multiple peaks in their PSTH that lock to the stimulus frequency; they are most likely to be SBCs or GBCs but possibly other cell types), PLN units (unfilled triangle: presumed to be GBCs), and histologically confirmed GBCs (filled triangle). E: Entrainment index. Symbols show fibers recorded in the trapezoid body (TB, diamonds: most likely to be SBCs or GBCs but possibly other cell types) and histologically confirmed GBCs (filled triangle). Red and blue lines in A-E show AN and GBC model responses, respectively. The empirical data reported in [25] were used for the AN upper bound of VS (dashed purple curve in D) and data points in D-E.
Fig 5.
Dimensional stacking plots of the model parameter space.
A: Schematic presentation of dimensional stacking. Six-dimensional parameters are nested into three layers of two-dimensional spaces. Each rectangular cell of the outer layer, which corresponds to a specific combination of two parameters (number of inputs ME and adaptation strength SA, in this case), represents a middle layer (input amplitude AE and refractory period TR), which, in turn, consisted of the cells of the inner layer (coincidence window WE and adaptation time constant TA). B: Steady-state driven rates (7000 Hz pure tone, 70 dB SPL) of all 567,000 model instances shown with dimensional stacking. The spiking rate threshold of 150 (spikes/sec) is shown in black: instances that had lower spiking rates than this threshold (white-orange) were discarded, while instances with spiking rates equal to or higher than this threshold (dark to light blue) were adopted for further analyses. C: Degree of phase locking quantified as VS (350 Hz pure tone, 70 dB SPL) of all model instances. The VS threshold of 0.9 is shown in black: instances that showed lower VS than this threshold (white-orange) were discarded and those with higher (blue) were accepted. D: Steady-state driven rates of model instances that satisfied all the criteria of spontaneous activity, high-frequency sound-driven activity (rate and coefficient of variation) and low-frequency sound-driven activity (phase-locking and entrainment). E: Steady-state driven rates of model instances that satisfied all the criteria in D and additionally showed PLN-type PSTHs. In D and E, discarded model instances are shown in white, while accepted models are shown by the same color code as in B.
Fig 6.
Representative high-frequency tone-driven PSTHs (7000 Hz, 70 dB SPL) and low-frequency tone-driven ISIHs (350 Hz, 70 dB SPL, insets) of model instances.
A-E: PLN instances that were included in our GBC model population. The second peak may be large (A) or small (B). Steady-state rates may also be high (C) or low (D), with other properties being almost identical. Some PLN instances show a very large peak in ISIH, indicating pronounced phase-locking and entrainment (E). F-J: Non-PLN instances including: OnL units having pronounced onset activity followed by a low sustained rate (50–150 spikes/sec, F-G), chopping response with multiple onset peaks and notches (H), and a dip in PSTH having a long (> 2 ms) pause after the first (I) or second onset peak (J). Parameter values used for each panel are shown in Table 2.
Table 2.
Parameter values used for Fig 6.
Fig 7.
Distribution of parameters for accepted GBC models.
A-F: PLN-type instances. G-L: OnL-type instances. A,G: Number of excitatory inputs ME. B,H: Coincidence window WE. C,I: Excitatory input amplitude AE. D,J: Refractory period TR. E,K: Time constant of adaptation TA. F,L: Strength of adaptation SA. The total number of model instances in each plot is 7520 (PLN) or 4094 (OnL).
Fig 8.
Distribution and correlations of pairs of parameters for accepted PLN- and OnL-type GBC model instances.
(Upper triangle) Distribution of parameter pairs. Each square panel presents the distribution of a specific parameter combination. The vertical or horizontal axis of each panel spans the range of a corresponding parameter (Table 1). Ascending parameter values are arranged from down to up (along the vertical axis) or from left to right (along the horizontal axis). The area of each filled circle shows the count of accepted parameter pairs. (Lower triangle) Correlation coefficients of parameter pairs. Distributions and values are marked in blue (positive correlation) or purple (negative correlation) for parameter pairs with an absolute correlation coefficient over 0.2 (and with p < 10−9 for 7520 + 4094 = 11614 instances), and in gray otherwise (i.e., with absolute correlation coefficients below 0.2).
Fig 9.
Distribution of output measures for accepted GBC model instances.
A-E: PLN-type instances. F-J: OnL-type instances. A,F: Spontaneous rate (SR). B-C,G-H: High-frequency tone-driven responses (7000 Hz, 70 dB SPL): steady-state rate (DR: B and G) and modified coefficient of variation of ISIs (CV': C and H). D-E,I-J: Low-frequency tone-driven responses (350 Hz, 70 dB SPL): vector strength (VS: D and I) and entrainment (EI: E and J). The accepted ranges were: 0–30 spikes/sec (A,F); >150 spikes/sec (B) or 50–150 spike/sec (G); 0.65–0.95 (C,H); >0.9 (D,E,I,J). The total number of model instances in each plot is 7520 for PLN and 4094 for OnL.
Fig 10.
Polynomial fit of model parameters to output measures in accepted PLN- and OnL-type GBC model instances.
R2-values for the linear, quadratic, and cubic regressions are displayed. For the linear and quadratic fits, relative contribution of each parameter (or each parameter pair) is shown by the area of a filled circle. Circles are colored dark blue for 12% or more variance explained, light blue for over 3% of variance explained, or gray otherwise. ME, number of inputs; WE, coincidence window; AE, excitatory input amplitude; TR, refractory period; TA, adaptation time constant; SA, adaptation strength.
Fig 11.
Frequency-dependent temporal properties of the GBC population.
A-B: Vector strength (VS) of PLN (A) and OnL (B) population plotted along the scale of log(1-VS). C-D: Entrainment index (EI) of PLN (C) and OnL (D) population. Solid curves in each panels show the lower and upper bounds of VS and EI calculated for the modeled GBC population (7520 PLN-type or 4094 OnL-type instances). Symbols in A-D show empirical data [25] (see legend of Fig 4 for the description of these data points).
Fig 12.
A: PSTHs of the AN (top, red) and GBC (bottom, blue) models driven by 500-Hz tone at 95 dB SPL (see Table 1 for the parameter values used). B-C: Level-dependent spiking rates (B) and phase-locking (C) of the same AN (red) and GBC (blue) models as in A. GBC responses to CF-tonal stimulation (dashed purple) are also shown for comparison. In A-C, the characteristic frequency (CF) of the models is 7000 Hz. Filled circles in B-C indicate the responses with which the model was further tested. D-F: CF-dependent tail-sync phase-locking of the PLN model population (D: blue curves), OnL model population (E, green curves), and the AN model (F: red curve) to 500-Hz tone at 95 dB SPL. Symbols in D and E show phase-lockers (PhL, diamond: most likely to be SBCs or GBCs but possibly other cell types), PLN units (unfilled triangle: presumed to be GBCs), and histologically confirmed GBCs (filled triangle), reported in [63]. Data points in F show empirical data of cat AN fibers [63]; different symbols correspond to different spontaneous rate (SR) categories.
Fig 13.
A: PSTHs of the AN (top, red) and GBC (bottom, blue) models driven by 100-Hz SAM tone at 20 dB SPL (see Table 1 for the parameter values used). Characteristic frequency (CF) of the models is 7000 Hz. B-C: Level-dependent spiking rates (B) and envelope phase-locking (C) of the same AN (red) and GBC (blue) models as in A. Filled circles in B-C indicate the condition for which the model showed the maximum VS. D-F: CF-dependent envelope phase-locking of the PLN model population (D: blue curves), OnL model population (E: green curves), and the AN model (F: red curve) to 100-Hz SAM tone at 20 dB SPL. In D-E, the maximum VS was computed for sound intensities between 10 and 45 dB SPL (5 dB steps). Triangles in D and E indicate empirical single-unit data of presumed GBCs, and hollow diamonds in F show AN fiber data recorded in cats [66].
Fig 14.
Simulated frequency response areas (FRAs).
A: FRA of the AN model. B: FRA of the default GBC model (see Table 1 for the parameter values used). CF of both models was fixed to 3500 Hz. The level and frequency of the stimulus tone were varied and the resulting spiking rates of the models were calculated and plotted in color. In both panels, the same color code was used for the range of 60–220 (spikes/sec) to facilitate comparisons. White curves show the estimated rate-level threshold at each frequency, defined as the sound pressure level at which the spiking rate was 10 (spike/sec) higher than the spontaneous rate.
Table 3.
Computational time required for calculating 500-sec responses.
"AN exact" and "AN approx" refer, respectively, to the exact and approximate calculation of the power law adaptation of the modeled AN (see [59]). Computations were carried out with Matlab 2018a (MathWorks) on a desktop computer (Dell Precision T1700) with 64 bit Windows 7 Professional Operating System, Intel Xeon CPU E3-1270 v3 (4 core, 3.5 GHz) and a 16 GB memory.