Fiber responses were obtained in cats (N animals = 5 for TB data and 2 for AN data) under general anesthesia. All procedures were approved by the K.U. Leuven Ethics Committee for Animal Experiments and were in accordance with the National Institutes of Health Guide for the Care and Use of Laboratory Animals. The recording procedures have been described before (Louage et al., 2005, 2006). In short, cats with normal eardrums and middle ears were brought under anesthesia with acepromazine (0.2 mg/kg) and ketamine (20 mg/kg). Sodium pentobarbital i.v. was used for maintenance of anesthesia. The animal was placed on a heating pad in a double-walled soundproof room. The pinnas were removed and the bullas exposed and vented. For TB experiments, the basioccipital bone was exposed and a longitudinal slit drilled close to the medial wall of the bulla. Since the TB contains crossing fibers, both ipsi- and contralateral fibers can be recorded from a single location. The AN was exposed via a posterior fossa craniotomy. Glass micropipettes filled with 3M NaCl or KCl were positioned in the TB or AN under visual control. Sounds were presented through speakers attached to ear bars that were tightly inserted into the ear canals. Sound stimuli were compensated for the acoustic transfer function measured with a probe tube near the ear drum. The neural signal was amplified, filtered and the action potentials were timed with 1 μs resolution.

The characteristic frequency (CF) of single fibers was determined using an automated threshold-tracking routine. Responses to short tone bursts at CF were obtained to allow classification of the fiber according to the shape of the peristimulus time histogram (Pfeiffer, 1966; Smith et al., 1991, 1993). Broadband noise stimuli were used with different inter-token correlation values. These were generated by mixing independent tokens of broadband noise with the same bandwidth (Robinson and Jeffress, 1963; Louage et al., 2006). The noise had a high-pass cut-off frequency of 50 or 100 Hz and a low-pass cut-off frequency between 8000 and 32000 Hz. Inter-token correlation values were (1; 0.99; 0.96; 0.91; 0.84; 0.76; 0; −1). Sound intensity was 70 dB SPL. The number of repetitions obtained per noise token varied from 20 to 75. Stimulus duration was either 600 ms (most AN fibers) or 1000 ms (all TB and some AN fibers) and interstimulus interval was between 1000 and 1400 ms.

Only fibers with a CF up to about 1500 Hz were selected, which covers the range important for fine-structure-based sensitivity to ITDs and enhanced synchronization of TB fibers relative to AN fibers (Joris et al., 1994a; Joris, 2003; Louage et al., 2005). For the TB fibers, we further limited the sample to fibers showing a “phase-locked” or “primary-like” type of response in the peristimulus time histogram. Fibers with a “primary-like” type of response are most likely to be SBCs, and fibers with a “phase-locked” type of response can either be SBCs or GBCs (Smith et al., 1991, 1993). The goal of this selection was to increase the proportion of SBCs as much as possible in our sample of TB fibers, since these are the excitatory inputs onto MSO neurons. With these constraints, 15 TB fiber datasets (CF: range 230–1518 Hz) and 16 AN fiber datasets (CF: range 136–1306 Hz) were selected for the current analysis.

In order to simulate a binaural coincidence response to broadband noise, we used responses of a monaural (AN or TB) fiber to broadband noise as “ipsi-” and “contralateral” inputs to the binaural cell. These responses mimicked the input of several SBCs onto one MSO cell for a particular binaural noise stimulus.

The first step in the simulation was to select input spike trains. For each neuron recorded, each stimulus was repeated N_tot times. In order to simulate a single “run,” we randomly selected a number of spike trains (N) as ipsilateral inputs, and an identical number of different spike trains as contralateral inputs. The ipsi- and contralateral spike trains could be in response to the same stimulus (correlated noise), e.g., when calculating a noise delay function (NDF), or to different stimuli (partially correlated or uncorrelated noise), e.g., when calculating a rate interaural correlation function (rICF). A given spike train was never present more than once in a single run of a binaural simulation. For a given simulation, all spike trains were responses of the same fiber. Note that once they are pooled, the monaural input spike trains for each side lose their “identity”: the input now consists of a collection of spike times which are treated equally without regard to their origin.

In a second step of the simulation, the operation of the binaural cell was simulated by counting coincident input events, i.e., events occurring within the same predefined time interval, the coincidence window (cw), centered at spike occurrence. This process of counting coincidences is illustrated in Figure 1A. In this example the number of inputs is 4 per side. From a collection of spike trains recorded from one (TB or AN) fiber in response to the same noise, 4 spike trains have been randomly selected as “ipsilateral” inputs, and 4 other spike trains as “contralateral” inputs. Then, coinciding spikes are counted across the input spike trains of one side (monaural coincidences) and across input spike trains from both sides (binaural coincidences). This is done by grouping all spike times from all relevant spike trains (monaural or binaural), sorting them in time, and counting, for each spike, the number of following spikes that fall in the same cw. We used a monaural coincidence threshold (thr_mon) and a binaural coincidence threshold (thr_bin) to decide whether respectively monaural or binaural coincident input spikes are effective (result in output spikes). If the number of coinciding spikes for one unilateral group of spike times exceeded thr_mon (set at 3 in the example in Figure 1A), a monaural coincidence was counted, and timed at the last coinciding spike in that window. In Figure 1A, there is one effective monaural coincidence for the ipsilateral inputs (red, case 2, indicated by numbers at the bottom) and one effective monaural coincidence for the contralateral inputs (case 4, green). Similarly, if the total number of coinciding spikes in the total (binaural) pool of spike times exceeded thr_bin (set at 2 in Figure 1) and each side contributed at least one spike, a binaural coincidence was counted (blue trace). This coincidence was again timed at the last coinciding spike. The total output spike train was calculated by adding (logical “OR” operator) the monaural and binaural coincidences (Figure 1A, magenta trace).

Note that these steps contain the inherent compression mentioned in the Introduction: only one effective coincidence could be counted per cw (monaurally and binaurally) even if there are multiple coincidences. For example, in the fictional example shown here, the binaural output is 1 for a bin with 2 coincident spikes (case 5) as well as for one with 4 coincident spikes (case 1). Note that cw was always equal for monaural and binaural coincidences (see also Discussion).

The total output spike train for many runs (not shown) gave a probability density function of coincidences for a given AN or TB fiber but could contain unphysiologically small intervals. We therefore imposed a refractory period of 1 ms by sequentially running through the total spike train and removing all spikes that occur within 1 ms after the previous spike. The length of the refractory period was determined by evaluating the interspike interval (ISI) histogram of noise responses from chinchilla lateral lemniscus (LL) fibers [Figure 3F, data from (Bremen and Joris, 2013)]. Only LL fibers where the number of false triggers—defined as ISIs < 0.5 ms—was below 0.01% were included in Figure 3F. The result after removal of “refracted” spikes was the final output (black trace). As is standard in physiological analyses, we summed the total number of coincidences or “spikes” over the stimulus duration and expressed this number as a rate (“spike rate” in spikes/s). This spike rate was used in all further analyses unless specified otherwise. The entire procedure was then repeated with newly drawn spike trains. We repeated this process three times for each choice of parameters and averaged the output spike rates.

We varied the following parameters of the coincidence process: the number of inputs per side (N), the monaural coincidence threshold (thr_mon), the binaural coincidence threshold (thr_bin), and the duration of the coincidence window (cw). Unless stated otherwise, N was varied from 1 to 10 (which covers the presumed range of MSO inputs per side, see Discussion), thr_mon from 2 to 11, thr_bin from 2 to 11 and cw was 50, 150, or 250 μs.

From the responses to noise, we computed two “pseudobinaural” functions which together provide a characterization of the influence of two basic binaural parameters: ITD and interaural correlation (Mc Laughlin et al., 2008). By time-shifting the “ipsilateral” relative to the “contralateral” inputs different ITDs were mimicked. When the output was represented as spike rate, a pseudobinaural noise delay function (NDF) was generated (Figure 1B). In addition to a NDF to correlated noise, where ipsi- and contralateral input spike trains were taken from responses to the same noise token, we also obtained a NDF to anticorrelated noise i.e., where the responses of one side are in response to the same noise waveform as on the other side but inverted in polarity. Obtaining responses to two opposite polarities is a standard procedure in our laboratory: it allows a fuller description and analysis of both monaural and binaural responses (Joris, 2003; Louage et al., 2004).

A second fundamental binaural parameter is ρ (Blauert, 1970; Yin et al., 1987; Albeck and Konishi, 1995; Shackleton et al., 2005; Coffey et al., 2006; Louage et al., 2006). To simulate interaural correlation sensitivity, we used spike trains in response to noises that had varying degrees of correlation. Responses were paired in all permutations possible (of the 8 noise tokens presented) (Louage et al., 2006). This was always done at an ITD of 0 ms. Plotting the output spike rate as a function of ρ resulted in a pseudobinaural rate-interaural correlation function (rICF) (Figure 1C).

For each fiber and for each set of model parameters we calculated a NDF (for correlated and anticorrelated noise) and a rICF. To explore the parameter space, we assessed the physiological plausibility of the final output functions (corresponding to the black trace in Figures 1B,C) by comparing their features to data from actual binaural cells. The features that were examined for physiological plausibility are spectral bandwidth (BW) and dominant frequency (DF); the steepness of the rICF; the maximal rate of the NDF (peak rate); the ITD modulation of the NDF and the halfwidth of the main peak of the NDF. For each of these features we bracketed the range of acceptable values, based on the same features in data from actual binaural responses as obtained in the cat IC (Mc Laughlin et al., 2008) or in axons from presumed MSO neurons of chinchilla (Bremen and Joris, 2013). A particular simulation was only accepted if it passed all “acceptance criteria.” Below we describe every acceptance criterion. Collectively they allowed us to identify model parameters that generate plausible output. We tested a range of criteria: none of our conclusions critically depend on the details of the definition of criterion boundaries.

Group data are reported as mean ± standard deviation (SD) unless stated otherwise. Group data between conditions are compared with an unpaired or paired t-test as indicated. Statistical significance is defined as a p-value ≤ 0.05.

Coincidence rates obtained with the AC analysis are much lower (Louage et al., 2005) than spike rates from actual binaural neurons (Figure 3C). Therefore we first determined the required N to get a physiologically plausible output rate. For this analysis, we chose the lowest possible thr_bin, i.e., 2, to maximize the probability of obtaining high output rates. To maximize ITD sensitivity, we chose a low value for cw: 50 μs. N was varied from 1 to 10 and thr_mon was varied from 2 to 11. Simulation output for these parameters are shown in Figures 4A,B for one TB fiber (CF = 559 Hz; spontaneous rate (SR) = 1.27 spikes/s). Figure 4A shows NDFs to correlated noise for different N, varied in the vertical direction, and for different thr_mon, varied from left to right. The ordinate is from 0 to 170 spikes/sec for all subplots while the abscissa is always from −2.94 to 2.94 ms. For most parameter choices, the NDF shows a clear, damped oscillatory shape. The black traces in Figure 4 correspond to the simulation results that passed all acceptance criteria, while the results in red did not. The maximal value for thr_mon was N + 1, because cases with higher thr_mon would be identical (effective monaural coincidences were absent for thr_mon ≥ N + 1).

Increase of N at a fixed thr_mon (e.g., vertical column at thr_mon of 2, Figure 4A), indeed caused an increase in spike rate, but also a general “DC shift upwards” of the entire NDF with a decrease of modulation depth (MD), i.e., a decrease in ITD sensitivity. Such shifted functions are not typically seen in binaural neurons and are caused by an increasing number of monaural coincidences. For higher values of thr_mon, e.g., 6, there is hardly any such upward shift of the NDF with increasing N. On the other hand, when increasing thr_mon (i.e., moving horizontally to the right in Figure 4A) for a fixed N (e.g., 7), the entire NDF moved down, because of a decreasing number of monaural coincidences for higher values of thr_mon. Note that, beyond a certain value (e.g. 4, for 7 inputs), further increases in thr_mon had no further effect on the NDF, which indicates that it was rare to have that large a number of input spikes from one side coinciding in one cw.

Thus, a first conclusion is that increasing N can bring a simple coincidence detector to physiological output rates, but this needs to be accompanied by a mechanism to prevent monaural coincidences from being too effective in generating spike output.

Figure 4B shows the corresponding rICFs for the same TB fiber and the same parameter variations. The ordinate is normalized for all subplots from the minimal to the maximal spike rate for that subplot. The abscissa ranges from interaural correlation −1 to +1. The spike rate increases in every subplot with increasing ρ. For a low thr_mon, e.g. 3, higher N (e.g., compare 9 inputs to 2 inputs) resulted in a relatively larger spike rate for lower values of interaural correlation. This resulted in an overall less expansive shape of the rICF for higher N. This could be due to the “binaural compression” mentioned in the Introduction (but see section Effect of Input Convergence on Expansiveness of rIC Function). Note however that this decrease in expansion of the rICF was rather subtle except for a large change in N.

Figure 5A shows NDFs for an AN fiber with similar CF (544 Hz, SR = 65.2 spikes/s). As in the TB fiber, increase of N, while keeping thr_mon low, caused an unphysiological upward shift of the entire NDF (thr_mon = 2). This was easily offset by a slight increase in thr_mon to 3 or higher. The rICF (Figure 5B) was much more linear than for the TB fiber, and this was the case for virtually all parameter combinations, except for the functions for which thr_mon = 2. Maximal spike rates were lower than for the TB fiber, both for the NDFs (note the difference in ordinate between Figures 4A and 5A) and the rICFs (not visible due to normalization). Another difference with the simulation results for the TB fiber is that a somewhat higher number of fibers was necessary to have accepted results (5 for AN, vs. only 3 for the TB).Twelve out of 15 TB datasets (80.0%) and 12 out of 16 AN datasets (75.0%) generated at least one accepted simulation for the chosen simulation parameters. The average required N was 3.92 ± 1.62 (mean ± SD) for the TB datasets, and 4.17 ± 0.72 for the AN datasets, but this trend was not significant (two-sample one-tailed t-test p = 0.685).

We investigated why low N did not result in accepted simulations, for simulations with thr_bin = 2. In this analysis, columns where all cases were rejected (e.g. leftmost column in Figure 4) or accepted (columns for thr_mon > 4 in Figure 4) were not informative and therefore not included. First, we determined the lowest N for accepted cases, for all values of thr_mon. For the example in Figure 4, the minimal N for thr_mon 3 and 4 are respectively 3 and 4. Then, for each value of thr_mon, the failed case was identified that had just one input less. In Figure 4, these are the cases with thr_mon = 3 and N = 2; thr_mon = 4 and N = 3. Next, for each of these unaccepted cases, we listed the failed criteria. For the example in Figure 4, the case with thr_mon = 3 and N = 2 failed because the p of the associated rICF was too high. The case with thr_mon = 4 and N = 3 failed for the same criterion. This information was then summarized per dataset by dividing the number of times a particular acceptance criterion failed, by the number of failed cases. For the example in Figure 4, this resulted in 1 (2 times divided by 2 cases) for the acceptance criterion “upper limit of power.” None of the other acceptance criteria failed in these cases and thus score 0. This procedure was performed for each TB dataset, and pooled and summarized for all datasets as boxplots in Figure 6A. The most frequent reason why simulations with low N were not accepted is that they resulted in a NDF peak spike rate that was too low. This was also the case for the AN (Figure 6B).

In summary, for cw = 50 μs and thr_bin = 2, a minimum of 4 input fibers from each side was typically necessary, primarily to achieve physiologically plausible output rates. To obtain physiologically plausible output for higher N, it was critical that monaural coincidences are suppressed (see section Possible Values of thr_mon).

Next we investigated whether binaural compression could lower the power p of the rICF. This power is indeed too high in AC analyses of TB fibers where coincidences are counted for a single input from each ear (triangles on Figure 7C; Mc Laughlin et al., 2008, 2014). For the analysis in this section, p was removed from the acceptance criteria. Figure 7A shows the value of p (background color of each subpanel) for multiple simulations of the same TB dataset as in Figure 4, with varying thr_mon (different columns) and N (different rows). Again, thr_bin = 2 and cw = 50 μs. Note that single inputs (lower left) resulted in a very high p value of ~7.4 (brownish color), consistent with the earlier AC analysis. In that case there are basically no coincidences at all for ρ < 0.5. Increase of N, keeping the same thr_mon (=2, leftmost column), caused a precipitous drop to p < 3: this is due to coincidences that now occur at ρ < 0.5. A similar but much more gradual tendency was present at other values for thr_mon: the value of p shows a gentle decrease with increasing N—from yellow to greenish colors. The reverse—a gentle increase in p with increase in thr_mon—is generally present within each row corresponding to a fixed N. There was a large increase in p for an increase in thr_mon from 2 to 4, but effects of further increases in that threshold were modest.

Figure 7B shows the change in p for the entire group of TB datasets. Gray circles indicate p for simulations with just one input on each side, i.e. the single input correlogram for each dataset. The resulting p values could be very high and were often outside the boundaries (red lines) observed in binaural neurons of the IC. This is consistent with the very expansive rICFs reported previously (Mc Laughlin et al., 2008, 2014). Magenta, green and blue lines are fitted through accepted simulations for the parameter combinations indicated by the respective colored arrows in Figure 7A: the diagonal arrow corresponds to simulations without effective monaural coincidences, because thr_mon is always higher than N; for the vertical arrows, thr_mon is fixed, and the number of monaural coincidences increases with N. Clearly, the largest effect on p occurred by increasing N from 1 to >1, and this effect was larger when p for N = 1 was very high. Overall there was a slight further power decrease with increasing N. The average slope of the simulations without monaural coincidences (blue lines; mean −0.075; one-sample one-tailed t-test p = 0.001) and of the simulations with thr_mon = 4 (green lines; mean −0.11; one-sample one-tailed t-test p = 0.016) were statistically significantly lower than 0, whereas the average slope of the simulations with thr_mon = 3 (magenta lines; mean −0.031; one-sample one-tailed t-test p = 0.32) was not. Therefore, having more than one input on each side could dramatically decrease the power, but the additional effects of increases of N > 2 were very small.

For a more direct comparison, Figure 7C shows the range of powers for the accepted model simulations and compares them with the values of IC neurons (circles) and the single input correlograms of TB fibers (solid triangles). The powers corresponding to the accepted simulations of TB fibers (red stars) were generally below the high values previously reported for the AC analysis (Mc Laughlin et al., 2014) and closer to those of binaural IC neurons, but the mean value was still higher than for the IC dataset (mean ± SD, N for simulations: 3.12 ± 1.65, 13 vs. IC: 2.05 ± 1.58, 29, one-tailed t-test, p = 0.027). Thus, convergence of inputs capped the power to values within the range of experimental binaural recordings, but they were still in the upper part of that distribution.

Previously, in Figure 4, we highlighted the increase in number of monaural coincidences with increase in N (thus along the magenta and green vertical arrows in Figure 7A). However, the occurrence of monaural coincidences was not required for a decrease of rICF expansiveness with a higher N, because it could also be seen for conditions without monaural coincidences (blue arrow in Figure 7A and blue lines in Figure 7B). Example rICFs are shown in Figure 8A for a TB fiber dataset (CF = 559 Hz): black circles correspond to 3 and black crosses correspond to 8 inputs, respectively. In both cases, thr_mon > N, so there were no effective monaural coincidences. The power (indicated in the caption) is lower for higher N: 4.1 for 8 inputs vs. 4.8 for 3 inputs. More inputs means a larger probability of having a (binaural) coincidence: having more inputs at both sides increases the chance that any two of them will fire at the same time, because the number of combinations of 2 out of N inputs increases with N. This effect is larger if there are not yet that many coincidences, i.e., for suboptimal values of ρ, but diminishes when there are many coincidences because there can only be one output spike per cw. Thus the lowering in p here could reflect the “binaural compression” mentioned in the Introduction. The red traces correspond to the black traces, but without application of the refractory period. Now there is no decrease in p anymore for the higher number of inputs. This suggests that the main reason for a lower p is not binaural compression, but rather spike rate saturation due to the refractory period. In Figure 8B, rICFs are shown for a fixed thr_mon = 4. This threshold enabled monaural coincidences for a high N (8) but not for a low N (3). The effect is a lowering of p for N = 8 (from 4.8 to 3.8). Now there is also a lowering of p without the refractory period (from 5.3 to 4.8, with increase in N from 3 to 8), in which perhaps “binaural compression” plays a role.

To explore this possibility more directly, we selected the five TB fiber datasets with the largest decrease in p for increasing N, for either thr_mon = 3 or 4. The accepted rICFs with the lowest and highest p were identified and are analyzed in Figure 8C. Here, the ordinate shows the average number of coinciding input spikes per output spike (for the non-refracted spikes, thus corresponding to the red functions in Figure 8B), and this average is shown as a function of ρ. The prediction is that “binaural compression” is highest for high ρ and for the largest N. We find that for negative ρ, the average ratio of input to output spikes is close to thr_mon. This is not surprising: at these ρ values, the vast majority of coincidences are monaural, and therefore the ratio is close to thr_mon (3 or 4). For ρ > 0, the ratio of input to output spikes is lower than for ρ < 0, reflecting the increasing number of binaural coincidences, for which the threshold is lower (trh_bin = 2). If “binaural compression” would be responsible for the lower p in case of high N, the average number of input events per spike would increase toward ρ = 1, and more so for high N than low N. However, it can be seen that this pattern is not consistently present. Instead, the average number of coinciding events is low, and does not markedly increase toward ρ = 1. We conclude that there are usually too few coinciding input events to get significant “binaural compression.”

To summarize, the convergence of multiple inputs softened the extreme expansiveness seen in the single-input rICF of many TB fibers but the influence was minor beyond a convergence ratio of 2 fibers per side. The main factor appears to be refractoriness (i.e., a ceiling on firing rate per cycle) rather than “binaural compression” (i.e., the sublinear relation between N and output spike rate that is inherent to the coincidence mechanism).

We already mentioned the fact that a higher thr_mon helped to prevent an unphysiological upward shift of the NDF for more inputs (Figure 4). In this section we explore the values of thr_mon for accepted simulations. Again cw was set at 50 μs. The other simulation parameters were varied. As was the case for the TB simulations shown in Figure 4, there was typically a minimal thr_mon associated with acceptable simulations. For the fiber of Figure 4 the minimal thr_mon was 3 for N = 3, 4, or 5, and 4 for higher N, in each case keeping thr_bin = 2. The layout of Figure 9A is similar to that of Figure 4 but now the abscissa shows the minimal thr_mon of accepted simulations for TB datasets. Different datasets are indicated with different symbols, and different colors indicate different values for thr_bin. So, for example, the data shown in Figure 4 are for a thr_bin of 2 and are shown in Figure 9A as the black circles filled with red. These are the most leftward accepted (black) combinations in Figure 4. Figure 9B shows the same data for AN datasets.

The general message from Figures 9A,B is that thr_mon almost invariably needed to be higher than thr_bin in order to obtain acceptable simulations. Across all simulations for thr_bin = 2 (red symbols), we found that the mean minimal value of the thr_mon for accepted simulations was 3.08 ± 0.29 for TB fibers and 3 ± 0 for AN fibers. For thr_bin = 3 (green symbols), the mean of the minimal thr_mon increased to 4.5 ± 1.85 for TB, and 4.13 ± 0.83 for AN. When the minimal thr_mon was divided by thr_bin, we obtained a mean of 1.42 ± 0.23 for TB datasets and 1.38 ± 0.18 for AN datasets. For thr_bin = 4 (blue symbols), there were few accepted simulations (none for AN), even for the maximal number of inputs tested (N = 10).

Because the number of monaural coincidences increases with N (Figure 4), it is expected that the minimal thr_mon will rise with N (Figure 9). It is therefore useful to express the minimum thr_mon as a fraction of N. This average was 0.42 ± 0.14 for TB and 0.30 ± 0.0096 for AN fibers (two sample t-test p = 0.0061; thr_bin = 2). This means that the requirement of thr_mon was more stringent for TB fibers than for AN fibers. Indeed, in the example in Figure 5 (thr_bin = 2), the simulations for thr_mon = 3 were acceptable for 6 inputs or more: for lower N, the output spike rate was too low. In contrast, the same parameters for the TB fibers of Figure 4 gave rise to a “DC problem” once N exceeded 5. This suggested that the enhancement of temporal coding between AN and TB (Joris et al., 1994a,b; Louage et al., 2005) comes at the cost of increasing the probability of effective monaural coincidences, unless this enhancement is accompanied by an increase of thr_mon (relative to thr_bin).

The boxplots in Figures 6C,D show the acceptance criteria determining the minimal thr_mon for simulations with thr_bin = 2. The most important criterion was the lower limit of the MD, both for TB datasets (panel C, N = 12 datasets) and for AN datasets (panel D, N = 12 datasets). This again indicated that a high thr_mon (relative to thr_bin) was necessary for physiological ITD sensitivity via suppression of monaural coincidences.

As mentioned in the Introduction, the sensitivity of a neuron to binaural coincidences vs. monaural coincidences has been addressed before. Colburn et al. (1990) proposed that the mere phenomenon of convergence would be an explanation for the fact that binaural coincidences are promoted relative to monaural coincidences, because the number of combinations of inputs that could possibly generate a binaural coincidence is higher than that of monaural inputs. The number of combinations NC_b that results in x inputs spiking at the same time in the binaural situation with N inputs on each side is given by (4)NCb = (2N)!((2N−x)! x!) The number of combinations NC_m of having x monaural inputs spiking at the same time is given by twice the number of combinations for each side separately, or (5)NCm = 2 N!((N−x)! x!) Figure 10A shows the number of possible combinations in the binaural and monaural situation, NC_b (black circles) and NC_m (green circles), resulting in 4 simultaneous spikes (x = 4), for different numbers of input N. For example, if each side receives 5 inputs (N = 5) and 4 spikes are required to exceed threshold, there are 210 possible binaural combinations to achieve this but only 10 possible monaural combinations. Because NC_b = 210 for N = 5 whereas NC_m is only 10, Colburn et al. (1990) concluded that just this difference in the number of possible permutations might explain the larger sensitivity of MSO neurons for binaural over monaural coincidences. Indeed this translates into a coincidence probability that rises much faster for the binaural situation than for the monaural situation: Figure 10B shows the probability of having 4 monaural coincidences in one coincidence window (green circles), as a function of N, assuming that the chance of having 1 event on 1 input is p_s = 0.0075 (corresponding to a spike rate of 150 Hz if cw = 50 μs). The blue circles indicate this probability for strictly binaural situations (where every side contributes at least one event), and the black circles show the total binaural probability of having 4 coincidences (adding the strictly binaural and the strictly monaural probabilities). Clearly, the probability that output spikes are driven by binaural coincidences is much higher than that of monaural coincidences.

This theoretical consideration contrasts with our finding that thr_mon needed to be higher than thr_bin (Figures 4,5). This means that the combinatorial effect pointed out by Colburn et al. (1990) does not suffice for all parameter conditions to promote binaural coincidences over monaural coincidences. The discrepancy is explained by the fact that this combinatorial binaural advantage is very dependent on the number of events required to coincide (x). Figure 10A (crosses) shows the number of combinations for x = 2 coinciding input events. Now the difference between NC_b (black) and NC_m (green) is much smaller, simply because the number of combinations of 2 out of 2N grows not that much faster with N than twice the number of 2 out of N. For example, for x = 2 and N = 5, NC_b = 45, and NC_m = 20. This leads to a coincidence probability that is not that much larger binaurally relative to monaurally (Figure 10C). Clearly, the probability of having 2 monaural coincident spikes (green) is almost the same as that of having a binaural coincidence (blue).

Of course, for a coincidence detector with thr_bin = 2, there will be some epochs where the effective number of coinciding input events is >2, and for which binaural coincidences are more likely than monaural ones. Note, however, that the absolute probability of having x = 4 coinciding events is several orders of magnitude smaller than the absolute probability of having x = 2 coinciding events (compare ordinate of panel B vs. panel C in Figure 10), so that such epochs are a small fraction of the total number of epochs that generates an output spike. In summary, requiring a relative large number of coincident spikes (e.g., x = 4) has the advantage of automatically favoring binaural over monaural coincidences, but has the drawback of yielding a low probability of coincidences. We explore this further below (section Maximal Value of thr_bin).

If thr_mon > N (i.e., the diagonal in Figures 4, 5, 7A), monaural coincidences are effectively removed. Such simulations are shown on the diagonal of these figures (thr_mon = N + 1). They illustrate that these parameter combinations can result in physiologically plausible input. One may argue that a complete absence of monaural coincidences is a requirement for an “ideal” binaural coincidence detector.

In summary, for physiological plausible output, it is important to have a thr_mon that is not too low, because monaural coincidences strongly decrease binaural sensitivity. This lower limit increased with N. Interestingly, thr_mon was especially stringent for TB fibers, because the high synchronization in their spike trains made the number of monaural coincidences rise fast with increasing N.

The sensitivity of the coincidence detector to input events was studied by varying thr_bin. Figures 11A,B shows simulations for which thr_bin and N were varied for one TB dataset (CF 456 Hz; SR 89.9Hz; thr_mon = 4). The simulation output decreased drastically when thr_bin was increased above 2. Functions for thr_bin > 4 are not shown because they are almost completely flat at 0 spikes/s. Note that there were two conflicting demands at work, with little margin. If the thr_bin was minimal (2), there was a steady increase in output rate with increasing N, as expected (Figure 11, leftmost column). For the highest N tested (9 and 10) the output was not accepted because the spike rate at the peak was too high. Increasing the thr_bin just by one gave the opposite result: except for one parameter value (N = 8), the accepted simulations of the leftmost column were now unacceptable and vice versa. In this case, the problem was the reverse one: the output rate was too low. This suggests that, given the firing rates and temporal characteristics of real TB fibers, and the low number of input fibers per side (see Discussion), permissible values of thr_bin are severely restricted.

Population data are shown in Figure 12 for all TB (A) and AN (B) datasets, with cw = 50 μs. Each symbol marks the maximal possible thr_bin (indicated on the abscissa) for a particular N (ordinate) and thr_mon (color of symbols). As in Figure 9A, different symbols indicate different datasets. The dataset shown in Figure 11A is shown in Figure 12A by black triangles filled with green. For the population, possible values for thr_bin were strikingly limited to only 2, 3 or sometimes 4, both for TB and AN. Thus, even for 10 perfectly matched (in terms of CF, properties, delay) input fibers from each side, a requirement of 3 binaurally coinciding input spikes, i.e., two from one side and one from the other, was practically the maximum threshold that can be imposed. With further increases in N (>10, not tested here), it is likely that a higher maximal thr_bin will generate acceptable simulations. Note however that the increase in N needed to be accompanied by a sizeable increase in thr_mon (colors).

The boxplots in Figure 6E (TB) and Figure 6F (AN) show which acceptance criteria failed for simulations for which thr_bin was just one step higher than the highest value for accepted simulations. The main limiting factor was the minimal value of the peak rate.

Thus, for the range of simulation parameters explored here, the coincidence detector has to be sensitive to very few (2–3) coinciding input spikes: if too many input spikes were required, the output spike rate was too low.

The final parameter explored is cw. Besides the standard value of 50 μs, we evaluated coincidence simulations with cw = 150 and 250 μs. In Figure 13 NDFs are shown for TB datasets for these different cw. Again, thr_bin was fixed at 2. For every dataset, the combination of lowest N and thr_mon was chosen that resulted in accepted simulations for a cw of 50 μs (corresponding to the blue NDFs). The abscissa ranges from −4 to 4 ms, and the ordinate from 0 to 150 Hz. These parameters were then kept the same for simulations with cw 150 (green) and 250 (red) μs, respectively. It can be seen that an increase of cw led to a general increase in spike rate, without a drastic change in shape of the functions. However, for many datasets the increase in spike rate was accompanied by a decrease in MD, again due to an upward “DC-shift.” The larger cw resulted in accepted simulations for 10 out of 12 TB datasets. In the remaining 2 datasets (indicated with asterisks), the cause of rejection of the simulation was a MD below the lower limit (both for cw = 150 μs as for cw = 250 μs).

We have shown before (Figures 6A,B) that the main factor setting a lower limit on the N is the spike rate at the peak. A longer cw increased spike rate, and can therefore lower the minimal N to get acceptable simulations. For longer cw (50-150-250 μs), the mean ± SD of the minimal N per dataset decreased from 3.92 ± 1.62 to 2.84 ± 1.63, and further to 2.46 ± 1.45.

For the same reason, a longer cw could also extend the range of possible values for thr_bin. Figures 12C,D shows the maximal values of thr_bin for TB datasets, for a cw of 150 μs (panel C) and 250 μs (panel D). The range of maximal thr_bin was extended to respectively 6 and 8, but it can be seen that in the majority of cases it was still limited to ~2–5, except when thr_mon was raised to high levels.

Thus, a longer cw helped achieving higher rates and therefore allowed lower N and higher thr_bin. This came at the cost of a decrease in MD. Another effect is that a longer cw inherently causes temporal integration and will ultimately limit binaural temporal sensitivity.

Because TB fiber spikes display enhanced temporal structure relative to the AN (Joris et al., 1994a,b; Louage et al., 2005), the probability of spikes from separate inputs occurring in the same cw will be higher. Therefore we hypothesized that the N required to get a given number of output spikes is lower for TB fibers. We compared the lowest N that generated acceptable simulations for TB and AN datasets, using cw of 50 μs. As mentioned above (section Estimation of Minimal N), for thr_bin = 2, the minimal N was slightly lower for TB fibers (mean ± SD 4.08 ± 1.68) than for AN fibers (4.17 ± 0.72), but the difference did not reach statistical significance (two sample t-test p = 0.876). For thr_bin = 3 however, the difference increased, with mean ± SD for TB 7.5 ± 1.41 and for AN 9.1 ± 1.46 (two sample t-test p = 0.04). Therefore TB fibers can be seen as more robust than AN as input fibers to the binaural neuron: the necessary increase in N for a less than perfect coincidence detector (one that fails to detect a single spike on both sides coinciding, i.e., thr_bin > 2) is larger for AN fibers than for TB fibers, because the latter have more tightly synchronized and more reproducible firing patterns. Therefore TB fiber inputs provide a larger safety factor for the detection of binaural coincidences: if a particular input spike is missed, there is a larger probability that there are others in the same cw present in order to reach spike threshold. This can be seen as well in the range of possible values for thr_bin, for TB inputs vs. AN inputs, where less AN than TB fibers could fulfill a requirement of 3 binaural coincidences (compare Figures 12A,B).

Next we compared the simulation output for TB and AN datasets, for the same simulation parameters. Here, thr_bin = 2. Figure 14 groups the accepted simulations for each combination of N (varied along the ordinate) and thr_mon (varied along the abscissa), using a cw of 50 μs. The background color of each subplot indicates the average of the peak spike rate for all accepted simulations with those parameters. As expected, and in agreement with the examples in Figures 4, 5, it can be seen that the maximal spike rate increased with increasing N, in both TB (panel A) and AN (panel B). Comparing the two panels, we observe that for low N, i.e., 3–7, the peak rate was higher in the TB case than in the AN case (i.e., “cooler” colors lower half of figure for AN). Surprisingly, for N > 7, the peak rate of AN simulations was larger than of TB simulations: the gradient in spike rate with increasing N was much more distinct in the AN than in the TB. Thus, in addition to the other (coding) advantages that TB fibers have above AN fibers (see Discussion), TB fibers allow sparser connectivity than AN fibers: it takes less inputs to get similar spike rates, at least within the lower range of N, and the output spike rate is also less dependent on the exact N. This is again a consequence of enhanced synchronization, because it takes fewer inputs to get enough spikes to reach coincidence threshold when there is consistent spiking with low temporal jitter. Increasing the N even more hardly increases the spike rate further, because the number of spikes is limited to maximally one spike per cw and per effective stimulus cycle.

The number in each subplot of Figures 14A,B indicates the percentage of datasets contributing at least one accepted simulation for that set of parameters. It can be seen that for low N (2–4), relatively more TB fibers contributed good simulations than AN fibers. For higher N, the opposite was true.

There is also a downside of having tight and consistent synchronization: the proportion of monaural coincidences rises fast with N. Figure 15A is organized like Figure 14A, but now the colors indicate the average MD for accepted simulations with TB datasets. For low thr_mon (e.g., 3 or 4), MD decreased rapidly with increasing N, due to a “DC-shift” (cf. Figure 4). This effect was counteracted by increasing thr_mon. For simulations with AN datasets (Figure 15B), the average MD showed milder dependence on N. The different requirements for the minimal thr_mon between AN and TB were already illustrated above (Figure 9).

Thus, TB fibers as inputs to the binaural coincidence detector are advantageous in that a low number of fibers can already result in a physiologically realistic output spike rate. The downside is that monaural coincidences will increase as well with more inputs, so that a mechanism to suppress the efficacy of monaural coincidences becomes more critical for an increasing number of well-synchronized inputs.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.