Exponential Stein model.

In the original type of the Stein model, which we here call the ‘exponential Stein model’, each synaptic input is converted into an exponential function (Fig 5A). Excitatory and inhibitory inputs are represented as positive and negative changes in the virtual membrane potential, respectively (Fig 5B). An output spike is generated when the virtual membrane potential reaches or exceeds the threshold. The amplitude and the time constant of exponential curves for unitary synaptic inputs were tuned to fit the output criteria of monaural and binaural responses (Fig 5C, right; Fig 5D, right; and Fig 5E, main panel). It should be noted, however, that the threshold level found in our parameter search (5.5 inputs) was considerably lower than the corresponding empirical data (9.6 ± 2.8 inputs in gerbils [77]) (see Materials and methods for more detail on parameter selection). This discrepancy was primarily due to the unrealistic shape of the modeled synaptic input that is extremely steep at the onset and decays rather slowly afterwards. Due to this sharp onset, the simulated traces are jagged with synaptic inputs (Fig 5C and 5D, left).

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Fig 5. Exponential Stein model of LSO.

A: Schematic drawing of the exponential Stein model. Each vertical bar corresponds to a spike (red: inhibitory inputs; blue: excitatory inputs; black: internal state (i.e., virtual membrane potential) of the model; green: spike output of the model). Synaptic inputs are modeled as exponentially decaying functions and linearly summed to produce the internal state of the model. An output spike is generated when the sum of inputs reaches or exceeds the threshold. The internal state is reset to and fixed at zero during the refractory period after each spike. B: Excitatory and inhibitory synaptic inputs modeled as exponentially decaying functions with different amplitudes and time constants. C: (Left) Traces of the internal state of the model driven by binaural AM tones with two different input phase differences. (Right) Output rates of the model in response to binaural AM tones with varied input phase differences. Bold numbers show the peak and trough rates. D: (Left) Traces of the internal state driven by binaural unmodulated tones with two different ILDs. (Right) Output rates of the model in response to binaural unmodulated tones with varied ILDs. Bold numbers show the rates at -45 dB and +15 dB. In panels C (Left) and D (Left), horizontal dotted gray lines and broken black lines indicate the zero input level and the threshold, respectively. At each threshold crossing, a vertical line was manually added to show the generation of an output spike. E: Monaural AM-tuning curve (rate-MTF) of the exponential Stein model. Bold numbers show the peak rate and the rate at 1200 Hz. (Inset) Monaural phase-locking (synch-MTF) of the model. Blue rectangular shading in C-E indicates the targeted ranges, while green shading in C-D shows the accepted ranges. F: Binaural AM phase-tuning curves of the model at three modulation frequencies. G: Binaural ILD-tuning curves of the model at five ipsilateral sound levels.

https://doi.org/10.1371/journal.pcbi.1005903.g005

Simulated monaural and binaural tuning curves (Fig 5E–5G) were roughly similar to empirical data (Fig 1B–1D). However, some of the tuning properties were different from the coincidence counting model. The rate-MTF (Fig 5E) for the exponential Stein model decreased slightly more slowly with the modulation frequency than for the coincidence counting model. The synch-MTF (Fig 5E, inset) did not decay at the modulation frequencies below 1 kHz. The trough positions of the binaural phase tuning curves did not perfectly align (Fig 5F) because of the temporally asymmetric synaptic inputs. Peak spike rates of the binaural level tuning curve (Fig 5G) tended to be higher than for the coincidence counting model. This is explained by the temporal summation of synaptic inputs: i.e., in the coincidence counting model, the effects of each excitatory synaptic inputs lasts only for the time window of W_ex = 0.8 ms (Fig 4B), whereas, in the exponential Stein model, synaptic inputs may sum up for a considerably longer time scale (Fig 5B). These simulation results suggest that the modeled synaptic inputs of the exponential Stein model may have to be revised to achieve a better fit with empirical data, although the general principle of the excitatory-inhibitory interactions is captured by the model [54]. This motivated us to introduce the alpha Stein model (next section).

Alpha Stein model.

In the alpha Stein model, unitary synaptic inputs are formulated as an alpha function (Fig 6A and 6B), while the other operations remain unchanged from the exponential Stein model (see Materials and methods). The selected threshold of this model (7.3 inputs) was higher than that of the exponential Stein model (5.5 inputs) and better matched corresponding empirical data (9.6 ± 2.8 inputs [77]). Simulated traces (Fig 6C and 6D, left) were smoother than those of the coincidence counting and exponential Stein models, although the simulated monaural and binaural tuning curves did not greatly differ (Fig 6C–6E). The synch-MTF started to decay at a lower modulation frequency than for the exponential Stein model (Fig 6E, inset). Because the modeled synaptic input has a steeper onset than the decay, there was a slight offset between the trough positions (Fig 6F). Relatively higher peaks for high-intensity inputs (Fig 7G) were found, as similarly seen in the exponential Stein model.

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Fig 6. Alpha Stein model of LSO.

A: Schematic drawing of the alpha Stein model. Each vertical bar corresponds to a spike (red: inhibitory inputs; blue: excitatory inputs; black: internal state (i.e., virtual membrane potential) of the model; green: spike output of the model). Synaptic inputs are modeled as alpha functions and linearly summed to produce the internal state of the model. An output spike is generated when the sum of inputs reaches or exceeds the threshold. The internal state is reset to and fixed at zero during the refractory period after each spike. B: Modeled excitatory and inhibitory synaptic inputs. Excitatory and inhibitory synaptic inputs are both converted into alpha functions, but with different amplitudes and time constants. C: (Left) Traces of the internal state of the model driven by binaural AM tones with two different input phase differences. (Right) Output rates of the model in response to binaural AM tones with varied input phase differences. Bold numbers show the peak and trough rates. D: (Left) Traces of the internal state driven by binaural unmodulated tones with two different ILDs. (Right) Output rates of the model in response to binaural unmodulated tones with varied ILDs. Bold numbers show the rates at -45 dB and +15 dB. In panels C (Left) and D (Left), horizontal dotted gray lines and broken black lines indicate the zero input level and the threshold, respectively. At each threshold crossing, a vertical line was manually added to show the generation of an output spike. E: Monaural AM-tuning curve (rate-MTF) of the alpha Stein model. Bold numbers show the peak rate and the rate at 1200 Hz. (Inset) Monaural phase-locking (synch-MTF) of the model. Blue rectangular shading in C-E indicates the targeted ranges, while green shading in C-D shows the accepted ranges. F: Binaural AM phase-tuning curves of the model at three modulation frequencies. G: Binaural ILD-tuning curves of the model at five ipsilateral sound levels.

https://doi.org/10.1371/journal.pcbi.1005903.g006

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Fig 7. Passive integrate-and-fire model of LSO.

A: Circuit diagram of the passive IF model. Θ denotes the threshold crossing detector. B: Membrane impedance of the model. C: Current-potential (I-V) relationship of the model. D: Model responses to step current input with three varied sizes. E: Membrane responses to modeled excitatory and inhibitory synaptic inputs. F: (Left) Modeled membrane potentials driven by binaural AM tones with two different input phase differences. (Right) Output rates of the model in response to binaural AM tones with varied input phase differences. Bold numbers show the peak and trough rates. G: (Left) Modeled membrane potential driven by binaural unmodulated tones with two different ILDs. (Right) Output rates of the model in response to binaural unmodulated tones with varied ILDs. Bold numbers show the rates at -45 dB and +15 dB. In panels F (Left) and G (Left), horizontal dotted gray lines indicate the resting potential. In panels D-G, vertical bars were manually added to show the generation of an output spike at each spike crossing. H: Monaural AM-tuning curve (rate-MTF) of the passive IF model. Bold numbers show the peak rate and the rate at 1200 Hz. (Inset) Monaural phase-locking (synch-MTF) of the model. Blue rectangular shading in F-H indicates the targeted ranges, while green shading in F-G shows the accepted ranges. I: Binaural AM phase-tuning curves of the model at three modulation frequencies. J: Binaural ILD-tuning curves of the model at five ipsilateral sound levels.

https://doi.org/10.1371/journal.pcbi.1005903.g007

Passive integrate-and-fire model.

In the passive IF model, the subthreshold response of the membrane is described as an RC circuit (Fig 7A). Therefore its impedance profile is low-pass (Fig 7B), and its I-V relationship is linear (Fig 7C). In response to step (rectangular) current injections, the model produced either no spikes (Fig 7D, left) or repetitive spikes (Fig 7D, middle and right), depending on the amplitude of the current. Summation of inhibitory synaptic inputs (Fig 7E, bottom) was sublinear because its reversal potential was only about 15 mV below the resting potential (i.e., E_inh = -75 mV). In contrast, since the excitatory reversal potential was far above the resting potential (E_ex = 0 mV), excitatory inputs summed almost linearly (Fig 7E, top).

Responses to modeled synaptic inputs in the passive-IF model (Fig 7F and 7G, left) resembled those in the alpha Stein model, but slightly less noisy because the low-pass membrane filtered out high-frequency fluctuations. Simulated monaural and binaural tuning curves (Fig 7F and 7G, right; Fig 7H) also had similar shapes. The trough positions of the phase-tuning curves (Fig 7I), however, showed a better alignment than the Stein models, since the low-pass membrane resulted in somewhat more symmetrical shapes in synaptic potentials. The level-tuning curves (Fig 7J) were almost identical to the alpha Stein model, with peaks higher than empirical data (Fig 1D), since the model lacks mechanisms to compress high intensity inputs (see Discussion).

Active integrate-and-fire model.

In addition to the membrane capacitance and the linear leak conductance, the active IF model has a non-linear low-voltage-activated potassium (KLVA) conductance (Fig 8A). KLVA channels are commonly found along the auditory pathway and play an important role in determining the temporal response properties of auditory neurons (see [81,82] for reviews). The impedance profile of the membrane was still low-pass (Fig 8B), whereas a much larger amount of KLVA conductance in other auditory coincidence detector models was found to make the membrane weakly band-pass [83,84]. KLVA current is activated near or slightly above the resting potential, resulting in lower membrane impedances at depolarized potentials (Fig 8C) and a phasic (onset) spiking response to a step current of an intermediate amplitude (Fig 8D, middle). The transition from phasic to tonic spiking was seen in slice recordings from rat LSO [85] corresponding to the tonotopic distribution of KLVA conductance [86].

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Fig 8. Active integrate-and-fire model of LSO.

A: Circuit diagram of the active IF model. Θ denotes the threshold crossing detector and spike current generator. B: Membrane impedance of the model. C: Current-potential (I-V) relationship of the model. D: Model responses to step current input with three varied sizes. E: Membrane responses to modeled excitatory and inhibitory synaptic inputs. F: (Left) Modeled membrane potentials driven by binaural AM tones with two different input phase differences. (Right) Output rates of the model in response to binaural AM tones with varied input phase differences. Bold numbers show the peak and trough rates. G: (Left) Modeled membrane potential driven by binaural unmodulated tones with two different ILDs. (Right) Output rates of the model in response to binaural unmodulated tones with varied ILDs. Bold numbers show the rates at -45 dB and +15 dB. In panels F (Left) and G (Left), horizontal dotted gray lines indicate the resting potential. H: Monaural AM-tuning curve (rate-MTF) of the active IF model. Bold numbers show the peak rate and the rate at 1200 Hz. (Inset) Monaural phase-locking (synch-MTF) of the model. Blue rectangular shading in F-H indicates the targeted ranges, while green shading in F-G shows the accepted ranges. I: Binaural AM phase-tuning curves of the model at three modulation frequencies. J: Binaural ILD-tuning curves of the model at five ipsilateral sound levels.

https://doi.org/10.1371/journal.pcbi.1005903.g008

Instead of the simple potential reset, an additional spike-associated current was injected after each threshold crossing (Fig 8E) in our active IF model. This spike-mimicking current was introduced to make the simulated potential traces (Fig 8F and 8G, left) more realistic than other simpler models. Monaural (Fig 8H) and binaural responses (Fig 8F and 8G right) of the active IF model were all within the targeted ranges (Fig 3). The rate-MTF (Fig 8H, main panel), synch-MTF (Fig 8H, inset) and binaural phase-tuning curve (Fig 8I) resembled those of the passive IF model. The binaural level-tuning curve (Fig 8J), however, showed lower peak rates especially with high ipsilateral levels, better replicating the empirical data (Fig 1D). This was achieved by the KLVA conductance, which compressed the accumulated input by making the input resistance low at depolarized membrane potentials (Fig 8C).

Original Wang-Colburn model.

In the original Wang-Colburn model, the values for the membrane capacitance and the leak conductance were taken from their prior study using an IF-type model [51], and were considerably larger than the values used for the other models in our present study. The resulting membrane constant of the Wang-Colburn model was 1.0 ms, comparable to the empirical values in gerbils (1.1 ± 0.4 ms: [77]), while the input resistance of 21.1 MΩ was relatively lower than the measured values in the same species (42 ± 21 MΩ: [77]). Because of the large amount of potassium conductances, the impedance was highest at 100–200 Hz, although the overall shape of the impedance curve was nearly low-pass (Fig 9B). The I-V curve showed an outward rectification (Fig 9C). Due to the activation of KLVA and inactivation of Na conductances near the resting potential, the excitability of this model was substantially lower than other models, showing phasic spiking even in response to a large step current (Fig 9D, middle).

Since the resting potential (around -65 mV) was close to the inhibitory reversal potential of this model (E_in = -70 mV), simulated inhibitory inputs (Fig 9E, bottom; unitary amplitude = 0.6–0.7 mV) were much smaller than the measured inhibitory inputs (1.6–8.3 mV: [77]). Spike peaks of the original Wang-Colburn model reached +10 to +30 mV (Fig 9E, top; Fig 9F, left; Fig 9G, left), resembling LSO neurons in mice [88] and rats [85], rather than gerbil LSO neurons that rarely showed overshooting [77]. Because of the weak inhibitory inputs, the output rate did not fall below 20 spikes/s, even when inhibitory input arrived in-phase with excitatory inputs (Fig 9F, right). Furthermore, because of the low excitability of the model neuron, the simulated maximum rate was only about 60 spikes/sec when driven by non-phase-locked excitatory inputs (Fig 9G, right).

As reported previously [53], simulated monaural AM tuning curve showed a peak around 200 Hz (Fig 9H). Synch-MTF decreased more rapidly even below 1 kHz (Fig 9H, inset) than other models. The binaural AM phase tuning curves roughly aligned at the troughs (Fig 9I), but the depths were generally shallow. The most notable difference from other models (and from empirical data) was the low output rates in the binaural level tuning curves (Fig 9J). This was due to the large potassium conductance in combination with the strong sodium inactivation, both of which hinder repetitive spiking when driven by continuous inputs, as seen with step currents (Fig 9D, middle).

Adjusted Wang-Colburn model.

The major discrepancies found between the original Wang-Colburn model and empirical data necessitated a revision of the model. We thus made the following changes to obtain an adjusted version of the Wang-Colburn model. First, we shifted the voltage dependence of the ion channel kinetics to reduce the effects of KLVA activation and Na inactivation. Next, we revised the membrane capacitance, leak and KLVA conductances, and reversal potentials to gain a better fit to corresponding experimental data. Finally, the values of Na and KHVA conductances were selected to achieve our output criteria of monaural and binaural tunings. More detailed descriptions of the model adjustment are provided in Materials and Methods.

The adjusted Wang-Colburn model has the same type of ion channels (Fig 10A) as the original version with a voltage shift of +5 mV. The membrane capacitance and resistance were reduced to fit corresponding data [77]. The membrane impedance profile (Fig 10B) and I-V curve (Fig 10C) resembled those for the active IF model. The adjusted model showed tonic (repetitive) spiking for a lower step current input (Fig 10D, right) than the non-adjusted model. Shapes of inhibitory inputs (Fig 10E, bottom) better resembled the in vitro data [77]. The model still shows large, overshooting spikes (Fig 10E, top), as the reversal potential of the sodium conductance was unchanged from the original version (see Discussion). With these revisions, subthreshold traces were not greatly altered (compare traces in Fig 10F and 10G with traces in Fig 9F and 9G), but the binaural tuning curves yielded a larger modulation depth in phase coding (Fig 10F, right) and a higher peak for level coding (Fig 10G, right).

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Fig 10. Adjusted Wang-Colburn model of LSO.

A: Circuit diagram of the adjusted Wang-Colburn model. B: Membrane impedance of the model. C: Current-potential (I-V) relationship of the model. D: Model responses to step current input with three varied sizes. E: Membrane responses to modeled excitatory and inhibitory synaptic inputs. F: (Left) Modeled membrane potentials driven by binaural AM tones with two different input phase differences. (Right) Output rates of the model in response to binaural AM tones with varied input phase differences. Bold numbers show the peak and trough rates. G: (Left) Modeled membrane potential driven by binaural unmodulated tones with two different ILDs. (Right) Output rates of the model in response to binaural unmodulated tones with varied ILDs. Bold numbers show the rates at -45 dB and +15 dB. In panels F (Left) and G (Left), horizontal dotted gray lines indicate the resting potential. H: Monaural AM-tuning curve (rate-MTF) of the adjusted Wang-Colburn model. Bold numbers show the peak rate and the rate at 1200 Hz. (Inset) Monaural phase-locking (synch-MTF) of the model. Blue rectangular shading in F-H indicates the targeted ranges, while green shading in F-G shows the accepted ranges. I: Binaural AM phase-tuning curves of the model at three modulation frequencies. J: Binaural ILD-tuning curves of the model at five ipsilateral sound levels.

https://doi.org/10.1371/journal.pcbi.1005903.g010

Simulated monaural AM tuning curve (Fig 10H), binaural phase-tuning curves at different modulation frequencies (Fig 10I), and binaural level tuning curves with varied ipsilateral levels (Fig 10J) all resembled empirical data (Fig 1B–1D), although there were still some minor discrepancies. The monaural spike rate at 1.2 kHz (Fig 10H), for example, was higher than most units in cat LSO (Fig 1B). The peak of the monaural tuning curve (Fig 10H) was steeper than other models. Furthermore, the peak height for the binaural level tuning curves had a ‘ceiling’ at about 130 spikes/s (Fig 10J), because the outward rectification and Na inactivation still had considerable effects in the adjusted model.

Comparison of model with empirical knowledge.

Discrepancies between model outcome and empirical observations indicate that some of the original model assumptions are inappropriate and suggest what additional factors need to be considered in a revised model [4,5]. In our LSO modeling framework, the structure and most of the parameters of each model were determined from corresponding empirical data that were taken in prior physiological (mostly in vitro) studies. Parameters for which no experimental data were available were subject to fitting, in which simulation outcome was compared against known spiking output of LSO neurons recorded in vivo. For example, the original Wang-Colburn model was capable of reproducing a monaural AM tuning curve (Fig 9H) but not a binaural level tuning curve (Fig 9J). However, by adjusting the model parameters and ion channel kinetics, an improved fit to empirical results was achieved (Fig 10H–10J). To test whether the adjustments of ionic conductances and channel kinetics are supported by empirical observations will be a subject of future studies.

The plausibility of a model is generally judged by the agreement between its underlying assumptions and corresponding empirical knowledge, but biological realism does not always equate with a good predictive performance. Models that were proven to be constructed on wrong or inappropriate assumptions often produce good (or sometimes excellent) predictions [65], which may be attributed to "trade-offs among model components" [2]. Many auditory periphery models, for example, (incorrectly) assume that different frequency channels operate independently of each other, but still generate more than reasonable results [67]. The spike generation and succeeding potential reset of the IF model is far from empirical reality, but nevertheless the IF model and its variations have served as an important means to study the spiking behaviors of a wide variety of neurons [9]. In our modeling framework, the coincidence counting model naively counts the number of incoming synaptic inputs for a fixed duration of time with equal weights and discards all other inputs (Fig 4A and 4B). This assumption is biologically unrealistic, but the model still predicts both monaural and binaural properties well (Fig 4E–4G).

Comparison between models.

In the study of complex systems, an increased confidence on the underlying, universal mechanisms is achieved when multiple models yield similar or identical predictions, whereas differences in model predictions implicate the necessity of further investigations to falsify some of the model assumptions. Such comparisons were performed, for example, with models of visual cortical maps [95], with myelinated auditory nerve fiber models [96], with basilar membrane models of the cochlea [67], and with compartmental models of neocortical pyramidal neurons [66]. A comparison of the leaky IF model with exponential IF and HH-type models suggested that interspike intervals driven by fluctuating inputs can be described by a gamma distribution, independently of detailed spike generation mechanisms [97], whereas a variation of IF model was shown to reproduce spike initiation of cortical neurons more realistically than detailed HH-type models [98].

Among our LSO models, the lack of compressive factor in the passive (linear) models led to a higher output rate when driven by high intensity inputs (Figs 6G and 7J) than in models with low voltage activated currents that lowers the membrane impedance at high membrane potentials and thereby reduces the effect of summed synaptic inputs (Figs 8J and 10J). This comparison (as shown in Fig 11D) suggests that the nonlinear suppression may stabilize the ILD processing of LSO by restricting the range of output spike rates. A prior study using a KLVA channel model showed that the dynamic properties of KLVA current are indeed important for auditory coincidence detection [99]. These results demonstrate that models with biophysical details (such as HH-type) can be useful in identifying important mechanisms that cannot be captured in simplified models.

The limit frequencies of phase-locking to monaural AM sounds differed between models: synch-MTFs of the HH-type models decayed at lower modulation frequencies (Figs 9H and 10H, inset) than those of other simpler models (Figs 5E, 6E, 7E and 8E, inset), even though the rate modulation transfer functions were similarly tuned. Future experimental studies may prove which model predictions on synch-MTFs are more realistic, whereas further theoretical investigations are needed to reveal the underlying mechanisms causing these discrepancies.

Simple vs. complex models.

The balance between simplicity and reality is always a crucial issue in modeling studies (e.g., [5,7,9,10,107]). A biophysical model needs to be detailed enough to assure biological fidelity (and generalizability), while it should be sufficiently simple (or abstract) to allow for theoretical analyses or interpretations. This trade-off is also described as the philosophical debate between reductionism and holism [2]: a reductionist approach, frequently seen in applied mathematics and physics, aims at decomposing the system into parts to identify essential features, while a holist approach tries to include (almost) every component to replicate the emerging integrative behaviour of the complex system. These two philosophies, however, are not mutually exclusive, but complement each other to advance our scientific knowledge [7]. The simplistic coincidence counting model highlights the LSO neuron as a coincidence detector [40], while the detailed Wang-Colburn model describes how each biophysical element may play a role in temporal processing of auditory signals [53], together revealing the computational principles of LSO from different aspects.

In constructing a model, a general rule-of-thumb is that the complexity of a model should not exceed the complexity of the question that the model is addressing [8]. And according to the "principle of parsimony" (e.g., [1,92]), simpler models are favored over complex models if they have the same predictive capabilities, because simple models are more transparent and easier to communicate, and usually require less effort for parameter optimizations and model validation [2]. The meaning of each parameter of a simple model, however, tends to be more abstract and thus harder to relate with its underlying processes. The coincidence window in the coincidence counting model, for example, is determined by complex interactions of several biophysical factors, such as the membrane and synaptic time constants and nonlinear activation kinetics of ion channels [40]. The contribution of each of these factors cannot be easily evaluated separately from each other. Furthermore, it is difficult to directly incorporate measured nonlinear conductances into the shot-noise models, since these models lack detailed membrane dynamics.

In theoretical studies, finding the minimal description of a system is important for revealing its essential components [8,107]. The coincidence counting model, which simply counts the number of synchronized inputs and generates a spike, is (close to being) minimal. Several earlier studies mathematically formulated the spike statistics of the exponential Stein model driven by random synaptic inputs [75,76,108,109]. The passive IF model has been widely used in various fields of theoretical and mathematical neuroscience, including analyses of neurons that receive phase-locked inputs [69,70].

Simple models, however, are usually less flexible than detailed models. In the coincidence counting model, for example, the spike count is always an integer, and therefore any adaptive mechanisms that require a finer tuning of the threshold cannot be represented by this model. Furthermore, the simple shot-noise models cannot be expanded with additional ion channels. Related to this, the Stein models and the passive IF model are incompatible with subthreshold nonlinearity, resulting in higher spike rates than more detailed models when driven by high intensity synaptic inputs (Fig 11D). Detailed conductance based models are required, for example, to investigate possible effects of neuronal morphology, ion channel distribution, and locations of synaptic terminals on resulting binaural computation (e.g., LSO: [45]; MSO: [99]).

In light of applications, ‘medium-sized’ models are practically useful, because "they are complex enough to have nontrivial and nonobvious results, but simple enough so that the implications of the assumptions are readily apparent" [110]. Such an intermediate model should retain essential features of more detailed models to gain important insights into the modeled system [111]. Our active IF model (Fig 8) is an example of such medium-sized models, which ignores complex spike-generation mechanisms but still faithfully reproduces known sub- and suprathreshold characteristics of LSO neurons. In order to reduce the computational costs of the active IF model, the spike-associated current can be replaced by a simple potential rest and a succeeding absolutely refractory period (as in the passive IF model), when the entire spike waveform is not important.

Which model to choose?

As already seen in Introduction, no model is perfect. Therefore the question that a model user should ask is not "What is the best model?" but "What is the special value of each model, and what is its advantageous applicability?" (see [90] for a similar statement on selecting scientific methods). Table 2 summarizes major strengths and weakness of the seven LSO models, which have been generalized and validated with the common set of criteria. Here we note our suggested use of these models: in order to simulate the input-output relationship of an LSO neuron without considering its biophysical details such as membrane potential, we recommend the coincidence counting model; its computational efficiency also suits large-scale simulations that involve thousands of neurons or engineering applications that require real-time computation. To theoretically formulate the function of a LSO neuron, shot noise models (coincidence counting and Stein models) or the passive IF model would be suitable, as there are number of relevant studies and techniques available. In order to study how specific membrane properties and their interactions (e.g., ion channel kinetics and conductance densities) may play a role in the excitatory-inhibitory interaction of an LSO neuron, the adjusted Wang-Colburn model is clearly advantageous over other simpler models, because of its well-founded biophysical details and potential expandability with additional ion channels and neuronal morphology. For a general purpose, or when the user does not have a strong preference on which model to choose, the active IF model would be a good compromise among biological reality, computational efficiency, and predictive credibility.

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Table 2. Summary of model characteristics.

https://doi.org/10.1371/journal.pcbi.1005903.t002

Parameter selection.

For a systematic searching of model parameters, a number of computerized methods have been invented [66,112–115]. Selecting the parameters of a nonlinear dynamical system, however, requires attention to various types of potential pitfalls. A small change of a parameter may cause dramatic changes in the resulting behavior of the model, referred to as a bifurcation (e.g., [116]). Conversely, different parameter sets sometimes lead to practically indistinguishable results [113,117,118]. This is because the model can be highly sensitive to parameter changes in certain directions and not to those in other directions, and because changes in some model components can compensate each other [119] (see [120] for a review of these issues). Furthermore, averaging ‘good’ parameter combinations occasionally yield poor results [121], since the distribution of good parameters can be highly skewed in the parameter space (i.e., mathematically speaking, the distribution is non-convex). As the number of unconstrained parameter increases, the volume of the parameter space to search rapidly increases, making the parameter search inefficient, which is so-called the ‘curse of dimensionality’ [8,66]. In addition, since it is impossible to test a model completely with an infinite amount of empirical data, the modeler needs to define the desired range of applicability and to leave the model for further tests in domains where it was not optimized for [2,66].

For our conductance-based LSO models (i.e., the passive and active IF models and the adjusted Wang-Colburn model), we adopted a multi-step procedure for parameter tuning (as suggested, for example, in [122]) instead of using a fully automated parameter optimization technique, because corresponding empirical data were sparse and taken from various independent (and qualitatively different) sources. First, we focused on the subthreshold (non-spiking) responses of the neuron and tuned corresponding parameters (i.e., membrane conductances, reversal potentials, synaptic time constants and amplitudes, etc.). We then selected the parameters for suprathreshold spiking responses (e.g., thresholds of the IF models, and Na and KHVA conductances of the adjusted Wang-Colburn model) driven by simulated synaptic inputs, to meet the output criteria for monaural and binaural tuning curves, which we considered the most important response characteristics of the LSO neuron. After parameter selection, all models except the original Wang-Colburn model satisfied the nine output criteria (i.e., within all the acceptable ranges shown in Table 1). It should be noted, however, that we did not try to find the ‘best’ parameter combination for each model. In our preliminary simulations, multiple parameter combinations resulted in virtually identical response properties for the reasons discussed above. The model parameters we adopted in this study should thus be regarded just as a reasonable option (see Materials and methods for more details about our parameter selection).

Availability of empirical data.

In this study, we selected the parameters of each model so that it replicates the three major response features of monaural AM tuning, and binaural phase and intensity tunings (Fig 11B–11D). For conductance-based models, sub- and suprathreshold responses were additionally used to calibrate their membrane properties. We focused on these three response features for two reasons: First, considering the binaural nature of this nucleus [12] and its behavioral relevance [123], we presumed that ITD and ILD tunings are the most prominent characteristics of LSO, and that monaural AM stimulus should be used to further characterize its responses to temporally structured excitatory inputs. Second, currently there is only limited information available that can be used for constructing and validating an LSO model. Except for the stimulus characteristics we adopted in this study, we practically do not have sufficient amount of quantitative data taken from LSO and its primary input sources (VCN and MNTB). It should also be noted that, in none of previous LSO experiments, were the three response features altogether tested in the same single neuron. Therefore, our criteria of simultaneously sufficing the nine targeted ranges (Table 1) might be too strict for a single neuron model. As more empirical data become available in future measurements, the models should be re-evaluated with a wider variety of stimuli (see also "Possible applications" below).

Earlier studies presented a ‘chopping’ behavior in the peristimulus time histogram (PSTH) to be a distinctive response feature of LSO [30,41,124], guiding subsequent modeling studies to replicate these activity patterns (e.g., [43,51]). However, we did not use PSTHs to tune or evaluate LSO models, because the observed chopping patterns were found to be influenced by anesthesia [125,126] and may further affected by time-delayed ipsilateral inhibition [126], which is beyond the scope of our present study.

Difference across species.

In order to sufficiently constrain the LSO models, we had to borrow empirical data from various experimental studies. Most in vivo data that we used to measure the output performances of the models had been obtained in cats, while most in vitro data that we adopted to tune the subthreshold membrane properties had been taken from rodents. This type of problem has been "well known among old-timers" [127], and is still common when a modeler tries to relate different levels of description: e.g., connecting in vitro slice recording data with in vivo network dynamics; or connecting neuronal activity with behavioral outcome. Moreover, cellular properties may be considerably different even between closely related species and sometimes incompatible with each other. Measured membrane resistances of LSO neurons of rodents, for example, were 42 ± 21 MΩ in gerbils [77] and 15–53 MΩ in mice [88], while they were much higher in guinea pigs (73 ± 17 MΩ: [48]) and in rats (109 ± 64 MΩ: [85]). Possibly related to this, action potentials in rats’ LSO neurons usually overshot with amplitudes of 81.3 ± 15.3 mV [85], whereas those in gerbils were substantially smaller (41 ± 14 mV: [77]). Furthermore, for technical reasons, in vivo recordings are performed normally with adult animals, while in vitro patch clamp recordings are done with juvenile animals whose cellular and synaptic properties may still be under development [128–130]. These discrepancies suggest that our models could represent an LSO neuron of a ‘chimera’ that may not exist on earth, although the predictions of the models were at least qualitatively consistent with known empirical data from real animals. Further experimental recordings and model optimizations would therefore be necessary to create an animal-specific LSO model.

Difference within species.

Even within the same species, LSO neurons show considerable variations in monaural and binaural responses [37–39,124]. Previous physiological and anatomical studies of LSO found tonotopic gradients in the number of inhibitory synaptic receptors [131], the morphology of dendritic arbors [132], the KLVA conductance and resulting spiking patterns [86], and the resonance property of the membrane [48]. These tonotopic differences may explain some of the observed unit-to-unit variations among LSO cells. Our previous modeling study demonstrated that variations of the parameters for coincidence detection indeed affect the shape of the monaural and binaural tuning curves [40]. Nevertheless, earlier physiological studies of LSO, either in vivo or in vitro, presented their data often without the information of the estimated characteristic frequencies of the recorded units. Due to such lack of data, we did not take tonotopic variations into account when we tuned the models. Systematic recording, analysis and optimization will thus be needed for constructing frequency-specific LSO neuron models that can be used as a building block for future binaural network simulations involving across-frequency integration.

Possible extensions.

In this study, we examined seven single-compartment LSO models with varied levels of complexity (Table 2). Our list, however, is far from exhaustive. Between the passive IF and HH-type models, for example, there would also be a number of nonlinear IF models, including the quadratic IF [133,134], exponential IF [135] and generalized IF models [136], which we expect to show similar response properties to our active IF model. In order to make the model biologically more plausible, various types of ion channels that were found in LSO and other auditory areas may be added to the Wang-Colburn model, such as persistent and resurgent Na⁺ channels [137], a number of different low- and high-voltage-activated K⁺ channels [138], and hyperpolarization-activated cation channels [139,140].

The spike initiation site of a principal neuron in the gerbil MSO is located in the axon at a distance of several tens of micrometers away from the cell body, where backpropagating, small spikes were observed [141]. Considering the morphological similarities between LSO and MSO neurons [13,132,142,143], remotely initiated action potentials could also explain non-overshooting spikes observed in whole-cell recordings of gerbil LSO neurons [77]. In order to simulate such backpropagating spikes, the model needs to have at least two compartments (e.g., [83,107]). Segregation of the synaptic integration site from the spike initiation site may also improve the biological plausibility of the model by reducing the (unrealistically) large sodium conductance of the cell body in the Wang-Colburn models (see [66] for related discussion with cortical neuron models). Furthermore, to incorporate the effects of dendritic integration of synaptic inputs mediated by active conductances, multicompartment models should be considered, as was done with other auditory coincidence detector neurons (e.g., [99,144]).

Possible applications.

In this study we focused on the steady-state responses of LSO models driven by Poissonian input spikes. By replacing the input stage of the modeling framework with a more detailed model (e.g., [145,146]), responses to transient stimuli (such as click trains) or naturalistic sounds (such as vocalization and speech) may be simulated [48]. In such cases, the output stage (i.e., LSO model) might also have to be revised by incorporating an adaptive threshold, so that the model could more faithfully compute the temporal information of sounds [147,148]. These modifications would allow us to predict and examine how LSO neurons may respond to more natural stimuli that simultaneously contain both ITD and ILD information [36,37]. Further generalizability of the LSO models should be tested, when sufficient empirical data become available in future physiological measurements with more complex sound stimuli.

As discussed earlier, modeling enables us to simulate hypothetical situations that cannot be easily reached with experimental approaches. Predicting the effects of pathological changes in the auditory brainstem, for example, would be a subject of such modeling studies. Age-related loss of inhibitory neurons is commonly found in various animal species (see [149,150] for reviews); synaptic and membrane properties are also affected by aging [151]. Degraded localization performance for high-frequency sounds in aged animal models [152] and humans [153] suggests that altered inhibitory inputs from the MNTB may affect the function of LSO. A modeling approach with a varied number and amplitude of synaptic inputs would reveal how binaural phase- and level-tuning curves may be influenced by the loss of synaptic inputs, and provide fundamental insights relating prior anatomical [154] and physiological [155,156] findings in aged animals. For another example, potential effects of altered connectivity in a disease model mouse on binaural coding [157] could also be simulated by incorporating observed changes in the modeling framework.

Binaural information processing is critical for efficient acoustic communication both in animals and in humans, especially in a noisy environment [11,54]. How to preserve and use binaural cues in cochlear implant users has been actively investigated (see [158] for a review). Combining electric hearing models with binaural neuron models [58,59] would be useful not only to predict how bilateral implantation may or may not contribute to the improvement of perceptual outcome, but also to highlight future directions towards the refinement of coding technologies. Such an application would constitute an important step for a neuronal modeling study, since it provides an opportunity to make ‘out-of-domain’ predictions beyond the initial scope of the original modeling endeavor [122].

Monaural AM frequency coding.

An LSO neuron changes its spike rate according to the modulation frequency f_m of ipsilateral AM tones (shown as an ‘AM-tuning curve’ in Fig 1B). This rate-f_m relation is also called the ‘rate modulation transfer function’ (rate-MTF) [169,170]. We calculated the rate-MTF of LSO models at modulation frequencies between 50 and 1200 Hz. Monaural AM coding can additionally be examined by the phase-locked output [34]. The ‘modulation gain’ was defined by 20log₁₀(2R), where R is the VS of the LSO output spikes at the modulation frequency f_m. This gain-f_m relation is also called the ‘synchrony modulation transfer function’ (synch-MTF) [170]. Since only insufficient empirical information is available, we did not use the synch-MTF for optimizing the model parameters, but used it as one of the output measures to characterize the models.

Binaural AM phase coding.

When simulated binaurally, LSO neurons change their spike rates according to the interaural time (or phase) difference of the envelopes of the AM tones (shown as a ‘phase-tuning curve’ in Fig 1C). The interaural time difference is modified and compensated by several transduction factors, resulting in the time difference of bilateral synaptic input at each LSO neuron (e.g., [11,171]). We simulated this time difference at the LSO synapse by changing the relative phase of simulated synaptic inputs. As stated above, excitatory inputs were assumed to be locked to one particular phase of the envelope of the stimulus AM tone, while inhibitory inputs are locked to another phase. We varied the difference of these locking phases, and calculated the output spike rates of the models, first at the modulation frequency of 300 Hz (for parameter fitting) and then at 150 and 450 Hz (for further characterization). We defined a positive phase difference as inhibitory inputs preceding excitatory inputs as in previous experimental studies [36,37].

Binaural intensity coding.

Because of the excitatory-inhibitory interaction, LSO neurons are generally sensitive to ILDs (shown as an ‘ILD-tuning curve’ in Fig 1D). Note that, by convention, the ILD is defined as the sound level at the contralateral ear minus that at the ipsilateral ear: i.e., a negative ILD means that the ipsilateral sound is louder. To simulate this ILD coding, we changed the binaural stimulus sound levels (of non-modulating sound) which drive excitatory and inhibitory inputs to the LSO model (Fig 2C). For parameter fitting, the ipsilateral level was first fixed at +35 dB and the contralateral level was varied between -10 and +50 dB, resulting in an ILD range between -45 and +15 dB. We then calculated the ILD-tuning curves with ipsilateral levels of +25, +30, +40, and +45 dB for further characterization of the model.

Membrane impedance.

Input impedance of the model membrane was calculated by applying sinusoidal currents with a fixed amplitude of I_app = 10 pA and varied frequencies of 0.01–40 kHz. The membrane potential was clamped at -60 mV by an additional constant DC current. At each frequency f_app, the maximum (V_max) and minimum (V_min) values of the steady state response (typically at >100 ms after the current onset) of the oscillating membrane potential were measured and the impedance was obtained as R(f_app) = (V_max-V_min)/(2I_app). In previous in vitro measurements in rats and guinea pigs [48], most LSO neurons had low-pass impedance profiles, while some other neurons that are found in the lateral (low-frequency) edges of the LSO showed weakly band-pass properties.

Current-voltage relationship.

Current-voltage relationship (I-V curve) of the model membrane was obtained by applying step currents with varied amplitudes I_app between -0.5 and 1.5 nA. The membrane potential was clamped at V_clamp = -60 mV by an additional constant DC current. The spike generator (threshold-crossing detector) of the IF models and sodium channels of the Wang-Colburn models were disabled to avoid spiking responses. For each step current, the steady state response V_app (typically at >100 ms after the current onset) of the membrane potential was measured to plot an I-V curve. The DC input resistance of the membrane was calculated as R_DC = (V_app-V_clamp)/I_app. with I_app = +10 pA.

In previous in vitro recordings, LSO neurons showed outward rectification [77,85]; i.e., the membrane resistance was lower at holding voltages above the resting potential than at or below. In the passive and active IF models and the adjusted Wang-Colburn model (see following sections for the definitions), we tuned the leak and other conductances so that the model membrane had an input resistance of 37–40 MΩ at the holding potential of -60 mV. This range corresponds to the measured values of 42 ± 21 MΩ in gerbils [77] and 15–53 MΩ in mice [88].

Step-current response.

Step-current responses of the model membrane were obtained by applying rectangular currents with a fixed duration of 30 ms and varied amplitudes of 0 to 1 nA. The spike response was characterized as ‘phasic’ when the membrane stopped spiking before the offset of the step current, and as ‘tonic’ when the spiking activity lasted until the end of the step current.

Previous slice recordings in mice [88] and gerbils [77] reported that LSO neurons show tonic spiking with step currents of a few hundreds of pA; LSO neurons in rats showed a combination of tonic and phasic spiking responses depending on the size of the injected current [85]. This makes a contrast to other well-known auditory coincidence detectors of octopus cells in the posteroventral cochlear nucleus and principal neurons in the MSO, which have a large amount of low-threshold conductance and thus typically show only phasic spiking in response to step current injections [82].

Model structure.

The coincidence counting model of LSO is an extension of spike-count-based model of auditory coincidence detectors [172,173], and was fully described in [40]. In brief, the model compares the weighted numbers of excitatory and inhibitory inputs in a pre-set time window and generates an action potential when the total number reaches the threshold. More particularly, a coincidence window of size W_ex (Fig 4A, vertical gray band) slides along the time axis, and the number of incoming excitatory synaptic inputs (Fig 4A, blue bars) in this window is counted. If the number of input spikes in the window reached the pre-set threshold θ (Fig 4A, black bar), an output spike is generated (Fig 4A, green bar). If multiple threshold crossings happen within the pre-set refractory period T_ref, then only the first one leads to an output spike and the others are discarded (Fig 4A, small arrow). When the model receives an inhibitory input, the threshold is elevated by a fixed amplitude H for the time length of W_inh (Fig 4A, dotted vertical rectangle). In other words, each inhibitory input subtracts the coincidence counts by H during the inhibitory time window of W_inh. Note that the threshold θ is always an integer (i.e., non-integer values of θ do not make sense), as the model simply counts the number of synchronized inputs in the coincidence window.

Parameter selection and justification.

The coincidence counting model has five free parameters (Table 4). The values we used were the same as the default values used in our previous study [40], which examined how each parameter affects the output of the model. In previous physiological experiments, measured refractory periods T_ref were 1.1–2.8 ms in cats [124]; measured length of inhibition window W_inh were 0.8–2.0 ms (rats) [174]; and measured thresholds θ were 9.6 ± 2.8 in gerbils [77]. In young gerbils [77], measured membrane time constants were 1.1 ± 0.4 ms and minimum durations of excitatory synaptic inputs were 1.5 ± 0.8 ms, both of which are expected to limit the maximum width W_ex of the coincidence window. Furthermore, based on the measured durations of excitatory and inhibitory inputs in gerbils [77], we assumed that the inhibition window W_inh was twice as long as the coincidence window W_ex. Based on these experimental constraints, we determined the set of parameter values (Table 4) to satisfy the targeted criteria (Table 1) of output rates driven by simulated monaural and binaural inputs (Fig 4C–4E). Since there was no measurement available that showed how many excitatory inputs are cancelled by an inhibitory input, we choose the value of the inhibition amplitude H such that the binaural phase-tuning curve yielded a sufficient modulation depth (see Figs 11 and 12 in [40] for simulation results).

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Table 4. Parameters for the coincidence counting model.

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Model structure.

The exponential Stein model was adopted by Colburn and Moss [47] to study the excitatory-inhibitory interaction of LSO neurons (see [54] for a review of relevant earlier studies). In this model, each input gives rise to an exponentially decaying response of the virtual membrane potential (Table 5). The response is positive for excitatory inputs and negative for inhibitory inputs (Fig 5A); excitatory and inhibitory inputs generally have different amplitudes and decay time constants. When the potential reaches or exceeds the pre-set threshold θ, an output spike is generated. After each spike generation, the potential is reset and fixed to zero for the pre-set refractory period of T_ref. Respecting the pioneering work by Stein [75,76], this type of model is called the ‘Stein model’ [109]. The term ‘exponential’ was added to clarify that each synaptic input is converted into an exponential function. This model was also called the ‘shot-noise model’ (e.g., [54]), but we leave this term for a more general category including the coincidence counting model (Fig 11A).

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Table 5. Equations and parameters for the exponential Stein model.

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Parameter selection and justification.

The exponential Stein model has five free parameters (Table 5). As in other models of this study, we used the refractory period T_ref = 1.6 ms, which was comparable to the measured values of 1.1–2.8 ms in cats [124]. Other parameters were selected to fit the targeted monaural and binaural tuning curves. In our parameter selection, we initially explored the meshed four dimensional parameter space of θ (range: 6.8–12.4 inputs; step: 0.1), τ_ex (range: 0.5–2.0 ms; step 0.1), H (range: 1.0–3.0 inputs; step 0.1), and τ_inh (range: 1.0–3.0 times τ_ex; step 0.2). The range of the model threshold θ was determined from the mean ± SD of measured threshold values in gerbils [77], whereas other parameter ranges were determined rather arbitrarily since there were no directly relevant data available for exponentially decaying synaptic inputs. In this parameter space, however, we did not find a set of parameters that satisfied all the nine accepted output criteria, primarily due to the low excitability of the model. Therefore we then expanded the range of the threshold as 5.4–13.8 inputs, which corresponded to the mean ± 1.5SD of measured values [77], and found parameter combinations (Table 5) that reproduced empirical monaural and binaural tuning curves (Fig 5C–5E). Although the shapes of inputs were considerably different between the model (exponential function) and in vitro slice recording data, the time scales of the modeled inputs (shown in Fig 5B) were roughly comparable to measured data (duration of excitatory inputs: 1.5–4.2 ms; duration of inhibitory inputs: 3.2–8.1 ms: [77]).

Model structure.

The alpha Stein model is a direct modification of the exponential Stein model by replacing the exponential function with an alpha function (Table 6, Fig 6A), which is often used for modeling synaptic inputs [78–80]. In contrast to the exponential function that has a peak at the onset (i.e., at time zero) irrespectively to its decay time constant, the alpha function with a time constant τ reaches its peak at time τ. The model synaptic inputs (Fig 6B) showed similar shapes to empirical results [77,175]. Thus this modification makes the model biologically slightly more realistic at some additional computational costs (Tables 1 and 2). For other operations (such as the spike generation and refractoriness) of the alpha Stein model, the same rules apply as the exponential Stein model.

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Table 6. Equations and parameters for the alpha Stein model.

https://doi.org/10.1371/journal.pcbi.1005903.t006

Parameter selection and justification.

The alpha Stein model has five free parameters (Table 6). As in other models, the refractory period T_ref was fixed to 1.6 ms. We then determined the excitatory and inhibitory time constants (τ_ex, τ_inh) of the modeled synaptic inputs so that their time courses became similar to empirical data [77] and to simulated synaptic inputs in more detailed conductance-based models (compare Figs 6B and 7E). Finally we selected the threshold θ (range: 6.8–12.4 inputs; step: 0.1) and inhibitory amplitude H (range: 1.0–3.0 inputs; step 0.1) from the two dimensional parameter space, so that the model output satisfied the accepted output measures of monaural and binaural tunings (Fig 6C–6E).

Model description.

In the conductance-based (IF and Wang-Colburn) models presented below, alpha functions were commonly used to simulate synaptic conductances. Each synaptic input was converted into an alpha function (α_ex, α_inh), summed into the total conductance (g_ex, g_in), and then multiplied by the driving voltage to yield the resulting synaptic current (I_ex, I_inh) (Table 7). As noted before, modeled synaptic inputs are assumed to be injected directly into the soma.

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Table 7. Equations and parameters for excitatory and inhibitory synaptic inputs commonly used with conductance-based models.

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Parameter justification.

The modeled excitatory E_ex and inhibitory E_inh reversal potentials were determined, respectively, from the standard driving voltage of AMPA receptors [78] and from the measurement of inhibitory postsynaptic potentials in gerbil LSO [77]. To calibrate the amplitudes and time constants, we used a simple RC membrane that has similar subthreshold electrical properties to gerbil LSO neurons [77]. The equation and parameters of the RC model was the same as the passive IF model (Table 8) but without any spiking thresholds. With the parameters shown in Table 7, simulated unitary amplitudes of excitatory and inhibitory postsynaptic potentials were about 2.3 mV and 2.7 mV, respectively, which were both within the measured ranges of 1.5–8.7 mV and 1.6–8.3 ms [77]. See Fig 7E for the shapes of modeled synaptic inputs.

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Table 8. Equations and parameters for the passive IF model.

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Sanes [77] also measured the duration of synaptic inputs, which was defined as the time length "from the rising latency to the time at which the signal returned to the baseline noise level". Since our model lacks intrinsic noise sources, we calculated the difference of the two time points where the simulated unitary postsynaptic potential crossed the 5% line of the peak amplitude, and used it as a rough estimation of the duration. With the model parameters in Table 7, the simulated durations of excitatory and inhibitory synaptic inputs were about 3.5 ms and 4.1 ms, respectively. These values corresponded to the measured durations of 1.5–4.2 ms (excitatory) and 3.2–8.1 ms (inhibitory inputs) [77]. It should be noted that these measured amplitudes and durations of synaptic potentials showed considerable variations across LSO neurons, suggesting that the model parameter values we chose may not be optimal. In our preliminary simulations, however, we found that small variations in these input parameters (up to a few tens of percents) did not greatly alter the output of the LSO model, if other parameters were re-adjusted with the new input. Nevertheless, we also note that the duration of inhibitory synaptic potential needed to be around 3–5 ms or less, to fully account for the binaural phase-tuning curves.

Model structure.

The passive integrate-and-fire (IF) model is the same as the standard leaky IF model used commonly in theoretical neuroscience [78–80]. We use the term ‘passive’ to make a contrast to the active IF model described below. The dynamics of the subthreshold membrane response is formulated as a linear (passive) RC circuit (Table 8, Fig 7A) with a low-pass impedance profile (Fig 7B) and a linear I-V relationship (Fig 7C). When the membrane potential V(t) reaches or exceeds the pre-set threshold V_θ, an output spike is generated. After each spike generation, the membrane potential is fixed to the reset potential V_reset for a refractory period of T_ref. It should be noted that, despite its name ‘integrate-and-fire’, an IF model with a short membrane time constant behaves rather like a coincidence detector than as an integrator of synaptic inputs [176].

Parameter selection and justification.

The values of the six free parameters (Table 8) of the passive IF model were selected with a two-step procedure. First, experimental data were used to tune subthreshold membrane properties. As stated in the subsection titled "Current-voltage relationship", we determined the leak conductance g_L = 26.4 nS to obtain an input resistance of R_memb = 1/g_L = 37.9 MΩ. We then fixed the membrane capacitance C = 24 pF, based on previous physiological data [86], to obtain a membrane time constant of τ_memb = R_membC = 0.91 ms, which was comparable to the measured time constant of 1.1 ± 0.4 ms in gerbil slices [77]. The reversal potential E_L of the leak current was set at -60 mV, based on empirical resting potentials [77,85,86]. As the second step of parameter fitting, we searched for the value of the threshold V_θ, so that the model output satisfied the criteria for monaural and binaural tunings (Table 1, Fig 7F–7H). The reset potential V_reset was the same as the resting potential E_L. The length of the refractory period T_ref was the same as the shot-noise models, comparable to the measured value in cats [124].

Model structure.

The active integrate-and-fire model has a KLVA conductance (Fig 8A), which is the major source of the subthreshold nonlinearity of LSO neurons [86]. The term ‘active’ indicates the existence of voltage-dependent subthreshold currents. The model comprises two differential equations that describe the dynamics of the membrane potential V(t) and the KLVA activation variable d(t) (Table 9). Since the KLVA conductance increases at depolarized membrane potentials, the I-V curve shows an outward rectification (Fig 8C). As in the passive IF model, an output spike is initiated when the membrane potential reaches or exceeds the pre-set threshold V_θ. After each spike (at time t_sp), the model is in a refractory period of length T_ref, during which the model generates no more spikes. In addition, a spike-associated current I_spike(t-t_sp) is injected to create a spike-like trajectory in the membrane potential to make the simulated responses somewhat more realistic than a simple potential reset [68].

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Table 9. Equations and parameters for the active IF model.

https://doi.org/10.1371/journal.pcbi.1005903.t009

Parameter selection and justification.

The activation α_d(V) and deactivation β_d(V) rates determine the steady state activation d_∞(V) and the time constant τ_d(V) of the KLVA conductance (Table 9). Based on previous recordings from auditory neurons in various animals (mice: [177]; rats: [178]; guinea pigs: [179,180]; chickens: [181]), we defined the activation and deactivation rates using simple (and symmetric) exponential functions. As no data were available for the value of potassium reversal potential E_K of LSO neurons in vivo, we rather arbitrarily set it as -75 mV, which nevertheless corresponded to the values used in previous modeling studies of auditory neurons (e.g., -70 mV: [87,182]; -80 mV: [45]; -90 mV: [183]). The values of the leak g_L and maximum KLVA conductances g_KL were determined to reproduce empirical I-V curves [77], with a resulting input resistance of ~38.2 MΩ at the holding potential of -60 mV. The leak reversal potential E_L was determined so that the resulting resting membrane potential was between -61 and -60 mV.

After all the parameters for the subthreshold response were fixed, we selected the value of the threshold V_θ to attain the targeted rates of monaural and binaural tuning curves (Table 1, Fig 8F–8H). The shape of the spike-associated current I_spike(t) was determined to mimic empirical spike shapes [77]. We used a sum of two exponential curves for I_spike(t) (Table 9), because an exponential function enables exact calculations at discrete time steps [184]. The first and second terms of the spike-associated current are responsible for the depolarization and repolarization of the membrane potential, respectively. The length of the refractory period T_ref was the same as in the other models with thresholds.

Model structure.

Wang and Colburn [53] employed a conductance-based model to study the biophysical mechanisms of monaural frequency tuning of LSO neurons. Here we use the term ‘original’ to indicate that the model equations and parameters were unchanged from their original publication [53], except for the sodium reversal potential and the KLVA conductance (see below). The Wang-Colburn model has leak, KLVA, high-threshold-activated potassium (KHVA), and fast sodium (Na) conductances (Fig 9A). The kinetic equations for these conductances (Table 10) were directly taken from the Rothman-Manis model of guinea pigs’ VCN neurons [87]. The Wang-Colburn model is a variation of the HH model, and hence its spike threshold is not explicitly represented as a parameter but determined by a nonlinear interaction of ionic conductances. In our simulations, we counted a spike when the membrane potential depolarized above -30 mV and successively repolarized below -45 mV.

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Table 10. Equations for the original and adjusted Wang-Colburn models.

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Adopted parameters.

Wang and Colburn [53] tuned the parameters of their LSO model, focusing primarily on the monaural AM frequency coding (Fig 1B). The value of the maximum KLVA conductance g_KL was presented as "varied" in their report [53]. We adopted a value of g_KL = 85 nS, which was one of the values that had been reported to approximate empirical monaural tuning curves (Fig 9H). The sodium reversal potential E_Na was revised from 0 mV to +50 mV, because we found that the model with the old value failed to reproduce their reported results and later confirmed that the new value was indeed correct (Le Wang, personal communication, 2017). All other parameters shown in Table 11 are the same as the published values [53]. No further optimizations were performed with the original Wang-Colburn model.

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Table 11. Parameters for the original Wang-Colburn model.

https://doi.org/10.1371/journal.pcbi.1005903.t011

Model structure.

Since the original Wang-Colburn model failed to reproduce binaural tuning curves (Fig 9F and 9G), we revised the model by shifting the voltage dependence of the channel kinetics and re-tuning the parameters. The term ‘adjusted’ indicates the revision from the original model. The model equations are the same as the original Wang-Colburn model (and thus the same as the Rothman-Manis model), except that an additional term V_shift was introduced to simultaneously adjust the voltage dependence of the sodium and potassium kinetics (Table 10). In our preliminary simulations, we found that, in the original Wang-Colburn model, activation of the large KLVA conductance in combination with the inactivation of the Na conductance prohibited repetitive spiking, resulting in low output rates (Fig 9J). In the adjusted model, we shifted the tuning curves of the ionic conductances by V_shift = +5 mV to reduce these effects. As noted above, we counted a spike when the simulated membrane potential exceeded -30 mV followed by an after-spike repolarization below -45 mV.

Parameter selection and justification.

The sodium reversal potential E_Na and the temperature factor Q₁₀ were the same as the original Wang-Colburn model (Table 12). For the membrane capacitance C and the potassium reversal potential E_K, we adopted the corresponding values from the active IF model (Table 9). The synaptic reversal potentials E_ex and E_in were the same as the IF models (Table 7). The leak g_L and maximum KLVA conductances g_KL were determined to mimic empirical I-V curves [77] with a resulting input resistance of ~38.4 MΩ at -60 mV. Note that the outward rectification observed at or above -40 mV (Fig 10C) was also due to the large KHVA conductance. As in the active IF model, the leak reversal potential E_L was adjusted to obtain a resting membrane potential between -61 and -60 mV.

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Table 12. Parameters for the adjusted Wang-Colburn model.

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After fixing the above parameters, we selected the values of the sodium g_Na and KHVA conductances g_KH to satisfy our rate criteria of monaural and binaural tuning curves (Table 1, Fig 10F–10H). The enormous value of the Na conductance (compared to other conductances) was necessary to achieve a sufficient excitability of the model neuron, while the large value of KHVA conductance was required to ensure the repolarization after each action potential generation.