To analyze signal processing in the locust ear, we performed intracellular recordings in vivo from single receptor-cell axons in the auditory nerve. The stimuli consisted of two short clicks. The clicks were sound-pressure pulses with peak amplitudes *A*_{1} and *A*_{2}, respectively, and were separated by a short time interval, Δ*t* (Figure 2A; see also Figure S1 for microphone recordings). For such stimuli, the receptor cell fired at most one action potential per double click; stimulus intensity hardly influenced spike timing, but strongly affected spike probability, as shown in Figure 2B. The response strength may thus be described by the probability that a spike occurs within a certain time window after the two clicks.

For fixed time interval Δ*t,* an iso-response set consists of those combinations of *A*_{1} and *A*_{2} that lead to the same predefined spike probability *p*. Since the spike probability increases with the click amplitudes, *A*_{1} and *A*_{2} can easily be tuned during an experiment to yield the desired value of *p* (see Materials and Methods). The tuning scheme was applied for stimulus patterns with different relative sizes of the two clicks, so that a multitude of different combinations of *A*_{1} and *A*_{2} corresponding to the same *p* was obtained. Rapid online analysis of the neural responses and automatic feedback to the stimulus generator made it possible to apply this scheme despite the time limitations of the in vivo experiments.

Figure 3 shows typical examples of such iso-response sets, measured for different time intervals Δ*t*. For each of the three cells displayed, two distinct values of Δ*t* were used. The sets can be used to identify stimulus parameters that govern signal processing at a particular time scale. Most importantly, the iso-response sets exhibit specific shapes that vary systematically with Δ*t*. For short intervals (below approximately 60 μs), the sets generally lie on straight lines, at least for low-frequency receptor cells. High-frequency receptor cells do not display straight lines even at the smallest Δ*t* used in the experiment (40 μs) for reasons that will become apparent later. For long intervals (between approximately 400 and 800 μs, depending on the cell), the iso-response sets fall onto nearly circular curves. Note that in Figure 3C, the iso-response set for Δ*t* = 500 μs deviates from the symmetry between *A*_{1} and *A*_{2}. In Figure 3D, the inter-click interval of Δ*t* = 120 μs fell in neither of the two regimes discussed above, and the corresponding iso-response set shows a particularly bulged shape. Recordings from a total of eight cells agree with the observations from the three examples displayed in Figure 3.

The two prominent shapes of the iso-response sets—straight lines and circles—reflect two different processing steps in the auditory transduction chain. A straight line implies that the linear sum, *A*_{1}**+** *A*_{2}, of both click amplitudes determines the spike probability and demonstrates that the sound pressure is most likely the relevant stimulus parameter. Such linear summation of the pressure on short time scales is not surprising, considering the mechanical properties of the eardrum; owing to its mechanical inertia, rapidly following stimuli can be expected to superimpose. This interpretation is in agreement with laser-interferometric and stroboscopic observations of the eardrum, which have demonstrated that it reacts approximately linearly to increases in sound pressure [3,19].

Each of these four steps transforms the signal in a specific way, which may be nearly linear (as for the eardrum response) or strongly nonlinear (as for spike generation, which is subject to thresholding and saturation). In general, the illustrated steps may contain further sub-processes such as cochlear amplification or synaptic transmission between hair cells and auditory nerve fibers. For the auditory periphery of locusts investigated in the present study, this schematic picture resembles anatomical findings [18], which reveal that the receptor neurons are directly attached to the eardrum and that they send their action potentials down the auditory nerve without any further relay stations.

For the longer intervals, on the other hand, the iso-response sets are circles to good approximation, indicating that the quadratic sum, or *A*_{1}^{2}**+** *A*_{2}^{2}, now determines the spike probability. It follows that the sound energy, which is proportional to the squared pressure, is the relevant stimulus parameter on this time scale. This quadratic summation represents a fundamentally different way of stimulus integration from that of the linear summation on short time scales and indicates the involvement of a different biophysical process. A process that can mediate stimulus integration over longer intervals is the accumulation of electrical charge at the neural membrane. According to this explanation, the electrical potential induced by a click is proportional to the click's energy; contributions from consecutive clicks are then summed approximately linearly because of the passive membrane properties. This is in accordance with earlier investigations for stationary sound signals that revealed an energy dependence of the neurons' firing rate [20]. We conclude that in between the mechanical vibration of the eardrum and the accumulation of electrical charge at the neural membrane, there is a squaring of the transmitted signal. This squaring may be attributed to the core process of mechanosensory transduction, i.e., the opening of ion channels by the mechanical stimulus.

The above findings motivate the following mathematical model, which describes how a stimulus consisting of two sound clicks is transformed into a spike probability. Within the model, a single click of amplitude *A* generates a vibration of the tympanum with strength *X = c*_{1}*·A,* i.e., linear in the amplitude with a proportionality constant *c*_{1}. This mechanical vibration leads to a membrane potential, whose effect on the generation of the spike some time *T* after the click is given by *J = c*_{2}·*X*^{2}*= c*_{2}*·*(*c*_{1}*·A*)^{2}, i.e., quadratic in the amplitude with an additional proportionality constant *c*_{2}. The square follows from the circular shape of the iso-response sets for longer time scales, which indicated that a quadratic operation must take place before the accumulation of charge at the neural membrane. Finally, the spike probability *p* is given by a yet unknown function *p = g*(*J*). As *J* is the relevant quantity determining spike probability, we also refer to it as “effective stimulus intensity.” The model contains a freedom of scaling; any proportionality constants in *J* can be absorbed into the function *g*(*J*). To simplify the notation, we thus set *c*_{1} = *c*_{2} = 1 and obtain *X = A* for the strength of the mechanical vibration and *J = X*^{2}*= A*^{2} for the effective stimulus intensity in response to a single click.

Note that in this picture, the mechanical vibration and the membrane potential are each captured by a single quantity that does not describe the time course of the corresponding processes, but rather their integrated strength in response to a click. In general, the conversion of the mechanical vibration into a membrane potential as well as the spike generation are dynamical processes that do not happen at a single moment in time. For simplicity, however, one may think of *X* as describing the velocity of the mechanical vibration immediately after the click and *J* as capturing the membrane potential at the time of spike generation.

For the two-click stimulus with amplitudes *A*_{1} and *A*_{2}, respectively, we choose the first click to be small enough so that it does not lead to a spike by itself. The measured action potential is thus elicited at some time *T* after the second click. To derive the model equation for this experimental situation, we divide the time from the first click to spike generation into the period between the two clicks and the period following the second click.

Let us start by focusing on the inter-click interval. After the first click, the mechanical vibration has the strength *X*_{1} = *A*_{1}. However, how much electrical charge accumulates during the inter-click interval to influence spike generation at time *T* after the second click depends on the length Δ*t* of the inter-click interval. This effect is incorporated by a Δ*t*-dependent scaling factor *Q*(Δ*t*) into the model and results in a first contribution from the first click to spike generation given by *J*_{1}*= A*_{1}^{2}*·Q*(Δ*t*). Since *Q*(Δ*t*) denotes the effect of the first click within the inter-click interval only, it should vanish in the limit of very small Δ*t*.

Let us now consider the remaining time before spike generation. After the second click, the mechanical vibration is due to a superposition of both clicks. For short inter-click intervals, the straight iso-response lines suggest a simple addition of the two click amplitudes; in general, however, the contribution of the first click to the membrane vibration after the second click will again depend on the inter-click interval Δ*t*. This is modeled by a scaling factor *L*(Δ*t*), i.e., the vibration after the second click has a strength *X*_{2} = *A*_{1}· *L*(Δ*t*) + *A*_{2}. Accordingly, the effect of the two-click vibration on the membrane potential at time *T* after the second click is *J*_{2}*=* (*X*_{2})^{2}*=* (*A*_{1}*· L*(Δ*t*) + *A*_{2})^{2}. For very small Δ*t, L*(Δ*t*) should approach unity to account for the equal contribution of both clicks for vanishing inter-click intervals. The total effective stimulus intensity is then given by

This quantity determines the spike probability *p* via the relation *p* = *g*(*J*).

How does this model explain the particular shapes of the iso-response sets in Figure 3? The linear and the circular iso-response sets apparently correspond to the two special cases: (1) *L*(Δ*t*) = 1 and *Q*(Δ*t*) = 0 (straight line) and (2) *L*(Δ*t*) = 0 and *Q*(Δ*t*) = 1 (circle).

We can therefore regard equation 1 as a minimal model incorporating linear as well as quadratic summation, as suggested by the measured iso-response sets. Based on the experimental data, we expect that the first case is approximately fulfilled for small Δ*t* and the second case in some range of larger Δ*t*. In our biophysical interpretation, the first case means that the two clicks are added at the tympanic membrane (*L*(Δ*t*) 1), but the short interval between the two clicks prevents a substantial accumulation of charge from the first click alone (*Q*(Δ*t*) 0), as already discussed above. The second case may be found for Δ*t* long enough that the mechanical vibration has already decayed (*L*(Δ*t*) 0). The two clicks are then individually squared, i.e., they independently lead to two transduction currents. The currents add up if the time constant of the neural membrane is significantly longer than the inter-click interval (*Q*(Δ*t*) 1).

In the two limiting cases, equation 1 is symmetric with respect to *A*_{1} and *A*_{2}, reflecting the symmetry of, e.g., the data in Figure 3B. However, for values of Δ*t* where neither of the two cases is strictly fulfilled, this symmetry of the iso-response sets will be distorted, as is noticable for the longer Δ*t* in Figure 3C. Other sets of values for *L*(Δ*t*) and *Q*(Δ*t*) may lead to very different iso-response shapes, as in Figure 3D.

Equation 1 presents a self-contained model for click stimuli and is sufficient to analyze the temporal characteristics of the individual steps. It can be interpreted as a signal-processing cascade that contains two summation processes, one linear in the click amplitudes and one quadratic. For click stimuli, the functions *L*(Δ*t*) and *Q*(Δ*t*) are thus filter functions associated with the linear and quadratic summation, respectively.

Despite the simple structure of the model, the filters *L*(Δ*t*) and *Q*(Δ*t*) can be expected to retain the salient features of the underlying biophysical processes such as frequency content and integration time. In Protocol S1, we show that equation 1 can be obtained in an a posteriori calculation from a generalized cascade model and that this derivation leads to an interpretation of *L*(Δ*t*) as the velocity of the mechanical vibration and of *Q*(Δ*t*), at least for large enough Δ*t,* as the time course of the membrane potential following a click. In this generalized model, the input signal is an arbitrary sound pressure wave *A*(*t*), and the effective stimulus intensity is a continuous function of time, *J*(*t*), which is given by

Here, the input *A*(*t*) is first convolved with a temporal filter, *l*(*τ*), the result is squared and subsequently convolved with a second filter, *q*(*τ*), as depicted in Figure 4. The filters *l*(*τ*) and *q*(*τ*) have characteristics similar to the click-version filters *L*(Δ*t*) and *Q*(Δ*t*), but are not identical to them. Their relations follow from the calculation in Protocol S1. As we here focus on click stimuli, we will use the simpler equation 1 to evaluate the temporal structures of *L*(Δ*t*) and *Q*(Δ*t*).

Note that we interpret equation 1 to yield the spike probability after the second click. If the first click is large and the second small, however, the first click alone may account for some of the observed spikes; clearly this is the case when the second click vanishes. This is not captured by equation 1, and one might expect that, for large values of *A*_{1}, these additional spikes lead to measured values of *A*_{2} that are slightly smaller than expected for a circular iso-response set. The data in Figure 3, however, suggest that this effect is small and not picked up by our experiment. Nevertheless, for the following quantitative study, we will keep the first click always on a level where the click by itself does not contribute substantially to the spike probability.

The previous experiment showed that the separate effects of the two summation processes can be discerned for short and long time intervals. For intermediate Δ*t,* however, their dynamics may largely overlap. Is it nevertheless possible to design an experiment that directly reveals the whole time course of the mechanical vibration *L*(Δ*t*) and the electrical integration *Q*(Δ*t*)? This would provide a parameter-free description of both processes and advance the quantitative understanding of the auditory transduction dynamics. To reach this goal, we again measure iso-response sets. As before, we exploit that for fixed Δ*t,* any pair of click amplitudes (*B*_{1}, *B*_{2}) should result in the same spike probability *p* as the pair (*A*_{1}, *A*_{2}) as soon as *J*(*A*_{1}, *A*_{2}) = *J*(*B*_{1}, *B*_{2}). It is this straightforward relation that allows us to determine both *L*(Δ*t*) and *Q*(Δ*t*) independently of each other. In fact, some appropriate set of measurements that fulfill the iso-response relation is all that is needed to calculate *L*(Δ*t*) and *Q*(Δ*t*). Illustrating this concept, we now proceed with a particularly suited choice of stimulus patterns, which keeps the mathematical requirements for the calculation at a minimum. For each Δ*t,* we measure two different iso-response stimuli, and as a key feature, one of these has a “negative” second click, i.e., a sound-pressure pulse pointing in the opposite direction as the first click, as depicted in Figure 5A. Mathematically, this choice of stimulus patterns leads to two simple equations for the two unknowns *L*(Δ*t*) and *Q*(Δ*t*), which can be solved explicitly, as explained in Materials and Methods. By repeating such double measurements for different values of Δ*t,* the whole time course of *L*(Δ*t*) and *Q*(Δ*t*) is obtained.

Figure 5 shows examples of *L*(Δ*t*) and *Q*(Δ*t*) for three different cells. *L*(Δ*t*) displays strong oscillatory components, as was observed for all cells. This property presumably reflects the eardrum's oscillation at the attachment site of the receptor cell. The detailed temporal structure of *L*(Δ*t*) now allows us to investigate the salient features of this oscillation. To quantify our findings, we fit a damped harmonic oscillation to the measured data for *L*(Δ*t*) and extract the fundamental frequency as well as the decay time constant. We can use these values to predict the neuron's characteristic frequency (the frequency of highest sensitivity) and the width of its frequency-tuning curve. Figure 6 shows the comparison of these predictions with traditional measurements of the tuning curves for all 12 cells measured under this experimental paradigm with sufficient sampling to extract *L*(Δ*t*). The remarkable agreement confirms that the new analysis faithfully extracts the relevant, cell-specific properties of the transduction sequence. The correspondence between the tuning characteristics and the filter *L*(Δ*t*) also explains why high-frequency receptor cells do not feature straight lines for their iso-response sets even at the shortest inter-click interval (40 μs) used in the experiment. For those cells, *L*(Δ*t*) decays rapidly, thus not allowing access to the region where *L*(Δ*t*) 1.

The short initial rise phase of the measured *Q*(Δ*t*) in Figure 5E and 5F illustrates the rapid buildup of the membrane potential after a click. The exponential decay following this phase suggests that the accumulated electrical charge decays over time owing to a leak conductance. Previously, the time constant could not be measured because of difficulties in obtaining recordings from the somata or dendrites of the auditory receptor cells. Using our new method, we find time constants in the range of 200 to 800 μs. These values are small compared to time constants in more central parts of the nervous system, reflect the high demand for temporal resolution in the auditory periphery, and explain the high coding efficiency of the investigated receptor neurons under natural stimulation [21].

In most of our recordings, the temporal extent of the filter *L*(Δ*t*) was considerably smaller than that of *Q*(Δ*t*). This usually leads to a region around a Δ*t* of 400–800 μs, depending on the specific cell, where *L*(Δ*t*) 0 and *Q*(Δ*t*) is still near unity. These findings correspond to the circular iso-response sets of the initial experiment.

Towards very small Δ*t,* on the other hand, the data show that *Q*(Δ*t*) usually decreases strongly. As explained earlier, this is expected from the linear iso-response sets, and it is observed exemplarily in the data shown in Figure 5E and 5F. In addition, the first few 100 μs of the data may show considerable fluctuations of *Q*(Δ*t*) for some recordings, as in Figure 5G. Different effects may influence this early phase of *Q*(Δ*t*). (1) The electrical potential might be shaped by further dynamics in addition to the low-pass properties of the neural membrane, such as inactivation of the transduction channels or electrical resonances as found in some hair cells [6]. (2) The fluctuations could reflect the oscillatory influx of current following from the oscillation of the eardrum. In other words, the low-pass filtering of the neural membrane may not be strong enough to quench all oscillatory components of the transduction currents. The resulting effect on the filter *Q*(Δ*t*)—though too small to be picked up reliably by the present experiments—can be observed in simulations of the processing cascade, see Figure S2. At present, we cannot distinguish between these two interpretations. More detailed future experiments, however, may allow a quantitative test of these hypotheses.

Measuring the mechanical and electrical response dynamics,

*L*(Δ

*t*) and

*Q*(Δ

*t*), completes the model. In order to test its validity and suitability to make quantitative predictions, we investigated the model's performance on a different class of stimuli, namely combinations of three short clicks. Having measured the required values for

*L*(Δ

*t*) and

*Q*(Δ

*t*) with two-click stimuli as in the previous experiment (see

Figure 5), we now ask the following question: if we keep the first two clicks small enough that they do not lead to a spike response, can we predict the size of the third click required to reach a given spike probability? We can use the measured values of

*L*(Δ

*t*) and

*Q*(Δ

*t*) to calculate these predictions and experimentally test them by performing a series of three-click iso-response measurements. This experiment was performed on three different cells; one cell featured an unusually high response variability, and results from the other two cells are shown in

Figure 7. The agreement between the predicted and the true click amplitudes shows that the model yields quantitatively accurate results.