We have presented a novel technique to disambiguate single processing steps within a larger sensory transduction sequence and to analyze their detailed temporal structures. Our approach is based on measuring particular iso-response sets, i.e., sets of stimuli that yield the same final output, and on specific quantitative comparisons of such stimuli to dissociate the individual processes. For the investigated auditory transduction chain in the locust ear, this strategy led to a precise characterization of two consecutive temporal integration processes, which we interpret as the mechanical resonance of the eardrum and the electrical integration of the attached receptor neuron. The method revealed new details of these processes with a resolution far below 1 ms. The results for the time course of the mechanical resonance agree with traditional measurements of tuning curves and show the decay of the oscillation with a temporal precision much higher than expected from the jitter of the measured output signal, the spikes. The time constants of the electrical integration that were extracted from the data had not been accessible by other means.
The analysis resulted in a four-step model of auditory transduction in locusts. The model comprises a series of two linear filters and two nonlinear transformations. The quadratic nonlinearity that separates the two linear filters suggests that the mechanosensory transduction can be described by an energy-integration mechanism, as the squared amplitude corresponds to the oscillation energy of the tympanum. This quadratic form was derived from the circular shape of the iso-response sets for longer time scales Δt and is in accordance with the energy-integration model that was found to capture the sound-intensity encoding of stationary sound signals in these cells . Furthermore, the direct current component of the membrane potential in hair cells is also proportional to sound energy , and in psychoacoustic experiments, energy integration accounts for hearing thresholds [23,24,25,26]. However, a recent analysis of response latencies in auditory nerve fibers and auditory cortex neurons in cats suggests an integration of the pressure envelope for determining thresholds . This effect may be attributable to the synapse between the hair cell and the auditory nerve fiber in the mammalian ear. In the locust ear, this synapse does not exist, as the fibers are formed by the axons of the receptor neurons themselves.
Although the quadratic nonlinearity is fully consistent with our data, there is a second possibility within the general cascade model, equation 2
, namely, squaring after rectification. From a biophysical point of view, this would be expected if the mechanosensory ion channels can only open in one direction. Based on the current data, we cannot distinguish between these two possibilities. As the two scenarios should lead to slightly different response characteristics, future high-resolution experiments should be able to resolve this question.
The linear filters L(Δt) and Q(Δt) were interpreted as the mechanical oscillation of the tympanum and the electrical integration at the neural membrane. Their oscillatory and exponential decay characteristics, respectively, support this view. In principle, however, other processes may well contribute to these characteristics, e.g., electrical resonances as seen in hair cells of the turtle and bullfrog [6,28]. These electrical amplification processes would be expected to influence the filter Q(Δt), but our data generally provide little evidence for such effects. Deviations from the exponential decay characteristics in Q(Δt) may in part be attributable to the oscillatory influx of charge resulting from the tympanic vibration. This may lead to a small oscillatory component in the early phase of the filter (cf. Protocol S1; Figure S2).
The mechanical coupling in the first step of our model is linear. This is in accordance with mechanical investigations of the tympanum using laser interferometry [3
] and stroboscopic measurements [19
]. As the short clicks used in our study produce reliable spiking responses only at high sound pressure, however, we cannot exclude the influence of nonlinear coupling at low sound pressure, which has been hypothesized on the basis of distortion-product otoacoustic emissions [29
]. In addition, the mechanical properties of the tympanum seem to change slightly under prolonged stimulation and give rise to mechanical adaptation effects with time scales in the 100-ms range [30
]. Spike-frequency adaptation also adds a nontrivial feedback term to the minimal feedforward model of Figure 4
. Similarly, specific potassium currents and sodium-current inactivation induced by sub-threshold membrane potential fluctuations may complicate the transduction dynamics for more general inputs, but do not leave a signature in the present click-stimulus data.
The model was quantitatively investigated by using combinations of short clicks. The particular structure of these stimuli allowed a fairly simple mathematical treatment. The derivation of equation 1 relied on capturing the mechanical vibration and the membrane potential, respectively, by single quantities in each time period following a click. This was possible because of the expected stereotypic evolution of the dynamic variables during the “silent phases” between and after the clicks. A generalization to arbitrary acoustic stimuli would require a more elaborate model in the form of equation 2 as well as extensions that account for neural refractoriness and adaptation.
Besides its applicability under in vivo conditions, the presented framework has several advantageous properties. First, the method effectively decouples temporal resolution on the input side from temporal precision on the output side by focusing on spike probabilities. In all our measurements, for example, spike latencies varied by about 1 ms within a single recording set owing to cell-intrinsic noise (see Figure 2). Still, we were able to probe the system with a resolution down to a few microseconds. This would not have been possible using classical techniques such as poststimulus time histograms, reverse correlation, and Wiener-series analysis. All these methods are intrinsically limited by the width of spike-time jitter and thus cannot capture the fine temporal details of rapid transduction processes. With our method, the resolution is limited only by the precision with which the sensory input can be applied. For the investigated system, the achievable temporal resolution thus increases by at least two orders of magnitude.
Second, the method is robust against moderate levels of spontaneous output activity, as this affects all stimuli within one iso-response set in the same way. Methods that require measurements at different response levels, on the other hand, are likely to be systematically affected because the same internal noise level may have a different influence at different levels of output activity.
Finally, in many input–output systems, the last stage of processing can be described by a monotonic nonlinearity. Here, this is the relation between the effective stimulus intensity J and the spike probability p = g(J), which includes thresholding and saturation. By always comparing stimuli that yield the same output activity, our analysis is independent of the actual shape of g(J). Preceding integration steps may thus be analyzed without any need to model g(J). This feature is independent of the specific output measure and applies to spike probabilities, firing rates, or any other continuous output variable.
Let us also note that the method does not require that the time scales of the individual processes be well separated. For the studied receptor cells, mechanical damping was on average about two times faster than electrical integration, and even for cells with almost identical time constants, iso-response measurements led to high-quality data and reliable parameter fits. Nor is the method limited to particularly simple nonlinearities. All that is needed are solid assessments of the iso-response sets. Mathematically, it is straightforward to substitute some or all of the analytical treatments of this work by numerical approaches, if required by the complexity of the identified signal-processing steps. This extension allows one to use a general parametrization of the full processing chain when the nonlinear transformation cannot be estimated from iso-response sets at large and small Δt. Instead, performing more than the two measurements at each intermediate Δt in the second experiment (see Figure 5) will provide additional information that can be exploited to improve the numerical estimates of the nonlinearity.
As in many other approaches of nonlinear systems identification, the development of a quantitative model relies on the prior determination of the appropriate cascade structure. Unfortunately, there is no universal technique for doing so. In many cases, intuition is required to find suitable models, which should eventually be tested by their predictive power. In the present case, the findings of characteristic shapes of the iso-response sets gave a clear signature of two distinct linear filters with a sandwiched quadratic nonlinearity. In addition, this structure was supported by its amenability to straightforward biophysical interpretation. Generalizing our results, specific iso-response sets may aid structure identification in conjunction with a priori anatomical and physiological knowledge. Once the cascade structure is established, the individual constituents can be quantitatively evaluated by specific comparisons of iso-response stimuli. Comparing responses to clicks in positive and negative directions as in this study is in essence similar to the approach used by Gold and Pumphrey , who evaluated the perceptual difference between short sine tones with coherent phase relations and sine tones that contained phase-inverted parts in order to estimate the temporal extent of the cochlear filters.
A yet open problem is the inclusion of feedback components. The present approach relies on the feedforward nature of the system to disentangle the individual processing steps. In particular cases, however, the iso-response approach may also aid in separating feedforward and feedback contributions, namely, when the feedback depends purely on the last stage of the processing cascade . In this situation, iso-response measurements lead to a constant feedback contribution, and the analysis of the feedforward components may be carried out as in the present case. The experiment may then be repeated for different output levels to map out the feedback characteristics.
The feedforward model that we have proposed here for the auditory transduction chain has the form of an LNLN (where “L” stands for linear and “N” stands for nonlinear) cascade, composed of two linear temporal integrations and two nonlinear static transformations . Similar signal-processing sequences combining linear filters and nonlinear transformations are ubiquitous at all levels of biological organization, from molecular pathways for gene regulation to large-scale relay structures in sensory systems. In neuroscience, applications range from the sensory periphery, including frog hair cells , insect tactile neurons , and the mammalian retina [35,36,37], over complex cells in visual cortex [38,39], to psychophysics . These studies are restricted to models that contain a single nonlinear transformation, corresponding to NL, LN, or LNL cascades [32,41]. An extension of these analyses was presented by French et al. , who derived an NLN cascade for fly photoreceptors.
Complementary to the correlation techniques underlying the parameter estimations in those models, the method presented in this work provides a new way of quantitatively evaluating and testing cascade models. The increased complexity of the LNLN cascade identified in the present case was made accessible by invoking particular iso-response measurements, and a higher temporal resolution was achieved by focusing on how spike probabilities depend on the temporal stimulus structure instead of relying on temporal correlations between stimulus and response.
Our experimental technique will be most easily applicable to systems whose signal processing resembles the cascade structure investigated here. The general concept of combining different measurements from within one iso-response set covers, however, a much larger range of systems. With increasingly available high-speed computer power for online analysis and stimulus generation, this framework therefore seems well suited to solve challenging process-identification tasks in many signal-processing systems.