Disentangling sub-millisecond processes within an auditory transduction chain

Abstract

Every sensation begins with the conversion of a sensory stimulus into the response of a receptor neuron. Typically, this involves a sequence of multiple biophysical processes that cannot all be monitored directly. In this work, we present an approach that is based on analyzing different stimuli that cause the same final output, here defined as the probability of the receptor neuron to fire a single action potential. Comparing such iso-response stimuli within the framework of nonlinear cascade models allows us to extract the characteristics of individual signal-processing steps with a temporal resolution much finer than the trial-to-trial variability of the measured output spike times. Applied to insect auditory receptor cells, the technique reveals the sub-millisecond dynamics of the eardrum vibration and of the electrical potential and yields a quantitative four-step cascade model. The model accounts for the tuning properties of this class of neurons and explains their high temporal resolution under natural stimulation. Owing to its simplicity and generality, the presented method is readily applicable to other nonlinear cascades and a large variety of signal-processing systems.

Figure 1. Sequential Processing in the Auditory Transduction Chain

A sequence of several steps transforms an incident sound wave into a neural spike response. (1) Mechanical coupling. The acoustic stimulus induces vibrations of a mechanical membrane (basilar or tympanic membrane). (2) Mechanosensory transduction. The deflections cause the opening of mechanosensory ion channels in the membrane of a receptor neuron. Many details of this transduction process are still unknown. The depicted schematic coupling follows the gating-spring model proposed for mechanosensory transduction in hair cells [43]. (3) Electrical integration. The electrical charge due to the transmembrane current accumulates at the cell membrane. (4) Spike generation. Action potentials are triggered by voltage-dependent currents. 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.

Figure 2. Receptor Neuron Responses for Two-Click Stimuli

(A) Stimulus parameters. Acoustic stimuli consisted of two short clicks with amplitudes A ₁ and A ₂, respectively, separated by a peak-to-peak interval Δt. The clicks were triangular and had a total width of 20 μs. The peak-to-peak interval was generally less than 1.5 ms. (B) Raster plots of spike responses. Spike times obtained from a single receptor neuron with four different peak intensities (83–86 dB SPL) are shown for 30 runs each. For the different intensities, both click amplitudes were varied while their ratio was kept fixed, with intensity values referring to the larger click amplitude. The inter-click interval in this example was 40 μs. The values of p denote the measured spike probabilities. The inset displays spike times from the strongest sound stimulus at higher magnification. All spikes fall in a temporal window between 4.5 and 5.5 ms after stimulation. Spike times were recorded with a temporal resolution of 0.1 ms. These data illustrate that the response of the receptor cell is well described by the occurrence probability of a single spike in a rather broad time window, for example, between 3 and 10 ms after stimulus presentation. As is often observed for these receptor cells, there is virtually no spontaneous activity.

Figure 3. Measurements of Iso-Response Sets and Identification of Relevant Stimulus Parameters

(A) Acoustic stimuli. The stimuli consisted of two short clicks with amplitudes A ₁ and A ₂ that were separated by a peak-to-peak interval Δt, here shown for Δt = 40 μs (upper trace) and Δt = 750 μs (lower trace). (B–D) Examples of iso-response sets from three receptor cells. Here, as throughout the paper, iso-response sets correspond to a spike probability of 70%. Each panel shows iso-response sets from a single receptor cell for two different values of Δt, one smaller than 100 μs (filled circles) and one larger (open squares). The solid lines denote fits to the data of either straight lines or circles. The values for Δt used in the experiments are indicated in the respective panels. All error measures display 95% confidence intervals. For the short intervals, the data are well fitted by straight lines (A ₁ + A ₂ = constant). For the long intervals in (B) and (C), circles (A ₁ ² + A ₂ ² = constant) yield good fits; a slight asymmetry is clearly visible in (C). The data for the intermediate inter-click interval Δt = 120 μs in (D) are not well fitted by either of these shapes. Here, the measured points are connected by a dashed line for visual guidance. Note that in (B) the overall sensitivity of the neuron seems to have changed; the intersections of the straight line and the circle with the x- and y-axis do not match exactly although the stimulus in these cases is the same, a single click. The reason may be either a slow adaptation process or a slight rundown of the recording over the experimental time of around 30 min. However, this does not account for the more prominent differences in shape of the two iso-response sets. These examples demonstrate that on different time scales, different stimulus parameters are relevant for the transduction process, the amplitude A of a sound stimulus for short times and its energy A ² for long times.

Figure 4. Generalized Cascade Model of the Auditory Transduction Chain

The model is composed of a sequence containing two linear temporal filters, l(τ) and q(t), and two static nonlinear transformations, namely a quadratic nonlinearity and an output nonlinearity g˜(·), which may differ from the nonlinearity g(·) of the click-stimulus model (see Protocol S1). First, the stimulus A(t) is convolved with the filter l(τ) (linear integration). Second, the result is squared (nonlinear transformation). Third, the result of the previous step is convolved with the filter q(τ), yielding the effective stimulus intensity J(t) (linear integration). Fourth, a final transformation g˜ of J(t) (nonlinear transformation) determines the response, which in this generalized model is the time-dependent firing rate r(t). The model thus corresponds to an LNLN cascade. This abstract structure directly follows the sequential configuration of the biophysical processing steps shown in Figure 1.

Figure 5. Temporal Structure of the Mechanical Oscillation and Electrical Integration

(A) Stimulus patterns. Two clicks were presented, separated by a time interval Δt. The first click (amplitude A ₁) was held constant throughout this experiment. The second click was presented in the same direction as the first click (solid line, amplitude A ₂) or in the opposite (“negative”) direction (dashed line, amplitude à ₂). The click amplitudes A ₂ and à ₂ were adjusted to fall in the desired iso-response set. (B–G) Mechanical oscillation and electrical integration of a high-frequency (B and E) and two low-frequency (C and F, and D and G, respectively) receptor neurons. (B–D) Time course of the eardrum vibration. The individual values (circles) were calculated from the measured values of A ₂ and à ₂ for each Δt. The results are compared with a theoretical curve from a damped harmonic oscillator (solid line) with fundamental frequency f and decay time constant τ _dec fitted to the data. (E–G) Time course of the electrical integration process. The measured data are compared to an exponential fit (solid line) with a time constant τ _int.

Figure 6. Predictions of Tuning Characteristics

(A) Tuning curves for the same two cells as in Figure 5B and 5E, and 5C and 5F, respectively. The data show the intensity required to drive a receptor cell at a firing rate of 150 Hz for different sound frequencies in the range of 1 to 40 kHz. The characteristic frequency f _CF is determined as the minimum of the tuning curve, and the tuning width Δf _3dB as the width of the curve 3 dB above the minimum value. (B) Comparison of the predicted and measured characteristic frequency and the tuning width. The predictions were obtained from the fundamental frequency and decay time constant of the measured filter L(Δt); the measured values are taken from the tuning curves as in (A) (n = 12). The encircled data points correspond to the three examples shown in Figure 5. The width of the tuning curves is notoriously difficult to assess quantitatively, as it depends sensitively on an accurate determination of the intensity minimum of the tuning curve. This contributes strongly to the differences of the tuning-width values.

Figure 7. Model Predictions for Three-Click Stimuli

(A) Stimulus patterns. The stimuli consisted of three clicks with amplitudes A ₁, A ₂, and A ₃ that were separated by time intervals Δt ₁ and Δt ₂, respectively. The second and third clicks were either given in the same or opposite (“negative”) direction as the first click. A ₁ and A ₂ were set equal and held constant, and A ₃ was adjusted to yield a spike probability of 70%. The following pairs of time intervals (Δt ₁, Δt ₂) were applied: (100 μs, 100 μs), (100 μs, 200 μs), and (200 μs, 100 μs). (B and C) Predicted and measured amplitudes of the third click for two different cells. Predictions were made after L(Δt) and Q(Δt) had been measured with two-click experiments such as in Figure 5. The comparison between predicted and measured values for A ₃ therefore contains no free parameters. The model equation for three-click stimuli is presented in Materials and Methods. As demonstrated by these data, the model allows quantitatively accurate predictions.