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Materials and Methods
- Disentangling Sub-Millisecond Processes within an Auditory Transduction Chain

Electrophysiology We performed intracellular recordings from axons of receptor neurons in the auditory nerve of adult Locusta migratoria. Details of the preparation, stimulus presentation, and data acquisition are described elsewhere [20]. In short, the animal was waxed to a Peltier element; head, legs, wings, and intestines were removed, and the auditory nerves, which are located in the first abdominal segment, were exposed. Recordings were obtained with standard glass microelectrodes (borosilicate, GC100F-10, Harvard Apparatus, Edenbridge, United Kingdom) filled with 1 mol/l KCl, and acoustic stimuli were delivered by loudspeakers (Esotec D-260, Dynaudio, Skanderborg, Denmark, on a DCA 450 amplifier, Denon Electronic, Ratingen, Germany) ipsilateral to the recorded auditory nerve. The reliability of the sound signals used in this study was tested by playing samples of the stimuli while recording the sound at the animal's location with a high-precision microphone (40AC, G.R.A.S. Sound and Vibration, Vedbæk, Denmark, on a 2690 conditioning amplifier, Brüel and Kjær, Langen, Germany). See Figure S1 for example recordings.

Spikes were detected online from the recorded voltage trace with the custom-made Online Electrophysiology Laboratory software and used for online calculation of spike probabilities and automatic tuning of the sound intensities. The measurement resolution of the timing of spikes was 0.1 ms. During the experiments, the animals were kept at a constant temperature of 30 °C by heating the Peltier element. The experimental protocol complied with German law governing animal care.

Measurement of iso-response sets Since the spike probability p of the studied receptor neurons increases monotonically with stimulus intensity, parameters of iso-response stimuli corresponding to the same value of p can be obtained by a simple online algorithm that tunes the absolute stimulus intensity. For fast and reliable data acquisition, we chose p = 70%. The response latency of the neurons varied by 1–2 ms, so that spike probabilities could be assessed by counting spikes over repeated stimulus presentations in a temporal window from 3 to 10 ms after the first click.

In the first set of experiments, stimulus patterns were defined by fixed ratios of A1 and A2, and the tuning was achieved by adjusting the two amplitudes simultaneously. The ratios were chosen so that the angles α in the A1A2 plane given by tanα = A2/A1 were equally spaced. In the second set of experiments, A1 was kept fixed, and only A2 was adjusted; similarly, in the three-click experiments, only A3 was adjusted. In the following, the intensity I always refers to the peak amplitude Amax of the stimulus pattern, measured in decibel sound pressure level (dB SPL),

For each stimulus, the absolute intensity I70 corresponding to a spike probability of 70% was determined online in the following way. Beginning with a value of 50 dB SPL, the intensity was raised or lowered in steps of 10 dB, depending on whether the previous intensity gave a spike probability lower or higher than 70% from five stimulus repetitions. This was continued until rough upper and lower bounds for I70 were found. From these, a first estimate of I70 was obtained by linear interpolation. Seven intensity values in steps of 1 dB from 3 dB below to 3 dB above this first estimate were then repeated 15 times. From the measured spike probabilities, a refined estimate of I70 was obtained by linear regression. Nine intensities from 4 dB above to 4 dB below this value were repeated 30 times (in some experiments 40 times). The final estimate of I70 was determined offline from fitting a sigmoidal function of the form

with parameters α and β to these nine intensity-probability pairs. This relation between p and I was then inverted to find the intensity and thus the absolute values of the amplitudes that correspond to p = 0.7.

Extraction of Lt) and Qt) from iso-response sets The response functions Lt) and Qt) can be obtained independently of each other by combining the results from different measurements within one iso-response set. Here, we derive explicit expressions based on a specific choice of stimuli that are particularly suited for our system. Two measurements are needed to obtain both Lt) and Qt) for given time interval Δt. Each stimulus consists of two clicks. The first click has a fixed amplitude A1; the amplitude A2 of the second click at time Δt later is adjusted so that a predefined spike probability p is reached. For the second measurement, the experiment is then repeated with a “negative” second click, i.e., a click with an air-pressure peak in the opposite direction from the first click. The absolute value of this click amplitude is denoted by Ã2. We thus find the two pairs (A1,A2) and (A1, Ã2) as elements of an iso-response set. Since the spike probability increases with the effective stimulus intensity J, equal spike probability p implies equal J. The two pairs (A1,A2) and (A1, Ã2) therefore correspond to the same value of J. According to the model, equation 1, the click amplitudes thus satisfy the two equations

Setting the two right sides equal to each other, we obtain


The first solution of this mathematical equation, Ã2 = − A2, does not correspond to a physical situation as both A2 and Ã2 denote absolute values and are therefore positive. The remaining, second solution reads

Solving for Lt), we obtain

Substituting Lt) from equation 10 in equation 5 or equation 6, we find

This yields

with c = J/A12. As we keep A1 and J constant throughout the experiment, this determines Qt) up to the constant c. It can be inferred from an independent measurement with a single click: by setting A1 = 0 in equation 5, we see that J corresponds to the square of the single-click amplitude that yields the desired spike probability. Alternatively, c can be estimated from the saturation level of Qt) for large Δt, as was done in the present study.

The specific form of the effective stimulus intensity, equation 1, led to particularly simple expressions for the response functions Lt) and Qt); see equation 10 and equation 12, respectively. Other nonlinearities may result in more elaborate expressions or implicit equations, but this technical complication does not limit the scope of the presented approach.

Data fitting The datasets for Lt) were fitted with velocity response functions of a damped harmonic oscillator

where ω and δ were optimized for minimizing the total squared error. From these, the fundamental frequency f and the decay time constant τdec were determined as f = ω/() and τdec = 1/δ. A simpler fit function of the form

led to essentially indistinguishable results for f and τdec.

The resonance frequency, which corresponds to the characteristic frequency, fCF, of the tuning curve, and the tuning width, Δf3dB, can be predicted from the fitted values of ω and δ according to the theory of harmonic oscillators:

The datasets for Qt) were fitted with an exponential decay

where the parameters a, τint, and c were adjusted. Here, only data points for Δt > 150 μs were taken into account, as Qt) initially shows a rising phase. The obtained value for c was used to determine the constant J/A12 in equation 12.

For comparing these predicted values with measurements, the minimum and width of the tuning curves (see Figure 6A) were determined by fitting a quadratic function to the five data points closest to the data point with smallest intensity.

Model predictions for three-click stimuli For stimuli consisting of three clicks with amplitudes A1, A2, and A3 that are separated by time intervals Δt1 and Δt2, respectively (see Figure 7A), an approximate equation for the effective stimulus intensity J can be derived in the following way: The first click induces a tympanic vibration proportional to A1 and a membrane potential proportional to A12. Following the second click, the tympanic deflection has become A1· Lt1) and is augmented by A2. This yields a membrane potential proportional to (A1·Lt1) + A2)2. After the third click, the tympanic deflection has evolved to A1·Lt1 + Δt2) + A2·Lt2) so that the membrane potential is increased by (A1·Lt1 + Δt2) + A2·Lt2) + A3)2. Summing up the different contributions and approximating the influence of the inter-click intervals on the membrane potential by appropriate factors of Q, we find for the effective stimulus intensity

The value of J for a predefined spike probability can be measured from a single-click experiment by setting A1= A2 = 0 and tuning A3 until the desired spike probability is reached. After having measured Lt) and Qt) from two-click experiments, the above equation can be used to predict the amplitude A3 needed to reach this predefined spike probability for any combination of A1, A2, Δt1, and Δt2.

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