Analysis of experimental data

Rate-level functions

The data related to the experimental estimates of rate-level functions were kindly provided by Peter Heil, and were the same as in (Heil et al., 2011). In total, RLFs from 72 SGNs from 3

work, an empirical model was introduced which provides very good fits to the experimental spike rate vs sound level relations. Thus, instead of considering the raw data, we used the best fit parameters of the empirical model in this dissertation. Exemplary RLFs shown in Fig. 4.11 were generated by using the best fit parameters of the corresponding experimentally estimated RLFs and adding the Gaussian noise with zero mean. The standard deviation of the noise was set to the mean square root deviation between the experimentally measured RLFs and their best fits with the empirical model of (Heil et al., 2011).

To select SGNs which innervate IHCs of similar cochlear tonotopic position (section 4.4.4), we had to determine the relative positions of the innervation points along the organ of Corti.

This was done by using an empirical expression by (Liberman, 1982)

x= log₁₀(CF/456 + 0.8)/2.1, (4.55)

where CF is the characteristic frequency in Hz, andx is the proportion of the total normalized length from the apex of the cochlea to the innervation point.

In order to estimate the distribution of optimal values of max[V_dc]/∆_V[R_s] due to the noise in the experimental data, we performed bootstrapping analysis (section 4.4.4). Each RLF was generated by using the empirical model of (Heil et al., 2011) with the optimal fit parameters and Gaussian noise of zero mean added on the top. The standard deviation of the noise term was set to the mean square root deviation between the experimentally measured RLFs and their best fits with the empirical model. Then, the RLFs were refitted over 1000 different realizations of the noise terms. This fitting procedure was repeated at different values of max[V_dc]/∆_V[R_s], which ranged from 0.1 to 5 by a step of 0.1. The three fitting parameters were V_0.5−V_rest, A_s, and P₀. The fitting was performed by using the optimization method described above (section 4.5.2).

ISI distributions

ρ_ISI(t) functions corresponding to the SGNs from guinea pigs (Fig. 4.12B – D) were taken from Fig. 2 in (Prijs et al., 1993). All other ρISI(t) functions considered in this work were based on the recordings from mouse SGNs and were estimated by us. One part of the data was taken from (Pangrˇsiˇc et al., 2015), the original article of chapter 3 of this dissertation. Another part of the data, based on the same experimental procedures as in (Pangrˇsiˇc et al., 2015), was kindly provided by Nicola Strenzke and her group, InnerEarLab, University of Goettingen.

SGN responses to 100 ms long pure tone bursts at CF, with the presentation rate of 2 s⁻¹, were recorded. The stimulus intensity was set to 30 dB above the spiking threshold, which drove SGNs to maximal or nearly maximal spiking rates (Taberner & Liberman, 2005). The number of stimulus bursts per SGN varied between 200 and 600 with average of 260. Only SGNs with CF > 4 kHz were selected for analysis to make sure that the AC component of the receptor potentials of IHC is negligible. In fact, CFs of 80% of the SGNs studied were higher than

The details of the experimental procedures used to collect the data from mice SGNs can be found in (Taberner & Liberman, 2005).

ISIs were estimated from the last 50 ms of responses to the repetitive, 100 ms long stimuli.

The number of the resulting ISIs per each neuron varied from 1200 to 8900 with average of 3200. Only responses with sufficiently stationary spike rates throughout the whole stimulation time were considered. In order to evaluate the stationarity of ISIs of a particular SGN, we calculated a normalized cumulative ISI time as a function of the number of the ISIs: ¯c(l) = Pl

i=1t_ISI(i)/PnISI

i=1 t_ISI(i). Then, we compared the resulting cumulative function with ˜c(l) =l.

The later would correspond to ideally stationary sequence ISIs of fixed length. Only SGNs with max_l|¯c(l)−c(l)|˜ <0.04 and 1/lPnISI

l=1 |¯c(l)−c(l)|˜ <0.02 were selected for further analysis. For 72 out of 77 SGNs which were selected, the average max_l|¯c(l)−˜c(l)| and 1/lP

l|¯c(l)−˜c(l)|

were equal to 0.016 and 0.007 respectively.

The bin sizes, n_bin, of the three ρ_ISI(t) functions from guinea pigs shown in Fig. 4.12B – D were 69, 63, and 63, respectively. The binning was not uniform, with more points in the regions with faster variations of the functions. For allρISI(t) functions from mice,nbin was 100.

The experimental ρ_ISI(t) functions were fitted with the models by minimizing the following discrepancy function

where superscripts e and m denote, respectively, the experimental and the model estimates.

ISI correlations

ISI correlations were estimated from the same data samples of recordings of mouse SGNs as the ρ_ISI(t) functions described above. The following estimator of the values of the serial correlation function was used: where n_rep is the number of tone burst applications for each SGN, n_ISI,i is the number of ISIs for SGN response to the i-th repetition of the tone burst.

To test the hypothesis that estimates of ¯C_ISI(1) could result from spike trains of uncorrelated events, we had to calculate the distributions of the ¯C_ISI(1) based on the uncorrelated spike trains. This was done in the following way. First, we fitted the experimental ρ_ISI functions with the Poisson model (including the convolution with the refractoriness, see section 4.4.3) for all SGNs studied. As we found in 4.4.3, this model could reproduce theρ_ISI functions of mouse SGNs well. Next, based on the best fitting parameters, we performed stochastic simulations and

method. N was set to match the number of the tone burst applications used in the experiments for each SGN individually. Then, we used the simulated data to calculate ¯C_ISI(1) based on Eq. (4.57). The same procedure was repeated 10⁵ times, which resulted in the distribution of the estimator ¯C_ISI(1).