## 3 Method
### Post-Processing of Cochlea Model Output  
The output of the cochlea model is spikes at certain time points along the BM in the Bark bands. From here the interspike intervals (ISIs) are calculated as the time interval between two spikes at one nodal point, so two for each Bark band. Accumulating the ISIs over a time interval and transforming the ISIs into frequencies like f=1/ISI, an ISI histogram of the frequency of appearance or the amplitude of certain frequencies or periodicities present in all Bark bands results. Figure 1 gives an example of the method. At the top, an arbitrary sound file, a sawtooth wave, is used as input to the cochlea model. The model output is shown in the middle plot of spikes occurring at Bark bands displayed vertically over time; it is shown horizontally. The model output is further processed using the ISI intervals between two adjacent spikes and ordered in a histogram for each Bark band, again vertically, over time. The histogram is calculated for the time windows and a 10 cent frequency discretization. The accumulation interval was chosen according to logarithmic pitch perception to be 10 cent, where one octave is 1200 cent. Just-noticeable differences (JNDs) in pitch perception are frequency-dependent and strongly differ between adjacent and simultaneous sound presentation. The value of 10 cent was found to be reasonable in terms of the used sounds, where smaller values lead to too few incidences to accumulate and larger values would blur the results. To simulate the coincidence detection found in neural nuclei subsequent to the cochlea, the ISI histograms as shown above are post-processed in three stages. First, the histograms of all time steps are summed up to build one single histogram for each sound. As shown in Figures 2 and 3 the histograms change over time, mainly due to the temporal decay of the sound. They also show inharmonicities during the initial transient. After temporal integration, the ISI histogram is divided by the amplitude of its largest peak, making the amplitude of this peak one. A threshold of th = 0.01 is applied, making all peaks smaller than th zero. Additionally, all frequencies above 4 kHz are not taken into consideration, as pitch perception is only present up to this upper frequency. This is not considered a serious bias for roughness estimation as roughness, different from sharpness perception, is mainly based on low and mid-frequencies close to each other, leading to musical beatings, and therefore, to roughness. Strong energy in high frequencies is merely perceived as brightness or sharpness and does not contribute to roughness perception in the first place. Secondly, Gaussian blurring of the spectrum is performed, where the spectrum is blurred using a Gauss shape with standard deviation \(\sigma\). This sums peaks which are very close to each other, and therefore, performs coincidence detection of periodicities close to each other. As discussed in the Introduction, the very nature of coincidence detection is to make a burned neural spike temporally more concise. Therefore, using Gauss blurring allows for the detection of how concise the spike bursts are. In case of a large \(\sigma\), coincidence is weak, as a large time interval is needed to align the peaks and vice versa. In a second step, only peaks with a certain sharpness s are used, where s = 0 selects all peaks and s > 0 selects only peaks where the negative second derivative of the spectrum is larger than s. This prevents very broad peaks still being taken as peaks after blurring of the spectrum. Toestimate the amount of coincidence detection in the listening test both values are varied in 21 steps between \(0\leq\sigma\leq 2\) and \(0\leq s\leq 0.002\). Figure 1: Example of sound processing. Top: Sound wave. Middle: Cochlea model output of spikes in respective Bark bands (vertical axis) at respective time points (horizontal axis). Bottom: ISI histogram of occurrences of spikes with \(\text{f}=1/\text{ISI}\) (vertical axis) over time (horizontal axis).