Various tone bursts

Two animal studies were selected to test the predicted latency pattern to various tone bursts: one in chinchillas (Temchin et al. [88]) and the other in barn owls (Koppl [25]). Chinchilla is a representative species for the across-species similarity to tone bursts [22] in Figure 4, and was selected to represent the latency pattern at low frequencies (0.1 to 3 kHz). Barn owl is a species that demonstrates significant phase locking at high frequencies up to 10 kHz [25].

The mean first-spike latency of an auditory unit to tone bursts in these animal studies was calculated from the slope of regression fit of the unit’s phase-versus-frequency function, which was derived from the mean phase in period histogram to various tone bursts around the neuron’s characteristic frequency (see 2.2.1). Therefore, strong phase locking in period histograms is important and the latency pattern of barn owls was selected to represent the similarity of first- spike latencies at high frequencies (up to 9 kHz).

3.3.2.1 Latency pattern in animal auditory nerve fibers

The neural latency patterns included the individual points of the raw data and the regression curves in both chinchillas and barn owls. The raw data in chinchillas were 40 latencies of 40 auditory nerve fibers whose characteristic frequencies ranged from 100 to 2934 Hz. The raw data in barn owls were 40 latencies of 40 auditory nerve fibers whose characteristic frequencies ranged from 355 to 8895 Hz. The two sets of raw data were obtained by personal communication with the authors of the two animal studies. Although no regression curve was provided in the study in chinchillas, the power fit regression curve was calculated according to the raw data the authors provided. The regression curve in barn owls to tone bursts was:

(where CF is in Hz). 706 . 0 205 833 . 0 (ms)  CF 

As mentioned above (3.1.3), when the regression curve in the animal study was used, the neural latency pattern was defined as the points on the regression curve when using the Fourier band frequencies as the characteristic frequencies. Due to the characteristic-frequency range of the raw data, a total of 27 Fourier band frequencies ranging from 0.2 to 2.8 kHz every 0.1 kHz were used to test the correlation with the chinchillas’ regression curve, and a total of 71 Fourier band frequencies ranging from 1 to 8 kHz every 0.1 kHz were used to test the correlation with the barn owls’ regression curve (also see 3.3.2.2), because there was only one datum below 1 kHz.

3.3.2.2 Latency pattern in Fourier analysis

The sampling rate and window size were set to 40,000 Hz and 400 points, respectively, to provide the frequency resolution of 0.1 kHz and the frequency range of 0.1 to 20 kHz to compare to the animal studies (see 3.2.4). These two parameters were the same as those used for

predicting the latency patterns to clicks in 3.3.1.2, to compare the predicted patterns to various stimuli (a rarefaction click, a condensation click, and tone bursts).

Input signals in Fourier analysis

In the animal studies, the above neural latency pattern to tone bursts (see 3.3.2.1) was calculated from period histograms, which were generated through the use of the positive zero crossing of the stimulus waveform [25] to trigger a fast data acquisition unit, which marked the occurrence of each action potential with reference to a single cycle of the stimulus frequency. Therefore, the input signals for calculating the predicted latency pattern should start with a positive zero crossing as in the animal studies. The tone bursts were therefore generated by the equation ( ) sin(2 ) s f n f n

x    , where = 0.1, 0.2… 10 kHz, f 1 n 400, (sampling rate) = 40,000. All these 100 tone bursts started with a positive zero crossing that was used as the reference in the animal studies.

s

f

The prediction of the first spike latency for a tone burst was calculated by the steps below. The frequency of 1 kHz is chosen arbitrarily for demonstration. Predictions for other frequencies can be calculated in the same manner.

The input signal is the 1-kHz tone burst shown in Figure 40. The input is sampled with a moving window process wherein the window is moved forward in a point-wise manner (see 3.2). There are totally 799 windows (i.e., 400×2-1) that contain at least one non-zero point of the 400- point stimulus.

Figure 40. The 1-kHz tone burst

Calculate predicted latencies at the stimulus frequency

Calculate the predicted latency at the stimulus frequency (i.e., 1 kHz in this example) for each window and plot them together.

For example in the single window 0 (the first window, Figure 41A), the marking latency is 40000 0 1000 2 1571 . 0 2     

= 0.000975 seconds (or 0.975 ms). Adding 1 ms (the synaptic and neural transmission; see 2.2.7), the predicted marking latency for window 0 is 1.975 ms. In the single window 99 (chosen arbitrarily for demonstration, Figure 41B), the marking latency is

40000 99 1000 2 1.7279 2     

= 0.0032 seconds (or 3.2 ms). The predicted latency for window 99 is 3.2 + 1 = 4.2 ms. Calculate predicted latencies in all windows and plot them together (Figure 41E).

Figure 41. The predicted latencies at 1 kHz to the 1kHz tone burst. A predicted latency is the marking latency in a single window with the correction to reference (window 0) and the additional 1 ms. A. Input of window 0. B. Input of window 99. C. Marking of the first amplitude maximum of 1k-Hz component for window 0. D. Marking of the first amplitude maximum of 1k-Hz component for window 99. E. Distribution of all 799 markings at 1 kHz, including those from window 0 and 99 (indicated with big blue markings).

Figure 41E shows a periodic distribution that is related to the period of the stimulus (1 ms). So it is possible to calculate the theoretical points where these markings tend to be

distributed. The can be achieved by computing a modulus after dividing the latency by the stimulus period.

The two steps below are to compute the central tendency of the marking distribution, to find a theoretical first latency from the central tendency and then define the predicted first-spike latency.

Convert the predicted latencies to latencies between 0 and the period

Calculate the modulus after dividing the predicted latency by the stimulus period.

The modulus is defined as X - n×Y where X is the predicted latency, Y is the stimulus

period, and n is the nearest integer less than or equal to Y

X . The predicted latencies are calculated as described in the previous step, as shown in Figure 41E. If the marking distribution in Figure 41E is viewed as a prediction of peristimulus-time histogram, this step is like “folding” the peristimulus-time histogram cycle-by-cycle to generate a period histogram.

For example, the converted latency for window 0 is the modulus after dividing 1.975 by 1, i.e., 0.975 ms. In the same way, the converted latency for window 99 is the modulus after dividing 4.2 by 1, i.e., 0.2 ms. Figure 42 shows the distribution of the converted latencies for all windows. The two big blue markings from windows 0 and 99 are indicated by arrows.

Figure 42. Distribution of the converted latencies between 0 and 1 ms to the 1-kHz tone burst. The two from windows 0 and 99 are indicated by big blue markings. If the markings are split (latency of window 0 is larger than that of window 798) as in this example, a cut line (average of these two latencies, i.e., 0.7125 ms) is set to move one part of the markings and merge all of them to calculate the mean latency.

Merge makings to calculate the mean converted latency

Assuming the markings are synchronized to a single point in one cycle, the mean converted marking latency should be calculated only when the markings are distributed in one group. If the markings are split (latency of window 0 is larger than that of window 798) as in this example, a cut line (average of these two latencies in the first and the last windows, i.e., 0.7125 ms) is set to move one side of the markings to the other and merge all of them to calculate the mean latency. Then make the mean converted latency range between 0 and the period (by adding or subtracting one period).

For instance, Figure 43 shows the merged markings made by moving the markings above 0.7125 ms in Figure 42 to the left (subtracting latencies by the period 1 ms). The mean converted latency is defined as the average of latencies of the merged markings, with the range between 0 and the period. It equals 0.20031 ms in this case.

Figure 43. The merged markings made by moving the markings above the cutline to the left (subtracting latencies by the period). The mean converted latency is defined as the average of latencies of the merged markings, with the range between 0 and the period. The markings from window 0 and 99 are indicated by arrows and big blue symbols. The marking from window 0 is moved from 0.975 to -0.025 ms.

The mean converted latency of 0.20031 indicates that the predicted markings tend to be synchronized at theoretical points of 0.20031 + a multiple of the period (1 ms), i.e., 0.20031 ms, 1.20031 ms, 2.20031 ms, etc. These points are defined as predicted synchronization points for the 1-kHz tone burst.

Calculate the predicted first-spike latency to the tone burst

Comparing the predicted synchronization points (whose latencies are the mean converted latency + a multiple of the period, Figure 44B) with the predicted markings (Figure 44A). The latency of one of the predicted synchronization points (Figure 44B) whose latency is within the first marking group is defined as the predicted first-spike latency.

Figure 44. The predicted markings to the 1-kHz tone burst tend to be synchronized at specific points. The predicted first-spike latency is defined as the synchronization point whose latency is within the predicted

There are no markings around the first and second predicted synchronization points in Figure 44A. The latency of the third predicted synchronization point (arrow in Figure 44B) is within the first marking group (arrow in Figure 44A). So it is defined as the predicted first- spike latency, which equals 2.20031 ms.

Calculate the predicted first-spike latency to all tone bursts

In the same manner, use tone bursts of 0.1, 0.2… 10 kHz and repeat the above steps to calculate the predicted first-spike latencies to all tone bursts.

The predicted latencies and the synchronized points for each stimulus were manually verified again that they matched well while calculating the predicted first-spike latencies. The predicted first-spike latencies would then be compared to response latencies in animal studies to investigate their correlation.

When the individual points of the raw data in the animal study are used as the neural latency pattern, a total of 80 neurons’ characteristic frequencies were used to calculate the predicted latencies to compare with chinchillas’ (40) and barn owls’ (40) raw data.

3.3.2.3 Correlation between Fourier and response latencies to various tone bursts

The Pearson correlation coefficient was calculated and statistically tested to show the strength and direction of the correlation between the measured and predicted latencies to tone bursts. It was computed for both regression curves and raw data in both chinchillas and barn owls. The coefficient of determination (r2) was calculated to show what percentage of the variance in the neural latency pattern can be explained by the Fourier latency pattern. When using the points on the regression curve as the neural latency pattern in chinchillas and barn owls,

n was 27 and 71, respectively (see 3.3.2.1). And when using the individual points of the raw data as the neural latency pattern in chinchillas and barn owls, n was 40 and 40, respectively (see 3.3.2.1).