Conversion of presynaptic vesicle fusion events into spikes

As we found in section 4.4.2, vesicle fusion at presynaptic AZs is almost always well approxi-mated by a Poisson process ifN_V is in the physiological range. The only exception is the regime of bursty vesicle release. Thus, in order to understand how refractoriness of SGNs transforms presynaptic fusion into postsynaptic spike trains we needed to concentrate on these two limiting cases only. Of course, when considering the ISI correlations, even relatively small deviations from the Poisson process had to be taken into account explicitly.

Inter-spike interval distributions

In order to model ISI distributions of high-CF SGNs, the homogeneous Poisson trains convolved with refractoriness have been intensively applied (Teich & Khanna, 1985; Young & Barta, 1986; Li & Young, 1993; Pangrˇsiˇc et al., 2015). As it is shown in Supplementary Material 2.5, the expressions of the ISI interval probability density function (ρISI(t)) and the coefficient of variation (CV_ISI) are then

ρ_ISI(t) =H(t−t_A)· Rr

1−τ_R·R_r ·

e^{−Rr·(t−tA)}−e^{−(t−tA)/τR} , CV_ISI(t) =

p1 + (R_r·τ_R)² 1 +R_r·(t_A+τ_R),

(4.51)

where H(. . .) is the Heaviside step function. Two properties relevant for our further analysis follows from Eq. (4.51). First, the spiking becomes more regular when t_A is increased, as higher and higher fraction of each ISI corresponds to the t_A, which is constant. Second, not taking into account the rightward shift byt_A,ρ_ISI(t) varies between an exponential distribution and a gamma distribution with shape parameter 2 (Fig. 4.8A). The former is observed when τ_R hti_IEI orτ_R hti_IEI. The latter is observed whenτ_R =hti_IEI. A large fraction of SGNs in vivo indeed demonstrate distributions in the range between the two shown in Fig. 4.8A, as discussed in detail in section 4.4.4.

When presynaptic vesicle fusion is bursty, fusion events falling in the refractory periods

interburst IEIs. Thus, the resulting spike bursts produced by SGNs have fewer events than the vesicle fusion bursts, even if the numbers of bursts are similar. In turn, the spike trains are more regular than the patterns of the corresponding presynaptic vesicle release. Of course, when p_o and N_V are large, bursts from several release sites overlap and the probability of at least one active site at any moment is high. In these cases, the refractoriness preferentially reduces the number of events within those time periods when many of the vesicular release sites are active. The regularizing effect of SGN refractoriness is illustrated in Fig. 4.8B, where the dependencies ofCV_ISI on the overall refractory period (t_A+τ_R) are plotted. Here, we considered an AZ with 2-state release sites in Ca²⁺-nanodomain coupling with 2-state Ca²⁺ channels. The kinetic parameters were set to reproduce an experimentally observedρ_ISI(t) function shown in Fig. 4.12D withN_V = 10. As we see,CV_IEI initially rapidly decreases witht_A+τ_R. CV_IEI = 1 is reached below (t_A+τ_R)/hti_IEI = 0.5 for allN_V considered (up to 15). At very long refractory periods, the spike times are mainly determined byt_Aandτ_Rindependent of the statistics of the underlying presynaptic release process. Thus, at large (t_A+τ_R)/hti_IEI,CV_ISI approaches those of the Poisson process convolved with the refractoriness (dashed grey line in Fig. 4.8B). Black dot in Fig. 4.8B (CV_ISI ≈ 1 and (t_A+τ_R)/hti_IEI ≈ 0.33) corresponds to the ISI distribution shown in Fig. 4.12D. In this case, in spite ofCV_ISI ≈1, the spike process is still quite different from the Poisson process convolved with the refractoriness. This is clearly visible from the shape of ρ_ISI functions shown in Fig. 4.12D. Note that t_A = τ_R was assumed in Fig. 4.8B.

We obtained very similar results when one of t_A and τ_R was set to zero (Fig. S6.3). Only the asymptotic values ofCV_ISI in the limit of larget_A+τ_R, determined by Eq. (4.51), are different in all these cases.

The extent of the reduction ofCV_IEI by the refractoriness depends on the exact statistics of the bursty vesicle release patterns. Given the same average presynaptic fusion rates, patterns with a higher fraction of periods when bursts are on would be more resistant to the refrac-toriness (compare curves for differentN_V in in Fig. 4.8B). The example considered in Fig. 4.8B corresponds to one of the most bursty fusion, i.e., with the highest CV, in the literature. Thus, it can be used to judge the upper limit for how much the refractoriness regularizes the vesi-cle release patterns. In mice, the average value of t_A+τ_R was 1.0 ms (Pangrˇsiˇc et al., 2015).

tA+τR = 1.2 ms (median value) was reported in cats by (Heil et al., 2007). We reanalyzed data from (Pangrˇsiˇc et al., 2015) and found that the average value of the ratio (t_A+τ_R)/ht_IEIi for the maximally driven responses was 0.46 with the 5 – 95 percentile range of 0.25 – 0.85.

Compared to the modeling results shown in Fig. 4.8B, these estimates suggest that the regu-larizing effect of the refractory period of the bursty spike trains can be considerable for the maximally driven responses in vivo. For moderate spike rates, the opposite is expected. For example, (t_A+τ_R)/ht_IEIi is only 0.05 for t_A+τ_R= 1.0 ms and R_s = 50 s⁻¹.

Spike rate dependence on the presynaptic membrane potential

As shown in section 4.4.1, the dependence of R_r onV_m is very well described by a Boltzmann function within the entire space of the kinetic parameters. In addition, we showed that the dependence of the spike rate onVmis also described by a Boltzmann function if the presynaptic vesicle release follows a Poisson process (Supplementary Material 6.2),

R_s(V_m) = As

1 +e^{−(Vm−V0.5−Sr−Ss)/k0V}, (4.52) with

A_s= A_r

1 +A_r·(t_A+τ_R) ≤A_r, S_s=−k_V⁰ ·ln 1 +A_r·(t_A+τ_R)

≤0. (4.53)

Here,Rsis the spike rate, As is the maximum spike rate,Ssis the shift of the Rs vsVm relation with respect to the Rr vs Vm dependence along the Vm axis. Thus, Rs(Vm) functions are left-shifted and scaled down versions of theR_r(V_m) functions, with identical dynamic ranges. This finding shows that, in principal, the variability of the refractory properties could contribute to the heterogeneity of SGN responses to sound stimuli. Recent experimental data, however, suggest that variability of the refractory times in cats and mice is small. Indeed, (Heil et al., 2007) reported that spontaneousR_s from different SGNs of cats could be well reproduced with fixed t_A = 0.59 and τ_r = 0.65, by varying only R_r. In accordance, the average of the total refractory period tA+τR was 1.0 ms with the 5 – 95 percentile range of 0.7 – 1.7 ms⁻¹ for maximally driven SGNs in mice (see Fig. S5 in Pangrˇsiˇc et al., 2015). We performed additional analysis of the data from (Pangrˇsiˇc et al., 2015) by using Eq. (4.53) and found that the average ofS_swas only−0.08·∆_V[R_s], with the 5 – 95 percentile range of−0.14·∆_V[R_s] –−0.05·∆_V[R_s].

Here, we used the fact that, by definition, ∆_V[R_s] = k_V ·ln(81). Accordingly, the measured R_s values were tightly correlated with the modeled ones when assuming t_A+τ_R fixed to 1.0 ms (Fig. S6.2A). Differently, the match between the measured and modeledR_swas very poor when fixing the R_r to the average over the whole population (Fig. S6.2B). Thus, we conclude that the heterogeneity of Rs of SGNs is mainly due to the heterogeneity of Rr, with only a minor contribution by the variability of t_A+τ_R.

In the case of bursty presynaptic vesicle release, the R_r dependence on V_m results purely from the modulation of p_o. Indeed, the release rate within bursts does not depend on V_m. On the other hand, the postsynaptic refractoriness primarily reduces the number of events within bursts without affecting the burst length at any V_m in this release scenario. Thus, we can conclude that the R_s vs V_m relation is a scaled down version of the corresponding R_r vs V_m relation, described by a Boltzmann function (section 4.4.1). Of course, when NV is so large that the vesicular release is approximated by a homogeneous Poisson process, a leftward shift of R_s vs V_m with respect to R_r vs V_m is expected. However, as shown above, physiological values of the refractory times t_A and τ_R are too small to cause any considerable shift. To

relations with bursty vesicle release patterns identified in section 4.4.1 (Fig. 4.3). As shown in Fig. S6.3 (2-state vesicular release sites) and Fig. S6.4 (7-state vesicular release sites), the normalized R_r vs V_m and R_s vs V_m relations indeed coincide well even when (t_A+τ_R)/hti_IEI is as high as 0.5. It has to be noted that the amplitude reduction of the R_s vs V_m relations by the postsynaptic refractoriness in the case of bursty release is larger than in the case of a more regular release (Fig. S6.5). Indeed, intraburst IEIs form the majority of all IEIs and they are considerably shorter than the average IEI. Thus, more of these events fall in the refractory periods of SGNs compared to more regular vesicle release patterns.

Inter-spike interval correlations

In our model, each ISI is equal to the corresponding IEI or a sum of subsequent IEIs which fall in the time window of the ISI due to the refractoriness. Thus, if the IEIs are uncorrelated so are the ISIs. In other words, the refractoriness itself does not introduce correlations between ISIs. If IEIs are correlated, then correlations between the merged events at particular laglare smaller in magnitude than the correlations between the original events at the same lag number. This is a natural consequence of the Markovian nature of the presynaptic vesicle release process. Longer refractory periods result in larger numbers of subsequent IEIs merged, and thus, in smaller correlations between the subsequent ISIs. Fig. 4.8C compares C_IEI vs l and the corresponding C_ISI vs l relations for an AZ built of 7-state vesicular release sites with Ca²⁺-microdomain topography scenario. There, the kinetic parameters of the model were fixed as reported by (Beutner et al., 2001),krep was set so that the replenishment stage took half of the time of the whole vesicle cycle on average. The ratio (t_A+τ_R)/hti_IEI was set to 0.46, the average value in mice SGN for maximally driven responses as shown in section 4.4.3. The reduction of ISI correlations was very pronounced forN_V = 2 (C_IEI(1)/C_IEI(1)∼20) and considerably milder for N_V = 10. As we showed in section 4.4.2, the IEI correlations decay faster with the lag numberl for AZ with smaller than larger N_V for AZs with Ca²⁺-microdomain coupling regime or in the limit of fast channel gating. That is why correlations between the subsequent ISIs, which result from combining subsequent IEIs, are reduced more for AZs with smallerNV. The situation is different in the regime of bursty vesicle release. Indeed, CIEI(l) decreases slowly with increased l in this case (see section 4.4.2) and, thus, the reduction of the correlations between ISIs is expected to be considerably weaker even at small N_V. This is exactly what we see in Fig. 4.8D, where AZ with 2-state vesicular release sites in Ca²⁺-nanodomain coupling with 2-state Ca²⁺ channels were examined. The kinetic parameters there were set to the values which produce the maximum peaks of theC_IEI vs l relations found in section 4.4.2 (Fig. 4.7B).

Figure 4.8: Impact of the refractoriness on the statistics of SGN spike trains. (A) Two lim-iting cases ofρ_ISI(t) when the underlying presynaptic vesicle fusion is approximated by a homogeneous Poisson process. (B) The dependence ofCV_ISI on the overall refractory period in the case of a bursty vesicle release (Ca²⁺-nanodomain coupling). Grey dashed line shows the CV_ISI vs (t_A+τ_R)/hti_IEI relation for the presynaptic release described by a homogeneous Poisson process. (C) Influence of the refractoriness on the serial correlation functions of ISIs (Ca²⁺-microdomain coupling, kinetic param-eters taken from Beutner et al., 2001). Thin dashed lines show the correspondingCIEI vs lrelations.

(D) Influence of the refractoriness on the serial correlation functions of ISIs (Ca²⁺-nanodomain cou-pling, bursty release). Thin dashed lines show the correspondingC_IEI vs l relations.

4.4.4 Analysis of experimental data: the origin of SGN response