Experimental σ-T av relation

(4.12) The analysis of experimental data show that T_av,σand T_detvary between individual cells of the same cell type, as we will illustrate in the next section.

4.2 Experimental results

To establish whether intracellular Ca²⁺ oscillations are noisy limit cycle oscillations based on deterministic dynamics or repeated stochastic waves, we performed com-prehensive measurements of ISI series in different cell types. In each case, IP₃Rs evoke the Ca²⁺ oscillations as shown in [72, 67, 210] for astrocytes, microglia and HEK cells, as well as in Figure A.1 for PLA stem cells from human adipose tis-sue. Figure 4.3 shows representative time series of global oscillations in astrocytes, microglia, PLA and human embryonic kidney (HEK) cells.

These examples show that ISIs are not regular, as can be seen in the lower panels in which each dot denotes the time from the actual spike to the next one, the following ISI. The length of the ISIs changes randomly, although in different extent.

For the spontaneous oscillations in microglia the ISIs vary between 40 s and 400 s, whereas stimulation induced oscillations in HEK cells do only differ in the range from 50 s to 80 s. The ISIs derived from records like those were used to characterize the oscillation mechanism by determining the serial correlation coefficientsρ_k, mean values T_av and the standard deviations σ of the corresponding spike trains.

4.2.1 Experimental σ-T

relation

For a systematic analysis we determine the mean and the standard deviation of many cells for each cell type. The individual cells may differ in a variety of properties.

Parameters that act locally on the IP₃R clusters are the IP₃ and the Ca²⁺ base level concentrations that determine the probability of the initiating event P_puff of a first puff. The strength of Ca²⁺ extrusion is a main property determined by the expression level of SERCAs and PMCA. Experiments with Ca²⁺ free buffers do not exhibit a significant change of the oscillatory behavior indicating that cells can in general regulate their Ca²⁺ base level sufficiently to low levels in the range of tens of nM. Hence, we expect a similar low Ca²⁺ base level in different cells, which prevents toxic effects.

0

Figure 4.3: IP₃R-mediated Ca²⁺ spikes in various cell types. Representative exam-ples of oscillation time series for astrocytes (A), microglia (B), PLA cells (C), and HEK cells stimulated with 30 µM CCh (carbamyl choline) (D). Lower panels show the ISI following each spike, with ISI defined as the interval between consecutive fluorescence maxima.

We expect a larger cell diversity caused by different cell arrangements. First of all, the expression level of IP₃R as well as their spatial arrangement and the local-ization of the ER have a huge impact. The spatial separation of clusters determines the communication between adjacent clusters. If the release sites are close to each other, a triggering puff can most likely induced a global signal. If the IP₃R clusters are more widely spread the probability of a triggering eventP_trig is lower, since the nucleation probability of a waveP_waveis low. Ca²⁺has to diffuse over long distances, where it is pumped back into the ER. Other global properties are therefore the pump strength and the buffer amount because they decrease the spatial communication and suppress a cooperative behavior.

The cell individuality leads to different parameters of the waiting time den-sity (4.3) and hence to different mean periods T_av and standard deviation σ. This is shown in Figure 4.4 for the four different cell types, where each dot corresponds to the spike train of a single cell.

The standard deviation σ increases with the mean period T_av for all cell types and all different stimulation strengths. Another obvious characteristic is the offset of data points on the T_av-axis. That indicates the deterministic time T_det predicted by our first hypothesis in Section 4.1.1. Further we see that T_avandσexhibit indeed

0 400 800

0 400 800

astrocytes ρ=0.87

0 300 600

0 300 600

microglia ρ=0.85

0 125 250

0 200 400

HEK ρ=0.86

0 250 500

0 350 700

PLA ρ=0.89

Average interspike interval T_av (s)

SD of interspike intervals σ (s)

B

D C

A

Figure 4.4: Ca²⁺ spikes occur randomly. Dependence of the standard deviationσof ISI on the average ISI T_av for 366 astrocytes (A), 224 microglia (B), 270 PLA cells (C) and 137 HEK cells stimulated with 30 µM CCh (D). σ and T_av were obtained from time series of single cells by temporal averaging. The correlation coefficientρ shows thatσ and T_av are highly correlated in all four cases. A further example for hepatocytes is shown in Figure A.4A.

a linear relation as indicated by the linear correlation coefficientρ which is close to 0.9 for all cell types.

Standard deviations are of the same order of magnitude as averages for most points, and for large T_av,σand T_av are similar. Hence, the uncertainty in predicting the occurrence of a spike is of the same order of magnitude as the mean ISI: Ca²⁺

spikes are random events. If spikes represented the active phase in a deterministic oscillation, the standard deviation of ISI would instead be in the range of the global interpuff interval, i.e. between a few hundred milliseconds for short T_av and a few seconds for long T_av.

Comparison with the predicted σ-T_av relation in Figure 4.2B exhibits a good agreement if we include a deterministic time T_det. A good test of the model is the slope of theσ-T_av dependence.

The time-dependence of the global nucleation rate differs between the different cell types as revealed by the slopes of the relationship between σ and T_av. Fig-ure 4.5A shows values of the parameters of the time-dependent global nucleation

rate (Equation 4.2) for the different cell types. The spontaneous oscillations in as-trocytes and microglia have a slope of theσ-T_av-relation close to 1 and an asymptotic nucleation rateλmuch smaller than the relaxation rateξ. Despite the relatively fast recovery from the previous spike, these oscillations are neither fast nor regular since the asymptotic nucleation rate λ of these cells is small. The stimulated oscillations in HEK cells show the inverse relation between λ and ξ.

Recovery from the previous spike is relatively slow in HEK cells. This corre-sponds to the initial decline in spike amplitudes in Fig. 4.3 during which the cell reaches a state corresponding to incomplete recovery from a spike during each ISI.

One possibility is that insufficient recovery from inhibition or incomplete refilling of the ER causes the decline. We might then suggest that immediately after stimula-tion, Ca²⁺ re-uptake during the ISI fails to keep pace with release during a spike and successive Ca²⁺ transients decrease in amplitude. But during the stationary phase of oscillations, the two fluxes balance such that Ca²⁺ uptake by the stores during the ISI matches the amount released during the preceding spike. The time-dependent nucleation rate Λ(t) reaches only 60 % of λ during an average ISI due to the small value of ξ. However, since the asymptotic nucleation rate λ is rather large, the oscillations in HEK cells have smaller σ than astrocyte oscillations. The values for λ and ξ of PLA cells lie between those for astrocytes and HEK cells.

With the fitted rates λ and ξ we can calculate the mean and the standard devi-ation by Equdevi-ations (4.7) and (4.11) and determine the expected CV. In Figure 4.5 CV is shown in dependence onλ and ξ. We see howCV decreases with decreasing ξ and increasing λ. For the fitted rates we obtain values, which correspond to the experimental population slopes in Figure 4.4.

4.2.2 Correlations

The findings above suggest a Poisson process. Especially theσ-T_avrelation of spon-taneous oscillations in glia cells substantiate this hypothesis, since they exhibit a slope of 1, which corresponds to the CV of a pure Poisson process. But also the lower slope for the HEK cells is in agreement with the time dependent Poisson pro-cess. Thus we expect that the uncertainty in spike timing is also visible in vanishing correlations between consecutive ISIs. To test that, we calculate the serial corre-lation coefficient ρ_k defined by Equation (3.39) of the ith and (i+k)th ISI of cell individual spike trains. Figure 4.6 depictsρ_k averaged over a cell population of HEK cells. Similar results are obtained for PLA cells and astrocytes in Figure A.2.

The results demonstrate that consecutive intervals are not correlated, indicating the randomness of the spike-generating mechanism. To confirm that the vanishing correlations are not caused by annulling of opposite correlations of individual cells we merge the ISIs of all cells by normalize them to the corresponding mean period T_av. With the resulting spike train we can perform second order statistic. Therefore

0

Figure 4.5: Fitted values λ and ξ of the time-dependent global nucleation rate Equation 4.2 for the different cell types and the correspondingCV. A: The values were obtained by fitting σ and T_av of individual time series to Equations (4.9) and (4.11) to obtain λ and ξ, which were then averaged across all cells of the same type. We have approximated T_det by the smallest value of T_av observed for the corresponding cell type in the fitting procedure. Error bars show standard errors (s.e.). B: Dependence of CV on the parameters of the waiting time density (4.3).

The cell specificCV determined by the fitted values forλandξfromAare indicated by the arrows and exhibit a good agreement with the population slopes of theσ-T_av relation, which correspond to CV without T_det.

we calculate the joint distribution density P2(ISI_i,ISI_i+1) shown in the first row of Figure 4.7 for the different cell types.

The joint probability distribution exhibits only very small deviations from white correlation, which corresponds to a circular dependence. Moreover, we see that the width is in the range of T_av, as expected by the estimated coefficient of variations CV. In this context the narrower distribution of the HEK cells is related to the smallerCV.

Another property we can proof byP₂(ISI_i,ISI_i+1) is the renewal assumption, i.e.

that spikes occur independently from the previous one. Therefore, we calculate the correlation map defined by the difference of the joint distribution density and the product of the single densities

P₂(ISI_i,ISI_i+1)− P(ISI_i)P(ISI_i+1) (4.13) shown in the second row of Figure 4.7. Although we obtain a structure of the second order correlation their values are infinitesimally small, namely in the range of 10⁻⁷. The influence of the population averaging on the serial correlation coefficient is shown in the last row of Figure 4.7, where the distribution of the first correlation coefficient ρ₁ is shown. We see that the correlations are centered around zero and

-0.5

Figure 4.6: Successive ISIs are not correlated in HEK cells. A-D:Serial correlation coefficient ρ_k between theith and (i+k)th ISI averaged across time series for HEK cells before (red) and after (blue) loading with additional Ca²⁺ buffer. The first and second periods of recording each lasted 45 min and were interrupted for 10 min to load cells with the concentrations of BAPTA-AM shown within the triangles and then 5 min to allow hydrolysis of the ester. CCh concentrations are shown beneath the black bars. The number of time series measured for the first and the second periods were, respectively: A: n₁ = 35, n₂ = 35; B: n₁ = 35, n₂ = 28; C: n₁ = 31, n₂ = 22 D: n₁ = 29, n₂ = 18. Error bars denote s.e. for averaging over n time series. Correlations for PLA cells and astrocytes are shown in Figure A.2.

that more than 80 % of ρ₁ have an amplitude smaller than 0.25. This does also hold for higher k as can be seen in the representative example of 35 HEK cells in Figure A.3 where the first six serial correlation coefficient are shown color coded for individual cells.

Hence, the vanishing correlations are not predominantly caused by averaging but by the vanishing correlations between single ISIs indicating the stochastic nature of Ca²⁺ oscillations as predicted by our third hypothesis in Section 4.1.1.

Thus we conclude that global oscillations result from a sequence of randomly occurring global Ca²⁺ spikes. The data show also that σ almost vanishes at the smallest values of T_av, indicating that almost regular oscillations with ISIs close to T_det do exist. The results are similar for spontaneous oscillations in astrocytes, microglia and PLA cells, and for oscillations evoked by different levels of stimulation in HEK cells, suggesting that IP₃-evoked Ca²⁺oscillations are sequences of stochastic spikes in many cell types.

-5 10

astrocytes PLA cells HEK cells

Figure 4.7: Second order statistics for astrocytes (first column), PLA cells (sec-ond column) and HEK cells (third column). The joint distribution density P₂(ISI_i,ISI_i+1) of two successive ISIs (first row) for merged spike trains normal-ized by T_av exhibit the structure of white, i.e. not correlated data for all cell types.

Also the correlation map (second row) indicates only very small dependences. This is confirmed by the distribution of the correlation coefficientsρ₁ (third row) obtained from individual spike trains. This does also hold for higherk as can be seen in the representative example of 35 HEK cells in Figure A.3.

Our conclusion is compatible with the idea that each Ca²⁺ spike reflects the passage of a Ca²⁺wave across the cell driven by successive activation of IP₃R clusters by Ca²⁺ diffusing between them [135, 23, 136]. This mechanism can generate the spectrum of observed shapes of oscillations [62, 60]. The results show that waves initiate randomly. The time of initiation is not set by a deterministic process, such as recovery from Ca²⁺-inhibition or a progressive sensitization of IP₃Rs by Ca²⁺.

Stochastic models of repetitive waves show that if σ is of the same order as T_av, it is dominated by the probability P_trig of triggering a wave after the cell has recovered from the previous one [60]. The smaller the value of P_trig, the longer it takes on average for the next wave to occur, and the larger is the value ofσ. For such repetitive triggering of waves,σ increases linearly with T_av, and σ= 1/P_trig= T_stoch holds for large values of T_av. Even ifP_trig relaxes exponentially from 0 immediately

after a spike to an asymptotic value, the linear relation between σ and T_av for large T_av still holds and it has a slope < 1 (see Fig. 4.2 and Fig. 4.4, D). Such a relaxation ofP_trig appears to apply also to wave initiation reported in [136]. Almost regular oscillations arise when P_trig is very large, because then as soon as the cell has recovered from one spike, the next one is triggered. We refer to the length of the ISI of these regular oscillations as the deterministic part T_det of the ISI. It might be set by a variety of processes depending on the cell type, for example store refilling, IP₃R inhibition or Ca²⁺-feedback to the IP₃ concentration. Each of these processes may also cause a time-dependence ofP_trig such as that described in a simplified way by the relaxation with time constant ξ in Equation 4.2.