Simon Hess

Christophe Pouzat

Lars Paeger

Andreas Pippow

Peter Kloppenburg
email: peter.kloppenburg@uni-koeln.de

Analysis of neuronal Ca 2+ handling properties by combining perforated patch clamp recordings and the added buffer approach

Keywords: added buffer approach, -escin, calcium-binding ratio, calcium buffering, cerebellum, fura-2, midbrain, perforated patch clamp

Ca 2+ functions as an important intracellular signal for a wide range of cellular processes.

These processes are selectively activated by controlled spatiotemporal dynamics of the free cytosolic Ca 2+ . Intracellular Ca 2+ dynamics are regulated by numerous cellular parameters.

. Since the added buffer approach typically employs the standard whole-cell configuration for concentration-controlled Ca 2+ indicator loading, it only allows for the reliable estimation of the immobile fraction of intracellular Ca 2+ buffers.

Furthermore, crucial components of intracellular signaling pathways are being washed out during prolonged whole-cell recordings, leading to cellular deterioration. By combining the added buffer approach with perforated patch-clamp recordings, these issues are circumvented, allowing the precise quantification of the cellular Ca 2+ handling properties, including immobile as well as mobile Ca 2+ buffers.

Introduction

This study combined perforated patch clamp recordings with ratiometric Ca 2+ imaging and the 'added buffer approach' to achieve significant improvements in the quantitative analysis of neuronal Ca 2+ handling. In neurons, Ca 2+ serves as a ubiquitous signal in a wide range of cellular processes, including the synaptic release of neurotransmitters, membrane excitability, enzyme activation, and activity-dependent gene activation [START_REF] Augustine | Local Calcium Signaling in Neurons[END_REF][START_REF] Berridge | Neuronal Calcium Signaling[END_REF][START_REF] Clapham | Calcium Signaling[END_REF][START_REF] Gilabert | Cytoplasmic Calcium Buffering: An Integrative Crosstalk In: Islam MdS[END_REF]. These processes are selectively regulated by controlled spatiotemporal dynamics of the free cytosolic Ca 2+ [START_REF] Berridge | Calcium microdomains: Organization and function[END_REF][START_REF] Berridge | The versatility and universality of calcium signalling[END_REF]. Intracellular Ca 2+ dynamics are regulated by numerous cellular parameters (Ca 2+ handling parameters) including Ca 2+ influx through voltage-and ligandgated Ca 2+ channels, Ca 2+ release from intracellular stores, the location of the Ca 2+ source, Ca 2+ buffering by mobile and immobile buffers, Ca 2+ extrusion, locally changing diffusion coefficients as well as the geometry of the cell [START_REF] Helmchen | Ca2+ buffering and action potential-evoked Ca2+ signaling in dendrites of pyramidal neurons[END_REF][START_REF] Kloppenburg | Highly Localized Ca2+ Accumulation Revealed by Multiphoton Microscopy in an Identified Motoneuron and Its Modulation by Dopamine[END_REF][START_REF] Lin | Dynamics of volume-averaged intracellular Ca 2+ in a rat CNS nerve terminal during single and repetitive voltage-clamp depolarizations: Presynaptic [Ca 2+ ] i dynamics at the rat calyx of Held terminal[END_REF][START_REF] Maravall | Estimating Intracellular Calcium Concentrations and Buffering without Wavelength Ratioing[END_REF][START_REF] Neher | Calcium gradients and buffers in bovine chromaffin cells[END_REF][START_REF] Sabatini | The Life Cycle of Ca2+ Ions in Dendritic Spines[END_REF][START_REF] Tank | A quantitative analysis of presynaptic calcium dynamics that contribute to short-term enhancement[END_REF]. Impairment of this finely tuned system can cause neuronal dysfunction and even neurodegeneration [START_REF] Berridge | Calcium Signalling and Alzheimer's Disease[END_REF][START_REF] Chan | Calcium homeostasis, selective vulnerability and Parkinson's disease[END_REF]. Neuronal Ca 2+ handling properties have been, up to now, analyzed by combining electrophysiological recordings and Ca 2+ imaging into what is referred to as the 'added buffer' approach [START_REF] Neher | Calcium gradients and buffers in bovine chromaffin cells[END_REF].

The added buffer approach is based on a single-compartment model with the following rationale. During the measurement of intracellular Ca 2+ dynamics with Ca 2+ chelator-based indicators, the amplitude and time course of the Ca 2+ signals (free cytosolic Ca 2+ concentration) are influenced by the concentration of the Ca 2+ indicator (e.g., fura-2), since the indicator acts as an exogenous (added) Ca 2+ buffer and competes with the endogenous buffer(s). Accordingly, increasing indicator concentrations reduce the amplitude and prolong the decay time constants of free intracellular Ca 2+ fluctuations [START_REF] Helmchen | Ca2+ buffering and action potential-evoked Ca2+ signaling in dendrites of pyramidal neurons[END_REF][START_REF] Neher | Calcium gradients and buffers in bovine chromaffin cells[END_REF][START_REF] Pippow | Differences of Ca2+ handling properties in identified central olfactory neurons of the antennal lobe[END_REF]. If the buffering capacity (Ca 2+ binding ratio) of the added buffer is known, the decay time constant can be used to estimate the Ca 2+ signal kinetics in the absence of exogenous buffer. The model of the original added buffer approach [START_REF] Neher | Calcium gradients and buffers in bovine chromaffin cells[END_REF] provides estimates of Ca 2+ handling parameters such as the free cytosolic Ca 2+ concentration ([Ca 2+ ] i ), the Ca 2+ extrusion rate (γ), the Ca 2+ binding ratio ( B ), and the endogenous decay time constant ( endo ).

The added buffer approach relies on information about the precise exogenous Ca 2+ buffer concentration in the neuron. Typically, the Ca 2+ indicator concentration can be determined at any time during the experiment from loading curves. The neurons are loaded via the patch pipette, whose solution contains a known Ca 2+ indicator concentration. During loading, fluorescence is monitored continuously, and it is assumed that when the fluorescence reaches a plateau, the indicator concentrations in the cell and the pipette are equal. Loading is usually performed in the whole-cell patch clamp configuration since this recording configuration provides low access resistance, ensuring that the solution in the patch pipette can freely interchange with the cytoplasm. Thus, the whole-cell configuration is ideally suited to "control" the composition of the intracellular milieu, including the Ca 2+ indicator (added Ca 2+ buffer) concentration (what is in the patch pipette ends up being what is in the neuron).

However, the free exchange of molecules between cytoplasm and patch pipette disturbs intracellular signaling by washing out components of the cytosolic signaling system, including, for example, mobile cytosolic Ca 2+ buffers. This not only impairs neuronal function but also causes estimation errors of the Ca 2+ handling properties (Akaike, 1994;[START_REF] Akaike | Nystatin Perforated Patch Recording and Its Applications to Analyses of Intracellular Mechanisms[END_REF][START_REF] Delvendahl | Reduced endogenous Ca 2+ buffering speeds active zone Ca 2+ signaling[END_REF][START_REF] Lindau | IgE-mediated degranulation of mast cells does not require opening of ion channels[END_REF][START_REF] Matthews | Tuning Local Calcium Availability: Cell-Type-Specific Immobile Calcium Buffer Capacity in Hippocampal Neurons[END_REF].

For pure electrophysiological recordings, this issue can be minimized or even ignored by using the perforated patch clamp configuration originally introduced by Horn and Marty [START_REF] Horn | Muscarinic activation of ionic currents measured by a new whole-cell recording method[END_REF]. In this recording configuration, electrical access between the recording pipette and the intracellular space is mediated by perforating agents that form channels or pores in the membrane. The washout of cytosolic components and the disruption of intracellular signaling pathways are then drastically reduced compared to the whole-cell configuration [START_REF] Korn | Influence of sodium-calcium exchange on calcium current rundown and the duration of calcium-dependent chloride currents in pituitary cells, studied with whole cell and perforated patch recording[END_REF]. The original and most commonly used perforating agents have been the antibiotic polyenes nystatin and amphotericin B, and the antibiotic polypeptide gramicidin (Akaike, 1994;[START_REF] Akaike | Nystatin Perforated Patch Recording and Its Applications to Analyses of Intracellular Mechanisms[END_REF][START_REF] Hess | The fat mass and obesity associated gene (Fto) regulates activity of the dopaminergic midbrain circuitry[END_REF][START_REF] Horn | Muscarinic activation of ionic currents measured by a new whole-cell recording method[END_REF][START_REF] Klöckener | High-fat feeding promotes obesity via insulin receptor/PI3K-dependent inhibition of SF-1 VMH neurons[END_REF][START_REF] Könner | Role for Insulin Signaling in Catecholaminergic Neurons in Control of Energy Homeostasis[END_REF][START_REF] Kyrozis | Perforated-patch recording with gramicidin avoids artifactual changes in intracellular chloride concentration[END_REF]. While polyenes and the peptide exhibit differences in their pore-forming mechanisms and ion selectivities (De [START_REF] Kruijff | Polyene antibiotic-sterol interactions in membranes of Acholeplasma laidlawii cells and lecithin liposomes. I. Specificity of the membrane permeability changes induced by the polyene antibiotics[END_REF][START_REF] Myers | Ion transfer across lipid membranes in the presence of gramicidin A[END_REF][START_REF] Russell | Effects of nystatin on membrane conductance and internal ion activities inAplysia neurons[END_REF][START_REF] Tajima | Perforated patch-clamp recording in cardiac myocytes using cation-selective ionophore gramicidin[END_REF], their pores share key common properties: they are permeable to small molecules with a molecular weight up to ~200 Da, including monovalent ions [START_REF] Kruijff | Polyene antibiotic-sterol interactions in membranes of Acholeplasma laidlawii cells and lecithin liposomes. I. Specificity of the membrane permeability changes induced by the polyene antibiotics[END_REF][START_REF] Kyrozis | Perforated-patch recording with gramicidin avoids artifactual changes in intracellular chloride concentration[END_REF][START_REF] Urry | The Gramicidin A Transmembrane Channel: A Proposed π(L,D) Helix[END_REF], but they are neither permeable to divalent ions like Ca 2+ nor to Ca 2+ indicators like fura-2 (whose molecular weight is >600 Da). While the combination of perforated patch clamp recordings and the added buffer approach would be ideal, it cannot be implemented with the previous standard small pore-forming perforating agents.

To alleviate the "small pore size" problem, we tried -escin as the perforating agent. escin is a saponin [START_REF] Sirtori | Aescin: pharmacology, pharmacokinetics and therapeutic profile[END_REF] and is structurally not related to the previous perforating agents. The action of saponins on biological membranes is concentration-dependent and complex. Most models of its action propose that membrane permeabilization is mediated by cholesterol complexation (formation of saponin-cholesterol complexes), which disturbs the normal membrane structure or, at high concentrations, removes the cholesterol from the membrane [START_REF] Bangham | Action of Saponin on Biological Cell Membranes[END_REF][START_REF] Böttger | The influence of saponins on cell membrane cholesterol[END_REF]. Since the pore size is generally larger and accordingly less selective compared to the ones of nystatin, amphotericin B and gramicidin, molecules of higher molecular weight (e.g., fluo-3, MW 770 Da) can pass through the cell membrane [START_REF] Fan | Perforated Patch Recording with β-escin[END_REF][START_REF] Orta | Human spermatozoa possess a calciumdependent chloride channel that may participate in the acrosomal reaction[END_REF]. Nevertheless, -escin permeabilized membrane still constitutes an effective diffusion barrier to larger molecules like Ca 2+ binding proteins [START_REF] Fan | Perforated Patch Recording with β-escin[END_REF][START_REF] Sarantopoulos | β-escin diminishes voltagegated calcium current rundown in perforated patch-clamp recordings from rat primary afferent neurons[END_REF]. In previous studies, -escin has been successfully used to permeabilize smooth muscle cells [START_REF] Kobayashi | Cytosolic heparin inhibits muscarinic and alpha-adrenergic Ca2+ release in smooth muscle. Physiological role of inositol 1,4,5-trisphosphate in pharmacomechanical coupling[END_REF] and -escin based perforated patch clamp recordings have been performed on cardiac myocytes [START_REF] Fan | Perforated Patch Recording with β-escin[END_REF] and different neuron types [START_REF] Inoue | Ionic channel mechanisms mediating the intrinsic excitability of Kenyon cells in the mushroom body of the cricket brain[END_REF][START_REF] Sarantopoulos | β-escin diminishes voltagegated calcium current rundown in perforated patch-clamp recordings from rat primary afferent neurons[END_REF].

Here, we combined -escin based perforated patch clamp recordings with the added buffer approach to measure intracellular Ca 2+ handling properties. A thorough quantitative analysis of the data showed that this approach leads to a significantly improved quantification of the Ca 2+ handling properties of individual neurons from brain slices of adult mice. The core experiments were performed on substantia nigra (SN) dopaminergic (DA) neurons from brain slices of adult mice. SN DA neurons are endogenous pacemakers and rely very heavily on intact, precisely orchestrated intracellular pathways, especially cytosolic Ca 2+ signaling.

Additionally, we performed a set of immunohistochemical experiments in cerebellar Purkinje cells, a cell type which is known for containing very high concentrations of calbindin D-28k, a mobile Ca 2+ binding protein [START_REF] Abe | Glycine responses in rat CNS neurons studied with gramicidin perforated patch recording[END_REF].

Results

The goal of this study was to implement the 'added buffer approach', a powerful method to determine neuronal Ca 2+ handling properties, in combination with perforated patch clamp recordings, which allow for electrophysiological recordings without compromising intracellular signaling. Since the typical perforating agents, nystatin, amphotericin B, and gramicidin, do not allow controlled loading of Ca 2+ indicators, we explored the suitability of β-escin for this purpose. This study was performed in brain slices of mice, mainly on DA SN neurons. These neurons are particularly well suited to test this experimental approach since they are endogenous pacemakers (Fig. 1) that strongly rely on intact, precisely controlled Ca 2+ signaling [START_REF] Bean | The action potential in mammalian central neurons[END_REF][START_REF] Lacey | Two cell types in rat substantia nigra zona compacta distinguished by membrane properties and the actions of dopamine and opioids[END_REF][START_REF] Richards | Electrophysiological and immunocytochemical characterization of GABA and dopamine neurons in the substantia nigra of the rat[END_REF].

The first part of the study shows that β-escin enables long-lasting, high quality perforated, patch clamp recordings with minimal rundown. The quality of these recordings is comparable to amphotericin B perforated recordings. In the second part, we show that β-escin perforated patch clamp recordings allow controlled Ca 2+ indicator loading and can significantly improve the added buffer approach to analyze the Ca 2+ handling properties of individual neurons. We achieved controlled fura-2 loading with a minimized impact on the integrity of the cells' cytoplasmic components. A thorough quantitative analysis of the data shows that this approach significantly improves the quantification of Ca 2+ handling properties in single neurons of adult animals compared to conventional whole-cell recordings. The versatility of this new approach is complemented by immunohistochemical stainings of cerebellar Purkinje cells, which possess very high concentrations of the mobile Ca 2+ -binding protein calbindin D-28k. Here we could show that calbindin D-28k is retained during -escin perforated patch clamp recordings, which is in stark contrast to results obtained during wholecell patch clamp recordings.

β-escin as perforating agent for perforated patch clamp recordings

After GΩ-seal formation, the perforation of the cell membrane was monitored by continuously measuring the membrane potential and the amplitude of the action potentials.

Initially, 'on-cell' action potentials were observed in the cell-attached configuration (Fig. 2A).

Diffusion of β-escin into the membrane gradually changed the recording to the perforatedpatch configuration, decreasing the series resistance and increasing the action potential amplitude (Fig. 2A from left to right). The perforation process was considered fully established when action potential amplitude and series resistance (R S ) had stabilized. Under our experimental conditions, we observed that indicator loading started within 5 min of seal formation, and perforation was fully established after ~20 min (Fig. 2B).

Comparison with other patch clamp configurations

To compare the β-escin mediated recordings with other configurations of patch clamp recordings, we analyzed whether and how important intrinsic neurophysiological parameters such as spontaneous or evoked action potential firing, membrane potential, and resting Ca 2+ levels ([Ca 2+ ] i ) are affected by the different recording configuration (whole-cell recordings, and amphotericin B-and β-escin perforated patch recordings). After establishing recordings with stable access (as judged by the series resistance), spontaneous and evoked action potential firing was monitored for at least 30 min. During whole-cell recordings, we observed a dramatic rundown of these parameters. As expected, they did not change during amphotericin B perforated-patch recordings. β-escin perforated-patch recordings gave results similar to amphotericin B mediated recordings, and only a small decline of the spontaneous firing was observed (Fig. 3A-D).

More specifically, during whole-cell recordings, more than 80% of the cells stopped spontaneous firing within 10 min after break-in (Fig. 3A,C). This loss of activity was accompanied by a ~4 fold increase in the resting [Ca 2+ ] i levels after 30 min of recording time (Fig. 4). This substantial rise in [Ca 2+ ] i was a clear and direct indicator for the deteriorating physiological state of the neurons [START_REF] Kuchibhotla | Aβ Plaques Lead to Aberrant Regulation of Calcium Homeostasis In Vivo Resulting in Structural and Functional Disruption of Neuronal Networks[END_REF][START_REF] Mattson | Calcium and neurodegeneration[END_REF] and showed directly that the Ca 2+ handling properties change during whole-cell recordings. In contrast, [Ca 2+ ] i remained stable during -escin perforated patch recordings over the same time (Fig 4).

Taken together, β-escin perforated patch clamp recordings significantly outperformed wholecell recordings, and β-escin perforated patch clamp recordings were of similar quality to amphotericin B mediated recordings. Overall, the data are in line with many earlier studies (e.g. [START_REF] Horn | Muscarinic activation of ionic currents measured by a new whole-cell recording method[END_REF][START_REF] Husch | Long-duration perforated patch recordings from spinal interneurons of adult mice[END_REF][START_REF] Lindau | IgE-mediated degranulation of mast cells does not require opening of ion channels[END_REF])) which clearly show the impact of whole-cell recordings on the cytosolic integrity and to what extend perforated patch clamp can minimize this impact (rundown), and thus are suitable to analyze 'true' intrinsic neurophysiological parameters over long periods of time.

Permeability of β-escin mediated pores

Next, we tested if the β-escin perforated membranes are permeable to fura-2 and to molecules with a MW in the range of Ca 2+ buffering proteins, like calbindin D-28k. To clarify this question, tetramethylrhodamine-dextran (TRD-40kDa, MW 40 kDa) or fura-2 was added to the pipette solution, and the fluorescence in the cell body was monitored during the recording. During β-escin perforated-patch recordings, the TRD-40kDa remained in the recording pipette, while it diffused into the neuron during whole-cell recordings (Fig 5). Furthermore, immunohistochemical staining showed that the endogenous calbindin-D28k remains in the cell during β-escin perforated patch clamp recordings but is washed out during whole-cell recordings (Supplement Fig. 1). In contrast, the loading curves for fura-2 (Fig. 6A) confirmed that fura-2 could diffuse readily through the β-escin perforated membrane into the cell. These results suggested that β-escin is well suited to implement the added buffer approach.

Ca 2+ handling properties

Given the stable and reliable recording conditions provided by β-escin, we combined βescin perforated patch clamp recordings with ratiometric fura-2 Ca 2+ imaging to analyze Ca 2+ handling properties. Using the added buffer approach, we determined the resting Ca 2+ concentration ([Ca 2+ ] i ), the Ca 2+ binding ratio κ S, and the extrusion rate γ of SN DA neurons, which display Ca 2+ dependent endogenous pacemaking properties [START_REF] Liu | Edaravone Protects Neurons in the Rat Substantia Nigra Against 6-Hydroxydopamine-Induced Oxidative Stress Damage[END_REF][START_REF] Neuhoff | Ih Channels Contribute to the Different Functional Properties of Identified Dopaminergic Subpopulations in the Midbrain[END_REF][START_REF] Smits | Molecular Marker Differences Relate to Developmental Position and Subsets of Mesodiencephalic Dopaminergic Neurons[END_REF]. Neurons were held at -70 mV to avoid spontaneous spiking.

Clear and reproducible Ca 2+ influx could be induced by trains of action potentials elicited by current pulses (500 pA, 500 ms) (Fig. 6B). The resulting intracellular Ca 2+ signals were monitored by ratiometric Ca 2+ imaging (Fig 6C ), and the data quality was compared with data acquired with conventional whole-cell patch clamp recordings.

Fura-2 loading: The intracellular concentration of the exogenous buffer at any time during the experiment was determined from the loading curves of the calcium indicator (Fig. 6A). DA SN neurons were loaded via the patch pipette with solutions containing 100 -200 µM fura-2. During dye loading, the fluorescence of the cell body was monitored in 30 s intervals using excitation at 360 nm (isosbestic point) to determine the dye concentration.

After the -escin perforated-patch configuration was fully established, fluorescence appeared in the cell bodies, and fluorescence intensity increased until it reached a stable plateau. The intracellular fura-2 concentration was estimated for different times during the loading curve, assuming that cells were fully loaded when the fluorescence reached a plateau and the fura-2 concentration in the cell and the pipette is equal at this point (for details see Methods).

Ca 2+ handling properties: If the buffer capacity of the added buffer is known, the time constant of the transient decay (τ transient ) can be used to estimate by extrapolation the Ca 2+ signal to conditions, with only endogenous buffers present (-κ B = 1 + κ S ). The model used for this study (Eq. ( 14), [START_REF] Neher | Calcium gradients and buffers in bovine chromaffin cells[END_REF] assumes that the decay time constants τ transient are a linear function of the Ca 2+ binding ratios (κ B and κ S ). κ S was determined from the negative x-axis intercept of a τ transient over κ S plot (Fig. 6D). The slope of the straight-line fit is the inverse of the extrusion rate (γ). The point of intersection of the linear fit with the y-axis denotes the endogenous decay time constant τ endo (no exogenous Ca 2+ buffer in the cell). This is demonstrated in Fig. 6 with an example of a DA SN neuron with following Ca 2+ handling parameters: τ endo : 0.7 ± 0.3 s, γ: 91.5 ± 10.4, s -1 , κ s : 66.4 ± 24.7.

Comparison of data quality between whole-cell and β-escin perforated patch clamp recordings

The data quality of the Ca 2+ imaging experiments used to determine the Ca 2+ handling properties was markedly better for the β-escin perforated-patch configuration compared to whole-cell recordings. Obvious evidence of "low-quality" data were implausible results such as negative Ca 2+ binding coefficients or Ca 2+ transients that did not return to baseline as frequently observed by us and previous studies [START_REF] Fierro | High endogenous calcium buffering in Purkinje cells from rat cerebellar slices[END_REF][START_REF] Pippow | Differences of Ca2+ handling properties in identified central olfactory neurons of the antennal lobe[END_REF].

For data sets that did not show such obvious flaws, an analysis on a finer scale confirmed the significantly higher data quality of the β-escin perforated records. For this purpose, parameters, which characterize the quality of data sets of individual neurons, were quantified. This included the goodness of fit for each Ca 2+ transient (a mono-exponential fit) and the linear regression that was used to fit the τ transient -κ B relation.

First, we tested the goodness of the mono-exponential fit for each Ca 2+ transient. Since our analysis method yields a meaningful standard error for the estimated [Ca 2+ ] i (see Methods), a fit obtained with a weighted nonlinear least-square produces standard errors for the estimated parameters (basal [Ca 2+ ] i , Δ[Ca 2+ ] i and, most importantly here, τ transient ) as well as residuals following a known distribution if the data do indeed follow the model; namely the residual sum of squares (RSS) should be the realization of a draw from a χ 2 distribution with a known number of degrees of freedom, and the residuals should not be correlated. We decided to keep a transient for further analysis if both the p-values of the RSS and the auto-correlation at lag 1 were larger than 0.01 (giving a probability of false-negative of 0.02). Since the decay fits were started once the transient had lost 50% of the peak increase (see Materials and Methods), the number of data points used for each fit is not fixed, meaning that the number of degrees of freedom of the χ2 distribution associated with each RSS is also changing. A proxy of the RSS that is easy to compare from transient to transient is the RSS per degree of freedom (that should be close to 1 for a good fit) and is illustrated in Fig. 7C. This goodness of fit criterium was fulfilled by 75% (24/32) of the transients measured in the whole-cell configuration compared to 92% (67/73) of the transient measured in the perforated configuration (Fig. 7D; using an exact test, based on the hypergeometric distribution, the probability that these two samples arise from the same underlying distribution is found to be smaller than 0.02). We then used the data sets with at least three 'good' transients to perform the linear regression (τ transient -κ B relation) and found that 15/15 (100 %) data sets with perforated recordings lead to an acceptable regression but only 2/6 (33 %) of those with whole-cell recordings (Fig. 7G). Thus, while experiments in the whole-cell configuration can yield results comparable to β-escin experiments (Fig. 8A [right panel]; κ s : 19.9 ± 25.1, τ endo : 0.4 ± 0.5 s, γ: 55.2 ± 10.4 s -1 ), it is evident that in many cases the whole-cell results were implausible (Fig. 8A, middle and left panel ). In these cases, the linear regression fit of the data intercepts the x-axis at positive value leading to a negative Ca 2+ binding ratio κ s and a negative endogenous decay time constant τ endo , respectively (see [START_REF] Delvendahl | Reduced endogenous Ca 2+ buffering speeds active zone Ca 2+ signaling[END_REF]))(κ s :

-72.9 ± 14.3, τ endo : -17.9 ± 2.7 s, γ: 4.0 ± 0.5 s -1 , middle panel; κ s : -3.1 ± 10.3, τ endo : -0.04 ± 0.2 s, γ: 58.0 ± 4.4 s -1 , left panel). It is important to note that the implausibility of the results from individual cells can be masked if pooled data sets from multiple neurons are analyzed (Fig. 8C andD).

In summary, the overall quality of the data obtained in β-escin perforated patch clamp experiments is tremendously improved compared to experiments in the whole-cell configuration. This improvement results from two factors. The perforated configuration preserves the general integrity and functionality of the cytoplasmic pathways, including the Ca 2+ handling machinery. In addition, it optimizes the measurements directly by preventing an uncontrolled change in mobile Ca 2+ buffer during the recording. Ultimately, this resulted in a far more reliable and better-defined estimation of the Ca 2+ handling properties of single neurons.

Discussion

Using β-escin as a perforating agent, we showed that it is possible to combine longlasting perforated-patch recordings with controlled Ca 2+ indicator loading and the 'added buffer approach' to analyze neuronal Ca 2+ handling properties. We consider this a substantial progress, and of particular importance for the analysis of intracellular Ca 2+ handling since the perforated-patch clamp configuration allows stable recordings with minimized impact on the intracellular signaling system, including mobile components of the Ca 2+ handling machinery, i.e., mobile Ca 2+ buffers. β-escin perforated recordings performed similarly well as amphotericin B, gramicidin, or nystatin perforated patch recordings, which are the 'gold standards' for stable long-term recordings [START_REF] Horn | Muscarinic activation of ionic currents measured by a new whole-cell recording method[END_REF][START_REF] Husch | Long-duration perforated patch recordings from spinal interneurons of adult mice[END_REF][START_REF] Rae | Low access resistance perforated patch recordings using amphotericin B[END_REF]. β-escin perforated patch recordings clearly outperformed conventional whole-cell recordings, which becomes especially apparent by monitoring the Ca 2+ levels. In contrast to the whole-cell configuration, the resting Ca 2+ concentration remained stable, indicating healthy recording conditions.

Typically, the 'added buffer' approach has been used in combination with conventional whole-cell patch clamp recordings, which allows the controlled loading of cells with Ca 2+

indicator dyes (e.g., fura-2) via the patch pipette [START_REF] Delvendahl | Reduced endogenous Ca 2+ buffering speeds active zone Ca 2+ signaling[END_REF]Lee et al., 2000a;[START_REF] Lin | Dynamics of volume-averaged intracellular Ca 2+ in a rat CNS nerve terminal during single and repetitive voltage-clamp depolarizations: Presynaptic [Ca 2+ ] i dynamics at the rat calyx of Held terminal[END_REF][START_REF] Neher | Calcium gradients and buffers in bovine chromaffin cells[END_REF]. In this context, the whole-cell configuration has the obvious advantage that it allows for easy and reliable loading of the cell with Ca 2+ indicators. However, this advantage is offset by the disadvantage that intracellular components are being washed out, leading to a gradual change of cellular properties over time. This causes not only general damaging effects on the physiology and Ca 2+ handling machinery of the cell, but it also introduces a systematic error to the analysis, since in the whole-cell configuration endogenous mobile Ca 2+ buffer is leaving the neuron at an unknown rate during loading of the 'added' buffer (here fura-2). Although this is not a new finding, since Neher and Augustine had already explicitly mentioned this matter in their original paper [START_REF] Neher | Calcium gradients and buffers in bovine chromaffin cells[END_REF], and although many studies have applied the added buffer approach in combination with whole-cell patch clamp recordings, this issue has not been addressed directly. In a strict sense, the added buffer approach in combination with the wholecell configuration can only be used to analyze the immobile fraction of Ca 2+ buffers and only if the measurements start after the mobile Ca 2+ buffers have diffused out of the cell and do not interfere with the analysis. Accordingly, this approach can be explicitly used to analyze the immobile Ca 2+ buffers as convincingly shown by [START_REF] Müller | Endogenous Ca2+ buffer concentration and Ca2+ microdomains in hippocampal neurons[END_REF] and [START_REF] Matthews | Tuning Local Calcium Availability: Cell-Type-Specific Immobile Calcium Buffer Capacity in Hippocampal Neurons[END_REF] in hippocampal neurons.

Employing β-escin as perforating agent addressed this issue and allowed to successfully combine perforated patch clamp recordings with controlled Ca 2+ indicator loading, and thus the added buffer approach. While our and other's whole-cell experiments [START_REF] Delvendahl | Reduced endogenous Ca 2+ buffering speeds active zone Ca 2+ signaling[END_REF][START_REF] Matthews | Tuning Local Calcium Availability: Cell-Type-Specific Immobile Calcium Buffer Capacity in Hippocampal Neurons[END_REF][START_REF] Müller | Endogenous Ca2+ buffer concentration and Ca2+ microdomains in hippocampal neurons[END_REF] showed that molecules with a molecular weight comparable to Ca 2+ binding proteins started being washed out immediately after establishing the whole-cell configuration, our permeability measurements showed that the βescin perforated membranes retained TRD-40kDa in the recording pipette and calbindin D-28k in the neuron, indicating that the perforated membrane is impermeable for the relative large endogenous mobile Ca 2+ buffers. Fura-2, on the other hand, with its low molecular weight, could easily permeate through the perforated membrane.

For practical purposes, it is important to note that the permeability of the perforated membrane is concentration-dependent. Thus, the β-escin concentration might have to be adjusted to different experimental conditions and cell types. Here, with β-escin concentrations of 10-15 µM, the perforated membrane was permeable to fura-2 but not to TRD-40kDa and calbindin D-28k. In permeabilized muscle cells, β-escin concentrations of 50 -100 µM render cell membranes permeable to molecules ranging from 10 to 150 kDa MW [START_REF] Iizuka | Receptordependent G protein-mediated Ca2+ sensitization in canine airway smooth muscle[END_REF][START_REF] Iizuka | Introduction of high molecular weight (IgG) proteins into receptor coupled, permeabilized smooth muscle[END_REF][START_REF] Konishi | Resting cytoplasmic free Ca2+ concentration in frog skeletal muscle measured with fura-2 conjugated to high molecular weight dextran[END_REF]. In recordings of cardiac myocytes with 50 µM β-escin, the membrane was permeable for fluorescent markers with a MW up to 10 kDa after 20 -25 min recording time [START_REF] Fan | Perforated Patch Recording with β-escin[END_REF].

Taken together, this study shows that the newly established approach is suitable to improve the quantitative analysis of intracellular Ca 2+ handling parameters markedly. The combination of perforated patch clamp recordings, which preserve the integrity and functionality of the intracellular milieu, with the 'added buffer approach', is a powerful tool to analyze mobile and immobile components of the intracellular Ca 2+ handling machinery.

Materials and Methods

Animals and brain slice preparation

All animal procedures were conducted in compliance with protocols that were approved by local government authorities (LANUV NRW, Recklinghausen, Germany). Mice were housed at 22 -24 ºC with a 12-hour light/12 hour dark cycle. Animals had access to water and chow ad libitum. Experiments were performed on brain slices from 9 -11 weeks old male C57BL/6 mice. Preparation of brain slices, recordings from DA SN and Purkinje neurons, labeling, and identification of neurons was performed as previously described [START_REF] Hess | The fat mass and obesity associated gene (Fto) regulates activity of the dopaminergic midbrain circuitry[END_REF][START_REF] Könner | Role for Insulin Signaling in Catecholaminergic Neurons in Control of Energy Homeostasis[END_REF][START_REF] Murru | Astrocytespecific deletion of the mitochondrial mAAA protease reveals glial contribution to neurodegeneration[END_REF]. The mice were lightly anesthetized with isoflurane (B506; AbbVie Deutschland GmbH & Co KG, Ludwigshafen, Germany) and subsequently decapitated. Coronal slices (250 -300 µm) containing the SN or sagittal slices containing the cerebellum (300 µm) were cut with a vibration microtome (HM-650 V; Thermo Scientific, Walldorf, Germany) under cold (4°C), carbogenated (95% O 2 and 5% CO 2 ), glycerol-based modified artificial cerebrospinal fluid (GaCSF; [START_REF] Ye | Patch-clamp studies in the CNS illustrate a simple new method for obtaining viable neurons in rat brain slices: Glycerol replacement of NaCl protects CNS neurons[END_REF]).

GaCSF contained (in mM): 250 Glycerol, 2.5 KCl, 2 MgCl 2 , 2 CaCl 2 , 1.2 NaH 2 PO 4 , 10 HEPES, 21 NaHCO 3 , 5 Glucose, adjusted to pH 7.2 with NaOH. Brain slices were transferred into carbogenated artificial cerebrospinal fluid (aCSF). aCSF contained (in mM): 125 NaCl, 2.5 KCl, 2 MgCl 2 , 2 CaCl 2 , 1.2 NaH 2 PO 4 , 21 NaHCO 3 , 10 HEPES, 5 glucose, adjusted to pH 7.2 with NaOH. First, they were kept for 20 min in a 35°C 'recovery bath' and then stored at room temperature (~24°C) for at least 30 min prior to recording. During the recordings, the slices were continuously superfused with carbogenated aCSF. To block synaptic currents, aCSF contained 10 -4 M PTX, 5 x 10 -5 M D-AP 5 , and 10 -5 M CNQX. All recordings were carried out at 24°C.

Electrophysiological recordings

DA SN neurons and the cerebellar Purkinje cells were recorded using whole-cell and perforated patch clamp recordings. Neurons were visualized and imaged with a fixed stage upright microscope (BX51WI, Olympus, Hamburg, Germany) using a 20× water-immersion objective (XLUMPLFL, 0.95 numerical aperture, 2 mm working distance, Olympus) with a 4× magnification changer (U-TVAC, Olympus), infrared differential interference contrast optics [START_REF] Dodt | Visualizing unstained neurons in living brain slices by infrared DIC-videomicroscopy[END_REF]) and fluorescence optics. Neurons were pre-identified by the size and location of their somata. This pre-identification was verified in each case by a physiological characterization during the recording and post hoc by immunohistochemistry for cell type-specific markers. DA SN neurons were recorded in the ventral tier SN. They have a low frequency, regular firing patterns with broad action potentials, and generate a clear I h -dependent "sag"-potential upon hyperpolarization (Fig. 1) [START_REF] Lacey | Two cell types in rat substantia nigra zona compacta distinguished by membrane properties and the actions of dopamine and opioids[END_REF][START_REF] Richards | Electrophysiological and immunocytochemical characterization of GABA and dopamine neurons in the substantia nigra of the rat[END_REF][START_REF] Ungless | Single cocaine exposure in vivo induces long-term potentiation in dopamine neurons[END_REF]. Purkinje cells were identified by their distinct morphology and localization within the cerebellar cortex and by their high-frequency spontaneous firing and very narrow action potentials (~200 µs; Supplement Fig. 1A) [START_REF] Bean | The action potential in mammalian central neurons[END_REF]McKay andTurner, 2005, 2004)). During the experiments, biocytin (0.1%; B4261; Sigma) diffused through the broken or perforated membrane into the recorded cell. Post recording, the identity of DA SN neurons and cerebellar Purkinje cells was confirmed by their immunoreactivity for the dopamine transporter (Fig. 1) [START_REF] Ciliax | The dopamine transporter: immunochemical characterization and localization in brain[END_REF] or the calcium-binding protein calbindin D-28k (Supplement Fig. 1) [START_REF] Abe | Glycine responses in rat CNS neurons studied with gramicidin perforated patch recording[END_REF][START_REF] Fournet | Selective localization of calciumbinding protein in human brainstem, cerebellum and spinal cord[END_REF], respectively. Biocytinstreptavidin labeling combined with dopamine transporter or calbindin D-28k immunohistochemistry was performed as previously described [START_REF] Hess | The fat mass and obesity associated gene (Fto) regulates activity of the dopaminergic midbrain circuitry[END_REF][START_REF] Murru | Astrocytespecific deletion of the mitochondrial mAAA protease reveals glial contribution to neurodegeneration[END_REF].

Whole-cell patch and perforated patch-clamp recordings

Electrodes with tip resistances between 3 and 5 MΩ were fashioned from borosilicate glass (0.86 mm inner diameter; 1.5 mm outer diameter; GB150-8P; Science Products, Hofheim, Germany) with a vertical pipette puller (PP-830; Narishige, London, UK).

Recordings were performed with an EPC10 patch-clamp amplifier (HEKA, Lambrecht, Germany) controlled by the program PatchMaster (version 2.32; HEKA) running under Windows. Data were sampled at 10 kHz and low-pass filtered at 2 kHz with a four-pole Bessel filter. The calculated liquid junction potential between intracellular and extracellular solution was also compensated (14.6 mV for normal aCSF, calculated with Patcher's Power Tools plug-in from http://www.mpibpc.mpg.de/groups/neher/index.php?page=software for IGOR Pro 6 [Wavemetrics, Lake Oswego, OR, USA]).

Whole-cell recordings were performed with pipette solution containing (in mM): 141 Kgluconate, 10 KCl, 10 HEPES, 0.1 EGTA, 2 MgCl 2 , 3 K-AT P, 0.3 Na-ATP and adjusted to pH 7.2 with KOH. For perforated patch clamp recordings and Ca 2+ imaging, ATP and GTP were omitted from the pipette solution since ATP might distort the measurement of Ca 2+ handling properties [START_REF] Matthews | Tuning Local Calcium Availability: Cell-Type-Specific Immobile Calcium Buffer Capacity in Hippocampal Neurons[END_REF] and to prevent uncontrolled permeabilization of the cell membrane during perforated patch clamp recordings [START_REF] Lindau | IgE-mediated degranulation of mast cells does not require opening of ion channels[END_REF]. For Ca 2+ imaging, EGTA was replaced by 100 or 200 µM fura-2 (pentapotassium salt, F1200, Molecular Probes, OR, USA). Perforated patch recordings were performed using protocols modified from Fan and Palade [START_REF] Fan | Perforated Patch Recording with β-escin[END_REF] and Sarantopoulos [START_REF] Sarantopoulos | β-escin diminishes voltagegated calcium current rundown in perforated patch-clamp recordings from rat primary afferent neurons[END_REF]. The patch pipette was tip filled with internal solution and backfilled with β-escin (~30 -60 μgml -1 ; E1378; Sigma) or amphotericin B (~200 μgml -1 ; A4888, Sigma) containing internal solution to achieve perforated patch recordings. β-escin was prepared as a stock solution (25 mM in internal solution) and stored up to 1 week at -20°C protected from light. Shortly before the experiments, the stock solution was diluted in intracellular saline to a final concentration. Amphotericin B was dissolved in dimethyl sulfoxide (DMSO; final concentration 0.4 -0.5%; D8418, Sigma) as described previously [START_REF] Hess | The fat mass and obesity associated gene (Fto) regulates activity of the dopaminergic midbrain circuitry[END_REF][START_REF] Kyrozis | Perforated-patch recording with gramicidin avoids artifactual changes in intracellular chloride concentration[END_REF][START_REF] Rae | Low access resistance perforated patch recordings using amphotericin B[END_REF] and added to the modified pipette solution shortly before use. The amphotericin B solution was freshly prepared every day. Ionophore containing solutions were stored on ice and replaced every 4 hours as needed. After obtaining a GΩ seal, the perforation process was monitored by continually measuring the access resistance (R a ). Experiments were started after R a had reached steady-state, and the action potential amplitude was stable (20 -30 min). A rupture of the membrane and the resulting conversion to the whole-cell configuration was indicated by abrupt changes in R a or action potential amplitude. Such experiments were rejected.

Imaging set up for fluorometric measurements

The imaging setup for fluorometric measurements consisted of an Imago/SensiCam CCD camera with a 640x480 chip (Till Photonics, Gräfelfing, Germany) and a Polychromator IV (Till Photonics) that was coupled via an optical fiber into the upright microscope (microscope details are provided above). The camera and the polychromator were controlled by the software Vision (version 4.0, Till Photonics) running under Windows. The excitation light from the polychromator was reflected onto the cells with a dichroic mirror, and emitted fluorescence was detected through an emission filter. The specific filter and mirror settings for the different experiments are given below. Data were acquired as 80x60 frames using 8x8 onchip binning. Images were recorded in arbitrary units (AU) and stored and analyzed as 12-bit grayscale images. For analyzing the fluorescence dynamics, the mean AU values of regions of interest (ROIs) from the middle of the cell bodies were used. 'Background subtraction' for the time series was performed with equivalent time series of neighboring ROIs.

Rhodamine-Dextran loading.

Loading experiments with fluorescent tetramethylrhodamine-dextran (TRD-40kDa¸ perforated patch configuration. The neurons were loaded with TRD-40kDa via the patch pipette (0.08%) and excited at 557 nm (bandpass: 562 nm, 562/40 BrightLine HC, Art-no. F39-562, AHF Analysentechnik AG, Tübingen, Germany; beamsplitter: 593 nm, HC BS 593, Art-no. F38-593, AHF). Emission was detected through a 593 nm long-pass filter (593/LP BrightLine HC, Art-no. F37-594, AHF). TRD-40kDa -fluorescence is reported as F/F 0 , where F is the measured fluorescence, and F 0 is the averaged fluorescence of 5 consecutive images in on-cell mode (before break-in or perforation of the cell membrane). Frames were taken in intervals of 30 s.

Fluorimetric Ca 2+ measurements with fura-2

Intracellular Ca 2+ concentrations were measured with the Ca 2+ indicator fura-2, a ratiometric dye suitable for absolute Ca 2+ concentration determination once calibrated [START_REF] Grynkiewicz | A new generation of Ca2+ indicators with greatly improved fluorescence properties[END_REF][START_REF] Poenie | Alteration of intracellular Fura-2 fluorescence by viscosity: A simple correction[END_REF]. DA SN neurons were loaded with fura-2 (pentapotassium salt, F1200, Molecular Probes) via the patch pipette (100 or 200 µM). Fura-2 was excited at 340 nm, 360 nm, or 380 nm (410 nm dichroic mirror; DCLP410, Chroma, Rockingham, VT, USA). Emitted fluorescence was detected through a 440 nm long-pass filter (LP440).

Calibration and 'added buffer approach'

The free intracellular Ca 2+ concentrations were determined as described by [START_REF] Grynkiewicz | A new generation of Ca2+ indicators with greatly improved fluorescence properties[END_REF], and the added buffer approach to analyze intracellular Ca 2+ handling properties was performed as previously described [START_REF] Neher | Calcium gradients and buffers in bovine chromaffin cells[END_REF][START_REF] Paeger | Energy imbalance alters Ca2+ handling and excitability of POMC neurons[END_REF][START_REF] Pippow | Differences of Ca2+ handling properties in identified central olfactory neurons of the antennal lobe[END_REF]. For completeness and notation clarification, these methods are described in detail in the Appendix. In brief: The added buffer approach was used in combination with whole-cell and perforated patch clamp recordings and optical Ca 2+ imaging. The added buffer approach is based on a single-compartment model with the rationale that for measurements of intracellular Ca 2+ concentrations with Ca 2+ chelator-based indicators, the amplitude and time course of the signals depends on the concentration of the Ca 2+ indicator (here fura-2). The indicator acts as an exogenous Ca 2+ buffer and competes with the endogenous Ca 2+ buffer(s).

The kinetics of cytosolic Ca 2+ signals strongly rely on the endogenous and exogenous (added) Ca 2+ buffers of the cell. The amplitude and decay rate of the free intracellular Ca 2+ change with increasing exogenous buffer concentration: The amplitude of free Ca 2+ decreases, and the time constant  transient of the decay increases. If the buffer capacity of the added buffer is known, the time constant of decay ( transient ) can be used to estimate, by extrapolation, the Ca 2+ signal to conditions, with only endogenous buffers present (-κ B = 1 + κ S ). The model used for this study (Eq. 4) assumes that the decay time constants  transient are a linear function of the Ca 2+ binding ratios (κ B and κ S ). κ S is determined from the negative x-axis intercept of this plot. The slope of the fit is the inverse of the linear extrusion rate (γ). The point of intersection of the linear fit with the y-axis denotes the endogenous decay time constant  endo (no exogenous Ca 2+ buffer in the cell). A detailed description of how to estimate the variance of these parameters is provided in the Appendix.

Tools for data analysis and statistics

Data analysis was performed with Igor Pro 6 (Wavemetrics), Spike2 (CED, Cambridge, UK), R (R Development Core Team [2009], http://www.R-project.org) and Graphpad Prism (version 8; Graphpad Software Inc., La Jolla, CA, USA). All calculations for the determination of EGTA purity, its dissociation constant, and the free Ca 2+ concentrations in the calibration solutions were performed in R. If not stated otherwise, all calculated values are expressed as means ± SEM (standard error of the mean). Box plots were constructed as follows: The '+' signs show the mean, the horizontal line the median of the data. Whiskers indicate the minimum and maximum values of the data. For pairwise comparisons of dependent and independent normal distributions, paired and unpaired t-tests were used, respectively. For pairwise comparisons of independent, not normal distributions, Mann-Whitney U-tests were used. For multiple comparisons, ANOVA was performed; post hoc pairwise comparisons were performed using t-tests as indicated in the figures. For comparison of the resting Ca 2+ levels ([Ca 2+ ] i ) over time under whole-cell and β-escin perforated-patch clamp conditions, the regression slope, after linear fitting, was also used. Tests were executed using GraphPad Prism 8 (GraphPad Software Inc., La Jolla, CA, USA). A significance level of 0.05 was accepted for all tests. Significance levels were: * p < 0.05, ** p < 0.01, *** p < 0.001. In all figures, n-values are given in brackets. Exact p-values are reported if p > 0.0001. Scaling, contrast enhancement, and z-projections of images were performed with ImageJ v2.0.0. The final figures were prepared with Affinity Designer (Ver. 1.6.1, Serif Ltd).

APPENDIX

Measurement of free intracellular Ca 2+ concentration

The free intracellular Ca 2+ concentrations were determined as in [START_REF] Grynkiewicz | A new generation of Ca2+ indicators with greatly improved fluorescence properties[END_REF]:

(1) [Ca 2+ ] i is the free intracellular Ca 2+ concentration for the background-subtracted fluorescence ratio R from 340 nm and 380 nm excitation. R min and R max are the ratios at a Ca 2+ concentration at virtually 0 M (i.e., ideally, no fura-2 molecules are bound to Ca 2+ ) and at saturating Ca 2+ concentrations (i.e., ideally, all fura-2 molecules are saturated with Ca 2+ ), respectively. K d,Fura is the dissociation constant of fura-2. F 380,min /F 380,max is the ratio between the emitted fluorescence of Ca 2+ free dye and the emitted fluorescence of Ca 2+ saturated dye at 380 nm excitation, reflecting the dynamic range of the indicator.

) ( ) ( ] Ca [ max min max , 380 min , 380 Fura d, i 2 R R R R F F K   
The term K d,Fura (F 380,min /F 380,max ) in Eq. ( 1) is dependent on the dye concentration and is substituted with the effective dissociation constant K d,Fura,eff , which is independent of the dye concentration and specific for each experimental setup (Neher, 1989):

(2)

We used in vitro calibration (in solution). For calibration K d,Fura,eff was determined by measuring fura-2 fluorescence ratios for R max , R min , and R = R def . R def is the ratio for a defined [Ca 2+ ] i , which was set to 0.35 µM (see below for the preparation of the solutions). K d,Fura,eff was then calculated from Eq. (2). To account for environmental differences between the cytoplasmic milieu and in vitro conditions, we used a correction factor (P) for R max and R min , as suggested by [START_REF] Grynkiewicz | A new generation of Ca2+ indicators with greatly improved fluorescence properties[END_REF], see also [START_REF] Pippow | Differences of Ca2+ handling properties in identified central olfactory neurons of the antennal lobe[END_REF]:

(3) P was determined as described by [START_REF] Poenie | Alteration of intracellular Fura-2 fluorescence by viscosity: A simple correction[END_REF][START_REF] Zhou | Mobile and immobile calcium buffers in bovine adrenal chromaffin cells[END_REF]. First, the fluorescence (peak) deflection at 340 nm was divided by that at 380 nm excitation from voltage-induced intracellular calcium transients (R d,cell ). Second, the ratio (R d,vitro ) from pairs of calibration solutions was determined by dividing (F 340,max -F 340,min )/(F 380,min -F 380,max ).

The correction factor P is the fraction of R d,cell /R d,vitro .

The fluorescence ratio R of an intracellular transient can then be converted to [Ca 2+ ] i using:

(4)

To determine the endogenous Ca 2+ binding ratio, the dissociation constant (K d,Fura ) has to be known. To formulate a relationship between K d,Fura and K d,Fura,eff that is independent of the dye concentration, an isocoefficient α was introduced by [START_REF] Zhou | Mobile and immobile calcium buffers in bovine adrenal chromaffin cells[END_REF]):

) ( ) ( ] Ca [ min max i 2 eff Fura, d, R R R R K     ) ( ) ( ] Ca [ min max i 2 eff Fura, d, P R R R P R K     ) ( ) ( ] Ca [ max min eff Fura, d, i 2 R P R P R R K   
(5) α is the isocoefficient, for which the sum [START_REF] Harrison | The effect of temperature and ionic strength on the apparent Ca-affinity of EGTA and the analogous Ca-chelators BAPTA and dibromo-BAPTA[END_REF] is independent of Ca 2+ concentration. F 340 and F 380 are the background-subtracted fluorescence signals measured during a brief Ca 2+ transient. Once the isocoefficient has been determined, K d,Fura can be calculated from Eq. ( 5).

Calibration solutions: The free Ca 2+ concentrations of the calibration solutions were adjusted by using appropriate proportions of Ca 2+ and the Ca 2+ chelator EGTA. The ability of EGTA to bind calcium is highly dependent on the environmental conditions such as ionic strength, temperature, pH, and the concentrations of other metals that compete for binding (Harrison andBers, 1989, 1987). In theory, the necessary amount of Ca 2+ and EGTA to set the free Ca 2+ concentration for the experimental conditions can be computed [START_REF] Patton | Some precautions in using chelators to buffer metals in biological solutions[END_REF].

However, small variations in the parameters such as pH, temperature, impurities of chemicals, pipetting, or weighing errors can lead to considerable errors in the estimate of the free Ca 2+ in EGTA-buffered Ca 2+ solutions with computer programs [START_REF] Mcguigan | Comparison between measured and calculated ionised concentrations in Mg2+ /ATP, Mg2+ /EDTA and Ca2+ /EGTA buffers; influence of changes in temperature, pH and pipetting errors on the ionised concentrations[END_REF]. To account for such variations, we determined the free Ca 2+ concentrations in our calibration solutions by using a Ca 2+ selective electrode following the guide from [START_REF] Mcguigan | Calcium buffer solutions and how to make them: A do it yourself guide[END_REF] and Pippow [START_REF] Pippow | Differences of Ca2+ handling properties in identified central olfactory neurons of the antennal lobe[END_REF] as described in the supplemental. Calibration solutions were prepared as follows (in mM): R max : 140 KCl, 2.5 KOH, 15 NaCl, 1 MgCl 2 , 5 HEPES, 10 CaCl 2 and 0.05 fura-2; R min : 129.5 KCl, 13 KOH, 15 NaCl, 1 MgCl 2 , 5 HEPES, 8 EGTA and 0.05 fura-2; R def : 129.5 KCl,13 KOH,10.3 NaCl,4.7 NaOH, 1 MgCl 2 , 5 HEPES, 4 EGTA, 2.7 CaCl 2 and 0.05 fura-2, yielding a free Ca 2+ concentration of 0.35 µM. All solutions were adjusted to pH 7.2 with HCl.

The concentrations of free Ca 2+ were determined from the ratioed imaging signals applying the approach from Grynkiewicz et al. [START_REF] Grynkiewicz | A new generation of Ca2+ indicators with greatly improved fluorescence properties[END_REF] (Eq. ( 2)). The

α ( ) α ( max min eff Fura, d, Fura d,    R R K K ) ( α ) ( ) ( 380 340 i t F t F t F  
calibration was performed in vitro (in solution). For calibration, a drop of each calibration solution (200 µl, R min : no Ca 2+ , R def : 0.35 µM Ca 2+ , R max : 10 mM Ca 2+ ) was placed on a sylgard coated recording chamber. Ratio images (from 340 nm and 380 nm excitation) were acquired for each solution. The acquired values were: correction factor [START_REF] Poenie | Alteration of intracellular Fura-2 fluorescence by viscosity: A simple correction[END_REF]) P = 0.78 ± 0.03, resulting in: R max = 1.267 ± 0.055 (n = 9); R min = 0.116 ± 0.004 (n = 9); R def = 0.277 ± 0.004 (n = 7); K d,Fura,eff = 1.048 ± 0.052 µM and isocoefficient α = 0.192 ± 0.003.

According to Eq. ( 5) the K d for fura-2 was K d,Fura = 0.222 ± 0.082 µM.

Single compartment model of calcium buffering: the 'added buffer approach'

For measurements of intracellular Ca 2+ concentrations with Ca 2+ chelator-based indicators, the amplitude and time course of the signals are dependent on the concentration of the Ca 2+ indicator (in this case, fura-2) that acts as an exogenous (added) Ca 2+ buffer and competes with the endogenous buffers [START_REF] Helmchen | Calcium dynamics associated with a single action potential in a CNS presynaptic terminal[END_REF][START_REF] Helmchen | A Single-Compartment Model of Calcium Dynamics in Nerve Terminals and Dendrites[END_REF][START_REF] Neher | Calcium gradients and buffers in bovine chromaffin cells[END_REF][START_REF] Tank | A quantitative analysis of presynaptic calcium dynamics that contribute to short-term enhancement[END_REF]. With the 'added buffer approach' [START_REF] Neher | Calcium gradients and buffers in bovine chromaffin cells[END_REF], the capacity of the endogenous Ca 2+ buffer in a cell is determined by measuring the decay of the Ca 2+ signal at different concentrations of 'added buffer' and by extrapolating to conditions in which only the endogenous buffer is present. Strictly speaking, the added buffer approach in combination with the whole-cell configuration can only be used to analyze the immobile fraction of Ca 2+ buffers and only when the measurements start after the mobile Ca 2+ buffers have diffused out of the cell and do not interfere with the analysis. In contrast, ß-escin perforated recordings can be used to measure the total endogenous Ca 2+ buffer capacity. The added buffer approach is based on a single-compartment model assuming that a big patch-pipette with a known (clamped) concentration of a Ca 2+ indicator (fura-2) is connected to a spherical cell. The model assumes that the cell contains an endogenous Ca 2+ buffer (S), of which the total concentration, dissociation constant K d,S , and its on (k + ) and off (k -) rates are unknown. When the cell membrane is perforated by β-escin, fura-2 (in its bound and unbound forms) starts to diffuse into the cell, which is considered small enough to be always in a 'well mixed' state (i.e., there are no gradients of chemical species within the cell body). In this model three cellular parameters determine the cytosolic Ca 2+ dynamics: (1) Ca 2+ sources, (2) Ca 2+ binding ratio and (3) Ca 2+ extrusion.

Ca 2+ sources: In the model, two forms of Ca 2+ sources are defined: (1) the Ca 2+ influx from the pipette (j in,pipette ) and ( 2) the Ca 2+ entering the cell via voltage-activated channels from a brief Ca 2+ pulse, which can be described by a delta function (j in,stim ) [START_REF] Helmchen | A Single-Compartment Model of Calcium Dynamics in Nerve Terminals and Dendrites[END_REF][START_REF] Neher | Calcium gradients and buffers in bovine chromaffin cells[END_REF]):

(7) (8)
[BCa] is the concentration of the exogenous buffer (fura-2) in its Ca 2+ bound form (the subscript p is for quantities within the pipette), D B is the diffusion coefficient of the exogenous buffer, ρ is the specific resistance of the pipette filling solution, R P is the pipette resistance, n Ca is the Ca 2+ influx induced by the stimulus, δ(t-t stim ) is the delta function with t stim as the time of stimulus, Q Ca is the Ca 2+ charge influx, and F is the Faraday's constant.

Ca 2+ binding ratio: In the cell, Ca 2+ is bound to the Ca 2+ buffers fura-2 and to the endogenous buffers, which are both assumed to be always in equilibrium with free Ca 2+ and not saturated. The ability of the experimentally introduced exogenous buffer to bind Ca 2+ is described by its Ca 2+ binding ratio that is defined as the ratio of the change in buffer-bound Ca 2+ over the change in free Ca 2+ : (9) [B T ] is the total concentration of the exogenous buffer B and K d,B is its dissociation constant for Ca 2+ . An analogous expression exists for the Ca 2+ binding ratio of the endogenous buffer S (κ S ).

p B p pipette in, ]) BCa [ ] [BCa ( R ρ D j   ) ( 2 ) ( stim Ca stim Ca stim in, t t δ F Q t t δ n j     2 B d, i 2 B d, T i 2 B ) ] Ca ([ ] B [ ] Ca [ ] [BCa K K d d   
Ca 2+ Extrusion: the model assumes a linear extrusion mechanism for Ca 2+ [START_REF] Neher | Calcium gradients and buffers in bovine chromaffin cells[END_REF]: (10) [Ca 2+ ] i,∞ is the steady-state intracellular Ca 2+ concentration [Ca 2+ ] i , to which a transient decays. According to Neher and Augustine [START_REF] Neher | Calcium gradients and buffers in bovine chromaffin cells[END_REF], γ 'reflects the combined action of pumps, exchange carriers, and membrane conductances and has the dimension l s -1 . In the following, we refer to it as γ᷈ . In more recent publications γ is defined as the extrusion rate for Ca 2+ measured in s -1 [START_REF] Helmchen | Calcium dynamics associated with a single action potential in a CNS presynaptic terminal[END_REF][START_REF] Helmchen | Ca2+ buffering and action potential-evoked Ca2+ signaling in dendrites of pyramidal neurons[END_REF]Lee et al., 2000aLee et al., , 2000b;;[START_REF] Lips | Endogenous calcium buffering in motoneurones of the nucleus hypoglossus from mouse[END_REF][START_REF] Palecek | Calcium dynamics and buffering in motoneurones of the mouse spinal cord[END_REF][START_REF] Sabatini | The Life Cycle of Ca2+ Ions in Dendritic Spines[END_REF][START_REF] Vanselow | Calcium dynamics and buffering in oculomotor neurones from mouse that are particularly resistant during amyotrophic lateral sclerosis (ALS) related motoneurone disease[END_REF]. Accordingly, we use (Jackson and Redman, 2003): (11) where V is the volume of the cell.

Dynamics of Ca 2+ transients: the mechanisms described in Eqs. 9 and 10 are combined to the model that describes the dynamics of the Ca 2+ decay after a brief Ca 2+ influx: (12) Substituting the time dependent terms for the buffers B and S with their Ca 2+ binding ratios κ B and κ S , respectively, [BCa] with its equilibrium value

[BCa] = [Ca 2+ ] i [B T ]/(K d +[Ca 2+ ] i ) and
τ loading = (VR P )/(D B ñ) we get [START_REF] Neher | Calcium gradients and buffers in bovine chromaffin cells[END_REF]):

(13) Neher and Augustine [START_REF] Neher | Calcium gradients and buffers in bovine chromaffin cells[END_REF] linearized Eq. ( 13) to simplify the solution for this equation for the case in which the baseline for [Ca 2+ ] i is constant, the concentration of the exogenous buffer [B T ] is constant and does not influence the baseline Ca 2+ , and when the

) ] Ca [ ] Ca ([ ] Ca [ i, 2 i 2 i 2 out          γ j V    V j j j dt d dt d dt d out stim in, pipette in, i 2 ] [SCa ] BCa [ ] Ca [       ) 1 ( ) ] Ca [ ] Ca ([ ) ] BCa ([ ] Ca [ S B ) ( i 2 i, 2 ] Ca [ ] B [ ] [Ca p 1 loading i 2 0 i 2 d T i 2                       V t t n V K Ca dt d
amplitudes of the Ca 2+ transients are small [START_REF] Neher | Calcium gradients and buffers in bovine chromaffin cells[END_REF]. Where 'small' is defined as less than 0.5 K d,Fura . When the Ca 2+ pulse is short compared to the decay time constant, the delta function equals zero. Under these conditions, the decay of a voltageinduced Ca 2+ transient can be described as:

(14)
 transient is proportional to  B , and a linear fit to the data has its negative x-axis intercept at 1 +  S , yielding the endogenous Ca 2+ binding ratio of the cell. The slope of this fit is the inverse of the Ca 2+ extrusion rate  (Eq. ( 11)). The y-axis intercept yields the time constant  endo for the decay of the Ca 2+ transient as it would appear in the cell without exogenous buffer.

Analysis of calcium buffering

Dye loading was monitored at 360 nm excitation, the isosbestic point of fura-2. Frames were taken at 30 s intervals (3 ms exposure time). We estimated the intracellular fura-2 concentration for different times during the loading curve, assuming that cells were fully loaded when the fluorescence reached a plateau and the fura-2 concentration in the cell and in the pipette is equal (100 or 200 µM) (Lee et al., 2000b).

During fura-2 loading, voltage-activated Ca 2+ influx was induced by short spike trains, which were elicited by current pulses (500pA, 500ms) from a holding potential of about -70 mV. To monitor the induced Ca 2+ transients ratiometrically, pairs of images at 340 nm (12 ms exposure time) and 380 nm (4 ms exposure time) excitation were acquired at 10 Hz for ~20 s.

Typically 3 -4 Ca 2+ transients were induced during loading at different intracellular fura-2

concentrations. i 2 T ) 1 ( S B transient ] Ca [ ] B [ : 1 loading S          small const const baseline for      

Ratiometric Data Analysis

Getting the standard error of the ratiometric estimator Given measurements at two excitation wavelengths (340 and 380 nm), 𝐴𝐷𝑈 ̂340 and 𝐴𝐷𝑈 ̂380 (these are pooled photon counts over the P pixels of the ROI) and concomitant measurements 𝐴𝐷𝑈 ̂𝐵,340 and 𝐴𝐷𝑈 ̂𝐵,380 (pooled photon counts over the 𝑃 𝐵 pixels of the "background region"), what we refer to as the "classical" ratiometric estimator at time 𝑡 𝑖 , is [START_REF] Joucla | Quantitative estimation of calcium dynamics from ratiometric measurements: a direct, nonratioing method[END_REF] (Eq. 8):

𝐶𝑎 ̂(𝑡 𝑖 ) = 𝐾 𝑒𝑓𝑓 𝑟(𝑡 𝑖 ) -𝑅 𝑚𝑖𝑛 𝑅 𝑚𝑎𝑥 -𝑟(𝑡 𝑖 ) , (1)
where 𝐾 𝑒𝑓𝑓 , 𝑅 𝑚𝑖𝑛 and 𝑅 𝑚𝑎𝑥 are calibrated parameters (assumed exactly known for now) and where (Joucla et al., 2010) (Eq. 6):

𝑟(𝑡 𝑖 ) = (𝑇 𝑒,340 ) -1 (𝐴𝐷𝑈 ̂340 /𝑃 -𝐴𝐷𝑈 ̂𝐵,340 /𝑃 𝐵 ) (𝑇 𝑒,380 ) -1 (𝐴𝐷𝑈 ̂380 /𝑃 -𝐴𝐷𝑈 ̂𝐵,380 /𝑃 𝐵 ) , (2)
where 𝑇 𝑒,𝜆 is the flash duration at wavelength 𝜆. Our model for the fluorescence intensity at each wavelength is [START_REF] Joucla | Quantitative estimation of calcium dynamics from ratiometric measurements: a direct, nonratioing method[END_REF] (Eq. 2a and 2b):

𝐹 340 = { [𝐹𝑢𝑟𝑎] 𝑡𝑜𝑡𝑎𝑙 𝜙 𝐾 𝐹𝑢𝑟𝑎 + [𝐶𝑎 2+ ] (𝑅 𝑚𝑖𝑛 𝐾 𝑒𝑓𝑓 + 𝑅 𝑚𝑎𝑥 [𝐶𝑎 2+ ]) + 𝑠 𝐵,340 } 𝑇 𝑒,340 𝑃, (3)
and

𝐹 380 = { [𝐹𝑢𝑟𝑎] 𝑡𝑜𝑡𝑎𝑙 𝜙 𝐾 𝐹𝑢𝑟𝑎 + [𝐶𝑎 2+ ] (𝐾 𝑒𝑓𝑓 + [𝐶𝑎 2+ ]) + 𝑠 𝐵,380 } 𝑇 𝑒,380 𝑃, (4)
where 𝑠 𝐵,𝜆 is the auto-fluorescence at wavelength 𝜆 -assumed homogeneous among the 𝑃 pixels of the ROI-, 𝐾 𝐹𝑢𝑟𝑎 is a calibrated parameter, [𝐹𝑢𝑟𝑎] 𝑡𝑜𝑡𝑎𝑙 𝜙, is the total (bound plus free) concentration of fura-2 in the cell multiplied by a dimensionless experiment specific parameter, 𝜙, lumping together the quantum efficiency, the neurite thickness, etc.

Under our assumptions [START_REF] Joucla | Quantitative estimation of calcium dynamics from ratiometric measurements: a direct, nonratioing method[END_REF] (Eq. 5) we have (we don't use a " ̂" here since we are dealing with a random variable, not a realization of it):

𝐴𝐷𝑈 𝜆 (𝑡 𝑖 ) = 𝐺𝐹 𝜆 (𝑡 𝑖 ) + 𝜖𝐺√𝐹 𝜆 (𝑡 𝑖 ) + 𝜎 𝑟𝑒𝑎𝑑-𝑜𝑢𝑡 2 , ( 5
)
where 𝐺 is the camera gain, 𝜎 𝑟𝑒𝑎𝑑-𝑜𝑢𝑡 2 is its read-out variance, 𝐹 𝜆 (𝑡 𝑖 ) is given by Eq. 3 and 4

and where 𝜖 ∼ 𝒩(0,1) (𝜖 is a Gaussian random variable with mean 0 and variance 1). In words: 𝐴𝐷𝑈 𝜆 (𝑡 𝑖 ) has a Gaussian distribution with mean 𝐺𝐹 𝜆 (𝑡 𝑖 ) and variance 𝐺 2 (𝐹 𝜆 (𝑡 𝑖 ) + 𝜎 𝑟𝑒𝑎𝑑-𝑜𝑢𝑡 2 ).

To determine the variance of 𝐴𝐷𝑈 𝜆 (𝑡 𝑖 ) we need to know 𝐹 𝜆 (𝑡 𝑖 ) and for that, we need to know

[𝐶𝑎 2+ ](𝑡 𝑖 )
precisely what we want to estimate. But 𝐸𝐴𝐷𝑈 𝜆 (𝑡 𝑖 ), the expected value of 𝐴𝐷𝑈 𝜆 (𝑡 𝑖 ), is 𝐺𝐹 𝜆 (𝑡 𝑖 ) so we can use the observed value 𝐴𝐷𝑈 ̂𝜆(𝑡 𝑖 ) of 𝐴𝐷𝑈 𝜆 (𝑡 𝑖 ) as an estimate for 𝐺𝐹 𝜆 (𝑡 𝑖 ) -in Eq. 5 we plug-in 𝐴𝐷𝑈 ̂𝜆(𝑡 𝑖 ) for 𝐺𝐹 𝜆 (𝑡 𝑖 )leading to:

𝜎 ̂𝐴𝐷𝑈 𝜆(𝑡 𝑖 ) 2 = 𝐺𝐴𝐷𝑈 ̂𝜆(𝑡 𝑖 ) + 𝐺 2 𝜎 𝑟𝑒𝑎𝑑-𝑜𝑢𝑡 2 ≈ 𝜎 𝐴𝐷𝑈 𝜆 (𝑡 𝑖 ) 2 , (6)
In practice, when we are dealing with several pixels making a ROI, we must multiply the . We could use the propagation of uncertainty (or error propagation) together with Eq. 2 and 1 for that, but we can also use a Monte Carlo approach: we draw a thousand quadruple of vectors (𝐴𝐷𝑈 340 [𝑗] (𝑡 𝑖 ), 𝐴𝐷𝑈 380 [𝑗] (𝑡 𝑖 ), 𝐴𝐷𝑈 𝐵,340 [𝑗] (𝑡 𝑖 ), 𝐴𝐷𝑈 𝐵,380 [𝑗] (𝑡 𝑖 )) (𝑗 = 1, … ,1000) from four independent Gaussian distributions of the general form: plug-in these quadruples into Eq. 2 giving us 1000 𝑟 [𝑗] (𝑡 𝑖 ) before plugging in the latter into

Eq. 1 leading to 1000 𝐶𝑎 ̂[𝑗] (𝑡 𝑖 ). The empirical variance of these simulated observations will be used as 𝜎 ̂𝐶𝑎 ̂(𝑡 𝑖 )

2

.

The method proposed here is slightly less rigorous than the "direct approach" of [START_REF] Joucla | Quantitative estimation of calcium dynamics from ratiometric measurements: a direct, nonratioing method[END_REF], but it is far more flexible since it does not require an independent estimation or measurement of [𝐹𝑢𝑟𝑎] 𝑡𝑜𝑡𝑎𝑙 . In the present study we also chose to consider the calibrated parameters 𝐾 𝑒𝑓𝑓 , 𝑅 𝑚𝑖𝑛 and 𝑅 𝑚𝑎𝑥 as known. Note that the values for these parameters are only estimates (necessarily imprecise since the calibration procedure is subject to errors). However, since the same batch of fura-2 was used for all experiments, they should all have the same (systematic) error. We are not trying to get the exact standard error of the calcium dynamics parameters but to show the difference between two protocols ("classical whole-cell" versus beta-escin perforated whole-cell). Ignoring the uncertainty on the calibrated parameters makes our estimates less variable and helps to make the difference if it exists, clearer. For a true calcium buffering capacity study, the errors on the calibrated parameters should be accounted for, and it would be straightforward to do it with our approach, simply by drawing also 𝐾 𝑒𝑓𝑓 , 𝑅 𝑚𝑖𝑛 and 𝑅 𝑚𝑎𝑥 values from Gaussian distributions centered on their respective estimates with a SD given by the standard errors.

Objective or "cost" function

We know now how to get both sequences of [𝐶𝑎 2+ ] estimates (Eq. 1) and standard errors from sequences of 𝐴𝐷𝑈 measurements. We also consider a calcium dynamics model (Eq. 8- The next task is to find the value of 𝜃 ̂ that makes Ω as small as possible; this a classical nonlinear weighted least squares problem (see [START_REF] Nielsen | Introduction to Optimization and Data Fitting[END_REF]). The advantage of this setting is that if we can find 𝜃 ̂ (that is, if the minimum exists and is unique), we also get:

1. Confidence intervals for each element of 𝜃 through their standard error (in fact, more than that, since we get the full covariance matrix).

2. The distribution from which Ω(𝜃 ̂) should have been drawn if the model, 𝐶𝑎(𝑡; 𝜃 ̂), is indeed correct: a 𝜒 2 distribution with 𝑁 -#𝜃 degrees of freedom, where #𝜃 stands for the number of elements of 𝜃 that are set from the fitted data.

Point 2 gives us an objective basis to accept or reject a fit, an essential ingredient for the procedure presented here.

Before going further, we should consider that 𝜃 in the form just defined contains 7 elements among which, 2 are supposed to be known, 𝐾 𝐹 and 𝐹 𝑡𝑜𝑡𝑎𝑙 . That leaves 5 values to extract from the transient. This is not straightforward since the latter seems, at least at first glance, close to a mono-exponential: 𝐶𝑎(𝑡; 𝐶𝑎 𝑠𝑡𝑒𝑎𝑑𝑦 , ∆, 𝜏) = 𝐶𝑎 𝑠𝑡𝑒𝑎𝑑𝑦 + Δ exp (-𝑡 𝜏

) (from which only 3 parameters can be obtained). Therefore, we follow [START_REF] Neher | Calcium gradients and buffers in bovine chromaffin cells[END_REF] The fit in point 2 is done with weighted non-linear least squares using 𝐶𝑎(𝑡; 𝐶𝑎 𝑠𝑡𝑒𝑎𝑑𝑦 , ∆, 𝜏) (Eq. 12) instead of 𝐶𝑎(𝑡; 𝜃) in the cost function definition (Eq. 10). Each fit gives 𝜏(𝜅 𝐹 ) and 𝜎 ̂𝜏 ̂(𝜅 𝐹 ) together with a goodness of fit criterion. The (𝜏(𝜅 𝐹 ), 𝜎 ̂𝜏 ̂(𝜅 𝐹 ) ) are used in point 3 for a weighted linear least squares fit that also leads to confidence intervals on the estimated parameters and to a goodness of fit criterion.

The codes doing the "heavy" part of the analysis were written in C. Some higher-level scripts were written in Python 3. The codes, their documentation, the data, and the way the codes were applied to the data can be found on GitHub (https://github.com/christophe-pouzat/hesset-al-beta-escin-2019) as well as on zenodo (https://zenodo.org/record/3238248). The weighted linear and non-linear least squares routines, as well as the random number generators, are the ones of the Gnu Scientific Library (GSL). The Lenvenberg-Marquardt implementation of the GSL was used for the non-linear least squares. Optimization was ended based on convergence criteria looking at the relative change of each estimated parameter value and at the gradient norm (see the GSL documentation); as a safeguard, a maximum number of iterations was set, but all the optimizations did actually stop because the convergence criteria were met.

Following [START_REF] Neher | Calcium gradients and buffers in bovine chromaffin cells[END_REF], our analysis of the calcium transients relies on the linearization (https://en.wikipedia.org/wiki/Linearization) of an assumed calcium dynamics (Eq. 9 to 15, [START_REF] Neher | Calcium gradients and buffers in bovine chromaffin cells[END_REF]). The key hypothesis that is made in all linearizations is that the dynamical system under investigation (here the calcium concentration) is submitted to a small perturbation from its resting state. What constitutes a "small enough" perturbation for the linearization to be valid in practice depends, in the specific case considered here, on parameters we do not know (total endogenous buffer concentration, endogenous buffer dissociation constant, etc.), but we can be sure that the closer we get to baseline, that is the later we look during the transient, the more likely the "small enough" requirement is to be met. If the linearization is valid, the transient (more precisely the portion of the transient that is fitted) must exhibit a mono-exponential decay, something we can rigorously test. In order to get a procedure usable on every transient and considering the fact that the peak response of the first transient can be large -6 times the baseline value, not really a "small" perturbation -we decided, before seeing the data, to start all the fits from the time at which the transient had lost 50% of its peak value. This is obviously arbitrary but it does not affect our results as the reader can check by downloading our data and our code before rerunning the whole analysis using a different location from which to start the fit. Accordingly, the number of data points used for each fit varies a bit from transient to transient. This choice of fitting from the "half-peak time" was made before seeing the data. It is important to realize that once such a choice has been made, we are constrained, the fit is done automatically, and the residual sum of squares (RSS) is compatible or not with the theoretical 𝜒 2 distribution.

Since we used a triple wavelength (340, 380 and 360) protocol for each point of the transient, we have access to an independent estimate of [𝐹𝑢𝑟𝑎] 𝑡𝑜𝑡𝑎𝑙 .We saw, as in [START_REF] Joucla | Quantitative estimation of calcium dynamics from ratiometric measurements: a direct, nonratioing method[END_REF] (Fig. 5), that the latter is not constant during the transients. When doing the 𝜏 versus 𝜅 𝐹 linear regression we therefore computed systematically three regressions using the minimum, the mean and the maximum observed values of [𝐹𝑢𝑟𝑎] 𝑡𝑜𝑡𝑎𝑙 during the transient. We always report the results of regression leading to the smallest RSS (best fit). As the reader can check from our GitHub repository, none of our conclusions depend on this. We emphasize again that since our method generates standard errors for 𝜏, the linear regressions are accepted or rejected on an objective basis (the RSS is compatible or not with the relevant theoretical 𝜒 2 distribution). (D) The decay time constants were plotted against the Ca 2+ binding ratios of fura-2 (κ B ). κ B was calculated from the intracellular fura-2 concentration, the K d of fura-2, and the resting concentration of free intracellular Ca 2+ . The solid red line is the linear fit to the data. An estimate of κ S was obtained as the negative x-axis intercept. The Ca 2+ extrusion rate is estimated from the slope of the fit and the endogenous decay time constant from the intercept with the y-axis. The resulting Ca 2+ handling parameters of this example are: τ endo : 0.7 ± 0.3 s, γ: 91.5 ± 10.4, s -1 , κ s : 66.4 ± 24.7.

Figures

)

  read-out variance by the number of pixels used. With the resulting 𝜎 ̂𝐴𝐷𝑈 𝜆(𝑡 𝑖 ) 2 we can work with, to get 𝜎 ̂𝑟(𝑡 𝑖 ) 2 and 𝜎 ̂𝐶𝑎 ̂(𝑡 𝑖 ) 2

Figure 1 .

 1 Figure 1. Basic identifying electrophysiological and morphological properties of

Figure 2 .

 2 Figure 2. Perforated patch recordings of dopaminergic substantia nigra neurons using -escin

Figure 3 .

 3 Figure 3. -escin mediated perforated patch clamp recordings allow stable recordings. (A,B)

Figure 4 .

 4 Figure 4. Intracellular Ca 2+ concentrations remain stable during -escin mediated perforated patch clamp recordings. Relative changes in intracellular Ca 2+ concentration during the time

Figure 5 .Figure 6 .

 56 Figure 5. Large molecules diffuse very slowly through the perforated membrane during patch

Figure 7 .

 7 Figure 7. The goodness of fit comparison between Ca 2+ transients recorded in the whole-cell

  where 𝐶𝑎(𝑡) stands for [𝐶𝑎 2+ ] 𝑓𝑟𝑒𝑒 at time 𝑡, 𝑣 is the volume of the neurite-within which where 𝐹 stands for the fluorophore and 𝐸 for the endogenous buffer. 𝐾 𝐸 , 𝐹 𝑡𝑜𝑡𝑎𝑙 , 𝐾 𝐹 , 𝐶𝑎(𝑡 = 0)) are known, we can substitute these values in Eq. 8, solve it (numerically) and predict the time course of calcium at any positive time. Let us write 𝐶𝑎(𝑡; 𝜃) as this solution of Eq. 8. In addition to this model, we have used the results of the previous section, on a set of 𝑁 time points 𝑡 𝑖 , a collection of (𝐶𝑎 ̂(𝑡 𝑖 ), 𝜎 ̂𝐶𝑎 ̂(𝑡 𝑖 ) ). We can then define a first objective or cost function:

	diffusion effects can be neglected -and
		𝑗(𝐶𝑎(𝑡)) ≡ 𝛾(𝐶𝑎(𝑡) -𝐶𝑎 𝑠𝑡𝑒𝑎𝑑𝑦 ),	(9)
	is the model of calcium extrusion -𝐶𝑎 𝑠𝑡𝑒𝑎𝑑𝑦 is the steady-state [1] [𝐶𝑎 2+ ] 𝑓𝑟𝑒𝑒 -and:
	𝜅 𝐹 (𝐶𝑎(𝑡)) ≡	𝐹 𝑡𝑜𝑡𝑎𝑙 𝐾 𝐹 (𝐾 𝐹 + 𝐶𝑎(𝑡))	2	𝑎𝑛𝑑 𝜅 𝐸 (𝐶𝑎(𝑡)) ≡	𝐸 𝑡𝑜𝑡𝑎𝑙 𝐾 𝐸 (𝐾 𝐸 + 𝐶𝑎(𝑡))	2 ,
	If the values of 𝜃 ≡ (𝐶𝑎 𝑠𝑡𝑒𝑎𝑑𝑦 , , 𝐸 𝑡𝑜𝑡𝑎𝑙 , Ω(𝜃) ≡ ∑ ( 𝛾 𝑣 𝑁 𝐶𝑎 ̂(𝑡 𝑖 ) -𝐶𝑎(𝑡 𝑖 ; 𝜃) 𝜎 ̂𝐶𝑎 ̂(𝑡 𝑖 ) 𝑖=1	2 )	.	(10)
	10):
	𝑑𝐶𝑎(𝑡) 𝑑𝑡	(1 + 𝜅 𝐹 (𝐶𝑎(𝑡)) + 𝜅 𝐸 (𝐶𝑎(𝑡))) +	𝑗(𝐶𝑎(𝑡)) 𝑣	= 0,	(8)

  and consider an approximation of Eq. 8 where 𝐶𝑎(𝑡) can be written as 𝐶𝑎 𝑠𝑡𝑒𝑎𝑑𝑦 + 𝛿𝐶𝑎(𝑡) and where 𝛿𝐶𝑎(𝑡) is small enough for a first order expansion of 𝐶𝑎(𝑡) in the vicinity of 𝐶𝑎 𝑠𝑡𝑒𝑎𝑑𝑦 to be valid. If we do this first order expansion, neglecting the communication between the pipette and the cytosol during the transient we get (setting 𝜌 = 0) in[START_REF] Neher | Calcium gradients and buffers in bovine chromaffin cells[END_REF] the dependence of the 𝜅 on 𝐶𝑎 𝑠𝑡𝑒𝑎𝑑𝑦 has been made implicit. We see that if we can obtain several transients with different values of [𝐹𝑢𝑟𝑎] 𝑡𝑜𝑡𝑎𝑙 leading to several values of 𝜅 𝐹 , we can exploit the linear dependence of 𝜏 on both 𝜅 to estimate 𝜅 𝐸 by extrapolation. This is the essence of the added buffer approach. The implementation of the latter requires:1.Get several (known) values of [𝐹𝑢𝑟𝑎] 𝑡𝑜𝑡𝑎𝑙 and therefore 𝜅 𝐹 in the recorded neuron.2. Trigger a calcium entry at each of these 𝜅 𝐹 values and fit the tail of the transient with a mono-exponential (Eq. 12)-fitting the tail ensures that the linear approximation of Eq. 8 by Eq. 11 is valid-.3. Plot 𝜏 versus 𝜅 𝐹 and fit a straight line to these points. The slope gives

						𝑣 𝛾	and the x-axis
	intercept gives 𝜅 𝐸 .
	(Eq. 15):
	𝛿𝐶𝑎(𝑡) 𝑑𝑡	= -	𝛾 𝑣 1 + 𝜅 𝐹 (𝐶𝑎 𝑠𝑡𝑒𝑎𝑑𝑦 ) + 𝜅 𝐸 (𝐶𝑎 𝑠𝑡𝑒𝑎𝑑𝑦 )	𝛿𝐶𝑎(𝑡).	(11)
	Leading to:
	𝐶𝑎(𝑡; 𝐶𝑎 𝑠𝑡𝑒𝑎𝑑𝑦 , ∆, 𝜏) = 𝐶𝑎 𝑠𝑡𝑒𝑎𝑑𝑦 + Δ𝑒𝑥𝑝 -	𝑡 𝜏	,	(12)
	where the time constant 𝜏 is defined by:
			𝜏(𝜅 𝐹 ) ≡	1 + 𝜅 𝐹 + 𝜅 𝐸 𝛾	,	(13)
				𝑣