Marco Canepari
email: marco.canepari@univ-grenoble-alpes.fr

Is Purkinje neuron hyperpolarisation important for cerebellar synaptic plasticity? A retrospective and prospective analysis

Keywords: Purkinje neuron, membrane potential, calcium channels, synaptic plasticity, Climbing fibre, Parallel fibres

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Introduction

In the vertebrate nervous system, the cerebellum integrates the incoming motor and sensory information to produce a feedback output from the cerebellar nuclei to the motor system that controls movements [START_REF] Eccles | The cerebellum as a neuronal machine[END_REF]. This output is precisely shaped by the inhibitory input to cerebellar nuclei neurons from cerebellar Purkinje neurons (PNs) in the cerebellar cortex, which also processes incoming motor information. Specifically, PNs fire action potentials at variable frequencies and these frequencies are modulated by the motor information received from the mossy fibres and processed by cerebellar granule cells (CGCs) that send excitatory parallel fibre (PF) inputs to PNs. The PN firing frequency can either increase or decrease during motor activity according to the balance of excitation versus feed-forward inhibition by molecular layer interneurons (MLIs) also excited by the same PF inputs [START_REF] Jelitai | Dendritic excitation-inhibition balance shapes cerebellar output during motor behaviour[END_REF]. But the whole cerebellar system also receives a large sensory excitatory input through the climbing fibres (CFs) from the brainstem inferior olive targeting both PNs and cerebellar nuclei neurons. According to the early theory proposed by Marr [START_REF] Marr | A theory of cerebellar cortex[END_REF] and Albus [START_REF] Albus | A theory of cerebellar function[END_REF], the principal role of the CF input is to provide an "error" signal that weakens the concomitantly active PFs input, in this way permitting a continuous learning of fine motor coordination. This theory was finally validated by the discovery that pairing PF and CF activity can induce long-term depression (LTD) of activated PF synapses [START_REF] Ito | Climbing fiber induced depression of both mossy fiber responsiveness and glutamate sensitivity of cerebellar Purkinje cells[END_REF]. Since then, it became evident that physiological PF synaptic depression is complex and highly heterogeneous in the way it is triggered throughout the cerebellar cortex [START_REF] Suvrathan | Depressed by Learning-Heterogeneity of the Plasticity Rules at Parallel Fiber Synapses onto Purkinje Cells[END_REF], and that PF plasticity must be bidirectional with synaptic weakening compensated by counterbalancing mechanisms of long-term potentiation (LTP) [START_REF] Jörntell | Synaptic memories upside down: bidirectional plasticity at cerebellar parallel fiber-Purkinje cell synapses[END_REF][START_REF] Vogt | On the induction of postsynaptic granule cell-Purkinje neuron LTP and LTD[END_REF].

All CF-mediated forms of homosynaptic plasticity in PNs, although different, require the ability of the CF input to target a triggering signal exclusively to the concomitantly activated PF synapses. This signal that coincidently detects the occurrence of PF and CF inputs and that can initiate a biochemical process leading to a change in the postsynaptic strength is a transient elevation of intracellular Ca 2+ concentration [START_REF] Wang | Coincidence detection in single dendritic spines mediated by calcium release[END_REF]. More precisely, when the occurrence of a CF excitatory postsynaptic potential (EPSP) is preceded by PF activation, the Ca 2+ transient associated with the CF-EPSP, at the location of PF synapses only, is larger than the Ca 2+ transient associated with the CF-EPSP without PF activation (unpaired CF-EPSP), and it is referred as "supralinear" Ca 2+ signal [START_REF] Brenowitz | Associative short-term synaptic plasticity mediated by endocannabinoids[END_REF][START_REF] Ouares | The Origin of Physiological Local mGluR1 Supralinear Ca 2+ Signals in Cerebellar Purkinje Neurons[END_REF]. Whereas CGCs form a complex circuit of ~175,000 contacts in the dendritic spines of PN dendrites [START_REF] Napper | Number of parallel fibre synapses on an individual Purkinje cell in the cerebellum of the rat[END_REF], the CF input is a unique synapse with over 500 realising sites [START_REF] Silver | Locus of frequency-dependent depression identified with multipleprobability fluctuation analysis at rat climbing fibre-Purkinje cell synapses[END_REF] targeting the soma and the initial dendritic segment and generating a large depolarisation throughout the dendritic arborisation leading to a widespread Ca 2+ transient [START_REF] Knöpfel | Climbing Fibre Responses in Olivo-cerebellar Slice Cultures. II. Dynamics of Cytosolic Calcium in Purkinje Cells[END_REF][START_REF] Miyakawa | Calcium transients evoked by climbing fiber and parallel fiber synaptic inputs in guinea pig cerebellar Purkinje neurons[END_REF][START_REF] Canepari | Dendritic spike saturation of endogenous calcium buffer and induction of postsynaptic cerebellar LTP[END_REF][START_REF] Ouares | Two Distinct Sets of Ca 2+ and K + Channels Are Activated at Different Membrane Potentials by the Climbing Fiber Synaptic Potential in Purkinje Neuron Dendrites[END_REF]. The mechanisms permitting the CF-EPSP to mediate localised Ca 2+ signals are therefore fundamental for PF synaptic plasticity. A supralinear Ca 2+ signal, i.e. a larger increase in intracellular Ca 2+ concentration can be generated by a larger flux of Ca 2+ into the cytosol, but also by an increase of free Ca 2+ due to a transient saturation of the endogenous Ca 2+ buffer (ECB). PN dendrites have an exceptionally high ECB [START_REF] Fierro | High endogenous calcium buffering in Purkinje cells from rat cerebellar slices[END_REF] formed mainly by two mobile high-affinity Ca 2+ binding proteins: Calbindin-D28k and Parvalbumin [START_REF] Bastianelli | Distribution of calcium-binding proteins in the cerebellum[END_REF][START_REF] Schmidt | Mutational analysis of dendritic Ca2+ kinetics in rodent Purkinje cells: role of parvalbumin and calbindin D28k[END_REF]. It has been demonstrated that PN-ECB can be saturated by Ca 2+ photolysis [START_REF] Maeda | Supralinear Ca2+ signaling by cooperative and mobile Ca2+ buffering in Purkinje neurons[END_REF] or by fast Ca 2+ influx mediated by the PF local depolarisation [START_REF] Canepari | Dendritic spike saturation of endogenous calcium buffer and induction of postsynaptic cerebellar LTP[END_REF]. More interestingly, it was recently shown that the slow Ca 2+ influx induced by PF-activated type-1 metabotropic glutamate receptors (mGluR1s) can transiently saturate the ECB [START_REF] Ouares | The Origin of Physiological Local mGluR1 Supralinear Ca 2+ Signals in Cerebellar Purkinje Neurons[END_REF]. Thus, PF inputs through mGluR1 activation can determine the condition to target a CF-mediated Ca 2+ signal to the synaptic sites where synaptic plasticity has to occur by locally amplifying the Ca 2+ transient associated with the CF-EPSP. The source of this Ca 2+ transient, however, is not the same at any membrane potential (V m ) [START_REF] Ouares | Two Distinct Sets of Ca 2+ and K + Channels Are Activated at Different Membrane Potentials by the Climbing Fiber Synaptic Potential in Purkinje Neuron Dendrites[END_REF]. When the PN is at a quiescent hyperpolarising state, the CFmediated depolarisation transient activates T-type voltage-gated Ca 2+ channels (VGCCs) and A-type voltage-gated K + channels that limit the activation of P/Q-type VGCCs preventing dendritic Ca 2+ spikes.

When instead the PN is at a more depolarised firing state, both T and A channels inactivate and the CFmediated depolarisation activates P/Q-type VGCCs enabling Ca 2+ spikes. Notably, both Ca 2+ sources are potentiated by PF-mediated mGluR1 activation, but not in the same way. Ca 2+ spikes via P/Q channels are boosted by mGluR1-mediated inactivation of A channels [START_REF] Otsu | Activity-dependent gating of calcium spikes by Atype K+ channels controls climbing fiber signaling in Purkinje cell dendrites[END_REF] whereas T-channels are directly potentiated by mGluR1s at hyperpolarised states [START_REF] Hildebrand | Functional coupling between mGluR1 and Cav3.1 T-type calcium channels contributes to parallel fiber-induced fast calcium signaling within Purkinje cell dendritic spines[END_REF][START_REF] Isope | Contributions of T-type voltage-gated calcium channels to postsynaptic calcium signaling within Purkinje neurons[END_REF]. Notably, the potentiation of T-type VGCCs is mediated by the same mGluR1 pathway responsible for the slow mGluR1-mediated Ca 2+ influx [START_REF] Canepari | Evidence for protein tyrosine phosphatase, tyrosine kinase, G-protein regulation of the parallel fiber metabotropic slow EPSC of rat cerebellar Purkinje neurons[END_REF] suggesting a possible co-localisation of the two processes. Thus, beside the voltage independent amplification through local saturation of the ECB, the supralinear Ca 2+ signal associated with a CF-EPSP includes an increase in Ca 2+ influx through the plasma membrane and has a different origin depending on whether the initial V m is at a hyperpolarised or at a firing state. This fact suggests that the standing V m of PNs plays a pivotal role in synaptic plasticity and cerebellar learning. This review is a retrospective and prospective analysis of the possible role of PN V m in synaptic plasticity. In particular, I focus on the putative role of PN hyperpolarisation since this state is likely episodic while the supralinear Ca 2+ signal appears to specifically target activated PF synapses. In the following sections, I first review in detail the biophysical origin of mGluR1-dependent supralinear Ca 2+ signals. Then, I review some of the most important findings on in vivo activity and provide interpretations in light of a possible role of the PN V m , I analyse the possible activation of K + channels that may generate episodes of hyperpolarisation and I speculate on how episodic hyperpolarisation can relate to the rules that govern synaptic plasticity. Finally, I propose a perspective of future research directions to tackle the different hypotheses analysed in this review.

Overview of the origin of mGluR1 supralinear Ca 2+ signals in Purkinje neurons

I start this analysis with an overview of the recent findings that motivated this review. The first findings concern the biophysical mechanisms governing the dendritic V m and Ca 2+ transients associated with the CF-EPSP at different initial V m [START_REF] Ouares | Two Distinct Sets of Ca 2+ and K + Channels Are Activated at Different Membrane Potentials by the Climbing Fiber Synaptic Potential in Purkinje Neuron Dendrites[END_REF]. Modifying a previous model [START_REF] Anwar | Controlling Ca2+-activated K+ channels with models of Ca2+ buffering in Purkinje cells[END_REF] to match combined V m and Ca 2+ imaging data, the response of a dendritic compartment to a CF depolarising transient was explained in detail with the synergistic activation of six channels. These channels are divided in two different sets that are selectively activated at different initial V m . By focussing on the two types of VGCCs and on the A-type K + channel (Fig. 1a), when the dendrite is hyperpolarised a CF-EPSP activates T-type VGCCs and A-type K + channels channels that limit the dendritic depolarising transient below ~10 mV, preventing the activation of P/Q-type VGCCs. When in contrast the dendrite is depolarised, T and A channels are inactivated and the CF-EPSP can eventually activate P/Q channels eliciting Ca 2+ spikes. A second study [START_REF] Ouares | The Origin of Physiological Local mGluR1 Supralinear Ca 2+ Signals in Cerebellar Purkinje Neurons[END_REF] has unravelled the mechanisms that allow the Ca 2+ transient associated with a CF-EPSP, following PF activity, to be amplified locally at the sites of pre-activated PF synapses (Fig. 1b). A first mechanism, independent of the initial V m , is determined by the Ca 2+ influx through an mGluR1-activated non-selective cation conductance [START_REF] Canepari | The conductance underlying the parallel fibre slow EPSP in rat cerebellar Purkinje neurones studied with photolytic release of l-glutamate[END_REF][START_REF] Canepari | Ca2+ ion permeability and single-channel properties of the metabotropic slow EPSC of rat Purkinje neurons[END_REF]. This channel is believed to be the C3-type transient receptor potential (TRPC3) [START_REF] Hartmann | TRPC3 channels are required for synaptic transmission and motor coordination[END_REF], but it was also shown that mGluR1s can activate GluD2 delta "orphan" glutamate receptors [START_REF] Ady | Type 1 metabotropic glutamate receptors (mGlu1) trigger the gating of GluD2 delta glutamate receptors[END_REF]. During the precise time window of mGluR1 action following PF bursting activity, Ca 2+ entering the cell through this channel binds to ECBs locally lowering the buffer capacity of the cell. Since this is produced by synaptically activated receptors, it can be segregated to activated synapses, i.e. it can co-localise with activated spines [START_REF] Schmidt | Spine neck geometry determines spino-dendritic cross-talk in the presence of mobile endogenous calcium binding proteins[END_REF] depending on their geometry [START_REF] Schmidt | Spino-dendritic cross-talk in rodent Purkinje neurons mediated by endogenous Ca2+-binding proteins[END_REF]. The mechanism of local ECB saturation amplifies both the Ca 2+ signal mediated by T-type VGCCs at hyperpolarised states and the Ca 2+ signal mediated by P/Q-type VGCCs at depolarised states combining, in the two cases, with two distinct further mechanisms contributing to the supralinear Ca 2+ signals. The mechanism of boosting P/Q channel Ca 2+ influx, due to the larger mGluR1-dependent inactivation of A channels [START_REF] Otsu | Activity-dependent gating of calcium spikes by Atype K+ channels controls climbing fiber signaling in Purkinje cell dendrites[END_REF] is not local. In contrast, the mechanism of boosting T channel Ca 2+ influx, due to the direct mGluR1-dependent potentiation of these channels [START_REF] Hildebrand | Functional coupling between mGluR1 and Cav3.1 T-type calcium channels contributes to parallel fiber-induced fast calcium signaling within Purkinje cell dendritic spines[END_REF], is likely occurring exclusively at activated PF synapses. Altogether, these findings lead to the question on whether the V m is actually a key parameter in the induction of synaptic plasticity and, in particular, whether the apparently rare states of V m hyperpolarisation are fundamental episodes necessary for cerebellar learning. In the next section I review the state-of-the art literature on the V m in vivo in light of the possible occurrence of hyperpolarised states.

Evidence of physiological hyperpolarisation of Purkinje neurons in vivo

Direct measurement of V m by in vivo patch clamp recordings in PNs were obtained both from anesthetised [START_REF] Loewenstein | Bistability of cerebellar Purkinje cells modulated by sensory stimulation[END_REF][START_REF] Kitamura | Targeted patch-clamp recordings and single-cell electroporation of unlabeled neurons in vivo[END_REF][START_REF] Arlt | Microcircuit Rules Governing Impact of Single Interneurons on Purkinje Cell Output In Vivo[END_REF] and awake animals [START_REF] Jelitai | Dendritic excitation-inhibition balance shapes cerebellar output during motor behaviour[END_REF]. In anesthetised rodents, independently of the anaesthesia procedure, PN V m is characterised by bistability consisting on the ability of alternating between hyperpolarised quiescent states and depolarised firing states associated with simple spikes [START_REF] Loewenstein | Bistability of cerebellar Purkinje cells modulated by sensory stimulation[END_REF], similarly to what was reported in vitro [START_REF] Williams | Membrane potential bistability is controlled by the hyperpolarization-activated current I(H) in rat cerebellar Purkinje neurons in vitro[END_REF]. While PN bistability is due to intrinsic membrane properties (see for example [START_REF] Yuen | Bistability in cerebellar Purkinje cell dendrites modelled with high-threshold calcium and delayed-rectifier potassium channels[END_REF]), a CF input can switch the V m from one state to the other [START_REF] Loewenstein | Bistability of cerebellar Purkinje cells modulated by sensory stimulation[END_REF]. Notably, the Ca 2+ transient associated with a CF-EPSP depends on the PN state [START_REF] Kitamura | Targeted patch-clamp recordings and single-cell electroporation of unlabeled neurons in vivo[END_REF], in the same way as it depends on the initial V m in brain slices [START_REF] Ouares | Two Distinct Sets of Ca 2+ and K + Channels Are Activated at Different Membrane Potentials by the Climbing Fiber Synaptic Potential in Purkinje Neuron Dendrites[END_REF]. Yet, the ability of PNs to switch between two states, supposedly associated with the possibility to activate either T-type or P/Q-type VGCCs, does not answer the question on how frequent the hyperpolarised state occurs. Less invasive in vivo extracellular recordings have shown that the frequency of simple spike firing is on average higher in awake mice with respect to anesthetised mice, ranging from ~30 Hz to ~100 Hz [START_REF] Arancillo | In vivo analysis of Purkinje cell firing properties during postnatal mouse development[END_REF]. The variability in spiking frequency reflects the cerebellar modules where modules expressing zebrin exhibit lower frequencies [START_REF] Zhou | Cerebellar modules operate at different frequencies[END_REF] and this specificity is mediated by TRPC3 channels [START_REF] Wu | TRPC3 is a major contributor to functional heterogeneity of cerebellar Purkinje cells[END_REF]. In all cases, these relatively high spiking frequencies indicate that hyperpolarising states lasting more than 100 ms must be, under physiological conditions, only episodic. This first conclusion leads to the second important question, i.e. how synaptic activity may lead to these episodic hyperpolarised states. While a CF input alone can theoretically switch from one PN state to another, the occurrence of complex spikes in vivo, indicative of a CF input, does not silence simple spiking activity (see for example [START_REF] Arancillo | In vivo analysis of Purkinje cell firing properties during postnatal mouse development[END_REF]) since the transient hyperpolarisation mediated by K + channels activation, in particular Ca 2+ -activated BK channels [START_REF] Rancz | Dendritic calcium spikes are tunable triggers of cannabinoid release and short-term synaptic plasticity in cerebellar Purkinje neurons[END_REF][START_REF] Ouares | Two Distinct Sets of Ca 2+ and K + Channels Are Activated at Different Membrane Potentials by the Climbing Fiber Synaptic Potential in Purkinje Neuron Dendrites[END_REF] is lasting only a few milliseconds. Thus, episodes of sustained hyperpolarisation can be only induced either by PF activity alone or by concomitant PF activity and CF inputs. A possibility is that hyperpolarisation episodes have a network origin being caused by outward Cl - currents through GABA A receptors activated by cerebellar interneurons. PF axons projecting onto PNs also excite MLIs producing feed-forward inhibition to the same PNs and this mechanism regulates behaviours associated with synaptic plasticity and spiking activity in PNs [START_REF] Wulff | Synaptic inhibition of Purkinje cells mediates consolidation of vestibulo-cerebellar motor learning[END_REF][START_REF] Blazquez | GABA-A Inhibition Shapes the Spatial and Temporal Response Properties of Purkinje Cells in the Macaque Cerebellum[END_REF]. Furthermore, the contribution of feed-forward inhibition in determining the firing rate is enhanced during self-paced locomotion where some PNs increase their spiking frequency whereas others decrease it [START_REF] Jelitai | Dendritic excitation-inhibition balance shapes cerebellar output during motor behaviour[END_REF]. This finding leads to a straightforward hypothesis illustrated in Fig. 2. In PNs where the firing frequency increases, and therefore the standing V m becomes more depolarised, a CF-EPSP would likely activate P/Q-type VGCCs whereas in PNs where the firing frequency decreases, and the standing V m becomes more hyperpolarised, a CF-EPSP would more likely activate T-type VGCCs. But how MLI inputs to PNs affect the CF response and synaptic plasticity? These questions were investigated by Rowan et al. [START_REF] Rowan | Graded Control of Climbing-Fiber-Mediated Plasticity and Learning by Inhibition in the Cerebellum[END_REF] who showed that concomitant optogenetic stimulation of MLIs reduced the CF-mediated Ca 2+ transient and when this protocol was paired with PF stimulation it changed the polarity of PF long-term synaptic plasticity from depression to potentiation (i.e. from LTD to LTP). In addition, while CF stimulation combined with head rotation produced an adaptive increase in the vestibulo-ocular reflex, concomitant optogenetic stimulation of MLIs reversed the effect into an adaptive decrease. The straightforward possible interpretation according to the scheme of Fig. 2 would be that MLIs drive the PN to a hyperpolarised state changing the polarity of PF synaptic plasticity. However, activation of molecular layer inhibitory synapses does not only hyperpolarise the V m , but also increases the dendritic membrane conductance decreasing the propagation of the CF depolarising transient throughout the dendrites. Hence, while molecular layer "disinhinibition" increases the CF-mediated Ca 2+ transient during behaviourally-induced PF activity, MLI activation suppresses local supralinear Ca 2+ signals [START_REF] Gaffield | Inhibition gates supralinear Ca 2+ signaling in Purkinje cell dendrites during practiced movements[END_REF], a result in striking contrast with the large supralinear Ca 2+ signal observed with hyperpolarised states obtained by somatic current injection [START_REF] Ouares | The Origin of Physiological Local mGluR1 Supralinear Ca 2+ Signals in Cerebellar Purkinje Neurons[END_REF]. Thus, while MLIs may in principle allow, in a portion of PNs, the activation of T-type VGCCs, the results obtained in these studies can be alternatively attributed to a shunting inhibition of the CF depolarising wave. The specific investigation of the occurrence of hyperpolarising episodes in vivo is only at a preliminary stage. It must be taken into account that T-type VGCCs recover from inactivation in a few milliseconds [START_REF] Kuo | Recovery from inactivation of t-type ca2+ channels in rat thalamic neurons[END_REF] and, therefore, an hyperpolarisation episode synchronising with a CF-EPSP can be as short as 100 ms to allow a local supralinear Ca 2+ signal mediated by these channels. For example, within feed-forward inhibition, a spike from one single MLI can alone transiently inhibit PN firing [START_REF] Arlt | Microcircuit Rules Governing Impact of Single Interneurons on Purkinje Cell Output In Vivo[END_REF], but MLIs can be also excited by glutamate spillover from a CF input [START_REF] Szapiro | Multiple climbing fibers signal to molecular layer interneurons exclusively via glutamate spillover[END_REF]. The hypothesis that episodes of PN hyperpolarisation playing a role in synaptic plasticity are determined by activation of Cl -implies that the underlying mechanisms are mainly governed by the cerebellar network. Yet, an alternative hypothesis is that hyperpolarisation episodes are determined by the history of the PN signals and governed by intrinsic mechanisms leading to activation of K + currents. This hypothesis is analysed in the next section.

Mechanisms of episodic hyperpolarisation of Purkinje neurons mediated by K + channels

A prolonged activation of K + conductance can drive PN V m towards the reversal potential of K + . This can be produced by a large Ca 2+ transient activating Ca 2+ -gated K + channels or by K + channels activated by other mechanisms. Among the different types of Ca 2+ -gated K + channels, the "small-conductance" K + (SK) channel is reported to play a role in different cellular processes in PNs, including in the calcium transients in dendritic spines [START_REF] Hosy | SK2 channel expression and function in cerebellar Purkinje cells[END_REF]. Although the expression of this channel decreases with development [START_REF] Cingolani | Developmental regulation of smallconductance Ca2+-activated K+ channel expression and function in rat Purkinje neurons[END_REF], it was recently reported that SK channels are highly expressed in dendritic spines co-localising with mGluR1s [START_REF] Luján | SK2 Channels Associate With mGlu 1α Receptors and Ca V 2.1 Channels in Purkinje Cells[END_REF] and that they are down-regulated during cerebellar learning increasing PN excitability [START_REF] Titley | Intrinsic Excitability Increase in Cerebellar Purkinje Cells after Delay Eye-Blink Conditioning in Mice[END_REF].

In general, SK channels can provide a medium-slow afterhyperpolarisation after PF bursting. The other type of Ca 2+ -gated K + channel expressed in PNs is the "big-conductance" K + (BK) channel that co-localise with P/Q-type VGCCs [START_REF] Indriati | Quantitative localization of Cav2.1 (P/Q-type) voltage-dependent calcium channels in Purkinje cells: somatodendritic gradient and distinct somatic coclustering with calcium-activated potassium channels[END_REF] and is activated by dendritic Ca 2+ spikes [START_REF] Rancz | Dendritic calcium spikes are tunable triggers of cannabinoid release and short-term synaptic plasticity in cerebellar Purkinje neurons[END_REF][START_REF] Ouares | Two Distinct Sets of Ca 2+ and K + Channels Are Activated at Different Membrane Potentials by the Climbing Fiber Synaptic Potential in Purkinje Neuron Dendrites[END_REF]. But BK channels are also activated by Ca 2+ influx via inositol 1,4,5-triphosphate (InsP3) receptors activated by mGluR1s with a faster kinetics with respect to the slow Ca 2+ influx [START_REF] Canepari | Kinetic, pharmacological and activity-dependent separation of two Ca2+ signalling pathways mediated by type 1 metabotropic glutamate receptors in rat Purkinje neurones[END_REF]. Notably, since InsP3 receptors are highly expressed in PN dendrites [START_REF] Maeda | Structural and functional characterization of inositol 1,4,5-trisphosphate receptor channel from mouse cerebellum[END_REF], several studies have linked PF activation of mGluR1s to Ca 2+ release from internal stores via these receptors [START_REF] Finch | Local calcium signalling by inositol-1,4,5-trisphosphate in Purkinje cell dendrites[END_REF][START_REF] Takechi | A new class of synaptic response involving calcium release in dendritic spines[END_REF] leading to the hypothesis that this signal is the trigger for PF-LTD [START_REF] Khodakhah | Induction of long-term depression and rebound potentiation by inositol trisphosphate in cerebellar Purkinje neurons[END_REF][START_REF] Inoue | Type 1 inositol 1,4,5-trisphosphate receptor is required for induction of long-term depression in cerebellar Purkinje neurons[END_REF]. While Ca 2+ release from internal stores does not contribute to the CF dependent supralinear Ca 2+ signal [START_REF] Ouares | The Origin of Physiological Local mGluR1 Supralinear Ca 2+ Signals in Cerebellar Purkinje Neurons[END_REF], InsP3 receptor activation requires dendritic depolarisation [START_REF] Canepari | Kinetic, pharmacological and activity-dependent separation of two Ca2+ signalling pathways mediated by type 1 metabotropic glutamate receptors in rat Purkinje neurones[END_REF] raising the hypothesis that this event may play a physiological role in generating hyperpolarisation short episodes.

Finally, Ca 2+ release from stores activating Ca 2+ -gated K + channels can also occur via ryanodine receptors following an intense stimulation [START_REF] Kano | Ryanodine receptor-mediated intracellular calcium release in rat cerebellar Purkinje neurones[END_REF], a mechanism that was also associated with the induction of plasticity [START_REF] Kohda | Ca2+ release from Ca2+ stores, particularly from ryanodine-sensitive Ca2+ stores, is required for the induction of LTD in cultured cerebellar Purkinje cells[END_REF]. Altogether, the ensemble of possible mechanisms leading to activation of Ca 2+ -gated K + channels is summarised in Fig. 3a. In addition to the pathways analysed above, K + channels can be activated by other synaptic mechanisms. PF activity activates type-7 metabotropic glutamate receptors (mGluR7) and this mechanism can transiently silence PN activity after cerebellar learning [START_REF] Johansson | Activation of a Temporal Memory in Purkinje Cells by the mGluR7 Receptor[END_REF]. Specifically, mGluR7s couple with Kir3/GIRK K + channels [START_REF] Niswender | A novel assay of Gi/o-linked G protein-coupled receptor coupling to potassium channels provides new insights into the pharmacology of the group III metabotropic glutamate receptors[END_REF] that are expressed in PNs [START_REF] Aguado | Cell type-specific subunit composition of G protein-gated potassium channels in the cerebellum[END_REF]. The same channels are activated by inhibitory synapses by coupling with metabotropic GABA B receptors [START_REF] Tabata | GABAergic activation of an inwardly rectifying K+ current in mouse cerebellar Purkinje cells[END_REF]. Altogether, the two mechanisms leading to activation of Kir3/GIRK channels are summarised in Fig. 3b. In light of this general analysis on what can produce hyperpolarisation transients during PF or CF activity, it is crucial to analyse, next, how these mechanisms may relate to the different protocols leading to synaptic plasticity. This issue is addressed in the next section.

How hyperpolarisation can relate to the induction of synaptic plasticity

In the classical theory, the CF input combined with PF activity induces PF-LTD, a prediction that found numerous experimental conformations in brain slices experiments (see for example [START_REF] Wang | Coincidence detection in single dendritic spines mediated by calcium release[END_REF][START_REF] Konnerth | Brief dendritic calcium signals initiate long-lasting synaptic depression in cerebellar Purkinje cells[END_REF][START_REF] Karachot | Stimulus parameters for induction of long-term depression in in vitro rat Purkinje cells[END_REF][START_REF] Safo | Endocannabinoids control the induction of cerebellar LTD[END_REF]). In the opposite direction, repetitive PF activity has been associated with induction of PF-LTP (see for example [START_REF] Canepari | Dendritic spike saturation of endogenous calcium buffer and induction of postsynaptic cerebellar LTP[END_REF][START_REF] Wong | A new form of cerebellar long-term potentiation is postsynaptic and depends on nitric oxide but not cAMP[END_REF][START_REF] Coesmans | Bidirectional parallel fiber plasticity in the cerebellum under climbing fiber control[END_REF]), although strong PF stimulation can alone induce PF-LTD [START_REF] Hartell | Strong activation of parallel fibers produces localized calcium transients and a form of LTD that spreads to distant synapses[END_REF]. These findings led to the initial hypothesis that the amplitude of a Ca 2+ transient determines the direction of the PF long-term synaptic plasticity according to an "inverse BMC rule" [START_REF] Jörntell | Synaptic memories upside down: bidirectional plasticity at cerebellar parallel fiber-Purkinje cell synapses[END_REF] with respect to the equivalent BMC rule believed to govern cortical synaptic plasticity [START_REF] Bienenstock | Theory for the development of neuron selectivity: orientation specificity and binocular interaction in visual cortex[END_REF]. Yet, later, it became clear that the induction of PF synaptic plasticity is more complex [START_REF] Vogt | On the induction of postsynaptic granule cell-Purkinje neuron LTP and LTD[END_REF] depending on many parameters. For instance, mGluR1-dependent PF-LTD [START_REF] Conquet | Motor deficit and impairment of synaptic plasticity in mice lacking mGluR1[END_REF][START_REF] Aiba | Deficient cerebellar long-term depression and impaired motor learning in mGluR1 mutant mice[END_REF][START_REF] Hartell | Induction of cerebellar long-term depression requires activation of glutamate metabotropic receptors[END_REF] is also critically dependent on the geometry of activated PF fibres [START_REF] Marcaggi | Short-and long-term depression of rat cerebellar parallel fibre synaptic transmission mediated by synaptic crosstalk[END_REF] since mGluR1s, located at the periphery of synaptic spines [START_REF] Baude | The metabotropic glutamate receptor (mGluR1 alpha) is concentrated at perisynaptic membrane of neuronal subpopulations as detected by immunogold reaction[END_REF], are activated only by glutamate spillover by adjacent terminals [START_REF] Marcaggi | Endocannabinoid signaling depends on the spatial pattern of synapse activation[END_REF] pointing out a potential importance of the arrangement of activated PFs forming "beams" capable of activating spines in close proximity [START_REF] Bower | The organization of cerebellar cortical circuitry revisited: implications for function[END_REF]. The current opinion, however, is now questioning whether general rules of synaptic plasticity induction physiologically apply. First, PF-LTD is not always necessary for vestibular-ocular reflex adaptation [START_REF] Welsh | Normal motor learning during pharmacological prevention of Purkinje cell long-term depression[END_REF][START_REF] Schonewille | Reevaluating the role of LTD in cerebellar motor learning[END_REF], being replaced by a rebound potentiation of inhibitory synapses to PNs [START_REF] Tanaka | Long-term potentiation of inhibitory synaptic transmission onto cerebellar Purkinje neurons contributes to adaptation of vestibulo-ocular reflex[END_REF]. Conversely, tactile stimulation in vivo can evoke a long-lasting increase in PN firing while synergistically reducing the MLI firing thanks to multiple plasticity mechanisms occurring at different cellular levels [START_REF] Ramakrishnan | Tactile Stimulation Evokes Long-Lasting Potentiation of Purkinje Cell Discharge In Vivo[END_REF]. Second, PF-LTP can be expressed either at presynaptic terminals [START_REF] Salin | Cyclic AMP mediates a presynaptic form of LTP at cerebellar parallel fiber synapses[END_REF][START_REF] Qiu | An NMDA receptor/nitric oxide cascade in presynaptic parallel fiber-Purkinje neuron longterm potentiation[END_REF] or at postsynaptic spines [START_REF] Canepari | Dendritic spike saturation of endogenous calcium buffer and induction of postsynaptic cerebellar LTP[END_REF][START_REF] Coesmans | Bidirectional parallel fiber plasticity in the cerebellum under climbing fiber control[END_REF], and the different forms of PF long-term synaptic plasticity, including PF-LTD, involve presynaptic mechanisms such as NMDA receptor activation [START_REF] Casado | Involvement of presynaptic N-methyl-D-aspartate receptors in cerebellar longterm depression[END_REF][START_REF] Bouvier | Burst-Dependent Bidirectional Plasticity in the Cerebellum Is Driven by Presynaptic NMDA Receptors[END_REF] triggering release of nitric oxide [START_REF] Ogasawara | Nitric oxide regulates input specificity of long-term depression and context dependence of cerebellar learning[END_REF] and endocannabinoid receptor activation [START_REF] Carey | Presynaptic CB1 receptors regulate synaptic plasticity at cerebellar parallel fiber synapses[END_REF]. Yet, these presynaptic mechanisms are regulated by retrograde postsynaptic mechanisms [START_REF] Kreitzer | Retrograde inhibition of presynaptic calcium influx by endogenous cannabinoids at excitatory synapses onto Purkinje cells[END_REF][START_REF] Maejima | Presynaptic inhibition caused by retrograde signal from metabotropic glutamate to cannabinoid receptors[END_REF][START_REF] Wang | Long-term potentiation at cerebellar parallel fiber-Purkinje cell synapses requires presynaptic and postsynaptic signaling cascades[END_REF], indicating that the synergy between presynaptic activity, controlled only by the principal motor input, and postsynaptic signalling, controlled also by the CF input, is a crucial determinant of synaptic plasticity. Finally, it was found that the rules for induction of PF plasticity are different in the flocculus and in the vermis [START_REF] Suvrathan | Timing Rules for Synaptic Plasticity Matched to Behavioral Function[END_REF], in the first case being narrowly tuned for a precise interval associated with oculomotor learning, while in the second case being highly variable possibly matching single cell functions. Thus, within the actual complicated scenarios of multiple rules of synaptic plasticity induction [START_REF] Hirano | Regulation and Interaction of Multiple Types of Synaptic Plasticity in a Purkinje Neuron and Their Contribution to Motor Learning[END_REF], the ability of V m to rapidly switch between two states characterised by different Ca 2+ sources and pathways may be a critical factor and it may allow the CF input to drive synaptic plasticity in different directions [START_REF] Zang | Climbing Fibers Provide Graded Error Signals in Cerebellar Learning[END_REF].

Further theoretical studies on cerebellar plasticity have challenged the concept of "motor error", since the CF input carries a sensory and not a motor information [START_REF] Dean | Visual awareness and the cerebellum: possible role of decorrelation control[END_REF]. Since a "sensory error" must be fully uncorrelated from the motor information in order to instruct the motor system in an independent way, the classical learning rule based on simple concomitant occurrence of PF and CF inputs has been replaced by a decorrelation learning rule based on positive or negative correlation of PF and CF activities [START_REF] Dean | Decorrelation control by the cerebellum achieves oculomotor plant compensation in simulated vestibulo-ocular reflex[END_REF]. This alternative learning theory brings up an intriguing question: can the PN V m detect the type of correlation?

A recent theoretical and experimental analysis has suggested that correlation detection is possible with two consecutive CF inputs separated by 100 ms [START_REF] Bouvier | Cerebellar learning using perturbations[END_REF]. In addition, in a more recent study, PF-LTD induction required the occurrence of two consecutive CF inputs at similar intervals [START_REF] Titley | Complex spike clusters and false-positive rejection in a cerebellar supervised learning rule[END_REF]. Under certain circumstances, it is plausible that the first CF input during or just after PF activity can switch the standing V m from a depolarised state to an hyperpolarised state, priming the dendrite for the second CF input according to a scheme shown in Fig. 4a. The biophysical mechanism underlying this scenario can be the activation of Ca 2+ -activated K + channels by the first paired CF input that activates P/Q channels. An alternative scenario, illustrated in Fig. 4b, can occur if the PF activity alone hyperpolarises the V m of the PN, for instance through MLI-mediated feed-forward inhibition or by activation of mGluR7s. Thus, the first CF input would activate T channels while the second CF input, occurring when the V m of the PN is restored to a standing depolarised state, would activate P/Q channels. Notably, the Ca 2+ transient associated with the second CF input would be amplified by the mGluR1-mediated saturation of the ECB, regardless of whether the Ca 2+ influx is mediated by the T-type or the P/Q-type VGCC. Interestingly, these hypotheses are consistent with the idea of "intrinsic" learning and memory in the cerebellum recently proposed in two perspective articles [START_REF] Titley | Toward a Neurocentric View of Learning[END_REF][START_REF] Johansson | Intrinsic memory of temporal intervals in cerebellar Purkinje cells[END_REF]. While these hypotheses are both, at this stage, only pure speculations, the careful analysis of the V m should be included in all future investigations of synaptic plasticity induction protocols.

Perspectives

To answer the general question formulated in the title of this review, three specific questions must be addressed by future experimental investigations: 1. What can cause activity dependent hyperpolarisation? 2. When does hyperpolarisation occur in vivo? 3. How does hyperpolarisation correlate with induction of synaptic plasticity? While activation of K + channels can be in principle still investigated in brain slices, these questions must be addressed by measuring dendritic V m in vivo. Interestingly, Roome and Kuhn [START_REF] Roome | Simultaneous dendritic voltage and calcium imaging and somatic recording from Purkinje neurons in awake mice[END_REF] achieved simultaneous dendritic V m and Ca 2+ imaging using an organic voltage indicator, combined with somatic electrical recording from spontaneously active PNs in awake mice. The preliminary results reported in this study suggest that, at a fine time scale, the spatiotemporal processing in PN dendrites is characterised by a high degree of complexity. The most promising approach is the one proposed by [START_REF] Villette | Ultrafast Two-Photon Imaging of a High-Gain Voltage Indicator in Awake Behaving Mice[END_REF], permitting to record dendritic activity for long periods and to associate these recordings with protocols of cerebellar learning. In summary, the recent findings on the critical role in V m in selecting Ca 2+ sources associated with a CF input pave the way for future investigations using the information on V m as a possible key for cerebellar synaptic plasticity.

Villette et al. based on genetically encoded indicators

Fig. 1 .

 1 Fig. 1. Ca 2+ sources responsible for Ca 2+ transients associated with CF-EPSPs and mechanisms underlying supralinear amplification. a Example of CF Somatic V m associated with CF inputs at initial hyperpolarised V m (black trace), activating T and A channels, and at resting V m (gray trace), activating P/Q channels. b At both states, the slow Ca 2+ current (I Ca ) gated by mGluR1 activation following PF activity saturates the ECB amplifying the CF Ca 2+ transient. At the hyperpolarised state (top), this mechanism combines with the co-localised mGluR1-dependent potentiation of T-type VGCCs activated by the CF. At the resting state (bottom), this mechanism combines with the boosted CF activation of P/Q-type VGCCs, but the two mechanisms are not co-localised.

Fig. 2 .

 2 Fig. 2. Hypothesis on the role of feed-forward inhibition in determining the Ca 2+ source associated with the CF input. On the left, two PNs (PN1 and PN2) firing at the same frequency when the mouse is at rest; on the right, PN1 increases the firing frequency with movement following excitation provided by CGC inputs while PN2 decreases the firing frequency following a dominance of MLI feed-forward inhibition; PN1 would favour activation of P/Q channels following a CF input while PN2 would favour activation of T channels following a CF input.

Fig. 3 .

 3 Fig. 3. Possible activity dependent pathways activating K + channels and producing hyperpolarising episodes. a Possible pathways leading to activation of Ca 2+ -activated K + channels. PF activity can activate SK channels either by directly inducing Ca 2+ influx or through ryanodine receptors (RyR). PF activity can activate BK channels through RyR activation or through activation of mGluR1s that trigger Ca 2+ release from stores. BK channels can be also activated by Ca 2+ influx induced by the CF input either alone or paired with PF activity (PF+CF). b Possible pathways leading to activation of Kir3/GIRK K + channels. PF activity can activate mGluR7s that activate Kir3/GIRK channels. PF activity or CF inputs (via glutamate spillover) can activate MLIs that activate GABA B receptors leading to activation of Kir3/GIRK channels.

Fig. 4 .

 4 Fig. 4. Possible scenarios of plasticity induction protocols based on two consecutive CF inputs. a PF activity further depolarises PN V m and the first CF input activates P/Q channels leading to an episode of hyperpolarisation (for instance by activating Ca 2+ -activated K + channels); the second CF input occurring when PN V m is hyperpolarised activates T channels. b PF activity hyperpolarises PN V m (for instance by MLI-mediated feed-forward inhibition or by mGluR7s) and the first CF input activates T channels; the second CF input occurring when PN V m is restored to a depolarised state P/Q channels. In both cases the Ca 2+ transient associated with the second CF input is locally amplified by mGluR1-mediated ECB saturation, but the sequence of the two Ca 2+ signals is different for the first and second CF input.

This work was supported by the Agence Nationale de la Recherche through the Labex Ion Channels Science and Therapeutics: program number ANR-11-LABX-0015.


Karima Ait Ouares

Nadia Jaafari

Nicola Kuczewski

Marco Canepari
email: marco.canepari@univ-grenoble-alpes.fr

Imaging Native Calcium Currents in Brain Slices

Keywords:

Introduction

Optical measurements have been historically designed to monitor the electrical activity of the nervous system, a task where the use of electrode techniques has clear limitations [START_REF] Braubach | Historical Overview and General Methods of Membrane Potential Imaging[END_REF]. In the last two decades, the development of new organic voltage sensitive dyes (VSD), in parallel with the progress of devices to excite and detect fluorescence [START_REF] Davies | Light sources and cameras for standard in vitro membrane potential and high-speed ion imaging[END_REF], allowed optical recordings of sub-cellular membrane potential (V m ) changes < 1 mV with a signal-to-noise ratio (S/N) comparable to that of patch clamp recordings [START_REF] Canepari | Imaging Inhibitory Synaptic Potentials Using Voltage Sensitive Dyes[END_REF]. This achievement suggested that voltage imaging can be used to investigate voltagedependent proteins, in particular voltage-gated ion channels, in their physiological environment. The principal function of an ion channel is to allow an ion flux through a membrane, i.e. to produce an ionic current. Thus, the study of the biophysics of ion channels is routinely performed by measurements of ionic currents in single-electrode or two-electrode voltage clamp [START_REF] Sakmann | Patch clamp techniques for studying ionic channels in excitable membranes[END_REF]. A way to investigate the biophysics of isolated native ion channels is to perform excised patches from ex-vivo membranes [START_REF] Gray | Rectification of single GABA-gated chloride channels in adult hippocampal neurons[END_REF].

Alternatively, ion channels can be expressed in foreign cells such as oocytes or mammalian cell lines (6) and studied by using patch clamp techniques (7). Yet, the physiological role and function of voltage-gated ion channels must be investigated in their natural environment, i.e. in their native cellular compartment and during physiological changes of V m . To this purpose, the voltage clamp electrode approach has serious limitations for several reasons. First, the ionic current is measured by maintaining the cell at a given artificial V m and even if the cell is dynamically clamped the V m change is never a physiological signal [START_REF] Antic | Simultaneous recordings of voltage and current waveforms from dendrites[END_REF]. Second, the current measured with the electrode is the summation of the filtered currents from all different cellular regions, including remote regions where V m is unclamped, and no information is available on the site of origin of the current [START_REF] Williams | Direct measurement of somatic voltage clamp errors in central neurons[END_REF]. Third, different ionic currents contribute to the physiological change of V m producing a functional coupling among the different ion channels [START_REF] Hodgkin | Currents carried by sodium and potassium ions through the membrane of the giant axon of Loligo[END_REF]. Thus, a single native ionic current must be pharmacologically isolated from the total current mediated by the other channels, but the block of these channels will make the V m change non-physiological.

In the last few years, we designed a novel approach to measure physiological Ca 2+ currents from neurons in brain slices [START_REF] Jaafari | Imaging Fast Calcium Currents beyond the Limitations of Electrode Techniques[END_REF]. The method is based on fast Ca 2+ optical measurements using lowaffinity indicators that can be combined with sequential [START_REF] Canepari | Combining voltage and calcium imaging from neuronal dendrites[END_REF] or simultaneous [START_REF] Vogt | High-resolution simultaneous voltage and Ca 2+ imaging[END_REF] V m optical recordings. The latter measurements can be calibrated in mV [START_REF] Vogt | Combining membrane potential imaging with L-glutamate or GABA photorelease[END_REF] using cell-specific protocols.

Individual cells are loaded with Ca 2+ and V m indicators using a patch clamp recording. In contrast to voltage-clamp current measurements, the current approach permits independent recordings of the V m change and of the Ca 2+ influx, i.e. the study of voltage gating during physiological V m changes. Since the Ca 2+ current is reconstructed by the measurement of Ca 2+ locally binding to an indicator, this approach provides information on channels in different areas of the cell with a spatial resolution as good as the optical recording allows. Finally, the Ca 2+ current is recorded without blocking all Na + and K + channels that are necessary to produce the physiological V m change. The principle of obtaining an optical measurement of a fast Ca 2+ current is based on the analysis of the dye-Ca 2+ binding reaction in a cell, a scenario initially studied by Kao and Tsien (15). According to their theoretical estimates and to our recent empirical measurements [START_REF] Jaafari | Using simultaneous voltage and calcium imaging to study fast Ca2+ channels[END_REF], the relaxation time of the dye-Ca 2+ binding reaction is less than 200 µs for low-affinity indicators with equilibrium constant (K D ) ≥ 10 µM such as Oregon Green BAPTA-5N (OG5N, K D = 35 µM, 17) or Fura-FF (K D = 10 µM, 18). Therefore, a fast Ca 2+ current with duration of a few milliseconds can be reliably tracked by low-affinity indicators if fluorescence is acquired at sufficiently high speed. The goal of this methodological article is to provide an exhaustive tool for those scientists aiming at performing this type of measurement. The next section addresses in detail the problem of extracting the Ca 2+ current kinetics from Ca 2+ fluorescence measurements under different cellular buffering conditions. The following section is devoted to the technical aspects of how to set up combined V m and Ca 2+ optical measurements and to calibrate V m signals on an absolute scale. We then illustrate some examples of combined V m and Ca 2+ current measurements and we finally discuss how to correctly interpret the results and how to use this information to significantly advance our knowledge on Ca 2+ channels function. All data shown here were from experiments performed at the Laboratoire Interdisciplinaire de Physique and approved by the Isere prefecture (Authorisation n. 38 12 01). These experiments were performed at 32-34°C using brain slices from 21-40 postnatal days old C57Bl6 mice of both genders.

Extracting Ca 2+ current kinetics from Ca 2+ fluorescence measurements

Biophysical foundations of Ca 2+ currents imaging

An optical measurement of a Ca 2+ signal is ultimately a measurement of the Ca 2+ indicator bound to Ca 2+ ions, which is proportional to the Ca 2+ fractional change of fluorescence (∆F/F 0 ) if the indicator is not saturated. If the kinetics of the Ca 2+ -binding reaction of the indicator is slower than the kinetics of the Ca 2+ source, and imaging is performed at higher rate, the time-course of Ca 2+ ∆F/F 0 essentially tracks the kinetics of the chemical reaction. Alternatively, if the kinetics of the Ca 2+ -binding reaction is faster than the kinetics of the Ca 2+ source, the Ca 2+ ∆F/F 0 signal tracks the kinetics of the Ca 2+ source. It follows that the equilibration (or relaxation) time of the Ca 2+ -indicator binding reaction is a crucial variable to use the technique to investigate the biophysics and the physiology of the Ca 2+ source. The relaxation of the Ca 2+ -binding reactions for early indicators was studied by Kao and Tsien [START_REF] Kao | Ca2+ binding kinetics of fura-2 and azo-1 from temperature-jump relaxation measurements[END_REF] who established that the rate of association for all these molecules is limited by diffusion leading to an association constant of ~6•10 8 M -1 s -1 . Thus, both the equilibrium constant (K D ) and the equilibrium time are determined by the dissociation constant, i.e. the lower is the affinity of the indicator the shorter is its equilibrium time. We have empirically demonstrated that indicators with K D ≥ 10 µM such as OG5N or FuraFF have relaxation time < 200 µs [START_REF] Jaafari | Using simultaneous voltage and calcium imaging to study fast Ca2+ channels[END_REF]. Since the kinetics of activation and deactivation of voltage-gated Ca 2+ channels (VGCCs) during physiological changes of V m (for instance action potentials), is governed by the kinetics of the V m transient, it follows that the relaxation time for those indicators is shorter than the duration of the Ca 2+ influx. Hence, since Ca 2+ binds to the indicator linearly in time, the Ca 2+ ∆F/F 0 is proportional to the integral of the Ca 2+ influx, i.e. to the integral of the Ca 2+ current. In the cell, however, Ca 2+ simultaneously binds to proteins that form the endogenous buffer and this binding is competing with the binding to the indicator. An endogenous buffer can be, in principle, at least as fast as the indicator in equilibrating. In this case, only a fraction of Ca 2+ is bound to the indicator, but this fraction is proportional to the total Ca 2+ entering the cell and therefore to the integral of the Ca 2+ current. Alternatively, an endogenous buffer can equilibrate over a time scale that is longer than the duration of the Ca 2+ current. In this case, Ca 2+ first binds to the dye and later to the endogenous buffer, implying that part of Ca 2+ moves from the indicator to the endogenous buffer during its relaxation time. Under this condition, the Ca 2+ ∆F/F 0 is not linear with the integral of the Ca 2+ current over this time scale. To clarify this important concept we make use of two simple computer simulations shown in Fig. 1, produced by a model that takes into account the chemical reactions as well as an extrusion mechanism re-establishing the initial Ca 2+ conditions over a time scale >100 ms. We analyse what hypothetically can happen if a Ca 2+ current with Gaussian shape occurs in a cell filled with 2 mM OG5N. In the first simulation (Fig. 1a), the cell has only 1 mM of a fast endogenous buffer behaving with the same association constant of the indicator and K D = 10 µM. In the second simulation (Fig. 1b), the cell has additional 400 µM of a slower endogenous buffer with association rate ~3 times slower than that of the indicator and K D = 0.2 µM. In the first case, the time derivative of the Ca 2+ ∆F/F 0 signal matches the kinetics of the Ca 2+ current (Fig. 1a). In contrast, in the presence of the slower buffer, the time derivative of the Ca 2+ ∆F/F 0 signal has a negative component and does not match the kinetics of the Ca 2+ current (Fig. 1b). In the next two paragraphs, we present the analysis strategies that can be applied to extract the kinetics of Ca 2+ currents from Ca 2+ imaging recordings.

The case of linearity between Ca 2+ influx and Ca 2+ fluorescence changes

The proteins expressed in a cell determine whether or not the time course of the Ca 2+ ∆F/F 0 signal is linear with the kinetics of the Ca 2+ current. As previously demonstrated [START_REF] Jaafari | Imaging Fast Calcium Currents beyond the Limitations of Electrode Techniques[END_REF], in the case of linear behaviour, the Ca 2+ ∆F/F 0 signal must reach its peak and remain constant for a few milliseconds afterwards, i.e. for the entire duration of the current. As shown in the simulation of Fig. 1a, the kinetics of Ca 2+ extrusion producing a slow decrease of the Ca 2+ ∆F/F 0 signal has negligible effect on the time derivative. Thus, the estimate of the Ca 2+ current kinetics is reliably obtained by the calculation of the time derivative of the Ca 2+ ∆F/F 0 signal. This calculation, however, requires the signal noise to be smaller than the signal change between two consecutive samples. The classical way to achieve this necessary condition is to apply to the Ca 2+ ∆F/F 0 signal a "smoothing algorithm", i.e. a temporal filter that reduces the noise of the signal with minimal distortion of its kinetics. At 20 kHz acquisition rate, we have found that the Savitky-Golay algorithm ( 19) is an optimal filtering tool permitting noise reduction of the signal without significant temporal distortion using time-windows of up to 20-30 samples [START_REF] Jaafari | Imaging Fast Calcium Currents beyond the Limitations of Electrode Techniques[END_REF]. The applicability of this strategy has however limitations, i.e. if the signal or the region of measurement are too small, or if the light is too dim, the smoothing of the signal might not be sufficient to reduce the noise down to the level permitting calculation of the time derivative. In this case, the alternative strategy to apply consists in fitting the raw or the filtered Ca 2+ ∆F/F 0 signal with a model function obtaining a noiseless curve that mimics the time course of the Ca 2+ ∆F/F 0 signal. A simple choice of function that resembles the time course of the Ca 2+ ∆F/F 0 transient is the sigmoid. In particular, we found that the product of three sigmoid functions always provides an excellent fit of the Ca 2+ ∆F/F 0 signal associated with a backpropagating action potential in CA1 hippocampal pyramidal neurons [START_REF] Jaafari | Using simultaneous voltage and calcium imaging to study fast Ca2+ channels[END_REF]. As shown in the example of Fig. 2a both strategies are faithful in correctly calculating the time derivative of the ∆F/F 0 signal. In this example, a CA1 hippocampal pyramidal neuron was filled with 2 mM OG5N and the dendritic Ca 2+ ∆F/F 0 signal associated with a backpropagating action potential was recorded at 20 kHz and averaged over 16 trials. This high sampling frequency was necessary to avoid signal aliasing and therefore distortion of the kinetics of the current. The filtering strategy is the straightforward approach that enables the calculation of the time derivative, but it produces a curve with noise. The noise can be reduced (if possible) by increasing the number of trials to average or by enlarging the dendritic area from where fluorescence is averaged. The fitting strategy is less direct but it produces a noiseless curve and it is therefore the only possible approach when the noise of the Ca 2+ ∆F/F 0 signal is above a certain level, as quantitatively estimated in an original report [START_REF] Jaafari | Using simultaneous voltage and calcium imaging to study fast Ca2+ channels[END_REF]. In particular, this is the case when the current must be extracted from single trials or when the recording is obtained from small or relatively dim regions.

The case of nonlinearity between Ca 2+ influx and Ca 2+ fluorescence changes

The method of estimating the kinetics of a Ca 2+ current by calculating the Ca 2+ ∆F/F 0 time derivative fails when Ca 2+ unbinds from the indicator over a time scale that is longer than the current duration, but sufficiently short to distort the estimate of Ca 2+ influx dynamics by fluorescence measurement. In other words, this method fails when the Ca 2+ ∆F/F 0 signal decays rapidly, after correction for bleaching, generating a negative component in its time derivative. Such a situation occurs, for example, where slow buffering is produced by Calbindin-D28k [START_REF] Nägerl | Binding kinetics of calbindin-D(28k) determined by flash photolysis of caged Ca(2+)[END_REF][START_REF] Airaksinen | Ataxia and altered dendritic calcium signalling in mice carrying a targeted nullmutation of the calbindin D28k gene[END_REF] and Parvalbumin [START_REF] Lee | Kinetics of Ca2+ binding to parvalbumin in bovine chromaffin cells: implications for [Ca2+] transients of neuronal dendrites[END_REF][START_REF] Schmidt | Mutational analysis of dendritic Ca2+ kinetics in rodent Purkinje cells: role of parvalbumin and calbindin D28k[END_REF]. As shown in the example of Fig. 2b, the Ca 2+ ∆F/F 0 signal associated with a climbing fibre excitatory postsynaptic potential (EPSP), recorded at 5 kHz from a dendritic region and averaged over four trials, decays rapidly after its maximum resulting in a negative component of its time derivative.

The distortion from the linear behaviour produced by the slow buffers can be compensated by taking into account the kinetics of Ca 2+ unbinding from the indicator. We have recently developed a successful method to achieve this goal [START_REF] Ouares | A generalised method to estimate the kinetics of fast Ca 2+ currents from Ca 2+ imaging experiments[END_REF]. The strategy is based on fitting the decay time of the Ca 2+ ∆F/F 0 signal with the result of a computer simulation of a model with a slow buffer. Initially the input current is the Gaussian function fitting the rising phase of the time derivative (that is still a good approximation of the initial part of the current). The kinetic parameters and the concentration of the slow buffer are set to obtain the best fit of the decay phase of the Ca 2+ ∆F/F 0 signal. Then, the kinetics of the Ca 2+ current is obtained as summation of four Gaussian functions that maximise the match between the result of the computer simulation and the experimental Ca 2+ ∆F/F 0 signal. Although this new method provides only an indirect approximation of the kinetics of the Ca 2+ current, this information is crucial at understanding the activation and deactivation of different types of VGCCs. In the dendrites of PNs, for instance, different Ca 2+ current kinetics components are associated with the activation of P/Q-type VGCCs [START_REF] Usowicz | P-type calcium channels in the somata and dendrites of adult cerebellar Purkinje cells[END_REF] and T-type Ca 2+ channels (26) that can be in principle separated by pharmacological block of one component. Thus, the extrapolation of a curve that approaches the kinetics of the Ca 2+ current can be used to quantitatively investigate the variability of channels activation at different dendritic sites, the modulation of channel activation due to physiological activity or to pharmacological action. Finally, it is important to say that such a strategy can be extended to estimate slower Ca 2+ currents where the fitting procedure can be applied to the slower decay time due to Ca 2+ extrusion [START_REF] Miyakawa | Calcium transients evoked by climbing fiber and parallel fiber synaptic inputs in guinea pig cerebellar Purkinje neurons[END_REF].

Combining membrane potential and Ca 2+ Imaging

Setting up combined voltage and Ca 2+ fluorescence measurements

To combine V m and Ca 2+ optical measurements, the VSD and the Ca 2+ indicator must have minimal overlap in the emission spectra. Water soluble voltage indicators with different excitation and emission spectra have been recently developed [START_REF] Yan | Palette of fluorinated voltage-sensitive hemicyanine dyes[END_REF]. In particular, the red-excitable and IR emitting VSD ANBDQPTEA (or PY3283) is suitable for coupling with other optical techniques [START_REF] Willadt | Combined optogenetics and voltage sensitive dye imaging at single cell resolution[END_REF].

Nevertheless, the most used VSDs for single cell applications are still JPW3028 [START_REF] Antic | Action potentials in basal and oblique dendrites of rat neocortical pyramidal neurons[END_REF] and the commercially available JPW1114 [START_REF] Canepari | Dendritic Spike Saturation of Endogenous Calcium Buffer and Induction of Postsynaptic Cerebellar LTP[END_REF]. These indicators have wide excitation spectrum in the blue/green region and they emit mainly in the red region. We have previously demonstrated that both indicators can be optimally combined with Fura indicators that are excited in the UV region and emit in the short green region [START_REF] Canepari | Combining voltage and calcium imaging from neuronal dendrites[END_REF]. In this case, VSDs were excited at 543 nm using a laser and Fura indicators were excited at 385 nm using a light emitting diode (LED) as shown in the scheme of Fig. 3a. Alternatively, simultaneous voltage and Ca 2+ imaging can be achieved using Oregon Green, Calcium Green or Fluo Ca 2+ indicators using blue light (470 -490 nm) to excite both VSDs and Ca 2+ indicators [START_REF] Bullen | Indicators and optical configuration for simultaneous high-resolution recording of membrane potential and intracellular calcium using laser scanning microscopy[END_REF]. Simultaneous imaging of JPW1114 and Oregon Green BAPTA-5N (OG5N) was adopted to obtain the first combined measurement of V m and Ca 2+ currents using the configuration of Fig. 3b. This type of measurement, however, has several disadvantages. First, OG5N fluorescence has a small tail component in the red region [START_REF] Bullen | Indicators and optical configuration for simultaneous high-resolution recording of membrane potential and intracellular calcium using laser scanning microscopy[END_REF] which can be negligible or not depending on the ratio of the two dyes at each site as well as on the ratio between the two signals. Thus, for example, it works in proximal dendrites of CA1 pyramidal neurons for signals associated with action potentials [START_REF] Jaafari | Imaging Fast Calcium Currents beyond the Limitations of Electrode Techniques[END_REF], where V m fluorescence is stronger than Ca 2+ fluorescence, but it does not in distal dendrites of cerebellar Purkinje neurons (PNs, data not shown), where V m fluorescence is weaker than Ca 2+ fluorescence. A second disadvantage is that the JPW1114 signal at 470 nm excitation is ~4 times smaller than that at 532 nm excitation. If simultaneous recordings are not critical, one can replace them with sequential recordings obtained by alternating 470 nm and 532 nm excitation as used in a recent study [START_REF] Jaafari | Functional coupling of diverse voltage-gated Ca(2+) channels underlies high fidelity of fast dendritic Ca(2+) signals during burst firing[END_REF]. Finally, a third disadvantage is that JPW1114 absorbs more in the blue range than in the green range, i.e. it exhibits toxic effects after fewer exposures. A crucial technical aspect to take into consideration while setting up combined voltage and Ca 2+ fluorescence measurements is the ability to record the two signals simultaneously at high speed. To this purpose, the company RedShirtImaging (Decantur, GA) has developed a dual-head version of the SMQ NeuroCCD (Fig. 3c). This camera permits simultaneous image acquisitions from both heads at 5-20 kHz, i.e. at the required speed. A demagnifier developed by Cairn Research Ltd (Faversham, UK) allows adjusting the size of the image before it is split in two images at the emission wavelengths of the two dyes.

Thus, the alignment of the two heads of the camera allows obtaining, at each precise region of interest, the V m and the Ca 2+ signal.

Calibrating membrane potential fluorescence transients

The calibration of V m optical signals on an absolute scale (in mV) is crucial to analyse the gating of Ca 2+ channels at the same locations where Ca 2+ recordings are performed. This is not, however, straightforward. Indeed, the fractional change of VSD fluorescence is proportional to V m [START_REF] Loew | Charge-shift probes of membrane potential: a probable electrochromic mechanism for p-aminostyrylpyridinium probes on a hemispherical lipid bilayer[END_REF], but the linear coefficient between these two quantities depends on the ratio between the inactive dye and the active dye that varies from site to site. The inactive dye is bound to membranes that do not change potential and contributes only to the resting fluorescence, while the active dye is bound to the plasma membrane and contributes to the resting fluorescence, but also carries the signal. In particular, in experiments utilising intracellular application of the dye, inactive dye is the dye that binds to intracellular membranes and organelles. Since the sensitivity of recording varies from site to site, a calibration can be achieved only if a calibrating electrical signal that has known amplitude at all locations is available. Such a signal is different in different systems. In mitral cells of the olfactory bulb, the amplitude of an action potential is the same in the whole apical dendrite and it can be used to create a sensitivity profile of the measuring system [START_REF] Djurisic | Voltage imaging from dendrites of mitral cells: EPSP attenuation and spike trigger zones[END_REF]. Another type of calibrating electrical signal can be a slow electrical change spreading with minimal attenuation over relatively long distances. Such a signal can be used to reliably calibrate VSD signals in PNs [START_REF] Canepari | Dendritic Spike Saturation of Endogenous Calcium Buffer and Induction of Postsynaptic Cerebellar LTP[END_REF]. An example of this type of calibration is reported in Fig. 4a. Starting from the resting V m , that we assume nearly uniform over the entire cell, long current hyperpolarising or depolarising current pulses are injected to the soma via the patch pipette and the change in V m is recorded. As shown by direct dendritic patch recording, the dendrite is hyperpolarised by the same amount of the soma [START_REF] Roth | Compartmental models of rat cerebellar Purkinje cells based on simultaneous somatic and dendritic patch-clamp recordings[END_REF]. Thus, the measurement of somatic hyperpolarisation can be used as voltage reference to calibrate the dendritic VSD fractional change of fluorescence (VSD ∆F/F 0 ) optical signal, as shown in Fig. 4a. In contrast, a depolarisation step attenuates along the dendrite. A third type of calibrating signal is a uniform depolarisation over the entire dendritic tree using L-glutamate photolysis from 4-Methoxy-7-nitroindolinyl-caged-L-glutamate (MNI-glutamate) [START_REF] Vogt | Combining membrane potential imaging with L-glutamate or GABA photorelease[END_REF]. This calibration procedure is applicable to all membrane expressing a relatively large number of glutamate receptors, i.e. to dendrites with high densities of excitatory synapses. The calibration is based on the principle that if the ionotropic glutamate receptor becomes the dominant conductance in a particular neuronal compartment, its reversal potential will determine the membrane potential of the compartment. Thus, in the area where dominance of glutamate receptor conductance is obtained, the resulting V m change will be the same and can be used to calibrate VSD signals. An example of this protocol to calibrate backpropagating action potentials in CA1 hippocampal pyramidal neurons is shown in Fig. 4b. The VSD ∆F/F 0 signal associated with the backpropagating AP at different sites of the apical dendrites is variable and cannot be directly correlated with the absolute change of V m . In the presence of 1 µM TTX, to block action potentials, L-glutamate is photoreleased to saturate glutamate receptors over the whole field of view. Since the recording is performed starting from the resting V m , the size of the VSD ∆F/F 0 corresponds to this potential in the whole illuminated area where V m reaches the reversal potential of 0 mV. Thus, this information is used to extrapolate the V m at each dendritic site.

Examples of combined voltage and Ca 2+ current imaging 4.1 Ca 2+ currents associated with backpropagating action potentials in CA1 hippocampal pyramidal neurons and in olfactory bulb mitral cells

In many neurons, action potentials generated in the axon hillock adjacent to the soma do not only propagate along the axon to reach neurotransmitter release terminals, but also backpropagate throughout dendrites to signal cell activation at the sites where the neuron receives the synaptic inputs. At least part of this information is given by the fast Ca 2+ transients produced by activation of VGCCs caused by the dendritic depolarisation associated with the action potential. The analysis that can be performed using the present imaging method is therefore crucial at understanding signal processing in individual neurons, as well as the specific role and function of the diverse VGCCs activated in dendrites. The propagation of the action potential and the consequent activation of VGCCs may be very different in different neuronal systems. In CA1 hippocampal pyramidal neurons, action potentials attenuate along the dendrite and activate both high-voltage activated (HVA) and lowvoltage activated (LVA) VGCCs [START_REF] Spruston | Activity-dependent action potential invasion and calcium influx into hippocampal CA1 dendrites[END_REF][START_REF] Canepari | Dendritic signals from rat hippocampal CA1 pyramidal neurons during coincident pre-and post-synaptic activity: a combined voltage-and calciumimaging study[END_REF]. We have very recently demonstrated that HVA-VGCCs and LVA-VGCCs operate synergistically to stabilise Ca 2+ signals during burst firing [START_REF] Jaafari | Functional coupling of diverse voltage-gated Ca(2+) channels underlies high fidelity of fast dendritic Ca(2+) signals during burst firing[END_REF]. Somatic and dendritic action potentials, at nearly physiological temperature, have 1-4 ms duration as in the example shown in Fig. 5a. In agreement with this, the kinetics of the Ca 2+ current is similar to that of the action potential, with a peak delayed by a few hundred milliseconds from the peak of the action potential. In total contrast to the CA1 hippocampal pyramidal neuron, in olfactory bulb mitral cells the action potential does not attenuate along the dendrites [START_REF] Bischofberger | Action potential propagation into the presynaptic dendrites of rat mitral cells[END_REF]. In addition, as shown in the representative example of Fig. 5b, the somatic and dendritic action potential at near physiological temperature (32-34 °C) has duration < 1 ms. Thus, in this system, the activation and deactivation of VGCCs is also faster leading to a Ca 2+ current with shorter duration and shorter delay from the V m waveform peak. This preliminary comparison between the two cases indicates that the role of VGCCs, activated by the action potential, is different in different systems. For example, Ca 2+ currents with delays of ~100 µs in presynaptic terminal where the function of this signal is to trigger neurotransmitter release [START_REF] Sabatini | Timing of neurotransmission at fast synapses in the mammalian brain[END_REF]. Here, the kinetics of the Ca 2+ current was obtained by calculating the time derivative of the Ca 2+ ∆F/F 0 signal fit [START_REF] Jaafari | Using simultaneous voltage and calcium imaging to study fast Ca2+ channels[END_REF]. VGCCs contribute to the shape of the action potential directly and indirectly by activating K + channels, but also provide a precise time-locked Ca 2+ transient capable to select fast-activated Ca 2+ binding proteins. The possibility to locally investigate, using combined V m and Ca 2+ current optical measurements, the physiological occurrence of Ca 2+ signals mediated by VGCCs will contribute enormously, in the near future, to the understanding of complex signal processing in neurons.

Ca 2+ currents associated with climbing fibre EPSPs in cerebellar Purkinje neurons

In contrast to pyramidal neurons of the cortex and hippocampus, and to olfactory bulb mitral cells, somatic/axonal action potentials in PNs do not actively propagate in the dendrites [START_REF] Stuart | Initiation and spread of sodium action potentials in cerebellar Purkinje cells[END_REF]. The dendrites of PNs, however, express P/Q-type HVA-VGCCs [START_REF] Usowicz | P-type calcium channels in the somata and dendrites of adult cerebellar Purkinje cells[END_REF] and T-type LVA-VGCCs ( 26) that are activated by the dendritic depolarisation produced by climbing fibre EPSPs. As shown in the example of Fig. 6, the shape of the dendritic V m , calibrated in Fig. 4a is quite different in the soma and in the dendrite, mainly reflecting the absence of Na + action potentials in the dendrite. In this system, the low-affinity Ca 2+ indicator used to estimate the Ca 2+ current was Fura-FF, since the larger Ca 2+ signal produced by OG5N contaminated the optical V m measurement. The prominent dendritic depolarisation produces a biphasic Ca 2+ current, which is in this case obtained by applying our recent generalised method [START_REF] Ouares | A generalised method to estimate the kinetics of fast Ca 2+ currents from Ca 2+ imaging experiments[END_REF]. The fast and sharp component is nearly concomitant to the short period in which V m > -40 mV and it is therefore likely mediated by HVA-VGCCs. The slower and more persistent component is instead mostly concomitant to the whole depolarisation transient and is therefore likely mediated by LVA-VGCCs, as demonstrated by selectively blocking T-type VGCCs (unpublished data not shown).

The analysis of Ca 2+ signalling associated with the climbing fibre EPSP is crucial for the understanding of synaptic plasticity in PNs [START_REF] Vogt | On the induction of postsynaptic granule cell-Purkinje neuron LTP and LTD[END_REF]. Yet, while the role of the Ca 2+ transient associated with the climbing fibre EPSP has been postulated to be auxiliary to the principal Ca 2+ signal mediated by parallel fibre EPSPs, these first measurements of the Ca 2+ current kinetics elucidate a quite precise timing of occurrence of the Ca 2+ source that may eventually provide a less ambiguous explanation of the precise function of this spread dendritic depolarisation. In summary, the examples illustrated in this section show the potentials of this novel optical method in providing physiological information not available by using electrode techniques.

Data interpretation and future prospective

The method described here has been developed to overcome the limitations of single-electrode or two-electrode voltage clamp techniques permitting the analysis of physiological Ca 2+ currents of native Ca 2+ channels. Indeed, in contrast to patch-clamp recordings, these Ca 2+ optical currents can be measured in conditions of a physiological change of V m and the measured currents are confined to the sites where they are recorded, as shown in apical dendrites of hippocampal pyramidal neurons [START_REF] Jaafari | Functional coupling of diverse voltage-gated Ca(2+) channels underlies high fidelity of fast dendritic Ca(2+) signals during burst firing[END_REF]. The additional information on local V m change, necessary to correlate the behaviour of the conductance with its biophysical properties, is obtained by combining VSD imaging. In cases of linear behaviour between Ca 2+ influx and Ca 2+ fluorescence changes the kinetics of the Ca 2+ current can be extracted by calculating the time derivative of the Ca 2+ ∆F/F 0 signal using low-affinity Ca 2+ indicators [START_REF] Jaafari | Imaging Fast Calcium Currents beyond the Limitations of Electrode Techniques[END_REF][START_REF] Jaafari | Using simultaneous voltage and calcium imaging to study fast Ca2+ channels[END_REF]. In the case of nonlinear behaviour between Ca 2+ influx and Ca 2+ fluorescence, produced by Ca 2+ -binding proteins with slower kinetics with respect to the Ca 2+ current, the kinetics of the Ca 2+ current can be still correctly estimated by taking into account the faster unbinding of Ca 2+ from the low-affinity indicator [START_REF] Ouares | A generalised method to estimate the kinetics of fast Ca 2+ currents from Ca 2+ imaging experiments[END_REF]. In this last section we address the question of how data, obtained using this technique, should be interpreted. In Ca 2+ current recordings from channels expressed in heterologous systems using voltage clamp, V m is controlled artificially and its change is therefore independent of the channel deactivation. Under physiological conditions, Ca 2+ channels contribute to the V m change directly, through the ion flux, and indirectly by regulating other conductances. It follows that the channel deactivation changes the V m waveform. We have shown that in CA1 hippocampal pyramidal neurons this phenomenon produces a modulation of LVA-VGCCs by HVA-VGCCs [START_REF] Jaafari | Functional coupling of diverse voltage-gated Ca(2+) channels underlies high fidelity of fast dendritic Ca(2+) signals during burst firing[END_REF]. More in general, a Ca 2+ current mediated by diverse VGCCs is always the result of a synergy among all different ion channels contributing to the V m waveform. It follows that in a Ca 2+ current optical measurement, a single component of the current cannot be extracted simply by blocking the underlying channel, since this block may affect the residual current as well. This evidence has important implications in the study of transgenic animals carrying Ca 2+ channel mutations. In this case, a certain phenotype is likely to result from the combined modification of function of many different channels, rather than from the specific Ca 2+ influx component, making the study of these animals as models for disease challenging. In summary, the investigation of the role and function of individual Ca 2+ channels must be performed in the global context of activation of all channels participating to the local V m waveform.

Another important aspect of data interpretation is the relation of the kinetics of Ca 2+ current with the putative molecular targets of Ca 2+ ions entering the cell. While importance is normally given to possible molecular coupling between the Ca 2+ channel and the Ca 2+ binding protein, the kinetics of the Ca 2+ current can be a potent selector of the molecular pathway which is activated. To illustrate this important concept we make use of computer simulations using the same theoretical framework for simple Ca 2+ -binding dynamics that we already used in the past [START_REF] Canepari | Imaging neuronal calcium fluorescence at high spatio-temporal resolution[END_REF]. We imagine the possible activation of two proteins: a "fast" protein with K ON = 5.7•10 8 M -1 s -1 and K D = 10 µM, expressed at the concentration of 500 µM; and a "slow" protein with K ON = 4•10 8 M -1 s -1 and K D = 0.4 µM, expressed at the concentration of 100 µM. In the first case, shown in Fig. 7a, the cell is receiving a fast Ca 2+ current with ~ 2 ms total duration which binds first to the fast protein and later to the slow protein. In the second case, shown in Fig. 7b, the cell is receiving a slower Ca 2+ current that is smaller in amplitude but that carries approximately the same amount of Ca 2+ . In this case the slow protein binds to Ca 2+ with a slower kinetics but the amount of the fast protein binding to Ca 2+ is less than half with respect to the first case. These simulations indicate that the ability to activate for a molecular pathway triggered by the fast protein strongly depends on the kinetics of the Ca 2+ current. Thus, the approach described here should drastically improve our understanding of the physiological function of Ca 2+ channels by providing the possibility to explore the biophysics of native channels during physiological activity locally within the complex neuronal architecture. The examples of combined V m and Ca 2+ current optical measurements from CA1 hippocampal pyramidal neurons, olfactory bulb mitral cells and PNs reported here are representative of the types of exploration that can be performed using this novel approach. ∆F/F 0 signal (green traces) does not match the kinetics of the current. To estimate the kinetics of the current we use a strategy that consists in matching the result of a computer simulation to the Ca 2+ ∆F/F 0 signal using an optimised two-buffer model [START_REF] Ouares | A generalised method to estimate the kinetics of fast Ca 2+ currents from Ca 2+ imaging experiments[END_REF]. We start from the Gaussian function fitting the rising phase of the ∆F/F 0 time derivative (top-right black trace). We then correct the current with three additional Gaussian components until a match of the computer simulation with the Ca 2+ ∆F/F 0 signal is obtained (process indicated by the arrow. The curve producing this match (I Ca , bottom-right black trace) is the estimate of the Ca 2+ current kinetics. Data, recorded at 5 kHz, were from averages of 4 trials.

Fig. 1 .

 1 Fig. 1. Simulation of hypothetical Ca 2+ ∆F/F 0 signals from 2 mM OG5N in the presence of endogenous Ca 2+ buffers. (a) Simulation of ∆F/F 0 signal (top trace) following a Ca 2+ current with simple Gaussian kinetics (I Ca , gray trace on the bottom) in the presence of 1 mM of a fast endogenous Ca 2+ buffer with same association constant of the Ca 2+ indicator (5.7•10 8 M -1 s -1 ) and K D = 10 µM. The kinetics of the ∆F/F 0 time derivative (middle trace) matches the kinetics of the Ca 2+ current. (b) Same as in the previous panel but in this case in the presence of additional 400 µM of a slower buffer with association constant equal to 2•10 8 M -1 s -1 and K D = 0.2 µM. The kinetics of the ∆F/F 0 time derivative (middle trace) does not matches the kinetics of the Ca 2+ current.

Fig. 3 .

 3 Fig. 3. Configurations and camera for combined voltage and Ca 2+ fluorescence measurements. (a) Schematic drawing of the apparatus for simultaneous voltage and Ca 2+ imaging using the VSD JPW1114 and the Ca 2+ indicator FuraFF (13); 385 nm LED light via the epifluorescence port of a commercial microscope is reflected by a 506 nm long-pass dichroic mirror (dic1); 543 nm laser light via the top of the microscope is reflected by a dual-band dichroic mirror transmitting wavelengths between 493 nm and 530 nm and wavelengths longer than 574 nm (dic2); the fluorescence images of the two dyes are demagnified and separated by a 565 nm long-pass dichroic mirror (dic3); The V m image and the Ca 2+ images are filtered by a 610 nm long-pass (em1) and by a 510±42 nm band-pass filter (em2) respectively, then acquired by two CCD cameras; the emission spectra of FuraFF (green) and JPW1114 (red) are shown on the top-right. (b) Schematic drawing of the apparatus for simultaneous voltage and Ca 2+ imaging using the VSD JPW1114 and the Ca 2+ indicator OG5N (11); 470 nm LED light via the epifluorescence port of a commercial microscope is reflected by a 495 nm long-pass dichroic mirror (dic4); the fluorescence images of the two dyes are demagnified and and processed as in the previous configuration; the emission spectra of OG5N (green) and JPW1114 (red) are shown on the top-right. (c) The dual NeuroCCD camera designed by RedshirtImaging for this type of measurement.

Fig. 5 .

 5 Fig. 5. Ca 2+ currents associated with backpropagating action potentials in CA1 hippocampal pyramidal neurons and in olfactory bulb (OB) mitral cells (a) On the left, fluorescence image of CA1 hippocampal pyramidal neuron filled with JPW1114 and 2 mM OG5N with a region of the apical dendrite outlined. On the right, somatic V m change associated with an action potential (black trace) in the soma and in the dendritic region. The associated Ca 2+ current kinetics, obtained with the fitting strategy from the raw Ca 2+ signal (dashed gray trace), is shown in the bottom. (b) On the left, fluorescence image of OB cell filled with JPW1114 and 2 mM OG5N with a region of the principal dendrite outlined. On the right, somatic V m change associated with an action potential in the soma (black trace) and in the dendritic region (red trace). The associated Ca 2+ current kinetics, obtained with the fitting strategy from the raw Ca 2+ signal (dashed gray trace), is shown in the bottom (blue trace) superimposed to the dendritic action potential (red trace). Data, recorded at 20 kHz, were from averages of 4 trials. All experiments were performed at 32-34 °C.

Fig. 6 .

 6 Fig. 6. Ca 2+ currents associated with climbing fibre EPSPs in cerebellar Purkinje neurons. On the left, fluorescence image of PN filled with JPW1114 and 2 mM Fura-FF with a region of the dendrite outlined. On the right, somatic V m change associated with climbing fibre EPSP (black trace) in the soma and in the dendritic region. The associated Ca 2+ current kinetics, obtained with the generalised method (24), is shown in the bottom. Data, recorded at 5 kHz, were from averages of 4

Fig. 7 .

 7 Fig. 7. Simulated activation of two different Ca 2+ -binding proteins by Ca 2+ currents. (a) In a cell containing 500 µM of a fast (F) buffer with K ON = 5.7•10 8 M -1 s -1 and K D = 10 µM, and 100 µM of a slow (S) buffer with K ON = 4•10 8 M -1 s -1 and K D = 0.4 µM, the curves on the right report the binding to Ca 2+ of the F and S proteins following the fast Ca 2+ current reported on the left. (b) Same as in the previous panel but following the fast Ca 2+ current reported on the left.

Acknowledment

This work was supported by the Agence Nationale de la Recherche through three grants: 1. Grant WaveFrontImag, program number ANR-14-CE17-0006-01; 2. Labex Ion Channels Science and Therapeutics, program number ANR-11-LABX-0015; 3. National Infrastructure France Life Imaging "Noeud Grenoblois" ; and by the Federation pour la recherché sur le Cerveau (FRC) through the grant Espoir en tête (in partnership with Rotary France).