In cortical networks, GABAergic inhibitory interneurons play a crucial role in the development and maturation of neural circuits, regulation of synaptic plasticity and rhythmogenesis (Cossart, 2011; Lehmann et al., 2012; Allen and Monyer, 2014). These cells form a highly heterogeneous population with distinct gene expression as well as morphological, molecular, and physiological properties that fit interneuron type-specific function (Klausberger and Somogyi, 2008; Kepecs and Fishell, 2014; Paul et al., 2017; Pelkey et al., 2017). The synaptic organization of inhibitory interneurons, with distinct transmission properties and plasticity mechanisms operating at individual synapses, adds another level of complexity in understanding interneuron function. Therefore, recent efforts have been undertaken to demystify the cell- and synapse-type specific forms of postsynaptic signaling and to identify the common rules important for interneuron learning.

Among ionotropic glutamate receptors, the GluA2-lacking Ca²⁺ -permeable alpha-amino-3-hydroxi-5-methyl-4-isoxazolepropionic acid (AMPA) receptors (CP-AMPARs) provide fast postsynaptic Ca²⁺ influx that is associated with the induction of distinct forms of synaptic plasticity in interneurons (Mahanty and Sah, 1998; Laezza et al., 1999; Liu and Cull-Candy, 2000; Goldberg et al., 2003a,b; Topolnik et al., 2005; Soler-Llavina and Sabatini, 2006; Lamsa et al., 2007; Oren et al., 2009; Croce et al., 2010; Nissen et al., 2010; Szabo et al., 2012; Camiré and Topolnik, 2014; Hainmuller et al., 2014; Lalanne et al., 2016). Parvalbumin (PV)-expressing fast-spiking (FS) cells, including basket cells in neocortical (Goldberg et al., 2003a,b) and hippocampal (Camiré and Topolnik, 2014; Hainmuller et al., 2014) regions and cerebellar stellate cells (Soler-Llavina and Sabatini, 2006) are particularly enriched with CP-AMPARs, which play a critical role in dendritic Ca²⁺ compartmentalization in the absence of dendritic spines. Specifically, the interaction between CP-AMPARs, the Ca²⁺ binding protein PV, and the membrane Na⁺/ Ca²⁺ exchanger (NCX) have been involved in the formation of highly localized dendritic Ca²⁺ microdomains during basal synaptic activity (Goldberg et al., 2003a; Soler-Llavina and Sabatini, 2006). The inward rectification of the CP-AMPAR-mediated current due to increased blockade by endogenous polyamines at depolarized levels of membrane potential (Koh et al., 1995) is associated with a strong attenuation of the CP-AMPAR Ca²⁺ influx (Topolnik et al., 2005) and dictates the rules for the induction of synaptic plasticity at excitatory inputs to interneurons (Lamsa et al., 2007). Indeed, CP-AMPAR-mediated Ca²⁺ influx is involved in the induction of anti-Hebbian long-term potentiation (LTP) under basal levels of synaptic activity in interneurons at rest (Camiré and Topolnik, 2014). Increasing activity levels during synchronous theta-bursting of CA1 pyramidal cells leads to additional activation of Ca²⁺ -induced Ca²⁺ release (CICR) via ryanodine receptors (RyRs) and generation of supralinear Ca²⁺ signals in dendritic microdomains (Camiré and Topolnik, 2014). As a result, this CP-AMPAR-triggered CICR is involved in the induction of long-term depression (LTD) (Camiré and Topolnik, 2014). Thus, CP-AMPARs are able to play a dual role as the triggering factor in both LTP and LTD induction, which indeed was observed in different cell types and synapses (Nissen et al., 2010; Hainmuller et al., 2014).

Interestingly, the hippocampal CA1 oriens/alveus (O/A) FS cells, including basket and bistratified cells, switch from LTP to LTD following transition from small to supralinear Ca²⁺ signals via CP-AMPARs in basal dendrites. Whether the apical dendrites of these cells may demonstrate supralinear Ca²⁺ signals triggered upon CP-AMPAR-dependent CICR remains unknown. To understand the input-specific organization of dendritic Ca²⁺ signaling in FS interneurons, here we compared the summation of postsynaptic Ca²⁺ transients (post-CaTs) between apical and basal dendrites, and further explored the common mechanisms that may shape the supralinear post-CaTs in dendritic compartments of FS cells. Our data indicate that CP-AMPAR-mediated Ca²⁺ influx is a common mechanism operating in FS interneuron dendrites, regardless of the input-specific innervation patterns. We also predict that the subcellular distribution and functional state of Ca²⁺ sources and Ca²⁺ extrusion mechanisms will have a major impact on supralinear Ca²⁺ dynamics and, accordingly, on the direction of synaptic plasticity in CP-AMPAR-expressing interneurons.

To investigate the postsynaptic Ca²⁺ dynamics along the dendritic tree of hippocampal CA1 FS interneurons, we performed whole-cell current clamp recordings in combination with two-photon dendritic Ca²⁺-imaging (Figure 1). Putative FS interneurons were selected for patch-clamp recordings using Dodt infrared-scanning gradient contrast as cells that had large-size round somata adjacent to the CA1 pyramidal layer (Figure 1A). Only cells that exhibited FS firing pattern (180–200 Hz at 32°C; Figure 1E) were included in this study. During recordings, FS cells were filled with biocytin to allow their unambiguous morphological identification. Most recorded cells showed axonal arborizations restricted to the pyramidal layer (Figure 1D) or to the RAD and O/A, suggesting that they belong to populations of basket and bistratified cells, respectively.

To compare the summation of postsynaptic Ca²⁺-transients (post-CaTs) in dendritic sub-regions receiving anatomically and functionally distinct excitatory inputs (Takács et al., 2012; Lee et al., 2014), we examined the dynamics of post-CaTs in dendritic segments within str. radiatum (apical) vs. those within str. oriens/alveus (basal). Dendritic excitatory synapses were activated locally using bipolar stimulation through a theta-glass microelectrode positioned near the dendrite of interest (Figures 1B,C, top). Recordings were performed in current-clamp at −70 ± 5.0 mV in the presence of the GABA_A and GABA_B receptor antagonists gabazine and CGP55845, respectively. Postsynaptic CaTs (post-CaTs) evoked by minimal electrical stimulation in apical and basal dendrites had similar amplitude, but the signal's area under the curve (cumulative charge) was higher in apical dendrites, consistent with a slower, longer-lasting CaT than in basal dendrites (Figures 1F,G,H). The postsynaptic depolarization associated with CaTs was of the same amplitude regardless of whether apical or basal dendrites were stimulated (Figures 1H,I). In addition, we found no correlation between the peak amplitude of post-CaTs and that of postsynaptic depolarization for both inputs (Figure 1I), consistent with the local nature of post-CaTs generated in distal dendrites of FS cells.

To compare the rules of summation of post-CaTs between apical and basal dendrites, we gradually raised the stimulation intensity up to 200% of minimal stimulation. For both inputs, the recordings were obtained in distal dendrites at similar distance to the soma (apical, distance from soma: 139.8 ± 13.1 μm, n = 10; basal, distance from soma: 112.5 ± 5.3 μm, n = 22), and in dendrites having a similar diameter (apical, 1.33 ± 0.14 μm, n = 9; basal, 1.24 ± 0.07, n = 19; P = 0.253). In line with previous findings in basal dendrites (Camiré and Topolnik, 2014), increasing the stimulation intensity in apical dendrites resulted in an all-or-none supralinear Ca²⁺-response (Figure 2A), indicating that supralinear Ca²⁺ integration is a common property of FS cell dendrites, regardless of the input activated. The Ca²⁺ threshold for the generation of supralinearity was similar in both apical and basal dendrites (apical ΔG/R, 0.082 ± 0.011; basal ΔG/R, 0.080 ± 0.009, P = 0.449). Supralinear post-CaTs also had similar amplitude in apical and basal dendrites (apical ΔG/R: 0.26 ± 0.03, n = 10; basal ΔG/R: 0.21 ± 0.02, n = 22; P = 0.0703; Figures 2A–D). The CaNL area under the curve, however, was significantly higher in apical dendrites than in basal ones (apical, 29 ± 4 ΔG/R^*ms, basal, 19 ± 3 ΔG/R^*ms, P = 0.0264; Figures 2C,D, middle), similar to the data obtained for the minimally evoked post-CaTs (Figure 1H). In line with this finding, the decay time constant of supralinear post-CaTs, was also significantly slower in apical dendrites (apical, 249 ± 76 ms, basal, 144 ± 16 ms, P = 0.034; Figures 2C,D, right). Importantly, whilst the degree of supralinearity during summation of post-CaTs was similar between the two dendritic regions resulting in a 3- to 4-fold increase in Ca²⁺-concentration when stimulation intensity was raised by 50% from the minimal level (Figures 2A,B), the overall gain in dendritic Ca²⁺-signal showed a strong degree of variability across the dendritic tree (apical, range: 3.7–11.6, variance: 7.9, n = 10; basal, range: 4.6–16.8, variance: 14.8, n = 21), indicating that dendritic site-specific mechanisms may shape the summation of post-CaTs in individual postsynaptic microdomains.

Furthermore, the supralinear post-CaTs in apical dendrites were associated with a significantly larger number of spikes (AP number to the 3rd stimulus, apical: 15.3 ± 7.3, n = 9; basal: 2.1 ± 0.3, n = 22; P = 0.0016, Mann-Whitney test; Figure 2E) and a longer lasting after-depolarization (apical, decay τ: 88.7 ± 17.7 ms, n = 9; basal, decay τ: 49.5 ± 4.2 ms, n = 22; P = 0.0039, unpaired t-test; Figure 2E). These data indicate that apical dendrites may have specific mechanisms for Ca²⁺-signal integration and modulation of dendritic excitability.

To explore this possibility experimentally, we next compared the pharmacological profile of supralinear post-CaTs in different dendritic regions (Figure 3). As previous studies identified CP-AMPA receptors, NMDA receptors, L-type Ca²⁺-channels and intracellular Ca²⁺-release as four major sources of post-CaTs in FS interneurons (Goldberg et al., 2003a,b; Camiré and Topolnik, 2012, 2014; Chiovini et al., 2014; Hainmuller et al., 2014), we focused on these Ca²⁺-mechanisms by comparing their contribution to supralinear post-CaTs in different dendritic regions (Figures 3A–D). Our data showed that the supralinear post-CaTs recorded in apical dendrites were sensitive to CP-AMPA receptor inhibitor NASPM (CP-AMPAR-mediated Ca²⁺-response component: 39.0 ± 5.6%, n = 15 segments/5 cells; P = 0.0003; paired t-test; Figures 3A–C), although the CP-AMPAR component was significantly smaller in apical than in basal dendrites (P = 0.001; unpaired t-test; Figure 3C), pointing to the input-specific distribution of CP-AMPARs. Furthermore, the NMDAR contribution to supralinear Ca²⁺-responses in apical dendrites was not significant (NMDAR-mediated Ca²⁺-response component: 17.7 ± 9.5%, n = 13 segments/4 cells; P = 0.345; Wilcoxon signed-rank test; Figures 3A–C). We have noticed a high variability in the NMDAR component between individual dendritic segments in both dendritic regions (apical, range: 0–68%, n = 13 segments/4 cells; basal, range: 0–77%, n = 12 segments/8 cells), consistent with previous ultrastructural data on the highly variable NMDAR distribution in dendrites of PV+ cells (Nyíri et al., 2003). One bistratified cell showed an unexpected increase in basal post-CaT amplitude in the presence of AP5 via a yet unknown mechanism (Dore et al., 2017), and was not included in the current analysis.

In addition, the sensitivity of supralinear CaTs to the SERCA pump inhibitor CPA (30 μM) in apical dendrites was also highly variable. In 5 dendritic segments from 2 cells, application of CPA resulted in a significant decrease (to 66.2 ± 9.4% of control CaT; P = 0.043; Wilcoxon signed-rank test) of supralinear post-CaTs, pointing to the role of Ca²⁺-release in dendritic Ca²⁺ amplification (Figures 3A–C). However, three other cells (6 dendritic segments) showed an increase and prolongation of supralinear post-CaTs (to 142.6 ± 9.8% of control CaT; P = 0.028; Wilcoxon signed-rank test), consistent with the inhibition of ER uptake (Figure 5D). Taken together, these data indicate that intracellular Ca²⁺-stores, through microdomain-specific CICR or Ca²⁺ uptake, play an important role in supralinear Ca²⁺-dynamics across the dendritic tree of FS interneurons.

We also examined the contribution of the L-type voltage-gated Ca²⁺ channels (VGCCs) to the supralinear post-CaTs in apical vs. basal dendrites (Figures 3A–C). In line with previous findings on the role of this channel in the generation of dendritic Ca²⁺-spikes in apical dendrites of CA1 basket cells (Chiovini et al., 2014), our data showed that blocking L-type VGCCs with nifedipine (10 μM) or nimodipine (10 μM) had a variable, albeit statistically significant, effect on supralinear post-CaTs in apical dendrites of FS cells (L-type-Ca²⁺-response component, 13.4 ± 2.6%, n = 43 segments/6 cells, P = 0.000005; Wilcoxon signed-rank test; Figures 3A–C). The latter was likely due to the low channel density and variable distribution in PV+ dendrites, as revealed with immunolabeling for the Cav1.3 subunit (Figure 3E, upper). Also, we were unable to detect the R-type VGCC Cav2.3 (Figure 3E, middle) and the T-type VGCC Cav3.1 (Figure 3E, lower) subunits in PV+ apical dendrites, indicating that these types of VGCCs do not contribute to dendritic Ca²⁺-dynamics in FS cells. Taken together, the results of pharmacological experiments indicate that similar mechanisms shape postsynaptic Ca²⁺-signaling across the entire dendritic tree of FS cells, with a major role for CP-AMPARs in Ca²⁺-signal amplification, and highly variable and site-specific involvement of other mechanisms. Then, why do supralinear post-CaTs exhibit slower decay time in apical dendrites and why are they associated with a stronger somatic depolarization?

Previous studies indicated that multiple factors can shape postsynaptic Ca²⁺ dynamics in neuronal dendrites, including the surface-to-volume ratio of dendritic compartments, the number of inputs activated synchronously during synaptic activity and the spatial distribution of Ca²⁺ sources and extrusion mechanisms (Topolnik et al., 2005; Losonczy and Magee, 2006; Scheuss et al., 2006; Branco and Häusser, 2011; Anwar et al., 2014; Bloss et al., 2018). To examine how these elements may impact the summation of post-CaTs in FS interneurons, we complemented experimental observations with a computational model of dendritic Ca²⁺-dynamics. We constructed a model based on realistic FS cell dendritic morphology (Figure 4A) with passive and active properties similar to those used previously for FS interneurons (Goldberg et al., 2003b; Nörenberg et al., 2010; Chiovini et al., 2014), which could reproduce experimental data on FS cell membrane properties, synaptic currents and Ca²⁺-dynamics (Camiré and Topolnik, 2014). To simulate minimally evoked and supralinear post-CaTs (Figure 4B), we used a simplified one-compartmental model, which is justified by the fact that, in FS interneurons, post-CaTs are generated in isolated dendritic microdomains (~5–10 μm) and, thus, weakly depend on somatic dynamics (Goldberg et al., 2003a; Camiré and Topolnik, 2014). Parameter optimization was performed to determine the values of biophysical parameters, which best fit the properties of the experimentally observed Ca²⁺-signals. We found that parameter values best fitting experimental recordings were in the following ranges: maximal RyR flux: 6.5 ± 0.5 ^* 10⁻³/ms, half-activation concentration K_RyR: 0.9 ± 0.1 μM, Ca²⁺ decay time constant: 250 ± 60 ms, maximal rate and half-activation concentration of Ca²⁺ pumping from ER: 1 ± 0.3 ^* 10⁻² μM/ms and 1.35 ± 0.2 μM, correspondingly. Both minimally evoked and supralinear post-CaTs were accurately captured by the model incorporating Ca²⁺-influx through CP-AMPA receptors and intracellular Ca²⁺-release with optimized parameters (Figure 4C).

First, we performed a set of simulation experiments that were structured to determine the impact of the surface-to-volume ratio on supralinear post-CaTs. When simulating post-CaTs in dendritic branches of different diameter (range, 0.5–4.0 μm), we found that generation of these signals is facilitated in thin dendrites (range, 0.5–1.5 μm; Figure 5A). However, no significant correlation between the diameter of dendritic branches in which Ca²⁺ imaging data were obtained and the amplitude of evoked CaTs was found in our experiments for both inputs (apical: 1.33 ± 0.14 μm; n = 10; r = −0.382; Pearson correlation; basal: 1.24 ± 0.07; n = 19; r = 0.081; Pearson correlation; Figure 5B).

Next, we estimated the number of synapses necessary for the generation of minimal and supralinear post-CaTs, and explored how it can affect the amplitude of these signals (Figures 5C,D). As the CP-AMPARs made a major contribution to supralinear post-CaTs (Figures 3A–C), at first, we considered the CP-AMPAR-containing synapses only. Using phase plane analysis, assuming that the cytosolic resting Ca²⁺ is fixed (70–100 nM; Aponte et al., 2008), we found that there exists a critical value for the resting endoplasmic reticulum (ER) Ca²⁺-level (ER-Ca²⁺) below which supralinear post-CaTs cannot be generated (Figure 5C). The amplitude of the Ca²⁺-event increased quasilinearly with the ER-Ca²⁺ concentration and the threshold for generation of these events stayed approximately constant in the concentration ranges of interest (threshold values equal to 372 ± 26 nM). For the optimal model parameter values, the system behaved linearly as soon as Ca²⁺ concentration stayed below the non-linearity threshold. The Ca²⁺ flux depends quasilinearly on the Ca²⁺ concentration, thus subthreshold Ca²⁺ dynamics may be approximated by a simple exponential relaxation process with a different effective Ca²⁺ decay time constant. The effective time constant found by least-squares fitting to Ca²⁺ flux vs. concentration dependence appeared to be in the range of 50 ± 15 ms. The linear relaxation approximation allowed analytical estimation of the minimal number of synapses activated to generate a supralinear post-CaT (Figure 5D). We found that, independent of the dendritic region, four synapses were activated during local synaptic stimulation to trigger a minimally evoked post-CaT observed typically in our experiments, and seven synapses were required to generate a supralinear signal (Figure 5D, left). Further increase in the number of synapses activated had no significant effect on the amplitude and decay time of post-CaT (Figure 5D, left), as the activation of RyRs always produced Ca²⁺-elevations of approximately the same amplitude and width. Collectively, these data indicate that differences in the decay time of supralinear post-CaTs observed are likely associated with site-specific distributions of kinetically slow Ca²⁺-sources and extrusion mechanisms.

As NMDARs could also make an important, albeit highly variable, contribution to supralinear post-CaTs, we also examined their impact on the number of synapses required for the generation of these signals (Figure 5E). In simulations where excitatory synapses with different CP-AMPA to NMDA receptor conductance ratio were activated, supralinear post-CaTs showed different threshold of induction and distinct decay kinetics (Figure 5E, left). In particular, increasing the NMDA receptor contribution resulted in larger amplitude and slower Ca²⁺-events that were now triggered by a lower number of synapses (Figure 5E, right). Therefore, including NMDA receptors in the model allowed us to decrease the minimal number of synapses necessary for the generation of supralinear Ca²⁺-elevations and increase their magnitude and duration, which could contribute to the experimentally observed high heterogeneity in Ca²⁺-signal gain across the dendritic tree.

In addition to the CP-AMPARs, the CICR via RyR activation represents a second important component of the supralinear post-CaTs in basal dendrites of FS cells (Camiré and Topolnik, 2014). To estimate whether this component may be responsible for the dendritic region-specific differences observed in the current study, we next explored the role of the RyR modulation on the magnitude of supralinear Ca²⁺-elevations. Consistent with experimental observations in basal dendrites of FS cells (Camiré and Topolnik, 2014), we found that supralinear post-CaTs were reduced to the minimally evoked Ca²⁺ level when simulating the effect of RYR antagonist ryanodine, and was potentiated in the model simulating the effect of caffeine (Figure 5F, left). In addition, varying the speed of Ca²⁺-flux through RyRs had a major impact on the amplitude and threshold of supralinear post-CaTs (Figure 5F, right), thus highlighting RyR modulation as a critical component in the Ca²⁺-signal summation. Next, given the important role of Ca²⁺ extrusion mechanisms in shaping dendritic Ca²⁺ signals (Scheuss et al., 2006), we also explored whether the slower supralinear post-CaTs in apical dendrites may result from different time constants of Ca²⁺ removal from the cytoplasm. By simulating different rates of activation of the Na⁺-Ca²⁺ exchangers (NCX) and membrane Ca²⁺-pumps (Figure 5G), we found that the activation of these mechanisms was efficient in controlling the width of supralinear post-CaTs, as well as the threshold for their induction and their amplitude (Figure 5G). Thus, our model predicts that the dendritic site-specific distribution and functional state of RyRs and Ca²⁺ extrusion mechanisms may be among the most important factors responsible for the observed difference in the decay time of supralinear post-CaTs and their overall variability.

Given the highly compartmentalized nature of dendritic signaling in different types of neurons (Goldberg et al., 2003a; Losonczy and Magee, 2006; Branco and Häusser, 2011), we combined experimental and computational studies in PV+ FS cells to examine the properties and the mechanisms of Ca²⁺ signals associated with activation of synaptic inputs in apical vs. basal dendrites. Using a combination of two-photon Ca²⁺ imaging and whole-cell patch-clamp recordings, we found significant differences in supralinear summation of Ca²⁺ signals, with a highly variable gain function, between the individual dendritic sites, further emphasizing the synapse-specific nature of dendritic Ca²⁺ integration in interneurons.

The CP-AMPARs made a major contribution to supralinear post-CaTs in all dendritic branches tested, with a stronger contribution in basal dendrites, pointing to the input-specific distribution of these receptors in FS cells. Compared with other types of cortical interneurons, FS PV+ cells exhibit the highest levels of GluA1, GluA3, and GluA4 mRNAs (Paul et al., 2017), pointing to the relative abundance of CP-AMPARs at their excitatory synapses. Furthermore, our data indicate that the CP-AMPAR–RyR interaction was critical for generation of supralinear post-CaTs in PV+ basket and bistratified cells, and in apical and basal dendrites that receive distinct excitatory inputs. These data indicate that all CP-AMPAR-containing synapses, regardless of the cell- and input-type, are capable of generating supralinear Ca²⁺ signals via AMPAR–RyR interaction. Interestingly, GluA1, 2, and 4 exhibit the sex-specific expression in dorsal hippocampus in response to early life experience, with a higher GluA1 and lower GluA2 and GluA4 content in females (Katsouli et al., 2014). These data indicate that CP-AMPAR Ca²⁺ signaling may dominate in females and underlie the sex-specific differences in learning and memory processes (Dachtler et al., 2011). Whether the sex-dependent GluA2 subunit expression affects hippocampal interneurons remains unknown but will be interesting to explore in relation to CP-AMPAR-CICR-mediated Ca²⁺ signaling and synaptic plasticity.

The NMDARs were also involved in the generation of supralinear post-CaTs in FS cells, albeit with a high degree of intercellular and intracellular variability. These data are consistent with the large variability in synaptic density of NR1, the obligatory NMDAR receptor subunit, in CA1 PV+ interneurons (Nyíri et al., 2003), and the cell type- as well as synapse-specific contribution of NMDAR subunits to glutamate transmission and long-term synaptic plasticity in interneurons (Lei and McBain, 2002; Lamsa et al., 2005, 2007; Galván et al., 2015; Akgül and McBain, 2016). Importantly, our data indicate that, at a subset of synapses, NMDARs are co-expressed with CP-AMPARs. Given the significant difference in activation kinetics between these two receptors, fast Ca²⁺ influx through CP-AMPARs precedes the activation of NMDARs and is involved in NMDAR inactivation (Rozov and Burnashev, 2016). It is therefore possible that at synapses with high CP-AMPAR/NMDAR ratio, NMDARs remain functionally silent, and a higher synaptic NMDAR fraction will be required for substantial contribution of these receptors to post-CaTs. It should be noted that the AMPAR/NMDAR contribution to postsynaptic Ca²⁺ influx may change depending on the level of synaptic activity, membrane polarization and state-dependent intracellular modulation and trafficking of different glutamate receptor subunits. These factors may all contribute to variability in contribution of CP-AMPARs and NMDARs to post-CaTs and may explain the fluctuation in the Ca²⁺ gain function.

To explore the biophysical basis of supralinear Ca²⁺ events, we constructed a computational model of FS basket cell. Analysis of simulations confirmed our experimental findings that supralinear Ca²⁺ signals in dendrites of FS cells rely mainly on Ca²⁺ influx at resting membrane potential through activation of CP-AMPARs with a subsequent Ca²⁺ release, and do not require the interaction with active conductances. This way the CP-AMPAR-enriched synapses can compensate for the relatively low densities of active ion conductances in dendrites of some interneurons, including PV+ basket and cerebellar stellate cells (Hu et al., 2014; Tran-Van-Minh et al., 2016). In this case, the CP-AMPAR–CICR-mediated amplification of local Ca²⁺ signals via generation of supralinear post-CaTs represents a simple voltage-independent way of synapse-specific modification of incoming inputs.

In addition, we examined the role of structural (dendrite diameter, number of clustered synapses) and functional (RyR and Ca²⁺ extrusion modulation) parameters in shaping the time course and magnitude of supralinear Ca²⁺ elevations. The model predicted that thin dendrites will be particularly prone to generation of supralinear post-CaTs, and that the number of the spatiotemporally clustered synapses required for such event to occur will depend on the CP-AMPAR/NMDAR ratio, with NMDARs decreasing the number of synapses necessary for supralinear Ca²⁺ integration. These data are in line with theoretical and experimental observations obtained in principal neurons, indicating that the synapse location, clustering, and composition are the leading factors in dendritic input integration (Rall, 1962; Holmes, 1989; Katz et al., 2009; Branco and Häusser, 2011; Anwar et al., 2014; Bloss et al., 2018). For simplicity, the CP-AMPAR-dependent inactivation of the NMDAR current (Rozov and Burnashev, 2016) was not included in the model but may have significant impact on the number of synapses required for generation of supralinear post-CaTs.

Our model also predicts that the properties and high variability of supralinear post-CaTs may depend on the functional state of the Ca²⁺ release, uptake and extrusion mechanisms, such as the RyR, Na⁺/Ca²⁺ exchanger and SERCA pump. RyRs are the largest ion channels known, which are able to generate significant Ca²⁺ flux from the ER to cytoplasm (Van Petegem, 2016; Williams et al., 2018). The 3 RyR isoforms (RyR1–RyR3) are expressed in the brain and are activated by cytosolic Ca²⁺, with RyR1 having a particularly high expression in the CA1 area (Hertle and Yeckel, 2007). Recent high-resolution imaging data revealed a tendency toward formation of irregular RyR clusters in cardiomyocytes (Macquaide et al., 2015). Whether the same spatial arrangement of RyRs may occur in central neurons remains unknown but could explain the fluctuating Ca²⁺ gain function during summation of post-CaTs along the dendritic tree. In addition, large amplitude CICR events occurring synchronously in several discrete loci were involved in membrane depolarization and generation of action potentials (Capogrossi et al., 1987). A similar mechanism could operate in apical dendrites of FS cells, where supralinear post-CaTs were associated with higher-level somatic depolarization and firing. Furthermore, recruitment of additional Ca²⁺ sources, such as mGluR-mediated mechanisms, may occur during specific patterns of activity and be coupled to activation of specific types of the transient receptor potential (Trp)-channels (Topolnik et al., 2005, 2006; Camiré and Topolnik, 2014; Hainmuller et al., 2014). Further studies will be required to understand their functional role in synapse- and cell type-specific amplification of interneuron dendritic Ca²⁺-signals.

The SERCA pump and NCX are considered the major mechanisms of cytosolic Ca²⁺ extrusion and complement each other in terms of Ca²⁺ affinities and transport capacity. They are also involved in control of pre- and postsynaptic Ca²⁺ dynamics, with direct impact on Ca²⁺-dependent signaling and synaptic plasticity (Blaustein and Lederer, 1999; Jeon et al., 2003; Scheuss et al., 2006; Empson et al., 2007). Modulating the time course of Ca²⁺ extrusion in the model had a significant impact on the duration and magnitude of supralinear post-CaTs, and also decreased the threshold for their induction. It is unknown whether the dendritic distribution of these mechanisms is uniform in these cells, but differences in Ca²⁺ extrusion could account for the slower decay of Ca²⁺ signals seen in apical dendrites experimentally. We found that this slower decay in turn lead to an increase in cumulative charge in apical dendrites. While we did not observe a difference in the induction of supralinear CaTs, the higher cumulative charge could facilitate Ca²⁺-dependent signaling in apical dendrites. These observations indicate that the dynamics of Ca²⁺ extrusion may be another important factor in the regulation of synaptic plasticity in interneurons.

Overall, the model reveals additional mechanisms important for supralinear integration of dendritic Ca²⁺ signals in FS cells. However, it does not explore all possible scenarios that can be activated during ongoing synaptic activity and coordinate dendritic Ca²⁺ dynamics and synaptic function. For example, we did not consider the role of dendritic GABAergic inhibition in regulation of Ca²⁺ signals, as our experiments were performed in the presence of GABA receptor blockers to isolate excitatory responses. Therefore, all GABAA receptor currents, including those involved in tonic inhibition, were abolished. Tonic inhibition is functional in some hippocampal interneuron populations (Song et al., 2011; Ferando and Mody, 2014; Pavlov et al., 2014). It is unclear whether it also operates in CA1 FS cells, but if this is the case, it may affect the spatial and temporal extent of supralinear Ca²⁺ events. Therefore, given the use of GABAA receptor blockers in our study to isolate excitatory responses, the physiological extend of our findings under normal conditions remains to be explored.

In addition, it has been suggested that mitochondria may be involved in CICR in certain conditions (Ichas et al., 1997). Moreover, the store-operated Ca²⁺ entry (SOCE), the mechanism through which the ER triggers Ca²⁺ influx following the store depletion, may contribute to Ca²⁺ signaling in neurons. While there is evidence that SOCE is involved in neuronal function (reviewed in Majewski and Kuznicki, 2015), the relatively low Ca²⁺ conductance of SOCE-associated channels makes them unlikely contributors to post-CaTs. However, following a massive release of Ca²⁺, the implication of these channels may become more important. Interestingly, SOCE is linked to L-type VGCC inhibition (Park et al., 2010). If this mechanism operates also in FS interneurons, it may explain the low contribution of L-type channels during Ca²⁺ non-linearities in our experiments.

More broadly, CP-AMPAR-mediated Ca²⁺ signaling has been implicated in the pathophysiology of several brain diseases. For example, the expression of CP-AMPARs increases following induction of experimental seizures (Rajasekaran et al., 2012; Lippman-Bell et al., 2016). In addition, CP-AMPAR-mediated Ca²⁺ signaling regulates depression-like behavior in chronic neuropathic pain (Goffer et al., 2013), and is among the first to respond to traumatic brain injury (Bell et al., 2007). It is clear that imbalanced CP-AMPAR Ca²⁺ signaling in dendrites of interneurons in combination with other factors may be involved in abnormal cellular and network computations. From this perspective, the identification of the CP-AMPAR interacting partners, and how these affect neuronal computations, will likely open new therapeutic avenues toward the prevention and treatment of several devastating disorders.

OC, TG, and LT performed experiments. IL performed computational simulations. OC and LT analyzed the data and wrote the manuscript.