Merged biochemical and electrophysiological modeling

Our analysis of IP3 and calcium dynamics in SCA Purkinje neurons is based on several computational studies of normal physiology from our lab [32, 35, 36]; these will be briefly described.

The cerebellar Purkinje neuron geometry is complex. In order to combine detailed biochemistry with electrophysiology in tractable Virtual Cell models (VCell; http://www.vcell.org[37–41], we developed a geometry reduction algorithm preserved path reduction or PPR[36]. This algorithm simplifies the Purkinje neuron geometry to just the essential dendritic segments and branches in a path between a prototypical dendritic spine and the soma, while maintaining the dynamic electrical features of the full geometry in these respective compartments. The resulting model has only 17 compartments and allows us to readily couple the biochemistry in the spine compartment with the electrophysiology of the entire system. Reference [36] provides full details of the algorithm and the resultant geometry.

The baseline VCell model for normal physiology of the system was also previously published by our laboratory [35]. This compartmental model focused on IP3R1-mediated calcium signaling in the Purkinje cell, taking into account the increased abundance and decreased sensitivity of the Purkinje cell IP3R1 relative to other cell types. The input to the system is provided as a train of IP3 pulses (representing IP3 production from multiple parallel fiber stimuli) with variable parameters that control the number, amplitude and duration of the IP3 pulses. In addition, the model included influx of calcium representing the opening of voltage-gated calcium channels when the Purkinje neuron is stimulated by a climbing fiber. It therefore allowed the study of coincident activation of the Purkinje spine by multiple parallel fibers and a single climbing fiber. The model also included the downstream effects leading to IP3R1-mediated calcium release, as well as calcium buffering and SERCA, which resides on the sER. All of the spine cytosol species are free to diffuse through the spine neck into the adjacent dendrite. IP3 also diffuses in the cytosol to bind its calcium channel-coupled receptor on the sER membrane. Calcium is released from sER stores into the cytoplasm and also co-activates IP3R1, as well as binds to various buffers. Some of this calcium is also pumped back into the sER by SERCA. The model definition in VCell includes specifying reactions, membrane transport mechanisms, stoichiometry, and kinetic rate expressions that correspond with these processes.

We subsequently created a 3D spatial model that explored [42] the membrane signaling mechanisms upstream of IP3R. We distinguished between candidate sources of sufficient phosphatidylinositol-4,5-bisphosphate (PIP2) in the spine for the production of requisite IP3. Local PIP2 sequestration provided an efficient means of fine-tuning coincident activation of parallel fiber and climbing fiber stimuli in cerebellar Purkinje neuron spines. The results reported in this study concluded that local high PIP2 achieved through a sequestration mechanism could provide sufficient IP3 for supralinear calcium release.

We used optimized parameters [42] to study physiological effects of varying IP3R1 sensitivity and abundance with a physiologically appropriate IP3 signal amplitude and duration. To manipulate IP3R1 abundance and sensitivity, we adjusted the parameters Jmax and d_IP3, respectively, in our comprehensive model (see a table of parameters in [42]). Small Jmax values represent reduced IP3R1 abundance. The default value is from Hernjak et al. [35], and is 10x larger than the IP3R1 determined for neuroblastoma cells, to account for the increased levels of IP3R1 in Purkinje neurons. The d_IP3 parameter represents the dissociation constant of IP3 from IP3R1, so that high d_IP3 values in our model represent low affinity of IP3R for IP3. We use the apparent affinity of IP3R1 for IP3 as a measure of sensitivity, as in the experimental study by Tang et al. [43]. Manipulation of these IP3R1parameters was done in the context of a train of 4 parallel fiber (PF) stimuli followed by a single climbing fiber (CF) stimulus, following the optimal protocol designed by Wang et al. [44]. This allowed us to examine the effects of IP3R1 insufficiency and supersensitivity in the context of Purkinje spine stimulation patterns similar to those in published experimental studies [35, 42, 44–46].

Adapting the PPR model to the SCA model

In order to investigate how biochemical signaling at the spine could affect electrophysiology, we needed to allow calcium release to influence the voltage-gated K_Ca channels. In the PPR model, K_Ca channels were only influenced by calcium influx into a phenomenological submembrane shell. This allowed for very high transient concentrations of calcium (500 μM), as well as a high concentration of resting calcium (~ 4 μM) near the inner surface of the plasma membrane. Since such a high amount of basal calcium is not physiologically observed or observable (in the absence of Purkinje neuron stimulation by nerve endings from regulatory cells) and since calcium concentration beyond the submembrane shell was not explicitly considered in the PPR model, we felt it was important to model calcium less phenomenologically. Therefore, in this work, we removed submembrane shell calcium from the model only in the spine and the adjacent dendrite, where calcium signaling biochemistry is modeled explicitly. The adjacent dendrite was adjusted in tandem with the spine, because calcium and other molecules involved in biochemical processes are allowed to diffuse freely between the spine and the adjacent dendrite, restricted only by the geometry of the spine neck. In a previous study, Hernjak et al. [35] used a one-dimensional model to determine a dendritic distance, 28 μm from a stimulated spine, at which biochemical concentrations remained at steady state during synaptic activity. The model therefore assumes that there is no net molecular flow between compartments, except for the adjacent dendrite (given a length of 28 μm) and the spine.

In other compartments, calcium influx feeds only into submembrane shell calcium. In the spine and adjacent dendrite, calcium influx is allowed to directly alter concentration of calcium involved in biochemical signaling; we term this calcium ‘global calcium’. The concept of the submembrane shell in electrophysiological models [47–49] assumes a characteristic relaxation time of 2 ms. This relaxation time represents the time it could take for submembrane shell calcium to relax to the ‘global calcium’ steady state, 40nM in the model. The submembrane shell relaxation time conceptually encompasses: (i) diffusion from beneath the membrane into the rest of the compartment, for example, the spine head, (ii) diffusion into adjacent compartments, for example, from the spine to the adjacent dendrite, (iii) buffering by calcium binding proteins, for example, calbindin and parvalbumin, (iv) uptake into the smooth endoplasmic reticulum by SERCA, and (v) extrusion through the plasma membrane by the Na-Ca exchanger. In addition, calcium extrusion from the cell by the plasma membrane Ca-ATPase (PMCA, calcium pump) is modeled explicitly as a Michaelis-Menten expression.

In order to properly replace submembrane shell calcium with global calcium in the spine and the adjacent dendrite, (i) – (v) are all modeled explicitly. The model by Hernjak et al. [35] assumed instantaneous diffusion in the spine, and included diffusion between the spine and adjacent dendrite. It also included explicit buffering by calbindin and parvalbumin, in addition to calcium uptake by SERCA. That model did not include calcium extrusion from the spine or adjacent dendrite. Here, we add in calcium efflux by including the Michaelis-Menten Ca-ATPase from the Purkinje neuron electrophysiology models [48, 49] and a Na-Ca exchanger from a study on cerebellar granule cells [50] (a similar Na-Ca exchanger in a Purkinje neuron model was not available). The Na-Ca exchanger is not electrogenic and simply provides an additional means of calcium extrusion; its kinetics are unchanged in our model. The total concentration of the Ca-ATPase is increased to allow for calcium efflux appropriate for maintaining a physiological steady state calcium concentration in the Purkinje neuron spine. The kinetics for the voltage-gated calcium-activated potassium channels were adjusted to be influenced by global calcium. This reflects results from experimental studies which suggest that BK channels can be activated by global calcium concentrations less than 10 μM [51].

The ICpeptide study

where K _{f_IP3R1_Ct} and K _{r_IP3R1_Ct} represent the forward and reverse rate constants for PP1α binding the endogenous tip of the IP3R1 C-terminal, and IP3R1_Ct_PP1 represents PP1α bound to IP3R1 at the smooth ER (sER). Rate constants are provided in Table 2. It was assumed that PP1α bound the ICpeptides and the endogenous IP3R1 with the same affinity.

The C-terminal of IP3R1 can also bind mutant Ataxin. The affinity of the IP3R1 C-terminal tip was taken to be greater for the mutant protein than for PP1α. This was done to incorporate two phenomena into the binding reaction. First, planar lipid bilayer reconstitution experiments from Chen et al. and Liu et al. [6, 10] suggest that direct association of the mutant ataxin protein with the C terminal of IP3R1 leads to supersensitivity of the receptor. Second, the binding domain of the mutant protein [52] on IP3R1 overlaps the PP1α binding motif [52] on IP3R1. Mutant protein binding IP3R1 in situ (as in cerebellar slices from mouse models that can be used to test simulation predictions; see Discussion) may decrease PP1α interaction with the receptor. These two factors are effectively modeled by assigning a larger affinity for mutant protein binding to IP3R1 than PP1a binding the receptor. Affinities were captured by the rate constants, all available in Table 2.

SCA15/16: pathophysiology of IP3R1 insufficiency

In ataxias with reduced IP3R1, e.g., SCA15/16, receptor levels appear to be decreased by varying amounts, even in conditions of heterozygous gene deletion [3, 4, 14–16, 20]. We captured this variability in our model by adjusting the abundance parameter Jmax to decrease IP3R1 levels to 50%, 40%, 30%, 25%, and 10% of the original value. Figure 3a shows the supralinear calcium response with simulated wild type IP3R1 levels (labeled as ‘Original IP3R1 density’) when the Purkinje neuron spine is stimulated by a single climbing fiber at 0.15 s (labeled as ‘Ca2+ influx only’), by 4 pulses from a single parallel fiber at 0.1 s (labeled as ‘Ca2 release only’), or by coincident stimulation by 4 pulses from a parallel fiber at 0.1 s and a climbing fiber at 0.15 s. The simulation uses Magnesium Green as the calcium indicator (K_d = 19 μm) that fluoresces more brightly when bound to calcium [35, 44]. The figure indicates that the coincident calcium transient is greater than the sum of the individual calcium transients from stimulation of only the climbing fiber or only the parallel fiber [35, 42, 54]; in our simulations, calcium release alone produces a barely perceptible increase in free Ca²⁺, but when it occurs together with calcium influx, the free Ca²⁺ increase more than doubles compared to influx alone. The figure also shows that halving the density of IP3R1 (as in heterozygous mouse models and individuals) severely reduces supralinear calcium release. Thus, the model can be used to investigate the heterozygous human ataxia SCA15/16.

Application of ICpeptide to adjust sensitivity in SCA15/16

Modeling results from Hernjak et al. [35] showed that increasing the sensitivity of the IP3R1 by a factor of 10 did not compensate for a decreased the level of IP3R1 to 10% of the wild type value. They argued that the normally low sensitivity of IP3R1 to IP3 was required to assure that calcium release is confined to the activated dendritic spine while the normally high level of IP3R assures robust calcium release despite the low sensitivity. This result is consistent with the finding in Figure 3a, which shows that the calcium transient with 10% IP3R1 abundance is almost entirely due to calcium influx through voltage-gated calcium channels. Very little of the calcium transient is due to calcium release from the ER, through IP3R. It allows us to suggest that in Purkinje cells, with have much lower sensitivity of IP3R to IP3 than in other cell types, decreasing receptor density to 10% of the wild type levels (which amounts to replacing Purkinje neuron IP3R with levels found in other cell types) is as good as having no IP3R at all. Figure 4 shows that increasing sensitivity in the wild type context (by adding mutant Ataxin-2 to the model with normal IP3R levels) disrupts the IP3 response [35]. However, in the context of decreased Purkinje IP3R abundance, augmented sensitivity (whether comparable to other cell types or greater) could potentially restore normal IP3 response if the IP3R1 is not lowered as drastically as the factor of 10 used in the Hernjak [35, 54] model.

The simulations were repeated with the climbing fiber stimulus applied 150 ms after the start of the train of the PF stimuli, unlike the previous results that have a timing difference of 50 ms. A 150 ms timing between the CF and PF stimuli falls outside the optimal time window suggested by both experiments [44, 56] and modeling [42]. Figure 3b shows that with this 150 ms time difference, supralinear calcium release is lost in the wild type case, as well as in the heterozygous case. This suggests that restoration of normal calcium release in the model does not alter fine-tuning of coincidence detection suggested by Brown et al. [42]. Thus, increasing IP3R1 sensitivity could be explored as a therapeutic tactic in SCA15/16 mouse models, e.g. IP3R1^{delta18/delta18} or IP3R1^+/− mice.

SCA2: application of ICpeptide in ataxias with pathologically increased sensitivity

Tang et al. also found that the huntingtin binding site on IP3R1 is immediately upstream of the PP1α-binding motif on the tip of the IP3R1 C-terminal [52]. The predicted minimal region on IP3R1 for huntingtin polyglutamine repeat binding (F2627-G2736) actually overlaps a small portion (5 aa) of the minimal PP1-binding motif (R2731-A2749)[52]. The huntingtin protein binding IP3R1 may therefore physically occlude or partially hinder binding by PP1αlpha in vivo and in situ. This would reduce PP1αlpha dephosphorylation of the receptor, leaving it in a hyperphosphorylated and therefore supersensitive state. A similar mechanism should occur with mutant Ataxins in SCA2 [10] and SCA3 [6] and also SCA1 [8, 10, 57], given that the mutant Ataxin and huntingtin proteins all have similarly expanded CAG repeats.

Figure 4 shows that adding 2 μM mutant Ataxin, e.g. mutant Ataxin-2 in SCA2, to the wild type model triples the increase in simulated fluorescence of the calcium indicator. Furthermore, the IP3R1 supersensitivity allows calcium concentration to start rising even before coincidence with calcium influx. In addition, we simulated the application of a different ICpeptide in our model. This ICpeptide, IC-G2736X (D2590-G2736)(Table 3), was also created by Tang et al., and encompasses the part of IP3R1 that encodes the minimal binding region for the expanded polyglutamine repeats in huntingtin, and likely the mutant ataxin proteins [52]. Further, Tang et al. used a β–galactosidase assay to show that IC-G2736X strongly associates with the mutant polyQ huntingtin [52]. In fact, they also used viral infection to insert IC10 into striatum of ataxic HD mice, and observed consequent stabilized calcium signaling, as well as neuroprotective effects [53]. In our model, IC-G2736X competitively binds the mutant Ataxin protein and blocks mAtaxin from binding endogenous IP3R1. The normal sensitivity of IP3R1 to activation by IP3 is therefore restored, and IP3R1 is no longer supersensitive. Figure 4 shows the restoration of normal supralinear calcium release. This result allows us to predict that experimental application of such an ICpeptide in cerebellar slices from mice with ataxias associated with supersensitive IP3R1 and increased calcium release may similarly restore normal calcium response. We expect that various concentrations of ICpeptide could be needed to normalize IP3R1 sensitivity and consequently calcium release in SCA1, SCA2, or SCA3, based on the degrees to which IP3R1 sensitivity to IP3 is pathologically increased. This could depend on the concentration of the mutant protein in the spine, as well as other unknown factors.

SCA1 (& SCA3): compensation versus pathology

Although expression of IP3R1 and other calcium signaling proteins is reduced in SCA3 mice [7] and in SCA1 mice and humans [9, 11], reduced calcium release is not experimentally observed in either disease. It is therefore not understood what role reduced IP3R1 plays in the pathology of these polyQ diseases. Figure 3 suggested that in SCA15/16 and other ataxias with reduced IP3R1 abundance and consequently reduced calcium release, therapeutic strategies to increase IP3R1 sensitivity to activation by IP3 could restore normal calcium release. This begs the question of whether, conversely, the downregulation of IP3R1 in ataxias with supersensitive IP3R1 could partially or wholly restore normal calcium response. A glimpse into a compensatory effect could lie in examining actual experimentally observed relative expression levels in SCA1 [9, 11], and using those observed levels in the model with supersensitive IP3R1. This allows the modeling and simulations to be driven by experimental observations. Accordingly, a 1.8-fold decrease (approximated from Lin et al. and Serra et al.) [9, 11] was simulated in the model for SCA1 pathology.

Figures 4a and 4b show that tuning down IP3R1 decreases the hyperactive supralinear calcium release to levels closer to wild type. This suggests that experimentally observed reduced abundance of IP3R1 may partially compensate for supersensitivity in SCA1 mice. However, other calcium signaling molecules were also downregulated by at least 1.8-fold [9, 11]. To determine the individual effect of the downregulation of some of the molecules, each molecule was sequentially downregulated. Tuning down SERCA in addition to IP3R1 has no additive effect. Myosin Va (MyoVa) and Homer 3 also are decreased [9, 11]. MyoVa and Homer 3 are thought to potentially (i) guide sER into the spine, and (ii) anchor sER in the spine, respectively [58, 59]. Decreasing sER volume in the spine by 1.8-fold in the ataxia model reduces the hyperactive calcium release to levels similar to tuning down IP3R1 or IP3R1 and SERCA. Although the correlation between Homer and MyoVa expression and sER volume fraction is probably not linear, it is remarkable that reducing the volume fraction by 1.8-fold gives a result that is close to directly tuning down IP3R1 and SERCA, which reside on the sER. Given the critical importance of IP3R1 on the sER in normal Purkinje neuron function, reduced volume fraction of spine sER could contribute to the pathophysiology of IP3R1-associated ataxia. However, in the context of IP3R1 supersensitivity, Homer and MyoVa downregulation could be compensatory. Finally, tuning down mGluR (by tuning down pulsatile IP3 production) in addition to SERCA and IP3R1 further decreases hyperactive calcium release, so that the resulting supralinear calcium release in the ataxic model is even closer to wild type. This is remarkable, suggesting that mGluR downregulation [9, 11] is also compensatory, and additive to IP3R1 downregulation. Not surprisingly, Figure 5b suggests that the greater the concentration of mutant Ataxin, the less the compensation achieved by tuning down IP3R1, SERCA, mGluR, and Homer.

Expression of calcium-binding proteins, parvalbumin and calbindin, are also experimentally observed to be reduced [9, 11]. It is not known whether this contributes to pathology or compensation. Calbindin knockout mice crossed with SCA1 mice worsens the phenotype [9]. This suggests that low levels of these buffer proteins could accelerate pathology. It is possible that normal or high levels of buffer proteins may help to maintain homeostasis in the presence of supersensitive IP3R1 early on in the SCA1 disease, then, when their expression is reduced, homeostasis is lost. Modeling results support this suggestion. Tuning down the calcium buffers parvalbumin and calbindin in the model makes supralinear calcium more robust than with down regulation of IP3R1, mGluR and SERCA alone (Figure 5a,b). Accordingly, the downward trend in the peak concentration of calcium is reversed with down regulation of calcium buffers. This is consistent with studies in Purkinje cells from parvalbumin and calbindin knockout mice that are ataxic [60, 61], where peak amplitudes of calcium transients evoked by climbing fiber stimulation were double those in wild type mice [62]. Notably, it has been suggested that because reduced expression of buffer proteins is observed in SCA1 mice before onset of symptoms, the timing of buffer protein downregulation may help determine the onset of symptoms [63, 64].

Notably, Liu et al. report that mutant SCA1 pathologically associates with IP3R1 [10], as do SCA2 and SCA3 [6, 10], suggesting a common supersensitive IP3R1 pathology, but they do not actually measure or report changes in IP3R1 sensitivity. Thus, we thought it was important to consider changes in calcium resulting from signaling protein downregulation, even if hyperactive IP3R1 is not assumed. Figures 5c and 5d illustrate that calcium responses resulting from IP3R1 and other calcium signaling protein downregulation, but with normal IP3R1 sensitivity, are less robust than wild type. This indicates that downregulation of these proteins in the absence IP3R1 supersensitivity does not appear consistent with the elevated calcium response relative to wild type, as observed in SCA1 mice [8]. The model therefore supports the hypothesis of IP3R1 supersensitivity in SCA1.

Influence of calcium on voltage-gated calcium-activated potassium channels

To better understand the interplay between calcium signaling and electrophysiology, we investigated the relationships between calcium influx, IP3R1-induced calcium release, and the voltage-gated calcium-activated potassium (K_Ca) channels. Our model includes the BK channel (‘big’ conductance calcium-activated voltage-gated potassium channel) and IK (intermediate conductance calcium-activated voltage-gated potassium channel) channels. Since more is known about the BK channels, we focused our study design and interpretation on these channels (see schematic in Figure 1b). The BK channel in the cerebellum is thought to contribute to repolarization of membrane potential transients in the dendrites, particularly calcium spikes (transient depolarization of membrane potential in spines and dendrites) resulting from the opening of voltage-dependent calcium channels in the dendrites [49]. In addition, experimental results by Sausbier et al. using ataxic BK knockout mice suggest that these channels are also important for the afterhyperpolarization of action potentials at the soma [65]. BK channels contribute to precision timing, and are thus important for normal firing of the cerebellar Purkinje neurons [65–70].

IP3R1 interacts closely with the BK channel, in glioma cells [71]. The BK channels seem to be in lipid rafts in the plasma membrane apposed to the sER. It is thought that in other cell types, including some neurons, the BK channels may form physical complexes with various plasma membrane calcium channels, bringing them within a few nanometers of the calcium channel pores [72]. We suspect that in the cerebellar Purkinje neuron spines, BK activity is also closely linked with the IP3R1-mediated calcium release. This idea is supported by direct activation of the BK channel by IP3R1-mediated calcium release in pyramidal neurons [73] and cereberal artery smooth muscle cells [74]. We therefore set out to investigate possible activation of BK channels (and IK channels) by IP3R1-mediated calcium release.

In the model, we allowed calcium released from the sER in eleven spines to activate the voltage-gated K_Ca channels. A group of eleven spines represents a small portion of a dendrite branchlet. (The Purkinje neuron has ~ 14 spines per μm² (Harris and Stevens, 1988)). Figure 6a shows that a 20 nA current injected at the soma for 400 ms gives a single sodium action potential spike, followed by a gradual incline. When calcium influx due to CF stimulation is applied 40 ms after the start of the current injection, an identical result is obtained (data not shown). When calcium release due to PF stimulation is applied 20 ms before the start of the current injection, a single sodium action potential spike is obtained, followed by the gradual incline, but then terminates with a short action potential train (Figure 6b). These action potential oscillations are similar to tonic firing oscillations observed experimentally [65]. An identical result is obtained when calcium influx is also added in 40 ms after the start of current injection. Thus in contrast to Figures 2 and 3, where calcium release is stimulated in only one spine, the transient calcium influx may not significantly enhance the overall effect of calcium release in 11 spines. This point is corroborated by examination of the corresponding calcium transients shown in Figure 6c. With PF stimuli only, in the absence of current injection, the Ca transient is minimal. With current injection at the soma, in the absence of biochemical signaling, there is a significant calcium transient at the level of the spines, corresponding to that first membrane potential spike. During that spike, sodium channels open at the soma, and calcium channels open throughout the dendrite. That allows calcium to rush into the cell. The calcium concentration then gradually rises, corresponding to the membrane potential incline. With PF stimuli preceding current injection onset by 20 ms, IP3 is already bound to IP3R1 when calcium rushes in through voltage-gated channels in the spines (in response to the depolarization of the membrane during current injection). This calcium also binds to IP3R1 on the sER. Thus, we have IP3 and calcium bound to IP3R1, yielding coincident activation of the receptor. In this case, coincidence detection is not of PF and CF stimuli, but of PF stimuli and current injection. IP3R1 can therefore serve as a coincidence detector for any activation of the Purkinje neuron that leads to changes in intracellular calcium concentration and increases in IP3 concentration. The figure shows that the resulting calcium transient is supralinear. With the CF stimuli only, a sharp calcium transient is obtained. When the CF stimulus is combined with current injection, the calcium transient is simply additive. This is because neither CF stimulus nor current injection provides IP3, which is necessary for coincidence detection at IP3R1. The corresponding modulation in the conductance of the BK channels is shown in Figure 6d. The figure indicates that BK channel conductance begins to rise almost immediately after either calcium release or calcium influx or both. Not surprisingly, the supralinear calcium transients give the biggest increases in conductance of the BK channels. The influence of the calcium transients on BK channels alters the membrane potential of the Purkinje neuron. The figure shows that with the calcium release from PF stimulus only, termination of the current injection results in a short train of action potentials. As with the membrane potential in Figure 6a, this spike does not occur in the absence of the PF stimulus. With supralinear calcium release activating BK channels, this also gives a termination spike that is identical to that obtained with calcium release only. CF stimulus alone does not alter the membrane potential.

In Figure 7, current injection commences significantly before both the PF and CF stimuli: PF stimulus occurs 160 ms after current injection, but still occurs 60 ms (within the experimentally observed time window [56]) before the CF stimulus. In addition, the current injection now lasts for 800 ms, as opposed to 400 ms in Figure 6. Figure 7a shows that in the presence of a prolonged current injection, a train of action potentials can be induced in the absence of any calcium increase. Consistent with Figure 6, with PF stimulus only, action potential spikes at the soma appear earlier (Figure 7b, top panel) than in the absence of the resulting biochemical calcium transient (Figure 7a, top panel). The bottom panels in Figures 6a and 6b show the corresponding calcium potential spikes at the spine, with similar results. These calcium spikes occur once a depolarization threshold is reached in the dendrites and spines, in response to high frequency repetitive firing of the Purkinje neuron (corresponding to high amplitude current injection) [49]. The resulting change in membrane potential in the dendrites and the spines propagates to and is reflected at the soma as the transient plateaus that occur in the midst of the sodium spikes. The resulting calcium transients are similar to those in Figure 6, in that superposition of parallel fiber stimulus and current injection gives supralinear calcium release, and further superposition with CF stimulus gives a calcium transient that is additive (Figure 7e). Also similarly, the supralinear calcium transient from calcium release and current injection leads to earlier calcium signal oscillations than in the absence of the biochemical calcium signal (Figure 7e). Interestingly, the calcium obtained from CF stimulus superimposed with current injection results in earlier membrane potential spiking (Figure 7c) than the supralinear calcium release from PF stimulus and current injection superposition (Figure 7b). Virtually identical results are obtained with coincident CF and PF activation superimposed with current injection (Figure 7d) as in Figure 7c. The calcium release portion of the calcium signal falls off more quickly upon superposition of CF and PF stimuli than supralinear calcium release from PF stimulus alone (Figure 7e). This can be attributed to the IP3R1 bell-shaped curve response to calcium concentration [75], where the calcium concentration rises sufficiently during combine Ca²⁺ influx and current injection to significantly populate the inhibited state of IP3R1. Since the calcium response falls off during this sustained potion of the overall Ca²⁺ signal (Figure 7e), the BK conductance also falls off (Figure 7f), thus affecting the membrane potential. Figure 7 suggests that the overall contour of the calcium signal and the resulting K_Ca conductances changes may underlie the effect of calcium on membrane excitability.

Taken together, these results suggest that in Purkinje neurons, coincident activation of the Purkinje spine by current injection at the soma and IP3R1-mediated calcium release with or without concurrent CF stimulus can alter membrane excitability. The results indicate that K_Ca activity increased by calcium release led to earlier action potential spikes than in the absence of biochemically induced calcium transients. This is reminiscent of a study by Sausbier et al. [65] in which control mice with wild type levels of BK activation showed more spontaneous discharge relative to ataxic BK^−/− mice. This is presumably due to another important result from that study: BK activity generates afterhyperpolarization of sodium (action potential) spikes. Thus, augmented BK channel activity polarizes the plasma membrane and increases excitability by more quickly resetting the sodium channels.

Computational models that combine the cell biology of calcium signaling with membrane electrophysiology involving calcium channels and calcium-activated ion channels can reveal emergent properties that are not observed in models of purely biochemical or purely electrophysiological models. The models that are presented in this work can be useful for studying these emergent properties in normal physiological Purkinje neurons, as well as in ataxia. Future studies can be developed to study spontaneous Purkinje neuron firing rate or precision of spike timing and interspike intervals in various classes of ataxias with mutations in genes that code for calcium channels [9, 76–78] and potassium channels [65, 79], as well as IP3R1 or related proteins (including SCA1, SCA2, SCA3, SCA15/16) [3, 4, 6, 7, 9–16].