Feedback inhibition underlies new computational functions of cerebellar interneurons

We used mice that express the light-activated cation channel, ChR2, exclusively in PCs to enable selective photostimulation of PCs in vitro. These mice were obtained by crossing transgenic mice expressing Cre recombinase behind a PC-specific promoter (PCP2) (Barski et al., 2000) with another transgenic mouse line expressing ChR2-H134R behind a floxed stop cassette (Madisen et al., 2012). Similar results were obtained using a second PCP2-cre line (Zhang et al., 2004; Witter et al., 2016). In these mice, light pulses generate photocurrents sufficient to evoke action potentials in PCs (Asrican et al., 2013). We examined ChR2 expression by imaging the yellow fluorescent protein (YFP) fused to the ChR2; virtually all PCs strongly expressed ChR2 in their somata and dendrites, with no fluorescence evident in MLI somata within the molecular layer (Figure 1a). Whole-cell patch clamp recordings in cerebellar slices from these double transgenic mice indicated that photostimulation reliably evoked action potentials in every PC examined (20/20), but was incapable of evoking any action potentials in MLI (0/406).

Figure 1

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Source files for properties of inhibitory input from PCs to PC-MLIs.

MLIs, molecular layer interneurons; PCs, Purkinje cells.

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We characterized synaptic connectivity between PCs and MLIs in sagittal slices of the cerebellar vermis, based on previous anatomical studies indicating that PC axon collaterals are oriented in the sagittal plane (Watt et al., 2009; Witter et al., 2016). We started by using relatively large spots of blue light (~0.2 mm² area) to photostimulate a large number of PCs (Figure 1b, top), while measuring postsynaptic responses in MLIs. Inhibitory postsynaptic currents (IPSCs) were observed only in MLIs located in the inner one-third of the molecular layer, where basket cells are located (Figure 1b, center). IPSCs were abolished by GABA_A receptor blockers, such as bicuculline and gabazine, but were unaffected by the glutamate receptor blocker kynurenic acid, indicating that the IPSCs are monosynaptic and mediated by GABA_A receptors. Connectivity between PCs and MLIs was observed in 16% of the 371 MLIs examined in the inner third of the molecular layer, where basket cells are located, in mice aged 3 weeks or older. No connections were detected (n=35) between PCs and MLIs located in the outer two-thirds of the molecular layer, where stellate cells are located (Figure 1b, bottom). We refer to the subset of MLIs receiving PC inputs as ‘PC-MLIs’ (Figure 1b, middle). The somata of PC-MLI were 60±2.9 μm (mean ± SEM, n=59) above the PC layer and thus are not candelabrum cells, interneurons whose somata reside within the PC layer (Lainé and Axelrad, 1994; Osorno et al., 2022). Inhibitory feedback from PCs to PC-MLIs was also observed in the lateral cerebellum: a total of eight connections were detected among 34 recordings in the inner molecular layer of simplex, crus I, and crus II lobules. This 24% rate of connections compares favorably to the ~16% observed in 371 recordings in the vermis. Therefore, the PC-to-PC-MLI feedback circuit seems widely distributed throughout the cerebellum and may play a general role in cerebellar computation. To determine how feedback from PCs affects the activity of PC-MLI, we photostimulated PCs with brief light pulses (5 ms) while depolarizing PC-MLIs to evoke rapid action potential firing. Light-evoked synaptic input from PCs was capable of completely suppressing PC-MLI activity (Figure 1c).

We then used recordings from pairs of connected PCs and MLIs for detailed characterization of the inhibitory feedback circuit. Although recordings from connected pairs were rarely obtained (see below), seven successful experiments (six whole-cell recordings and one cell-attached recording from PCs) allowed us to directly define the functional attributes of this circuit. Single action potentials (simple spikes) evoked in presynaptic PCs induced IPSCs in postsynaptic MLIs (Figure 1d). IPSC latency was brief, with a mean of 1.15±0.02 ms (Figure 1e). IPSCs were quite variable in amplitude (Figure 1f, red), with a coefficient of variation (CV) of 0.39±0.08 and peak amplitude of 56.1±20.3 pA at –50 mV (n=7). This indicates a synaptic conductance of 2.13±0.61 nS. The mean amplitude of spontaneous (presumed miniature) IPSCs was 18.0±1.3 pA (Figure 1f, blue), indicating a quantal content (ratio of amplitudes of evoked/spontaneous IPSCs) of 3.0±0.9 for evoked IPSCs. Recalculating based on IPSC integral, rather than peak amplitude, raised the estimate to 6.7; this reveals a substantial contribution of asynchronous GABA release (Song and Augustine, 2016). These unitary synaptic properties are similar to those reported for synapses formed between PCs by PC axon collaterals (Orduz and Llano, 2007; Watt et al., 2009; Witter et al., 2016). However, unlike PC-PC synapses (Watt et al., 2009; Witter et al., 2016), the PC-to-PC-MLI synapse exhibited very few failures (Figure 1d). This is a consequence of the relatively high quantal content and large readily releasable pool (RRP) of synaptic vesicles (99±16 quanta; n=15; see Figure 2h below). From the ratio of quantal content and RRP size, we estimate a release probability of 0.07.

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Source files for properties of synaptic depression at PC-to-PC-MLI synapses.

MLI, molecular layer interneuron; PC, Purkinje cell.

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Because PCs fire repetitively at rates up to 100 Hz in vivo (Thach, 1968; Zhou et al., 2014), we determined how sustained activity affected transmission at the PC-to-PC-MLI synapse at near-physiological temperature (34–35°C). Trains of presynaptic PC activity (10–100 Hz) initially caused synaptic facilitation, which was followed by a mild depression of IPSC amplitude (Figure 2a–b and Figure 2—figure supplement 1). Depression followed an exponential time course, with the rate of depression depending upon stimulus frequency (Figure 2c and d). Remarkably, IPSC amplitude was sustained at approximately half of its initial strength even during prolonged activity, independent of stimulus frequency (Figure 2e). As a result, the steady-state amount of synaptic transmission – measured as total IPSC charge – paradoxically increased at higher frequencies (Figure 2f). By measuring cumulative synaptic charge during repetitive activity (Figure 2g), we determined the size of the RRP (Figure 2h) and the rate of mobilization of synaptic vesicles from a reserve pool (Figure 2i). This analysis revealed that while RRP size is independent of the rate of synaptic activity, PCs sustain their transmission via an activity-dependent acceleration of synaptic vesicle mobilization from the reserve pool to the RRP. This is similar to the behavior of synapses between PCs and deep cerebellar nuclear neurons (Turecek et al., 2016). Thus, PCs can inhibit PC-MLIs even during high-frequency activity, thereby maintaining an inverse relationship between the activities of these two neuron types.

The circuit between presynaptic PCs and postsynaptic MLIs was mapped by scanning small spots of laser light (405 nm, 4 ms duration, ~1 μm diameter in the focal plane) to evoke action potentials in presynaptic PCs expressing ChR2 (Wang et al., 2007; Kim et al., 2014). Under these conditions, 1.2 µW or more of laser power could reliably evoke action potentials in PCs in cerebellar slices from mice 3 weeks or older. Optimal spatially resolved photostimulation (Kim et al., 2014) occurred at a laser power of 3 µW: at this laser power, light spots were capable of evoking action potentials at any location on the dendrite or cell body of individual PCs (Figure 3a, position 2), aside from their axons. In contrast, positioning the light spot more than 30 μm away from the PC soma along the PC layer failed to evoke action potentials (Figure 3a, position 1 or 3).

Figure 3

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Source files for spatiotemporal organization of PC-to-PC-MLI circuit revealed by optogenetic mapping experiment.

MLI, molecular layer interneuron; PC, Purkinje cell.

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The spatial organization of the circuit was mapped by scanning the laser light spot to evoke action potentials in presynaptic PCs, while recording IPSCs in MLIs. A light-evoked IPSC indicated a synaptic connection between a PC photostimulated at that location and postsynaptic PC-MLI; thus, correlating light spot location with the amplitude of IPSCs mapped the position of presynaptic PCs (Wang et al., 2007; Kim et al., 2014; Figure 3b). Recordings from stellate cells in the outer molecular layer yielded blank maps (Figure 3c), again indicating an absence of connections between PCs and stellate cells (n=35). IPSCs were evoked in PC-MLI only when the light spot was positioned within the molecular and PC layers, where PCs are located. Presynaptic PC inputs to PC-MLI were most often revealed (59%) as a single cluster of pixels similar in size to the light-sensitive area of a single PC (Figure 3d1), occasionally (12%) as two discrete input areas (Figure 3d2) or a relatively large contiguous input area (29%; Figure 3d3). Within an input field, there were no obvious ‘hot spots’ of synaptic input, consistent with the homogeneous light sensitivity of individual presynaptic PCs (Figure 3a).

To estimate the number of PC inputs converging on a PC-MLI, we compared the area of the synaptic input field (Figure 3d) with the area over which light spots could evoke action potentials in single PCs (optical footprint, Kim et al., 2014; Figure 3a). Mean input field area was 1.5 times larger than the mean area of PC optical footprints, suggesting that on average a PC-MLI receives inhibitory input from only one or two converging PCs (Figure 3e). This is consistent with the observed spatial organization of input maps shown in Figure 3d. The spatial organization of the PC-to-PC-MLI circuit is different from that of the MLI-to-PC circuit (Kim et al., 2014): presynaptic PCs were largely found within 200 µm of their postsynaptic PC-MLI (Figure 3f) and were asymmetrically distributed, with most presynaptic inputs originating from either juxtaposed PCs or PCs located toward the apex of the lobule (Figure 3g). These observations are consistent with the anatomy of PC axon collaterals (Watt et al., 2009; Witter et al., 2016).

Synaptic inputs from PCs to PC-MLI could be detected as early as postnatal day 14 (P14; Figure 3h), a time when MLIs are still developing (Altman, 1972; Pouzat and Hestrin, 1997). However, the rate of connectivity at ages earlier than P21 (7.5%) was somewhat lower than at later ages (16%), suggesting that the circuit is established during the time that MLIs mature. After P21, the probability of detecting functional connections between PCs and PC-MLI was relatively constant up to P77 (Figure 3h), well after MLIs have fully matured (Altman, 1972; Pouzat and Hestrin, 1997). Thus, this synaptic circuit is functional into adulthood. No feedback inhibitory connections were observed between PCs and stellate cells at any time between postnatal weeks 3 and 9.

The large number of IPSCs collected during our mapping experiments enabled further characterization of synaptic transmission at the PC-to-MLI synapse. Within the PC input fields, IPSC amplitude varied substantially from pixel to pixel but this variability was not due to variations in photostimulation of presynaptic PCs (Figure 3a). The CV of light-evoked IPSCs was 0.28±0.02 in 18 experiments; this is similar to the CV of IPSCs observed in our paired recordings from PCs and PC-MLIs (Figure 1d) and provides a second indication of highly variable transmission at the PC-to-MLI synapse. Mean IPSC amplitude, measured at a holding potential of –40 mV, was 39.1±4.0 pA (Figure 3i; 2128 IPSCs in 18 recordings). This corresponds to a synaptic conductance of 1.15±0.12 nS, which is not significantly different from that measured in paired recordings (p=0.17, Welch’s t-test). IPSC rise time (20–80%) was relatively fast (1.69±0.13 ms, Figure 3j), similar to IPSCs produced at PC-to-PC synapses (Orduz and Llano, 2007; Watt et al., 2009; Witter et al., 2016) but more rapid than the MLI-to-PC connection (5.6±0.5 ms time-to-peak; see Figure 5c of Kim et al., 2014).

Given that MLIs provide feedforward inhibition to PCs, we next asked whether circuits between PCs and MLIs are reciprocal, that is, whether MLIs innervated by a given PC also inhibit the same PC. For this purpose, we combined paired electrophysiological recordings with optogenetic mapping. Due to the abundance of PCs, we used mapping to increase the probability of detecting functional PC-MLI connections: after finding a PC-MLI and identifying the location of its presynaptic PC by scanning the photostimulating laser spot, as in Figure 3b and d, recordings were made from the somata of PCs within the area where the light spot evoked IPSCs. Among 139 paired recordings made via this strategy, synaptic inputs from PCs to MLIs were detected in seven cell pairs. This 5% success rate is an underestimate of PC-to-PC-MLI connectivity; although our optogenetic mapping identified the volume in which presynaptic PCs were located, there was still a low likelihood of sampling the one or two presynaptic PCs within the dozens of PCs within this volume. The results from one successful experiment are shown in Figure 4a. Action potentials evoked in the PC by depolarizing current pulses elicited IPSCs in the postsynaptic PC-MLI (Figure 4a, upper traces). However, evoking action potentials in the PC-MLI did not elicit IPSCs in the PC, indicating that this pair did not have a reciprocal connection (Figure 4a, lower traces). This was the case for all coupled PC/PC-MLI pairs.

Figure 4

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Source file for temporal relationship in connected PC/PC-MLI pairs.

MLI, molecular layer interneuron; PC, Purkinje cell.

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As an alternative strategy, we randomly selected PCs within 100 µm of an MLI located in the inner third of the molecular layer. Synaptic inputs from MLIs to PCs could be detected as IPSCs evoked in PCs in response to MLI stimulation (Figure 4b, lower traces, 13/54 pairs). However, in such cases, action potentials in the postsynaptic PC invariably failed to evoke reciprocal IPSCs in the MLI (Figure 4b, upper traces). In 13 experiments where stimulation of an MLI in the inner one-third of the molecular layer evoked IPSCs in a PC, in no case did stimulation of the PC produce IPSCs in the MLI partner (Figure 4c). This includes three MLIs that were confirmed to be PC-MLIs, because they exhibited IPSCs in response to PC photostimulation. Overall, in a total of 194 paired recordings, only unidirectional connections (either PC-to-MLI or MLI-to-PC) or unconnected pairs were detected; in no case were reciprocally connected PCs and MLIs observed (Figure 4c). Although we cannot completely exclude the possibility of reciprocal connections, due to the limited number of recordings from connected pairs, our findings of no reciprocally connected cells in 10 connected PC/PC-MLI pairs indicate it is unlikely that these connections are reciprocal. Simulations indicate that our results would almost never be obtained (probability less than 0.001) if 50% of PC/PC-MLI connections were reciprocal, a 0.05 probability of observing no reciprocal connections in 10 trials if 20% of all connections are reciprocal and a probability greater than 0.9 if 1% or less of the connections are reciprocal. In summary, our data indicate that few, if any, PC/PC-MLI connections are reciprocal; thus, PCs will influence other PCs primarily through disinhibition, while not influencing their own behavior through recurrent inhibition (Figure 4d).

To describe how an action potential in a presynaptic PC influences spiking in postsynaptic PC-MLIs, we examined the temporal relationship between spikes in PCs and PC-MLIs (four cell pairs). This cross-correlation analysis revealed that inhibitory inputs from PCs to PC-MLIs produced a clear inverse correlation in the activity of these two cell types (Figure 4e), similar to the results shown in Figure 1c. Shuffling the timing of PC spikes eliminated this effect (Figure 4f), indicating that the inverse correlation normally observed after a PC action potential is genuine and is not an artifact of our analysis method.

To determine the role of this feedback circuit in cerebellar network function and behavior, we examined the activity of PCs and MLIs during eyelid conditioning, a cerebellar-dependent form of motor learning (Halverson et al., 2015). Eyelid conditioning involves pairing of eyelid stimulation, as an unconditioned stimulus (US), with a preceding tone that serves as a conditioned stimulus (CS). During repeated pairing of the CS with the US, the eyelid gradually produces a conditioned response (CR) to the CS. This paradigm allowed us to examine the relationship between PC and PC-MLI activity during a well-characterized and cerebellum-dependent behavioral task, both in silico and in vivo.

To provide a controlled experimental platform to predict how PCs and PC-MLIs interact in vivo – in the presence and absence of feedback inhibition from PCs to PC-MLIs (red in Figure 5a) – we used a previously established large-scale computer simulation (Medina and Mauk, 1999; Medina et al., 2000; Medina et al., 2001; Medina et al., 2002; Ohyama et al., 2010; Kalmbach et al., 2011; Khilkevich et al., 2018). The details of this simulation are provided in the Methods section. A variety of interstimulus intervals (ISIs) were used for training the simulation, to yield CRs that varied widely in their timing. After training, the activity of MLIs (stellate and basket cells) was not correlated with CRs in the absence of feedback inhibition (Figure 5b1). However, providing PC feedback inhibition to a subset of MLIs, to match the PC-to-PC-MLI connectivity observed in our slice experiments, yielded a bimodal distribution of correlations: a subpopulation of MLIs had activity that was highly correlated with eyelid CRs (mean r=0.60 ± 0.06 SEM), while the activity of other MLIs had a minimal correlation with CRs (mean r=–0.02 ± 0.01 SEM) (Figure 5b2). Thus, feedback inhibition causes PC-MLI activity to be time-locked to cerebellar output (i.e. eyelid CRs).

Figure 5

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The simulation provided three experimentally testable predictions regarding the activity of PCs and PC-MLIs during eyelid conditioning: (1) There should be a bimodal distribution of correlations between MLI activity and behavioral CRs, with the highly correlated group being PC-MLIs (as in Figure 5b2). Thus, MLIs whose in vivo activity is correlated with CRs are likely to be PC-MLIs. (2) Because the activity of a subset of PCs (called eyelid PCs) decreases during CRs (Heiney et al., 2014; Halverson et al., 2015), the simulation predicts that the PC-to-PC-MLI feedback circuit will cause the activity of PC-MLIs to increase (via disinhibition) as presynaptic eyelid PCs become less active during the CR. Further, if feedback inhibition of PC-MLIs is indeed responsible for the inverse activity of PCs and PC-MLIs, then the simulation also predicts that (3) decreases in PC activity should precede increases in MLI activity during CRs. The following sections describe experiments that test these predictions via in vivo recordings of cerebellar neuron activity during CRs.

To illustrate motor learning behavior, Figure 6a1 depicts eyelid responses in rabbits after conditioning. In this and subsequent figures, the gray shaded area represents the timing of the CS (tone). Anticipatory eyelid movements occur in response to the CS only after conditioning, indicated by upward deflections (dark gray) during the CS. In Figure 6a, these movements are sorted according to the time of eyelid closure relative to the start of the CS, with CR onset latency indicated by red points. Simultaneous measurements of action potentials (simple spikes) in an eyelid PC, identified by the presence of US-evoked complex spikes (Halverson et al., 2015), during these CRs are shown in Figure 6a2. A peristimulus time histogram (Figure 6a2, upper) illustrates that the activity of this eyelid PC decreased during the CRs. A raster plot (Figure 6a2, lower) reveals that these decreases in eyelid PC activity were highly correlated, on a trial-by-trial basis, with the time of onset of eyelid movement (red points). Previous work has established that eyelid CRs are driven by these learning-dependent decreases in eyelid PC activity (Heiney et al., 2014; Halverson et al., 2015).

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The simulation predicts that PC-MLI activity should be correlated with eyelid CRs. We tested this prediction by recording MLI activity during eyelid CRs. In extracellular recordings, MLI activity can be distinguished by an established algorithm that categorizes neurons according to their baseline firing properties (Simpson et al., 2005). We observed that a small subset of MLIs increased their activity at the time that CRs occurred. Figure 6b2 shows an example of such an MLI: the activity of this cell increased during the CR, evident in a peristimulus time histogram (upper) and in raster plots (lower) of action potential firing. The corresponding eyelid CRs are shown in Figure 6b1. The additional examples shown in Figure 6c and Figure 6—figure supplement 1b–d also illustrate striking correlations of MLI activity with the time of CR onset. Because only a subset of MLIs receive PC feedback, the model predicts that not all MLI responses should be well correlated with CRs (Figure 5b2); consistent with this, Figure 6c shows examples of MLIs whose correlation with CRs ranges from 0.2 to 0.91.

We analyzed the correlation between MLI activity and CR timing during eyelid conditioning for a total of 814 cells (10 rabbits), using three different ISIs. As in the simulation, the majority of MLIs had little or no correlation with the kinematics of CRs (Figure 5b3). This broad range of MLI activity could be fit by the sum of two Gaussian distributions (blue and red curves in Figure 5b3). The means of each MLI subpopulation (red: r=0.43 ± 0.01 SEM, blue: r=–0.05 ± 0.01 SEM) were similar to those predicted by the simulations that included PC feedback (Figure 5b2). The skewness (rightward shift) of the in vivo distribution (0.84 in Figure 5b3) also more closely matched the skewness (0.65) of the simulated distribution with PC feedback (Figure 5b2) than that of the simulated distribution without PC collaterals (0.43; Figure 5b1). These results are consistent with prediction (1) above.

The subset of MLIs showing a relatively high correlation with CRs during eyelid conditioning are likely to be PC-MLIs; thus, we will refer to them as putative PC-MLIs. We operationally defined putative PC-MLIs as cells whose activity was correlated with eyelid position with a correlation coefficient of 0.43 or greater. This criterion identifies MLIs with a 95% or greater probability of being within the high-correlation population (red curve) of MLIs that create the high skewness of the distribution shown in Figure 5b3. Most of the 71 putative PC-MLIs identified in this way (8.7% of all MLIs) came from sessions where the activity of nearby eyelid PCs was recorded simultaneously, to ensure that the PC-MLIs were nearby the eyelid PCs (within the detection range of the recording tetrodes, which is approximately 120 µm). These connectivity rates and distances are comparable to our observations in cerebellar slices, where we found that approximately 5–6% of MLIs receive PC feedback inhibition (Figure 1b) that extends over 200 μm or less (Figure 3). Correlations between the activity of all MLIs examined with eyelid position and velocity are shown in Figure 6—figure supplement 1a.

Recordings from the 71 putative PC-MLIs – including the examples shown in Figure 6b2, c2 and c3 – reveal two clear relationships between the activity of putative PC-MLIs and eyelid CRs: (i) there was little or no response in these neurons in trials where no CR occurred (bottom of the sorted raster plots) and (ii) there was a burst of activity in putative PC-MLIs that closely tracked the time of onset of CRs (red points) on a trial-by-trial basis. These results are consistent with prediction (2) of the simulation. In many MLIs – both putative PC-MLIs and other cells identified as MLIs – an additional increase in activity was often observed at the beginning of the CS. Examples of such responses are shown in Figure 6c2 and c3. Such responses were also observed in PCs and were not correlated with the occurrence of a CR; thus, they are sensory responses associated with detecting the sound stimulus.

The activity of putative PC-MLIs was also correlated with the kinematics of conditioned eyelid movements. Figure 7 compares the mean time course of the activity of all putative PC-MLIs (Figure 7c) to the position (Figure 7a) and velocity (Figure 7b) of eyelid CRs, sorted according to the ISI used for training. Despite the loss of temporal precision imposed by averaging, the firing rate of putative PC-MLIs corresponded well to both average eyelid position and velocity at each ISI. These high correlations are likely a result of activity driven by feedback from eyelid PCs, whose activity highly anti-correlates with both measures (Halverson et al., 2015). Additional analyses revealed that the activity of putative PC-MLIs closely tracked both the onset latency and amplitude of CRs for each ISI tested (Figure 7—figure supplement 1). In sum, we conclude that conditioning-related increases in PC-MLI activity are closely linked to eyelid CRs, consistent with prediction (2) of the simulation.

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To test prediction (3) of the simulation, we examined the relationship between the activity of PCs and putative PC-MLIs during CRs. For this purpose, we employed recordings of the activity of pairs of PCs and putative PC-MLIs detected by the same tetrode. We began by considering the baseline activity for 10 s prior to CS onset and calculated cross-correlograms of putative PC-MLI activity triggered by PC simple spikes. We were unable to evaluate the data at time 0 because simultaneously recorded action potentials from two units on the same tetrode merge into a signal that cannot be assigned to either unit during sorting (Gao et al., 2012). Following a PC spike, there was a significant reduction in putative PC-MLI firing in six PC/PC-MLI pairs (Figure 8a and Figure 8—figure supplement 1a–b). This relatively small and brief decrease in activity was specific to PC/PC-MLI pairs identified on the same tetrode. The pattern of cross-correlation between connected PCs and putative PC-MLIs was qualitatively similar to that observed in slices (Figure 4e) and was abolished when the timing of PC spikes was shuffled (Figure 8—figure supplement 1a–b).

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We also performed a similar cross-correlation analysis on PC/PC-MLI pairs in the cerebellar simulation. There was a qualitative match between the in vivo recordings and simulations: a transient reduction in PC-MLI firing rate was observed following a PC action potential (Figure 8b) and was abolished when PC spike activity was randomized. A weaker effect of single PC spikes in vivo, compared to the effects observed in simulations (Figure 8b) and in slices (Figure 4e), probably arises from the much larger number of spontaneous excitatory inputs to each neuron in vivo. Differences in the timescale of the responses between in vivo and brain slice data are likely due to the lower neuronal firing rate in slices (which yields longer time intervals between PC spikes) as well as temperature differences between the two experimental systems. In summary, cross-correlation analysis from all three data sets demonstrated an inverse relationship between PC and PC-MLI activities, which is consistent with prediction (3). This correspondence strengthens the conclusions that PCs provide inhibitory feedback to PC-MLIs in vivo and that putative PC-MLIs identified in vivo are equivalent to the PC-MLIs identified in cerebellar slices.

Because of the relatively modest effects of individual PC action potentials on PC-MLI activity observed in vivo (Figure 8a), we extended our cross-correlation analysis to the larger bouts of PC activity occurring during action potential bursts, as well as the large reductions occurring during pauses in PC spiking. These analyses were restricted to recordings done between conditioning trials, to avoid the large changes in activity that occur during CRs (e.g. Figure 6a and b and below). Pauses were defined as the largest 35% of inter-spike intervals, while bursts were the smallest 35% of inter-spike intervals (50% for simulations). When spikes for a PC/PC-MLI pair were aligned to the start of a pause in PC activity, we found that these pauses clearly increased activity in putative PC-MLIs (Figure 8c; average of 11/27 PC-PC-MLI pairs detected on same tetrode). Further, bursts of PC activity decreased putative PC-MLI activity (Figure 8d; average of 13/27 PC-PC-MLI pairs). Examples of responses of individual pairs of PCs and PC-MLIs during PC pauses and bursts also showed strongly anti-correlated activity (Figure 8—figure supplement 1c–d).

These results indicate that the inverse relationship between PC and PC-MLI activity observed in vivo is more prominent during large changes in PC activity, compared to single PC spikes. The need for relatively large changes in PC activity in vivo highlights the importance of the frequency-independent synaptic transmission at the PC-to-PC-MLI synapse illustrated in Figure 2. By demonstrating that the inverse relationship between PC and PC-MLI activity is driven by changes in PC firing, these results are consistent with prediction (3) of our simulation.

We further tested prediction (3) by simultaneously measuring the activity of PCs and putative PC-MLIs during eyelid conditioning at three different ISIs. Examples of such recordings are shown in Figure 9. Peristimulus time histograms for the paired activity of an eyelid PC (green) and a putative PC-MLI (black) are shown in Figure 9a and b. In each case the eyelid PC and the putative PC-MLI fired inversely, with decreases in PC activity and increases in putative PC-MLI activity both corresponding – on a trial-by-trial basis – with the eyelid CR (red points). For all three examples in Figure 9, eyelid PC activity decreases before both CR onset and the increase in putative PC-MLI activity on every trial when a CR occurred; this is clearest in superimposed plots of average firing rate of both cells (Figure 9c). This temporal pattern is blurred when calculating the average firing rate for all eyelid PCs and putative PC-MLIs (Figure 9—figure supplement 1a) highlighting the importance of investigating individual pairs to determine the sequence of firing of PCs and putative PC-MLIs. In general, eyelid PC activity not only declined faster than PC-MLI activity increased, but also reached a minimum earlier during the CS. These unambiguous results are consistent with prediction (3) of the simulation.

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Because of the complex kinetics of the responses of eyelid PCs and PC-MLI during CRs, we employed two alternative approaches to quantify the precise temporal relationship between these two responses. At the single-trial level, we measured the delay between the time at which PC activity reached 50% of its minimum during the CS and the time at which the activity of putative PC-MLIs reached 50% of its maximum for the pairs shown in Figure 9. For each ISI, whenever a CR was present (red points) PCs reached their half-maximal response before putative PC-MLIs did (Figure 9—figure supplement 1b). Importantly, this temporal relationship was not observed during trials without CRs (blue points). Next, we aligned firing rate to CR onset and examined this relationship for the entire ensemble of 136 pairs of simultaneous recordings of the activity of putative PC-MLIs and eyelid PCs during conditioning (Figure 9—figure supplement 2a). Individual pairs of PCs and putative PC-MLIs were examined by calculating a ratio representing the magnitude of changes in activity prior to CR onset, normalized to the peak amplitude of the change during the entire interval (Figure 9—figure supplement 2b). The distribution of these differences, color-coded by ISI, is shown in Figure 10. This distribution has a mean (white dashed line) that is greater than zero (black dashed line), indicating that eyelid PCs decreased their activity before putative PC-MLIs increased their activity in a majority of cases. A t-test (verified with Jarque-Bera, p=0.5) indicated that the combined distribution was significantly different (p=8 × 10^–15) from a normal distribution with a mean of 0. Individual ISI distributions were also skewed rightward (skewness = 0.27 for ISI 250, 0.36 for ISI 500, and 0.78 for ISI 750), indicating that, on average, eyelid PC activity decreased before putative PC-MLI activity increased.

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Figure 10—source data 1

Source files for ∆PC-∆MLI distribution between the responses of simultaneously recorded eyelid PC and eyelid PC-MLI during eyelid conditioning at different interstimulus intervals (ISIs).

Full and preCR PC normalized firing rate decrease (∆PC = preCR/full PC activity decrease) and full and preCR MLI normalized firing increase (∆MLI = preCR/full MLI activity increase) were measured to access the timing of PC and PC-MLI activity changes in recorded pairs (∆PC-∆MLI; see Figure 9—figure supplement 2b). MLI, molecular layer interneuron; PC, Purkinje cell.

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In summary, the activity of PCs and putative PC-MLIs is anti-correlated. This is true both during CRs and at other times, and is particularly evident during large changes in PC activity. Further, several different types of analysis all point to the conclusion that changes in PC activity precede both CR onset and changes in putative PC-MLI activity. All these results are consistent with prediction (3) of the simulation. Collectively, our measurements of the activity of PCs and putative PC-MLIs – and their relationship to CRs – are in line with all three predictions of the simulation. Given that the computational predictions require the presence of PC-to-PC-MLI feedback inhibition, we conclude that this feedback circuit likely operates in the cerebellum during motor learning.

Given the success of the computational simulation in predicting the in vivo properties of PC-MLIs, we further interrogated the simulation to understand the role of feedback inhibition in cerebellar circuit performance during conditioning.

First, we determined the origins of the temporal relationship between changes in eyelid PC activity and PC-MLI activity during conditioning. Because both PC and PC-MLI activity changes during eyelid conditioning, plasticity likely occurs at parallel fiber (PF) synapses somewhere within the circuit; these synapses are known to be plastic and potentially involved in associative learning (Ito, 1986; Lisberger, 2021). To deduce where this plasticity occurs, we tested simulations with different loci of learning-induced PF plasticity. Comparisons between the predicted and observed activity of PC-MLIs and PCs (Figure 10—figure supplement 1) indicated that the presence of plasticity at PF synapses onto PCs yielded changes in PC activity that preceded changes in PC-MLI activity, as observed in vivo (Figure 10—figure supplement 1a and b). However, if plasticity was present only at PF synapses onto MLIs, changes in firing rates of MLIs and PCs were approximately simultaneous, with changes in MLI activity slightly preceding changes in PC activity (Figure 10—figure supplement 1c and d). Including plasticity at both synapses produced too many MLIs that led PCs (data not shown), in clear contradiction to the in vivo data. In summary, only simulations with plasticity exclusively at PF excitatory synaptic inputs to PCs were capable of accurately predicting our observed results. Thus, it appears that learning-related plasticity may occur at the PF-to-PC synapse, rather than PF-to-MLI synapses.

Next, we considered the behavioral consequences of the feedback circuit from PCs to PC-MLIs via additional simulations that predicted the ability of the cerebellum to learn during eyelid conditioning. In these simulations, eyelid conditioning was mimicked by presenting mossy fiber and climbing fiber inputs over a wide range of ISIs. Examples of virtual eyelid responses, produced by simulations trained at an ISI of 200 ms, are shown in Figure 11a. It can be observed that while simulations with or without PC feedback learned, the simulation that included PC feedback performed better (Figure 11a, right). This was true at nearly all ISIs: for example, in Figure 11b and c only simulations with PC feedback were able to acquire CRs at the shortest (150 and 200 ms) and longest ISIs (ISI >700 ms). Even at intermediate ISIs, where simulations were able to learn without PC feedback, the presence of PC feedback improved the rate of acquisition and final asymptotic performance. This is evident from comparison of CR amplitude acquisition curves at ISI 500 ms in simulations with (black) and without (red) PC feedback (Figure 11b). These results indicate that PC feedback can increase the rate of learning, particularly improving performance on the temporal margins, where stimulus patterns otherwise make learning difficult.

Figure 11

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Finally, to better understand how PC feedback improves learning, from the simulation results we calculated training-induced changes at all 12,000 excitatory PF synapses between granule cells and PCs. In Figure 12a–d, synapses that underwent net LTD appear as negative changes in synaptic weight, while those showing LTP have positive changes in weight. While most synapses underwent little or no net change in strength, more PF synapses underwent LTD than LTP in simulations both with and without PC feedback (Figure 12a and b). However, comparing the outcome with and without feedback (Figure 12c) revealed that fewer PF synapses underwent LTD in the presence of PC feedback. Subtraction of the results from the two simulations reveals the net decrease in learning-dependent LTD of PF synapses resulting from PC feedback (Figure 12d). To compare this effect across all 11 ISIs, we converted different histograms, such as the one in Figure 12d, into cumulative distributions of changes in synaptic weights (Figure 12e); the blue line (ISI 500) is derived from the data shown in Figure 12d. For all ISIs, PC feedback is predicted to cause fewer synapses to undergo LTD (Figure 12e). In summary, simulations with PC feedback learned better and required less net LTD of PF synaptic efficacy, indicating a more efficient distribution of learning-induced changes in synaptic strength.

Figure 12

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It has long been known that PC collateral axons bifurcate and extend along the PC layer (Hámori and Szentágothai, 1968; Larramendi and Lemkey-Johnston, 1970; Chan-Palay, 1971; O’Donoghue and Bishop, 1990; Witter et al., 2016), to inhibit other PCs and to reach into the lower molecular layer and upper granule cell layer. In rodents, PC axon targets outside of the PC layer are mainly MLIs within the molecular layer as well as Lugaro cells and granule cells in specialized regions of the granule cell layer (Larramendi and Lemkey-Johnston, 1970; Chan-Palay, 1971; Guo et al., 2016; Witter et al., 2016). Here, we built upon this anatomical information, and limited physiological analysis done in vivo (Bloedel and Roberts, 1969; O’Donoghue et al., 1989) or in slices (Witter et al., 2016), to functionally characterize the monosynaptic connection between PCs and MLIs.

Connections between PCs and MLIs are selective: PCs apparently connect to only a fraction of MLIs within the inner one-third of the molecular layer. It remains to be determined whether the MLIs that receive PC feedback inhibition are basket cells or some other type of interneuron. Our connectivity rate is lower than a previous observation (Witter et al., 2016); this difference may arise from our more reliable (seven times larger) sample size. Although the fraction of MLIs receiving PC input could be underestimated, due to loss of connections during slicing of the cerebellum, our observations are consistent with anatomical observations of sparse representation of PC axons within the molecular layer (Chan-Palay, 1971; Watt et al., 2009). Our observation of no functional connectivity between PCs and stellate cells in the outer two-thirds of the molecular layer is also consistent with anatomical studies (Hámori and Szentágothai, 1968; Lemkey-Johnston and Larramendi, 1968). These connectivity features account for the difficulty in finding connected pairs of neurons, both in vitro and in vivo.

We were able to characterize the fundamental properties of the PC-to-PC-MLI circuit. Inhibitory synaptic responses evoked by PC stimulation were sufficiently strong to prevent action potential firing in postsynaptic PC-MLIs. These responses had a short onset latency and were unaffected by a glutamate receptor antagonist, consistent with a monosynaptic connection. Other noteworthy properties of IPSCs at this synapse include variable amplitudes and fast rise times. These properties generally are similar to those of IPSCs at the inhibitory synapse formed between PCs via axon collaterals (Orduz and Llano, 2007; Watt et al., 2009; Witter et al., 2016), aside from a lower rate of failures at the PC-to-PC-MLI synapse. The latter property arises from a relatively large RRP of synaptic vesicles. A remarkable activity-dependent enhancement of mobilization of reserve pool synaptic vesicles allows this synapse to sustain transmission even at the high rates of activity typical of PCs.

Our high-speed mapping visualized the spatial organization of PC inputs onto a postsynaptic PC-MLI and revealed one or two discrete areas of input within the molecular and PC layers, where PCs are located. The presence of PC-MLI receiving two discrete areas of presynaptic PC input (e.g. Figure 3d2), as well as a mean presynaptic input area 1.5 times larger than the size of the optical footprint of a single PC (Figure 3e), indicates that on average one or two PCs converge onto a postsynaptic MLI. This is in line with the relative abundance of PCs and PC-MLIs: our measurements indicate that approximately 1/18 of all MLIs receive PC feedback, while previous estimates suggest that the number of PCs is approximately 1/10 that of MLIs. Thus, PCs are 1.8 times more abundant than PC-MLIs, which is in reasonable agreement with the 1.5-fold convergence between PCs and PC-MLI that we observed. Because PCs are the sole output of the cerebellar cortex, any cell that influences PCs will influence behavior; the comparable numbers of PCs and PC-MLIs indicates high potential for such influences. The relative abundance of PCs and PC-MLI also suggests that nearly every PC provides feedback inhibitory drive to PC-MLIs. Our observation that PC-MLIs sometimes receive two convergent PC inputs (Figure 3d2) contrasts with a previous suggestion that only single PCs innervate MLIs (O’Donoghue et al., 1989). PC-MLI connections were mostly located within 200 µm from postsynaptic PC-MLI somata, consistent with the reported sagittal extent of PC axon collaterals (O’Donoghue and Bishop, 1990; Watt et al., 2009; Witter et al., 2016). We also found that a majority of presynaptic PCs were located on either the apical or juxtaposed side relative to postsynaptic MLIs, verifying anatomical observations (Watt et al., 2009; Witter et al., 2016).

Despite using two experimental strategies that optimized detection of either PC-to-MLI or MLI-to-PC connectivity, in no case were reciprocally connected pairs of PCs and MLIs observed (Figure 4c). Although it was previously assumed that PC-to-PC-MLI circuits are non-reciprocal (O’Donoghue et al., 1989; Guo et al., 2016), our results provide the initial experimental test of this important point. Such an arrangement allows feedback inhibition of PC-MLIs by PCs to disinhibit neighboring PCs (Figure 4d), which may help to synchronize PC activity in the sagittal plane. It will also serve to prevent network oscillations, which are often the outcome of reciprocal inhibitory circuits (Satterlie, 1985).

This interconnectivity between PCs and MLIs gave rise to specific and testable predictions about the activity of PC-MLIs in vivo. MLIs that receive input from PCs should fire inversely with PCs (Figure 4e) and, like PCs, their activity should correlate on a trial-by-trial basis with cerebellar-mediated behavioral responses. We tested these predictions using eyelid conditioning because: (1) the role of the cerebellum in this task is well established; (2) the activity of eyelid PCs is well characterized and appears to control the timing and amplitude of the behavioral CRs; and (3) computer simulations of the cerebellum allowed us to explore the plausibility of alternative explanations for why MLI activity correlates, on a trial-by-trial basis, with PC activity and behavioral responses.

Recordings from MLIs during conditioned eyelid responses supported these predictions. A small fraction of MLIs showed unusually high trial-by-trial correlations with the eyelid responses. Computer simulations predicted such high correlations only when PC-to-PC-MLI connections were included, even when these simulations were constructed to exaggerate contributions from other sources. Simultaneous recordings showed that changes in eyelid PC activity tended to precede, on a trial-by-trial basis, that of putative PC-MLIs (Figures 9a–c–10). Cross-correlation analysis revealed similar inverse activity correlations between PCs and PC-MLIs across all three platforms we examined: cerebellar slices (Figure 4e), in vivo recordings (Figure 8a), and computational modeling (Figure 8b). Finally, we observed that computer simulations trained with eyelid conditioning procedures learned better when PC-to-MLI connections were included (Figure 11), despite these connections yielding less net synaptic plasticity (Figure 12). Even though previous versions of our simulation that did not include PC-to-MLI feedback could exhibit learning (Medina and Mauk, 2000; Li et al., 2013), simulations that included PC-MLI connections performed better, particularly in difficult learning conditions.

These results suggest that in networks where PCs inhibit MLIs, each PC acts to make other PCs fire more like itself through disinhibition (Figure 4d). In this way, PC-MLIs can exert strong influence over cerebellar processing and learning despite being a small fraction of MLIs. Because PC-MLIs are both inhibited by PCs and inhibit other PCs, their connectivity confers upon them particularly strong influence over PC activity and thence over the output of the cerebellar cortex. We observed inverse activity between PCs and PC-MLIs during instances of large increases or decreases in PC activity, both during large pauses in PC activity controlling expression of CRs and during the largest changes in PC activity during the inter-trial interval (pause/burst-triggered averages). As a consequence, PC feedback to PC-MLIs could serve to selectively communicate large changes in PC activity within a parasagittal strip through PC-MLIs.

Inhibitory neurons are ubiquitous throughout the brain and are often connected to each other by inhibitory synapses. Therefore, our results suggest a computational principle: activation that involves disinhibition – as occurs between PCs and PC-MLIs – could synchronize the activity of inhibitory neurons. Here, we use ‘synchrony’ as a broad term that includes PC pause synchronization. Pause synchrony can be exerted as diverse forms (pause beginning, pause ending, pause overlapping synchrony) for several milliseconds, which results in a variety of forms of modulation of neuronal activity and may lead to a broad range of behavioral outcomes (Sudhakar et al., 2015; Hong et al., 2016). A transient increase in PC synchrony recently reported to occur during saccades (Sedaghat-Nejad et al., 2022) also occurs over a timescale that is compatible with our observations of PC feedback and feedforward inhibition.

One apparent consequence of disinhibition-based synchronization of learning-related activity in PCs suggested by our simulations is to require a smaller net change in excitatory synaptic input for PCs to decrease their activity to the level required to produce a well-timed CR in response to the CS. Such economy of plasticity could be greatly beneficial for faster learning, better learning at difficult ISIs, and increased capacity to store learned responses. While it is well established that plasticity occurs at PF synapses between GCs and PCs (Ito, 1986; Lisberger, 2021) (blue in Figure 5a), such plasticity cannot be directly proven based on measurements of learning-related changes in PC activity. For example, these changes could instead result from plasticity at synapses onto MLIs (green in Figure 5a). However, such plasticity has only been inferred from learning-related changes in MLI activity and from MLI receptive field expansion (Jörntell and Ekerot, 2002; Jörntell et al., 2010). Indeed, previous evidence of learning-dependent changes in MLI activity during eyelid conditioning was also interpreted to result from plasticity at MLI synapses (ten Brinke et al., 2015). Our results substantially limit such inferences by showing that learning-dependent changes in MLI activity could instead come from interactions between PCs and PC-MLIs. While our results do not completely exclude the possibility of learning-related changes at PF synapses innervating MLIs, they do demonstrate that learning-related changes in the activity of MLIs (or PCs) do not necessarily indicate changes in the strength of synapses onto MLIs (or PCs). Further work will be required to fully sort out the contributions of MLI synaptic plasticity to learning. Independent of detailed mechanism, we nonetheless conclude that feedback inhibition from PCs to PC-MLIs contributes to more efficient cerebellar-mediated motor learning.

We employed a computational simulation intended to emulate the synaptic organization and physiology of the cerebellum (Eccles et al., 1967; Hull and Regehr, 2022; Figure 5a). This simulation has been employed over many years and has been optimized to successfully model many aspects of cerebellar processing, including bidirectional learning, adaptive timing, and roles for multiple sites of plasticity, short-term plasticity, and recurrent feedback (Medina and Mauk, 1999; Medina et al., 2000; Medina et al., 2002; Ohyama et al., 2010; Kalmbach et al., 2011; Khilkevich et al., 2018). The details of this simulation have been presented elsewhere (Medina and Mauk, 1999; Medina and Mauk, 2000; Medina et al., 2000; Medina et al., 2001; Medina et al., 2002; Ohyama et al., 2010; Kalmbach et al., 2011; Li et al., 2013; Khilkevich et al., 2018) and source code is available at https://github.com/mauk-lab-utexas/CBMSim, (Halverson, 2022; copy archived at swh:1:rev:6f592845581c0f06bd80a9bd43dabba2000965bf). Here, we will briefly summarize the simulation. As an approximation to the ratio of cell types within the cerebellum, the simulation implemented the following neuronal elements: 600 mossy fibers, 12,000 granule cells, 900 Golgi cells, 24 PCs, 8 deep cerebellar nuclei cells and 4 climbing fibers. (Figure 14a). Further, connections between PCs and MLIs were simulated via 96 basket cells (including 9 designated as PC-MLIs) and 240 stellate cells (Figure 14b). These neurons were interconnected to emulate a parasagittal stripe, where all PCs receive input from climbing fibers of the same type – that is, those activated by an eyelid US and that control eyelid responses with their output. Each neuron was represented as a conductance-based calculation of membrane potential, with spikes determined by thresholds that varied according to recent spiking activity to implement the net effects of active conductances. Using 1 ms time steps, the change in membrane potential was calculated according to synaptic and leak conductances and to membrane capacitance. Spikes were initiated when membrane potential exceeded threshold. Threshold increased for each spike and returned exponentially to the baseline value. All synaptic conductances were based on time constants derived from in vitro studies; the threshold properties of each representation was tuned to produce behavior that matched the in vivo characteristics of the target cell type.

Figure 14

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The simulation represented the geometric relationships and divergence and convergence ratios of synaptic connections within the cerebellum by generating a two-dimensional array of granule cells, Golgi cells, and mossy fiber glomeruli. For each type of connection an eligibility span was specified to represent the region of the array that the presynaptic neuron could potentially make contact with a postsynaptic target. These spans are based on published accounts of the spatial relationships of connections within the cerebellum. For example, since the axons of granule cells run transversely through the cerebellar cortex, the contact area for granule cells was a narrow rectangle. While this area constrained the range over which cells could make connections, the connections were determined randomly in a way that produced the known divergence and convergence ratios (e.g. each granule cell could only receive four mossy fiber inputs).

Separate rules for plasticity were implemented at two types of synapses within the simulation: (1) the granule cell-to-PC synapses and (2) the mossy fiber-to-DCN synapses. A granule cell-to-PC synapse underwent LTD or LTP every time it fired a threshold burst of spikes, LTD occurred when this burst fell within a window between 300 and 100 ms prior to a climbing fiber input to the PC, otherwise LTP occurred. Mossy fiber-to-DCN synapses active within a time window of an abrupt pause in PC activity underwent LTD whereas those active during strong Purkinje activity underwent LTP.

Eyelid conditioning was simulated by presenting to the simulation mossy fiber and climbing fiber inputs based on empirical recordings during eyelid conditioning. Each mossy fiber was assigned a background firing rate between 1 and 40 Hz. To mimic activation of mossy fibers as a CS, a randomly selected 3% of the mossy fibers were designated phasic CS mossy fibers and were active for a brief 100 ms burst at CS onset. Another randomly selected 3% were designated tonic CS mossy fibers and were activate at a rate between 80 and 100 Hz throughout the duration of the CS. All mossy fiber activity was stochastic with the target firing rate for any given time used to determine the probability of activating an excitatory conductance in ways that made the actual activity noisy. The activation of an excitatory conductance for the four climbing fibers served to mimic the presentation of the US. The averaged and smoothed activity of the eight deep nucleus neurons was used to represent the output of the simulation and the predicted ‘eyelid response’ of the simulation. These simulations also included excitatory recurrent collaterals from the deep cerebellar nuclei, which deliver to the cerebellar cortex – as mossy fiber input – a copy of output that can be used for sequence learning (Giovannucci et al., 2017; Khilkevich et al., 2018). These collaterals were included in the same proportion as mossy fibers representing the CS: 5.2% of all mossy fibers (Houck and Person, 2015). To calculate the distribution of cross-correlations between PCs-basket cells activity, we repeated 1000 random draws of the same number of PC-BC pairs from the simulation, as we have total (64) in experimental data for ISIs 500, 700, and 750. These randomly drawn pairs were used to calculate cross-correlation values between PCs and basket cells average firing rates in the same fashion as with real data. For statistical comparison between correlation distributions of real data and simulations, we drew 1000 times from the simulation the same amount of PC-PC-MLI pairs as in real data and made a comparison using either paired Student’s t-test or two-sample Kolmogorov-Smirnov test for each draw.

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

We thank K Chung for excellent technical assistance and P Teo and M Yeow for administrative support.

All mouse procedures were conducted according to the Institutional Animal Care and Use Committee guidelines of the Biopolis Biological Resource Center (AUP 120707) and the Nanyang Technological University (AUP 18095). Treatment of rabbits and surgical procedures were in accordance with National Institutes of Health guidelines and an institutionally approved animal welfare protocol (AUP 2015-00137). All surgery was performed under anesthesia and every effort was made to minimize suffering.

1. Received: February 4, 2022
2. Preprint posted: March 3, 2022 (view preprint)
3. Accepted: December 7, 2022
4. Accepted Manuscript published: December 8, 2022 (version 1)
5. Version of Record published: December 21, 2022 (version 2)

© 2022, Halverson, Kim et al.

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