Due to the positioning of the SNr within the BG, synaptic integration of GABAergic projections from the direct (Str) and indirect (GPe) pathways in the SNr is likely a critical factor in BG function. Nonetheless, the effects of these two pathways on SNr activity are not well understood. Complicating matters, GPe and Str inputs form synapses on disparate locations on SNr neurons, undergo distinct short-term synaptic plasticity and likely have differing susceptibilities to breakdown of EG⁢A⁢B⁢A, mediated by the C⁢l- load. Therefore, to investigate synaptic integration of GPe and Str GABAergic inputs to the SNr in more detail, we constructed a conductance-based neuron model with somatic and dendritic compartments (Figure 1). The two compartments are electrically coupled and intracellular Cl⁻ concentration ([C⁢l-]i) is maintained in each compartment by the potassium-chloride co-transporter (KCC2). The baseline firing rate (≈ 10 Hz) and action potential peak of the model are tuned to match experimental data from in vitro mouse and rat slice recordings (Richards et al., 1997; Atherton and Bevan, 2005; Yanovsky et al., 2006; Zhou et al., 2008; Ding et al., 2011), while the AHP is tuned to match data presented in Higgs and Wilson, 2016. For a full model description see Materials and methods.

The GABAergic synapses from the GPe and Str neurons undergo short-term synaptic depression and facilitation, respectively. To decide how to implement and tune these effects in our model, we turned to the experimental literature. Two studies reported on short-term plasticity of GPe and Str projections in in vitro slice preparations (Connelly et al., 2010; Lavian and Korngreen, 2016). Because this data was averaged over multiple neurons and trials, we incorporated an established mean-field model of short-term synaptic depression/facilitation (Abbott, 1997; Dayan and Abbott, 2001; Morrison et al., 2008) into our simulated synaptic currents to capture short-term synaptic dynamics in our simulations.

Interestingly, the two experimental papers reported results that superficially appear to be at odds with each other. In Connelly et al., 2010, the magnitude of synaptic depression and facilitation of synapses onto SNr neurons was found to be largely independent of the tested stimulation frequencies (10 Hz, 50 Hz, 100 Hz). In contrast, in a BG output nucleus analogous to the SNr, the entopeduncular nucleus (EP), a similar characterization of the short-term synaptic dynamics of GPe and Str projections found that short-term depression and facilitation are highly frequency-dependent (Lavian and Korngreen, 2016). Moreover, the magnitude of synaptic facilitation of Str projections was shown to decrease in the EP for simulation frequencies above 10 Hz.

A critical distinction between these studies is that data was collected under a voltage-clamp configuration in Connelly et al., 2010 and under a current-clamp configuration in Lavian and Korngreen, 2016. Under current-clamp, the membrane potential (V_m) is free to change. Consequently, stimulation of GPe or Str projections hyperpolarizes V_m towards the GABAergic reversal potential (EG⁢A⁢B⁢A), which reduces the GABAergic driving force (Vm-EG⁢A⁢B⁢A) and ultimately decreases the magnitude of the inhibitory postsynaptic potential (IPSP). In contrast, the GABAergic driving force does not change under voltage-clamp, as V_m is fixed. In both voltage- and current-clamp EG⁢A⁢B⁢A may also be considered fixed due to the whole cell configuration and free ionic diffusion between the cell and recording pipette. Based on these considerations, we tuned our model to match the voltage-clamp data from Connelly et al., 2010, as it is likely a better representation of the underlying short-term synaptic dynamics of GPe and Str inputs (Figure 2). Interestingly, with this tuning, the short-term GPe and Str synaptic dynamics in our model when tested under current-clamp also reproduces the synaptic dynamics reported in Lavian and Korngreen, 2016. Specifically, GPe synaptic depression and Str synaptic facilitation are strongly frequency dependent, and the magnitude of synaptic facilitation in Str synapses decreases for stimulation frequencies above 10 Hz (Figure 3). These results demonstrate the importance of considering the differences between voltage- and current-clamp recordings when characterizing short-term synaptic dynamics. Additionally, these findings suggest that short-term synaptic dynamics of inputs from GPe and Str in the EP are tuned in a similar way to those in the SNr.

Next, we used our model to consider effects of variability of the GABA reversal potential on SNr responses to its GABAergic inputs. Maintenance of the C⁢l- gradient is largely determined by a neuron’s ability to preserve a low intracellular chloride concentration ([C⁢l-]i), which in turn depends on the balance of the neuron’s capacity for C⁢l- extrusion by the potassium-chloride co-transporter KCC2 (Doyon et al., 2011; Raimondo et al., 2012; Doyon et al., 2016; Mahadevan and Woodin, 2016) and the C⁢l- influx into the neuron that occurs through C⁢l--permeable ion channels that contribute to IG⁢A⁢B⁢A.

Due to the importance of C⁢l- regulation in GABAergic synaptic transmission, we first characterized the relationship among a conductance associated with a tonic chloride load (gG⁢A⁢B⁢AT⁢o⁢n⁢i⁢c), the C⁢l- extrusion capacity (gK⁢C⁢C⁢2), and EG⁢A⁢B⁢A in the somatic compartment of our model (Figure 4A). We found that EG⁢A⁢B⁢A may vary from approximately –80 mV with very low net C⁢l- influx to approximately – 45 mV with high gG⁢A⁢B⁢AT⁢o⁢n⁢i⁢c and low C⁢l- extrusion capacity; note that the level of depolarization of EG⁢A⁢B⁢A is also influenced by the H⁢C⁢O3- concentration gradient across the cell membrane (Kaila and Voipio, 1987; Kaila et al., 1989; Staley et al., 1995; Staley and Proctor, 1999; Raimondo et al., 2012; see Materials and methods, Equation 23). Importantly, depending on gG⁢A⁢B⁢AT⁢o⁢n⁢i⁢c and gK⁢C⁢C⁢2, EG⁢A⁢B⁢A can vary over ranges that correspond to excitatory, inhibitory and shunting effects of the resulting GABAergic current even for relatively small gG⁢A⁢B⁢AT⁢o⁢n⁢i⁢c.

Next, we investigated the effect of simulated somatic GABAergic projections from the GPe on the firing rate of the model SNr neuron. This was achieved by simulating optogenetic stimulation of the model’s somatic synapses at 40 Hz for 1 s. Four distinct types of SNr firing rate responses were observed: ‘complete inhibition’, ‘no effect’, ‘excitation’, and ‘partial inhibition’ (Figure 4B–E). Additionally, two sub-types of partial inhibition occurred: (1) deletion of one or a few spikes followed by a step reduction in firing rate (Figure 4E1) and (2) complete inhibition followed by a late escape and continuation of spiking (Figure 4E2). The type of response in the model depends on the magnitude of EG⁢A⁢B⁢A relative to V_m at the start of the stimulation, and, in the case of the second type of partial inhibition, the slow depolarizing drift of EG⁢A⁢B⁢A that is the result of intracellular C⁢l- accumulation. The effects of the short-term synaptic depression at these synapses on most of the SNr responses turns out to be minimal. This lack of effect arises because these synapses reach steady-state level of depression after approximately five stimulus pulses, which occurs after just 125.0 ms when stimulating at 40 Hz. The one exception occurs with the first type of partial inhibition, for which gG⁢A⁢B⁢A is large enough at the start of the stimulation window to cause an early spike deletion, after which depression can allow the reduced-rate firing to emerge.

We next performed a parallel analysis of the effects of simulated optogenetic stimulation of Str GABAergic projections in the dendritic compartment of the SNr model under the same stimulation protocol. As with the somatic compartment, we first characterized the relationship among gG⁢A⁢B⁢AT⁢o⁢n⁢i⁢c, gK⁢C⁢C⁢2 and EG⁢A⁢B⁢A in the dendritic compartment and found that EG⁢A⁢B⁢A varies over a comparable range (–80 mV to –45 mV), depending on the balance of C⁢l- influx and extrusion rates (Figure 5A). Stimulation of the dendritic GABAergic synapses resulted in the same four response types seen in the somatic compartment with an additional ‘biphasic inhibitory-to-excitatory’ response and a slightly different pair of ‘partial inhibition’ responses (Figure 5B–F), one mediated by the short-term facilitation of direct pathway synapses. Specifically, with repeated stimulation, the strengthening of these synapses can induce a gradual slowing in the SNr firing rate throughout the simulation, which may eventually stop neuronal spiking (Figure 5E1). Despite this facilitation, a form of partial inhibition consisting of an initial pause in SNr spiking followed by a recovery of spiking can also occur in the model with direct pathway stimulation, mediated by a sufficiently large C⁢l- accumulation to allow the effects of dynamic EG⁢A⁢B⁢A to dominate the post-synaptic response (Figure 5E2). The biphasic inhibitory-to-excitatory response type is an extreme case of the partial inhibition shown in Figure 5E2. This biphasic response occurs when EG⁢A⁢B⁢A is initially hyperpolarized relative to V_m, the Str GABAergic conductance is strong and the C⁢l- extrusion capacity is weak, which allows for unusually rapid C⁢l- accumulation and subsequent depolarization of EG⁢A⁢B⁢A near or above the action potential threshold. The biphasic response type is more likely to occur with Str rather than GPe stimulation, due to the larger surface area-to-volume ratio and concomitant increased susceptibility to C⁢l- accumulation in the dendritic compartment that Str inputs target, relative to the soma. In our model biphasic responses to GPe inputs can be elicited under some conditions, see Figure 4—figure supplement 1.

Our simulations in the previous sections predict that GABAergic inputs from the GPe and Str may produce a diverse range of effects on SNr activity depending on EG⁢A⁢B⁢A and [Cl−]i levels and dynamics. To test these predictions, we optogenetically stimulated the synaptic terminals from D1 striatal neurons of the direct pathway and from GPe neurons of the indirect pathway in the SNr for 10 s periods. During stimulation, we performed patch clamp recordings of SNr activity. Experiments were conducted in in vitro slice preparations and patch clamp recordings were performed in cell attached mode to avoid perturbing the intracellular C⁢l- concentration critical for GABAergic signaling. In response to optogenetic stimulation, we found a wide array of SNr response types, which we classified into five categories: (1) complete inhibition - cessation of spiking; (2) partial inhibition - sufficient reduction of firing rate with or without a pause; (3) no effect - no change in firing rate; (4) excitation - sufficient increase in firing rate; and (5) biphasic - decrease or pause in spiking followed by an increase in firing rate above baseline. This heterogeneity in SNr responses may relate to differences in slicing-induced damage and corresponding baseline EG⁢A⁢B⁢A values or to other local factors. Example traces for the response types observed with GPe and Str stimulation are shown in Figure 6 A1 and B1, and the frequencies of occurrence for these responses are quantified in Figure 6 A2 and B2; see also Figure 6—figure supplement 1 and Figure 6—figure supplement 2 for raster plots and firing rate time courses for all stimulation frequencies tested. All response types could be induced by optogenetic stimulation of the GPe or the Str projection; however, with GPe stimulation, biphasic responses were slower to emerge (see Figure 6—figure supplement 1) and less common overall than with Str stimulation, consistent with the absence of biphasic responses in our 1 s simulations of GPe inputs and with slower C⁢l- accumulation, over several seconds, in the soma than in the dendrite. Biphasic responses do emerge with simulation of longer GPe stimulation in our model, see Figure 4—figure supplement 1. In a portion of the neurons partially inhibited by GPe or Str stimulation, the duration of the pause in spiking is longer than can be explained by short-term synaptic dynamics. Additionally, the number of partially inhibited neurons with a ‘long pause’ increases with stimulation frequency (GPe: 10 Hz, 1/25; 20 Hz, 8/26; 40 Hz, 13/29; 60 Hz, 16/24; Str: 10 Hz, 1/25; 20 Hz, 8/26; 40 Hz, 13/29; 60 Hz, 16/24). These findings, in addition to the observation of biphasic responses, are consistent with gradual C⁢l- accumulation and depolarization of EG⁢A⁢B⁢A during the stimulation period.

Previous computational modeling studies that have shown that, due to the larger surface area-to-volume ratio of dendrites relative to the soma, C⁢l- accumulation and depolarization of EG⁢A⁢B⁢A is faster in dendritic compared to somatic compartments (Doyon et al., 2011; Ratté and Prescott, 2011), and this result could explain why biphasic responses were almost never seen with GPe stimulation below 60 Hz. Nonetheless, C⁢l- accumulation and depolarization of EG⁢A⁢B⁢A may still arise, on a slower time scale, with stimulation of the indirect pathway. If slow C⁢l- accumulation and depolarization of EG⁢A⁢B⁢A are indeed occurring, then the strength of inhibition should slowly weaken during stimulation, which will result is a slow increase in firing rate during the stimulation period.

Measurements of spiking frequency relative to baseline during and after stimulation of the GPe and Str projections as a function of stimulation frequency (Figure 6 A3 and B3) support the idea that EG⁢A⁢B⁢A dynamics may contribute to synaptic integration within the SNr. For this analysis, we divided the stimulation period into 1 s intervals in order to assess any dynamic changes in the strength of the input over the course of stimulation. We found that Str projections are initially more effective at inhibiting SNr spiking relative to GPe projections (Str: 72.3–76.7% peak reduction, GPe: 43.1–61.9% peak reduction). Interestingly, for both GPe and Str projections, the strength of inhibition decreases on average during the stimulation period, consistent with slow accumulation of intracellular chloride. Moreover, the loss of firing rate reduction was most prominent for Str stimulation at high frequency, despite short-term synaptic facilitation known to occur at these synapses (Connelly et al., 2010; Lavian and Korngreen, 2016), consistent with the emergence of some excitatory and biphasic SNr responses in that regime.

The diversity of experimental responses to GPe and Str stimulation seen in Figure 6 support the idea that GABAergic synaptic transmission in the SNr is not purely inhibitory and may even be excitatory in some neurons. In the following sections we return to our computational model to explore the functional significance of this finding in physiologically relevant settings.

In addition to receiving GABAergic projections from the GPe and Str, SNr neurons interact locally through GABA_A-mediated synaptic transmission (Mailly et al., 2003; Brown et al., 2014; Higgs and Wilson, 2016). The role of these synapses is unclear; however, they have been proposed to regulate synchronization of SNr activity (Higgs and Wilson, 2016). Levels of EG⁢A⁢B⁢A will affect the strength and polarity (inhibitory, shunting, excitatory) of these interactions. Therefore, we next used our computational model to characterize how variations in EG⁢A⁢B⁢A, potentially due to differences in GPe firing rates, affect these local SNr interactions.

On average, a given SNr neuron receives GABAergic synaptic projections from 1 to 4 neighboring SNr neurons (Higgs and Wilson, 2016). Consequently, synaptic interactions between SNr neurons result in brief synaptic transients that have been proposed to impact neuronal synchrony incrementally by changing the oscillatory phase of the post-synaptic neuron. Therefore, we first characterized how transient GABAergic stimulation modulates the phase of our model SNr neuron as a function of the phase of the SNr oscillation at which the stimulation occurs, using phase response curves (PRCs) (Ermentrout, 1996; Ermentrout and Terman, 2010) computed for an array of values of EG⁢A⁢B⁢A (see Figure 7). The PRCs that we obtained for hyperpolarized values of EG⁢A⁢B⁢A are qualitatively consistent with those found previously for mouse SNr neurons in brain slices with EG⁢A⁢B⁢A≈-65⁢m⁢V (Simmons et al., 2018).

PRCs can be used to predict the synchrony between two oscillating neurons that interact synaptically (Ermentrout, 1996; Jeong and Gutkin, 2007; Ermentrout and Terman, 2010; Smeal et al., 2010). We applied this idea with our computationally-generated PRCs to predict the synchrony in a network of two SNr neurons under two configurations, unidirectional and bidirectional synaptic connectivity (Figure 8). For the unidirectional case a first, presynaptic SNr neuron stimulates a second, postsynaptic one. Phases of the presynaptic neuron’s ongoing oscillation at which the firing of the postsynaptic neuron will become locked can be predicted by finding locations where the PRC crosses the horizontal (phase) axis. Although all crossings represent fixed points and hence phases at which locking can theoretically occur, only those with a positive slope are stable and are predicted to arise robustly and be observed in simulations (e.g., Ermentrout, 1996; Ermentrout and Terman, 2010). By tracking the fixed points, we found that in the unidirectional case, the locked phase relation between the two SNr neurons is predicted to go from synchrony, or phase 0, to progressively more asynchronous phase locking and then back toward synchrony again as EG⁢A⁢B⁢A depolarizes from —60 mV to –50 mV, with perfectly anti-phase spiking for EG⁢A⁢B⁢A≈-53⁢m⁢V (Figure 8A4). We also observe that phase locking is predicted to be unstable (indicated by open circles) for sufficiently negative EG⁢A⁢B⁢A (less than ≈ –57 mV).

To test these predictions computationally, we simulated the unidirectionally connected two-neuron network and recorded the timing of synaptic inputs in the phase of neuron 2 (Input Phase). We found that the predicted synchrony/asynchrony is in good agreement with our simulations, and is indicated by the distributions of the input phase histograms shown in Figure 8A4 (gray curves). Interestingly, for relatively hyperpolarized EG⁢A⁢B⁢A where synchronous phase locking is predicted to be unstable we observe that, instead of phase locking, slow oscillations in the phase of the postsynaptic neuron relative to that of presynaptic neuron begin to emerge. Correspondingly, the distribution of presynaptic neuron phases when the postsynaptic neuron fires spreads out across the [0,1] interval and the frequency of firing of neuron the postsynaptic neuron repeatedly drifts below that of the presynaptic neuron, at a rate of about 1 Hz (Figure 8A5). The mechanism underlying these slow oscillations will be discussed in more detail below.

For bidirectional connectivity, we no longer have a clear distinction between a pre- and a postsynaptic neuron, and instead we just refer to neuron 1 and neuron 2. Due to the symmetry of the network, we can plot the PRC for neuron 1 together with that of neuron 2 by reflecting the PRC for neuron 2 about the mid-point of the phase axis, 0.5 (see Materials and methods for more detail). Phase locking between the two neurons can then be predicted by finding the intersections (fixed points) of these two PRCs (Figure 8B1–B3). By symmetry, a value near 0.5 is always a fixed point in this case, and we found that this was the only fixed point for the bidirectional system and remained stable regardless of the value of EG⁢A⁢B⁢A (Figure 8B1-4); see also Figure 8—figure supplement 1 for a schematic illustration of how this anti-phase locking develops. Again, this prediction was tested by simulating the bidirectionally connected two-neuron network and recording the input phase in neuron 2. The predicted asynchrony between the two neurons is in good agreement with our simulations, in which the phase relationship between the two neurons remained tightly distributed around 0.5 for all values of EG⁢A⁢B⁢A tested (Figure 8B4).

The slow oscillations of approximately 1 Hz seen with unidirectional connectivity can be understood by taking a closer look at the PRCs calculated for EG⁢A⁢B⁢A less than approximately –57 mV in the undirectional case (Figure 8A). For these values of EG⁢A⁢B⁢A, the PRCs only have unstable fixed points. Under these conditions, the phase of neuron 2 relative to neuron 1 is delayed by different amounts across successive inputs from neuron 2 (or possibly advanced if inputs arrive during a specific narrow phase window). Moreover, based on the shape of the PRC, the magnitude of change in phase is large when phase is away from 0 and 1, such that spiking is asynchronous, and small when the phase of is nearly synchronous. As a result, the network remains close to synchrony most of the time but with approximately periodic asynchronous excursions, a phenomenon referred to as phase slipping (Thounaojam et al., 2014). We refer to the oscillations that arise through phase slipping as PS oscillations. The frequency of phase slipping is determined by the number of stimulus kicks needed for the phase to progress through one full cycle, which in turn is determined by the shape of the PRC. For example, one full phase slipping cycle is illustrated in Figure 9A–B for EG⁢A⁢B⁢A=-60⁢m⁢V. As previously mentioned, the slow oscillation in phase is also seen as a periodic negative excursion in the frequency of spiking (Figure 8A5).

Since the frequency of PS oscillations is determined by the shape of the PRC and the PRC is in part determined both by EG⁢A⁢B⁢A and by the weight/conductance of the synaptic projection from the other neuron (WG⁢A⁢B⁢AS⁢N⁢r), changes in EG⁢A⁢B⁢A or WG⁢A⁢B⁢AS⁢N⁢r should affect the phase slipping frequency. Therefore, we also characterized the relationship between EG⁢A⁢B⁢A and the frequency of the phase slipping for different values of WG⁢A⁢B⁢AS⁢N⁢r. In our simulations, we found that phase slipping oscillations begin at approximately EG⁢A⁢B⁢A=-56⁢m⁢V and linearly increase in frequency as EG⁢A⁢B⁢A is held at progressively more hyperpolarized values (Figure 9C). The hyperpolarization of EG⁢A⁢B⁢A leads to stronger inhibition and hence a larger PRC amplitude, which allows for the postsynaptic neuron to progress through the full phase range on fewer cycles (i.e., at a higher frequency). Moreover, the magnitude of the slope of the linear relationship between EG⁢A⁢B⁢A and frequency increases/decreases with increases/decreases in the strength of WG⁢A⁢B⁢AS⁢N⁢r due to similar effects. We also simulated SNr neurons with different levels of applied current, leading to different firing rates, but this variability did not strongly impact resulting oscillation frequencies.

Next we investigated how changing the presynaptic neuron’s firing rate affects the relationship between EG⁢A⁢B⁢A and SNr synchrony and the emergence of slow oscillations. Under normal conditions in in vitro slice preparations, SNr neurons have been measured to spike at 10.4 ± 0.2 Hz (Zhou et al., 2008) and 10.7 ± 0.9 Hz (Zhou et al., 2009). We increased/decreased the firing rate of the presynaptic neuron over a wider range, from below 10 Hz to above 11 Hz, via current injection (Figure 10A) and examined the resulting dynamics in the unidirectional network. These simulations show that the synchrony relationship between the neurons is maintained regardless of the presynaptic neuron’s firing rate. For each choice of presynaptic firing rate, we observe a range of EG⁢A⁢B⁢A values supporting pure phase locking and another range supporting PS oscillations. The slower the presynaptic firing rate, the more hyperpolarized EG⁢A⁢B⁢A needs to be for PS oscillations to occur (Figure 10C1-4). Moreover, a new feature that appears when the presynaptic firing rate is slowed is a second type of oscillations, which we term phase advancing (PA) oscillations. These arise when EG⁢A⁢B⁢A is relatively depolarized (Figure 10C1-2) and manifest as transient increases in the postsynaptic neuron’s firing rate, as shown in Figure 10D1,D2,E1,E2 and contrasting with the PS oscillations shown in Figure 10D3,D4,E3,E4.

We also systematically tested the robustness of the unidirectionally connected two-neuron network oscillations and phase locking predictions to several factors: (1) increased SNr neuron firing rates, (2) the presence of noise, (3) synaptic delays, and (4) the number of presynaptic neurons projecting to each postsynaptic target; see Figure 10—figure supplements 1–6. Robustness results were similar for mutually connected pairs. For (4), we consider (a) various numbers of presynaptic neurons projecting to a single postsynaptic cell (Figure 10—figure supplements 4–6) and (b) activity patterns within a specific three-cell motif (Figure 10—figure supplement 7). In general, we found that the predicted oscillations and phase locking as a function of EG⁢A⁢B⁢A in the model are extremely robust to these factors.

Finally, we simulated a network of 100 model SNr neurons each receiving synaptic inputs from between 0 and 8 other SNr neurons (Figure 11A). The heterogeneity in inputs led to variability across individual neurons’ mean firing rates (Figure 11B) and spike rate CV (Figure 11C) at hyperpolarized EG⁢A⁢B⁢A, with more uniform spiking at more depolarized EG⁢A⁢B⁢A (Figure 11D). Interestingly, strong oscillations in the 0–4 Hz frequency range were prevalent in many cells within the network at relatively hyperpolarized and relatively depolarized EG⁢A⁢B⁢A (Figure 11E), consistent with the smaller circuit results and with the identified framework of PS and PA oscillations. Examples of the power spectra and firing rate time courses for oscillating neurons at EG⁢A⁢B⁢A=-60⁢m⁢V and EG⁢A⁢B⁢A=-50⁢m⁢V are shown in Figure 11F; the upward and downward deviations from baseline (≈ 10 Hz) in these plots support the suggestion that PS and PA oscillations persist in the larger SNr network. Overall, these results represent a strong indication of the robustness of our findings.

Slow oscillations have been reported in the SNr in vivo under dopamine depleted (DD) conditions in lightly anaesthetized (Walters et al., 2007) and awake behaving animals (Whalen et al., 2020). Our simulations predict that similar slow oscillations will occur when EG⁢A⁢B⁢A is equal to or hyperpolarized relative to the membrane AHP. Assuming that these oscillations are driven by the mechanism described in Figure 9, manipulations that depolarize EG⁢A⁢B⁢A should reduce and stop such oscillations. As illustrated in Figure 4, changing the tonic C⁢l- conductance to the soma is one way to depolarize EG⁢A⁢B⁢A. This could be achieved by increasing the firing rate of GPe neurons. Therefore, next we examined if these slow oscillations are suppressed by optogenetic stimulation of GPe neurons in the SNr. Consistent with previous descriptions (Walters et al., 2007; Whalen et al., 2020), under DD conditions we found slow oscillations in the firing rates of SNr neurons (Figure 12A–C). The frequency of the oscillations was characterized by finding the peak in the power spectral density (PSD) as described in Whalen et al., 2020 and shown in Figure 12B. We identified five oscillatory units with frequencies ranging from 1.46 Hz to 1.95 Hz (mean ± SD = 1.7 ± 0.204 Hz, Figure 12C). In these units, optogenetic stimulation of GPe terminals in the SNr had limited effect on SNr firing rates during a 30 s stimulation period (Figure 12D; see Materials and methods for a full description of the experimental preparation and stimulation protocol). Yet, stimulation of GPe terminals in the SNr significantly reduced the power in the PSD in the 0.25–4.0 Hz band (Figure 12E). The impact of GPe stimulation on oscillations but not firing rate in the SNr is consistent with our simulations and suggests that slow oscillations in the SNr seen under DD conditions may be due to the phase slipping mechanism described in Figure 9. These data also suggest that a role of the GPe may be to tune SNr dynamics by modulating the tonic C⁢l- conductance and EG⁢A⁢B⁢A in the SNr.

In tasks involving perceptual decision-making, visual motor responses (saccades) are thought to be triggered when evidence accumulates above some threshold level. Experiments suggest that the BG is involved is regulating the dynamics of these visual motor responses (Basso and Wurtz, 2002; Basso et al., 2005; Shires et al., 2010; Sato and Hikosaka, 2002). In the BG, evidence accumulation is thought to be represented by a ramping increase in the firing rate in striatal neurons of the direct pathway (Ding and Gold, 2010) that, above some threshold, generates a pause in SNr activity (Wei et al., 2015; Dunovan et al., 2019). The pause in SNr spiking disinhibits downstream motor targets and allows the initiation of a selected action. As we have shown above, the effect(s) of striatal inputs on the firing rate and pattern of SNr neurons is highly dependent on EG⁢A⁢B⁢A, which, in turn, is determined by the tonic chloride conductance and the C⁢l- extrusion capacity of the KCC2 pump. Therefore, changes in EG⁢A⁢B⁢A are predicted to modulate the threshold at which ramping striatal activity will generate a pause in SNr firing.

In the previous section we argued that the tonic GABAergic input from GPe neurons of the direct pathway may provide a mechanism to tune EG⁢A⁢B⁢A in the soma of SNr neurons. Assuming that the coupling between the somatic and dendritic compartments is sufficiently strong, the tonic somatic C⁢l- conductance provided by GPe inputs may also tune EG⁢A⁢B⁢A in the dendritic compartment. To illustrate this idea, we first constructed a population of 100 SNr neurons with a baseline firing rate turned up to ≈ 25 Hz in order to better represent in vivo conditions (Freeze et al., 2013; Mastro et al., 2017; Willard et al., 2019; Figure 13A,B). In this set of simulations [Cl−]i in the somatic and dendritic compartments interact by the addition of a coupling term (see Materials and methods for a full description). Next, we characterized EG⁢A⁢B⁢A in the somatic and dendritic compartments as a function of the tonic somatic C⁢l- conductance (representing the tonic GABAergic GPe input). As expected, EG⁢A⁢B⁢A depolarizes in both compartments as the somatic chloride conductance is increased (Figure 13C). In the dendritic compartment in particular, EG⁢A⁢B⁢A ranges from just below –75 mV with no chloride conductance to approximately –57 mV with a 1.0 nS/pF Cl−conductance in the soma.

Finally, we characterized the relationship between tonic C⁢l- conductance and the time required to decrease the mean SNr population firing rate below thresholds of 1 Hz, 5 Hz, and 10 Hz in response to a ramping striatal input (Figure 13D–F). As the tonic C⁢l- conductance increases, EG⁢A⁢B⁢A becomes less hyperpolarizing (Figure 5) and hence more time is needed to push SNr activity below threshold; for high enough C⁢l- conductance, the ramping striatal input is unable to suppress SNr firing below 1 Hz. These simulations illustrate a plausible mechanism through which the tonic C⁢l- conductance provided by the level of GPe activity may be able to tune dendritic (and somatic) EG⁢A⁢B⁢A, altering SNr responses to direct pathway striatal inputs and, ultimately, the response times in perceptual decision-making tasks. Qualitatively, these results are unchanged if the dendritic compartment is divided into multiple thin dendrites as opposed to one lumped dendrite, although this modification can hasten the rise and decay of EG⁢A⁢B⁢A after the onset and offset of a stimulus, respectively; see Figure 13—figure supplements 1 and 2.

Model SNr neurons were developed that each feature both a somatic and a dendritic compartment and incorporate Hodgkin-Huxley style conductances adapted from previously described models and/or experimental data (Xia et al., 1998; Zhou et al., 2008; Corbit et al., 2016; Doyon et al., 2015). The membrane potentials for the somatic (V_S) and dendritic compartments (V_D) are given by the following differential equations:(1)CS⁢d⁢VSd⁢t=-IN⁢a-IN⁢a⁢P-IK-IC⁢a-IS⁢K-IL⁢e⁢a⁢k-IG⁢A⁢B⁢AS-ID⁢S+IA⁢P⁢P(2)CD⁢d⁢VDd⁢t=-IT⁢R⁢P⁢C⁢3-IG⁢A⁢B⁢AD-IS⁢Dwhere CS=100⁢p⁢F and CD=40⁢p⁢F are the capacitances for the somatic and dendritic compartments. The currents in each compartment are represented by I_i where i denotes the current type. The somatic compartment features the essential spike generating currents as well as several others: fast Na⁺ current (IN⁢a), persistent Na⁺ current (IN⁢a⁢P), delayed rectifying K⁺ current (I_K), Ca²⁺ current (IC⁢a), Ca²⁺-activated K⁺ current (IS⁢K), and leak current (IL⁢e⁢a⁢k) as well as a synaptic current which represents the GABAergic input from the GPe neurons of the indirect pathway (IG⁢A⁢B⁢AS). IA⁢P⁢P denotes an applied current injected from an electrode. The dendritic compartment contains a current from a transient receptor potential channel 3 (TRPC3) (IT⁢R⁢P⁢C⁢3) and a synaptic current (IG⁢A⁢B⁢AD), which represents the GABAergic input from the striatal neurons of the direct pathway. IT⁢R⁢P⁢C⁢3 contributes to depolarization of the SNr neuron (Zhou et al., 2008) and was included in anticipation of future work to consider the dopamine depleted regime, in which this channel may be altered (Zhou et al., 2009). The two additional currents ID⁢S and IS⁢D are coupling terms that represent the current from the dendrite into the soma and from the soma into the dendrite, respectively. The currents are defined as follows:(3)IN⁢a=gN⁢a⋅mN⁢a3⋅hN⁢a⋅sN⁢a⋅(VS-EN⁢a)(4)IN⁢a⁢P=gN⁢a⁢P⋅mN⁢a⁢P3⋅hN⁢a⁢P⋅(VS-EN⁢a)(5)IK=gK⋅mK4⋅hK⋅(VS-EK)(6)IC⁢a=gC⁢a⋅mC⁢a⋅hC⁢a⋅(VS-EC⁢a)(7)IS⁢K=gS⁢K⋅mS⁢K⋅(VS-EK)(8)IL⁢e⁢a⁢k=gL⁢e⁢a⁢k⋅(VS-EL⁢e⁢a⁢k)(9)IG⁢A⁢B⁢AS=gG⁢A⁢B⁢AS⋅(VS-EG⁢A⁢B⁢AS)(10)ID⁢S=gCαC⋅(VS-VD)(11)IT⁢R⁢P⁢C⁢3=gT⁢R⁢P⁢C⁢3⋅(VD-ET⁢R⁢P⁢C⁢3)(12)IG⁢A⁢B⁢AD=gG⁢A⁢B⁢AD⋅(VD-EG⁢A⁢B⁢AD)(13)IS⁢D=gC1-αC⋅(VD-VS),where g_i is the maximum conductance, E_i is the reversal potential, and m_i and h_i are gating variables for channel activation and inactivation for each current I_i. sN⁢a is an additional inactivation term governing spike-frequency adaptation. The parameter αC=0.714 is the ratio of somatic and total capacitances. The GABAergic synaptic conductances gG⁢A⁢B⁢AS, gG⁢A⁢B⁢AD are variable and will be defined below. The values used for the g_i and E_i are given in Table 1.

Activation (m_i) and inactivation (h_i, s_i) of voltage-dependent channels are described as follows:(14)d⁢zid⁢t=zi∞-ziτzi,i={N⁢a,N⁢a⁢P,K,C⁢a},z={m,h,s}.

Steady-state (in)activation functions and their time constants (τzi) are described by:(15)zi∞⁢(V)=11+e-(V-z1/2i)/kzi,(16)τzi⁢(V)=τzi0+τzi1-τzi0e(τ1/2i-V)/σzi0+e(τ1/2i-V)/σzi1.

The parameters for these currents are given in Table 1 and were adapted from Corbit et al., 2016.

Activation of the small conductance calcium-activated potassium channels (SK) is instantaneous and depends on the intracellular calcium concentration ([C⁢a]i):(17)mSK([Ca]in)=(1+(kSK[Ca]in)nSK)−1,where kS⁢K represents the half-activation C⁢a2+ concentration and nS⁢K is the Hill coefficient. The parameters are given in Table 1 and were taken from Xia et al., 1998.

The intracellular calcium concentration is determined by the balance of C⁢a2+ influx carried by IC⁢a and efflux via the C⁢a2+ pump. In the model, IC⁢a and IS⁢K are only expressed in the soma and therefore [C⁢a]i⁢n dynamics is only simulated in the somatic compartment. The dynamics of [C⁢a]i⁢n is described by the following equation:(18)d⁢[C⁢a]i⁢nd⁢t=-αc⁢a⋅IC⁢a-([C⁢a]i⁢n-C⁢am⁢i⁢n)/τC⁢a,where αc⁢a=1.0⋅10-8⁢m⁢M/f⁢C is a conversion factor relating current and rate of change in [C⁢a]i⁢n, τC⁢a=250⁢m⁢s is the time constant for the C⁢a2+ extrusion and C⁢am⁢i⁢n=5.0⋅10-8⁢m⁢M is the minimum calcium concentration, where the C⁢a2+ pump turns off. Because of the balance between C⁢a2+ efflux from the pump and influx from IC⁢a activation, these parameters result in a typical value for [C⁢a]i⁢n of about 2.5⋅10-4⁢m⁢M in our simulations.