Abstract
Striatal oscillatory activity is associated with movement, reward, and decision-making, and observed in several interacting frequency bands. Local field potential (LFP) recordings in rodent striatum show dopamine- (DA-) and reward-dependent transitions between two states: a “spontaneous” state involving β (~15-30 Hz) and low γ (~40-60 Hz), and a state involving θ (~4-8 Hz) and high γ (~60-100 Hz) in response to DAergic agonism and reward. The mechanisms underlying these rhythmic dynamics, their interactions, and their functional consequences are not well understood. In this paper, we propose a biophysical model of striatal microcircuits that comprehensively describes the generation and interaction of these rhythms, as well as their modulation by DA. Building on previous modeling and experimental work suggesting that striatal projection neurons (SPNs) are capable of generating β oscillations, we show that networks of striatal fast-spiking interneurons (FSIs) are capable of generating θ and γ rhythms. Our model consists of three interconnected populations of single or double compartment Hodgkin-Huxley neurons: a feedforward network of FSIs exhibits a D-type potassium current as well as DA-modulated gap junctional and inhibitory connectivity, and two networks of SPNs exhibit an M-type potassium current and express either excitatory D1 or inhibitory D2 DA receptors. Under simulated low DAergic tone the FSI network produces low γ band oscillations, while under high DAergic tone the FSI network produces high γ band activity nested within a θ oscillation. SPN networks produce β rhythms in both conditions, but under high DAergic tone, this β oscillation is interrupted by θ-periodic bursts of γ-frequency FSI inhibition. Thus, in the high DA state, packets of FSI γ and SPN β alternate at a θ timescale. In addition to a mechanistic explanation for previously observed rhythmic interactions and transitions, our model suggests a hypothesis as to how the relationship between DA and rhythmicity impacts motor function. We hypothesize that high DA-induced periodic FSI γ-rhythmic inhibition enables switching between β-rhythmic SPN cell assemblies representing the currently active motor program, and thus that DA facilitates movement by allowing for rapid, periodic shifts in motor program execution.
Author summary Striatal oscillatory activity is associated with movement, reward, and decision-making, and observed in several interacting frequency bands. The mechanisms underlying these rhythmic dynamics, their interactions, and their functional consequences are not well understood. In this paper, we propose a biophysical model of striatal microcircuits that comprehensively describes the generation and interaction of striatal rhythms, as well as their modulation by DA. Our model suggests a hypothesis as to how the relationship between DA and rhythmicity impacts the function of the motor system, enabling rapid, periodic shifts in motor program execution.
Introduction
As the largest structure of the basal ganglia network, the striatum is essential to motor function and decision making. It is the primary target of dopaminergic (DAergic) neurons in the brain, and its activity is strongly modulated by DAergic tone. Disorders of the DA and motor systems, such as Parkinson’s, Huntington’s, Tourette’s, and many others, result in abnormal network activity within striatum [1]. Rhythmic activity is observed in both striatal spiking and local field potential, and oscillations in the striatum are correlated with voluntary movement, reward, and decision-making in healthy individuals [2–4], while disruptions of these rhythms are biomarkers of mental and neurological disorders [1,5,6]. However, the mechanisms of these oscillations, and their role in motor behavior and its dysfunctions, remain poorly understood.
Four oscillatory bands in particular are frequently observed in striatal local field potential: θ (4-7 Hz), β (8-30 Hz), low γ (50-60 Hz), and high γ (70-80 Hz) [2,7,8]. Power in these bands consistently correlates with responses to task parameters such as motor initiation, decision making, and reward [2–4,6]. Power in the β band is elevated in Parkinson’s disease and correlates with the severity of bradykinesia [1]. In the healthy basal ganglia, β and γ activity are inversely correlated and differentially modulated by slower basal ganglia rhythmic activity, suggesting that the balance of these distinct oscillatory dynamics is important to healthy motor function [8]. In rat striatum in vivo, spontaneous β and low γ oscillations transition to θ and high γ dynamics upon reward receipt and with administration of DA agonist drugs [2]; similarly, in rat caudate and putamen, DAergic agonists produce robust low-frequency modulation of high γ amplitude [7].
In this paper, we propose a biophysical model of striatal microcircuits that comprehensively describes the generation and interaction of these rhythms, as well as their modulation by DA. Our simulations capture the dynamics of networks of striatal fast-spiking interneurons (FSIs) and striatal projection neurons (SPNs), using biophysical Hodgkin-Huxley type models. SPNs, responsible for the output of the striatum, make up 95% of striatal neurons in rodents [9]. SPN firing is regulated by relatively small populations of striatal interneurons, including fast spiking interneurons (FSIs), which strongly inhibit SPNs. Our model FSIs exhibit a D-type potassium current [10], and our model SPNs exhibit an M-type potassium current [11]. Both cell types are modulated by DAergic tone: FSIs express the excitatory D1 DA receptor [12], while two distinct subpopulations of SPNs express exclusively the D1 or the inhibitory D2 receptor subtype. We modeled both SPN subpopulations, with high simulated DAergic tone increasing and decreasing D1 and D2 SPN excitability, respectively. To model DA effects on the FSI network, we simulated three salient experimentally observed effects: increased excitability [12] and gap junction conductance [13], and decreased inhibitory conductance [12]. Both gap junctions and inhibition are known to play a role in the generation of rhythmic activity [14].
Our previous experimental and modeling work suggests that striatal SPN networks can produce a β (15-25 Hz) oscillation locally [15]. Our current model demonstrates that FSI networks can produce θ, low γ, and high γ oscillations. A fast-activating, slow-inactivating potassium current (the D-type current) allows FSIs to produce γ and θ rhythms in isolation, and network interactions make these rhythms, otherwise highly susceptible to noise, robust. In our simulations, DA induces a switch between two FSI network states: a low DA state exhibiting persistent low γ rhythmicity, and a high DA state in which a θ oscillation modulates high γ activity. As a result of FSI inhibition of SPNs, DA induces a switch in striatal dynamics, between a low DA state in which low γ and β rhythms coexist, and a high DA state in which bursts of FSI high γ and SPN β rhythms alternate, nested within (and appearing at opposite phases of) an FSI θ rhythm.
Thus, our model generates a hypothesis as to how observed relationships between DA and rhythmicity impact the function of the motor system. Namely, DA appears to encourage or permit periodic motor program switching, by allowing the emergence of an FSI-mediated θ-modulated γ rhythm that breaks up the “stay” signal mediated by SPN β rhythms [16].
Results
Single model FSIs produce θ-nested γ rhythms whose power and frequency is modulated by excitation
We modified a previous single-compartment striatal FSI model [17] by adding a dendritic compartment (shown to be an important determinant of gap-junction mediated synchrony [18]) and increasing the conductance of the D-type K current to 6 mS. Previous work showed that two characteristic attributes of FSI activity in vitro – stuttering and γ resonance (defined as a minimal tonic firing rate in the γ frequency range) – are dependent on the D-current. Our modified FSI model successfully reproduced these dynamics as well as revealing other dynamical behaviors (Fig. 1).
With increasing levels of tonic applied current (Iapp), our model FSI transitions from quiescence to (periodic) bursting to periodic spiking. The bursting regime, of particular interest in this work, is dependent on the level of tonic excitation (Fig. 1A,C,D), the level of noise in excitatory drive (Fig. 1B & 2C), and, centrally, the D-current conductance (Fig. 1C,D). With lower levels of D-current (as used in previous FSI models [10,17,19]), bursting is aperiodic. For sufficiently large D-current conductance, FSI bursting occurs for a broad range of applied currents (Iapp over 8 uA/cm2, Fig. 1C,D).
The frequency of bursting depends on the decay time constant of the D-type potassium current (τD); in the absence of noise, it is in the δ frequency range for physiologically relevant τD (<~200 ms, Figure 1A,C, 2B). Note that τD changes the inter-burst interval without changing the timing of spikes within a burst (Fig. 2B). Burst frequency also increases with increasing tonic excitation (Fig. 1,A,C), and increases to θ frequencies with small amounts of noise (Fig. 1B, 2C), which decrease the interburst interval. However, large amounts of noise abolish rhythmic bursting altogether (at least in single cells, Fig. 1B).
As shown previously [17], the FSI model’s γ rhythmic intraburst spiking arises from its minimum firing rate, which is also set by the D-current conductance (Fig. 2A); when this conductance is zero, the model has no minimum firing rate (Fig. 2A). As this conductance is increased, the minimum firing rate also increases. Thus, our choice of D-current reflects not only our interest in the bursting regime, but also our desire to match experimental observations of striatal γ frequency [2,17]. FSI spiking frequency also increases with tonic drive (Figure 1A,D). Since simulated DA acts on our FSI model by increasing tonic excitation, DA causes a switch in model FSI spiking from low γ rhythmicity to δ-modulated high γ rhythmicity. Below, we demonstrate that the FSI γ is determined by this single-cell rhythmicity and is mostly independent of the timescale of inhibitory synapses.
In summary, a single model FSI displays low-frequency-nested γ oscillations, dependent on the D-type current, under a wide range of tonic excitation levels. Both low frequency power and γ frequency increase with tonic excitation. While noise increases the frequency of the slower rhythm from δ to θ, it also diminishes the power of this rhythm in the single cell. Below we demonstrate that all of these effects are also present in a network of FSIs, with the key difference that the θ rhythm becomes robust to noise.
FSI networks produce DA-dependent θ and γ rhythms
To determine if θ and γ oscillations persist in networks of connected FSIs, and how DA could modulate network dynamics, we simulated a network of 100 model FSIs connected randomly (with an independent connection probability of 0.3 for each type of connection) by both inhibitory synapses and gap junctions. We also implemented three salient and experimentally observed effects of DA on FSI networks: increased tonic excitation of individual FSIs [12], increased gap junction conductance between FSIs [13], and decreased inhibitory conductance between FSIs [12] (see Methods).
Unlike in single cells, FSI network θ rhythmicity is dependent on sufficient levels of tonic excitation: at low levels of tonic input (Iapp <≈ 1μA/cm2), the FSIs do not attain enough synchrony for a strong network θ (Fig. 3Aii). As in single cells, FSI network θ power increases with tonic input strength (Fig. 3Aii). Sufficiently strong gap junction coupling is also a requirement for the FSI network to attain sufficient synchrony to produce θ rhythmicity (Fig. 3Cii), protecting the FSI network θ rhythm from the effects of noise (as in [14]). Finally, inhibitory synaptic interactions between FSIs have a desynchronizing effect that interferes with network θ, and increasing inhibitory conductance within the FSI network decreases power in the θ band (Fig. 3Bii).
FSI network γ power and frequency both increase with tonic input strength (Fig. 3Ai), and, like the network θ, the network γ rhythm is dependent on sufficient gap junction conductance and is disrupted by inhibition (Fig. 3B & C, i).
To explore FSI network dynamics that might be observed during normal fluctuations in DA during goal-directed tasks [20], we simulated FSI network activity under two conditions, simulated low (or baseline) and high DAergic tone. During simulated low DAergic tone, characterized by low levels of FSI tonic excitation and gap junction conductance, and high levels of inhibitory conductance, the network produces a persistent low frequency γ oscillation (~ 55 Hz) in the model LFP (the mean voltage of the FSI network, Fig. 4Bi-Di). The raster plot of FSI spike times (Fig. 4Eii) shows that individual FSIs exhibit sparse spiking in the low DA state. Although individual FSIs exhibit periodic spike doublets or bursts (γ-paced and entrained to the network γ) that recur at θ frequency, the timing of these bursts is independent (Fig. 4Ei). Therefore, while θ power is present at the level of individual FSIs, there is not sufficient synchrony for it to appear in the network (Fig. 4Di).
During simulated high DAergic tone, characterized by high levels of tonic excitation and gap junction conductance and low levels of inhibitory conductance, network activity is much more structured: a strong 70 Hz γ rhythm, phase-modulated by a 5 Hz θ rhythm, are evident in both the simulated LFP and network raster plots (Fig. 4Bii-Eii, right). In this state, active FSIs spike at the same phase of both θ and γ, producing dual (and nested) network rhythms (Fig. 4ii).
To explore whether the γ rhythms observed in the FSI network are generated by inhibitory interactions, we examined the dependence of γ frequency on the time constant of GABAA inhibition, as the characteristic frequency of canonical interneuron network γ (ING) has been shown to depend on this time constant [21]. The frequency of the γ rhythm produced under low DA conditions decreased with increases in the GABAA time constant (Fig. 3D), suggesting this rhythm is ING-like. However, the γ produced under high DA conditions had a frequency that was not highly dependent on the inhibitory time constant, suggesting that this γ rhythm is mechanistically different from previous ING models, being generated by synchronous γ frequency bursts in individual cells, as opposed to inhibitory interactions.
SPN networks generate DA-dependent β oscillations
Previous work by our group found that robust β oscillations can emerge from inhibitory interactions in networks of model striatal SPNs [15]. The interaction of synaptic GABAA currents and intrinsic M-currents promotes population oscillations in the β frequency range; their β timescale is promoted by the M-current, which allows rebound excitation at ~50 ms in response to synaptic inhibition. Excitation of these neurons increases β power (see Methods). This previous work explored the transition from a healthy to a parkinsonian state with pathologically low levels of striatal DA. To explore the generation of β rhythmicity during normal fluctuations in DAergic tone, we simulated two independent networks of 100 D1 receptor expressing (“direct pathway”) SPNs and 100 D2 receptor expressing (“indirect pathway”) SPNs. Model SPNs are single compartment cells expressing the Hodgkin-Huxley spiking currents and the M-type potassium current, interconnected all to all by inhibitory GABAA synapses (connection probability 1). We simulated the effects of DA on model D1 and D2 SPNs by increasing and decreasing their levels of tonic excitation, respectively (whether DA generates a positive or negative applied current was the only difference between D1 and D2 expressing SPNs in our model; Fig. 5i and ii; see Methods). Paradoxically, in the absence of FSI input, neither population was sufficiently excited to exhibit spontaneous spiking under low DA conditions (Fig. 5i); under high DA conditions, D1 SPNs exhibited persistent β rhythmicity at ~15 Hz (Fig. 5ii).
FSI network γ and θ oscillations rhythmically modulate SPN network β oscillations only in high DA state
To understand the interactions between FSI and SPN networks, and between β, γ, and θ rhythms, we simulated a combined FSI-SPN striatal microcircuit, in which 100 model FSIs randomly inhibited independent networks of 100 D1 and 100 D2 SPNs (connection probability from FSIs to D1 or D2 SPNs of 0.1). FSIs were interconnected by gap junctions and inhibitory synapses (connection probability 0.3 for each). D1 and D2 SPNs were connected by all to all inhibitory synapses within (connection probability 1) but not across populations. There were no connections from SPNs back to FSIs [22].
During simulated baseline DAergic tone, we modeled D1 and D2 SPNs as being equally excitable, with equal firing rates matching in vivo observations [23] while under the influence of FSI inhibition. The presence of FSIs is necessary for the SPNs to fire in the low dopamine state (Fig. 5iii); this paradoxical excitatory effect of GABAergic input arises because SPNs can be excited via post-inhibitory rebound, as demonstrated in previous work. [15]. Both SPN networks produce a β rhythm (15 Hz), while the FSI network produces a low (50 Hz) γ (Fig. 5iii & 6i). The generation of low γ rhythms by the FSIs and β by the SPNs matches observations of striatal rhythmicity in resting healthy animals in vivo [2]. Our model suggests that these γ and β rhythms are independently generated by FSI and SPN networks, respectively.
During simulated high DAergic tone, an FSI-mediated high (~70 Hz) γ and an SPN-mediated β are observed during opposite phases of an ongoing FSI network θ rhythm (Fig. 5iv & 6ii). During the peak of the θ, the incoming γ frequency input from the FSIs silences the SPNs. When the FSIs are silent during the θ trough, both D1 and D2 SPN populations are sufficiently excited to produce a β rhythm. Thus, while the SPNs cannot entrain to the γ frequency of FSI inhibition, they are modulated by the FSI-generated θ rhythm.
Discussion
Our model suggests that DAergic tone can produce a transition between two dynamical states in striatal GABAergic networks. In the baseline DAergic tone state, ongoing low γ (50-55 Hz) and β (15 Hz) oscillations are generated by striatal FSI network and SPN networks, respectively (Fig. 6i). In the high DAergic tone state, packets of FSI-mediated high γ (65-70 Hz) and SPN-mediated β (10-20 Hz) rhythms alternate at θ (~5 Hz) frequency (Fig. 6ii). Our results make predictions about the generation of striatal rhythms, have implications for the role of FSIs in regulating the activity of SPNs and suggest an underlying mechanism for the temporal dynamics of motor program selection and maintenance (Fig. 6E).
Mechanisms of γ and δ/θ oscillations in single FSIs
Prior work has shown γ oscillations in striatal FSIs arising from an interaction between the spiking currents and the spike frequency adaptation caused by the potassium D-current, which produces a minimum FSI firing rate in the γ range [17,24]. The frequency of the FSI γ depends on excitatory drive to the FSIs, which in our model leads to the modulation of γ frequency by DA, a phenomenon also observed in striatal γ oscillations in vivo [25–28].
Prior work has also suggested that the D-current is responsible for the bursting or stuttering behavior of FSIs, in which brief periods of high frequency activity are interspersed with periods of quiescence [10]. However, regularities in these periods of quiescence have not been previously observed. Thus, the present study is novel in its description of the generation of low-frequency rhythms by FSIs with high levels of D-current conductance; FSIs have previously been characterized solely as generators of γ oscillations. Our model predicts that FSI-mediated slow rhythms depend on a high level of D-current conductance. In our model, the D-current is activated by burst spiking, e.g., at γ frequency, and hyperpolarizes the cell for roughly a θ period due to its long time constant of inactivation. Though individual cells produce a δ rhythm, the frequency of the resulting θ oscillation in the network is robust to changes in excitatory drive. This transition to a higher rhythm in the network is likely a result of gap-junction induced synchrony driving burst frequency higher while maintaining robustness to noise. Notably, this study is also a novel demonstration of the generation of both θ and γ oscillations by a single membrane current.
Mechanisms of γ and θ oscillations in FSI networks
Our model FSI network produces qualitatively different dynamics at high and baseline levels of DA conditions. Under high dopaminergic tone, the FSI network produces high γ band (70 Hz) oscillations modulated by a θ (4-6 Hz) oscillation, while under low dopaminergic tone the FSI network produces low γ band (55 Hz) oscillations alone (Fig. 4). While both θ and γ are present at the level of individual cells, only in the high DA condition is bursting sufficiently synchronized that θ power is present in the network. The presence of θ at the network level can be attributed to the higher level of gap junction conductance in the high DA condition (Fig. 3Cii).
The ability of gap junctions to generate synchrony is well established in computational work [29]. Previous models from other groups suggest that gap junctions can enable synchronous bursting in interneurons, such that the burst envelopes are aligned, as in our model [14]. While a shunting effect of low conductance gap junctions can inhibit spiking [30], gap junctions with high enough conductances have an excitatory effect, promoting network synchrony [31,32]. Previous work has also shown the importance of gap junction connectivity in stabilizing network γ oscillations in silico [33], as well as network γ and θ oscillations in inhibitory networks in vitro and in silico containing noise or heterogeneity [32]. FSIs in vivo are highly connected by gap junctions as well as inhibitory synapses, [34] similar to the networks of inhibitory interneurons that produce ING rhythms [35]. Unlike ING, however, our FSI network γ is independent of GABAergic synapses: inhibitory conductance has only a small impact on γ frequency, and γ power is highest when inhibitory synapses are removed (Figure 3B). In slice, the γ resonance of striatal FSIs is dependent on gap junctions but not on GABA [36], suggesting that our model is an accurate representation of FSI γ.
It is important to note that while our model is conceived as a representation of the dorsal striatal circuit, physiologically similar fast-spiking interneuron networks are present in cortex [10]; therefore, the mechanisms described here may contribute to the generation of θ-modulated γ oscillations in cortex as well.
Support for striatal rhythm generation
Our model provides mechanistic explanations for all four oscillatory bands observed in ventral striatum in vivo (θ, β, low γ, and high γ) [37]. Previous modeling and experiments suggest β can be generated by striatal SPNs [15, 38, 39]. Our results suggest that FSIs generate striatal γ, and that motor- and reward-related increases in γ power reflect increased striatal FSI activity.
There is evidence to support the existence of a locally generated striatal γ oscillation that is not volume conducted and that responds to local DAergic tone [40, 41]. The FSIs of the striatum are the most likely candidate generator of this rhythm: they are unique among striatal cell types in preferentially entraining to periodic input (from each other and from cortex) at γ frequencies [42–46]. Different populations of striatal FSIs in vivo entrain to different γ frequencies, and FSIs entrained to higher frequencies are also more entrained to cortical input [25–28]. It is likely that different subpopulations of FSIs selectively entrain to specific γ frequencies depending on physiological differences, context, and neuromodulatory (e.g. DAergic) states; the frequency of γ may itself determine cell assembly size and membership [33].
Experimental evidence also supports striatal FSI involvement in a DA-modulated θ rhythm. FSIs phase lock to spontaneous striatal LFP oscillations at θ as well as γ frequencies [23,47–51]. In vivo, striatal θ power is modulated by task-related phenomena such as choice points and motor execution, as well as by reward and reward expectation, suggesting its responsiveness to DA (known to phasically increase in response to reward cues) [4,52–54]. θ has also been shown to modulate the response of SPNs to reward [55].
θ rhythmicity in striatal dynamics and movement
In vivo, striatal β power has a well established negative correlation with DA and locomotion in both health and disease, while striatal γ power has a positive correlation with both [1,3–6,56]. β oscillations in the basal ganglia are thought to provide a “stay” or “status quo” signal that supports maintenance of the currently active motor program [16], and they are causally implicated in motor slowing and cessation [8,56–60].
In our simulations of high DAergic tone, FSI spiking at high γ frequencies θ-periodically inhibits SPN-generated β oscillations, permitting SPN β only during the 150-200 millisecond θ trough corresponding to the FSIs’ interburst interval. We hypothesize that these periodic gaps between SPN β packets are necessary to terminate ongoing motor programs and initiate new motor programs, as represented by active SPN assemblies. During the θ trough, all SPN cell assemblies are simultaneously released from inhibition and viable to compete once again to determine the current motor program, with incoming input from cortex influencing this competition. Under this interpretation, our results predict that striatal networks oscillate between a “stay” or “program on” state marked by SPN β oscillations, and a “switch” or “program off” state marked by FSI high γ oscillations, and that the θ period limits the speed of sequential motor program execution (Fig. 6E).
In support of this hypothesis, striatal representations of behavioral “syllables” that can be combined to create motor programs are active for a maximum of ~200 ms [61], and the velocity of continuous motion is modulated intermittently at a θ frequency (~6-9 Hz) [62]. In healthy animals, the duration of β bursts has an upper limit of ~120 ms, about half a θ cycle [8], in agreement with our hypothesis of θ phase-modulation of β activity. Striatal γ has also been observed in transient (~150 ms) bursts that are associated with the initiation and vigor of movement [63]. Additionally, other biophysically constrained computational models have suggested that SPN assemblies fire in sequential coherent episodes for durations of several hundred milliseconds, on the timescale of one or several θ cycles [64]. Overall, evidence supports the hypothesis that β and γ oscillations in striatum in vivo, and therefore the motor states they encode, are activated on θ-periodic timescales.
Furthermore, β and γ power are anticorrelated in EEG and corticostriatal LFP [6,7,65], in agreement with our model’s prediction that these rhythms are coupled to opposite phases of ongoing θ rhythms. FSI and SPN firing are inversely correlated in vivo, entrained to θ, and they are active during opposite phases of θ, as observed in our model [23, 48, 66–68]. θ-γ cross-frequency coupling is observed in striatum and increases during reward, when DAergic tone is expected to be high [7,69–72]. Our model suggests that these cross-frequency relationships occur in part due to FSI inhibition of SPNs. Though FSIs are smaller in number, FSI-SPN synapses have a much stronger effect than SPN-SPN connections, with each FSI inhibiting many SPNs [22,73].
During baseline DAergic tone in our model, FSIs produce an ongoing low γ that does not effectively suppress SPN β activity (produced sporadically in both D1 and D2 SPN networks), and thus does not facilitate the switching of the active SPN assembly. Thus, our model suggests that at baseline levels of DA, switching between SPN assembles may be more dependent on cortical inputs or downstream BG circuit computations. Although the function of FSI low γ inhibition of SPN dynamics is unclear, it may facilitate striatal responsivity to cortical low γ input, which occurs in an afferent- and task-specific manner [37]. SPNs do not entrain to γ in our model, suggesting that γ oscillations are not transmitted to downstream BG structures.
In contrast, both the β and θ rhythms in our model entrain SPN networks and may be relayed to other basal ganglia structures. Intriguingly, alternation between β and γ on a θ/δ timescale has been observed in the globus pallidus in vivo, and DAergic tone modulates these oscillations and their interactions [7,74]. Thus, the mechanisms proposed in our model may also play a role in the oscillatory dynamics of other basal ganglia structures, through a combination of rhythm propagation and local rhythm generation by similar circuits. Similar pauses in FSI activity, allowing transient SPN disinhibition and production of β oscillations, occur in a recent computational model of GPe [19], also based on an earlier model of stuttering FSIs [10]. In contrast to this work, we emphasize the mechanisms producing β and the coordination of β and γ by θ, not addressed previously [19].
Implications for disease
In Parkinson’s disease, which is characterized by motor deficits and chronic DA depletion, β power is correlated with the severity of bradykinesia [1]. Parkinsonian β may be generated by striatal D2 SPNs [15,38,39]. Parkinsonian conditions also produce high cholinergic tone [75], known to decrease the conductance of GABAergic FSI-SPN synapses [76]. Thus, the failure of the FSI inhibition-mediated motor program switching described above may play a role in the motor deficits observed in Parkinson’s: if DA is low, and FSIs are unable to inhibit either D1 or D2 SPNs, θ modulation of SPN β rhythmicity will be supplanted by ongoing D2 β rhythmicity, impairing motor initiation by reducing the possibility of motor program switching in the Parkinsonian striatum.
In hyperkinetic motor disorders, γ and θ rhythms are potentiated: a mouse model of Huntington’s disease (HD) displays unusually high θ and γ band striatal LFP power [77–79]; and L-DOPA-induced hyperkinetic dyskinesia is also characterized by increased high γ and θ power and reduced β power in the striatal LFP [80–83]. As these rhythms are tied to FSI activation in our model, we suggest that hyperkinetic disorders may result from striatal FSI hyperfunction. Consistent with this hypothesis, in HD model animals, FSI to SPN connectivity is increased, and SPNs respond more strongly to FSI stimulation [84].
However, hypofunction of striatal FSI networks can also lead to hyperkinetic disorders, including Tourette’s syndrome, dystonia, and dyskinesias [81–83, 85–88]. Dystonia, which as a disorder of involuntary muscle activation is considered hyperkinetic, can also be characterized by rigidity and freezing due to activation of antagonistic muscles. Indeed, dystonia is characterized by an increase in SPN firing rate due to D2 receptor dysfunction. Our model suggests that FSI hypofunction may be to blame, resulting in excessive SPN β rhythmicity and decreased probability of motor program switching [89]. A reduction in theta/γ cross frequency coupling has been reported in L-DOPA-induced dyskinesia, suggesting that a chronic hyperkinetic high-DA state may also abolish the FSI-generated θ-coupled γ produced here, possibly by pushing the FSI out of its bursting regieme and into a tonic spiking mode [90]. These findings underscore the importance of balanced FSI inhibition of SPNs, exemplified by the periodic suppression observed in our model, which we suggest enables the flexible striatal network activity that allows for smooth, purposeful movements.
Caveats and limitations
Little experimental evidence on the striatal FSI D-current conductance exists. The level of D-current conductance we’ve chosen leads to γ frequencies and FSI firing rates that are more in line with experimental observations than previous models. This level of D-current also produces θ rhythmicity in FSI networks. Our parameter choices result in a model exhibiting a transition between “low DA” and “high DA” dynamic states that matches experimental observations and has powerful functional interpretations. Validating our results will require further experimental investigation of the D-current in striatal FSIs. Interestingly, DA has been shown to downregulate D-current conductance in prefrontal cortical FSIs. If striatal FSIs exhibited a similar DA-dependent D-current downregulation, our simulations suggest that the transition between high and low DA states could be different from that described in the current study. The existence and functional interpretations of other dynamic transitions are beyond the scope of this paper.
In general, many DA-dependent changes in striatal neurophysiology have been observed. For the sake of simplicity, most of these have been left out of our modeling. For example, D1 and D2 SPNs respond differently to adenosine [91] and peptide release [92], but we did not consider these significant factors in the production of striatal β oscillations.
We also omitted inhibitory connections between D1 and D2 SPN populations. The connectivity from D1 to D2 SPNs is very sparse (6 percent). Connections from D2 to D1 SPNs are more prevalent, but it seems unlikely that these projections would qualitatively alter our results: during the baseline state, the D1 and D2 SPNs are identical; during the high DA state, SPN inhibition tends to increase SPN β rhythmicity and spiking.
In our model the number of FSIs is small, so every FSI participates on every θ cycle; in vivo, the participation of multiple FSI populations is likely coordinated by cortex. Coordinated FSI activity has proven hard to observe over long periods in vivo [28,93]. However, FSIs form local functional circuits [94], and in vivo, striatal FSI assemblies exhibit transient gap-junction dependent synchronization [95], possibly resulting from brief bouts of correlated cortical or homogeneous DAergic input. Furthermore, different subpopulations of FSIs have strong preferences for projecting to either D1 or D2 SPNs, as opposed to the overlapping projections modeled in our current study, and these distinct populations respond differently to cortical oscillations [49]. Thus, local γ synchrony may exist in small striatal subnetworks and be amplified by DA or cortical input via the recruitment of multiple FSI subpopulations.
Finally, cortical input to both FSIs and SPNs was simulated as Poisson noise. In a sense, then, we simulated a model of striatum to which cortex is not providing informative input. It could be the case that this is a population that is not “selected” by cortex to take part in motor activity, a population that is in a “listening” state awaiting cortical input, or a population taking part in a learned behavior that can be executed without cortical input. However, cortical input is probably essential in determining which SPNs and FSIs take part in network oscillatory activity. If the FSIs play a role in organizing the response of the SPNs to cortical input, changing the properties of the simulated input may prove informative in terms of how this organization might take place. In particular, cortical inputs may be more correlated within certain FSI subpopulations than others. Previous modeling work has shown that networks of striatal FSIs can detect correlated input [30], a property that may play an important computational role in striatal function. Additionally, we can expect that input from cortex has oscillatory properties of its own. Exploring these complexities is an important direction for future research into the role of striatal GABAergic networks and rhythmic dynamics in motor behavior.
Materials and methods
All neurons (FSIs and SPNs) are modeled using conductance-based models with Hodgkin-Huxley-type dynamics. SPNs are modeled with a single compartment and FSIs have two compartments to represent the soma and a dendrite. The temporal voltage change of each neuron is described by (Eqn. 1):
Membrane voltage (V) has units of mV. Currents have units of μA/cm2. The specific membrane capacitance (cm) is 1 mF/cm2 for all FSIs and SPNs. Each model neuron has intrinsic membrane currents (Imemb) and networks of neurons include synaptic currents (Isyn). The applied current term (Iapp) represents background excitation to an individual neuron and is the sum of a constant and a noise term.
All membrane currents have Hodgkin-Huxley-type conductances formulated as:
Each current in Eqn.2 has a constant maximal conductance and a constant reversal potential (Eion). The activation (m) and inactivation (h) gating variables have nth and kth order kinetics, where n, k > 0. The dynamics of each gating variable evolves according to the kinetic equation (written here for the gating variable m):
The steady-state functions (m∞) and the time constant of decay (τm) can be formulated using the rate functions for opening (αm) and closing (βm) of the ionic channel by using the equations:
The specific functions and constants for different cell types are given below.
Striatal fast spiking interneurons
Striatal fast spiking interneurons (FSIs) were modeled as in Golomb et al., 2007 [10] using two compartments. The voltage in the somatic compartment (V) and in the dendrite (Vd) evolve according to:
Background excitation is represented by the term Iext, which is formulated as the sum of a tonic, DA dependent current and Poisson input. The units of Iext are in μA/cm2. The tonic, DA dependent current is discussed below. Each FSI receives independent, excitatory Poisson input with a rate of 2000 inputs per second.
The synaptic current (Isyn) is the sum of GABAA currents and electrical connections between FSIs (formulated below). The FSI membrane currents (Imemb) consisted of a fast sodium current (INa), a fast potassium current (Ik), a leak current (IL), and a D-current (ID). The formulations of these currents were taken from previous models of striatal FSIs. [10,17]
The maximal sodium conductance is and the sodium reversal potential is ENa = 50mV. The sodium current has three activation gates (n = 3) and one inactivation gate (k = 1). The steady state functions for the sodium current activation (m) and inactivation (h) variables and their time constants (τm and τh, respectively) are described by:
The maximal conductance for the fast potassium channel is and the reversal potential for potassium is EK = –90mV. The fast potassium channel has no inactivation gates but has four activation gates described by its steady state function (n∞) and time constant (τn):
The leak current (IL) has no gating variables. The maximal leak channel conductance is gL = 0.25mS and the leak channel reversal potential is EL = –70mV.
The D-current (ID) is described mathematically as in Golomb et al, 2007 [10] and has one activation (a) and one inactivation (b) gate. The steady state functions for the activation and inactivation gates are formulated as:
The time constant of the decay is 2 ms (τa) for the activation gate and 150 ms (τb) for the inactivation gate. The maximal conductance of the D-current is 6 mS.
Striatal spiny projection neurons
Spiny projection neurons were modeled with four membrane currents: a fast sodium current (INa), a fast potassium current (Ik), a leak current (IL), and an M-current (Im) [11]. We do not model SPN up and down states which are not prevalent in the awake state of striatum [96], the state being modeled, and therefore we do not include the Kir current in our model, which is active during the SPN down state.
The sum of all excitatory inputs from the cortex and thalamus and inhibitory inputs from striatal interneurons is introduced into the model using a background excitation term (Iapp). Iapp is the sum of a constant term and a Gaussian noise term. The Gaussian noise has mean zero, standard deviation one and an amplitude of where δt is the time step of integration. D1 and D2 SPNs were distinguished only by the value of tonic term of Iapp when DA levels were high. DA is excitatory to D1 receptors and inhibitory to D2 receptors [97]. Thus, we modeled D1 and D2 SPNs as having the same tonic Iapp at baseline DAergic tone state with Iapp = 1.19μA/cm2. To model the high DA state, let the tonic term of Iapp = 2.19μA/cm2 for the D1 SPNs and Iapp = 0.19μA/cm2 for the D2 SPNs.
Fast sodium current
The rate functions for the sodium current activation (m) and inactivation (h) variables are formulated as:
The maximal conductance of the sodium current is . The sodium reversal potential is ENa = 50mV. The sodium current has three activation gates (n = 3) and only one inactivation gate (k = 1).
Fast potassium current
The fast potassium current (IK) has four activation gates (n = 4) and no inactivation gates (k = 0). The rate functions of the activation gate are described by:
The maximal fast potassium channel conductance is . The reversal potential for potassium is EK = –100mV.
Leak Current
The leak current (IL) has no gating variables (n = 0, k = 0). The maximal conductance of the leak channel is gL = 0.1 mS. The leak channel reversal potential is EL = –67mV.
M-current
The M-current has one activation gate (n = 1) and no inactivation gate (k = 0). The rate functions for the M-current activation gate are described by:
We use a Q10 factor of 2.3 to scale the rate functions of the M-current since the original formulation of these kinetics described dynamics at 23°C [98]. Thus, for a normal body temperature of 37°C, the M-current rate equations are scaled by Qs, which is formulated as:
The maximal M-current conductance is .
Synaptic connectivity and networks
Networks of FSIs contained 100 neurons. For networks that additionally had SPNs, we modeled 100 D1 SPNs and 100 D2 SPNs. Due to computational constraints, we did not include enough SPNs to simulate a realistic ratio of interneurons to projection neurons. Although in rodents, interneurons consist of at most 5% of all cells in the striatum [9], interneurons account for at least 25% of neurons in the human striatum [99]. Thus, our networks consist of proportions of FSIs and SPNs that are likely closer to those found in humans than rodents.
The model synaptic GABAA current (IGABAA) is formulated as in McCarthy et al., 2011 [15] and is the only synaptic connection between SPNs and from FSIs to SPNs. The GABAA current has a single activation gate dependent on the pre-synaptic voltage.
The maximal GABAA conductance between FSIs is . The maximal GABAA conductance from FSIs to SPNs is and between SPNs was . These values are consistent with FSI to SPN inhibition being approximately six times stronger than inhibition between SPNs [9].
The gating variable for inhibitory GABAA synaptic transmission is represented by si. For the jth neuron (FSI or SPN) in the network:
The variable Sikij describes the kinetics of the gating variable from the kth pre-synaptic neuron to the jth post-synaptic neuron. This variable evolves in time according to:
The GABAA time constant of decay (τi) is set to 13 ms for SPN to SPN connections [97] as well as for FSI to FSI connections and FSI to SPN connections [30] The GABAA current reversal potential (Ei) for both FSIs and SPNs is set to −80 mV. The rate functions for the open state of the GABAA receptor (gGABAA(Vk)) for SPN to SPN transmission is described by:
The rate functions for the open state of the GABAA receptor (gGABAA (Vk)) for FSI to FSI and FSI to SPN transmission is:
The value of τr is 0.25 ms. FSIs were additionally connected by dendritic electrical connections. The electrical coupling for dendritic compartment i is denoted as Ielec, has units in μA/cm2 and is formulated as:
The value of the gap junction conductance ggap depended on DA level (see below). Within the 100-cell FSI network, each pair of FSIs had an independent 30 percent chance of a dendro-dendritic gap junction chosen from a uniform random distribution and an independent 30 percent chance of a somato-somatic inhibitory synapse also chosen from a uniform distribution. SPNs are connected with each other in a mutually inhibitory GABAergic network [100]. We modeled all to all connectivity of inhibitory synapses from any SPN to any SPN of the same receptor subtype.
DA
DA impacts both connectivity and excitability in the model networks. DAergic tone was simulated as having five components: direct excitation of FSIs [12], increased gap junction conductance between FSIs [13], decreased inhibitory conductance between FSIs [12], increased excitation to D1 SPNs, and decreased excitation to D2 SPNs. DA-induced changes to SPN excitation were discussed above. Excitation to FSIs was modeled as the sum of a tonic, DA dependent input current (Itonic) and a noise term. DA did not change the noise term in either SPNs or FSIs. The baseline DAergic tone state was modeled in FSIs using Itonic = 4μA/cm2, ggap = 0.05mS and the GABAA conductance between FSIs was gii = 0.1mS. The high DA state was modeled in FSIs using Itonic = 14μA/cm2, ggap = 0.2mS and gii = 0.005mS.
Local field potential
The local field potential (LFP) was calculated as the sum of all voltages in all cells. Stationarity of the network appears in the raster plots after about 500 ms. To eliminate transients due to initial conditions, our LFP is evaluated only after 1,000 ms of simulated time. We estimated the power spectral density of the simulated LFP using the multitaper method [101].
Simulations
All simulations were run on the MATLAB-based programming platform DynaSim, a framework for efficiently developing, running and analyzing large systems of coupled ordinary differential equations, and evaluating their dynamics over large regions of parameter space [102]. DynaSim is open-source and all models have been made publicly available using this platform. All differential equations were integrated using a fourth-order Runge-Kutta algorithm with time step was .01 ms. Plotting and analysis were performed with inbuilt and custom MATLAB code.
Acknowledgments
We thank Jason Sherfey and Erik Roberts for developing the DynaSim Toolbox used to run the simulations in this manuscript and assisting with debugging and visualization. In addition, we thank Sean Patrick for assistance in interpreting the behavioral implications of our model.
Footnotes
↵☯ These authors co-mentored this work
Figure rendering errors removed
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