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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptNIH Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
J Neurosci. Author manuscript; available in PMC Aug 17, 2010.
Published in final edited form as:
PMCID: PMC2833015
NIHMSID: NIHMS179263

Phase response curve analysis of a full morphological globus pallidus neuron model reveals distinct perisomatic and dendritic modes of synaptic integration

Abstract

Synchronization of globus pallidus (GP) neurons and cortically-entrained oscillations between GP and other basal ganglia nuclei are key features of the pathophysiology of Parkinson's disease. Phase response curves (PRCs), which tabulate the effects of phasic inputs within a neuron's spike cycle on output spike timing, are efficient tools for predicting the emergence of synchronization in neuronal networks and entrainment to periodic input. In this study we apply physiologically realistic synaptic conductance inputs to a full morphological GP neuron model to determine the phase response properties of the soma and different regions of the dendritic tree. We find that perisomatic excitatory inputs delivered throughout the inter-spike interval advance the phase of the spontaneous spike cycle yielding a type I PRC. In contrast, we demonstrate that distal dendritic excitatory inputs can either delay or advance the next spike depending on whether they occur early or late in the spike cycle. We find this latter pattern of responses, summarized by a biphasic (type II) PRC, was a consequence of dendritic activation of the small conductance calcium-activated potassium current, SK. We also evaluate the spike-frequency dependence of somatic and dendritic PRC shapes, and we demonstrate the robustness of our results to variations of conductance densities, distributions, and kinetic parameters. We conclude that the distal dendrite of GP neurons embodies a distinct dynamical subsystem that could promote synchronization of pallidal networks to excitatory inputs. These results highlight the need to consider different effects of perisomatic and dendritic inputs in the control of network behavior.

Keywords: dendrite, SK current, synchronization, oscillation, basal ganglia, Parkinson's disease

Introduction

The emergence of synchronous activity in neuronal networks can result from changes in the synaptic response function of component neurons described by the phase response curve (PRC). The PRC is constructed by plotting spike time shifts caused by inputs at different times within the spike cycle. Type I PRCs for excitatory (inhibitory) stimuli are composed predominantly of positive (negative) values indicating that inputs throughout the spike cycle advance (delay) the next spike. In contrast, type II PRCs contain both positive and negative regions indicating that excitatory inputs can either advance or, paradoxically, delay the next spike depending on input phase. Type II PRCs have been related extensively in vitro and in silico to synchronization of connected neuronal networks with particular architectures (Hansel et al., 1995; Ermentrout, 1996; Crook et al., 1998a, b; Ermentrout et al., 2001; Netoff et al., 2005a; Netoff et al., 2005b; Goldberg et al., 2007; Achuthan and Canavier, 2009; Bogaard et al., 2009) and of uncoupled neurons receiving correlated inputs (Galan et al., 2007a; Marella and Ermentrout, 2008; Abouzeid and Ermentrout, 2009).

Recent experimental and theoretical studies have demonstrated that passive dendritic filtering of inputs and the contributions of active membrane currents are critically involved in shaping the PRC and therefore network dynamics (Crook et al., 1998a, b; Gutkin et al., 2005; Goldberg et al., 2007; Stiefel et al., 2008; Stiefel et al., 2009). Sensitivity of the PRC to neuromodulation may underlie switching between different network functional states (Stiefel et al., 2008; Stiefel et al., 2009), and chronic alterations of functional network connectivity or removal of modulation are related by their effects on phase response properties to pathological network synchronization characteristic of epilepsy (Netoff et al., 2004; White and Netoff, 2008), which may similarly apply to Parkinson's disease (PD).

Synchronized oscillations and bursting in basal ganglia (BG) structures are key features of the pathophysiology of PD, and several physiological studies (Plenz and Kitai, 1999; Magill et al., 2000, 2001; Loucif et al., 2005) and network simulations (Terman et al., 2002) suggest that the GP-Subthalamic Nucleus (STN) feedback loop within the BG can promote oscillatory pattern generation (Bevan et al., 2002). Furthermore, recent evidence indicates an orchestrating role for GP in the β-frequency synchronization of BG activity in PD (Mallet et al., 2008). To elucidate how cellular properties of GP neurons may be involved in the emergence of synchronous states, it is important to examine what conditions can support biphasic PRCs. We used a well-characterized, full morphological GP neuron model (Gunay et al., 2008) to determine how PRCs of these neurons may depend on input characteristics and intrinsic cellular mechanisms. Using a model analysis allowed us to fully trace the parameter dependence of PRC shape and circumvented the experimental problems underlying accurate PRC estimation due to intrinsic spike cycle variability, which is prominent in GP (Deister and Wilson, 2008). We found that perisomatic PRCs in GP model neurons were type I, whereas distal dendritic excitatory inputs yielded type II PRCs due to local activation of the small conductance calcium-activated potassium current (SK) at the site of stimulation. The influence of this dendritic SK mechanism on spike timing is likely to promote GP synchronization and entrainment to oscillatory STN inputs that are prominent in PD.

Methods

Simulations were run on Emory University High Performance Compute Clusters (Sun Microsystems) using the GENESIS simulation platform (www.genesis-sim.org/GENESIS). Approximately 1.5 min of processor time was required to simulate one second of data with 20 μs time steps using the full 585-compartment GP neuron model. Custom routines were written using Matlab (The MathWorks, Natick, MA) for the analysis of voltage, current, conductance, and spike time data.

GP Neuron Model

Morphology and Passive Electrical Properties

The morphology and passive electrical properties of our baseline GP neuron model (GPbase) were determined as previously described (Hanson et al., 2004; Gunay et al., 2008). Briefly, the Neurolucida (MicroBrightField, Inc.) reconstruction of a GP neuron that showed electrophysiological properties (spike width, spike height, input resistance, spike adaptation) typical of GP neurons in our recorded population of GP neurons (Gunay et al., 2008) was converted to a GENESIS morphology file using CVAPP software (www.compneuro.org). The resultant somato-dendritic morphology contained a spherical soma and 511 dendritic compartments (Fig. 1A), each no more than 0.02 lambda in electrotonic length. Matching of the model to experimental voltage responses was achieved by setting passive biophysical parameters as follows: specific membrane capacitance (CM=0.024 F/m2), specific membrane resistivity (RM=1.47 Ωm2), & specific axial resistivity (RA=1.74 Ωm). To allow axonal spike initiation and realistic axonal current sources and sinks, a standard axon consisting of a highly excitable axon initial segment and nodes of Ranvier separated by myelinated inter-node segments, was adapted from Shen et al. (1999) and attached to the soma.

Figure 1
GP model morphology, conductances, and electrophysiological properties reproduce those of recorded GP neurons. A. Neurolucida (MicroBrightField, Inc) reconstruction of the GP neuron morphology used for all versions of the GP model. Somatic (S) and proximal, ...

Active Conductances

Based on experimental evidence of their presence in GP neurons, one calcium-activated conductance and eight voltage-gated membrane conductances were added to the soma and dendrite of the passive model. Because distinct electrophysiological cell types are not apparent in GP when examined with rigorous statistical tests (Gunay et al., 2008), we made a single active model resembling the mean behavior of recorded GP neurons. The voltage-gated conductances were modeled using standard Hodgkin-Huxley equations and included the following channel types: fast-transient and persistent sodium currents, NaF & NaP (Magistretti and Alonso, 1999; Magistretti et al., 1999; Raman and Bean, 2001; Magistretti and Alonso, 2002; Khaliq et al., 2003; Hanson et al., 2004; Mercer et al., 2007); fast and slow delayed-rectifier (Kdr) potassium currents, KV3 & KV2 (Baranauskas et al., 1999; Baranauskas et al., 2003); A-type potassium current, which we modeled as two channel populations in order to accurately match fast and slow components of the inactivation kinetics, KV4F & KV4S (Tkatch et al., 2000); M-type potassium current, KCNQ (Gamper et al., 2003; Prole and Marrion, 2004); hyperpolarization-activated mixed cation current, or h-current, which we also modeled as two channel populations, HCN1 & HCN2 (Wang et al., 2002; Chan et al., 2004); and a high-voltage-activated calcium current (CaHVA) representing a mixture of L-, N-, and P/Q-type currents. The calcium dependence of the small-conductance calcium-activated potassium current (SK) (Hirschberg et al., 1998; Xia et al., 1998; Hirschberg et al., 1999; Keen et al., 1999) was modeled using the Hill equation. Table 1 in the online supplemental materials lists the activation and inactivation parameters for the conductances implemented in GPbase, and Supp. Fig. 1 illustrates the steady-state voltage or calcium dependencies of activation and inactivation gates for each channel type.

Conductance Distributions and Density Tuning

Each conductance was distributed uniformly throughout the dendrite with the exception of CaHVA whose density was greater in thinner dendritic compartments (Hanson and Smith, 2002). Since dendritic diameter tapered further from the soma, the highest densities of CaHVA occurred at distal regions of dendritic branches.

During the original tuning process, somatic and dendritic conductance densities were determined using a semi-automated process comparing model behaviors with physiological recordings, and a thorough exploration of parameter space was performed (Gunay et al., 2008). For this study, we updated the model to incorporate recent data describing ion channel kinetics in GP (Mercer et al., 2007) and used conductance densities within the previously explored parameter space that provided a good match with physiological current clamp data (see supplemental tables 1 and 2 for final parameters). The resultant base model, GPbase, exhibited a spontaneous spike waveform closely matching the average spike from 50 recorded GP neurons (Fig. 1B) and fell well within the limits of physiological variability for spontaneous spike frequency (Supp. Fig. 3A), the somatic FI curve (Fig. 1D), spike frequency adaptation (SFA) and spike height attenuation (SHA) during positive current steps, voltage ‘sag’ (a consequence of h-current activation) during negative current steps, and latency to the first spike following the offset of negative current steps (Supp. Fig. 2). The KV2 and KV3 conductances, in the model as in GP neurons, were the primary contributors to the delayed rectifier current (Baranauskas et al., 1999), and the peak amplitudes of the respective currents were similar (Fig. 1C). The conductance density and somato-dendritic distribution of the small-conductance calcium-activated potassium current, SK, were the primary determinants of the depth of the mAHP, and together with NaP and KCNQ, contributed significantly to the spontaneous spike frequency (Supp. Fig. 3A). Interestingly, the slope of the somatic FI curve was also influenced by the density and distribution of SK conductance (Supp. Fig. 3B). To determine the robustness of our PRC findings with this model, we varied key conductance parameters through a large range deviating from the base model (see Results). These manipulations also serve as explorations of potential heterogeneity among pallidal networks and of possible effects of altered modulation of GP neurons. The full GENESIS model used in this study is available for download from ModelDB (http://senselab.med.yale.edu/ModelDB/).

Simulation Protocols

PRC Stimulation Protocol

In 72 separate simulations, inputs were delivered at each of 72 evenly-distributed time points within the first inter-spike interval (ISI) of the control spike train. The onset of the earliest input was timed to be coincident with the somatic spike delineating the start of the spike cycle. Spike times were recorded using the GENESIS spikehistory element which provided precision of 1e-6 s, and input-evoked shifts in spike timing were calculated relative to the spike terminating the first control spike cycle. PRCs were plotted as spike advances (in units normalized to the period of the control ISI) as a function of input phase, such that positive values reflected advancements of the spike cycle. Separate PRCs were constructed for somatic stimulation and for stimulation of 7 dendritic sites each composed of 25 contiguous compartments. The dendritic sites of stimulation were distributed across the three major branches of the dendrite (Fig. 1A) such that the first and second dendritic branches each had proximal and distal stimulation sites (D1P&D and D2P&D), while the longer third branch had proximal, mid, and distal stimulation sites (D3P,M,&D). In order to conduct PRC analyses during steady state behavior, we let the spontaneous activity of the model settle for a 10 s period and used the first spontaneous spike cycle after this period to apply PRC stimulation. To avoid repeating the simulation of 10 s settling time many times, we saved all state variables in a 'snapshot' file after conducting this simulation once, and used it for all subsequent simulations.

Inputs to the model

Stimuli were either 3 ms square-wave current injections or dual exponential conductance injections representing excitatory AMPAergic (1 ms rise time, 3 ms decay time, EAMPA = 0 mV) or inhibitory GABAA (1 ms rise time, 12 ms decay time, EGABA = −80 mV) synaptic inputs. Input strength was manipulated by varying current injection amplitude or the synaptic peak conductance. Current injections or synaptic conductance inputs to dendritic regions were divided evenly amongst the 25 component compartments such that each received 4% of the total reported current or conductance.

Faster Spiking and Higher-Order PRCs

Since GP neurons in vivo spike at approximately 30 Hz, we drove the GP model faster than the spontaneous spike frequency by applying tonic inward current to the soma. To test how increasing spike frequency affects the PRC, we derived somatic and dendritic PRCs using 2 nS AMPA inputs while the model spiked at specific rates between 10 and 60 Hz. Applied current strengths necessary to achieve the desired spike frequencies (± 0.5 Hz) were determined by precise interpolation from the FI curve.

Synaptic inputs to the model, particularly the dendrite, evoked membrane currents that contributed to shifts in spike timing. In many cases the time courses of these evoked currents were not limited to the ongoing ISI, however, so it was necessary to consider multiple spike cycles to capture to full effect of an input. Effects of inputs on the timing of the second, third, and fourth subsequent spikes were calculated to give the F2, F3, and F4 PRCs, respectively. By convention (Preyer and Butera, 2005), these higher-order PRCs were plotted to the left of the primary PRC (F1). The ‘permanent’ PRC reflects the ultimate advancement or delay of spiking measured after several spike cycles (Prinz et al., 2003a). The permanent PRCs depicted in this report were calculated by summing single-cycle PRCs F1 through F5.

Categorization of PRCs as type I or type II

The distinction between type I and type II PRCs can be somewhat ambiguous. Type I PRCs are characterized by being composed predominantly of positive values for excitatory inputs. However, there may be a small negative region early in phase corresponding to stimulus delivery during the down-stroke of an action potential, without disqualifying the corresponding PRC as type I (Rinzel and Ermentrout, 1998; Oprisan and Canavier, 2002). Using a morphological model compounds this ambiguity, because back-propagating action potentials in the dendrite are wider than somatic spikes, and dendritic stimuli lead to a correspondingly wide negative region in the PRC. To distinguish these effects from dendritic mechanisms responsible for significant delays of spike timing, we limited categorization of PRCs as type II to cases where delays of spike timing were not the consequence of direct interactions between PRC stimuli and somatic or back-propagating dendritic spikes. This was accomplished by categorizing PRCs using a variant of the r-value convention of Tateno and Robinson (Tateno and Robinson, 2007) as follows: The r-value of a PRC, a measure of how biphasic the PRC is, was defined for this study as the ratio of the negative area of the PRC (absolute value) to the positive area of the PRC (or the inverse of that ratio, whichever was smaller). PRCs were deemed type I when the r-value was less than 0.175, and PRCs were deemed type II when the r-value exceeded 0.175.

Membrane potential and channel current analysis

To characterize the size and spatial decay of voltage transients in response to inputs we recorded membrane voltage at the soma, axon, and dendritic sites in each simulation. Dendritic voltages were recorded from the middle and edge compartments of each of the 7 sites of dendritic stimulation (Fig. 1A). In addition we recorded current traces for each type of membrane conductance at the soma, axon, and the 7 dendritic sites used for stimulation. Current traces described in this report as ‘somatic’ reflect the sum of currents from the soma and hillock, and recordings of currents from any of the 7 dendritic sites reflect sums taken across the compartments composing that site.

To evaluate the contributions of membrane currents to input-evoked shifts in spike timing, we calculated ‘difference’ or ‘evoked’ currents by subtracting current traces from control simulations (without stimulus) from the corresponding stimulated traces. This subtraction isolated the evoked component of membrane current flows.

Results

Responses to Weak Phasic Inputs

Much of conventional PRC theory relies on the assumption that inputs to a neuron are weak such that multiple inputs within a spike cycle result in the linear sum of the corresponding individual phase perturbations. Within the ‘domain of weak coupling’, the PRC scales linearly in amplitude as a function of input strength, because two inputs arriving simultaneously have exactly twice the effect on spike timing as one input of the same strength. Thus, for sufficiently weak inputs the normalized PRC has a static shape called the ‘infinitesimal’ PRC (iPRC). However, realistic synaptic conductances can violate the assumption of weak coupling causing the measured phase response curve to deviate from the iPRC (Acker et al., 2003; Netoff et al., 2005a). To evaluate the range of weak input strengths for which the somatic PRC scales linearly in our GP model, we applied 3 ms square-wave somatic current steps of varying amplitude at different phases of the spontaneous spike cycle and measured the resultant shifts in spike timing. Depolarizing current pulses elicited advances of the spontaneous spike rhythm that persisted indefinitely (Fig. 2A&B). For these inputs (<=100 pA) there were no higher order effects of somatic stimuli on phase, i.e. all ISIs after the stimulated spike cycle were equal in duration to the spontaneous period (Fig. 2A). The magnitude of phase advancement varied smoothly with input phase (the PRC), showing greater sensitivity to inputs in the last quarter of the spike cycle and relative insensitivity to inputs arriving during the first half of the spike cycle (Fig. 2C). The somatic PRC was type I, because it was composed purely of positive values reflecting that depolarizing inputs at any phase of the spike cycle advanced the next spike. The positive peaks of normalized somatic PRCs obtained with stronger inputs occurred earlier in the spike cycle (Fig. 2C), indicating a deviation from linearity and thus the mathematical domain of weak coupling already at input amplitudes below 100 pA. Another non-linearity was evident when 3 ms negative current injection pulses were applied: Within the domain of weak coupling, the iPRC obtained using positive current pulses should be an exact mirror image of that obtained with negative pulses. We observed that only for very small stimuli was the spike delay caused at different input phases equal and opposite to the spike advance caused by positive current pulses of the same amplitude. Hyperpolarizing stimuli of increasing amplitude yielded PRCs that scaled much more linearly than those obtained with depolarizing inputs of the same amplitudes, suggesting that even small amounts of depolarization caused the activation of non-linear response properties. This leads to a fundamental difference in phase response properties between inhibitory and excitatory stimuli not predicted by PRC theory.

Figure 2
Physiologically realistic inputs to GP are outside somatic and dendritic domains of weak coupling. A. Voltage traces illustrating spontaneous 7.9 Hz spiking of the model and voltage deflections and spike advances elicited by 100 pA somatic stimuli delivered ...

To delimit the domain of weak coupling for somatic inputs in our GP model we examined the features of the PRC that were most sensitive to input strength and used them as a means to evaluate the convergence of PRC shape to the iPRC. The peak of the somatic PRC occurred later in the spike cycle for weaker depolarizing inputs (Fig. 2C) and converged to a phase of ~0.903 for inputs of 7.5 pA or less (Fig. 2G, solid black line). At this amplitude, PRCs also became symmetrical for depolarizing and hyperpolarizing stimuli (Fig. 2H), again indicating that 7.5 pA stimulus amplitude was the upper limit for linear phase response behavior in a strict sense.

Next, we compared the effects of phasic somatic inputs on output spike timing with inputs delivered to a distal dendritic site (D3D, Fig. 1). Because the local input resistance of small dendrites is much higher than that of the soma, the voltage response to current injection of equal amplitude is greater. For example a 10 pA 3 ms stimulus at 0.5 phase resulted in a 0.4 mV peak voltage deflection in the soma, whereas the amplitude of the local voltage deflection in the distal dendrite (D3D site) was 9.0 mV in response to the same stimulus. Therefore, the domain of weak coupling for dendritic stimuli was even smaller than for the soma, and divergence for our 2 measures of linearity (peak shift, Fig. 2G and peak symmetry Fig. 2H) was seen for stimuli exceeding just 1 pA amplitude. The resultant distal dendritic iPRC was type I (r-value = 0.066) (Fig. 2D&F and Supp. Fig. 4), but surprisingly, inputs delivered within a brief time-window just after a spontaneous spike resulted in slight delays of the subsequent action potential. These delays and the corresponding negative region early in the PRC (Fig. 2D, inset) were increased with higher input strengths (Supp. Fig. 4) and are analyzed in detail below.

Infinitesimal PRCs (1 pA stimuli) for the soma and 7 dendritic stimulation sites are shown in Figure 2I. More distal inputs yielded attenuated PRCs with peaks occurring earlier in phase as a partial consequence of passive filtering of inputs by the dendrite (Goldberg et al., 2007). Nonetheless, iPRCs for all regions of the GP model were type I, and they differed smoothly from one another as a function of distance of the stimulation sites from the soma.

Responses to Larger Somatic Current Injections and Realistic Synaptic Inputs

The peak conductance of a single AMPAergic synaptic event is on the order of 250 pS to 1 nS producing a peak synaptic current of approximately 15-60 pA (Hernandez et al., 2006). Thus, in our GP model currents due to the activation of even a single synapse lie outside the domain of weak coupling that provides a basis for analytical mathematical treatment of PRCs. To address the question of how the input phase of physiologically realistic synaptic input affects spike timing, we delivered stronger current injections or dual exponential AMPA-type or GABAA conductance inputs to somatic and dendritic sites of the model. Like the weaker inputs discussed above, somatic current pulses of up to 300 pA yielded type I PRCs, and the peak of the somatic PRC occurred earlier in the spike cycle for stronger depolarizing inputs (Fig. 3A). This pattern reflects: 1) that maximal sensitivity to weak somatic inputs occurs just before a spike, and 2) the theoretical limit of spike advancement for strong inputs (Fig. 3A, long-dashed black line) is reached when a spike is initiated immediately with delivery of the stimulus. Stronger somatic inputs caused the PRC to approach the limit of spike advancement over a larger range of phase. Thus, the peak of the PRC occurred earlier within the spike cycle and the region of the PRC just before the peak became increasingly steep (Fig. 3A).

Figure 3
PRC shape depends both on stimulus location and strength. A. Somatic PRCs are always Type I for square-pulse positive current injections and for simulated excitatory synaptic AMPA conductances. Somatic PRCs have peaks that scale approximately linearly ...

The phase shifts elicited by hyperpolarizing or inhibitory inputs are not bounded by a theoretical limit, since arbitrarily strong inhibitory inputs can theoretically delay subsequent spiking indefinitely. As a result, negative current injection delivered just prior to spike initiation resulted in maximal phase delays (Fig. 3D). PRCs resulting from negative current injection scaled nearly linearly with input strength and maintained the same qualitative shape (Fig. 3D).

AMPAergic synaptic inputs to the soma elicited a similar pattern of spike advances as did depolarizing current injections (Fig. 3A), and the spike delaying effects of somatic GABAergic inputs were similar to those of hyperpolarizing current injections to the soma (Fig. 3D). The only notable difference between peak-amplitude-matched somatic PRCs obtained using synaptic inputs and those obtained using current injections was that the peaks of synaptic PRCs occurred slightly earlier in the spike cycle, both for excitation and inhibition. This effect is likely due to the gradual onset and delayed peak of synaptic inputs compared to the sharp onset of square-pulse current steps, thus giving rise to responses resembling those elicited by current steps occurring slightly later in phase. GABAergic inputs of 0.7 nS to 2.9 nS yielded somatic PRCs that matched in amplitude the somatic PRCs obtained using current injections of −50 pA to −300 pA, whereas AMPAergic inputs of 0.5 nS to 4.2 nS were necessary to match the peaks of somatic PRCs for +50 pA to +300 pA current injections (Fig. 3A,D; Supp. Fig. 5A,B). Overall, we found good agreement between stimuli using current step inputs or conductance waveform inputs, indicating that probing PRCs with brief current steps does not limit the validity of PRC analysis.

Type II PRCs from Stronger, More Distal Dendritic Inputs

Globus pallidus neurons have long, thin dendrites that create a flat disc-like field up to 1 mm wide and oriented perpendicular to striatal inputs (Park et al., 1982; Yelnik et al., 1984; Kita and Kitai, 1994). Non-uniform dendritic distributions of membrane conductances (Hanson and Smith, 2002; Hanson et al., 2004) and higher local input resistance at thinner, more distal dendritic segments make it unlikely that inputs to different sites along the length of GP dendrites will evoke equivalent voltage transients. To test how the strength and timing of physiologically realistic inputs to different sites of the dendritic tree affect output spike timing, we constructed PRCs for seven dendritic sites using synaptic inputs of a range of peak conductance amplitudes. In each case PRCs for proximal dendritic inputs were qualitatively similar to somatic PRCs (data not shown). Surprisingly, larger AMPAergic inputs to the distal sites of the first and second dendrites and to the middle of the longest dendrite (D1D, D2D, & D3M) yielded PRCs with increasing type II character. (PRCs for D2D are shown in Figure 3B, and r-values quantifying the increase in type II character for stronger synaptic inputs are plotted in Supplemental Figure 6A.) Whereas D2D PRCs for excitatory inputs of 0.5 nS and 1 nS were type I (r-values of 0.0183 and 0.0449, respectively), inputs stronger than 1 nS significantly delayed the next spike when delivered any time during the first half of the spike cycle. For AMPAergic inputs of 1.5 nS to 3 nS to D2D, the negative regions of the corresponding type II PRCs (r-values of 0.24 to 0.71) were progressively deeper. Thus, paradoxically, strengthening ‘excitatory’ synaptic input to D2D changed the net effect of the input from phase-advancing to phase-delaying for a significant portion of the spike cycle. Increasing AMPA input strength also increased the proportion of the spike cycle for which inputs delayed the subsequent spike. The positive peaks of those PRCs were also increased for stronger excitatory inputs (Fig. 3B) resulting in biphasic, type II PRCs of increasing amplitude.

Excitatory synaptic inputs to the distal-most site of the longest dendrite (D3D) yielded type II PRCs for all input strengths tested (r-values >= 4.75 for inputs >= 0.5 nS) (Fig. 3C and Supp. Fig. 6B), but as was the case for D2D, stronger AMPA inputs to D3D increased the depth of the negative region in the PRC (Fig. 3C). (Note that in Fig. 2D, current injections of as little as 10 pA notably increased the type II character of the D3D PRC.) A prominent step in the depth of the negative region of the D3D PRC occurred between AMPA input strengths of 0.5 nS (Fig. 3C, red line) and 1 nS (Fig. 3C, magenta line), but strengthening AMPAergic synaptic inputs beyond 1 nS increased the amplitude of the D3D PRC without qualitatively affecting its shape (Fig. 3C and Supp. Fig. 6B). In general, inputs that were applied at distal dendritic sites shifted spike timing by smaller amounts than stimuli at more proximal sites. Comparing the D2D and D3D stimulation sites, the maximal spike advance for stimuli occurring late in phase was reduced approximately 10-fold, however, the spike delays reflected in the negative region of the PRC were of similar magnitude. This indicates that excitatory inputs even at very distal sites can contribute significantly to spike delays resulting from early-phase excitatory inputs.

GABAergic inputs to distal sites of the three major dendritic branches, D1D, D2D, & D3D, yielded monophasically negative, type I PRCs for all input strengths tested (Fig. 3E&F). Thus, inhibitory inputs to the distal dendrite delivered at any time within the spike cycle delayed the next spike, and there was no dependence of the qualitative shape of the PRC on the strength of those inputs. Since increasing the strength of inhibitory inputs did not qualitatively affect PRC shape, we hypothesized that intrinsic membrane currents activated more strongly by stronger depolarizing inputs were responsible for the increasing type II character of excitatory distal dendritic PRCs.

Strong AMPA Inputs to D2D Evoked Outward SK Current in Excess of Combined Inward Synaptic and Membrane Currents

Increasing the strength of excitatory inputs to D2D transitioned the PRC from type I to type II in a graded fashion (Fig. 3B). To evaluate which membrane currents contributed to this transition we compared the local voltage trajectories and patterns of locally-evoked currents for weak and strong inputs to D2D (Fig. 4). Each panel in Figure 4 shows voltage or current transients evoked by inputs at three different times within the spike cycle. Note that the control voltage traces from D2D (Fig. 4A&B, solid black lines) contain attenuated back-propagated spikes of ~5 mV amplitude which delineate the spike cycle, and red and blue dashed traces illustrate advancements or delays of the next spike, respectively.

Figure 4
Different input strengths evoke different contingents of membrane currents at the distal dendrite. Shown are stimulus-evoked voltage and current transients for 0.5 nS and 2 nS excitatory inputs to D2D. A. Voltage deflections elicited by 0.5 nS excitatory ...

The gross shape of the PRC for inputs to D2D can be inferred from inspection of the local voltage trajectories relative to the control. Following the depolarizing transients evoked by 0.5 nS AMPA input to D2D, the local voltage trajectories for stimulated trials converged back towards the control trajectory, but remained slightly depolarized relative to the control until the next spike event (Fig. 4A). Thus, 0.5 nS AMPA input at all phases of the spike cycle resulted in spike advances (Fig. 4A, red traces). In contrast, following the larger depolarization transients evoked by 2 nS AMPA to D2D, the local voltage trajectories for stimulated trials showed a larger initial depolarizing response but then hyperpolarized below the control until the time of the next spike (Fig. 4B). For the ‘late-phase’ stimulus, the next spike occurred almost immediately after the depolarization transient (Fig. 4B, red trace) corresponding to an advancement of the spike cycle. However, inputs to D2D delivered earlier in the spike cycle resulted in delays of the subsequent spike (Fig. 4B, blue traces) as the local membrane remained hyperpolarized below control traces.

Figure 4 (C&D) depicts the five intrinsic membrane currents evoked most strongly by distal dendritic synaptic inputs in the stimulated compartments. Local voltage transients in response to 0.5 nS and 2 nS inputs to D2D were approximately 5 mV and 25 mV, respectively (Fig. 4A&B). This supralinear relationship between input and response amplitudes was primarily due to stronger activation of inward NaF current by the stronger stimulus (Fig. 4C&D, red lines). The most pronounced effect, however, of strengthening synaptic input to the distal dendrite was the increased local activation of the small conductance calcium-activated potassium current, SK (Fig. 4C&D, green lines). The peak of the SK current evoked by the 2 nS input was fifteen-fold that of the 0.5 nS input. Though the peak amplitude of the evoked SK current was smaller than the peak evoked NaF current, the duration of SK activation was much longer lasting, resulting in local hyperpolarization and a net current sink in response to early-phase AMPA stimulation and explaining how excitatory inputs can yield phase delays.

Comparison of voltage-gated and synaptic currents explains somatic type I and dendritic type II PRCs

Whereas D2D PRCs transitioned from type I to type II for stronger synaptic inputs, somatic PRCs were type I and D3D PRCs were type II over the entire physiological range of stimulus strengths tested. Figure 5 (A-D) shows the local responses to 1 nS AMPA inputs to the soma and D3D allowing the comparison of voltage and current transients between morphological sites for which identical inputs yielded type I and type II PRCs, respectively. The voltage trajectories elicited by somatic inputs remained depolarized relative to the control until the next spike event (Fig. 5A). In contrast, inputs to D3D evoked larger transients of depolarization (~40mV) followed by deep hyperpolarizations of the local membrane (Fig. 5B), similar to the effect of strong AMPA inputs to D2D.

Figure 5
Targeting identical stimuli to different regions of the model evokes different contingents of membrane currents. Shown are stimulus-evoked voltage and current transients for 1 nS excitatory inputs to the soma vs. the most-distal region of dendrite, D3 ...

During the spontaneous spike cycle the somatic voltage traversed a much larger sub-threshold range (~10 mV) than did the distal dendritic voltage (<5 mV). Consequently, the contingent of currents evoked locally by somatic AMPA inputs depended more strongly on the phase of the input (Fig. 5C) than did the pattern of currents evoked by inputs to D3D. However, the somatic membrane currents evoked by 1 nS AMPA inputs were insignificant relative to the synaptic current (Fig. 5E), because the voltage deflection caused by the stimulus was small. Therefore, somatic inputs resulted in type I PRCs, because for stimuli at all phases of the spike cycle the inward synaptic current overshadowed the net-outward contribution of evoked membrane currents (Fig. 5E).

Because action potentials did not back-propagate efficiently into the distal dendrite in our GPbase model, the control voltage trajectory at D3D was relatively flat throughout the spike cycle (Fig. 5B). Consequently, the pattern of intrinsic currents evoked by D3D inputs was relatively independent of input phase. AMPA inputs of 1 nS to D3D evoked intrinsic membrane currents with a fast inward component and a slower outward component (Fig. 5D&F), as follows: Strong, ‘fast’ NaF current (~30 pA) amplified the immediate depolarization evoked by an input. In turn, higher calcium influx through the high-voltage-activated calcium channel (CaHVA, the calcium source for SK in the model) elicited greater activation of the long-lasting, ‘slow’ SK current (peak>20 pA). (Fig. 5D). As depicted in Figure 5F, the net outward charge transfer by local intrinsic membrane currents at D3D was sufficiently large to exceed the depolarizing charge carried by the inward synaptic current (Fig. 5F).

The Type II Character of Distal Dendritic PRCs Directly Reflects Local SK Conductance

The analysis of membrane currents elicited locally by strong synaptic stimulation of distal dendritic sites indicated that activation of the SK conductance by calcium influx played a predominant role in sculpting type II dendritic PRCs. To test this hypothesis directly we manipulated the conductance densities of CaHVA and SK in the distal dendrite. Importantly, the intrinsic spiking rhythm of the model was not disturbed, because this manipulation was limited to the 25 distal dendritic compartments composing the stimulated site (D2D or D3D). Local up- or down-regulation of CaHVA and SK in tandem directly scaled the magnitude of SK current evoked at the distal dendrite (Fig. 6A&C) and resulted in type II distal dendritic PRCs with deeper or shallower negative regions, respectively (Fig. 6B&D). Removal of either CaHVA or SK from D2D eliminated the type II character of the corresponding PRCs almost completely resulting in visually identical type I PRCs (r-values of 0.0351 and 0.0357, respectively) (Fig. 6B and Supp. Fig. 6C). However, despite the local removal of the CaHVA or SK conductance, these PRCs still contained shallow negative regions (Fig. 6B, asterisks) as a consequence of SK activation in compartments adjacent to the stimulated site. Activation of SK in compartments neighboring the D3D stimulation site was particularly strong, and D3D PRCs met our criterion for type II categorization even when CaHVA or SK was removed from the site (r-values of 0.78 and 0.80, respectively) (Fig. 6D and Supp. Fig. 6D). Models with different distributions of SK conductance between the soma and dendrite (Supp. Fig. 3) also exhibited type II distal dendritic PRCs for which the negative region scaled in depth with the dendritic density of SK irrespective of the somatic SK density (Supp. Fig. 7).

Figure 6
Tandem local up- and down-regulation of HVACa and SK modulates the depth of the negative peak of distal dendritic PRCs. A. SK difference currents evoked by 2 nS excitatory synaptic input to D2D in models where the densities of HVACa and SK have been dialed ...

Spike Frequency Dependence of Somatic and Dendritic PRCs

Whereas GP neurons in slice typically spike spontaneously at frequencies below 15 Hz, GP neurons in vivo spike with a mean frequency of approximately 30 Hz in rodents (Ruskin et al., 1999; Urbain et al., 2000). A change in PRC shape with increasing firing frequency has been previously described for a simple θ-neuron model and may shift synaptic integration from coincidence detection at low rates to acting as an integrator at high rates (Gutkin et al., 2005). To evaluate the dependence of PRC shape on spike frequency in our full morphological GP neuron model, we drove the model to 10-60 Hz firing frequency with 15-371 pA tonic somatic current injection.

During 7.9 Hz spontaneous spiking, only the ongoing ISI was affected by 2 nS AMPA input to the soma. At this spike frequency, the PRC for the 2nd interval (F2) was therefore flat (Fig. 7A3, Fig. 8A1) indicating the absence of higher-order effects of synaptic excitation.

Figure 7
Spike frequency affects somatic but not distal dendritic SK activation. A. Faster spiking elevates baseline levels and augments stimulus-evoked transients of somatic SK current. A1. When spiking was driven from 10 Hz (black trace) to 30 Hz (red trace) ...
Figure 8
Faster axo-somatic spiking changes the shapes of somatic and dendritic PRCs, and it is necessary to consider stimulus effects that outlast a single spike cycle. A1. When the model is driven faster (from 10 Hz to 30 Hz) by increasing applied current to ...

During 30 Hz spiking, however, the soma was depolarized by 5 mV on average compared to 10 Hz spiking (Fig. 7A1), and 2 nS AMPA inputs were capable of eliciting immediate spike initiation at earlier phases of the spike cycle than during 10 Hz spiking (Fig. 7A3&A4). This led to a left shift in the peak of the corresponding F1 PRCs (Fig. 8A1) for higher spike frequencies that was gradual between 10 and 60 Hz. During 30 Hz spiking, the activation profile of somatic currents was substantially changed. Notably, when spiking was driven from 10 Hz to 30 Hz, the baseline somatic SK current was elevated from ~50 pA to ~125 pA (Fig. 7A2) and stimulus elicited SK transients were additionally enhanced (Fig 7A2, asterisk). As a consequence, during faster spiking the somatic SK current activation accompanying spikes triggered by AMPA inputs was greater than the SK current activation accompanying control spikes. Therefore, the mAHP following an AMPA-evoked spike was slightly deeper than the mAHP during control spiking, and the subsequent spike cycle was longer than the control ISI (Fig. 7A4). While spike initiation immediately after AMPA stimulation corresponded to a large phase advance (F1, Fig. 8A1), a part of this advance was lost in the 2nd spike cycle following stimulation (F2, Fig. 8A1), because this 2nd ISI was longer than the baseline ISIs (Fig. 7A4). Therefore, at in vivo spike frequencies, an important effect of somatic SK conductance is to resist lasting advancements of the spiking rhythm. By lengthening the spike cycle following an AMPA-evoked spike, somatic SK makes the effect of the input more transient and significantly restores the original pattern of spiking. This effect of somatic SK conductance appears in the single-cycle somatic PRCs as a positive peak in the F1 that is nearly symmetrically opposed by a negative peak in the F2 (Fig. 8A1).

The so called ‘permanent’ PRC (Prinz et al., 2003a) consists of the sum of PRCs for the stimulated spike cycle and all subsequent spike cycles that show changed ISIs compared to baseline. Therefore, the permanent PRC reflects the lasting shift of the spike train (measured after several spike cycles) compared to the control oscillation. For somatic AMPA stimulation the permanent PRC remained type I for all spike frequencies from 10-60 Hz (Fig. 8A2), indicating that the spike delay during the F2 period never exceeded the original spike advance in the stimulated spike cycle.

Distal dendritic voltage and current transients elicited by AMPA stimuli were generally unaffected by faster somatic spike rates (Fig. 7B1-B3), e.g. the SK transient elicited by input to D3D was identical at 10 Hz and 30 Hz firing frequencies (Fig. 7B3). Nevertheless, the PRC for D3D stimulation showed a dramatic shift at faster somatic spike rates, because the elicited dendritic currents had a fixed time course that now showed a different phase relationship with respect to the somatic spike cycle. The PRC of the stimulated spike cycle showed a positive peak shifted to the left, which resulted in a type I primary PRC for spike frequencies above 30 Hz (Fig. 8B1). Because the waveform of AMPA-evoked SK lasted more than 50 ms (Fig. 7B3), at spike frequencies above 15 Hz the effect of SK current evoked by late-phase AMPA inputs to the distal dendrite impinged primarily on the 2nd spike cycle (F2) following the stimulus. Consequently, the 2nd ISI was prolonged, and the F2 PRC contained a significant negative peak for late-phase inputs (Fig. 8B1). In the case of D3D stimulation, the delayed spikes in F2 and F3 intervals more than compensated any spike advance occurring in the stimulated spike cycle. The corresponding permanent PRC became entirely negative at spike frequencies above 10 Hz, i.e. a strong AMPA stimulus at any phase of the spike cycle ultimately resulted in a delay of the spike oscillation. Therefore, our model predicts that at in vivo spike rates strong AMPA input to the distal dendrites of GP neurons will advance the first subsequent spike but ultimately reduce spike frequency.

Stimulus locations at intermediate electrotonic distance from the soma showed a combination of the effects seen at the soma and the distal dendrite. The voltage at these locations was somewhat more depolarized during faster spiking (as at the soma), but local SK transients evoked by AMPA inputs were still relatively invariant across spike frequencies (as at the more-distal dendrite). Like the D3D case, AMPA stimuli applied late in the spike cycle led to a negative region in the PRC that moved from the F1 to the F2 ISI for increasing spike rates (Fig. 8B1&C1, blue arrowheads). The lasting effects of AMPA inputs on spike timing were similar for D2D and D3D, such that the peak of the permanent PRC was shifted leftward and became flatter at higher spike frequencies (Fig. 8B2&C2). However, for mid-distal AMPA stimuli the PRC retained a positive region for all spike frequencies, and thus remained type II.

It will be of significant interest to evaluate how well the permanent PRC explains the emergence of synchronous states of network activity, or whether the development of network synchrony from asynchronous modes is better explained by the single-cycle primary and higher-order PRCs. It is important to note that the time-period during which the SK current evoked by excitatory dendritic stimulation is strongest lasts approximately 40 ms (~25 Hz), suggesting the possibility that dendritic SK current could cause GP neurons to resonate with β-frequency oscillatory excitation from STN.

Dendritic Conductance Densities Modulate Dendritic PRCs

The preponderance of synaptic inputs on GP neurons is located on dendrites, which are quite long, but thin and sparsely branching (Falls et al., 1983; Shink and Smith, 1995). To understand how dendritic inputs remote from the soma could influence spiking one needs to consider the local processing of inputs by active conductances in the dendrite. However, dendritic currents have not been recorded experimentally because the dendritic diameter is generally too thin for whole cell recording. Therefore dendritic conductance properties are not well constrained for GP neurons and the assumptions of our model with respect to dendritic conductance densities are tentative, and predominantly derived from matching somatic current clamp data. To examine the robustness of our findings with respect to parameter variations of dendritic conductance densities, we varied dendritic conductance densities over a large range and determined the consequences for PRCs. Specifically, we either adjusted the fast spike conductances (NaF, NaP, KV3, and KV2) together between 0 and 150% of their baseline values, or we separately adjusted the major slow dendritic inward (NaP) or outward (SK) currents.

We focus our analysis of how manipulating dendritic parameters affects PRCs on 2.5 nS AMPA stimulation of the D2D stimulation site, because this location showed a combination of local dendritic and somato-dendritic coupling effects in our preceding analyses (Fig. 8). To eliminate spike rate effects on PRC shape, we drove all models to 30 Hz spiking with somatic current injection to match in vivo rates (103-202 pA current depending on dendritic parameters). Varying the density of NaP alone or of the 4 spike conductances together had relatively slight effects on PRC shape that were similar between the two manipulations. Notably, for the largest values of NaP or the spike currents both the maximal spike advances in the stimulated spike cycle (F1) and the maximal spike delays in the subsequent spike cycle (F2) were increased (Fig. 9A1&B1). The combination of these effects led to permanent PRCs that showed an increased spike advance for AMPA stimulation at mid spike-cycle and increased spike delay for AMPA stimulation at the beginning or end of the spike cycle (Fig. 9A2&B2). These effects can be understood by examining the differences in dendritic membrane currents between models with different dendritic conductance densities (Supp. Fig. 8). Higher dendritic densities of NaP or of the spike conductances increased the positive peaks of the corresponding F1 PRCs by supporting larger inward current transients, and increased the negative regions of the corresponding F2 PRCs by amplifying outward evoked SK currents. Consequently the permanent PRCs showed increased positive and negative peaks while the basic shape was retained (Fig. 9A2). Overall, these manipulations show that the PRC results we have described are quite robust even for large changes in the density of persistent inward and spike currents. The dependencies of PRC shape on dendritic conductance densities also provide insight into important potential sources of heterogeneity in pallidal networks, and they illustrate potential mechanisms of modulatory influence on neuronal and network dynamics. Note that we kept the increase in spike currents below the threshold for full blown propagating dendritic action potentials. When spike conductance densities were sufficient to support dendritic spike initiation in response to AMPA input, the type II character of dendritic PRCs disappeared and spike advances were much more pronounced (Supp. Fig. 9). Thus, neurons or neuron models with spiking dendrites should be seen as integrating dendritic inputs in a completely different fashion resulting in type I PRCs.

Figure 9
Distal dendritic PRCs are robustly type II against significant variation of dendritic conductance densities. A. Single-cycle PRCs (A1: F1 through F4) and the permanent PRC (A2) for models with 0, 1, and 4 times the base dendritic density of NaP. For models ...

Dendritic SK conductance

As described above, we found that dendritic SK can powerfully alter PRC shape by converting the inward synaptic current of an AMPA input into a net outward membrane charge flow and resulting in spike delays for early-phase inputs. Not surprisingly, decreasing or increasing the dendritic SK density had important effects on dendritic PRCs (Fig. 9C). When SK conductance was removed from the dendrite, the permanent PRC became entirely positive (Fig. 9C2, red trace). In this case, activation of somatic SK still resulted in F2 spike delays, because the advancement of the first (F1) spike led to a larger somatic SK current that impacted the second (F2) spike cycle (Fig. 9C1, blue arrowhead). Increasing the density of dendritic SK smoothly affected the corresponding PRCs. Models with greater dendritic SK showed a diminished positive peak in the F1 PRC and a larger negative peak in the F2 PRC (Fig. 9C1, red arrowhead). The mechanism underlying this result is straightforward, as increased dendritic SK conductance led to increased outward current and greater spike delays in response to dendritic AMPA inputs (Supp. Fig. 8). Other effects of increasing dendritic SK were mild, mainly resulting in slight hyperpolarization of the dendrite during the control spike cycle and a concomitant decrease of other depolarization-activated currents. These results indicate that dendritic SK can be varied over a large range while maintaining a smoothly changing type II dendritic PRC, whereas a complete absence of dendritic SK would lead to type I phase response dynamics.

Type II Dendritic PRCs are robust within a large range of SK activation parameters

There is uncertainty in our model about kinetic parameter settings governing SK activation, because the precise calcium dynamics governing SK activation as well as SK activation time constants in GP neurons are not experimentally well constrained. Therefore, we performed a final set of simulation experiments to determine the dependence of our results on variables central to the mechanism of SK activation that could importantly affect phase response dynamics in response to dendritic excitation. SK was activated by inward calcium pulses carried by the CaHVA conductance. The transient spike in intracellular calcium elicited by an AMPA input to the dendrite lasted less than 10 ms in our simulations (Supp. Fig. 10), approximating the dynamics of the interaction of calcium channels with SK channels in membrane micro-domains. However, SK deactivated relatively slowly as a consequence of the default SK deactivation time constant of 76 ms in our simulations. This slow deactivation of SK in the model prolonged SK current transients that shaped the spontaneous mAHP to match experimental data (Gunay et al., 2008). Direct measurements of SK deactivation in oocyte expression systems indicate time constants between 22 and 38 ms (Xia et al., 1998), consistent with a relatively slow calcium dependent deactivation.

To determine the effects of key kinetic parameters affecting SK activation on PRC behavior we varied 1) the rate of intracellular calcium clearance between 0.25 and 2 ms, 2) the steady-state calcium dependence of SK activation between 0.175 and 1 μM, and 3) the time constant of calcium-dependent SK activation between 25 and 100 ms. (Supp. Fig. 11). Each of these manipulations affected the spontaneous spike frequency, so each model was again driven to 30 Hz by somatic applied current for PRC analysis. Permanent D2D PRCs were type II for all levels of each manipulation, although the amplitude and skewness were modulated by varying the activation and kinetic parameters for SK. Therefore, the dendritic mechanism that yields type II dynamics in our model was not a consequence of specific parameter choices and is likely to encompass the regime of dendritic processing of strong AMPA inputs in vivo.

Discussion

The primary result of our analysis of phase response curves of GP model neurons was that perisomatic and dendritic excitatory inputs yielded distinctly different PRCs. Perisomatic excitation at all times in the spike cycle led to spike advances and thus a type I PRC, whereas dendritic excitation led to spike delays for inputs early in the spike cycle thus yielding a biphasic type II PRC. We found that the type II character of dendritic PRCs was the consequence of stimulus-elicited calcium transients and subsequent SK activation that occurred even for stimuli simulating a single small distal synaptic input. Type II PRCs can enhance the ability of neurons to synchronize in connected networks (Hansel et al., 1995; Goel and Ermentrout, 2002; Galan et al., 2007b; Tsubo et al., 2007; Bogaard et al., 2009) and populations of neurons with type II PRCs synchronize more readily to correlated inputs (Galan et al., 2007a; Marella and Ermentrout, 2008; Abouzeid and Ermentrout, 2009). Because most excitatory input to GP neurons is on dendritic sites (Shink and Smith, 1995) our results suggests a novel cellular mechanism important in the integration of excitatory input in GP network activity in normal conditions and to the pathological synchronization of GP neurons in Parkinson's disease.

Phase Response Curve Analysis of a Morphological GP Neuron Model

We turned to computer modeling to examine detailed PRC behavior in GP neurons, as spike firing even in brain slices is not sufficiently regular or precise (Deister et al., 2009) to reliably quantify small spike shifts caused by individual synaptic inputs. In addition application of dendritic inputs of controlled amplitude and location is not currently experimentally feasible. These factors put severe limits on the experimental measurement of PRCs.

GP model construction

For this study we adapted a GP neuron model that we have previously used to investigate the influence of conductance densities on spiking properties (Gunay et al., 2008). We observed that most electrophysiological properties of the models were affected by multiple conductances, and each conductance affected multiple electrophysiological measures (Gunay et al., 2008), which is consistent with modeling studies of other types of neurons (Prinz et al., 2003b; Achard and De Schutter, 2006; Taylor et al., 2009). Adaptations to the model were primarily made to include new experimental findings on sodium channel properties in GP neurons (Mercer et al., 2007). The new base model was located in a broad basin of models with similar spiking properties when channel densities were varied, as described in our earlier publication (Gunay et al., 2008). Hence, it is important to note that the model used here does not represent a unique solution to GP spiking properties, but that all findings are robust against considerable parameter variations.

Dendritic properties

The architecture of dendritic trees and the complement of active dendritic conductances are critical features of the computational machinery employed by neurons with diverse functional objectives (Mainen and Sejnowski, 1996; Vetter et al., 2001; Krichmar et al., 2002; Stiefel and Sejnowski, 2007; Chen et al., 2009; Komendantov and Ascoli, 2009). We used full morphological reconstructions from biocytin filled neurons to obtain a correct representation of the thin, long and sparsely branched structure of GP dendrites (Falls et al., 1983). No dendritic recordings of GP dendrites exist, leading to considerable uncertainty about dendritic current densities in GP. Channel antibody staining has revealed the presence of voltage-gated sodium channels (Hanson et al., 2004) and calcium channels (Hanson and Smith, 2002) in GP dendrites, however. Somatic conductances in GP have been characterized in multiple studies (Baranauskas et al., 1999; Tkatch et al., 2000; Baranauskas et al., 2003; Chan et al., 2004; Mercer et al., 2007) and were matched to these results in our model (Gunay et al., 2008). The SK conductance has been found to govern the mAHP waveform in GP neurons (Deister et al., 2009) and in other cell types has been shown to be activated by calcium inflow through L-type (Marrion and Tavalin, 1998) or N-type (Hallworth et al., 2003; Goldberg and Wilson, 2005) HVA calcium channels. Immunocytochemistry showed an increased density of L- and N- type HVA calcium channels in GP dendrites (Hanson and Smith, 2002), which would support a high level of dendritic SK activation. Dendritic SK has not been examined in GP, but has been found to be present in other types of neurons with SK conductance (Sailer et al., 2002; Cai et al., 2004; Maher and Westbrook, 2005).

Dependence of PRC shape on physiological properties

Traditional PRC analysis relies on linear scaling of PRCs for positive and negative stimuli of varying amplitudes. We found that excitatory dendritic inputs in particular may not have a regime of weak coupling in GP, because the local input resistance is so high that any stimulus of physiological amplitude results in local depolarization that triggers changes in active conductances and hence response non-linearities. Dendritic depolarization following AMPA inputs leads to a transformation of a type I infinitesimal PRC for very small (<1 pA) stimuli to a type II PRC for physiologically sized stimuli due to SK activation. The property of symmetry for positive and negative stimuli was also lost, as GABA inputs were associated with type I PRCs at all amplitudes. These results were robust for a large range of CaHVA and SK conductance densities and different parameters for SK activation kinetics, though the precise shape of biphasic PRCs smoothly changed when parameters were varied. Another important observation was that when spike frequency was driven to frequencies typically observed in vivo, the negative region of biphasic PRCs was pushed into the F2 spike interval, because the time course of SK activation with AMPA stimulation exceeded the inter-spike interval at fast firing frequencies.

Implications of PRC Behavior for GP Network Interactions

To make PRC analysis analytically tractable, it is most commonly implemented in silico using brief, weak perturbations applied to simple point neuron models that lack a full contingent of membrane conductances. In such cases explicit predictions of phase-locked modes can be made for networks with well defined connectivity architectures, and type II PRCs have been associated with synchronization in networks coupled through excitation (Hansel et al., 1995; Goel and Ermentrout, 2002; Galan et al., 2007b; Tsubo et al., 2007). Further work recognized the importance of active conductances in shaping PRC behavior. Ermentrout et al (2001) and Gutkin et al (2005) evaluated the contribution of a spike frequency adaptation (SFA) potassium current to skewness of PRCs and the consequent emergence of network synchronization. Recently, Stiefel and colleagues (2008, 2009) demonstrated that in principle cholinergic modulation of cortical pyramidal neurons could switch the PRC from type II to type I as a consequence of SFA current (IM) down regulation.

Dendrites can change PRC shape through an attenuation and delay of an input's effect on somatic spiking and hence a left-shift of the dendritic PRC in the domain of weak coupling (Goldberg et al., 2007). It has also been shown that the electrotonic length of dendritic cables can determine the mode of synchronization in a reciprocal network of two distally coupled ball-and-stick neuron models (Crook et al., 1998a). These studies demonstrate that active dendritic conductances can sculpt effective input waveforms and compensate for the passive filtering properties of dendrites. Our own results indicate that active dendritic conductances can have far reaching consequences for dendritic PRCs, and that SK conductance in particular can lead to dendritic type II PRC behavior in neurons showing somatic type I PRCs. Thus, dendritic conductances can bestow phase response characteristics that are not predicted by somatic excitability, causing the dendrite to act as a distinct dynamical subsystem. This mechanism also gives excitatory input a special relevance for network synchronization, as type II phase response properties were not observed with inhibition.

Given that phenomena of network synchronization are highly sensitive to PRC shapes, our finding of type II dendritic PRCs in GP likely has important consequences for considering network activity in globus pallidus. GP neurons shift from an asynchronous firing mode to synchronous bursting in Parkinson's disease (PD) that is phase-locked to oscillations in other basal ganglia nuclei (Bergman et al., 1994; Nini et al., 1995; Raz et al., 2000). An important mechanism underlying synchronization in basal ganglia networks is believed to be given by STN-GP feedback activity (Plenz and Kitai, 1999; Bevan et al., 2002; Terman et al., 2002). For cortical slow wave oscillations elicited under anesthesia, STN fires bursts in synchrony with cortical oscillations in normal or dopamine depleted animals, whereas GP only shows robust synchronized bursting following dopamine depletion (Magill et al., 2001). This result suggests that modulation in the Parkinsonian state may induce a change in the propensity of GP neurons to engage in network synchronization. Synchronization of pallidal networks would in turn increase the effectiveness of GP inhibition to cause correlated pauses and rebound bursting in STN (Terman et al., 2002), and entrainment of GP neurons to β-frequency STN oscillations would promote resonance between these coupled nuclei. Previous work suggested that HCN channels play a major role in GP neurons in controlling firing regularity and bursting in PD (Chan et al., 2004). Our PRC analysis suggests that in addition dendritic SK may be a key determinant in controlling synchronization through changing the phase dependence of synaptic effects on spike timing. Dopamine modulates SK conductance through CaV2.2 down-regulation in STN neurons (Ramanathan et al., 2008). If this mechanism holds for GP neurons as well, a CaV2.2 upregulation and hence increased SK activation and an increase in type II phase response dynamics might be expected in PD based on our results.

Supplementary Material

Supp1

Acknowledgements

This project was supported by NINDS grant R01-NS039852. We thank Astrid Prinz for numerous productive discussions of PRC analysis.

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