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Computational Estimation of the Distribution of L-type Ca²⁺ Channels in Motoneurons Based on Variable Threshold of Activation of Persistent Inward Currents

Canadian Institute for Health Research Group in Sensory-Motor Systems, Department of Physiology, Centre for Neuroscience Studies, Queen’s University, Kingston, Canada

Submitted 22 June 2005; accepted in final form 17 September 2005

	ABSTRACT

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In the presence of neuromodulators such as serotonin and noradrenaline, motoneurons exhibit persistent inward currents (PICs) that serve to amplify synaptic inputs. A major component of these PICs is mediated by L-type Ca²⁺ channels. Estimates based on electrophysiological studies indicate that these channels are located on the dendrites, but immunohistochemical studies of their precise distribution have yielded different results. Our goal was to determine the distribution of these channels using computational methods. A theoretical analysis of the activation of PICs by a somatic current injection in the absence or presence of synaptic activity suggests that L-type Ca²⁺ channels may be segregated to discrete hot spots 25–200 µm long and centered 100–400 µm from the soma in the dendritic tree. Compartmental models based on detailed anatomical measurements of the structure of feline neck motoneurons with L-type Ca²⁺ channels incorporated in these regions produced plateau potentials resulting from PIC activation. Furthermore, we replicated the experimental observation that the somatic threshold at which PICs were activated was depolarized by tonic activation of inhibitory synapses and hyperpolarized by tonic activation of excitatory synapses. Models with L-type Ca²⁺ channels distributed uniformly were unable to replicate the change in somatic threshold of PIC activation. Therefore we conclude that the set of L-type Ca²⁺ channels mediating plateau potentials is restricted to discrete regions in the dendritic tree. Furthermore, this distribution leads to the compartmentalization of the dendritic tree of motoneurons into subunits whose sequential activation lead to the graded amplification of synaptic inputs.

	INTRODUCTION

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Motoneurons are the direct intermediary between the CNS and the skeletomuscular system. As a consequence, the firing behavior of motoneurons is a critical determinant in the control of motor actions by the nervous system. The interplay between two sets of factors regulates the firing behavior of motoneurons: the nature (e.g., inhibitory vs. excitatory) and activity of their synaptic inputs and the intrinsic properties of motoneurons. The latter play a major role in shaping the response to synaptic inputs and include passive membrane properties (Barrett and Crill 1974; Clements and Redman 1989; Fleshman et al. 1988; Perreault 2002; Rose and Vanner 1988; Segev et al. 1990; Ulrich et al. 1994), dendritic geometry (Bras et al. 1987; Bui et al. 2003; McDonagh et al. 2002; Rall 1959) and the type and location of voltage-gated channels (Berg et al. 2004; Campbell and Rose 1997; Carlin et al. 2000a; Lee and Heckman 2001; Lee et al. 2003; Li and Bennett 2003; Muennich and Fyffe 2004; Perrier and Tresch 2005; Powers and Binder 2003) and ligand-gated channels (Alvarez et al. 1997, 1999; Hornby et al. 2002; Perrier and Hounsgaard 2003).

In the presence of monoamines such as serotonin and noradrenaline, motoneurons exhibit persistent inward currents (PICs) that can lead to sustained depolarization (Hornby et al. 2002; Hounsgaard et al. 1984, 1988; Lee and Heckman 1996, 1998, 2000; Schwindt and Crill 1980; Svirskis and Hounsgaard 1998). These PICs are composed of a calcium current mediated by low-threshold slowly deactivating L-type Ca²⁺ channels (Carlin et al. 2000b; Lipscombe et al. 2004; Perrier and Hounsgaard 2003) and a sodium current mediated by persistent sodium channels (Lee and Heckman 2001; Li and Bennett 2003; Li et al. 2004). The activation of PICs by excitatory synaptic inputs and deactivation by inhibitory synaptic inputs represent a powerful means by which synaptic inputs can be amplified (Hultborn et al. 2003; Kuo et al. 2003; Lee and Heckman 2000; Lee et al. 2003).

To fully understand how PICs modulate the response of motoneurons to synaptic inputs requires an understanding of the spatial relationships between the channels responsible for PICs and the synaptic inputs to motoneurons. Delgado-Lezama et al. (1999) showed that by spatially restricting serotonergic activity, the synaptic amplification by PICs could be confined to inputs innervating specific regions of the dendritic tree. This clearly showed that amplification of synaptic inputs by PICs could be compartmentalized in the dendritic trees of motoneurons. The extent to which motoneurons can be compartmentalized is dependent in part on the location of the L-type Ca²⁺ channels. There is a wealth of electrophysiological data suggesting the presence of L-type calcium channels on dendrites of motoneurons (Carlin et al. 2000a; Hounsgaard and Kiehn 1993; Svirskis et al. 2001). However, immunohistochemical studies offer little consensus as to the precise location of these channels. Immunoreactivity for Ca_V1.3 channels, the member of the L-type Ca²⁺ channel family responsible for PICs in motoneurons (Heckman et al. 2003), has been observed at: the soma and proximal dendrites (Westenbroek et al. 1998), second- and third-order dendrites (Carlin et al. 2000a), large numbers of punctate patches over the entire somato-dendritic surface (Simon et al. 2003), and a small number of punctate beads at distances of 1,300 µm from the soma (Ballou et al. 2003). Recent calcium imaging studies reinforce the evidence for dendritic calcium channels but do not resolve the question of their precise location (Davenport et al. 2003).

Considering the variability of the reported distribution of the L-type Ca²⁺ channels produced by immunohistochemical analysis, our goal was to estimate the distribution of L-type Ca²⁺ channels on the dendrites of motoneurons using a strategy different from immunohistochemical analysis. To estimate the location of these channels, we used a computational analysis based on experimental observations of the varying somatic thresholds for PIC activation in the absence or presence of tonic excitatory or inhibitory synaptic activity (Bennett et al. 1998). We estimated L-type Ca²⁺ channels were located in discrete hot spots that were centered on the sites that shared a common membrane potential. Simulations using models with L-type Ca²⁺ channels placed at those estimated locations replicated the experimental observations of Bennett et al. (1998). Conversely, simulations using models with channels distributed uniformly throughout the dendritic tree were unable to replicate the same experimental observations. The distribution of L-type Ca²⁺ channels in hot spots led to the compartmentalization of the dendritic tree into computational units that could be sequentially recruited with increasing synaptic activity. Portions of this work have been presented in abstract form (Rose et al. 2002, 2003).

	METHODS

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Compartmental modeling

The construction of the compartmental models of feline neck motoneurons based on anatomical measurements has been described in detail in Bui et al. (2003). The value of the specific resistivity of the cytoplasm (R_i) selected for this study, 70 · cm, is based on calculations for motoneurons (Barrett and Crill 1974) and is close to the value for saline (Hille 2001). The value of the specific resistivity of the membrane (R_m) was 15,000 · cm².

Simulations were performed using Saber, a mixed-signal simulator software package (Synopsys, Mountain View, CA) (Carnevale et al. 1990). To calculate the total synaptic current that reaches the soma, the membrane potential of the cell body was clamped to –64 mV to simulate resting conditions and to –55 mV to simulate suprathreshold conditions. This is analogous to the experimental technique developed by Heckman and Binder (1988) and subsequently employed by Binder and colleagues (e.g., Lee and Heckman 1996, 1998, 2000; Powers and Binder 1995, 2000) to calculate the effective synaptic current generated by tonic activation of synaptic inputs to motoneurons.

Modeling synaptic conductance changes

Current injected by synaptic activation (i_syn) is a product of the conductance generated by channel opening and the driving potential, such that

(1)

where g is the conductance change, E_rev is the reversal potential, and V_m is the membrane potential. Under steady-state conditions, the synaptic conductance change can be time-averaged, , as described by Bernander et al. (1991). In this case, the synaptic conductance in each compartment can be calculated by the following equation

(2)

where g_peak is the peak conductance change, t_peak is the time-to-peak of the conductance change, f is the frequency of synaptic activity, and P is the probability of neurotransmitter release. The parameters for excitatory synapses on motoneurons were based on the experimental findings of Finkel and Redman (1983); E_rev, t_peak, and g_peak were assigned values of 0 mV, 0.2 ms, and 5.0 nS, respectively. The characteristics of inhibitory synapses were based on the experimental findings of Stuart and Redman (1990); E_rev, t_peak, and g_peak were assigned values of –81 mV, 0.65 ms, and 9.0 nS, respectively.

The distribution of excitatory synaptic inputs to neck motoneurons has been estimated to be approximately one synapse per 14.3 µm² of surface membrane area throughout the dendritic tree, corresponding to a uniform density of seven synapses per 100 µm² (Rose and Neuber-Hess 1991). Assuming that the ratio of excitatory to inhibitory synapses is 1:1, we distributed excitatory and inhibitory synaptic inputs at a density of 3.5 synapses per 100 µm², respectively. This represents 15,322, 15,966, and 13,756 total synapses for model motoneurons LAD5-4, LVN2-1, and LVN4-1, respectively.

Modeling L-type Ca²⁺ channels

The conductance change of L-type Ca²⁺ channels g_L,Ca was modeled as

Here g_L,Ca represents the maximal conductance per surface area. The variable m is a voltage- and time-dependent activation variable. In the case of the L-type calcium channel, there is very little voltage-dependent inactivation (Lipscombe et al. 2004). The activation variable is modeled by the differential equation

(4)

where the time constant of activation, _m, was assigned a value of 20 ms (Carlin et al. 2000a). The steady-state activation level, m, is given by

(5)

where the half-activation voltage, V_1/2, was –33 mV (in proximity to values used by Booth et al. 1997; Carlin et al. 2000a; and Svirkis et al. 2001) and the activation sensitivity, k, was assigned a value of –6 mV (Carlin et al. 2000a). Based on these values, m reached 0.1 at a membrane potential of –60.6 mV. E_rev for the calcium current was set at 60 mV (Carlin et al. 2000a).

	RESULTS

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Bennett et al. (1998) used the feline decerebrate preparation to determine the voltage threshold at which a somatic current injection activated PICs in motoneurons during tonic activation of excitatory synapses by stretch of the Achilles tendon (excitatory state), during tonic activation of inhibitory synapses through activation of the common peroneal nerves (inhibitory state), and in the absence of induced synaptic activity (resting state). In the excitatory state, the threshold measured at the soma was hyperpolarized (average: –56.4 mV) in comparison with the resting state (average: –50.6 mV). In the inhibitory state, the threshold measured at the soma was depolarized (average: –43.0 mV) in comparison with the resting state. QX-314 was used to block all sodium currents (Lee and Heckman 1999), and thus the current underlying the observed PICs was mediated by L-type Ca²⁺ channels. In the absence of sodium currents, the activation of these L-type Ca²⁺ channels produces an all-or-none depolarization termed plateau potentials. Based on these measurements, we can estimate the location of the L-type Ca²⁺ channels underlying the plateau potentials that were activated by the somatic current injection. We will first describe the theory on which our estimative methods are based.

Determination of L-type Ca² channel location using measurements of membrane potential throughout the dendritic tree

THEORY. Figure 1A shows the membrane potential along one dendrite, from soma to terminal end, in a hypothetical resting state at various levels of current injected at the soma (black lines). Initially, the membrane potential throughout the dendrite is at resting membrane potential. The membrane potential at the soma is raised by a somatic current injection (the direction of membrane polarization is indicated by the black arrow) to the threshold for activating PICs in the resting state (thick black line). In the presence of the somatic current injection, the membrane potential along the dendrite decays with distance away from the soma because of the cable properties of passive neurons (Rall 1977). In a hypothetical excitatory state, the membrane potential is raised throughout the dendrite by excitatory synaptic activity (top thin gray line). If the excitatory synaptic activity is sufficient to activate PICs, as observed in some instances by Bennett et al. (1998), the soma is hyperpolarized by a current injection (the direction of membrane polarization is indicated by the gray open arrow) to inactivate PICs. Subsequently, increasingly depolarized current is injected to raise the membrane potential at the soma (the direction of membrane polarization is indicated by the black arrow) to a threshold for activating PICs (thick gray line). At this threshold, the soma membrane potential is more hyperpolarized than in the resting state, in accordance with the observations made by Bennett et al. (1998) (Fig. 1B). The membrane potential in the more distal regions of the dendrites is more depolarized in the excitatory state than in the resting state because of the activation of excitatory synapses. Conversely, in a hypothetical inhibitory state, the membrane potential at the soma is raised by a somatic current injection (the direction of membrane polarization is indicated by the black arrow) to a threshold for activating PICs (thick black dashed line) that is more depolarized than in the resting state, again in accordance with the observations made by Bennett et al. (1998) (Fig. 1C). As the current injection becomes increasingly depolarized, the membrane potential at more distal regions of the dendrite is more hyperpolarized than in the resting state because of the activation of the inhibitory synapses. The membrane potential profiles at the threshold for activating PICs in the three states (Fig. 1, A–C, thick lines), represent three different snapshots of the same event, the activation of PICs by a somatic current injection. The common element between these three snapshots is the presence of a region in the dendrites where the membrane potential has been raised sufficiently to reach a threshold (_PIC) to initiate the activation of persistent inward currents.

View larger version (24K): [in this window] [in a new window]   FIG. 1. Hypothetical profiles of membrane potentials along a dendrite just subthreshold to activation of persistent inward currents (PICs) by somatic current injection. A: resting state (background synaptic activity). Thick black solid line (Vthreshold, resting state) represents the membrane potential profile of the dendrite just before activation of PICs. Thin black solid lines represent the membrane potential profile of the dendrite at various levels of current injection before activation of PICs. Arrow indicates direction of increasing depolarizing current. B: excitatory state (elicited excitatory synaptic activity). Thick gray solid line (Vthreshold, excitatory state) represents the membrane potential profile of the dendrite just before activation of PICs. Thin gray solid lines represent the membrane potential profile of the dendrite at various levels of current injection before activation of PICs. Bottom thin gray solid line represents the membrane potential profile of the dendrite with an initial hyperpolarizing current injection. Gray arrow indicates direction of the initial hyperpolarizing current injection. Black arrow indicates direction of increasing depolarizing current. C: inhibitory state (elicited inhibitory synaptic activity). Thick black dashed line (Vthreshold, inhibitory state) represents the membrane potential profile of the dendrite just before activation of PICs. The thin black dashed lines represent the membrane potential profile of the dendrite at various levels of current injection before activation of PICs. Arrow indicates direction of increasing depolarizing current. D: superimposition of membrane potential profile in the resting state (solid black line) and excitatory state (solid gray line). Vint is the membrane potential at the intersection point between the 2 profiles. Thin gray long-dashed lines represent possible values of PIC, the threshold of PIC activation. Thick black double-arrowhead line represents possible locations of the channels underlying PICs. E: superimposition of membrane potential profile in the resting state (solid black line) and inhibitory state (black short dashed line). Thin gray long-dashed line represents possible values of PIC, the threshold of PIC activation. Thick black double-arrowhead line to the right of the intersection of the membrane potential profiles in the resting state and inhibitory state represents possible locations of the channels underlying PICs. F: superimposition of membrane potential profile in the resting state, excitatory state, and inhibitory state. Gray long-dashed line represents possible values of PIC, the threshold of PIC activation. Black circle represents the possible location of the channels underlying PICs.

The superposition of the membrane potential profiles for the resting state and for the inhibitory state (Fig. 1E) yields complementary, but slightly different, interpretations. Assuming that _PIC is constant throughout the dendrite, three possibilities are shown: _PIC is higher than the membrane potential at the intersection point between the two profiles (V_int), _PIC is equal to V_int, and _PIC is lower than V_int. The cases where _PIC is greater than V_int or where _PIC is lower than V_int cannot exist. In these respective cases, the channels proximal to the left curves should have been activated in both states by lower current injections, the channels between the two curves should have been activated by lower current injections in the state represented by the right curve, and the channels more distal than the right curve should not have been activated in the state represented by the left curve. In the case where _PIC is equal to V_int, the intersection point of the resting state and the inhibitory state provides an indication of the most proximal point of the region where the channels could be located (marked by thick black double-arrowhead line). Channels that are located more proximally than this point should have been activated by a smaller current injection in both states. If the membrane potential profile of all three states intersect at the same point (Fig. 1F), the only region where channels could be located is the intersection point.

IMPLEMENTATION OF THE THEORY. For three compartmental models of feline neck motoneurons, we simulated three states of synaptic activity. In the feline decerebrate preparation used by Bennett et al. (1998), background synaptic activity is present throughout the experimental procedure. Thus in the resting state, we simulated background synaptic activity corresponding to the tonic activation of 8% of all excitatory synapses (firing at 100 Hz with a P of 0.50) and 10% of all inhibitory synapses (firing at 50 Hz with a P of 0.50). At –64 mV, the net current reaching the cell body produced by this synaptic activity in the absence of voltage-gated channels was negligible for all three motoneurons. In the excitatory state, we simulated synaptic activity corresponding to the tonic activation of 24% of all excitatory synapses (firing at 100 Hz with a P of 0.50) and 10% of all inhibitory synapses (firing at 50 Hz with a P of 0.50). The current reaching the cell body produced by this synaptic activity in the absence of voltage-gated channels was 9.4 (LAD5-4), 7.9 (LVN2-1), and 5.8 nA (LVN4-1), which is slightly higher than the amount of current produced by the coactivation of Ia afferents from lateral gastrocnemius, medial gastrocnemius and soleus nerves (Lee and Heckman 2000). In the inhibitory state, we simulated synaptic activity corresponding to the tonic activation of 4% of all excitatory synapses (firing at 100 Hz with a P of 0.50) and 20% of all inhibitory synapses (firing at 50 Hz with a P of 0.50). The net current reaching the cell body produced by this synaptic activity in the absence of voltage-gated channels was –6.7 (LAD5-4), –5.5 (LVN2-1), and –4.1 nA (LVN4-1), similar to the observed currents produced by activation of the common peroneal nerve (Powers and Binder 2000).

For every dendrite, we calculated the intersection point of the membrane potential profiles in the excitatory state and the resting state (Fig. 2). For the three motoneurons, the intersection points of the membrane potential profiles in the excitatory state and the resting state were distributed between 108 and 398 µm away from the soma. The median intersection points were, respectively, 253.6 (LAD5-4), 212.9 (LVN2-1), and 181.2 µm (LVN4-1) away from the soma (Fig. 2B, gray boxes). In comparison, the median lengths of the dendrites, from soma to terminal, were 1,085.7 (LAD5-4), 1,147.9 (LVN2-1), and 1,094.5 µm (LVN4-1), respectively (Fig. 2B, white boxes). In terms of branching order, the intersection points were located on dendritic branches whose branching orders were between 1 and 6, typically on second- to fourth-order dendrites (Fig. 2C, gray boxes). In comparison, the highest branching order of the terminal branches of motoneurons was 11 (Fig. 2C, white boxes). The intersection points were also calculated with an R_m value of 5,000 · cm². The median intersection points were shifted distally 30 µm at most (data not shown).

View larger version (27K): [in this window] [in a new window]   FIG. 2. Determination of the intersection points of membrane potential profiles for a dendrite of LVN2-1. A: left: reconstruction of the dendritic tree of LVN2-1 with the examined dendrite highlighted. Right: membrane potential profile in the resting state (gray line) and excitatory state (black line) of highlighted dendrite shown in the reconstruction. B: box plot of the intersection points for all dendrites of 3 motoneurons between the resting state and excitatory state (gray boxes) and of the length of all dendrites (from soma to terminal, white boxes). C: box plot of the branching order of the dendritic branches on which the intersection point lies (gray boxes) and of the terminal branches (white boxes). For LAD5-4, the 10th and 25th percentile intersection points are of branching order 2. For LVN2-1, the 50th and 75th percentile intersection points are of branching order 3.

L-type Ca²⁺ channels were incorporated into the three motoneuron models as discrete regions (hot spots). The center of these hot spots was selected to be the intersection point of the membrane potential profiles in the excitatory and resting state for each dendrite. If the intersection points of the membrane potential profiles for two soma-to-terminal trajectories are located on the same branch segment (between two branch points), a single hotspot was inserted in that branch segment and centered at the most proximal of the intersection points. Hotspots were extended into daughter branches or into parent branches whenever these cases arose. The length of the hot spots was arbitrarily set at 100 µm long. Triangular current ramps were injected at the soma at a speed of 5 nA/s. The threshold of PIC activation was measured as the membrane potential at which the derivative of the somatic membrane potential with respect to time was 0.1 V/s on the rising slope of the highest peak (Fig. 3). While this value may be arbitrary, visual inspection of the responses to the triangular current ramps verified that it corresponded approximately to the membrane potential at which a sharp inflection led to a plateau potential.

View larger version (30K): [in this window] [in a new window]   FIG. 3. Response to triangular current ramps (5 nA/s) of a motoneuron (LVN2-1) in the inhibitory state, resting state, and excitatory state with L-type Ca2+ channels distributed in 100-µm hot spots centered at the intersection points of the membrane potential profile of the resting state and excitatory state. A: reconstruction of the dendritic tree of LVN2-1 with Ca2+ channels distributed in 100-µm-long hot spots (red). Thicknesses of dendrites where the hot spots are located have been increased for illustrative purposes. B: somatic membrane potential. Somatic threshold of PIC activation was defined as the membrane potential at which the derivative of the somatic membrane potential with respect to time was equal to 0.1 V/s on the rising slope of the highest peak. Arrows mark the measured somatic threshold of PIC activation. C: derivative of the somatic membrane potential with respect to time. Dotted line marks a value of the derivative of 0.1 V/s.

We assessed the sensitivity of the LVN4-1 model to different parameters such as the length of the L-type Ca²⁺ hot spots, their location, and their distribution in discrete regions. For each change in the value of a particular parameter, we adjusted the conductance density to produce a plateau in the resting state with approximately the same (±1 mV) somatic threshold as calculated by Bennett et al. (1998) and then calculated the somatic threshold in the excitatory and in the inhibitory state (Table 2).

View larger version (14K): [in this window] [in a new window]   FIG. 4. Comparison of the shift in somatic threshold of PIC activation from the resting state to excitatory state and from the resting state to inhibitory state for motoneuron LVN4-1. A: with various locations of 100-µm hot spots. A plateau potential could not be elicited in the inhibitory state with 100-µm-long hot spots centered 200 µm away from intersection points. B: with various sizes of hot spots centered at intersection points of the membrane potential profiles of the resting state and excitatory state. C: with a decreased level of excitatory activity in the excitatory state. Intersection points were recalculated before adding L-type Ca2+ channels.

We also constructed a model of LVN4-1 where the L-type Ca²⁺ channels were distributed uniformly across the soma and the dendritic tree at a constant conductance density. The conductance density was adjusted to produce a plateau potential with a threshold of –50.0 mV in the resting state (Fig. 5A). The somatic threshold in the inhibitory state and in the excitatory state was –30.9 and –83.9 mV, respectively. However, inspection of the somatic membrane potential in response to the triangular current ramp in the excitatory state revealed no discernible plateau potential. Indeed, the presence of many small spikes in the derivative of the somatic membrane potential with respect to time (Fig. 5B) indicates the presence of numerous partial plateaus resulting from the activation of separate groups of L-type Ca²⁺ channels at membrane potentials as low as –150 mV. Similarly, a model of LVN4-1 where the L-type Ca²⁺ channels were distributed from the intersection point of the resting state and the inhibitory state to the terminal end of the dendrites could not produce plateaus in the inhibitory state because of the presence of many partial plateaus. In the excitatory state, the somatic threshold in the excitatory state was –107.5 mV.

View larger version (11K): [in this window] [in a new window]   FIG. 5. Response of a motoneuron (LVN4-1) to triangular current ramps in the inhibitory state, resting state, and excitatory state with L-type Ca2+ channels distributed uniformly throughout the soma and the dendrite at a constant density (conductance/surface area). A: somatic membrane potential. Somatic threshold of PIC activation was the membrane potential at which the derivative of the somatic membrane potential with respect to time was equal to 0.1 V/s on the rising slope of the highest peak. Arrows mark the measured somatic threshold of PIC activation. B: derivative of the somatic membrane potential with respect to time.

VARIATION IN INTENSITY OF EXCITATORY SYNAPTIC ACTIVITY. To assess the sensitivity of the somatic threshold of PIC activation to the particular level of excitatory synaptic activity that we selected to recreate an excitatory state, we reduced the level of excitatory synaptic activity in LVN4-1 by activating 16% rather than 24% of all available excitatory synapses. This decrease in excitatory activity in the excitatory state decreased the net current arriving at the cell body in the absence of any voltage-gated channels from 5.8 to 3.1 nA. The membrane potential along every dendrite during the resting state and during the excitatory state was measured along with the intersection points of these two membrane potential profiles. The decrease in excitatory synaptic activity shifted these intersection points 70 µm distally (median: 251 µm). The L-type channel hot spots were centered at the new intersection points, and the somatic thresholds for PIC activation were reassessed (Fig. 4C, square). In the excitatory state, the shift in somatic threshold for PIC activation was increased from –3.7 to –6.8 mV, closer to the experimental value measured by Bennett et al. (1998). In the inhibitory state, the shift in somatic threshold for PIC activation was increased from 9.6 to 10.9 mV, further from the experimental value measured by Bennett et al. (1998).

VARIATION IN CHANNEL PROPERTIES. We tested the sensitivity of the models to changes in the half-activation voltage (V_1/2) of the L-type Ca²⁺ channels. V_1/2 was increased or decreased by 5 mV, and we assessed the shifts in somatic thresholds in models with the different hotspot sizes and locations listed above (data not shown). When V_1/2 was set to –28 mV, models with hot spots from 25 to 200 µm were able to replicate the shifts in somatic thresholds reported by Bennett et al. (1998). Similar to when V_1/2 was set to –33mV, the model with 50-µm-long hot spots could best replicate the shifts in somatic thresholds. The model with hot spots based at the intersection points of the membrane potential profile of the resting state and the excitatory state could best replicate the shifts in somatic thresholds compared with models with hot spots located elsewhere.

When V_1/2 was set to –38 mV, only a few of the models could produce a clear all-or-none plateau potential (50- and 150-µm-long hot spots centered at the intersection points of the membrane potential profile of the resting state and the excitatory state and 100-µm-long hot spots centered 100 µm proximal to the intersection points) in the inhibitory state. The other models exhibited the presence of many partial plateaus.

Moreover, for both values of V_1/2, models with uniformly distributed L-type Ca²⁺ channels were unable to exhibit shifts in somatic thresholds from the resting state to the excitatory state that were within 100 mV of the value reported by Bennett et al. (1998). Therefore while the exact size and location of the L-type Ca²⁺ hot spots seems to depend on the activation properties of the channels, our findings still support the conclusion that the channels are distributed in spatially restricted regions.

Input–output properties of motoneurons in the presence of L-type Ca²⁺ channel hot spots

The firing frequency of motoneurons is predicted by the current arriving at the cell body (I_N) generated by synaptic activity (Powers and Binder 2000). To quantify the effect of distributing L-type Ca²⁺ channels in 100-µm hot spots centered at the intersection points of the membrane potential profiles in the resting state and the excitatory state, we measured the relationship between synaptic activity and current arriving at the cell body with and without these channels. Excitatory synapses were tonically activated (firing at 100 Hz with a P of 0.50), and the current arriving at the cell body was measured using a somatic voltage clamp of –64 mV (Fig. 6A). With passive dendrites, the relationship between synaptic activity and current arriving at the cell body in motoneurons is sublinear because of the loss of synaptic current by saturation of excitatory driving potential (Rose and Cushing 1999). In the presence of L-type Ca²⁺ channel hot spots, the relationship between synaptic activity and current arriving at the cell body was amplified. For two of the cells, LAD5-4 and LVN4-1, the relationship between synaptic activity and current arriving at the cell body consisted of two nearly linear phases. The initial phase was much steeper than that seen in the models without L-type Ca²⁺ channels. This phase was a direct consequence of a sequential recruitment of hot spots (Fig. 6B). The slope of the second phase was less than that seen for the first phase. Over this range of excitatory synaptic activity, few if any additional hot spots were activated. The input–output properties of LVN2-1 were more complex. Most of the hot spots were activated over a narrow range of excitatory synaptic activity. As a consequence, the current reaching the soma increased abruptly as the percentage of active synapses increased from 16 to 32%.

View larger version (21K): [in this window] [in a new window]   FIG. 6. A: current reaching the soma after tonic activation of excitatory synapses in the presence and absence of L-type Ca2+ channels distributed in 100-µm hot spots centered at intersection points. Membrane potential at the soma was clamped at –64 mV. B: number of active L-type Ca2+ channel hot spots in response to tonic activation of excitatory synapses. Maximum number of hot spots is marked by the dotted line.

During the repetitive firing of action potentials, the membrane potential at the soma enters a quasi voltage-clamp state set by the time-averaged membrane potential (Koch et al. 1995). Therefore we reassessed the current arriving at the cell body with a somatic voltage clamp of –55 mV to mimic these conditions (Fig. 7). Under these conditions, all three cells displayed the same input–output properties. There was an initial steep phase corresponding to the activation of L-type Ca²⁺ channels and a flatter phase representing the saturation of the activation of L-type Ca²⁺ channels. Compared with the models clamped at –64 mV, the more depolarized somatic voltage clamp reduced the level of synaptic activity required to activate the L-type Ca²⁺ channel hot spots, and activation occurred over a narrower range of synaptic activity.

View larger version (11K): [in this window] [in a new window]   FIG. 7. Current reaching the soma after tonic activation of excitatory synapses in the presence and absence of L-type Ca2+ channels distributed in 100-µm hot spots centered at intersection points. Membrane potential at the soma was clamped at –55 mV.

We repeated the simulations after adding tonically active inhibitory synaptic activity. Five or 10% of all inhibitory synapses were activated at 50 Hz with a probability of release of 0.5. The measured current reaching the cell body was smaller in the presence of inhibition (Fig. 8A). The input–output properties were approximately linear for all three cells at both levels of inhibitory synaptic activity. At intermediate levels of excitatory synaptic activity, the synaptic inhibition reduced the number of L-type Ca²⁺ channel hot spots activated, thereby reducing the magnitude of the PICs. This reduction in PICs serves as a means to amplify the effective inhibition produced by inhibitory synaptic activity as observed experimentally by Hultborn et al. (2003) and Kuo et al. (2003).

View larger version (25K): [in this window] [in a new window]   FIG. 8. A: current reaching the cell body after tonic activation of excitatory synapses and tonic activation of 5 or 10% of all inhibitory synapses. L-type Ca2+ channels were distributed in 100-µm hot spots centered at intersection point. Membrane potential at the soma was clamped at –64 mV. B: number of active L-type Ca2+ channel hot spots in response to tonic activation of excitatory synapses.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

Methodological considerations

INTERSECTION POINTS OF MEMBRANE POTENTIAL PROFILES. The distribution of L-type Ca²⁺ channels in hot spots was based on the changes in membrane potential along the dendrites of motoneurons when PICs are activated in the presence of different levels of synaptic activity (Bennett et al. 1998). The superimposition of the membrane potential profiles in different states was used to estimate the location of L-type Ca²⁺ channels that produced PICs in the particular experimental preparation of Bennett et al. (1998). In this analysis, we assumed that the cells were essentially passive before PICs were activated. This is likely an oversimplification given the presence of several types of Ca²⁺ channels and delayed rectifier K⁺ channels on motoneuron dendrites (Ballou et al. 2003; Carlin et al. 2000b; Muennich and Fyffe 2004; Powers and Binder 2003; Simon et al. 2003; Westenbroek et al. 1998) and the absence of studies of the distribution of other types of channels. As well, we assumed that the resting membrane potential was uniform throughout the dendrites. While deviations from any of these assumptions will change the shape of the membrane potential profiles, as long as the overall shape of the membrane potential along a dendrite in the resting state is essentially monotonically decreasing with increasing distance away from the cell body, and the membrane potential along a dendrite in the excitatory state is essentially monotonically increasing with increasing distance away from the cell body, the analysis of the superimposition of the membrane potential profiles will still restrict the region in which the channels underlying the PICs observed by Bennett et al. (1998) were located.

The membrane potential profiles, and subsequently, the exact location of the intersection point of the membrane potential profiles, depended on the level of excitatory and inhibitory synaptic activity that we simulated to represent the three different states of synaptic activity. Therefore we tested the sensitivity of the models to variations in the size and location of the hot spots. Within a certain range, the models showed shifts in somatic threshold for PIC activation that were within 5 mV of the average values calculated by Bennett et al. (1998). The model with L-type Ca²⁺ channels distributed uniformly across the soma and the dendrites exhibited shifts in somatic thresholds much larger than the values reported by Bennett et al. (1998), despite having adjusted the densities of the channel to produce a plateau in the resting state. This supports the conclusion that, in the preparation studied by Bennett et al. (1998), the distribution of L-type Ca²⁺ channels underlying PICs must be restricted to discrete regions located a finite distance from the cell body.

CONSTANT THRESHOLD OF PIC ACTIVATION. In our theoretical analysis, the assumption that the local threshold of PIC activation, _PIC, was constant throughout the dendrite (Fig. 1, D–F, gray dashed straight lines) led to the conclusion that L-type Ca²⁺ channels underlying plateau potentials were distributed in discrete regions within the dendritic tree. In a physiological setting, _PIC could be variable depending on the local densities and properties of various channels as well as the local electrotonic environment. A variable _PIC (Fig. 1, D–F, leading to gray dashed curves rather than straight lines) could lead to the theoretical conclusion that the L-type Ca²⁺ channels are distributed in a uniform manner or in multiple hot spots within the same dendrite.

However, it is critical to note that _PIC is not necessarily constant in our models; only the channel properties are set to be constant. _PIC in our models may well be variable as a result of heterogeneities in the local electrotonic environment set by local differences in cell geometry. While we have not considered the possibility of different hot spots within the same dendrite in our models, our simulations show that a uniform distribution of L-type Ca²⁺ does not reproduce the changing threshold of PIC activation with synaptic activity.

LACK OF PERSISTENT SODIUM CURRENT AND FURTHER EFFECTS OF QX-314. Two currents are known to underlie PICs in motoneurons (Heckman et al. 2003): a calcium current mediated by L-type Ca²⁺ channels and a persistent sodium current whose source has not been resolved as of yet. We did not include this current in our models because Bennett et al. (1998) used the sodium channel blocker QX-314 in their assessment of the somatic threshold of PIC activation. The sodium current plays a major role in spike initiation (Lee and Heckman 2001; Li and Bennett 2003; Li et al. 2004) and contributes one-third of the magnitude of sustained PICs (Li and Bennett 2003). The inclusion of such a current could increase the amount of current delivered by excitatory synaptic activity. Lee and Heckman (1999) showed that the use of QX-314 depolarizes the threshold of plateau initiation in low conductance cells but hyperpolarizes the threshold for high conductance cells so that on average the threshold of plateau initiation across all motoneurons. Thus assuming that Bennett et al. (1998) sampled motoneurons of various input conductance, the shift in PIC activation threshold should not change in the presence or absence of QX-314 on average. Therefore the shift in somatic threshold of PIC activation observed by Bennett et al. (1998) on which this study is founded should not be significantly altered by the fact that QX-314 was used in their experimental procedures.

MAGNITUDE OF CA²⁺ CURRENT. The size of the plateau potential and the magnitude of the persistent inward currents are larger than previously reported values (Bennett et al. 1998; Lee et al. 2003). Our primary objective in this study was to replicate the shifts in somatic threshold of PIC activation observed by Bennett et al. (1998) and not the exact size characteristics of PICs and plateau potentials. These latter characteristics could be influenced by the presence of potassium currents and the number of activated synapses. Furthermore, the compound QX-314 that was used by Bennett et al. (1998) to prevent the firing of action potentials has been shown to reduce the magnitude of the PICs (Lee and Heckman 1999). This may have reduced the size of the plateau potential recorded by Bennett et al. (1998).

L-TYPE CA²⁺ CHANNEL PROPERTIES. Our models behaved similarly in terms of ability to replicate the shifts in somatic thresholds of PIC activation reported by Bennett et al. (1998) with L-type Ca²⁺ channels whose half-activation voltage was –28 or –35 mV. These values are within the –30 to –35 mV ranges used in other models of L-type Ca²⁺ channel models in motoneurons (Carlin et al. 2000a; Svirkis et al. 2001). However, with a half-activation voltage value of –38 mV (closer to the value of –40 mV used by Booth et al. 1997), our models could not produce an all-or-none plateau potential in the inhibitory state for many of the models tested with various size of hot spots and locations. To our knowledge, the channel properties of L-type Ca²⁺ channels in motoneurons have not been characterized and may well depend on serotonin that seems to facilitate the activation of plateau potentials (Perrier and Hounsgaard 2003). Our study suggests that if the activation properties of L-type Ca²⁺ channels are more hyperpolarized than previously thought, the presence of other voltage-gated channels may be essential to the generation of plateau potentials.

SUITABILITY OF MOTONEURON MODEL. Our study is based on three neck motoneurons. As discussed previously (Bui et al. 2003), these motoneurons are morphologically similar in many respects to hindlimb motoneurons studied by Bennett et al. (1998). Furthermore, the variability in the dendritic tree structure of these three motoneurons is representative of the variability seen within motoneurons innervating the same muscle (Cameron et al. 1983; Cullheim et al. 1987; Kernell and Zwaagstra 1989; Moritani et al. 2003; Rose 1982). The synaptic innervation that was modeled was constrained by known anatomical observations of the innervation of neck motoneurons (Rose and Neuber-Hess 1991). However, inhibitory inputs to motoneurons from Ia inhibitory interneurons and Renshaw cells are distributed proximally (Burke et al. 1971; Fyffe 1991). Stimulation of the common peroneal nerves, as used by Bennett et al. (1998) to define their inhibitory state, activates these inputs. Thus it may be more appropriate to model the inhibitory state using a proximal bias for inhibitory inputs. However, stimulation of common peroneal nerves also excites group II afferents (Levin and Chapman 1987), and the distribution of synapses activated by these afferents is not known. Thus the best means of replicating the inhibitory state remains unclear, but further studies of the effects of activating predominantly proximal inhibitory inputs on the shift in somatic threshold of PIC activation and the input–output properties of motoneurons have merit.

Anatomical versus functional distribution of L-type Ca²⁺ channels

It was first suggested that L-type Ca²⁺ channels on motoneurons were distributed in dendritic regions in light of electrophysiological evidence that persistent inward currents were more efficiently activated by synaptic activity rather than somatic current injections (Bennett et al. 1998; Carlin et al. 2000a; Hultborn et al. 2003; Lee et al. 2003). Several anatomical studies, based on the distribution of immunoreactivity for Ca_V1.3, the primary alpha subunit of the L-type Ca²⁺ channel responsible for the Ca²⁺ current underlying PICs, have confirmed the dendritic location of these channels (Ballou et al. 2003; Carlin et al. 2000b; Simon et al. 2003; Westenbroek et al. 1998). However, because of methodological differences (e.g., species, scope of immunohistochemical analysis), the sum of these studies does not provide a definite description as to the specific distribution of these channels.

Several studies have described motoneuron models with plateau-like behaviors. A two-compartment model of motoneurons with L-type Ca²⁺ channels in the dendritic compartment (Booth et al. 1997) was able to produce the bistable firing patterns and the hysteresis in the relation between firing frequency and current injected that has been observed in the presence of serotonin (Lee and Heckman 1998; Schwindt and Crill 1980). Such a model presumes that L-type Ca²⁺ channels are distributed throughout the dendritic tree of motoneurons. A morphologically more accurate model with uniformly distributed L-like and N-like Ca²⁺ conductances was also able to replicate the hysteretic I-V relation (Carlin et al. 2000a). Svirskis et al. (2001) were also able to reproduce the hysteresis in the current-voltage curves of motoneurons by using models of turtle motoneuron with L-type Ca²⁺ channels distributed uniformly throughout the dendritic tree or restricted to some dendritic branches only. Taylor and Enoka (2004) were able to produce bistable firing patterns in motoneuron models with L-type Ca²⁺ channels segregated to the proximal portions of the dendrites.

Thus immunohistochemical and modeling studies have essentially estimated a wide range of possible locations or distributions of L-type Ca²⁺ channels in spinal motoneurons. Variations in the species and/or age of the animals studied may well have contributed to the observed variability in distribution. However, all of the distributions described need not be mutually exclusive. Distributions of L-type Ca²⁺ channels estimated using immunohistochemical approaches describe the anatomical distribution of L-type Ca²⁺ channels. Estimations of the distribution of these channels using an approach based on the analysis of electrophysiological data estimate the distribution of the L-type Ca²⁺ channels that mediate plateau potentials through the activation of PICs. This distribution is what we would call their functional distribution. The set of channels that comprise the functional distribution of L-type Ca²⁺ channels may consist of only a subset of all L-type Ca²⁺ channels. In other words, we are proposing that the L-type Ca²⁺ channels that were available for activation in the system studied by Bennett et al. (1998) were located in the hot spots described in our study, even though there may be L-type Ca²⁺ channels in different locations as well. The channels that are present in the anatomical distribution but not in the functional distribution could be excluded from the latter because of the lack of neuromodulatory coactivation, a localized reduced membrane excitability or a density too low to engage in the self-catalytic process that results in a persistent inward current. It is entirely possible that, in another set of circumstances, motoneurons may possess another functional distribution of L-type Ca²⁺ channels that may or may not contain the channels distributed in the hot spots that we estimated, leading to different electrical behaviors. From a single anatomical distribution of channels, different functional distributions may be produced through several mechanisms.

Mechanisms responsible for different functional distributions of L-type Ca²⁺ channels in motoneurons

ORGANIZATION OF NEUROMODULATORY SYNAPSES. Synapses that release neuromodulatory neurotransmitters such as serotonin, noradrenaline, acetylcholine, GABA, or glutamate may be distributed in an organized fashion (Fyffe 2001; Heckman et al. 2003). In the turtle, serotonergic axons that contact motoneurons are located mainly in the dorsolateral funiculus (Kiehn et al. 1992) such that serotonin modulates the activation of channels found mostly on the lateral dendrites of motoneurons (Delgado-Lezama et al. 1999). In fact, stimulation of the dorsolateral funiculus facilitates the activation of PICs by depolarization applied to the lateral side of the dendritic tree of motoneurons but not by depolarization applied to the medial side (Delgado-Lezama et al. 1999). Conversely, activity in the medial funiculus facilitates the activation of PICs (supposedly through the neuromodulatory action of a neuromodulator different from serotonin) by depolarization applied to the medial side of the dendritic tree of motoneurons but not by depolarization applied to the lateral side. Thus two different functional distributions of L-type Ca²⁺ channels were created by the activation of two distinct sets of neuromodulatory inputs. In cat hindlimb motoneurons, serotonergic synapses are distributed uniformly throughout the dendritic tree (Alvarez et al. 1999). Nonetheless, various functional distributions could be produced by the selective activation of subsets of axons belonging to a single neuromodulatory system or different neuromodulatory systems that innervate different regions of the dendritic tree.

ORGANIZATION OF NEUROMODULATORY RECEPTORS. A second way to create different functional distributions of L-type Ca²⁺ channels in motoneurons is through an organized distribution of neuromodulatory receptors rather than neuromodulatory synapses. In this manner, a uniform set of neuromodulatory inputs may still facilitate only a subset of all available L-type Ca²⁺ channels. Serotonergic receptors have different receptor subtypes producing different actions depending on the activated biochemical pathway (Hochman et al. 2001; Perrier and Hounsgaard 2003; Perrier et al. 2003; Rekling et al. 2000). Localized application of serotonin to the cell body, proximal dendrites, or distal dendrites of isolated turtle motoneuron can either facilitate persistent inward currents or inhibit the firing of action potentials (Perrier and Hounsgaard 2003). Facilitation of PICs through the activation of 5-HT₂ in turtle motoneurons was observed after application of serotonin to all areas of the neurons. Inhibition of spike generation was observed for application of serotonin to the cell