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1Wallace H. Coulter Department of Biomedical Engineering, Georgia Institute of Technology; and 2Department of Biomedical Engineering, Emory University, Atlanta, Georgia
Submitted 25 January 2007; accepted in final form 2 April 2007
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ABSTRACT |
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INTRODUCTION |
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; Hultborn et al. 2003; Lee and Heckman 2000) and bistability (Bennett et al. 1998; Hounsgaard et al. 1984; Lee and Heckman 1996, 1998a; for review of motoneuron dendritic processing, see Heckman et al. 2003). Motoneurons are not unique in this respect because the dendrites of many types of neurons take an active part in shaping synaptic input (London and Hausser 2005; Segev and Rall 1998; Shepherd et al. 1985). Experimentation has yielded substantial insight into the dendritic ionic conductances behind these behaviors. For example, L-type Ca2+ channels are known to play a major role generating the plateau potentials underlying bistability in motoneurons (e.g., Carlin et al. 2000; Hounsgaard and Kiehn 1985) and Na+ and K+ channels are known to be present in the dendrites of hippocampal CA1 cells that exhibit action potential back-propagation (Hoffman et al. 1997; Jung et al. 1997). However, a complete experimental description of dendritic currents and their role in neuronal function does not yet exist. Instead, these experimental glimpses into the inner workings of dendritic processing raise larger questions: How many types of active conductances are in the dendrites? What are their distributions? And, perhaps more important, what computational requirements dictate how type and distribution are determined and regulated? These questions are difficult to address in any neuron but are particularly vexing in motoneuron research because of their extensive dendritic arborization (Cullheim et al. 1987a,b; Fleshman et al. 1988).
In this study, we focus on the mechanisms underlying synaptic amplification and bistability. It is generally assumed that synaptic amplification arises from the same Ca2+-based plateaus associated with bistability and recent modeling efforts have all included exclusively Ca2+-based inward currents in the dendrites (Elbasiouny et al. 2005, 2006). However, amplification was previously observed even in the absence of bistability (Lee and Heckman 2000) and some aspects of amplification have time courses that would appear to be too rapid for these channels (Jones and Lee 2006). Additionally, the all-or-none nature of plateaus would seem to conflict with a graded behavior such as amplification. This opens the door to the possibility that two distinct mechanisms are involved, with Ca2+-based plateaus being responsible for bistability and a short-timescale mechanism being responsible for amplification. A candidate for that second mechanism is electrotonic length compression arising from active dendritic currents (i.e., a mechanism that acts to make the dendrites more electrically compact; Cook and Johnston 1997, 1999). The concept of electrotonic length compression first proposed by Cook and Johnston was based on the 
-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA) and the T-type Ca2+ current for CA1 hippocampal neurons. Here, we propose that it is fast Na+ currents, particularly the persistent component, and delayed-rectifier K+ (KDR) currents that serve to cancel dendritic passive membrane capacitance and conductance. The advantage of this current combination is that they fulfill this role even at the fastest timescales, making this principle effective for fast transient inputs in addition to slower and steady-state inputs (Jones and Lee 2006). Additionally there is some evidence for Na+ currents in the dendrites of sacral motoneurons (Li and Bennett 2003; Li et al. 2004) and in hypoglossal motoneurons (Powers and Binder 2003).
In an effort to rein in model complexity and provide a platform for easier theoretical exploration of motoneuron function, we have undertaken a "top-down" approach to examining the potential role of dendritic sodium currents as an amplifying mechanism. The intent of a top-down model is to more fully articulate the layers of embedded hypotheses that typically remain implicit and are therefore absent from the model description. In effect, this added structure attempts to encapsulate the why behind the what in a model specification, making the model as a whole more manageable. This type of top-down approach to systems analysis has a long track record in engineering fields (particularly in artificial intelligence), but its use in basic science research has but few examples (Cannon et al. 2003). We applied this approach as a four-tiered set of rules from which our motoneuron model was manually constructed. The plateau-based and electrotonic compression–based mechanisms have been codified as hypotheses/rules in the top-down motoneuron model description (along with others to specify additional behaviors such as action potential shape) and examined for their effect on overall model behavior.
Within this context, it is concluded that although these mechanisms can independently produce their desired effects, they are mutually exclusive because those effects overlap in voltage and therefore conflict. Here we show that this conflict can be resolved by adding another hypothesis governing channel distribution. The resultant motoneuron model exhibits all the known behaviors currently attributed to dendritic processing in motoneurons. Specifically this model exhibits hysteresis in terms of both frequency–current (F–I) and current–voltage (I–V) relations, dendritic plateau potentials, and synaptic amplification. Additionally, it predicts that synaptic amplification should occur for even the fastest inputs and that dendritic branches differentially favor either amplification or plateau potentials. Finally, it is suggested that differential distributions of Ca2+ currents can account for this differentiation of function with the added result that the onset of dendritic plateaus is broadened as is seen in the experimental record. Portions of this work were previously published in abstract form.
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METHODS |
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). It has been recast as a rule-based model that explains why and how specifications were made. Because this is a somewhat novel presentation format, the third and fourth tiers (i.e., the traditional model specification) are at a conceptual level with the detailed mathematical description presented in the APPENDIX. Rule-based model
The rule-based model is constructed in four tiers. The top tier represents hypothesized mechanisms underlying behavioral specifications. The second tier represents hypothesized regulatory actions for achieving those mechanisms. The third tier represents the transporters (i.e., ionic conductances, pumps, and buffers) and their mathematical description. This is the typical starting point for a traditional neuron model specification. Finally, the fourth tier represents the specification of all the parameters introduced in the third tier. These tiers are presented below.
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MECHANISMS TIER |
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5) A region of the neuron (presumably the axon hillock/ initial segment) where action potentials can be easily initiated. Initiation is presumed to require a local I–V relation that contains a negative-slope region at moderate to high frequencies, but not necessarily at steady state (Lee and Heckman 2001).
6) A somatic action potential–generating (as opposed to initiating) mechanism.
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REGULATORY TIER |
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), are arranged in such a way that the most proximal compartment forms the parent segment of two, distally projecting daughter branches. Daughter branches are divided into 14 uniform compartments (see Fig. 1A and APPENDIX). The parent segment has about twice the physical dimensions of any one compartment distal to the bifurcation so it retains the same electrotonic properties as the daughter branch compartments.
TRANSPORTER TIER.
The Na+ conductances used in this work contain some persistence (Kuo and Bean 1994) and are conceptually distributed based on the level of persistence (Safronov et al. 1997) (see APPENDIX). The KDR conductance could be described as a "typical" (i.e., voltage-dependent, Boltzmann-based conductance) version of such a current with the exception that it has been specifically tuned to adult cat lumbar motoneuron experimental data that exhibit nonconstant time constants (Barrett et al. 1980). L-type Ca2+ currents are modeled as two different channel types: one with a relatively high activation (CaV1.2) and one with a relatively low activation (CaV1.3). These two channels were previously immunolabeled in murine spinal motoneurons (Jiang et al. 1999) and could also be described as "typical" voltage-dependent channels. There are conflicting reports as to the time constant of L-type Ca2+ channels. We use the "traditional" slower basis, rather than the more recently reported faster data. We chose this basis based on preliminary examinations in both our model and in the Booth–Rinzel–Kiehn model (Booth et al. 1997) that indicated that a fast time constant basis was incompatible with bistable firing as observed in motoneurons; the fast time constant versions result in no slow "acceleration" of firing (Lee and Heckman 1998a; Schwindt and Crill 1982) as the plateau reaches maximum within one spike (see DISCUSSION).
The Ca2+-dependent K+ conductances modeled here are based on motoneuron experimental data (Barrett et al. 1981; Viana et al. 1993) and more general Ca2+-dependent models (Sah 1993). Finally, h-currents are modeled after previous work (Bayliss et al. 1994), with the exception that time constants have been dramatically reduced to fit spinal motoneuron behavior (Gustafsson and Pinter 1985).
PARAMETER TIER.
For the purposes of this work, the time constants, half-activation values, and so forth are simply made consistent with the published behavior of the currents in motoneurons and are not tuned in any way to fix overall behaviors (see DISCUSSION). For poorly specified parameters, values were obtained from our previous model version (Kuo et al. 2006). In contrast, maximal conductance values (GMax) were chosen to satisfy the mechanisms tier and regulatory tier specifications. The manner in which this was accomplished is described in the subsequent subsection.
Generation of the model
Rather than starting with a spiking soma model then adding a passive structure to represent pentobarbital-anesthetized dendritic behavior then adding plateau-generating currents as is typically done, we chose to start with the dendrites of the decerebrate preparation where all of the active dendritic processing is observed. The central hypothesis here is that Na+ channels in the dendrites generate a persistent current that is just sufficient to cancel the leak conductance, resulting in a voltage region of net zero conductance, thereby reducing the steady-state electrotonic length (i.e., Regulatory tier rule #1; see APPENDIX for further details). This provides the maximum possible amplification without creating a negative region that would be a potential site for rhythmic firing initiation in the dendrites (Lee and Heckman 1998b). To fully comply with the concept of reducing electrotonic length, it is also necessary to consider the need for a negative capacitance. The inactivating portion of the same dendritic Na+ channels can fulfill that role as can a KDR channel. Using both appears to generate the best steady-state I–V behavior. This I–V relationship is established for a single dendritic compartment (normalized to a leak conductance of 1). This compartment is then replicated to represent a "typical" cat lumbar motoneuron dendrite (Regulatory tier rule #3). To this dendrite, passive soma and initial segment compartments are added. (The passive specifications of the initial segment compartment partially satisfy Regulatory tier rule #5.)
Active conductances in the initial segment and soma were subsequently specified. First, the initial segment is populated with Na+ channels exhibiting approximately 2% persistence (Safronov et al. 1997; and Regulatory tier rule #5). It is worth noting that this produces an "A spike" consistent with that seen in experimental settings (Schwindt and Crill 1980). Next, Na+ channels with less persistence (Safronov et al. 1997) were added to the soma to generate a spike height of about 80 mV; KDR is added to narrow the spike (Regulatory tier rule #6; values based on experimental data are consistent with Kernell 1965, 1966).
Ca2+ and SK channels need to be added to generate the AHP as well as plateau potentials (Regulatory tier rules #2 and #4). However, the placement and makeup of these channels are still open to a great deal of uncertainty. Additionally, there are issues of Ca2+ pumping and buffering that are equally open to discussion. We have chosen to rely on only L-type Ca2+ (with no T- or N-type Ca2+) and SK channels and with sufficient Ca2+ pumping to control the intracellular Ca2+ concentration.
When both amplification mechanisms were co-localized, preliminary modeling studies using composite, high- and low-frequency synaptic inputs indicated that amplification occurred preferentially for the low-frequency signal components. The onset of a plateau potential sufficiently destabilized the local I–V relation to the extent that amplification of transient input events was obscured (i.e., the presence of transient inputs was distinguishable only as a more rapid formation of the plateau potential). Considering the likelihood that transient synaptic events occur amid a background level of excitation, we addressed this issue with a morphological dendritic bifurcation. This "split-dendrite" model was created with CaV1.3 channels absent from one branch (denoted as branch "B"). In this branch, the Ca2+ATPase maximal pumping rate is halved, whereas SK GMax is doubled, relative to the "A" branch, to meet the requirements of the regulatory rule governing plateau formation. The CaV1.3 and SK channel distributions are uniform, on a per-branch basis, with the exception of a substantial load of SK in the most proximal dendritic compartment. This increased load effectively produces the AHP as well as a large outward current that is known to appear experimentally just above threshold during voltage-clamped voltage ramps of the soma (Lee and Heckman 1998a). Consequently, although we have accomplished these tasks, it should be noted that the chosen distributions represent only one possible scenario rather than the solution to distributing these active mechanisms.
Validation of the model
One aspect of this top-down approach is that process of model validation is made more explicit. To satisfy the various rules, parameters directly linked with those rules are modified. As such, the model will exactly reproduce any output explicitly specified by a rule. In contrast, all remaining output metrics are simply a product of the other rules because no general parameter tuning is done to improve the overall conformance of the model to experimental values. Thus for example, whereas the electrotonic compression rule explicitly ensured a region of zero slope in the dendritic I–V relation, it did not specify the amplification factor. These unspecified output metrics serve a more traditional "validation" role when compared with experimental values.
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RESULTS |
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Figure 2 illustrates the basic behavior of the model to standard protocols. Standard I–V and frequency–current (F–I) properties were within normal bounds (for comparison to experimental data, see Lee and Heckman 1998a,b). F–I properties were assessed from firing evoked by a slow, 3-nA/s current ramp of 15-nA peak magnitude (Fig. 2A). The primary range F–I gain (i.e., slope of the best linear fit to equivalent current ranges per firing range) is 2.53 Hz/nA (Fig. 2B). (Acceleration and secondary range gain are highly variable experimentally and so are not evaluated here; see Lingering issues in DISCUSSION.) The current resulting from a slow (8 mV/s) voltage ramp (Fig. 2C) is parameterized by membrane potential to produce the I–V function (Fig. 2D). Arrows denote the ascending and descending phases of the I–V and F–I functions. The first point of zero slope on the rising phase (occurring at –53 mV and 3.8 nA) nominally corresponds to the plateau onset, whereas the offset of this persistent inward current (PIC) occurs at –66 mV and –3.8 nA, a more hyperpolarized potential than the onset value (
V = 13 mV; defined as the magnitude of I–V hysteresis; cf. Lee and Heckman 1998b). Also shown in Fig. 2D is the fast, persistent inward current (Ifast) associated with action potential initiation and rhythmic firing (Lee and Heckman 2001). The voltage onset for Ifast is –51 mV, after which it attains a peak amplitude of –2.2 µS (defined as the maximal slope of the leak subtracted current).
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). Examination of the dendritic compartments for an elicited spike reveals that some degree of back-propagation does occur (Fig. 3) in this model.
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One key parameter value not fully specified by the set of rules is the half-activation of the CaV1.3 channels responsible for the dendritic plateaus. The baseline value of –41 mV was chosen to match experimental values (DJ Bennett, unpublished observation). However, it is plausible that this half-activation voltage might resolve the preceding I–V hysteresis discrepancy (and thus generate another hypothesis/rule). Consequently, we investigated the effect of altering the CaV1.3 half-activation voltage (Fig. 4).
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V narrows from 16 to 13 to 5 mV for half-activation values –46, –41, and –36 mV, respectively. Significant transitions that accompany the narrowing are 1) the slope change of inward current on descent of the voltage command (leftward arrows in Fig. 4); 2) the increase in voltage range from rest to onset; and 3) the polarity reversal of offset current. These alterations are indicative of fundamental changes in the neural input–output relationship and manifest a shift in the "visibility" of plateau-generating conductances from the soma.
With the more physiologically accepted half-activation value of –41 mV, the model correctly exhibited bistable firing about zero input current as predicted by the I–V function (Fig. 5; cf. Lee and Heckman 1998b; Fig. 1A). The model could be transitioned from quiescence to sustained, rhythmic firing with a brief-duration, uniformly distributed excitatory synaptic input, which was subsequently withdrawn. Spiking persists in the absence of this input at both positively biased (+3 nA) and nonbiased current offsets, whereas a –15-nA bias current abolishes the tonic discharging.
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To assess synaptic amplification, a mock excitatory Ia synaptic input was applied uniformly across the dendrite to approximate that which is used experimentally (see Lee and Heckman 2000). Similar to Jones and Lee (2006), the composite input consisted of a dynamic component analog (which was exaggerated, for better illustration, as a 10-Hz, train of 3-ms-duration pulses of 160-nS total magnitude) superimposed on a steady background component (24-nS total conductance step). Recall that the overall dendritic space is subdivided into three discrete regions of differing functionality: 1) the Ca2+-plateau zone (nominally, "D2A–D8A"; see Fig. 1A) for amplification of low-frequency synaptic inputs, 2) the Ca2+-plateau–free zone (D2B–D8B) for high-frequency amplification, and 3) the proximal zone (D1) where AHP regulation and amplification-mode integration occurs.
With uniform application of the composite synaptic input described earlier, this regionalized Ca2+-plateau distribution permits local CaV1.3 activation in isolation from the amplification of high-frequency components occurring in the other dendritic branch. The synaptic input signal is differentially processed in dendritic compartments distal to the morphological bifurcation such that the low-frequency component of the input triggers a plateau potential in branch A, whereas excitatory postsynaptic potentials (EPSPs) elicited by the high-frequency inputs are locally amplified to a slightly greater extent (local amplification factor of 3.3 in D8B vs. 2.4 in D2B) in the most distal compartment of branch B (Fig. 6 A, bottom). These branchwise voltage asymmetries diminish with increased proximity to the soma (Fig. 6A, top). Each transient input event produces a rapid excitatory postsynaptic current (EPSC) resulting in a variable-amplitude, effective synaptic current (IN). The peak amplitude of the largest EPSC (recorded at –49 mV) is nearly double that for the EPSC elicited at rest (–60 mV); this corresponds to a synaptic amplification factor of 1.9 at the soma (Fig. 6B), consistent with experimental values for fast amplification (2.0 ± 0.7; Jones and Lee 2006).
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) facilitates plateau formation (PIC onset voltage is shifted to more hyperpolarized potentials) and results in an approximately nanoampere boost over the voltage range of –58 to –52 mV (Fig. 6D). It is important to note that the branchwise difference in dendritic voltage response illustrates the contrast between the fast and slow modes of amplification. Although the steady-state amplification derived from the Ca2+ plateau requires the local formation of a sustained voltage offset, fast/dynamic amplification demands that the local voltage closely tracks the time course of transient synaptic inputs. Therefore when specifying such differential responses from a top-down perspective, spatial separation of the mechanisms underlying each mode of amplification provides a sufficient means to ensure that the single-state variable (voltage) is not subjected to conflicting dynamics.
Fast synaptic amplification under unclamped conditions
The significant densities of fast Na+ channels in the dendrites required to achieve electrotonic compression raise the possibility of dendritic spikes. Dendritic spikes could be viewed simply as a more extreme version of amplification wherein fast synaptic transients elicit very large currents. Although the issue of whether this phenomenon should be labeled as "true" spiking versus fast amplification is open for debate, the model does exhibit this behavior (Fig. 7). Uniform application of a pulse train synaptic input while stepping the soma to progressively more depolarized levels results in larger EPSCs in the most distal dendritic compartment, D8B, of the Ca2+-plateau–free branch (Fig. 7A; comparison of EPSC amplitude at holding potentials of –55 and –49 mV gives an amplification factor of 5.5). Examination of the distal dendritic voltage illustrates that these dendritic spikes occur only in the Ca2+-plateau–free branch (Fig. 7B). Although these voltage-clamp simulations indicate that dendritic spikes are possible, voltage clamp of the soma is decidedly nonphysiological. Thus the open question is how these dendritic spikes will affect behavior during unclamped conditions when electrotonic compression would presumably tighten the somatodendritic coupling.
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To test the generality of the split-dendrite model, we next used it as a template to explore an assortment of morphological arrangements. We constructed a model with a 1:4 branching ratio (parent:daughters) incorporating all of the previously stated rules regarding dendritic conductance ratios and distributions. In this "four-dendrite base model" Ca2+-plateau conductances were equally distributed among three of the four equal-length daughter branches, as well as in the parent segment. Independent activation of plateau potentials could be achieved in this base model with an asymmetric distribution of synaptic input applied during voltage ramping (not shown). In the absence of synaptic input, variation of either the CaV1.3 GMax or the electrotonic length (L) in two of the three, Ca2+-plateau branches resulted in a gradation of the negative-slope region in the I–V for this model. Figure 9 B illustrates this gradation arising from a differential variation (i.e., negative covariation) of CaV1.3 GMax in two of the Ca2+-branches, whereas the third Ca2+-branch remains the same. Here, CaV1.3 GMax is increased in one branch and decreased in another branch by one third, one half, and two thirds of the four-dendrite base model (labeled as "Uniform L and GMax"); note that when CaV1.3 GMax was increased the SK GMax and Ca2+-ATPase IMax were decreased and increased, respectively, by the same proportion and vice versa when CaV1.3 GMax was decreased).
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Effect of combined variation
Simultaneous variation of both CaV1.3 GMax values and L allowed for independent activation of differently scaled plateau potentials resulting in an enhanced smoothing of the negative-slope region. The three plateaus (in dendrites 2, 3, and 4) of the Uniform L and GMax model activate simultaneously, resulting in a smooth but narrow in voltage, negative-slope region. The result of varying either L or CaV1.3 GMax shown above was to spread out the negative-slope region due to plateaus activating in a sequential fashion in the dendrites. Varying both L and CaV1.3 GMax in any given branch raises the question of whether they should be positively covaried, such that variation in CaV1.3 GMax could be thought to "compensate" for a changing L (compensated case in Fig. 10 A) or whether they should be negatively covaried such that variation in CaV1.3 GMax could "accentuate" the effect of changing L (accentuated case in Fig. 10A). If, as we suggested earlier, the negative-slope region of the base model is too narrow in voltage, then it would appear that the hypothesis regarding distribution of CaV1.3 in relation to L should be at least noncompensating.
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). Thus it would appear that, again, any hypotheses added to the model regarding synaptic distribution should be at least noncompensating. |
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DISCUSSION |
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Electrotonic compression versus plateau potentials
Herein, we focused on the mechanisms behind synaptic amplification and bistability in motoneurons. The widely held belief is that the same L-type Ca2+ conductance (CaV1.3) underlies both of these phenomena. In light of this belief, the most up-to-date motoneuron models include only the L-type Ca2+ conductance in the dendrites (Elbasiouny et al. 2005, 2006). These models exhibit what appears to be slow, but not fast, amplification. We have demonstrated a set of theoretical regulatory mechanisms tcath separate the physiological process underlying these behaviors. The hypothetical mechanism behind synaptic amplification (i.e., electrotonic compression theory) was shown here to fit well with experimental observations and this modeling study concludes with the suggestion that an empirical foundation be laid for such ratiometric arrangements of dendritic conductances conducive to this mechanism.
The additional degree of spatial separation needed to permit the combined action of fast amplification and plateaus is, itself, an interesting finding. Although there are no definitive published data, some preliminary immunolabeling results have shown a notable absence of CaV1.3-positive puncta on a number of dendritic branches (Powers and Binder and Grande and Rose, personal communications). Electrotonic compression demands that the membrane potential lies within close vicinity to the first zero-slope point of the I–V function, although Ca2+-plateau formation requires this fixed point to be unstable (i.e., adjacent to a negative-slope region in the I–V). It is this conflict that suggests the spatial separation of these amplification mechanisms used in our present model. Recent modeling studies of spinal motoneurons, which use reconstructed morphologies, suggest that nonuniform distributions of CaV1.3, localized as clusters or "hot spots," may play a functional role in Ca2+-plateau–based amplification (Bui et al. 2006). Our present results offer a rather different perspective on the spatial aspect of dendritic conductance compartmentalization, wherein the dendritic functional subunits are arranged with regard to the incompatibility between amplification mechanisms. It would therefore seem that the natural enforcement of this arrangement requires less global information (e.g., distance from the soma to the hot-spot center) and instead may rely on local rules governing channel distribution (e.g., CaV1.3 channels, or L-type Ca2+ transcripts, are delivered to only a fraction of the dendrites, whereas all branches maintain a conserved ratio of persistent Na+, KDR, and leak channels).
Top-down modeling
Although it has been a fruitful technique, the traditional "bottom-up" approach to constructing conductance-based models of neurons is fraught with issues once the parameter space grows beyond some threshold. Hand-tuning is a tractable task with highly reduced models but the inclusion of new physiological variables increases the dimensionality of the parameter space. Parameter estimation in these larger models has been approached with a variety of methods including automated parameter searches (Vanier and Bower 1999) and genetic algorithms (Taylor and Enoka 2004). However, these techniques are effective only at refining output metrics and fair rather poorly at finding emergent behaviors such as synaptic amplification. A further complication is the likelihood for parameter value nonuniqueness for a given set of model outputs (Bhalla and Bower 1993; Foster et al. 1993; Goldman et al. 2001; Prinz et al. 2004). The nonuniqueness problem can be mitigated by experimental data. However, in regions less accessible to experimental control, such as the large dendritic tree of mammalian spinal motoneurons, the parameter space is largely unconstrained save for a few rational assumptions.
In contrast, the top-down approach used here represents a rational solution for many of the complexity management issues raised earlier. Additionally, the concept that active conductances are organized to achieve specific goals has a certain teleological appeal and so is not new. For example, it has long been assumed that the ratio of fast Na+ currents to KDR currents was the important factor as opposed to simply the absolute level of conductance of each (Muller and Lux 1993; Wolff et al. 1998). Additionally, it is common for neuromodulators to directly and indirectly affect several types of ionic conductances simultaneously. For example 5-HT in motoneurons is known to simultaneously reduce TASK (Perrier et al. 2003; Talley et al. 2000), increase L-type Ca2+ (Perrier and Hounsgaard 2003), decrease AHP (Talley et al. 1997), and reduce N-type Ca2+ currents (Bayliss et al. 1997; Koike et al. 1994).
Another advantage of this type of model specification is that it tends to bring into stark relief the gaps in our understanding of neuronal function. For example, it was recently reported that CaV1.3 channels may have a time constant that is dramatically faster than previously thought (<3 vs. 24 ms; Helton et al. 2005). This new work is based on recombinant channels expressed in nonneuronal (tsA) cells versus previous work that was based on channel blockers. On one hand, channel blockers have become somewhat notorious for their unintended side effects; on the other hand, channels expressed in cell lines have been taken out of their normal context of subunit composition, modifying proteins, and posttranslational processes. Thus as is often the case, we are forced to choose between conflicting bases, neither of which is directly applicable. (The only thing worse, of course, would be no basis as all.) In this instance, experimental data from motoneurons would seem to favor the longer time constant as a fast time constant for CaV1.3 results in dendritic plateaus that fully activate with a single spike. In contrast, onset of plateau formation was previously observed in motoneurons under both current- and voltage-clamp conditions to be slower (Bennett et al. 1998). Thus without a theory to explain how fast channels can still result in a slow plateau onset (requiring another rule in our model) we chose to stick with the earlier, slower time constant basis. Another area of less-than-ideal basis is the specification of how Ca2+ currents interact with the Ca2+-dependent SK conductance. These currents regulate everything from plateau potentials to the AHP and are disturbingly underdefined. In this model we, somewhat arbitrarily, linked plateau potentials and the AHP. This worked reasonably well but is admittedly not perfect. Other models have separated the two (Elbasiouny et al. 2005, 2006; Taylor and Enoka 2004), solving some of our problems, yet introduced problems of their own (e.g., low I–V slope above plateau onset and excessive AHP-spike height sensitivity; see Mechanisms tier rule #4). We believe that only by a systematic, hypothesis-based exploration of possible mechanisms can we hope to resolve these issues.
Lingering issues
There are several areas in which the present model either falls short of realistically emulating behaviors of real motoneurons or otherwise raises issues worthy of further discussion. As mentioned in a previous section, the steepness of the I–V negative-slope region is excessive. This is explained by the sudden onset of the plateau, which in turn produces a rather severe acceleration in firing as seen in the F–I (Fig. 2). In fact, the very nomenclature presently used to describe this phase of the F–I is a subject of debate. Depending on the stability of the firing rate (i.e., whether the discharge rate can be held constant), this region of the F–I may be referred to as either the acceleration phase (Lee and Heckman 1998a) or the secondary range (Bennett et al. 1998). Further complicating the matter is the means by which the F–I relation is typically assessed; somatic activation of the dendritic plateau requires a greater current than would be the case with a local current source (i.e., a more "natural" synaptic activation of the plateau). The sudden jump in firing rate we have attributed to an all-or-none plateau onset, and dubbed F–I acceleration, may indeed be an incremental series of events when evoked in the opposite direction. In either case, we have demonstrated that the PIC onset may be graded by variation of the morphology, channel distribution, and/or synaptic density of secondary dendrites (see Figs. 9 and 10). Branchwise fractionation of the PIC may also address the issues with the Voffset and V measures, both of which fall outside of the 95% confidence interval for the values given in Table 1, with the early deactivation of those branches potentially hastening deactivation in other branches.
There are also remaining issues with both fast and slow amplification in the present model. Although the voltage range for fast amplification (–51 to –45 mV; see Fig. 6B) corresponds to the reduced ("zero") slope region of the local I–V for branch B compartments (see Fig. 1B), as expected; it also begins at the Ifast onset voltage. If we were to expect a correlation between Ifast and the dynamic component of Ia IN (referred to as Isyn,fast in Jones and Lee 2006), then the onset voltage for fast amplification should be more hyperpolarized (mean value of –58.3 mV; Jones and Lee 2006). However, the model presented here is not specifically tuned to match this behavior. Instead the correspondence between Ifast onset and fast amplification onset, albeit moderately depolarized, appears to be a secondary effect of electrotonic compression. Although it is not among our present set of model goals, we could incorporate fast amplification onset voltage into the scope of output metrics, although doing so would require amending Mechanism tier rule #1 to specify the voltage range for fast amplification and the addition of another Regulatory tier rule specifying the width of the local I–V zero slope region. Additionally, the steady component of Ia IN (Isyn,slow in Jones and Lee 2006) presents in a rather exaggerated form (compare Fig. 6D to Lee and Heckman 2000; Fig. 3B). The abrupt onset and offset of Ia IN (as well as the "afterdepolarization"-like bump) contrasts with the smooth, graded onset/offset seen in Lee and Heckman 2000; however, the current magnitude (4 nA) and the voltage range at peak (–58 to –52 mV) are in close agreement to the low input conductance cells presented by Lee and Heckman. We demonstrated in Figs. 9 and 10 that small amounts of anatomical heterogeneity among the secondary dendrites mitigate the PIC onset in a manner that is potentially suitable to grade the abrupt transitions of Ia IN seen here.
Finally, it is worth pointing out that the set of rules presented here is not complete. There are still currents and, more notably, parameter values in tiers 3 and 4 that are poorly connected to higher-level rules. Although these currents and parameters may have an experimental basis, they provide no insight into why they have the values that they do, how they are related to other mechanisms, and so forth. However, this approach is tolerant of this incompleteness and even facilitates the incremental inclusion of rules as greater insights are obtained.
Predictions
This model of motoneuron behavior results in several predictions. First, this model predicts that amplification, particularly fast amplification, and plateau potentials are mutually exclusive within a given dendritic branch. The exploration shown in Figs. 9 and 10 indicates that a well-varied density of Ca2+ currents along with variations in the length of individual dendrites would create this scenario and would have the additional effect of broadening the onset of the plateau like that seen experimentally. In the context of the rule-based model format presented here, these distributions would need to be noncompensating for one another.
Another prediction of this model is that the presence of dendritic Na+ currents suggests that motoneurons may be capable of action potential back-propagation and dendritic spiking akin to that seen in other cells of the CNS (e.g., Hanson et al. 2004; Stuart et al. 1997). It may also lead to direct-mode firing like that recently observed in hippocampal cells (Losonczy and Magee 2006). Unlike the previous prediction, which can be verified only by immunocytochemical/anatomical examination, demonstration of these features may be within the reach of electrophysiologists. If present, these behaviors might yet again expand our view of how motoneurons process synaptic information.
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APPENDIX |
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All model outputs were calculated using a custom built simulation environment. Compartmental voltages are computed by numerical integration of voltage state variable (n)
 | (A1) |
t) are calculated by a predictor–corrector scheme and values are kept within a logically relevant range (e.g., nonnegative concentrations of mobile charge carriers). Electrotonic architecture
The active somatic volume was taken to be 20% of the total volume occupied by a sphere of 60-µm radius; intracellular Ca2+ concentrations were tracked within this space. Radii and lengths of the cylindrical dendrites and initial segment were 25 x 6,000 µm (combined length of 11 branching levels before equivalent cylinder form) and 3 x 100 µm, respectively. Specific membrane capacitance, specific membrane resistivity, and axial resistivity were estimated as 1 µF/cm2, 10.3 k
cm2, and 70 cm, respectively, in accordance with electrotonic studies of spinal motoneurons (Clements and Redman 1989).
Detailed description of conductances, pumps, and buffers
All transmembrane conductances are based on a single archetype. Voltage-gating equations follow the general form of
 | (A2) |
 | (A3) |
x). In cases in which the time constant is nonconstant, but rather a bounded interval or composite function of time and/or voltage, the gating kinetics reflects those characteristics. The compartments composing the motoneuron model are constructed without regard to fine morphological detail; instead we target relevant magnitude and scaling of parameters. Therefore any subsequent reconstructions of the model that aim to take into account the detailed geometries of certain compartments would be advised to target the subsequently reported GMax values.
SYNAPSES. Dendritically based, excitatory synapses are modeled as generic transporters with a reversal potential of 0 mV (NMDA- or AMPA-like) and a uniform distribution across a given compartment. Except where otherwise noted, all synaptic conductance injections are uniformly distributed across the entire dendritic arbor and are reported as particular waveforms with total dendritic values.
SODIUM CHANNELS.
As detailed in our previous model (Kuo et al. 2006) the Na+ channel kinetics is based on a Markov model. The transition rates between states are adapted from the 12-state kinetic scheme of Kuo and Bean (1994) but in a more simplified form. For computational reasons, we reduce the model from 12 states to four states in accordance with the assumption that most of the intermediate transition rates occur at much faster timescales than the final transitions. Practically, this reduction is accomplished by assuming the intermediate activation states are fast and by setting their time constants to zero. Effectively the steady-state values of their gating variables are instantaneously reached. The reduced model has a pair of half-activation and half-deactivation potentials as well as time constants corresponding to the opened/closed and the deactivation/inactivation state transitions. These parameters for NaV1.2 and NaV1.6 are –32 mV, –45.5 mV, 0.005 ms, 0.2 ms; and –34.5 mV, –47 mV, 0.005 ms, 0.05 ms, respectively. NaV1.2 channels occur solely in the somatic compartment (GMax = 6.0 µS) and are responsible for generating the rapid depolarization underlying the somatic action potential. NaV1.6 channels display a 1.55% noninactivation, indicating that nearly 2% of a given compartmental population will fail to inactivate and NaV1.2 channels display a 0.8% noninactivation, together giving rise to a fast (i.e., Na+-based), persistent inward current (PIC). This feature underlies the macroscopic behaviors of rhythmic firing and amplification of rapidly changing inputs. The dendritic distribution of NaV1.6 is homogeneous across all compartments distal to the split (GMax = 0.765 µS; D1 GMax = 1.53 µS), with none occurring in the soma, and a second population in the initial segment (GMax = 5.0 µS).
CALCIUM CHANNELS, TRANSPORTERS, AND BUFFERING.
The voltage-gated Ca2+ channels included in the model are of the HVA L-type class (CaV1.2 and CaV1.3), differing only by their half-activation voltage value. CaV1.2 is present in only the first dendritic compartment (GMax = 0.86 µS), whereas the CaV1.3 ion channel is homogeneously distributed throughout branch "A" of the dendritic tree (GMax = 0.030 µS). Both channel types have a first-order voltage-dependent gating, GCaL = 
m. The half-activation and time constant values for CaV1.2 and CaV1.3 are –20 mV, 30 ms and –41 mV, 24 ms, respectively. The voltage sensitivity for both of these channels is 6 mV.
Sodium–calcium exchanger (NCX) imports 3 Na+ with each efflux of one Ca2+, thereby producing a net inward current. The rate of exchange, Flow(ion), is simply the instantaneous rate of change for each ion species
 | (A4) |
 | (A5) |
 | (A6) |
 | (A7) |
NCX) and an equilibrium constant K of 6 x 10–5 M. Calcium pumps (Ca2+ATPase) are modeled as an instantaneous outward current proportional to the cytosolic Ca2+ concentration, i.e., there is no activation time constant. The pump current is determined from free Ca2+ concentration as in Eq. A7 with the equilibrium constant K, 2 x 10–4 M, and Imax, 1 nA in compartments D1–D8A and 0.5 nA in D2B–D8B of the branched dendrite.
The net Ca2+ currents from CaV1.2, CaV1.3, Ca2+ATPase, and NCX are used to determine intracellular Ca2+ concentration
 | (A8) |
 | (A9) |
 | (A10) |
POTASSIUM CHANNELS.
All model compartments contain Hodgkin–Huxley (HH)-type delayed-rectifier K+ channels (KDR): (Initial Segment) GMax = 1.0 µS, (Soma) GMax = 4.0 µS, and (Dendrites) GMax = 0.4525 µS per compartment (0.905 µS in D1). The KDR channel kinetics follows the fourth-order activating, HH-like model: GKDR = n4 (Barrett et al. 1980)
 | (A11) |
This channel has a half-activation of –25 mV and voltage sensitivity of 20 mV. The time constant for the KDR conductance is modeled as a voltage-dependent Boltzmann equation
 | (A12) |
 | (A13) |
 | (A14) |
m is subject to the imposed boundary conditions of Max = 11.9 ms and Min = 1.4 ms.
SK-type Ca2+-activated K+ channels occur in two distinct populations. The distribution is homogeneous within a particular secondary branch of the dendrites, GMax = 0.05 µS in branch "A" and 0.1 µS in branch "B," whereas the most proximal dendritic compartment contains a more substantial load, GMax = 1.0 µS. The SK channel exhibits first-order gating
 | (A15) |
 | (A16) |
 | (A17) |
MISCELLANEOUS TRANSPORTERS. Na+/K+ leak is specified by a physiologically relevant constant value on a per-compartment basis; we estimated this to be 0.04525 µS (0.0905 µS in D1; 0.22308 µS in the soma; 0.00385 µS in the initial segment).
We used data from hypoglossal motoneurons (Bayliss et al. 1994) to approximate the h-current time constant. The activation time constant is sensitive to negative-voltage trajectories (Eq. A18) and deactivates with depolarization (Eq. A19)
 | (A18) |
 |
 | (A19) |
Generation of electrotonic compression
Central to the notion of electrotonic compression is the contour of the I-V relationship in a single dendritic compartment. The "ground level" for amplification (i.e., an amplification factor of 1) occurs when there is no current loss for distal inputs en route to the soma. This should hold true for steady-state up through higher-frequency inputs (Mechanism tier rule #1). Once the passive properties for the dendritic cylinder were established, we incrementally added NaV1.6 and KDR channels to a single, representative, dendritic compartment to attain a zero-slope conductance in the pseudosteady-state (8 mV/s ramp) I-V. Combined with the fast activation of NaV1.6, the restorative effect of KDR channels on the local membrane potential further augments the "high-pass filter"-like qualities of the electrotonic compression mechanism and prevents the membrane time constant from becoming infinite. The overall process is equivalent to balancing the dynamic and steady-state components of the passive compartment's current- and voltage-step response. The capacitive and leak currents are compensated with inactivating NaV1.6 (+ KDR) and noninactivating NaV1.6 currents, respectively; it is in this regard that the term "negative capacitance" arises.
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GRANTS |
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FOOTNOTES |
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Address for reprint requests and other correspondence: R. Lee, 313 Ferst Drive, Atlanta, GA 30332-0535 (E-mail: robert.lee{at}bme.gatech.edu)
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ABSTRACT
INTRODUCTION
7.8 nS per dendritic compartment) persisting for 1.5 s evokes a bistable firing response in all but the hyperpolarizing current case (–15 nA; bottom).
