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Department of Biology, Emory University, Atlanta, Georgia
Submitted 16 May 2008; accepted in final form 9 August 2008
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ABSTRACT |
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INTRODUCTION |
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). One of the aims of cellular-level neurophysiology is to explain how the different types of membrane currents interact to give rise to a neuron's electrical activity and what role particular membrane currents play in shaping that activity. A large body of both experimental and modeling work to address these questions has led to a number of insights about the function of particular current types, including current types that are involved in several different electrical activity types. For example, fast sodium and delayed rectifier potassium currents are known to be the major currents shaping fast action potentials (Hodgkin and Huxley 1952), hyperpolarization-activated cation currents contribute to a cell's postinhibitory rebound properties (McCormick and Pape 1990) and pacemaking activity (Difrancesco 1981), slowly inactivating potassium currents support cellular short-term memory (Turrigiano et al. 1996) and can extend the dynamic range of action potential firing in response to depolarizing current (Kupper et al. 2002), transient potassium currents facilitate slow repetitive action potential firing (Connor and Stevens 1971), and depolarizing leak currents determine neuronal excitability (Cymbalyuk et al. 2002). Most such studies about the role of particular channel types in neuronal activity seek to be generally applicable to the channel type in question rather than making fine distinctions regarding channel subtypes or the exact voltage dependencies and dynamical properties of a channel.
Many of these identified roles of particular channel types have repeatedly been observed in a number of different neuron types and thus appear to be quite general and to some extent independent of the particular voltage dependencies or dynamical properties of the channel types in questions. In contrast, other conclusions about the influence of a particular ion current type on neuronal activity are based on observations in only a few types or a single type of neuron. How reliably do such findings generalize to other types of neurons with their own specific complement of ion channels or channels subtypes? The answer to this question depends on just how different ion channels of the same type are in different classes of neurons. Most ion channel types [with the notable exception of gap junction channels (Panchin 2005)] are evolutionarily quite old and show great similarity even across different phyla (Hille 2001). Nonetheless due to differences in primary sequence, subunit composition, posttranslational modification, phosphorylation state, etc., ion channels of the same type can differ significantly between and even within different animals with respect to their voltage dependences and gating dynamics. This is the case not only for the inherently very diverse potassium-selective channels but also for the less diverse sodium, calcium, and chloride channels (Hille 2001). Neuronal activity patterns and response properties arise from the complex interplay of many different ion channel types in the same neuron, so the variability of membrane properties among neurons is likely to be reflected in the different roles specific conductances play in different neurons.
Furthermore, similar electrical activity can often arise through multiple different mechanisms involving different ratios of the same ionic currents (Prinz et al. 2003, 2004) or interactions between different types of ionic currents. For example, bursting activity can result from an interplay between the slow dynamics of the hyperpolarization-activated current Ih and the low-threshold calcium current IT (Buzsaki 2006;McCormick and Pape 1990) or from activation of the repolarizing calcium-dependent potassium current IKCa following calcium accumulation during the burst (Wilson and Goldberg 2006). An elegant overview of a variety of spike and burst mechanisms that each involve different types of currents is provided by Izhikevich (2007).
Here we examine the involvement of several different current types in tonic spiking in two well-established models of neurons from different phyla: a model crustacean stomatogastric (STG) neuron and a rodent thalamocortical (TC) relay neuron model. Tonic spiking occurs in both types of neurons in vivo. In TC neurons, tonic spiking functions to relay trains of action potentials from sensory systems to cortex during states of wakefulness (McCormick and Huguenard 1992) and in the STG several types of follower neurons spike tonically when isolated from rhythmic synaptic input (Golowasch and Marder 1992; Harris-Warrick et al. 1995a,b; Selverston and Miller 1980). Both model neurons contain similar sets of membrane currents, including sodium, calcium, potassium, and nonspecific cation currents of the same types. In the case of several of the membrane currents present in STG neurons, the underlying channels have been cloned, and significant sequence homology with corresponding rodent and/or fly channels has been demonstrated (Baro et al. 1996a,b). Specifically, most recently cloned crab STG channels show close to 90% amino acid identity with corresponding fly channels and 50–80% identity with human channels (D. J. Schulz, unpublished observations). Furthermore, both model neurons are capable of spontaneous tonic spiking in two distinct frequency ranges (fast and slow), depending on the maximal conductance levels of their membrane current types. Despite these similarities in the types of ionic currents they contain and in their functional behavior, our results indicate that the two classes of neurons "use" their membrane currents quite differently to arrive at similar electrical activity. This suggests that some conclusions about the role of particular current types in neuronal activity are not readily generalizable from one type of neuron to another, even if those neuron types rely on similar complements of ion channels.
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METHODS |
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The model neurons explored here are single-compartment models of crustacean STG neurons and mammalian TC neurons. Both models have been described in detail elsewhere (Huguenard and McCormick 1992; McCormick and Huguenard 1992; Prinz et al. 2003), and complete descriptions of the differential equations and parameters used in both models are provided in the supplemental material1 (with the exception of the maximal conductances and permeabilities listed in the following text). For comparison, the supplemental materials also contain graphs of the voltage dependences of steady-state activation and inactivation variables and time constants of all currents in the two models. Note that all currents in both model neurons were described with the Hodgkin-Huxley type conductance formulation (Hodgkin and Huxley 1952), and their prevalence in the membrane was described by maximal conductances, with the exception of the calcium currents in the TC model. These were instead modeled using the Goldman-Hodgkin-Katz constant field equation (Hille 2001), and their amplitude thus scaled with a permeability rather than a conductance (McCormick and Huguenard 1992). For simplicity, we will usually refer to both conductances and permeabilities as "conductances" in the rest of the paper.
Both models contain a fast sodium current INa, fast and slow calcium currents (STG: ICaT and ICaS, TC: IT and IL), a transient potassium current IA, a calcium-dependent potassium current (STG: IKCa, TC: IC), a potassium current (STG: IKd, TC: IK2), and a hyperpolarization-activated cation current Ih. In contrast to the single mixed-cation leak current Ileak of the STG model, the TC model contains separate sodium and potassium leak currents INaleak and IKleak and also has a persistent sodium current INaP. Because the permeabilities of IT and IL in the TC model were held at a fixed ratio of 1:2 in keeping with experimental results (McCormick and Huguenard 1992), the STG model had eight and the TC model had nine independent maximal conductances.
Database construction
Two model neuron databases were constructed by independently varying the maximal conductances of single-compartment STG and TC model neurons. The STG database, and the technique of database construction and analysis, are described in detail in Prinz et al. (2003), and the TC database was constructed in complete analogy to the procedures described there. In the model databases, each independent conductance could take one of six different, equidistant values that covered the physiological range (including 0 as the smallest value). In the STG database, these values were (given as increment/maximum, in mS/cm2): 100/500 for INa; 2.5/12.5 for ICaT; 2/10 for ICaS; 10/50 for IA; 5/25 for IKCa; 25/125 for IKd; 0.01/0.05 for Ih and Ileak. In the TC database, the values were (in µS unless otherwise noted): 3/15 for INa; 0.0028/0.014 for INap; 4·10–4/20·10–4 cm3/s and 2·10–4/10·10–4 cm3/s for the maximal permeabilities of IL and IT, respectively; 0.8/4 for IA; 0.6/3 for IK2; 0.009/0.045 for Ih; 0.006/0.03 for IKleak; and 0.0024/0.012 for INaleak. In the remainder of the text and in Figs. 2, 3, and 6, the six different conductance values explored for each membrane current are referred to as levels 0 through 5. For example, a conductance level of three for INa in the STG neuron refers to a maximal conductance of three times INa's conductance increment of 100 mS/cm2, which is 300 mS/cm2. All possible conductance combinations were simulated, resulting in 68 
1.7 million STG model neurons and 69 10 million TC model neurons. Spontaneous activity, that is, the electrical activity exhibited by the model neuron in the absence of any synaptic input or injected current, was simulated and saved for each conductance combination. An automated algorithm described in detail in Prinz et al. (2003) was used to detect spikes (defined as local voltage maxima) and compute interspike intervals. The algorithm classified a model as tonically spiking if its steady-state spontaneous activity showed inter-spike intervals of constant duration.
Dimensional stacking
To visualize model neuron database content, we used dimensional stacking, a visualization technique described in detail in Taylor et al. (2006) and briefly outlined here, using construction of a dimensional stack for the STG database as an example.
The dimensional stack for STG database visualization was constructed by assigning a color value to each model neuron in the database using the same assignment as in Figs. 1 and 2: orange for slow spikers with a narrow spike, blue for slow spikers with a broad spike, green for fast spikers, and white for all other models, which included silent, bursting, and irregularly firing models. Colored pixels representing each STG model neuron were then arranged in the dimension stack in Fig. 3 in the following manner: each of the 6 x 6 major blocks of pixels visible in the figure contains models with fixed maximal conductances for IKCa and IKd with the IKCa conductance increasing from block to block from bottom to top as indicated by the number at the left and the IKd conductance increasing from block to block from left to right, as indicated by the number at the bottom. Within each of these major blocks, STG models are arranged in 6 x 6 smaller blocks according to their conductances for ICaT (increasing from bottom to top) and INa (increasing from left to right), and so forth through two more levels of organization governed by the remaining four conductances as indicated by the bars at the bottom left of the figure, which show the pixel range over which any given conductance is constant.
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; Taylor et al. 2006). The edginess minimization method used here was described in detail in Taylor et al. (2006); briefly, the algorithm starts with a dimensional stack of random stacking order (the "parent") and determines its edginess by counting all edges between adjacent pixels of different color. Next, the algorithm generates all stacking orders ("children") that can be obtained from the original stacking order by swapping two conductances and determines the edginess for each of the children stacks. Out of the parent and the children, the method then selects the stacking order associated with the lowest edginess value and uses it as the parent stacking order for the next iteration. The algorithm stops once a stacking order has been identified that leads to lower edginess than all of its children; that stacking order is then taken to be the optimal one. For the model neuron databases used here, on the order of five to six iterations of the algorithm usually were sufficient to find an optimal stacking order. Although an optimal stacking order found with this method is strictly speaking only a local edginess minimum, repeated applications of the algorithm with different starting points reliably lead to the same optimal stacking order, suggesting that the method is successful at identifying a global edginess minimum (Taylor et al. 2006). Edginess minimization favors stacking orders that group pixels with the same color in contiguous regions of the stack. Alternatively, a dimensional stack that reveals the internal structure of a database can often be achieved by manually adjusting the stacking order to position conductances that are suspected to have little influence on a given behavior at lower levels of organization and conductances that are thought to be relevant for the behavior at higher levels of stack organization.
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RESULTS |
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We used two databases of model neurons constructed by systematically exploring the eight- and nine-dimensional maximal conductance spaces of an STG and a TC model neuron and searched the resulting databases for tonically spiking model neurons, which we defined as models whose spontaneous activity showed interspike intervals of constant duration. We found 586,966 tonic spikers (35%) in the STG database and 463,292 tonic spikers (5%) in the TC database with the remaining models in both databases consisting of silent, regularly bursting, or irregularly firing neurons.
The tonic spikers in the STG dataset were further separated into 322,679 spikers with a broad voltage shoulder after the spike and 264,287 tonically active models with a narrow spike. The broad STG spikers correspond to the "one-spike-bursters" previously described in (Prinz et al. 2003), where they were separated from the narrow spikers according to their integral area under the voltage peak between –40 and –15 mV (>0.5 mV for broad and <0.5 mV for narrow spikers) and their voltage peak (overshooting 0 mV for narrow spikers). Broad and narrow STG spikers were shown in (Prinz et al. 2003) to be distinct, nonoverlapping populations according to this criterion.
Figure 1 shows spike frequency distributions for the spikers in both databases. The STG spikers show two clearly separate subpopulations based on frequency, one with spike frequencies <11.5 Hz ("slow spikers") and one with frequencies >e 11.5 Hz ("fast spikers"). The slow STG spikers include the broad spikers (blue) and part of the narrow spikers (orange), whereas the fast STG spikers (green) are all narrow.
The overall frequency histogram of the TC spikers looked qualitatively similar to the STG histogram and also suggested two subpopulations, although they showed more frequency overlap than in the STG histogram and the slower TC subpopulation extends to somewhat higher frequencies compared with the slow STG population. Inspection of randomly selected TC example traces with high and low spike frequencies revealed that the two tentative subpopulations differed markedly in their time difference between a spike peak and the following most hyperpolarized point of the voltage trajectory (termed "peak-to-trough time" here). This is illustrated by a fast and slow example trace in the insets of Fig. 1B, where the spike peak and most hyperpolarized potential and the peak-to-trough time are indicated. A histogram of all peak-to-trough times (included in the supplementary materials) indicated a clear separation of the two subpopulations at a peak-to-trough time of 6.5 ms. TC spikers were classified as fast spikers if they had a peak-to-trough time <6.5 ms and as slow spikers if their peak-to-trough time was >6.5 ms. The TC spikers consisted of 80,981 fast and 382,311 slow spikers according to this definition. The two populations are shown in Fig. 1B in different colors.
Both model neuron databases show fast and slow subpopulations that are clearly separable based on spike frequency and/or other features of their voltage traces. To our knowledge, bimodal spike frequency distributions such as these have not been previously described, and the spike mechanisms underlying fast and slow spiking in the two model neurons are not a priori obvious.
Conductance statistics of fast and slow spiker populations
STG and TC model neurons, although based on experimental data from two different phyla, have related sets of membrane currents and show qualitatively similar spike frequency histograms, which both separate into fast and slow subpopulations. Do these similar distributions arise from similar combinations of membrane currents in the two types of neurons? To address this question, we first compared the distributions of the fast and slow spikers in both databases over the six different levels of each maximal membrane conductance probed during conductance space exploration. Results from this analysis are shown in Fig. 2.
The figure shows that the distributions of fast and slow spikers over the six different levels of a conductance are qualitatively similar for some currents, such as Ih in both databases, or the leak current in the STG database. In contrast, the distributions of fast versus slow spikers differ considerably for other conductances, most markedly for Na and Kd in the STG database and for K2 and C in the TC database. We take the similarity or difference of fast versus slow spiker distributions as a first indication of which membrane currents are likely to play a role in determining whether a model neuron spikes slowly or fast. Conductances for which the distributions of fast and slow models are very similar are unlikely to have a large effect on spike frequency of tonic spikers, whereas large differences in the distributions for a conductance would suggest that that conductance is differentially involved in the mechanisms underlying fast and slow spiking.
The distribution of slow TC spikers over the six conductance levels of the transient potassium current IA is relatively homogeneous, whereas the fast TC spikers show a preponderance of low levels of IA. This is consistent with the previously proposed role of IA retarding fast repetitive spiking in TC neurons (Connor and Stevens 1971). Similarly, the kinetically slower IK2, which was described in McCormick and Huguenard (1992) as modulating TC neuron firing rate through prolonged changes in excitability, tends to be large in slow TC spikers and small in fast TC spikers (Fig. 2B).
In the following sections, we will analyze the involvement of different membrane currents in fast and slow spiking separately in the STG and TC databases with the aid of dimensional stacking and comparison of representative voltage and current traces, dynamic variable ranges, and frequency-current relationships. A comparison of the results for the two types of neurons follows in the DISCUSSION.
Fast and slow spiking in STG model neurons
DIMENSIONAL STACK.
To understand the role different membrane currents play in fast versus slow spiking in STG neurons, we used a visualization technique called "dimensional stacking" to examine the location of the fast and slow spiker subpopulations in conductance space. Dimensional stacking displays high-dimensional parameter spaces in a two-dimensional (2D) plot without projecting or averaging. The technique is described in detail in Taylor et al. (2006) and briefly outlined in METHODS.
Figure 3 shows a dimensional stack of the STG database that uses the same color assignment as Figs. 1 and 2 to distinguish slow narrow, slow broad, and fast spikers from all other models in the database. The stacking order used in Fig. 3 was arrived at by assigning to the lowest levels of organization those conductances that were assumed to have little influence on spike frequency based on the similarity of their fast and slow spiker distributions in Fig. 2A, such as the Ih and Ileak conductances. Conversely, conductances that were assumed to have significant influence on spike frequency based on different distributions in Fig. 2A were assigned to higher levels of organization—examples are the conductances for IKCa and IKd. Furthermore, the stacking order was manually optimized to achieve good visual separation of the three spiker categories (slow narrow, slow broad, fast) in the 2D display.
Fast spikers (green) are completely absent from all locations in Fig. 3 in which the INa conductance or the IKd conductance is zero, consistent with the well-known role of these currents in spike generation. Furthermore, fast spiking models are located preferentially in a wedge-shaped region in the bottom right corner of Fig. 3, indicating that fast spiking in STG neurons requires nonzero delayed rectifier (IKd) conductance and low or zero (specifically, <4) levels of IKCa conductance simultaneously and that either one of these conditions alone is not sufficient for fast spiking in this model. Figure 3 also indicates that slow spiking with narrow spikes (orange) can only occur in STG neurons with lower than level 1 ICaT conductance because all orange pixels are located in the bottom sixth of the second vertical level of organization—this is consistent with the conductance statistics in Fig. 2A. Most other conductances seem to have little effect on the distribution of slow narrow spikers in conductance space with the exception of IKd, for which slow narrow spikers show a slight preference for high values because the horizontal stripes of orange models grow denser toward the right edge of the stack. Slow broad spikers (blue) require that either the IKd conductance is <1 (high concentration of blue pixels in the left 6th of the stack) or the INa conductance is <1 (vertical stripes of blue pixels in the left 6th at the 2nd level of horizontal organization) or that both conductances are small (
2, remaining blue pixels). This correlation between the levels of sodium and delayed rectifier conductance in the slow broad spikers was not obvious from Fig. 2A, which only shows a bias of the slow broad spikers toward small values of both conductances.
EXAMPLE TRACES. To analyze how the membrane currents in STG neurons interact to produce fast and slow spiking behavior, we randomly selected 10 fast and 10 slow spikers from the STG model neuron database. The latter group contained six models with broad and four models with narrow spikes, respectively, whereas the fast STG spikers, by definition, are all narrow. Voltage traces from all 20 models are shown in Fig. 4, where each trace is aligned to exhibit a spike peak at time 0. Although the example traces in the figure were randomly chosen from the three spiker categories, the groups show clearly distinct spike and inter-spike dynamics in Fig. 4, strengthening the suggestion that fast, slow narrow, and slow broad spikers are indeed distinct categories rather than extremes of a continuum of spike dynamics.
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The voltage ramp of slow broad spikers leading up to the following spike (the end of which is visible to the left of the aligned spike peaks in Fig. 4) is similarly shallow, but the period of the broad slow spikers is further increased by a very long spike duration, consistent with the fact that the broad spikers are located at the very left edge of the STG spike frequency distribution in Fig. 1. How are the membrane currents that were identified as potentially determining the spike frequency based on Figs. 2A and 3 involved in shaping the spikes and interspike intervals in the three spiker categories? To address this question. we generated plots of the relevant membrane currents in all three categories, using the same randomly selected models as in Fig. 4. These currents are shown in Fig. 5.
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5 ms after the spike peak (or less) in both fast and slow narrow spikers such that they do not contribute significantly to the steeper depolarization of fast spikers after the AHP. While the larger inward currents in fast spikers support the slightly slower repolarization before and the faster depolarization after the AHP, the contribution of the potassium currents IKd and IA to fast spiking is less obvious. Both currents are again larger on average in fast (green) than in slow (orange) narrow spikers; this somewhat counteracts the effects of the larger inward currents. In the case of IKd the larger current amplitude after the spike (time range: 4–15 ms) in fast spikers is not readily explained by the conductance distributions of fast and slow narrow STG spikers for IKd (which are similar, see Fig. 2A) but is rather due to the faster depolarization of the fast spikers in that time range, which keeps more IKd activated than in slow narrow spikers. The larger IA amplitudes of fast spikers during the spike are also not explained by the conductance distributions for that current because the distribution for fast spikers actually favors lower amplitudes, whereas slow narrow spikers show an approximately equal distribution over the six conductance values (Fig. 2A, row 4). The larger outward currents IKd and IA in fast spikers during and immediately after the spike are, however, more than compensated for by the larger inward currents described above. In summary, fast spiking in STG neurons is mainly supported by higher densities of inward sodium and calcium currents compared with slow narrow spikers.
Compared with both slow narrow (orange) and fast (green) spikers, the slow STG spikers with broad spikes (blue) stand out in Fig. 5 due to their much smaller or even absent INa and IKd currents, their larger total calcium influx due to longer-lasting ICaT and ICaS transients, and above all their larger IKCa and IA currents. With the exception of IA, all these differences are readily explained by the conductance statistics shown in Fig. 2A. The small amplitudes of INa and IKd, which are normally involved in fast action potential depolarization and repolarization, are due to the low conductance densities of these currents in slow, broad STG spikers, and their small amplitudes contribute to the broader spikes in these neurons. Similarly, the larger calcium influx in slow broad spikers is supported by the bias of the conductances for ICaT and (to a lesser extent) ICaS in these neurons toward higher values. Together with preferentially high densities of IKCa conductance, this large calcium influx favors a large IKCa current, which slows the depolarization in preparation for the next spike. The slow dynamics of the slow, broad STG spikers is thus dominated by calcium influx and the IKCa it brings with it, both hallmarks of so-called "calcium-spikers."
Several of the slow, broad STG spikers show a pronounced second INa peak that coincides with the action potential shoulder and is not seen in any other STG or TC spiker category (Fig. 5, top left). Analysis of the corresponding sodium conductance and activation and inactivation waveforms (not shown) shows that this second peak in the sodium current is caused by the broad action potential shoulder in two ways. One, the slow repolarization of the membrane potential over several tens of milliseconds during the shoulder (see blue traces in Fig. 4) causes a small but nonnegligible fraction of the INa conductance to de-inactivate while it is still fully or partially activated, thus allowing for a window current to flow. And two, the same slow repolarization is accompanied by an increase in the sodium driving force. Together, the de-inactivation and increased driving force lead to a larger INa flow toward the end of the shoulder (40–50 ms after the spike peak) compared with the beginning of the shoulder (first few milliseconds after the spike peak) and thus a second peak in INa.
Only the role of IA in slow broad spikers is not immediately obvious because this current shows a slight preference for small conductances in these neurons and yet causes larger IA out flux than in the two other STG spiker groups. The larger IA current is explained by the relatively slow activation and inactivation kinetics of IA compared with IKd. In the absence of significant amounts of IKd in most broad spikers, IA in the early parts and IKCa in the later parts of the spike are largely responsible for action potential repolarization, which contributes to the significantly broader spikes in these neurons.
Taken together, the slow dynamics and broad spikes of this third group of STG spikers are mainly caused by the fact that the fast action potential currents INa and IKd are at least partially replaced by calcium currents and IA and IKCa due to the conductance composition of slow broad spikers and the slower dynamics of IA compared with IKd.
Fast and slow spiking in TC model neurons
DIMENSIONAL STACK. Are the mechanisms underlying fast and slow spiking in TC neurons related to those in STG neurons? To address this, we constructed the dimensional stack of fast and slow TC spikers shown in Fig. 6. The stacking order was arrived at by edginess minimization (see METHODS). Because only 5% of all TC models are tonic spikers, we increased visibility of the spikers in the dimensional stack in two ways. First, we reduced the number of dimensions in the stack from nine to eight by limiting the stack to the subset of the TC database with the Ih conductance fixed at level two. This does not significantly influence the appearance of the stack because the distributions of the fast and slow spiker populations over the levels of all other conductances are almost identical for the entire TC database and for the subset with Ih at level 2 (data not shown). It does, however, increase the size of each pixel sixfold. Second, we further increased the size of the orange and green pixels by dilating them, i.e., by turning each colored pixel into a block of 5 x 5 pixels of that color. This pixel dilation comes at the expense of information derivable from the stack at the lowest levels of horizontal and vertical organization, in this case INaP and IA, but greatly increases spiker visibility overall.
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A additional pattern visible in the stack are the shallow orange "staircases" rising from the left to the right side of the stack. These indicate a positive correlation between the two leak currents, IKleak and INaleak, in slowly spiking TC neurons. Increasing levels of one of the leak currents apparently have to be balanced by increasing levels of the other leak current in order for slow spiking to occur. This interdependence presumably arises because tipping the balance too far in favor of one over the other leak current would tend to silence spiking by pulling the membrane potential toward the respective leak reversal potential.
EXAMPLE TRACES. Voltage and current traces of 10 randomly selected models from the fast and slow TC spiker categories are shown in Fig. 7. The voltage traces again illustrate that the fast and slow TC spikers can clearly be separated based on their peak-to-trough times, which are short in fast and long in slow spikers.
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, that IT contributes to the generation of single-spike activity in TC neurons, whether fast or slow. In contrast, the potassium currents IK2, IA, and IC show dramatic differences between the two groups. IK2 and IA are much larger and longer lasting in slow than in fast spikers, which in both cases is favored by the distributions of the fast and slow spikers over the different conductance levels of these currents (see Fig. 2B). Consistent with the proposition that IA may have considerable effects on the rising phase of TC neuron spikes (Huguenard and McCormick 1992), its amplitude is greatest during the early phase and voltage peak of both fast and slow TC spikes. The higher IK2 conductances in slow TC spikers, combined with the larger action potential amplitudes and slow kinetics of IK2, lead to substantial IK2 flow in slow spikers during repolarization and even after the spike has largely repolarized (later than 5–10 ms after the spike peak). In keeping with a prediction in Huguenard and McCormick (1992) and its confirmation in McCormick and Huguenard (1992), IK2 therefore contributes to the repolarization of slow TC spikers and offsets large parts of the depolarizing calcium currents IL and IT that continue to flow in both fast and slow spikers after spike repolarization. The net membrane current available to depolarize the neuron leading up to the next spike is therefore smaller in slow TC compared with fast TC spikers, which leads to their longer interspike intervals and helps make them slow spikers.
Consistent with the conclusions from Fig. 6, the traces for IC are systematically different in fast compared with slow TC spikers, suggesting that IC may play a role in setting spike frequency. The substantially larger IC currents in fast compared with slow TC spikers likely contribute to the difference in spike frequency by repolarizing fast spikers more quickly, thus shortening their spike duration. Furthermore, the fact that IC is the dominant potassium current during the spike in fast, but not in slow, TC spikers reproduces findings described in (Huguenard and McCormick 1992; Jahnsen and Llinas 1984b; McCormick and Huguenard 1992), again confirming that the fast spiker subpopulation more closely matches biological TC neurons than the slow population. However, it is not obvious from Fig. 2B what causes these larger IC amplitudes in fast spikers compared with slow spikers because the IC conductance distribution of slow spikers appears bimodal with maxima both at the high and at the low end of the IC conductance spectrum.
In summary, the longer spike period of slow TC spikers is supported by high densities of IK2 channels in these spikers, which due to the slow kinetics of IK2 leads to a long interspike interval. A further contribution to the frequency difference comes from the faster repolarization of fast spikers supported by high IC amplitudes. A role of INa in setting spike frequency, as suggested by Fig. 6, is not obvious in the current traces in Fig. 7.
Dynamic variable ranges of STG and TC spikers
Our analysis thus far shows three categories of STG spikers and two categories of TC spikers that are clearly distinct based on their spike frequencies, spike shapes, conductance composition, and membrane current involvement. This suggests that they may rely on substantially different spike mechanisms. Such different spike mechanism are likely to be reflected in different limit cycle shapes in the high-dimensional space of the neuron's dynamic variables, i.e., the periodic orbit in dynamic variable space traversed by the model once every spike cycle. In both the STG and the TC model, the dynamic variables are the membrane potential, the intracellular calcium concentration, and the activation variables m and inactivation variables h of all voltage-dependent membrane currents for a total of 13 dynamic variables in the STG model and 16 dynamic variables in the TC model. As a crude proxy for the shape of the high-dimensional limit cycle, we asked whether the ranges over which the dynamic variables vary (i.e., the maximum minus the minimum value of each dynamic variable during the limit cycle) are different for different spiker categories. Figure 8 shows three projections of the 13-dimensional space of dynamic variable ranges of the STG model neuron, and one projection of the 16-dimensional dynamic variable range space of the TC model neuron (bottom right). The STG projections were chosen to illustrate that the limit cycles of fast, slow narrow, and slow broad STG spikers indeed differ fundamentally. For example, the panel at the top left shows that the ICaT inactivation variable and the IA activation variable change by 
0.3 in all slow broad (blue) STG spikers. In contrast, both variables never change by >0.15 during an oscillation cycle of a fast (green) spiker. The limit cycle of slow broad spikers thus has much larger extent in the h(ICaT) and m(IA) plane than the limit cycle of fast spikers. Taken together, the STG panels in Fig. 8 illustrate that all three STG spiker subpopulations can easily be distinguished according to their limit cycles, supporting the notion that have different spike dynamics.
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Frequency-current relationships of STG and TC model neurons
The electrical identity of a neuron is characterized not only by its spontaneous activity in the absence of inputs but also by its behavior in response to inputs. To determine whether the fast and slow subpopulations of STG and TC tonically spiking neurons also differ in their response properties, we simulated the injection of ascending and descending current steps into a the same randomly selected subsets of fast and slow STG and TC model spikers use in Figs. 4, 5, and 7. Ascending and descending sequences of current steps were chosen as the stimulus for frequency-current (f-I) relationships to simultaneously test the model neurons for bistability because bistable behavior had previously been anecdotally observed in STG as well as TC model neurons (data not shown). The range of DC currents injected and the increment from step to step was chosen separately for each spiker category based on preliminary simulations. Ranges and increments were (in nA): –3–5 in steps of 1 for slow STG spikers, –10–2 in steps of 1 for fast STG spikers, and –2.5 to 2.5 in steps of 0.5 for both slow and fast TC spikers.
An example current injection simulation and f-I curves for all spiker categories are shown in Fig. 9. Fast versus slow STG and fast versus slow TC spikers clearly differed in their f-I curves and bistable properties, confirming again that these subpopulations are distinct and rely on different spike mechanisms and dynamics. The coarse-grained f-I curves of both slow narrow and slow broad STG spikers (Fig. 9B) at first glance appear to exhibit type 1 behavior as defined by Hodgkin (1948), meaning that they sensitively respond to depolarizing current injections of increasing amplitude with increasing spike frequency and that their f-I curve seems to show arbitrarily low firing frequencies. However, additional high-resolution f-I simulations around the transition point from silence to tonic spiking (Fig. 9B, inset) revealed the existence of a small step from silence to a finite minimal spike frequency in all tested f-I curves. This finite minimal spike frequency indicates that the slow STG spikers are in fact of type 2 as defined by Hodgkin rather than type 1. To determine what type of bifurcation is associated with the transition from silence to spiking in slow STG spikers, we additionally performed bifurcation analysis using the phase-plane analysis tool XPP-AUT (http://www.math.pitt.edu/bard/xpp/xpp.html). We found that in all example spikers tested, the transition occurred through a Hopf bifurcation, consistent with type 2 excitability in these neuron models. A representative bifurcation diagram for the slow STG spiker shown in cyan in Fig. 9B is included in the supplementary materials. With the current step protocol used here, no evidence of bistability was found in slow STG spikers.
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), i.e., they have a discontinuity between silence and tonic spiking with a lowest possible spike frequency significantly different from zero, and their spike frequency is relatively insensitive to the level of injection current above that discontinuity. Bifurcation analysis showed that all tested fast STG spikers transition exhibit a Hopf bifurcation as the injection current is increased and have type 2 excitability. An example of a fast STG spiker bifurcation diagram is included in the supplementary materials. Furthermore, fast STG spikers exhibit clear bistability, as evident from the loops in the f-I relationships in Fig. 9D and from the corresponding bifurcation diagram. As shown in the example trace in Fig. 9A, fast STG spikers switch from silence to spiking once a certain current threshold is crossed in the ascending part of the current steps but continue spiking beyond that threshold on the descending part, such that there is a range of injection currents for which the model can be either silent or tonically active, depending on its history. This kind of bistability is common in type 2 spiking neurons (Rinzel and Ermentrout 1998). Slow TC spikers are similar to slow STG spikers in that their f-I relationships also appear to be type 1 when simulated with 0.5 nA increments (Fig. 9C) but are revealed to in fact be type 2 with smaller current increments (Fig. 9C, inset). The presence of a Hopf bifurcation responsible for the silence-to-spiking transition was again confirmed using bifurcation analysis for all 10 example spikers, and a representative bifurcation diagram is included in the supplementary materials. Slow TC spikers also, like slow STG spikers, do not show the type of silent-versus-spiking bistability just described for fast STG spikers, although some slow TC spikers (4 of 10 in Fig. 9C) show a different type of bistability between tonic spiking and depolarization block for large depolarizing injection currents. However, because depolarizations large enough to push a living TC neuron into depolarization block seem physiologically unrealistic, we propose that for all practical purposes slow TC spikers show the same response properties as slow STG spikers, at least in a physiologically reasonable range of depolarizing input.
The final category of spiking models, fast TC spikers, are the most difficult to categorize in terms of their response properties. Like neurons from the other three categories, fast TC spikers are silent for a range of hyperpolarizing injection currents and spike tonically with increasing spike frequency for increasing depolarizing injection currents. The f-I curves of fast TC spiking models shown in Fig. 9E are similar to those reported for rodent TC neurons (Jahnsen and Llinas 1984a) in terms of current and frequency ranges observed and in terms of the near linearity of their f-I dependence in the range where tonic spiking occurs, again confirming that the fast TC spiker population described here closely matches the properties of biological TC neurons. However, for intermediate injection currents at the transition between silence and tonic spiking these models show bursting rather than spiking activity (thus the missing data points around –1 nA in Fig. 9E) and can therefore not be classified as type 1 or type 2 based on their f-I relationships or bifurcation analysis. Interestingly, this tendency to transition into bursting on injection of hyperpolarizing current is consistent with the converse transition, from bursting to tonic spiking, that was described in the preceding text as the TC cellular correlate of the transition from slow wave sleep to wakefulness and that can be reproduced in vitro by reducing the IKleak conductance (McCormick and Huguenard 1992). In essence, injection of hyperpolarizing current amounts to the opposite manipulation of decreasing the conductance for IKleak, and it is thus not surprising that it causes an opposite transition.
In addition, the randomly selected fast TC spikers sampled here show two types of bistability. A subset of them show a silent-versus-spiking bistability similar to the one found in fast STG spikers, while another subset shows a spiking-versus-spiking bistability in which the model can spike at two different frequencies for the same injection current (see Fig. 9E, inset).
To summarize the results from current injection simulations, both f-I relationships and bistable properties of STG and TC model neurons confirm the reality of the distinct fast and slow subpopulations previously identified on the basis of spontaneous spike frequency (Fig. 1). Compared between STG and TC model neurons, the slow subpopulations seem to be of the same type based on their response properties and lack of bistability in the physiologically relevant current injection range. The parallels between fast STG and fast TC spikers are less clear, although both categories show the same silent-versus-spiking bistability.
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DISCUSSION |
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The fact that fast and slow spiking neurons of both types, STG and TC, differ substantially in the role of several of their membrane currents in spike generation implies that a transition from slow to fast spiking cannot be attributed to any single model parameter changing but rather is a result of multiple parameters changing simultaneously. This means that in complex dynamical systems such as these neurons, transitions between different types of behavior—in this case different spike mechanisms—cannot necessarily always be interpreted in a satisfactory manner with dynamical systems tools such as bifurcation analysis.
While the fast STG and fast TC spiker subpopulations cover almost exactly the same spike frequency range, the slow STG spikers are confined to frequencies below 10 Hz while the slow TC spikers range in spike frequency from a few Hertz up to 30 Hz, including a substantial number of spikers <10 Hz. All the observations we have made about slow TC spikers also apply to those slow TC spikers with frequencies <10 Hz, so we are convinced that, despite the differences in overall spike frequency range between slow STG and slow TC spikers, a comparison between the two is meaningful.
A separation of TC spiking models into distinct subpopulations based on action potential frequency and shape has, to our knowledge, not been previously observed. The two subpopulations of TC spikers show some overlap in their frequency distributions (Fig. 1, right) and in some of their dynamic variable ranges (Fig. 8D) but are clearly separable based on the peak-to-trough times of their voltage traces (Fig. 7 and supplemental materials) and their distinct f-I curves (Fig. 9). While our results do not completely rule out the possibility that the fast and slow TC subpopulations are parts of a continuum of tonically spiking models, it therefore seems unlikely that they are. Regardless our main result of different involvement of related current types in fast and slow spiking in STG versus TC neurons holds whether the fast and slow TC spikers are completely distinct or part of a continuum.
Taken together, several of our findings suggest that the faster TC spiker subpopulation more closely resembles biological TC neurons than the slower subpopulation. For example, a comparison of the spike frequency ranges of the fast and slow TC subpopulations with those experimentally observed in rodent thalamocortical neurons (Huguenard and McCormick 1992; McCormick and Huguenard 1992) show that the fast TC model spikers match the TC neuron behavior observed in vivo better than the slow ones. Similarly, the involvement of the potassium currents IC, IA, and IK2 during and between action potentials in fast TC spikers fits that described in (Huguenard and McCormick 1992; McCormick and Huguenard 1992). A further correspondence of the fast TC spiker population with properties of TC neurons in vivo relates to the role of IKleak in the transition from rhythmic burst firing to tonic activity associated with the transition from slow wave sleep to waking or to rapid-eye-movement sleep (McCormick and Huguenard 1992). Decreasing IKleak in TC neurons in vitro through application of acetylcholine, norepinephrine, or histamine (McCormick 1992; McCormick and Prince 1988; McCormick and Williamson 1991) or through injection of depolarizing current (Jahnsen and Llinas 1984a) reproduces that transition, and correspondingly the fast spikers in the TC database tend to have little or no IKleak conductance (Fig. 2B), whereas similar models with higher IKleak conductance are more likely to be bursting (data not shown), presumably because the overall membrane hyperpolarization that results from higher levels of IKleak conductance activates larger currents through the low-voltage gated IT conductance and thus promotes bursting.
The fact that crustacean STG and rodent TC neurons can achieve very similar spike frequency distributions by using related currents in very different ways suggests that the basic biological mechanisms underlying ion conduction across membranes allow for more than one—and potentially very many— "solutions" for a given neuronal function, such as tonic spiking in a certain frequency range. This allows for considerable flexibility because it means that perturbations in one molecular component involved in electrical signaling, for example through mutation, can be compensated for by adjusting the properties of other components. It is tempting to speculate that in evolution different species that require similar functions from a subset of their neurons may have found different possible solutions in this manner.
The present study uses single-compartment models for both the STG and the TC neurons. In vivo both types of neurons have extensive arborizations, such that treating them as single electrical compartments amounts to a significant simplification. We propose that describing the neurons with multi-compartment models and realistic morphologies instead would probably even increase the flexibility and number of "solutions" described in the preceding text because it would potentially allow for perturbations to one part of the cell to be compensated for by other parts.
The notion of multiple solutions for a given function is a recurring theme in complex biological systems. We have previously demonstrated similar mechanistic flexibility at the level of single neurons with varying mixtures of membrane conductances (Prinz et al. 2003) and at the level of small networks of neurons (Prinz et al. 2004). At the level of single neurons, functionally equivalent electrical activity patterns can be achieved with different sets of maximal conductances of membrane currents the dynamics and voltage dependences of which are otherwise identical (Prinz et al. 2003). Within such a set of solutions that generate the same behavior, individual maximal conductances can vary several-fold, and similar variability in the maximal conductances of the same identified neurons in multiple animals has also been observed experimentally (Goldman et al. 2001; Golowasch et al. 1999, 2002). Similarly, at the small network level similar and functional network activity patterns can result from many different combinations of cellular properties and synaptic strengths (Prinz et al. 2004). These previous studies left the exact voltage dependences and gating dynamics of the underlying currents unchanged and therefore can be thought of as examining the possibility of similar neuron or network output from different molecular components within a species. The present study in a sense generalizes this theme by extending the analysis to a comparison between species of similar cellular behavior with different roles for related currents.
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GRANTS |
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ACKNOWLEDGMENTS |
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FOOTNOTES |
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1 The online version of this article contains supplemental data. 
Address for reprint requests and other correspondence: A. A. Prinz, Emory University, Dept. of Biology, O. Wayne Rollins Research Center, 1510 Clifton Rd., Atlanta, GA 30322 (E-mail: astrid.prinz{at}emory.edu)
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ABSTRACT
INTRODUCTION
, approximate location of example neurons in frequency histograms. Peak-to-trough times for TC example traces are indicated.
