Model properties

It has been suggested that IN firing under physiological conditions is predominantly controlled by synaptic input to the proximal dendrites [39]. In our simulations, IN responses (i.e., in the membrane potential, ) were instead evoked with somatic current injections, as this enabled us to compare our results with experimental literature using current-clamp recordings [15], [43], [45]. Experimentally, it has been found that activation of 1-4 synapses can evoke somatic excitatory post-synaptic current (EPSC) amplitudes between 30 and 600 pA in INs [43]. In all simulations presented in the following, the stimulus amplitude was within this physiologically realistic range. The IN model was able to qualitatively reproduce the two major types of response characteristics that have been observed in the IN, namely, 1) regular AP-firing, characterized by a steady stream of action potentials and 2) burst firing, characterized by bursts of APs that ride atop Ca²⁺-spikes [15], [23], [43], [45]–[47].

A prolonged current injection with a relatively low amplitude () evoked an initial phase of rapid AP-firing, followed by slower, regular AP-firing (Fig. 2A). The response to the prolonged stimulus resembled that observed experimentally [45]–[47]. Single APs successfully backpropagated into the distal dendrites of the IN (Fig. 3A), as has also been seen in experiments [43], [44]. A brief current injection with high amplitude () evoked a pronounced burst of APs (Fig. 2B). The response to the brief stimulus resembled that observed experimentally in [43]. The burst rode atop a Ca²⁺-spike mediated by T-channels [15], [23], [45]. The Ca²⁺-spike could be seen in isolation when AP-firing was suppressed by setting the Na⁺ conductances to zero (Fig. 2B, dashed lines). The Ca²⁺-spikes successfully invaded the dendritic arbor of the INs (Fig. 3B), as in [43].

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Figure 2. Response to somatic current injection.

(A) A prolonged (1000 ms) stimulus protocol, (A-iv), evoked an initial high AP-firing frequency phase, followed by a series of regularly spaced APs. (B) A brief (10 ms) stimulus protocol, (B-iv), evoked a burst of APs. When AP firing was suppressed by setting the Na⁺-conductance to 0, the Ca²⁺-spike underlying the burst was revealed (dashed lines). The somatic response for three distributions are shown: the soma distribution (A-i, B-i), the uniform distribution (A-ii, B-ii), and the distal distribution (A-iii, B-iii). (C) Number of regular APs elicited during the last 700 ms of the stimulus period as a function of prolonged stimulus amplitude () for all T-distributions. The regular AP-firing rate was facilitated by having T-channels in the proximity of the soma. (D) Number of APs elicited in bursts as a function of brief stimulus amplitude () for all T-distributions. The bursting propensity was facilitated by having T-channels in the proximity of the soma.

https://doi.org/10.1371/journal.pone.0107780.g002

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Figure 3. Regular APs and Ca²⁺-spikes invade distal dendrites.

Propagation illustrated here in branch 1 for the soma (red), the uniform (blue) and the distal (magenta) distribution. (A) Backpropagating, regular APs shown at the soma (point of origin) and at selected locations along a single dendritic branch. Trains of regular APs were generated by a prolonged stimulus protocol ( for 1000 ms) to the soma, and showed close-ups of the last AP in the train. (B) Ca²⁺-spikes shown at the soma and at selected locations along a single dendritic branch for selected distributions. Ca²⁺-spikes were evoked by a brief stimulus protocol ( for 10 ms), and with Na⁺-conductances set to 0 to suppress AP firing. Curves were graded from dark colours (close to the soma) to light colours (far from the soma).

https://doi.org/10.1371/journal.pone.0107780.g003

A broad range of stimulus amplitudes and different T-distributions produced qualitatively similar responses to those shown in the representative examples in Figs.2 and 3. These response patterns and their dependence on the T-distribution are further analyzed below.

Regular APs

Regular AP-firing (Fig. 2A) was evoked using a prolonged stimulus protocol, i.e., a 1000 ms current injection (Fig. 2A-iv). The initial, irregular high-AP-frequency phase of the response typically lasted between 100 and 200 ms, and was due to the activation of T-channels. This initial high frequency phase was ignored in the subsequent study where we focus on the regular AP-firing frequency (R1) and the dendritic backpropagation of a single, regular AP (R2). The same current injection was used in all simulations of AP-propagation to evoke regular AP-firing.

Successful AP-backpropagation does not depend on T-distribution.

In the following simulations, the same current injection was used in all simulations of AP-propagation to evoke regular AP-firing. We studied the effect of the T-distribution on the backpropagation of a single, regular AP (R2). In order to circumvent any distortion effect that might be related to the initial Ca²⁺-spike, we focused on the last AP in the series of regular APs (obtained by using the prolonged stimulus protocol) and observed its propagation along different dendritic branches.

Fig. 3A shows, in selected segments along a single branch, how the amplitude and shape of the (last) single AP evolved as it (back-) propagated from the soma towards the distal dendrites. The somatic AP-shapes obtained with , and (Fig. 3A(i–iii)) were as good as identical. AP-backpropagation was also very similar in the different distributions. In all cases, the AP experienced a broadening and a decrease in amplitude upon its propagation away from the soma.

Fig. 4 summarizes AP-backpropagation in three different dendritic branches, and for all T-distributions. Fig. 4(A–C) shows the AP-amplitude () as a function of distance from the soma in three different branches (Fig. 1A). The panels (A–C) in Fig. 4 represent the different branches, while the six curves in each panel represent different T-distributions. The exact evolution of the AP amplitude will depend on complex morphological features of the branching dendrites. We do not explore this in further detail here, but focus solely on how AP-propagation is influenced by T-distribution.

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Figure 4. Effect of T-distribution on AP backpropagation.

(A–C) In each segment of a selected branch, the peak amplitude of last action potential (Fig. 2A) (left panel) and (D–F) the time taken from soma to reach peak amplitude (right panel) are plotted as a function of distance from soma (peak time, , was plotted relative to the time of the somatic AP-peak ()). While the distribution does not have a significant effect on peak amplitude of last potential (A–C), it does affect the speed of propagation as shown in D–F. The signal travels faster in distributions where the density of channels increases in distal dendrites.

https://doi.org/10.1371/journal.pone.0107780.g004

The exact shape of the attenuation profile depends on branch-specific morphological features of the interneuron (see Fig. S3 for a detailed explanation). Despite morphological variation between the branches (compare panels A–C), AP propagation followed the same general trend. In the proximal dendrites, the AP amplitude decreased monotonically with distance from soma, from about 26.7 mV in the soma to between −10 mV and −15 mV at a distance of around from the soma. At intermediate distances , varied between −15 and 0 mV. In the distal dendrites , the AP amplitude increased with distance from soma, reaching an amplitude between 8 and 16 mV at the end point.

A comparison between -curves for each individual branch (A–C) shows that the T-distribution had a relatively small impact on . Also, there was no trend in how the T-distributions influenced the local . For example, gave the lowest at the endpoint of one branch (Fig. 4A), and the highest at the endpoint of another branch (Fig. 4C). We conclude that the T-distribution had no clear effect on the shape and amplitude of backpropagating APs.

Bursts and Ca²⁺-spikes

In this section, we study burst firing, evoked here with a brief (10 ms) current injection in the soma (Fig. 2B-iv). During the bursts, the AP amplitude varied, a phenomenon observed both experimentally and in previous modelling studies [15], [45]. We here focus on the effect of the T-distribution on the burst (R3), the magnitude of the Ca²⁺-spike, and the way that the Ca²⁺-spike is conveyed to distal dendrites (R4). For a quantitative measure of the effect of the T-distribution on the burst for a given stimulus, we use the number of APs riding the crest of the low-threshold Ca²⁺-spike (Fig. 2B).

Effect of T-distribution on somatic Ca²⁺-spikes.

It is known that Ca²⁺-spikes can trigger dendritic GABA-release in INs, even when AP-firing is suppressed by the Na⁺-channel blocker TTX [43]. GABA-release triggered by Ca²⁺-spikes has been found to have a longer duration compared to the (axonal and possibly dendritic) GABA-release triggered by single APs [43]. This implies that the Ca²⁺-spike has an important functional role beyond that of mediating bursts of APs, and motivated us to conduct a further study of the Ca²⁺-spike in isolation.

In the following simulations (Fig. 5), the Na⁺ conductance was therefore set to zero so as to suppress the APs in the burst. Brief current injections then evoked Ca²⁺-spikes. In order to gauge the effect of the T-channels on the response, we compared the somatic voltage response of the different model distributions with the response obtained when no T-channels were included in the model (referred to as the null-distribution, black line).

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Figure 5. Ca²⁺-spikes in the soma for all T-distributions.

(A–C) Ca²⁺-spikes, evoked by 10 ms somatic current injections are illustrated here for stimulus amplitudes, , of 100, 150 and 250 pA. AP firing was suppressed by setting the Na⁺-conductance to 0. The Ca²⁺-spikes varied significantly between the different T-distributions (differently coloured curves, as indicated by legend in D). (D) Magnitude of Ca²⁺-spikes as a function of the stimulus amplitude for all T-distributions. For technical reasons (a Ca²⁺-spike peak amplitude could not be defined for all cases), the response magnitude refers to the area under the response curve. For reference, a response curve for a null distribution (black curve), a model devoid of T-channels, was included in all panels.

https://doi.org/10.1371/journal.pone.0107780.g005

Simulations were run for different stimulus amplitudes. In all cases (Fig. 5A–C), the brief stimulus caused an initial peak in after 10 ms. After stimulus offset, initially started to decay towards the resting potential (). However, the relatively slow activation of the T-channels gave rise to an inward T-current which interfered with the passive decay. To describe the observed responses, we make a distinction between full and partial Ca²⁺-spikes. When the T-current proceeding from the stimulus was strong enough to evoke a second, post-stimulus depolarization, and thus a second local maxima in at some point after the stimulus offset (), we considered that we had a full Ca²⁺-spike. Full Ca²⁺-spikes were, in all T-distribution models, evoked only for strong stimulus amplitudes (Fig. 5C). For weaker stimulus amplitudes (Fig. 5A–B), only a subset of the distributions elicited full Ca²⁺-spikes. For example, with , only elicited a full Ca²⁺-spike (Fig. 5A). For the other distributions, the T-current merely slowed down the decay of towards the resting potential, compared to what we would expect from a purely passive response. We refer to this kind of T-channel mediated response as partial Ca²⁺-spikes.

For a quantitative measure of the magnitude of the Ca²⁺-spike, we chose to use the area under the response curve. This was partly practically motivated as, e.g., a quantification in terms of the peak amplitude could not be defined in the case of partial Ca²⁺-spikes. However, as we know that Ca²⁺-spikes trigger dendritic actions distinct from that of single APs [43], these actions likely depend on the time-course of the depolarization, and not only the peak amplitude. As the area under the response curve depends on both these aspects, we reasoned that it represented a functionally relevant quantification of the magnitude of the Ca²⁺-spike.

In all model versions, the Ca²⁺-spike area increased with increasing stimulus injection, but approached saturation and increased little when was increased beyond 300 pA (Fig. 5D). Interestingly, the T-distribution that gave the biggest Ca²⁺-spike response, depended on the stimulus injection. For a relatively weak stimulus (), gave the biggest response area. For stimuli at intermediate strength (), and gave the biggest response area. Finally, for strong stimuli (), , and gave the biggest response area. We do not go further into the details of this, but we make use of this analysis below, when we study how the Ca²⁺-spike is conveyed from the soma to distal dendrites.

T-distribution affects the way somatic Ca²⁺-spikes are conveyed to distal dendrites.

Next, we investigated the influence of the T-distribution on how well somatically elicited Ca²⁺-spikes were conveyed to distal dendritic sites (R4). In the case of AP backpropagation, it was clear that the AP originated in the soma, and then propagated towards the distal dendrites (Figs. 3 and 4). The time-aspect of dendritic Ca²⁺-spikes was more complex. As T-channels have much slower kinetics than Na⁺-channels, T-channels over the entire dendritic tree had overlapping activation time, and contributed simultaneously to the generation of Ca²⁺-spikes. In the case of Ca²⁺-spikes, it would therefore not be appropriate to speak of propagation. For instance, for , the Ca²⁺-spike reached its peak value earlier in the distal dendrite than in the soma, although the current injection was applied to the soma (Fig. 3D). We did not explore the time-aspect of Ca²⁺-spikes further, but from now on focus on their magnitude.

In Fig. 6A–C, we have plotted the Ca²⁺-spike magnitude as function of the distance from soma in three different branches. The local magnitude of the Ca²⁺-spikes depended on two factors: (i) the proximity to the injection site, and (ii) the local T-density. The first factor (i) explains the results obtained with (blue line). In that case, the Ca²⁺-spike-magnitude decreased with distance from soma, almost in parallel to how the passive response obtained with the null-distribution decreased with distance (black line). In case of and (red and yellow lines), the Ca²⁺-spike-magnitude decreased more steeply with distance from soma, due to the additional effect (ii) of decreasing T-density. In case of or (purple and green lines), the second factor (ii) dominated over the first (i), and the Ca²⁺-spike response magnitude increased with distance from the soma, giving a maximum value at dendritic endpoints, where the T-density was highest. Finally, in case of (cyan line), the Ca²⁺-spike response magnitude was either highest in the soma (Fig. 6(A,C)), where the stimulus was applied, or around from the soma, where the T-density was highest (Fig. 6B). For distances greater than from the soma, the Ca²⁺-spike-magnitude decreased slowly with distance from soma.

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Figure 6. Ca²⁺-spikes in the dendritic tree.

(A-C) Magnitude of Ca²⁺-spikes as a function of distance to the soma. The Ca²⁺-spike magnitude clearly reflected the underlying T-distribution (different coloured curves, as indicated by legend). Ca²⁺-spikes were evoked by brief (10 ms) current injections with stimulus amplitude (). (D-F) Magnitude of Ca²⁺-spikes at dendritic endpoints as a function of stimulus amplitude . The magnitude was always biggest for T-distributions with a high distal T-density. (A-F) In all cases, AP firing was suppressed by setting the Na⁺-conductance to 0. Response magnitude was defined as the area under the response curve. The different panels in the vertical direction represent three different dendritic branches.

https://doi.org/10.1371/journal.pone.0107780.g006

Next, we explored how the trends identified in Fig. 6A–C depended on the amplitude of the somatic current injection. As we were mainly interested in how Ca²⁺-spikes were conveyed to dendritic endpoints, i.e., the typical locations of triadic synapses, we limited the analysis to explore the Ca²⁺-spike-magnitude at the tips of the three dendritic branches (Fig. 6D–F). The trends that we identified in Fig. 6 A–C were robust for all stimulus injection above a certain threshold (). Below this threshold, the Ca²⁺-spike-magnitudes were quite similar in all model versions, except in , where the Ca²⁺-spike had a high amplitude in the distal dendrites, even for very weak stimuli (). However, we recall from Fig. 5 that, for , not all model versions elicited full Ca²⁺-spikes with well defined peaks in . We therefore regard the results obtained with stimuli above 200 pA as most relevant for our study of Ca²⁺-spikes.

We conclude that, for brief somatic current injections strong enough to evoke full Ca²⁺-spikes in all model versions, the Ca²⁺-spike-magnitude in the tips of the dendrites was facilitated by having a high distal density of T-channels ( and ). Similar results were found when we used another parametrization (P1) of the model (Figs. S5–S7).

Synaptic integration

Finally, we investigated the impact of the T-distribution on synaptic integration (R5). Excitatory postsynaptic potentials (EPSPs) were evoked by a AMPA synapse inserted at the end points in one of the three branches (1, 2 and 3 in Fig. 1A). The synapse was adapted to experimental data for AMPA-synapses on thalamocortical relay neurons [41], and is described in further detail in the Methods section. To isolate effects due to T-channels, the sodium conductance was blocked. However, qualitatively similar results were obtained when the sodium conductance was not blocked (Figs. S1 and S2).

Impact of the T-distribution on synaptic integration.

Synaptic activation evoked a local EPSP at the synaptic site, which eventually resulted in a voltage deflection in the soma. The evolution of the EPSP is illustrated for selected segments along branch 3 (Fig. 7) for the soma-, uniform-, linear-, and distal T-distributions. The T-distribution had a clear effect on the local, dendritic response. When all the T-channels were in the soma, the local dendritic EPSP to the synaptic input had a peak amplitude of about -20 mV (Fig. 7A). A non-zero T-density in the distal dendrite significantly prolonged the duration of the local EPSP-response at the synaptic site (Fig. 7B–D). A similar role for dendritic T-channels has been proposed in an earlier, general computational study [48].

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Figure 7. EPSPs obtained with different T-distributions.

(A-D) EPSPs at different locations along a single dendritic branch (branch 3), as a response to synaptic input () applied to the dendritic endpoint. Different panels represent different T-distributions, as indicated. EPSPs were attenuated upon propagation from the synapse (dark coloured curves) to the soma (light coloured curves). The Na⁺-conductance was set to 0 to suppress the dendritic Na⁺-spikes that would have been evoked in distributions with a high density of T-channels close to the synapse. The figure legends indicate distance from soma.

https://doi.org/10.1371/journal.pone.0107780.g007

The propagation of the EPSP towards the soma is summarized for all six distributions in Figs. 8A–C, which show how the peak amplitude of the signal vary with distance from the soma in 3 different branches. While the duration of the EPSPs (Fig. 7) varied with different T-distributions, their amplitudes at the synaptic sites were similar. The attenuation profiles of the signals as they propagated towards the soma were similar in appearance except for and . In the case of especially, there was a significant boost in the response partly caused by the presence of T-channels along the way. We note here that the signal boost is far more significant in branches 1 and 3 than in branch 2, showing that it is affected not only by the distribution of the T-channels, but also by morphological features. Branch 2 has fewer ramifications (i.e. smaller membrane area) and thus fewer T-channels in the distal regions than branches 1 and 3 where the signal is given an additional boost as a result of backpropagation from nearby dendritic branches. This explains the smaller difference between the linear and the distal distribution for this particular branch. In fact, when we used another parametrization P1 (see Methods), turned out to be more efficient than for this branch (see Fig. S9).

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Figure 8. EPSP amplitudes for different T-distributions.

(A-C) EPSP amplitude () as a function of distance from soma in 3 different branches (each panel represents a distinct branch). In each branch, synaptic input was applied to the dendritic endpoint. For all distributions, EPSP amplitudes decreased upon propagation towards the soma. (D-F) Somatic EPSP amplitude as a function of synaptic weight, . The somatic EPSPs had bigger amplitude for T-distributions with a high density of T-channels close to the synapse. The synaptic weight in (A-C) was . (A-F) Different coloured lines correspond to different T-distributions. Black lines (null distribution) correspond to the case without T-channels.

https://doi.org/10.1371/journal.pone.0107780.g008

Due to the low-pass filtering properties of the dendrites, the most enduring EPSPs experienced the least attenuation during propagation. This is apparent in Fig. 8D–F, which shows EPSP-amplitude in the soma as a function of the synaptic strength. The general trend was that the EPSP amplitude in the soma was higher when the T-density was high in the distal dendrites. This trend was independent of the synaptic strength.

It hence turned out to be more efficient to position the T-channels locally at the synaptic site, thus boosting the EPSP before integration than to have a high T-density in the soma thereby boosting the incoming EPSP response when it arrived.

R1: Somatic, regular AP-firing

It seems unlikely that the T-distribution should be adapted to yield a certain regular AP-firing rate in INs. In our simulations, the increase in firing rate gained from changing between the two extreme T-distributions (from to ) could also be achieved with a relatively small (20%) increase in the stimulus amplitude, suggesting that the AP-firing rate could be controlled by any depolarizing current. Based on general knowledge of neural firing properties, we would expect it to chiefly be controlled by Na⁺- and K⁺-channels [50]. Ca²⁺-dependent afterhyperpolarization currents may also be involved in reducing the IN firing rate [45].

Several other membrane mechanisms (not included in the current model) may also affect the AP-firing rate. For example, L-type Ca²⁺ channels () have been shown to activate through depolarizations positive to -35mV [26], [49]. This suggests that they should predominantly open during AP firing. Ca²⁺ influx through could then provide a source for activation of Ca²⁺-dependent afterhyperpolarization channels (), which could reduce the AP-firing frequency in thalamic neurons [45], [51].

R2: Backpropagation of APs into distal dendrites

We showed that AP-backpropagation could be speeded up with a high T-density in distal dendrites (Fig. 4). To our knowledge, this is the first study that has predicted an impact of T-channels on AP-propagation speed. It is however unlikely that its potential impact on the AP-propagation speed is the key role of the T-distribution. Generally, the most important prerequisite for AP-backpropagation is the dendritic Na⁺-conductance [44], [52]. Other conductances could play a role. For example, INs possess A-type K⁺-channels () [49]. This ion channel type has been found to influence AP-backpropagation in hippocampal neurons [53]. It also remains unclear whether backpropagating APs play a role in mediating IN output [43].

R3: Somatic burst firing

It has been established that T-channels are the main mediators of bursts in INs [8], [15], [21], [45], [49], although contributions from other ion channels have been found. Our simulations showed that, beyond a certain stimulus, the number of APs elicited during a burst tended to saturate towards a maximum value, which depended on the T-distribution (Fig. 2). This highlights the unique role of T-channels in burst generation. The T-distribution is thus likely to have adapted, at least partly, to regulate the somatic bursting pattern, as has been suggested for other thalamic neurons [11], [32].

As in TCs [32], a high T-density in the proximal dendrites resulted in a greater propensity for somatic bursting. However, in INs, the T-distribution may not be optimized solely to enhance the number of APs in a burst. Experimentally recorded IN bursts typically having a lower intraburst AP-frequency and a longer duration than TC bursts [15], [21], [46], [47], are closer in resemblance to thalamic reticular (RE) neurons. In RE neurons, which have been likened to INs [54], prolonged bursts have been explained by a high T-density in distal dendrites [11]. Our simulations also indicate that a reduction in the intraburst AP-firing frequency and a prolonged burst-duration may result from moving T-channels away from the proximal region towards the distal dendrites (Fig. 2). On the other hand, differences in burst shapes between INs and TCs may also be explained by differences in total T-conductance and T-channel kinetics [21]. In fact, characteristics of IN bursts and the numbers of APs therein tend to vary quite significantly between different INs [15], [49]. It is therefore difficult to use a quantitative comparison between the simulated bursts in Fig. 2 and experimentally observed bursts to deduce the most likely T-distribution.

Somatic burst firing may also depend on ion channels not included in this study. Ca²⁺-activated non-selective cation channels (), for example, have been found to prolong the duration of bursts in INs, and in some cases evoke plateau-potentials [15]. As in INs is thought to be activated predominantly by Ca²⁺ entering through T-channels [15], its presence could serve to amplify T-channel mediated effects, and make the functional consequences of the T-distribution even more pronounced. A hyperpolarization-activated non-specific cation channel () has been identified in INs [45], [55]. has been found to contribute to the generation of rebound bursts [45], i.e., bursts that the IN elicit when it is released from holding potentials far more hyperpolarized than the resting potential. However, due to the strong hyperpolarizations required to activate [45], it is unlikely that would be active in the situations that we studied in the current work.

Although most studies have indicated that burst firing in INs is predominantly mediated by T-channels [8], [15], [21], [23], [26], [46], [49], one experiment has observed bursts that could be abolished by the appliance of the L-type Ca²⁺ channels blocker nimodipine [43], suggesting that these Ca²⁺-spikes and bursts were mediated by . Nimodipine has also been found to have a strong impact on dendritic GABA-release [43], [56], and post-synaptic NMDA-responses [57] in INs suggesting that may have several functional roles in INs that may overlap with those of T-channels. The bursts observed in [43] were evoked by brief (10 ms) stimulus injections to the soma, and showed a striking similarity (in terms of amplitude, duration and number of APs) to the bursts that we obtained in our simulations when we used the same stimulus protocol (compare Fig. 2 with Fig. 1 in [43]). In our study, bursts were a robust response feature, generated by T-channels that were modelled with activation kinetics based on experimental data from INs [21], [45]. This raises questions regarding the distinct roles of T-channels and for IN signalling. Bursts mediated by also seem to somehow challenge the high depolarization required for activation in previous experiments [26], [49]. A potential resolution to these apparently conflicting findings could be that nimodipine also interacts with T-channels, and that some of the effects of nimodipine appliance can be ascribed to the suppression of T-channel mediated activity. Experimental evidence that nimodipine may act on T-channels is sparse, but a nimodipine-induced reduction of T-channel activity has been demonstrated in neurons of the lateral dorsal nucleus of the thalamus [58]. We do not know if this is a plausible hypothesis in INs, but rather wish to pose it as an open question to the research community.

R4: Dendritic spread of Ca²⁺-spikes

The Ca²⁺-spikes underlying the bursts, and the manner in which these are conveyed to distant sites in IN dendrites is of particular interest. Dendritic Ca²⁺-spikes have been known to trigger GABA-release from IN dendrites, even when AP-firing was suppressed by the Na⁺-channel blocker TTX [43]. Furthermore, GABA-release has been found to have a longer duration when triggered by Ca²⁺-spikes than when triggered by single APs [43]. The T-distribution could therefore have adapted in a way that guarantees that Ca²⁺-spikes are well conveyed to the distal dendrites. The T-distributions that were found to favour this function did not favour somatic bursting (R3) and vice-versa. On one hand, a high proximal T-density increases the number of APs in the burst (Fig. 2), and thus the axonal IN output. On the other hand, a high distal T-density increases the response magnitude of Ca²⁺-spikes in distal dendrites (Fig. 6), and putatively the dendritic output of INs. More detailed insight into the action of dendritic Ca²⁺-spikes will have to await experimental identification of the detailed mechanisms behind dendritic GABA-release, which are currently unknown. It has been suggested that GABA-release is controlled by a local voltage threshold [59]. A second possibility, that GABA-release rather depend on widespread (i.e., not highly localized to presynaptic terminals), intracellular Ca²⁺-dynamics, has also been discussed [43], [60]. Ca²⁺ entering through T-channels has been shown to be involved in exocytosis in some neurons [18], [19], yet it is unknown whether this is the case in thalamic neurons. Another possible action of dendritic Ca²⁺-spikes could be to provide depolarizations necessary for the relief of the voltage-dependent Mg²⁺-block of NMDA-receptors, which could induce changes in the post-synaptic strength [61], [62].

Note that we have focused primarily on signalling between the soma and most distal dendritic endpoints. However, it is possible that Ca²⁺-spike triggered GABA-release may occur throughout the dendritic tree. If we had looked at Ca²⁺-spikes from more proximal dendritic sites, our conclusions regarding the optimal T-distribution would have been different.

R5: Synaptic integration

The response to distal synaptic input is a complex topic. A high distal T-density was shown to boost and prolong the EPSPs in the soma (Fig. 8), thus enhancing the integration of distal synaptic input (Fig. 7). This has been proposed earlier as a key functional role of distal T-channels in RE neurons [63]. However, because the dendrites in INs are long and leaky, somatic activity of INs could be controlled more efficiently by synapses positioned at more proximal locations [39].

Also, triadic synapses in distal dendrites may release GABA as a response to local, synaptic input from the retina, i.e., without requiring involvement from the soma of INs [36], [39], [41], [43], [57], [59]. In fact, it might even be undesirable to have strong somatic EPSPs evoked by dendritic input to the triads, as the purpose of their distal location may be to ensure a certain electrical decoupling from the soma [40], [57], [59], [64]. The main function of distal, postsynaptic terminals may therefore be to trigger local output rather than to control somatic firing. T-channels could very well play a role in this localized release machinery, as a high distal T-density would increase the duration of the local postsynaptic depolarization (Fig. 7). The possible gain from this would, as we discussed for the Ca²⁺-spikes above, depend on the unknown local condition for dendritic GABA release. It is also possible that there may be multiple trigger mechanisms for local GABA-release, as has been observed, e.g., in granule cells in the olfactory system. There, Ca²⁺ entering through NMDA-receptors may trigger vesicle release in a highly localized fashion, whereas voltage-dependent vesicle release also may occur, but then typically triggered by backpropagating APs [65].

Data on synaptic activation in INs is sparse, but one experiment has shown the somatic EPSC-response to the activation of a single synapse with unknown location (see Fig. 4 in [43]). The experimental EPSC had a duration of about 5 ms, and amplitude of about 50 pA. In additional simulations (results not shown), we found that a distal synapse, no matter how fast and strong, could not produce EPSCs that matched both the amplitude and time course of the experimental EPSCs. This is due to the low-pass filtering properties of the long and thin dendrites, which would tend to broaden the somatic EPSCs. With the synapse model that we used, we found that the experimental EPSCs could only be reproduced if we placed it closer to the soma. For example, with a synaptic weight of 2 nS, and with a uniform T-distribution, we obtained EPSCs that agreed with the experimental recordings (amplitude of 50 pA and duration of about 5ms) when the synapse was placed from the soma (branch 1). The EPSCs in the experiments [43] were thus likely due to more proximal input.

Final remarks

A common conception in biology is that any phenotype, such as the T-distribution, has evolved or been adapted to optimize for a certain biological function. In modelling it is also common to use the reverse argument, i.e., if a theoretical study demonstrates that a specific (hypothetical) T-distribution is optimal for a certain function, it can be considered a prediction that this be the real T-distribution. Our simulations did not converge to any such prediction of the real T-distribution, as we found that different T-distributions were optimal for different IN properties (R1-R5). Since all the properties considered here are of putative importance for different aspects of the INs function within the LGN-circuitry, the real T-distribution is likely to reflect a compromise between the different properties (and thereby between different functions).

There is also a theoretical possibility that the effective T-distribution may vary between INs, or even vary dynamically within a single IN. It is known that some neurons may use activity sensors to dynamically regulate the density of ion channels on their membrane to maintain a stable pattern of activity and to compensate for ongoing, state dependent processes [66]. T-channels modulation by acetylcholine, dopamine and several other transmitter substances has also been demonstrated in several excitable cells [67]. Although response properties of INs are modulated by state dependent input from various regions of the brain [38], [40], [57], [68], there is no experimental evidence that T-channels are subject to dynamic regulation or modulation in INs. However, one could speculate that such regulation or modulation could change the effective distribution of T-channels over time. This could be one potential explanation of the discrepancy between the two experimental studies of the T-distribution in INs [26], [29]. It could also provide a mechanism for the IN to switch between a predominantly dendritic and a predominanly somato-axonal output regime.

The study that we have presented here was motivated by the complex role of IN dendrites within the LGN circuitry. As IN dendrites are long, thin and electrotonically non-compact, and as they serve dual roles as both input and output channels, we hypothesized that the subcellular T-distribution could play a particularly important role in this neuron type. However, although our simulations and motivations were cell specific, we believe that our results shed light on the way the T-distribution affects neural signalling in general.

Simulation

Simulations were run with the NEURON/Python simulating environment [69]. We used a reduced version of the previously developed multicompartmental model of the LGN IN [45], implemented in NEURON [70]. The original model was adapted to current clamp recordings from two different interneurons. It was presented with in two versions (parameterizations P1 and P2), which were able to capture the somewhat different response properties of the two neurons. The reduced model adopted the morphology and the passive properties from the original model [45]. Simulations based on a reduced version of parametrization P2 are presented in the main article. Simulations based on a reduced version of parametrization P1 are found in Figs. S4 - S9. We chose to use P2 for the main presentation. This was because P2 had a more hyperpolarized resting potential () compared to P1 (), and showed the most pronounced effects of T-channel activation. However, qualitatively similar results were obtained with both parameterizations.

Morphology

The morphology used in all simulations was based on a realistic, 3D reconstruction of a mouse IN (Fig. 1A). The model interneuron consisted of a soma and 104 dendritic sections, that were subdivided in smaller segments, resulting in a total of 330 segments. The total surface area of the model IN was ; the summed length of all dendrites was ; the longest dendrite was , and the mean somatodendritic diameter was about .

Passive properties

The axial (cytoplastic) resistivity, , the membrane capacitance, and the membrane resistance, were adopted from the original model [45] for both P1 and P2. The values that were used are indicated in Table 1.

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Table 1. Parameter sets P1 and P2.

https://doi.org/10.1371/journal.pone.0107780.t001

As T-channels have a nonzero activation level around the resting potential, changing the T-distribution could lead to changes in the resting potential. We prevented this from happening by adjusting the reversal potential of the passive leak current () so that the IN always had the same somatic resting potential (-69 mV in the P2 version, and -63 mV in the P1 version).

Ion channels

Of the seven active ion channels in the original model [45], only three were included in the reduced model. These were, in addition to the T-type Ca²⁺-channel, the traditional AP-generating Na⁺- and delayed-rectifier K⁺-channels (Na and K_dr). The conductances of , , and channels were set to 0.

We had several reasons for using a reduced version of the original multicompartmental model [45], excluding some of the ion channels (, , and -channels). Firstly, stripping the model down to the essentials allowed us to investigate the effect of the T-distribution in isolation, and made the results easier to interpret. Secondly, the subcellular ion-channel distribution is poorly known for most ion channels in INs. Adding additional mechanisms would have meant introducing additional unconstrained parameters to the model. It is uncertain whether this would have led to more realistic simulations, especially when it comes to dendritic signalling. Thirdly, the fact that the simplified model was sufficient to generate all the essential response features that we were interested in exploring (Fig. 2), validated our conclusions, at least on a qualitative level.

Na and K_dr-channels

The Na and K_dr-channels were described by the standard Hodgkin-Huxley formalism [71]:(1)(2)

We adopted the Na⁺ and K⁺ reversal potentials ( and ), and the kinetics of the gating variables , and from the original model [45]. However, in the reduced (new) model, we adjusted the Na-activation threshold and the maximum conductances ( and ) to account for recent experimental studies, which reported that (i) somatically generated APs successfully invade distal dendrites of INs, whereas (ii) synaptic input to distal dendrites was never observed to evoke APs that initiated locally in the dendrites [44].

In both P2 and P1, the sodium kinetics which was shifted by -0.2 mV (from -50.3 mV to -50.5 mV) relative to the threshold in the original model. These values were found to (i) ensure successful (back-)propagation of APs into distal dendrites, and (ii) reduce as much as possible the ability of dendritic compartments to evoke local APs.

T-channels

The model of the T-channel was adopted from the original model [45], but is described in detail here due to its high relevance for the current project. It was modelled using the Goldman-Hodgkin-Katz formulation [72]:(3)which describes the T-current () in terms of a maximum permeability for fully open channels ( with units cm/s). Like the conductances ( and ), the permeability essentially reflect the density of the respective ion channel type on the cellular membrane. The function(4)essentially accounts for the dependence of the Ca²⁺-reversal potential on intracellular () and the extracellular () Ca²⁺-concentrations. In Eq. 4, is the valence of the Ca²⁺-ions, is the gas constant, is Faraday's constant and is the absolute temperature. Note that the function (Eq. 4) has units , so that the product has the units () of a current density.

The extracellular Ca²⁺-concentration was assumed to be constant (). The intracellular Ca²⁺-dynamics was modelled as a leaky integrator:(5)with a ), and as in the original model [45].

The rationale behind using a different modelling scheme for T-channels is that intracellular Na⁺ and K⁺-concentrations typically are on the orders of several millimolars, and change little during normal neural activity. In contrast, the intracellular Ca²⁺-concentration is on the order of nanomolars and may experience substantial relative changes following Ca²⁺-influx.

The model of the T-channel was adopted from the original model, and we refer to that work for a description of the kinetics of the gating variables and . Only the distribution of was different from the current work, as described below (note that in the original model [45], was denoted as we mistook it for a conductance).

Synapse model

Synaptic input was modelled using a sum of two exponentials with rise and decay time constants of 0.5 ms and 2 ms, respectively, and with a reversal potential of 10mV. These values have properties that correspond to those found for the AMPA synapse in TC cells [41]. The synaptic weight was varied between 0 and 4 nS.

In the simulations shown in the Results section, the synapse was placed in the distal dendrites. Three different branches were investigated, and synapse locations in branch 1, 2 and 3 had distances , and from the soma, respectively.

Output

In all simulations the model output was the voltage response to a given input. Model output was as specified in the different subsections in the results section and included voltage amplitude, number of action potentials, and calcium spike magnitude. Calcium spikes typically had a duration of 100 - 200 ms and were followed by a small undershoot, i.e., a brief period with . Calcium spike magnitude was calculated as the integral of the function taken from the stimulus onset to the time where the voltage descended below the resting potential.