The capabilities and limitations of conductance-based compartmental neuron models with reduced branched or unbranched morphologies and active dendrites

The passive properties of a neuron model strongly influence its responses in the presence of active conductances (Rall et al. 1992). Therefore, we compared the passive responses between the full model and our branched or unbranched reduced models with different numbers of compartments (Fig. 2). Both 0 Hz (DC) and 1,000 Hz current injections were applied because the passive response properties of a neuron model depend on the frequency of inputs (Johnston and Wu 1995). Despite the use of the same soma and axon compartments for all models as well as the preservation of certain dendritic morphological properties (see Methods), passive responses to somatic current injection were affected by dendritic properties due to axial current flow and varied by model. Mismatches between the somatic input resistance (R_IN) of the reduced and full models ranged from 1% for the 98comp unbranched model to 23% for the 5comp unbranched model (Fig. 2(a)). In contrast, the 1,000 Hz input impedance (Z_IN) was a more challenging measure for reduced models with few compartments: the mismatches between reduced and full model somatic Z_IN ranged from only 2% for the 98comp unbranched model to 67% for the 5comp unbranched model (Fig. 2(d)). The quality of the match was greater for reduced models with mean compartment lengths which were more similar to that of the full model, whether or not the branching structure was preserved (Table 1, Fig. 2(a)). The increased R_IN and Z_IN of the models with the largest trunk compartments near the soma (5comp unbranched and 41comp branched models) is explained by the decreased axial current that can exit the soma because of the high axial resistance (R_A) of the first dendritic compartment, which lumps together the R_A of several compartments in the full model. Therefore the dendritic trunk is charged less and the soma more by somatic current injection. This effect is larger for 1,000 Hz input to the soma (Fig. 2(d)), and thus fast somatic currents such as those during an action potential (AP) can be expected to create larger somatic voltage deflections with a different time course in models with high R_A values in the initial trunk compartments. This dependence of fast transients on all axial currents in a neuron should be kept in mind, for example when examining the interaction between ion channel kinetics and morphology to produce smooth or kinky action potential onsets, an issue that has recently gathered much interest (McCormick et al. 2007; Naundorf et al. 2006).

Next we examined the reproduction of somatic responses to dendritic input in our reduced models. We found that the match to the median passive somatic response amplitude of the full model to dendritic current injections in all compartments (one by one) ranged from 2% for the 93comp branched model to 14% for the 5comp unbranched model (Fig. 2(b)). In contrast, with 1,000 Hz injection, we found that median response amplitude of the closest matching branched reduced model (93comp) was 15% smaller than that of the full model while the closest matching unbranched reduced model’s median response amplitude was 174% larger than that of the full model (Fig. 2(e)). This higher median response amplitude is due to the fact that many fewer compartments are located at distal positions in the unbranched models.

In contrast to the small differences between the somatic passive responses of the full, branched and unbranched reduced models to dendritic input, the local dendritic responses diverged between branched and unbranched model reductions. Indeed, mismatches between the median local dendritic DC R_IN of the branched reduced and full models ranged from only 7% to 12% while mismatches for the unbranched reduced models ranged from 40% to 52% (Fig. 2(c)). For the more challenging measure of median local dendritic Z_IN with 1,000 Hz input, the mismatches between the branched reduced and full models ranged from only 10% to 41% while unbranched reduced model mismatches ranged from 71% to 93% (Fig. 2(f)). The unbranched reduced models possessed lumped branches which were thicker than those of the full or branched reduced models and therefore exhibited smaller local R_INs and Z_INs because their dendritic compartments possessed much smaller median R_A relative to surface area (Table 1). Due to this unavoidable characteristic of unbranched models, current flowed away from the injection site more easily and did not charge the membrane locally as much as in the full or branched reduced models. This difference between branched and unbranched models could be quite important for modeling studies of excitatory postsynaptic potential (EPSP) amplification, for instance, because the amount of amplification depends non-linearly on the dendritic voltage reached (Golding and Spruston 1998).

Despite matching the median dendritic R_IN and Z_IN of the full model, the branched reduced models lacked dendritic R_INs or Z_INs as large as some present in the full model. Indeed, the full model possessed maximum dendritic DC R_INs and 1,000 Hz Z_INs which were respectively about twice and four times as large as any possessed by a branched reduced model (box plot whiskers in Fig. 2(c,f)). Furthermore, the full model possessed 105 compartments which had larger 1,000 Hz Z_INs than any exhibited by a branched reduced model compartment. The unbranched reduced models showed even less variability in local dendritic R_INs and Z_INs, due to their lack of small side branches.

In conclusion, the unbranched reduced models matched the full model’s passive somatic responses but failed to match its dendritic R_INs and Z_INs. In contrast, the branched reduced models matched both the somatic and median dendritic passive responses of the full model to current injection. However, the branched reduced models lacked a population of dendritic R_INs or Z_INs as large as many present in the full model. These similarities and differences in passive properties between the reduced and full models provide much of the basis for emerging similarities and differences in the active models.

The detailed spike shape is a frequently studied electrophysiological feature (Bean 2007) which can depend on a neuron’s morphology (McCormick et al. 2007). It is therefore an interesting question whether reduced computer models should be expected to faithfully reproduce this feature of the experimental data. The results shown above suggest that purely passive differences in axial current flow between full and reduced models could confound studies of how active conductances determine the detailed spike shape. Unfortunately, isolating differences in spike shape due to axial current flow in the presence of voltage-gated conductances is complicated due to the non-linear interactions between the membrane voltage trajectory and conductance activation. The injection of a fixed ‘mock AP current’ into the passive models allowed us to determine which differences in the active AP should be expected based on passive properties alone. For a mock AP, we used a square current pulse injection with 5 nA amplitude and 0.5 ms duration, which produced a voltage spike with an amplitude and rise time similar to a Na⁺ action potential. We found that the mock AP voltage deflection was within 1% of the full model’s value for the 98 comp unbranched model; however, as the average electrotonic size of compartments increased, the closeness of reduced model voltage deflections to that of the full model decreased (Table 1, Fig. 3(a)). Indeed, the mock AP voltage deflection showed a steeper onset and was 55% larger in the 5comp unbranched model than in the full model due to the reduced axial currents flowing into the high Ra dendritic trunk of the 5comp model (Fig. 3(b)). For the same reason, the mock action current led to a much decreased voltage deflection in the dendrites of this model (Fig. 3(c)). Note that the same factors contributed to dendritic mock AP size differences between the 14comp unbranched model and full model, and that even the branched reduced models showed smaller dendritic mock AP sizes due to larger Ra values of the trunk dendritic compartments. Overall, our results with mock AP current injection show that large dendritic trunk compartments can lead to differences in AP size. Additionally, decreased passive back propagation of the AP could influence the activation of dendritic voltage gated currents and therefore could further exacerbate differences in integrative properties for the most reduced models. Nevertheless, both branched reduced models and unbranched models with >10 compartments generally exhibited close matches to full model somatic mock AP size.

After isolating the influence of reduced model architectures on voltage transients for fixed current injection pulses, we added the full complement of ion channels back in to compare Na⁺ AP shapes between the full and reduced models. This set of simulations allowed us to determine the influence of the described passive mismatches on voltage-gated channel activation, which could further exacerbate spike shape mismatches.

Using the same active conductance parameter set that was shown to reproduce physiological GP neuron activity in the full model (see Methods) for all models, the active full and reduced models each exhibited tonic, regular spontaneous spiking (Fig. 4(a)). Spikes were always initiated in the axon compartment due to its high NaF conductance. Despite identical conductance densities, spike shapes differed somewhat depending on each model’s morphology. As predicted by the mock AP comparisons, the somatic spikes of the 5comp model were taller than those of the full model (by 7.9 mV). Spikes in the other reduced models were also taller than those in the full model, but not by as much, and they followed the same pattern as for the height of the mock AP (Fig. 4(b) left inset). It should be kept in mind, however, that the amount of variability in spike height between the full and reduced models was not large relative to the variability seen between different GP cells in vitro (standard deviation = 10.8 mV) (Gunay et al. 2008). Note that AP height depended on the average dendritic compartment length rather than on the preservation of the detailed dendritic branching structure (see Table 1). Spike height differences were due to variations in the amount of axial current leaving the soma during a spike (Fig. 4(d), left inset): models with less axial current exiting the soma during a spike (Fig. 4(d), compare positive peaks) possessed taller action potentials (Fig. 4(b), left inset) because more current remained in the soma to charge the somatic membrane. Models with less axial current flow also exhibited slightly more hyperpolarized spike initiation thresholds (Fig. 4(c), left inset) because less positive driving current was able to exit the soma into the dendrites during spike onset. The medium dendritic gNa level in our default full model was too low to propagate active dendritic APs, but a parameter set with high dendritic gNa allowing active dendritic spiking showed quite similar results for the dependence of spontaneous spike shape on model reduction (Supplemental Fig. S1, Online Resource 1), indicating that these findings generalize to spiking and non-spiking dendrites.

A direct consequence of smaller axial current flows into the dendrites during a mock AP was less depolarization of the dendrites (Fig. 3(c), passive case). Less depolarized proximal dendrites contributed less positive axial current back to the soma immediately following the spike during the fast afterhyperpolarization (fAHP) (Fig. 4(d), right inset). Indeed, fAHPs were deeper in models with smaller axial current flows back into the soma immediately following a spike (e.g. the 5comp model, Fig. 4(b) right inset, Fig. 4(d) inset). In contrast to the small spike height differences, the difference between the full and 5comp model fAHPs was 17.3 mV, about six times larger than the experimental variability for this measure (Gunay et al. 2008). Deeper fAHPs in the 5comp model lasted for many milliseconds following a spike (Fig. 4(c), right inset). In sum, axial current flow was a key determinant of spike shape differences: the reduced models that exhibited axial current flows most similar to those of the full model (e.g. the 50 and 98comp unbranched models) also provided the closest matches to the full model’s spike shape as measured by the mean absolute error (MAE, defined as the mean point by point absolute value difference between reduced and full model traces) (Fig. 4(e)). Branched model MAEs were large relative to those of the unbranched models with similar numbers of compartments. This was due to the fact that branched reduced model trunk dendritic compartments were still electrotonically longer than those of the unbranched models since a large number of compartments was necessary simply to reflect the entire branching structure of the full model (see Table 1). Indeed, we observed a strong positive correlation (r = 0.97, Pearson correlation coefficient) between spike shape MAE and mean compartment electrotonic length (Fig. 4(e), inset).

We considered the possibility that manipulation of conductance densities in the 5comp model could allow a closer match to the full model’s spike shape. The full model’s spike was shorter (by 7.9 mV) and its fAHP shallower (by 17.3 mV) than that of the 5comp model. By hand-tuning the 5comp model’s conductance densities, these mismatches in spike height and fAHP depth respectively could be decreased by 97% and 77%. This level of error reduction was achieved through a 4-fold decrease in the somatic and axonal NaF conductance densities and a 16-fold decrease in Kv2, Kv3, and Kv4fast/slow. While our hand-tuning allowed a much closer match of spike height and fAHP depth, these conductance density changes also caused the spike width at −20 mV to increase from 0.53 ms to 0.88 ms (the full model spike width was 0.58 ms). Because spike width was a key determinant of calcium entry through HVA channels in the model, increasing the spike width lead to an increase in SK channel activation that secondarily altered multiple model properties. Overall, hand-tuning conductance densities in reduced models away from the full model parameter set always produced trade-offs in our experience, and it was not possible to fully compensate for mismatches incurred by morphological differences. In addition, the channel densities that made the 5comp model spike shapes most similar to electrophysiological recordings were several-fold different from the original channel densities in the full morphology, and thus the biological interpretation of such highly reduced models with channel densities adjusted to match physiological recordings would need to take this fact into consideration.

A neuron model’s bAP amplitudes have important consequences for the modeling of spike time dependent synaptic plasticity (Caporale and Dan 2008). If a reduced model’s bAP amplitudes were appreciably different from those in a full model, coincident bAPs and EPSPs in the dendrites would fail to evoke similar local responses. To examine the effect of model reduction on bAP amplitudes during fast spiking activity as may be expected for GP neurons in vivo, 75 Hz spiking was elicited in each model by somatic DC injection. We normalized bAP amplitudes to somatic AP height so that models with taller somatic spikes did not mistakenly appear to have more effective back propagation. In most dendritic compartments of the full and reduced models, bAP amplitudes decayed at about the same rate with electrotonic distance from the soma (Fig. 5(a)). However, in the most highly reduced models even the trunk dendritic compartments were at a considerable electrotonic distance from the soma (Fig. 5(a)), and therefore the initial decrease in bAP amplitude was much more pronounced. A further difference between the reduced and full models was that variability in bAP amplitude for different dendritic branches at a given electrotonic distance from the soma was much larger in the full model. In particular, one full model branch had a greatly increased bAP amplitude (Fig. 5(a ₁), black arrow). Indeed, the bAP amplitude at the end of this branch (L ≈ 0.5) was 49% of the somatic AP height, while the largest bAP amplitudes observed at this electrotonic position in any branched or unbranched reduced models were respectively only 13% or 16% of somatic AP height. This large bAP disappeared in the full model without dendritic gNa (Supplemental Fig. S2A , Online Resource 1), which meant that the large amplitude was due to local amplification by gNa. Thus even when the branching structure of the full morphology was maintained, morphological reduction still resulted in significant functional changes in the presence of active conductances.

The primary cause of bAPs lies in the propagation of axial current. The peak axial current during a bAP quickly decayed with electrotonic distance at about the same rate in the full and reduced models (Fig. 5(b)) such that only a small fraction of the initial axial current at the soma boundary even reached intermediate dendritic positions at 0.5 L (Fig. 5(b ₁)). The peak axial current reaching distal locations (e.g. 1.0 L) was quite variable between dendritic branches (Fig. 5(c)) though the variability was diminished by either removing dendritic gNa or setting it to a high density that supported active spike back propagation (Supplemental Fig. S2, Online Resource 1). However, the peak axial current reaching any given compartment did not determine the local bAP amplitude. Instead, the amplitude was primarily determined by the local Z_IN at each location. For example, in the full model at 0.54 L, the branch with the largest bAP amplitude (1,000 Hz Z_IN of 304 MΩ) received only 35% of the axial current received by the branch with the smallest bAP (1,000 Hz Z_IN of 76 MΩ). Such a large spread in 1,000 Hz Z_IN was not observed in any of the reduced models, explaining the smaller spread of bAP amplitudes (Fig. 5(a)). While the branched reduced models possessed the same branches as in the full model, their 1,000 Hz Z_INs were smaller due to their larger compartment sizes, leading to the observed mismatches in bAP amplitudes. These differences between the full model, branched reduced models and unbranched reduced models should be considered when using modeling to study bAPs or employing them to study spike dependent plasticity rules.

The somatic fI curve is a basic neuronal input-output function which is frequently recorded by cellular electrophysiologists due to the important information that it yields about neuronal excitability. Possible mismatches in the somatic fI curve due to model reduction could result in consequential changes with respect to excitability in the presence of synaptic input. To assess the extent of such mismatches, we compared somatic fI curves between the full model and the branched or unbranched reduced models (Fig. 6(a)). All models exhibited spontaneous spiking which could be eliminated by injecting between −31 pA and −41 pA of DC into the soma. For positive current injection amplitudes between 0 pA and 500 pA, the full and reduced models exhibited nearly linear increases in spike rate. However, the somatic fI curve was steeper in some of the reduced models than in the full model. This slope mismatch was related to differences in somatic R_IN between the full and reduced models (compare Figs. 2(a) with 6(a)). Models with larger somatic R_IN allowed more somatically-injected current to charge the somatic and axonal membranes instead of flowing into the dendrites. This difference was most pronounced for the 5comp unbranched model, since this model had by far the highest somatic R_IN (Fig. 2(a)). Overall, the differences in spike rate between the full and reduced models were small (e.g. spontaneous rate varied by only 2.4 Hz) relative to the variability that is observed between different GP cells recorded in vitro (standard deviation for spontaneous rate = 5.9 Hz) (Gunay et al. 2008). Therefore our data indicate that somatic fI curve mismatches are not one of the important limitations of reduced models, with the possible exception of highly reduced models with severely limited axial currents such as our 5comp model.

We again considered the possibility that the somatic fI curve of the 5comp model could be improved relative to that of the full model by hand-tuning its conductance densities. Indeed, by decreasing axonal NaP by 12.5% and increasing somatic SK by 50%, the 5comp model’s somatic fI curve closely matched that of the full model. However, these manipulations did not appreciably decrease the discrepancy between the 5comp model’s spike rate responses to dendritic current injection relative to those of the full model: this discrepancy will be addressed in the next section.

While the somatic fI curve is an important measure of the excitability of a neuron, it does not take into account input processing taking place in active dendrites. Active dendrites are capable of many types of input processing, depending on their precise morphology and their complement of ion channels (London and Hausser 2005). As a first step towards characterizing the influence of model reduction on dendritic processing, we recorded the spike frequency with different dendritic DC injection amplitudes at a relatively distal location (~0.8 λ) in each model. As shown in Fig. 6(b), the dendritic fI curves were much steeper in the unbranched reduced models than in the full or branched reduced models. To test whether mismatches in dendritic fI curves depended on the particular dendritic location chosen, we injected 200 pA of DC into every dendritic compartment of each model and compared the distribution of spike frequencies between models (Fig. 6(c)). For all dendritic compartments, the median spike frequencies of the unbranched reduced models were about twice as large as those in the full or branched reduced models (Fig. 6(c)). In addition, the minimum spike frequency was much larger in the unbranched reduced models than in the full or branched reduced models (Fig. 6(c)). However, this difference was sensitive to the dendritic gNa level: with high dendritic gNa, the full model responded to dendritic current injection with high spike frequencies that matched the unbranched models, although the spread of spike frequency responses remained much higher in the full model (Supplemental Fig. S4C, Online Resource 1). Overall, these results showed that differences between full and reduced models in their dendritic fI curves were quite large, and furthermore depended on the complement of ion channels.

Next, we sought to understand more fully the relationship between dendritic current injection and somatic spike rate. To examine the possibility that different spike frequency responses were simply related to the electrotonic distance of each injected compartment to the soma, we compared the spike rate response during 200 pA DC dendritic injection to the electrotonic position of each injected dendritic compartment. Perhaps surprisingly, there was considerable variability between spike rate responses to current injections at matching electrotonic positions in different branches of each model (Fig. 7(a)). Because electrotonic position need not be directly correlated to local R_IN, we hypothesized that local dendritic R_IN could be a stronger predictor of spike frequency responses. Indeed, we found a very close relationship between high values of local dendritic R_IN at the site of current injection and low values of somatic spike frequency regardless of model reduction level (Fig. 7(b)). Since the unbranched models lacked compartments with high local R_INs due to their collapsed branching structures (Fig. 2(c)), their minimum spike frequency responses to dendritic current injections were higher (Fig. 6(c)).

To determine whether dendritic conductances were responsible for the differences in spike rates with dendritic DC injection, we removed all dendritic conductances and measured spike rates in response to the same stimuli used for Fig. 6(c). We found that passive dendrites allowed close matches between all models regarding spike rate responses to dendritic DC injections (Fig. 6(d)). The spike rates achieved with current injection into passive dendrites were much higher than those with active dendrites, indicating that activation of outward currents during dendritic depolarization primarily determined slow spike responses. Because dendritic depolarization with input is directly dependent on the local R_IN, the unavoidable reduction of local dendritic R_IN when multiple dendritic branches are collapsed in unbranched models leads to significant mismatches in the response to input when active conductances are present. These mismatches can lead to either decreased or increased spiking based on the relative predominance of inward or outward dendritic currents.

The integration of distributed synaptic inputs into spiking output is one of the fundamental computational functions of a neuron (Destexhe et al. 2003; Gulledge et al. 2005). Accurate synaptic integration relative to a full model or biological neuron is particularly important for a reduced model that will be incorporated into network simulations. A first order approximation of the input/output function of a neuron model is frequently represented by fF curves, which compare spike frequency vs. input frequency at different mixes of inhibition and excitation. We generated fF curves for all models by recording the spike frequencies for increasing rates of asynchronous excitatory input to all dendritic compartments at three levels of inhibition. We applied identical synaptic conductance activation patterns between all models (see Methods). We found that the fF curves of all reduced models matched the full model more closely than the fI curves for dendritic current injection had suggested (Fig. 8(a), compare to Fig. 6(b,c)). Nevertheless, the unbranched models showed a somewhat larger spike frequency increase with increasing excitatory input rates as suggested by the dendritic fI curves, whereas the 59comp and 93comp branched reduced models presented a close approximation to the fF curves of the full model. As we observed with dendritic fI curves, the degree to which the fF curves matched between the models was sensitive to the dendritic gNa level. With high dendritic gNa, no reduced model was able to match the steep fF curves of the full model due to differences in dendritic spike initiation and propagation (Supplemental Fig. S5, Online Resource 1).

To better understand the fF curve mismatches of the unbranched models with medium dendritic gNa, we studied the somatic and dendritic voltage fluctuations of the 93comp, 98comp and full models after removing their somatic and axonal active conductances (Fig. 8(b–d)). The somatic voltages were shifted relative to one another in a manner consistent with the relative firing rates in the fF curves. However, the magnitude of somatic voltage fluctuations was quite similar between the full and reduced models (Fig. 8(b)). In contrast, the dendritic voltage fluctuations in the unbranched 98comp model were much smaller than in the full or branched 93comp model at each level of inhibition (Fig. 8(c,d); note the close similarity between the local fluctuations of the full and 93comp branched models). We hypothesized that these different dendritic voltage fluctuations influenced dendritic conductances in a complex manner and prevented the 98comp model from precisely matching the fF curves of the full model. To test this, we removed all dendritic conductances from each model and reproduced the fF curves. As we predicted, the fF curves of the full and reduced models without dendritic conductances were much more similar to each other than in the case with dendritic conductances (Fig. 8(e)).

While synaptic rate coding is used for many neural computations, some circuits such as those involved with processing transient sensory inputs must use the information contained in the timing of the first spike (VanRullen et al. 2005). In network models of circuits employing precise spike timing as a coding mechanism, the exact firing times of the component neuron models therefore become important. If the spike times of a reduced branched or unbranched model differ from those of a biological neuron or full model, the degree of spike time preservation should be understood before a particular morphological reduction method is used to construct neurons for such networks. Therefore, we evaluated the preservation of precise full model spike times by branched or unbranched reduced models in the presence of identical synaptic inputs.

To study the precision of spike timing with synaptic inputs, we constructed a set of asynchronous distributed inputs where each of the 511 excitatory synapses was randomly activated with a mean rate of 5 Hz, and each of 511 inhibitory synapses with a mean rate of 10 Hz. After carefully examining the spike time preservation of each reduced model at each point on the fF curves shown in Fig. 8(a), we found that this combination of excitation and inhibition displayed a degree of spike time preservation that was representative of the range of synaptic backgrounds that we studied (e.g. Fig 8(a)). Therefore, we present the specifics of spike time preservation by the reduced models with this one synaptic background to avoid introducing redundant detail. Because input synchronization is generally thought to be an important aspect in neural coding and occurs naturally for example due to oscillatory activity (Ward 2003), we also constructed a synaptic stimulus in which clusters of synapses were activated synchronously (see Methods). Spike time preservation was defined as the percentage of the full model’s spikes that were generated by the reduced model within ±5 ms of the full model’s spike time, a method we previously employed in dynamic clamp studies to examine spiking precision (Gauck and Jaeger 2000). In our previous dynamic clamp study, we showed that about 60% of precise spike times were preserved within ±5 ms when repeatedly applying the same conductance activation pattern to neurons in slice. We consider this degree of preservation to be the limit of spike time accuracy due to intrinsic neural noise and thus any reduced models matching this level of precision present sufficient replication of full model activity.

We found that preservation of ±5 ms spike timing did not vary widely between the branched and unbranched reduced models (Fig. 9(a _1–3 )). Preservation was generally about 50% with asynchronous excitation (Fig. 9(a _1–3 )), and from 50% to 75% with synchronous excitation (Fig. 9(b _1–3 )), for branched or unbranched reduced models. Coincident excitatory inputs increased spike time preservation because powerful events were more likely to overcome small differences in activity between the models to drive spikes at a precise time. With spike time preservation, as with the other functional measures analyzed in this study, the dendritic gNa level was an important determinant of how well the reduced models matched the full model. Preservation of spike timing was greatly diminished by high dendritic gNa whether inputs were asynchronous or synchronous. With 10 Hz inhibition and 5 Hz excitation, spike time preservation for the high-dendritic-gNa models was often less than 20% due to differences in dendritic spike propagation and initiation with the full model (data not shown). Conversely, removing dendritic gNa altogether caused spike time preservation to improve by 5% relative to the case with medium dendritic gNa for the 93comp model, suggesting that dendritic sodium was amplifying coincident EPSPs differently in the different models (data not shown). Removing the remaining dendritic conductances did not further improve spike time preservation for the 93comp branched model, most likely because close somatic voltage fluctuation matches were observed even with all dendritic conductances present (e.g. Fig. 8(b)).

To understand why the 93comp model did not preserve 100% of precise full model spike times despite the lack of noise, identical synaptic activation patterns, and direct mapping of synaptic positions, we compared the somatic voltage responses of both models under two conditions. In, the first condition, somatic and axonal conductances were removed from both models, leaving just dendritic conductances (“nonspiking” in Fig. 9(c)). We observed a close match between the somatic responses of the full and 93comp models in the nonspiking case (Fig. 9(c), dashed traces). In the second condition, both models possessed the full complement of somatic, axonal, and dendritic conductances (“spiking” in Fig. 9(c)). In the spiking models, one model occasionally fired a spike while the other model did not quite reach threshold (e.g. Fig. 9(c), black arrow). If a spike-triggering synaptic event occurred for both models immediately after this event, then the model that just fired a spike did not fire again, while the other one did (e.g. Fig. 9(c), blue arrow). This kind of sequence tended to cause spikes in the subsequent few hundred milliseconds to be misaligned between the two models and accounted for most of the unpreserved spike times. Indeed, due to the fact that near-threshold events can lead to spike initiation in one condition but fail to cause spike initiation in a closely parallel condition, a fully precise control of spike timing by synaptic input may be impossible.

Overall, in the absence of dendritic spiking, the degree of precise full model spike time preservation by the reduced models was similar to the spike time preservation that we have observed between multiple applications of identical conductance activation patterns to slice neurons (Gauck and Jaeger 2000).

Researchers using full models commonly face the problem of hand-tuning conductance densities to match target output: this is always a laborious task and dissatisfying in that only a single—possibly arbitrary—solution is found. Therefore, there has been great interest in finding algorithms to optimize the conductance densities of a model automatically (Huys et al. 2006; Vanier and Bower 1999) and to determine the parameter space of similar solutions with database approaches (Gunay et al. 2008; Prinz et al. 2003; Prinz et al. 2004). However, even with such algorithms it is still demanding to find a suitable parameter set for a full model because of the long simulation times required to run thousands of parameter combinations in such models. This time requirement can be drastically decreased by applying the search algorithm to a reduced model, as long as the resulting conductance densities can be used in the full model to produce similar output.

To examine which reduction strategy could be used to efficiently search the parameter space of the full model, 100 random parameter sets were generated by varying each conductance density independently between 25% and 400% of its default value. These parameter sets produced widely varying spike shapes as well as somatic and dendritic fI curves (Fig. 10). Of these parameter sets, two that yielded different somatic and dendritic fI curves and spike shapes were chosen to illustrate a comparison of the output of the full and reduced models (Fig. 10, ‘RandSet1’ (colored solid lines) and ‘RandSet2’ (colored dashed lines); for conductance densities see Supplemental Table S3, Online Resource 1). This comparison showed that the distinct differences in the model output between the two parameter sets were generally preserved in both branched and unbranched models (Fig. 10). For instance, the somatic fI curves and spike shapes were closely preserved when these two parameter sets were used in the full, 41comp branched and 14comp unbranched models (Fig. 10(a,b)). Note that even the 5 ms ISI doublet firing of the full model with RandSet1 (colored solid lines) was preserved by both reduced models (Fig. 10(a)). In contrast, the dendritic fI curves were closely preserved by the 41comp branched model but not by the 14comp unbranched model, which generally fired faster with the same current injection input (Fig. 10(c)).

To quantify the ability of the reduced and full models to produce similar output with the same densities across all 100 parameter sets, we calculated the MAE between the full and reduced model output for each measure (6 rate points for each fI curve and 801 voltage points for each spike shape). This yielded a set of 100 MAEs for each measure in each reduced model. We observed that increasing the number of compartments provided significantly smaller somatic spike shape and somatic fI MAEs (p < 0.05 with Bonferroni correction using the Mann-Whitney U test) for both branched and unbranched reduced models in most cases (Fig. 10(a ₄ ,b ₄ )) see Supplemental Table S4 for all statistical comparisons, Online Resource 1). In contrast, increasing the number of compartments while keeping the reduced model type fixed significantly decreased the median dendritic fI MAE in only one instance (from 59comp to 93comp branched reduced, Supplemental Table S4, Online Resource 1). Instead of a compartment number relationship, we observed that preserved branching structure allowed much smaller MAEs for the branched reduced models than for the unbranched reduced models regarding the dendritic fI curve. For example, the median dendritic fI curve MAE exhibited by the 93comp branched reduced model was significantly (p = 1.9e-11, Mann-Whitney U test) smaller by 4.9 Hz than that exhibited by the 98comp unbranched reduced model, despite having fewer compartments (Supplemental Table S4, Online Resource 1). The differing ability of the branched and unbranched reduced models to reproduce full model dendritic responses with identical conductance densities was very similar to the differing ability of the two types of reduced models to match the dendritic responses of the full model with the tuned parameter set (Fig. 6). The similar deviations between models for these 100 randomly generated parameter sets support the general validity of our analysis tracing these differences back to axial current and dendritic R_IN mismatches.

Overall, we found that identical conductance densities allowed our unbranched reduced models to produce somatic input output dynamics similar to those of the full model, while the branched reduced models additionally reproduced the dendritic input driven somatic output dynamics of the full model.