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

Abstract

Conductance-based neuron models are frequently employed to study the dynamics of biological neural networks. For speed and ease of use, these models are often reduced in morphological complexity. Simplified dendritic branching structures may process inputs differently than full branching structures, however, and could thereby fail to reproduce important aspects of biological neural processing. It is not yet well understood which processing capabilities require detailed branching structures. Therefore, we analyzed the processing capabilities of full or partially branched reduced models. These models were created by collapsing the dendritic tree of a full morphological model of a globus pallidus (GP) neuron while preserving its total surface area and electrotonic length, as well as its passive and active parameters. Dendritic trees were either collapsed into single cables (unbranched models) or the full complement of branch points was preserved (branched models). Both reduction strategies allowed us to compare dynamics between all models using the same channel density settings. Full model responses to somatic inputs were generally preserved by both types of reduced model while dendritic input responses could be more closely preserved by branched than unbranched reduced models. However, features strongly influenced by local dendritic input resistance, such as active dendritic sodium spike generation and propagation, could not be accurately reproduced by any reduced model. Based on our analyses, we suggest that there are intrinsic differences in processing capabilities between unbranched and branched models. We also indicate suitable applications for different levels of reduction, including fast searches of full model parameter space.

Fig. 1

Methods of morphological reduction. To aid visualization, dendritic diameters are multiplied by 5 while soma diameters are multiplied by 2. Mean and maximum compartment electrotonic lengths (L) are provided for each model to facilitate comparisons of structure across model types. (a) The full GP model (green) has 513 compartments: these include 511 dendritic compartments, an axon compartment, and a soma. (b) To create the simplest ‘branched’ reduced model (41comp, orange), the soma and axon compartment were unaltered while each group of compartments in the full model’s dendritic tree not containing any branch points was collapsed into one dendritic compartment with the same total surface area and total electrotonic length (see Methods). The dendrites of the 41comp model were then divided lengthwise into equal parts until no dendritic compartment in the model was longer than 0.2 L (59comp, magenta) or 0.1 L (93comp, red). (c) To create the first ‘unbranched’ reduced model (5comp, cyan), each major branch of the full model’s dendritic tree was collapsed into a single dendritic compartment which preserved the total surface area and average soma-to-tip electrotonic distance for that major branch (see Methods). To create the 14comp (turquoise), 50comp (purple), and 98comp (blue) models, each dendritic compartment of the 5comp model was divided lengthwise into 4, 16, and 32 pieces, respectively. All conductance densities and passive parameters were preserved in each reduced model. The color scheme used in this figure was consistently applied to all remaining figures

Fig. 2

Passive somatic responses to DC or 1,000 Hz current injection into the soma or individual dendritic compartments. For panels C, E and F, a log scale was used for the y-axis to facilitate comparisons between models. (a) Voltage deflections (top, selected models) and R_IN (bottom, all models) are plotted for 50 pA current injections into the soma. R_IN was calculated by dividing the steady state voltage change by the DC amplitude. Note the slight increase in R_IN for the most reduced branched or unbranched models. (b) Each dendritic compartment was separately injected with 50 pA of DC. The box and whisker plots show the amplitudes of the resulting somatic voltage deflections: the middle horizontal bar represents the median while the upper and lower horizontal bars represent the 75th and 25th percentiles, respectively; the whiskers identify the most extreme values found. Box and whisker plots, bar graphs, and raster plots present the models in the same order consistently throughout the paper. Note that the median somatic voltage change for injection into different dendritic compartments was comparable for all models, but that branched reduced models showed slightly larger variability in somatic responses than the full model, whereas unbranched reduced models showed slightly lower variability. (c) Distributions of local dendritic R_IN values for each model. The median and quartile values but not whisker extents were similar between the branched reduced models and the full model, whereas the unbranched reduced models had lower median dendritic R_IN and showed much less variability between compartments. (d–f) Same plots as (a–c), except the injected current was a 1,000 Hz sinusoidal input with a peak-to-peak (P2P) amplitude of 50 pA. Input impedance (Z_IN) was calculated as P2P voltage deflection divided by 50 pA

Fig. 3

The passive somatic response amplitude to a mock action current depended on dendritic axial resistance. The stimulus was a brief (0.5 ms) large (5 nA) somatic current pulse which caused a voltage response in the passive soma that mimicked the amplitude and rise-time of an AP in the absence of voltage gated conductances (mock AP). (a) The mock AP was smallest in the full model and the most subdivided (50comp and 98comp) unbranched reduced models. The ability of the reduced models to match the full model mock AP response improved as compartments were divided. (b) Models with more finely-divided compartments allowed larger axial currents to exit the soma into the dendrites during the mock AP. (c) Models with larger axial currents exiting the soma exhibited larger average voltage deflections in their most proximal dendritic compartments (those directly connected to the soma)

Fig. 4

Shape analysis of spontaneous somatic APs in fully active models. Models at all levels of reduction possessed identical somatic, axonal and dendritic conductance densities. (a) Spontaneous spiking is shown for the full and reduced models. (b) The relative spike heights in the active models (left inset) corresponded directly to the relative mock AP response amplitudes shown in Fig. 3(a). Models with shorter spike heights possessed shallower fast afterhyperpolarizations (fAHPs) as well (right inset). (c) Expanded time scale shows steeper rise time to spike onset in the most reduced models (left inset) and extended differences in fAHP potential (right inset). (d) Axial current between soma and dendritic trunk compartments. The spike depolarization led to a large positive current into the dendrites; somatic spike size was increased in models with less axial current due to high Ra values. During the somatic fAHP, the axial current reversed, and current flowing back into the soma from the depolarized dendrites led to a decrease in fAHP amplitude. Therefore fAHP was largest in the most reduced models because they had the highest Ra values into the dendritic trunk compartments directly connected to the soma. (e) The quality of the match to the full model spontaneous spike shape shown in (b) was calculated for each model as the MAE of the voltage traces from 2 ms before to 6 ms after the spike peak. To calculate MAE, spikes were aligned so that they crossed 0 mV at the same time. MAE was highly correlated with mean compartment electrotonic length (‘mean comp L’; r = 0.97, Pearson correlation coefficient; see inset)

Fig. 5

Back propagating AP (bAP) amplitudes can be much larger in the full model than in the reduced models due to larger local dendritic high frequency Z_INs. All models were driven to fire at 75 Hz + −0.1% with DC somatic injection. (a) The bAP amplitude in each compartment was normalized to the soma spike amplitude and plotted against the electrotonic distance from the soma for the full model (a ₁), branched models (a ₂) and unbranched models (a ₃). Most models showed a similar decay in bAP amplitude with electrotonic distance. However, the 5comp and 14comp unbranched models showed much smaller bAP sizes in their proximal dendrites because these compartments were already electrotonically quite distant from the soma. In addition, the full model showed more variable bAP amplitudes in different branches (see black arrow) than even the most detailed branched reduced models. (b) The maximum amount of axial current flowing into each compartment during a spike (referred to simply as axial current) is plotted against electrotonic distance from the soma. (c) Log scale representation of the plots shown in (b). Note that the decay of axial current with electrotonic distance is approximately log-linear and is quite similar between the full and reduced models

Fig. 6

The spike frequency response to somatic and dendritic DC injection (fI curve) at different levels of model reduction. DC injection amplitudes ranged from −100 pA to +500 pA to elicit the full physiological range of spike rates. (a) fI curve of each model for somatic current injections. Models with higher somatic R_INs (see Fig. 2(a)) exhibited steeper somatic fI curves. Note that subdividing the compartments of the reduced models allowed closer matches to the somatic fI curve of the full model. (b) fI curve of each model for DC injection into a sample dendritic compartment at an electrotonic distance of 0.83 L from the soma. The injected compartment was positioned on the same sub-branch in the full and branched reduced models and on the same major branch in the unbranched reduced models. Note that the branched reduced models closely matched the dendritic fI curve of the full model while the unbranched reduced models did not provide close matches. (c) Box plots for distributions of spike frequency responses for 200 pA DC injections into each dendritic compartment of each model. The median spike frequency response to dendritic injection was much higher in the unbranched reduced models than in either the full model or the branched reduced models. The minimum spike frequency response to dendritic injection was also much higher in the unbranched reduced models than in either the full model or the branched reduced models. (d) Same as (c) except that all channels were removed from the dendrites. Note the much closer match between all models and the strong correspondence to the relationships shown in Fig. 2(b)

Fig. 7

R_IN was a stronger predictor than electrotonic position of the spike rate response to dendritic injection. The R_IN of a dendritic compartment was calculated as for Fig. 2. (a) The spike rate response during 200 pA DC injection into each dendritic compartment depended weakly on electrotonic distance from soma (L). (b) In contrast, the spike rate response during 200 pA DC injection into a dendritic compartment depended strongly on the injected compartment’s R_IN

Fig. 8

The spike frequency (f) responses of the full model to different event frequencies (F) of synaptic input (fF curves) were well matched by the divided branched reduced models but less well matched by the other reduced models. (a) With asynchronous excitation, the divided branched reduced models consistently provided the closest match to the fF curves of the full model (59comp model responses not shown, but similar to those of the 93comp model). Note that the fF curves of the unbranched models were generally steeper than those of the full model, but that the mismatch depended on the precise combination of excitation and inhibition. ‘No’, ‘Medium’, and ‘High’ inhibition respectively refer to average rates for each inhibitory synapse of 0, 10, and 20 Hz. Each fF curve was constructed from 100 s simulations for each data point. (b–d), somatic and axonal conductances were removed to prevent spiking. Event frequencies circle in panels a _1–3 were used for panels (b – d). (b) Short somatic voltage traces are shown in each type of model. Note that the full and branched somatic voltages were practically identical while the unbranched 98comp model exhibited slightly different fluctuations with no or medium inhibition. Further note the correspondence of these somatic voltage fluctuations to the relative firing rates of each model. (c) Short voltage traces are shown from an example dendritic compartment at approximately the same electrotonic position and branching structure (if possible) in each type of model. The full and branched dendritic compartments experienced larger voltage fluctuations than did the compartment in the unbranched model. Note the close correspondence of the full and 93comp branched reduced model’s dendritic voltage fluctuations. (d) The probability that a particular voltage was reached by any dendritic compartment at any time during synaptic input is plotted for each model. The ranges of voltages reached by the dendrites of the full and branched models were larger than the range of voltages reached by the dendrites of the unbranched model. (e) After removing all active dendritic conductances, the fF curves were reproduced. Note the much closer fF curve matches, particularly between the 98comp unbranched reduced model and the full model

Fig. 9

Precise full model spike times were often preserved by the reduced models in the presence of synaptic input. (a) Spike time preservation during asynchronous synaptic bombardment. A ₁ , Somatic voltage traces for the full model and 93comp branched reduced model are shown along with the time-varying combined synaptic reversal potential (Vsyn). As we have done previously (Gauck and Jaeger 2000), Vsyn was calculated by multiplying the reversal potential of each synaptic conductance type (AMPA, NMDA, and GABA) by the momentary whole cell synaptic conductance for that type and then dividing by the total synaptic conductance for all three types. V_syn was shifted down by 20 mV relative to voltage traces for easier visualization. A ₂, Raster plot showing the spike times for each model during the simulation segment in A ₁. A ₃ , The percent of preserved spikes was generally about 50% for all the reduced models. (b) Same as (a), but with synchronous excitation. Note the improved preservation percentages for each reduced model. Further note that none of the reduced models preserved more than about 75% of full model spike times. (c) Somatic voltage traces are shown in the full and 93comp models during a short period when precise full model spike times were not preserved. The dashed traces represent the somatic voltages of the models without somatic or axonal conductances (nonspiking, shifted down by 20 mV to aid visualization), while the solid traces are from the active models. Note the extremely close match of the nonspiking somatic voltage fluctuations, the point when the full model spikes but the 93comp model does not (black arrow), and the 93comp spike which follows soon after (blue arrow). Synaptic input distributions were identical to those used for Fig. 8. Medium excitation (5 Hz) and medium inhibition (10 Hz) were used for all panels of this figure. Preservation of full model spike times was calculated 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. The colors used in the raster plots in this figure were the same as those used for the rest of the figures

Fig. 10

Different model activity patterns generated by many random conductance density sets were matched between the full and reduced models. One hundred conductance density parameter sets were randomly generated by varying each conductance density between 25% and 400% of its original value. Two of these parameter sets that produced very different model outputs were selected to illustrate the responses of the different models with identical conductance densities (colored solid or dashed lines). The remaining 98 random parameter sets are displayed as solid light gray lines. Model spike shapes were compared during 500 pA somatic DC injection because this level of current injection caused every morphology to spike with each random parameter set tested. (a ₁ ,b ₁ ,c ₁), Each random parameter set was inserted into the full model. For each parameter set, the somatic spike shape (a ₁), somatic fI curve (b ₁) and dendritic fI curve (c ₁) are displayed. Note the wide range of spike shapes and fI curve gains. (a ₂ ,b ₂ ,c ₂), Same as the first column but for the 41comp branched morphology. (a ₃ ,b ₃ ,c ₃), Same as the first two columns but for the 14comp unbranched morphology. (a ₄ ,b ₄ ,c ₄), Box plots of the distribution of MAEs calculated for the 100 random parameter sets are shown for each reduced model. Note that dendritic fI curve MAEs were smaller for the branched reduced models than for the unbranched models