Contribution of morphology and membrane resistance to integration of fast synaptic signals in two thalamic cell types

Abstract

Thalamocortical cells (TCs) and interneurons (INs) in the lateral geniculate nucleus process visual information from the retina. The TCs have many short dendrites, whereas the INs have fewer and longer dendrites. Because of these morphological differences, it has been suggested that transmission of synaptic signals from dendritic synapses to soma is more efficient in TCs than in INs. However, a higher membrane resistance (R_m) for the INs could, in theory, compensate for the attenuating effect of their long dendrites and allow distal synaptic inputs to significantly depolarize the soma. Compartmental models were made from biocytin filled TCs (n = 15) and INs (n = 3) and adjusted to fit the current- and voltage-clamp recordings from the individual cells. The confidence limits for the passive electrical parameters were explored by simulating the influence of noise, morphometric errors and non-uniform and active conductances. One of the useful findings was that R_m was accurately estimated despite realistic levels of active conductance. Simulations to explore the somatic influence of dendritic synapses showed that a small (0.5 nS) excitatory synapse placed at different dendritic positions gave similar somatic potentials in the individual TCs, within the TC population and also between TCs and INs. A linear increase in the conductance of the synapse gave increases in somatic potentials that were more sublinear in INs than TCs. However, when the total synaptic conductance was increased by simultaneously activating many small, spatially distributed synapses, the INs converted the synaptic signals to soma potentials almost as efficiently as the TCs. Thus, INs can transfer fast synaptic signals to soma as efficiently as TCs except when the focal conductance is large.

The lateral geniculate nucleus (LGN) of the thalamus processes visual information using two main cell types; the excitatory thalamocortical cells (TCs) and the local inhibitory interneurons (INs). Interactions between morphology and membrane properties (passive and active) determine how these neurons integrate synaptic information.

As TCs and INs have strikingly different morphology and because the TCs have much lower somatic input resistance (R_in) than INs, at least during low network activity (Pape et al. 1994; Williams et al. 1996; Zhu & Uhlrich, 1997; Zhu et al. 1999a; Zhu & Heggelund, 2001; Briska et al. 2003; Perreault et al. 2003), their integration properties may be rather different. For instance, it has been suggested that synaptic signals originating in the dendrites are transmitted much more efficiently to soma in TCs than in INs (Bloomfield et al. 1987; Bloomfield & Sherman, 1989; Briska et al. 2003; Sherman, 2004).

The study of the conversion of synaptic conductances to somatic voltages relies on accurate neuronal models because direct voltage measurements at individual dendritic synapses are not yet available. The models that are needed must include a description of the cell morphology and the parameters describing the passive electrical behaviour of the cells (i.e. the membrane resistance (R_m), the cytoplasmic resistivity (R_i) and the membrane capacitance (C_m) (Barrett & Crill, 1974; Clements & Redman, 1989; Rall et al. 1992)).

In most types of neurons, the exact values of the passive parameters have been difficult to determine. Whereas C_m is thought to be uniform (Gentet et al. 2000), R_m and perhaps also R_i (Golding et al. 2005) may vary substantially within and between cell populations. The estimation of R_m in particular has been problematic for several reasons. First, R_m has often been assumed to be uniform over the cell membrane but this may not be the case. A non-uniform R_m may introduce a bias in the estimation of this parameter (London et al. 1999).

The presence of time- and voltage-dependent currents may also make it difficult to estimate R_m. Whereas this problem can be reduced by blocking the active currents (Major et al. 1994), this procedure may give an artificially high R_m. In the present study, we pursued a new trend whereby only a few (Stuart & Spruston, 1998; Roth & Häusser, 2001; Golding et al. 2005) or no active currents are blocked (this study and Briska et al. 2003). We use simulations to quantify the errors in R_m introduced by active currents as well as four additional types of models errors: inaccuracies due to the experimental noise, inaccuracies in the length and diameter of the dendrites (Roth & Häusser, 2001; Steuber et al. 2004; Szilagyi & De Schutter, 2004; Ambros-Ingerson & Holmes, 2005; Holmes et al. 2006), and violations in the assumption of a uniform R_m (London et al. 1999).

We found that R_m was larger in INs than in TCs and was accurately estimated at realistic levels of active conductance. Furthermore, the higher R_m of the INs compensated for the attenuating effect of their long dendrites. Because of this, distal synaptic inputs could significantly depolarize the soma. Parts of this work have been published previously in abstract form (Perreault et al. 2000).

Methods

Preparation

The experiments were performed using brain slices prepared from 21-day-old rats. Some of the present material (half of the cells) has been used in a previous study (Perreault et al. 2003). All procedures were approved by the National Animal Research Authority in Norway. The animals were deeply anaesthetized with halothane and decapitated. Blocks of brain tissue were quickly removed, submerged in ice-cold, oxygenated (95% O₂–5% CO₂) solution containing (mm): NaCl 125, KCl 2.5, NaH₂PO₄ 1.25, NaHCO₃ 25, MgCl₂ 1, d-glucose 25 and CaCl₂ 2 at pH 7.4 and placed on a cold plate of an HR2-microslicer (Sigmann Elektronik, Germany) where coronal slices (300 μm thick) containing the LGN, but not the reticular nucleus, were cut. The slices were then incubated at 37.0 ± 0.5°C for a recovery period of > 30 min before they were transferred to an immersion recording chamber. The oxygenated physiological solution in the recording chamber was kept at room temperature (23.5 ± 0.5°C). Neurons were visualized using a 40 × water immersion objective and infrared differential interference contrast (IR-DIC) video microscopy (Zeiss Axioplan-FS2, Germany).

Electrophysiology

Patch electrodes were made from borosilicate capillaries (Harvard Apparatus, UK) and had resistance of 5–7 MΩ when filled with a solution containing (mm): potassium gluconate 115, Hepes 10, MgATP 2, MgCl₂ 2, Na₂ATP 2, GTP 0.3 and KCl 20 at pH 7.3. Liquid junction potential was approximately −13 mV and was not included in the measurement of the resting membrane potential. Following gigaohm seals and electrode capacitance compensation, whole-cell recordings from the somata of INs and TCs were made in current-clamp (CC) and voltage-clamp (VC) modes using a HEKA 9 amplifier (Heka Elektronik, Germany). Voltage- and current-clamp recordings were made without serial resistance compensation and bridge balance. Instead the access resistance was included as a free variable in the simulations (see below). Stimulus delivery and data acquisition was performed using the Pulse software (Heka Elektronik). For CC experiments, we used 300 and 600 ms current steps that hyperpolarized the membrane less than −10 mV. For VC experiments, we used 600 ms long, −10 mV steps. Both CC and VC responses consisted of averages of three to six traces.

Histology and morphological reconstructions

All cells recorded were filled with biocytin (0.25%) added to the intracellular pipette solution. After recording, the slices were fixed in 0.1 m phosphate buffer containing 4% paraformaldehyde for up to 2 weeks. The slices were subsequently processed for biocytin labelling (Horikawa & Armstrong, 1988) using avidin–biotin peroxidase complex (ABC-Elite kit; Vector Laboratories, USA) and diaminobenzidine (DAB) reaction. Slices were mounted in the aqueous medium Mowiol (Hoechst AG, Frankfurt AM, Germany). Morphological reconstruction of the labelled cells were done using 40 × or 100 × water immersion objective and the computerized tracing program Neurolucida (MicroBright Field, USA).

Compartmental models

For each cell, the Neurolucida file containing the coordinates, diameters and branching of the dendritic and axonal processes was translated into NEURON program format (Hines, 1989; Hines & Carnevale, 1997) using custom-built software. These data were used to build a compartmental model for each neuron. The soma was modelled as a cylinder with diameter and height equal to the average between the maximal width and maximal length of the soma contour. The average number of compartments was 190 and the average compartment length 0.05 electrotonic units using R_m = 10 kΩ cm² and R_i = 100 Ω cm.

The distortion/shrinkage error associated with the present histological procedure (no re-slicing, air-drying or dehydrating) has been estimated to be < 5% (Roth & Häusser, 2001). In agreement with this estimate, we found a small and not significant increase of 3.3 ± 0.1% (P = 0.7, n = 12) in the diameters of TC primary dendrites (measured at a distance of 6.3 ± 0.9 μm from soma) when we compared the measurements taken from the living slices (2.9 ± 0.1 μm) with those taken after mounting the slices (3.0 ± 0.3 μm). The models were not corrected for this small distortion. Similarly, we did not correct for the possibility of an increased area due to dendritic spines because we found very few spines present on our cells. Similar observations have been made in the LGN of ferrets (82 spines per cell and 0.04 spines per μm, Rocha & Sur, 1995).

Simulations and fitting procedures

The responses produced by our models (simulated responses) were fitted to the experimental CC and VC traces. As with the experimental traces, the baseline of the simulated traces was adjusted to zero by subtracting the trace level before the steps (average over a period of 50 ms). The reversal potential for the passive leak conductance was also set to the actual value of the resting potential recorded during the experiments. The time steps used in the simulations were 0.3 ms and 0.1 ms for CC and VC, respectively. In the simulation with active membrane properties (see below) we also tested time steps ¹/₁₀ of these values. This gave a difference in the resulting parameter estimates which was < 2% in all individual cases.

We routinely fitted 300 and 50 ms of the CC and VC traces, respectively. For the CC traces, fitting started at the onset of the applied current pulse. For VC traces, fitting started 3 ms after the beginning of the voltage step to avoid electrode capacitance artifacts. To find the best fits, a simplex algorithm (Press et al. 1986) was used to minimize the squared error between the simulated and the experimental traces. The free variables in our models were R_m, C_m, R_i as well as the access resistance (R_acc). The latter was modelled either as an additional voltage step of amplitude equal to electrode resistance × injected current (CC) or as the electrode resistance (VC). We chose to include R_acc as a free variable because simulations (explained in the Results) showed that this strategy gave the most accurate estimates for R_i, with only a modest reduction of the accuracy of the other parameters. Moreover, the commonly used estimate based on the initial peak current in VC requires an accurate measurement of the somatic voltage (Jackson, 1992), which is not available with single-electrode voltage clamp.

Simulations of model errors

To explore the errors that may result from factors not accounted for in the original models, we performed a series of simulations where we modified the original models in the following ways (cf. ‘modified models’ in Fig. 3). We multiplied all diameters and lengths in the original models with a factor drawn from a uniform distribution between 0.8 and 1.2. We introduced active and non-uniform conductances (described separately below) also by drawing values for maximal conductances from the ranges given in the table in Fig. 3. We also added noise. This noise was a scaled version of noise recorded during the real experiments and was added arithmetically to the synthetic traces. The scaling was done so that the standard deviation of the noise covered the full range of standard deviations observed experimentally. Altogether, a data set of five factors describing the errors that distinguished these ‘modified models’ from the original models was obtained. Each factor was assigned 10 different values resulting in 10 modified models from each original model.

Figure 3. Simulating model errors.

A, illustration of the error simulation procedure. The table gives the ranges from which error values were picked to modify the original models so they contained a set of errors that distinguished them from the original models (details given in Methods). These modified models were then given a set of predefined values for R_m, C_m, R_i and R_acc, voltage and current responses were simulated, and the noise added (two topmost curves). The original models were then used to produce charging curves (two bottom curves) to fit the simulated responses by adjusting R_m, C_m, R_i and R_acc until the sum of squared differences were at the minimum (similar to Fig. 1). B, the inverse of the R_m error (quantified as 1/R_m,fit/1/R_m,predef) and the C_m error (C_m,fit/C_m,predef) correlated well with the surface error. C, we normalized the 1/R_m and C_m errors by dividing by the surface errors and were then left with the error not accounted for by the surface error. The remaining errors (distance from the correct value of 1.0) increased with increasing noise both in CC and VC. The dotted line indicates the average noise level recorded during the real experiments.

The modified models were given predefined values, R_m,predef, C_m,predef, R_i,predef, R_acc,predef, chosen to be compatible with our recordings and commonly used values in the literature: R_i = 100 Ω cm, C_m = 1.0 μF cm⁻², R_m = 35 and 100 kΩ cm² (TCs and INs, respectively) and R_acc = individual cell value. We could then produce VC and CC charging curves, which we call ‘synthetic curves’.

Finally, we used our original models to fit these synthetic curves and obtained parameters (R_m,fit, C_m,fit, R_i,fit, and R_acc,fit). These parameters, of course, were slightly different from the predefined values and these errors in the estimates were quantified by taking the ratio between the fit and predef values.

Non-uniform R_m

To simulate non-uniform R_m the passive membrane conductance per cm² (G_x) at a given compartment x was computed as:

where G_soma is the conductance at soma, d_x is distance between soma and compartment x, d_max is distance between soma and most remote dendritic compartment, and k is a constant giving the conductance gradient.

The average R_m (R_{m, avg}, used to compare with R_m,predef) was then calculated as:

Active membrane properties

We made R_m voltage sensitive by including, in soma, Hodgkin–Huxley-type models of fast Na⁺ and K⁺ currents and, in all compartments, hyperpolarization-activated cationic (I_h) and low-threshold calcium (I_t) currents and a Ca²⁺ extrusion process, as described by Destexhe et al. (1996). These types of conductances were chosen because they have been characterized in TCs (Jahnsen & Llinas, 1984a,b; Suzuki & Rogawski, 1989; Coulter et al. 1989; Crunelli et al. 1989; Hernandez-Cruz & Pape, 1989; Pape & McCormick, 1989; McCormick & Pape, 1990; Huguenard & Prince, 1992) and INs (McCormick & Pape, 1988; Pape et al. 1994; Pape & McCormick, 1995; Williams et al. 1996; Zhu et al. 1999a,b).

The model description files for the above conductances were downloaded from http://cns.iaf.cnrsgif.fr/abstracts/Dlgn96.html (HH2.mod, It.mod, Ih.mod (iar), and cadecay.mod (cad)). Temperature was set to 24°C.

Before voltage or current steps were applied to the modified models, a period of 6 s was used to let the membrane potential stabilize. The depolarizing effect of I_h was compensated so that the somatic membrane potential was similar to that recorded in the real experiments. This was done by adjusting (with a manual, interactive procedure) the reversal potential of the passive leak conductance. R_m,avg at the end of the steps was then estimated as described above.

Simulations of synaptic conductance

We simulated the activation of one excitatory synapse with small peak conductance (0.5 nS, reversal potential set to 0 mV). The synaptic conductance was modelled as an alpha function:

where G_max is the peak conductance and τ is the time constant of the conductance change. We monitored the voltage produced both locally and at soma. To study how synaptic position affected the somatic potential, the synapse was placed at 50 different dendritic locations with a uniform probability over the dendritic surface. Using uniform probability relative to dendritic length (rather than surface) did not influence the conclusions herein. In this series of simulations, we used R_i = 100 Ω cm, C_m = 1.0 μF cm⁻² and the R_m estimates from the individual cells. The R_m values were taken from the VC fits because simulations described in the results showed that these were more reliable than the values obtained from the CC fits.

Statistics

Differences between the means were tested with a Student's t test unless stated otherwise. The correlations between the best-fit parameters (Fig. 2) were examined using Pearson's correlation coefficient (r).

Figure 2. Best-fitting parameters for all recorded neurons.

A, the R_in values for the 15 TCs (median with 10%–90% confidence limits, 170–370 MΩ) were clearly smaller than the R_in values of the three INs (0.5–1.1 GΩ)(P < 0.001, Mann–Whitney U test), both for CC and VC recordings (measurements at steady state). B, R_m values for TCs (20–54 kΩ cm²) were outside the range of the R_m values for INs (83–124 kΩ cm²), both in CC and VC modes (median 33 and 106 kΩ cm² for TCs and INs, respectively, P < 0.001, Mann–Whitney U test). C, correlation between R_in and R_m in the combined TC and IN population (Pearson's r = 0.78, P < 0.001) and in the TC population alone (r = 0.55, P = 0.04), without correlation between R_in and area (r = −0.2, P = 0.51). D, TCs and INs had similar surface areas. E, both cell types had similar C_m both in CC and VC modes (median 1.2 μF cm⁻² in both cases). F, negative correlation between R_m and C_m (r = −0.75, P = 0.001). G and H, the R_i was similar in the two cell populations with a median of 79 Ω cm with 10%–90% confidence limits of 22–424 Ω cm for the TCs and a median of 113 Ω cm for the INs (VC mode). R_i probably contributed very little to R_in which was almost completely accounted for by the ratio between R_m and area.

To quantify differences in linearity in the current–voltage (I–V) plots (Fig. 4), we calculated an R² value from I–V data for all models. The R² values were not used the conventional way (i.e. to test the hypothesis that there is no correlation between two variables). Instead, we used R² only to quantify the deviation from a straight line and made our own estimates for how R² would be distributed if the models were passive. This was done by estimating R² for the I–V curves produced by the passive neuronal models with only experimentally recorded noise added. The resulting population of R² values represented the expectation for linearity. Then we compared this expectation with the R² values from non-linear models (and original data) to test whether the hypothesis of linearity could be rejected.

Figure 4. R_m and C_m errors due to active conductances.

A, I–V curves for the experimental data (seven TCs and one IN) were linear (visual assessment). B, the responses of a TC modified model with 0.2 mS cm⁻² of G_t and G_h showing obvious active membrane behaviours (sag and overshoot). Yet, the corresponding I–V curves (C) were surprisingly linear, both in CC and VC (measurements at steady state). D, to quantify how much the data deviated from a straight line we compared R² distributions (see Methods). The mean of the R² distribution for the purely passive models (Passive) was compared to the means for the models with 0.2 mS cm⁻² of G_t and G_h (0.4), models with 1 mS cm⁻² of G_t and G_h (2.0) and the real experiments (Exp.). The real experiments were not significantly different from purely passive models (Passive) or models with 0.4 mS cm⁻² total active conductance. E, the R_m and C_m errors increased with increasing G_t and G_h (the sum of them given at the x-axis), except for C_m in VC which had low error over the entire simulated range of active conductances. F, the sum of squared differences (SSq) between the best fitting charging curve and the charging curve of real or simulated experiments was a sensitive measure to detect the amount of G_t and G_h. The sum of G_t and G_h in 180 models (10 from each reconstructed neuron) is plotted along the x-axis. For comparison we plotted the SSq value from the real experiments (Exp.) and positioned it along the x-axis to fit in the rank of increasing SSq values.

Results

We recorded from 15 TCs and three INs (Fig. 1Aa and Bb, respectively), labelled them and used their morphology to construct compartmental models (see Methods). Somatic recordings were used to record their responses to current and voltage command steps in CC and VC modes, respectively. The steps were delivered when the somatic membrane potential was at rest (mean of −60.0 ± 0.9 mV for TCs and −59.8 ± 0.7 mV for INs).

Figure 1. Fitting model responses to experimental records.

Aa and Ba, biocytin-filled TC and IN with surface areas 19627 μm² and 20266 μm², respectively, were used to construct compartmental models. Axons are represented by dotted lines. Ab, the response recorded from the TC cell (continuous trace) was fitted by the model's CC response (grey trace) by minimizing the squared deviation. The difference between the two traces is shown on the top. The best fit was achieved with R_m = 59 kΩ cm², R_i = 10 Ω cm, and C_m = 0.91 μF cm⁻², giving R_in = 303 MΩ. Ac, similarly, the fit to a VC response was best with R_m = 61 kΩ cm², R_i = 200 Ω cm, and C_m = 0.85 μF cm⁻², giving R_in = 314 MΩ. Bb and c, fitting between the responses recorded from the IN (solid traces) and the model CC and VC response (grey traces). The best fits were achieved with: R_m = 78 and 107 kΩ cm², R_i = 113 and 87 Ω cm, and C_m = 0.84 and 0.91 μF cm⁻², giving R_in of 591 and 555 MΩ, for CC and VC responses, respectively. The vertical glitch on the residual is due to the fact that the first 3 ms were not fitted (see Methods).

The compartmental models were then used to produce responses which were fitted to the experimental responses. Optimal fits between the model-produced and experimental responses (grey and black traces in Fig. 1) were achieved by adjusting the free parameters R_m, C_m, R_i and R_acc. In the examples presented in Fig. 1, the differences between the simulated and experimental responses (residuals, plotted above the charging curves) were small, without obvious trends.

Parameter values giving optimal fits

Whereas the two neurons in Fig. 1 had clearly different morphology, they had similar total membrane surface. This means that the differences in steady-state responses shown in Fig. 1 could not be attributed to differences in membrane surface, but rather to differences in R_m, R_i and/or diameters of dendrites.

The best-fitting values for R_m were similar in CC and VC modes, but about 50% less in the TC compared to the IN. This difference was similar to the difference in R_in between the two cells, suggesting that R_m was the main factor determining R_in.

Whereas C_m was similar for the two recording modes and for the two cell types, the estimate for R_i differed considerably in CC and VC modes and between the two cells. Whether this indicates that R_i was not reliably estimated in one or both modes, or that incorrect model assumptions had a different impact in the two recording modes, was tested with simulations (next section).

The differences and similarities between the TCs and INs shown in Fig. 1 were supported by the results from all 15 TCs and three INs (Fig. 2). Figure 2 present the data from both CC and VC fits, given as medians with 10%–90% confidence intervals for the TCs, and the individual values for the three INs.

In TCs, both R_in and R_m values were distributed over a narrow range (Fig. 2A and B). The R_m values for INs were clearly outside this range. In agreement with the idea that the R_in difference between the two cell groups was mainly due to a difference in R_m, we found a clear correlation between R_in and R_m (Fig. 2C). We found no correlation between R_in and membrane surface areas (not shown). This was expected because membrane surface was similar in TCs and INs (Fig. 2D). It is interesting that R_m may also be the most important factor for the differences in R_in within the TC group, because R_in and R_m were correlated within this group, but R_in and membrane area were not.

TCs and INs had similar values for C_m (Fig. 2E). However, some TCs had C_m values > 1.5 μF cm⁻², which is unlikely for neuronal membranes (Gentet et al. 2000). C_m and R_m also correlated negatively (Fig. 2F), suggesting that the large C_m values may be due to reconstruction errors (Roth & Häusser, 2001). This possibility was tested in our simulation experiments (next section).

The R_i estimates for both TCs and INs overlapped (Fig. 2G). In TCs, however, the range was large, supporting the suggestion that R_i can not be accurately determined in the TCs because of their short dendrites (Briska et al. 2003). In both cell types, however, R_i probably added very little to the somatic R_in because dividing R_m by the surface area produced resistance values very close to the R_in values with only small deviations from the line of identity (Fig. 2H). Such tight linear relationship between R_in and R_m/area suggests that the cells were electrotonically compact at steady state.

The parameter values that gave optimal fits in this study are similar to those reported by Briska et al. (2003) in the adult rat. Moreover, our larger sample allows us to clearly show a separate distribution of R_m values between the two cell types and a very narrow distribution of R_in and R_m in the TC population.

Simulations to evaluate potential sources of errors in R_m, C_m and R_i estimates

Because models are only approximate representations of real neurons, we wanted to test the reliability of our parameter estimates. To do this we used our original models to perform a series of parameter fits to voltage and current responses that were produced by models that were slightly modified versions of the original models (‘modified models’ in Fig. 3A and see Methods for details).

Five types of errors were examined by including in the modified models: (1) inaccuracies in the length and (2) diameter of the dendrites; (3) inaccuracies due to the experimental noise; and (4) violations of the assumptions of linear and time-independent conductances with (5) uniform distribution. For each type of error, the ranges of values explored are indicated in the table at the top of Fig. 3A (for details see Methods). We describe the results of the simulations first for R_m and C_m and then for R_i separately.

Errors of R_m and C_m estimates due to surface errors and experimental noise

All error types were included in the simulations to allow interactions between the different model errors. Initially, we found only weak correlations between model errors and the errors of the estimated parameters. However, by multiplying the errors of the dendritic length and diameter to obtain a surface error, we found strong and almost linear correlations between the surface error and the errors of R_m and C_m estimates (Fig. 3B). Based on these correlations, we could estimate that the 3.3% larger diameter of TC we found by comparing measurements in living and mounted slices (see Methods) would give only a 3.3% underestimation of 1/R_m and C_m.

In Fig. 3B, the sensitivity of the parameter estimates to the remaining three model errors (noise, active and non-uniform conductances) can be assessed from the variability around the identity lines. This variability was clearly smaller in VC than CC, showing that the R_m and C_m estimates were less sensitive to these three model errors in VC compared to CC.

To find which of those three model errors contributed the most to errors in R_m and C_m estimates, we adjusted the R_m and C_m errors by normalizing them by the surface errors. In Fig. 3C we plotted these normalized R_m and C_m estimates as a function of the s.d. of the noise. The results show that the errors of the surface normalized R_m and C_m estimates (i.e. the spread around the 1.0 value) were small at low noise levels despite the presence of active and non-uniform conductances. This means that, over the parameter ranges explored, the C_m and R_m errors were considerably more influenced by the noise level than by active and non-uniform conductances. At the average noise levels that were recorded experimentally (vertical dotted line), the R_m and C_m errors were substantial in CC but only in the order of 5% in VC. Increasing the number of averaged traces would of course reduce the noise contribution to the parameter errors also in CC (see, e.g. Roth & Häusser, 2001)). However, with the same number of traces in the averages, VC showed smaller parameter errors.

Errors of the R_m and C_m estimates due to active conductances

A major concern when passive neuronal models are used for parameter estimation is that the charging curves from the real neurons are shaped by voltage- and time-dependent conductances. We studied the I–V response of our neurons in the voltage range where we applied the test pulses and, under simple visual inspection, these appeared linear and symmetrical around the resting potential (Fig. 4A). However, when a modified model with voltage- and time-dependent conductances (low threshold calcium conductance (G_t), 0.2 mS cm⁻²; hyperpolarization activated conductance (G_h), 0.2 mS cm⁻²; see Methods) was used to simulate similar voltage and current responses (Fig. 4B), we also obtained rather linear I–V curves (Fig. 4C).

We quantified deviations from perfect linearity using R² (see Methods), which gives the value 1.0 at perfect linearity. We compared the distributions of the R² values for the I–V curves produced by the passive models to which only experimental noise was added (‘passive’) to the R² values of the I–V curves produced by the modified models and the R² values of the experimental I–V curves (‘exp.’ in Fig. 4D). The means and s.d. for the various R² distributions are displayed in Fig. 4D. As shown, the R² obtained using the modified models with active conductances (G_t, 0.2 mS cm⁻² and G_h, 0.2 mS cm⁻²) was not significantly different from the passive models, whose R² value was a little less than 1.0 due to the experimental noise. In fact, we had to increase the active conductances to 2.0 mS cm⁻² (G.t, 1.0 mS cm⁻² and G.h, 1.0 mS cm⁻²) get an R² value that was significantly different from the passive models. This was true only for CC mode and in VC mode the R² value was not significantly different from the passive models even at this conductance level.

Although it was possible to find other combinations of G_t and G_h, and other holding potentials, which made the I–V plots less linear, we want to emphasize that the I–V plots could often be close to linear even in the presence of some active conductance. Therefore, an important question is whether the amounts of active conductances not detectable on I–V plots could give a serious error of R_m and C_m estimates.

In order to explore this question we tested a large range of G_t and G_h. Figure 4E shows the R_m and C_m errors as a function of the sum of G_t and G_h. The C_m errors were very little influenced by the active conductances in VC. In CC, however, the errors were larger, as indicated by the increasing deviation from 1.0 with increasing amount of active conductances. The R_m errors were generally larger than C_m errors. With 0.4 mS cm⁻² active conductance (amount where the I–V plots were not significantly non-linear, Fig. 4D), many R_m errors were > 10%.

Obviously, some of the CC charging curves in Fig. 4B cannot be well fitted with a purely passive model. Thus, in addition to the linearity of the I–V plots, we examined the fitting error (sum of squared deviations; SSQ) at different levels of G_t and G_h. The SSQ should be a more sensitive statistic because the shape of the charging curve is evaluated, not only the steady-state level. For all the neurons, we simulated charging curves similar to those shown in Fig. 4B. These synthetic curves were then fitted with our original models. An SSQ was obtained, which was normalized to the SSQ from the fit between the responses from the original models and the passive models with only experimental noise added. If the original models were correct, the latter SSQ should represent the best possible SSQ and give an expectation of 1.0 (indicated by a dotted line in Fig. 4F).

As shown in Fig. 4F, the relative SSQ errors increased with increasing active conductances (sum of G_t and G_h), and were significantly different from the passive models when the sum of active conductances was > 0.05 mS cm⁻². Also plotted for comparison, the average SSQ from the fit to the experimental data (arrow) was not significantly different from the value of 1.0. These results suggest that, with the type of currents included in this study, the SSQ error (fitting error) is more sensitive to active conductances than the linearity of the I–V plot. Furthermore, with these data we can estimate that at the conductance level of 0.05 mS cm⁻², the errors of C_m and R_m was < 5% (Fig. 4E, VC data). Thus, the errors in R_m and C_m estimates introduced by active conductances were small and not larger than the errors introduced by noise and dimensional errors.

Errors of the R_i estimate

The R_i estimate is very sensitive to errors in the determination of R_acc. We found that the commonly used approach to estimate R_acc (dividing the step command voltage by the initial peak current in VC) systematically overestimated R_acc by 4%–6% because of an underestimation of the peak current. This relatively small inaccuracy in R_acc measurement resulted in a large (30%–80%) underestimation of R_i. Therefore, we included R_acc as a free variable (together with R_m, C_m and R_i) in our models. As described below, by doing so we were able to obtain reliable R_i estimates for the INs.

In simulations with all model errors (as in Fig. 3), R_i errors correlated positively with length error and negatively with the error of the cross-sectional area (squared diameter) of the dendrites. When the R_i estimation error was displayed as a function of the summed errors in dendritic length and cross-sectional area (x-axis in Fig. 5), a clear correlation could be seen for the INs but not for the TCs. The fact that the R_i cannot be accurately estimated for TCs is attributed to their shorter dendrites and their lower total axial resistance.

Figure 5. R_i was sensitive to errors in model dimensions.

The error of the R_i estimate in INs (•) correlated well with errors of the reconstructed morphology, even with all the other model errors (see Fig. 3) present. For TCs, the R_i estimate was poorly constrained (○). The R_i error in INs increased almost linearly with the length error and inverse of the squared diameter error (i.e. the error of the inverse cross-sectional area of the dendrites).

The present simulations therefore suggest that for neurons with long dendrites such as INs, R_i can be determined with good accuracy, over a large range of noise levels and active and non-uniform conductances, as long as the morphology of the neuron is accurately reconstructed. Furthermore, using the results displayed in Fig. 5, we can estimate directly the errors of R_i estimates at least for INs. With dimensional errors of about 5%, the error on R_i would be < 10%.

Converting synaptic conductances into somatic voltages; role of R_m and morphology

The purpose of finding accurate parameter estimates was to investigate how short-lasting synaptic conductance changes are converted to somatic voltages. In this context, we asked one simple question: what would be the somatic voltage response to a single excitatory synapse? We measured the excitatory postsynaptic potentials (EPSPs) resulting from activation of one fast excitatory synaptic conductance (0.5 nS, see Methods) and repositioned the synapse at 50 different, randomly selected positions in the dendritic tree. Our previous simulations showed that the passive parameters were always more accurate in VC than in CC mode. Therefore, for the present series of simulations each cell was given their respective VC passive parameters.

Peak somatic EPSP

The amplitudes of the EPSPs were, of course, much larger at the synaptic site (Fig. 6A) than at the soma (Fig. 6B). We will focus on the somatic EPSP (EPSP_soma) because of its direct role in bringing the membrane closer to firing threshold. For each cell, the EPSP_soma from the 50 dendritic locations were first rank-ordered with respect to increasing amplitude. For each rank, the values were averaged and a population mean was obtained for TCs and INs separately. In Fig. 6C–F, the population means are shown as open circles for TCs and filled circles for INs.

Figure 6. Simulating dendritic synaptic conductances.

Fifty randomly chosen locations (○) along the dendrites of TCs and INs were tested with a 0.5 nS excitatory synapse (see Methods). As expected, the EPSPs at the synapse (A) were larger than the EPSPs measured at soma (B). Note, however, that the peak amplitudes of the EPSP_soma were relatively similar in the INs and the TCs. C, for each cell, the EPSP_soma from the 50 dendritic locations for the synapse were rank-ordered with respect to the peak amplitude. For each rank, a population average for all INs (•) and TCs (○) was obtained. The vertical bars give standard deviations. The right plot, which is the same as the left plot but with one-quarter of the membrane resistance, shows that the median peak amplitude is not very sensitive to membrane resistance in the tested interval. D, same as C, for the integral of EPSP_soma. The integral was larger for almost all dendritic positions in INs compared to TCs. The INs displayed larger EPSP_soma than the TCs even after a large reduction in R_m (right plot). E, for the cells illustrated in A, the EPSP_soma induced by a 20 nS synaptic conductance. F, at this conductance level, EPSP_soma was smaller in the INs than in the TCs at almost all dendritic locations. G, with further increase in the conductance of the synapse (data connected by solid lines), the median EPSP_soma still increased much less in INs (•) than in TCs (○). However, when the increases in total conductance were obtained by simultaneous activation of many small (0.5 nS), spatially distributed synapses, the EPSP_soma reached much larger amplitudes in both cell types (data connected by dashed lines).

In each TC, EPSP_soma from different dendritic locations had very similar peak amplitude with a coefficient of variation (CV) of only 12% on average (see also Neubig & Destexhe, 2001; Briska et al. 2003). Furthermore, the different TCs were also very similar to each other; the mean EPSP_soma varied only 24% between the different TCs. Because the INs have much longer dendrites than the TCs they have been thought to filter their EPSPs much more than the TCs. We found, however, only moderate differences between the TCs and INs, with an average EPSP_soma in INs only 12% smaller than the average EPSP_soma in TCs.

For the INs, the peak amplitude of the EPSP_soma from different dendritic locations varied with a CV of 36% on average. This may indicate that dendritic location is slightly more important for the EPSP_soma in INs than in TCs. However, the largest contribution to the variation in INs was from a few very large EPSPs, probably from synapses close to the soma (largest peak values in left panel of Fig. 6C). When the synapses located farthest away from the soma (10% smallest EPSP_soma) were considered, then the effect of dendritic location was similar in both cell types: the 10% smallest EPSP_soma were 85% and 69% of the median EPSP_soma for TCs and INs, respectively.

Finally, because R_m is the most flexible passive parameter and may vary with changes in, for example, temperature and network activity, we wanted to test how sensitive the peak amplitude of EPSP_soma was to changes in R_m. We tested a 75% reduction in R_m. As shown in Fig. 6C (right panel), such a reduction decreased the peak EPSP_soma by only 24% in TCs and 20% in INs. This result confirms that the peak amplitude of the EPSP_soma is relatively insensitive to changes in R_m (Spruston et al. 1993; Jaffe & Carnevale, 1999; Briska et al. 2003) suggesting that at a given synaptic conductance, the peak amplitude of the EPSP_soma in LGN cells is mainly determined by their cell geometry.

Integral of EPSP_soma

Because several EPSPs would be needed to depolarize the soma above the threshold for action potential, depolarization over time may be a more relevant measure than the peak amplitude for assessing the contribution of an EPSP to soma depolarization. Consequently we also measured the integral of the EPSP_soma.

We found that similar to the peak EPSP_soma, the integral (Fig. 6D, left panel) varied little in the individual TC (on average 11%) or between the TCs (26%). The CV of the EPSP_soma in the individual IN was a little larger than for the TCs with a value of 21%. For the INs, the integral measurements made it even more obvious than the peak amplitude measurements that the long dendrites of the INs did not prevent synaptic inputs from all dendritic locations from influencing somatic voltage. Indeed, the EPSP_soma in INs were typically larger than in TCs (compare the two curves in left panel of Fig. 6D). The 10% smallest EPSP_soma were 86% and 77% of the median EPSP_soma for TCs and INs, respectively.

As expected, the integrals of the EPSP_soma were more sensitive to changes in R_m than the peak amplitude. A decrease of 75% in R_m (Fig. 6D, right panel) reduced the integrals by 73% and 68% in TCs and INs, respectively. However, the difference between the median and the 10% smallest EPSP_soma remained small (the 10% smallest EPSPs_soma were 85% and 74% of the median for TCs and INs, respectively).

Voltage saturation at synaptic sites

In the simulations above we used a synaptic conductance of 0.5 nS because activation of such a small conductance resulted in a moderate reduction of the driving potential at the synaptic site. However, because of a lack of precise knowledge about the size of the unitary synaptic conductances and to account for possibility of several simultaneously activated synapses on the same dendrite, we also tested a range of synaptic conductances.

When the single synaptic conductances were increased by 100%, the peak EPSP_soma increased on average only by 82% and 70% in TCs and INs, respectively (not shown). This sublinearity, attributed to the reduction of the voltage driving force, was more pronounced for INs than for TCs. For example, with a synaptic conductance of 20 nS, most EPSP_soma were smaller in the IN than in the TC (Fig. 6E and F).

With even larger synaptic conductances (stepwise increase up to 100 nS in Fig. 6G), the loss of voltage driving force clearly put an upper limit to how large an EPSP_soma could be generated from a small dendritic segment; around 8 and 2.5 mV in TCs and INs, respectively (Fig. 6G data points connected with solid lines). This indicates that the INs were more susceptible to saturation.

Because the most remote synapses saturated at smaller conductances than the more proximal, the 10% smallest peak EPSP_soma were only 30% (TCs) and 19% (INs) of the median EPSP_soma with 100 nS conductance. Similar results (with slightly smaller differences between TCs and INs) were obtained when we measured the integral of the EPSP_soma rather than the peak (not shown).

For comparison we tested again the effect of a large increase in the total synaptic conductance but this time by simultaneous activation of many, small (0.5 nS) spatially distributed conductances. This resulted in a much more linear relationship between the total synaptic conductance and the peak EPSP_soma (Fig. 6G data connected with dashed lines), and in much larger EPSP_soma in both cell types. In fact, by spatially distributing many small synaptic conductances, the INs converted the synaptic signals to soma potentials almost as efficiently as the TCs. Our data also indicate that the INs were more sensitive to the spatial distribution of synaptic conductances than the TCs. With 100 nS total synaptic conductance, the EPSP_soma produced by the distributed synaptic conductances was six times larger than the single-site EPSP_soma for the TCs and 13 times larger for the INs.

Active dendrites

One problem in the study of integration properties is that we do not know in detail the amount and distribution of active conductances in the dendrites. When we tested large synaptic conductances (> 2.0 nS), or several synapses, together with a combination of 0.2 mS cm⁻² for G_h and G_t, we obtained results that differed in several ways from the passive simulation described above. More importantly, we found that the results were very sensitive to the exact combination of these factors (not shown).

However, as long as G_h and G_t were < 0.2 mS cm⁻² (i.e. below the values found to be incompatible with our charging curves, see Fig. 4F) and the synapses were small (1.0 nS), our main conclusions were not influenced by the active conductances. With 0.2 mS cm⁻²G_h and 0.2 mS cm⁻²G_t, the different synaptic locations gave very similar EPSP_soma, as for the passive models. The integral was similar in the individual TC (CV of 10% on average) and between the TCs (CV of 24%). For the INs, the CVs were slightly larger than for the TCs, with an average value of 20%. In both cell types, the smallest 10% of the peak and the integral EPSP_soma were > 72% of the median EPSP_soma.

Discussion

Conversion of small synaptic conductances to somatic voltages

Our study suggests that despite large differences in dendritic morphology, the TCs and INs have similar ability to convert small, short-lasting synaptic conductance changes to somatic voltages. This conclusion is different from other investigations that have emphasized the more local signal processing that may occur in the INs due to their long dendrites (Bloomfield & Sherman, 1989; Sherman, 2004). Part of the explanation may be attributed to differences in methodology and the use of tight-seal patch electrodes rather than penetrating electrodes. With tight-seal patch electrodes, higher R_m values are obtained (present study and Briska et al. 2003) and this may compensate at least partly for the attenuating effect of long dendrites. An additional explanation is that earlier investigations have often focused on voltage attenuation, usually defined as the ratio between somatic and dendritic voltage, whereas we focused on the absolute amplitude of the somatic potentials. We made this choice because the influence on somatic voltage is easy to interpret. The exact voltage at the synapse may be important, particularly if it reaches the equilibrium potential for the synapse or, as in the case of INs, can trigger transmitter release by the dendrites. The ratio between dendritic and somatic voltage, however, does not have any known physiological effect.

One main finding was that the integrals of the smallest EPSP_soma produced by single, small synaptic conductances were larger in the INs than in the TCs (Fig. 6D), illustrating clearly that long dendritic cables, like in the INs, do not prevent distal synapses from influencing the somatic voltage. Because INs have larger dendritic fields than TCs (Perreault et al. 2003), the fact that even distal synaptic inputs can significantly depolarize the soma suggests that INs have larger integration fields.

We also provide new data that suggest that TCs are a particularly homogeneous population with respect to integration properties. We found that the ability to convert synaptic conductances to soma potentials was remarkably similar among TCs, possibly giving all synapses on all TCs an equal weight. For the integral of the EPSP_soma, the low variation fits well with the previously described low variability in R_in (Zhu & Heggelund, 2001; Perreault et al. 2003). This is not so surprising because R_in is the main determinant of the EPSP integral. However, the variation in the peak amplitude between the different TCs was also small (24%) despite the fact that this measure depends mostly on dendritic diameter, branching pattern and R_i (i.e. factors that contribute little to R_in).

Interference with active membrane properties

It is important that our conclusions are limited to small, short-lasting synaptic conductance changes, because with larger and longer conductances, the results become very sensitive to the level of voltage- and time-dependent dendritic conductances. The knowledge about such dendritic properties is not sufficiently accurate to make model-based predictions of the voltage trajectories close to action potential threshold.

However, with small synaptic conductances our conclusions were almost unaffected by I_h and I_t as long as these were within levels compatible with the shape of our experimental charging curves. The main reason is probably that the synaptic voltage transients in the dendrites were too fast and local to influence I_h and I_t sufficiently to significantly change EPSP_soma. The linear behaviour of the membrane in response to single synaptic inputs is supported by results from other cell types (Langmoen & Andersen, 1983; Gasparini & Magee, 2006).

Accuracy of the models

The conclusions in this study depend on the accuracy of the models. Therefore an important part of this article was to explore the influence of different model errors on the accuracy of the parameter estimates. In addition to the finding that VC mode gave more accurate parameter estimates, several useful results came out of these simulations. These are discussed below.

R_m

Using morphology-based models together with tight-seal recording techniques, Briska et al. (2003) have recently suggested R_m values around 30 kΩ cm² for two TCs and 90 kΩ cm² for two INs. These values are considerably higher than earlier estimates based on more simplified morphologies (Bloomfield et al. 1987; Bloomfield & Sherman, 1989). In the present study, R_m was relatively similar within the population of 15 TCs, with 80% of the cells lying between 20 and 53 kΩ cm² (compatible with Briska et al. (2003)). By providing a range of R_m values within the TC population, we show that the INs had R_m values clearly outside this range.

Our biggest concern when estimating R_m was that our experimental curves were contaminated by active currents even when our I–V plots were close to linear. We found, however, that a passive model can capture R_m even in neurons with moderate amounts of active currents.

Our upper level of G_h and G_t that were compatible with the original charging curves (0.025 mS cm⁻² of G_h and 0.025 mS cm⁻² of G_t) were multiplied with the cell surfaces, obtaining 3.8 nS for INs and 3.2 nS for TCs, for both G_h and G_t. These values are compatible with the values from other reports (G_h, 1 nS for INs and 3–15 nS for TC; G_t, 8.4 nS for both cell types; Crunelli et al. 1989; Zhu et al. 1999a,b). So the range of the conductances we tested are relevant and would not severely bias our R_m estimates.

C_m

Our median value for C_m was 1.2 μF cm⁻² and our simulations indicated that this value was probably correct within 20% of the true value (Fig. 3C, VC data). This fits well with C_m values close to 1.0 μF cm⁻² from measurements in cells where surface area can be reliably estimated (Curtis & Cole, 1938; Hodgkin & Huxley, 1952; Gentet et al. 2000).

We found that the most important error in the estimation of this parameter was from error in the measurement of the membrane area. The relatively large range of C_m values for the TCs (Fig. 2G), may suggest that the surface area was inaccurate for some cells. Such a hypothesis would also fit with the negative correlation we found between C_m and R_m (Fig. 2H). However, as reported in the Methods, shrinkage did not significantly change the cell dimensions. An alternative possibility is that those few cells that display high C_m might have been connected to other TCs via gap junctions (Parenti et al. 2000; Hughes et al. 2002, 2004).

R_i

The only passive parameter whose accuracy was obviously cell-type dependent was R_i, which was more accurate for INs than for TCs. The R_i estimate in INs was influenced by inaccuracies in dendritic diameter and length. While this was expected from cable theory, we did not expect this influence to be so obvious when the active currents, non-uniform R_m and noise were present at the same time (Fig. 5). We thus conclude that the latter three factors introduced less error than morphometric inaccuracies, at least in the ranges explored in this study.

Is there an important role for dendritic voltage saturation?

INs and TCs differed in their ability to convert single large synaptic conductances to somatic voltage (see Fig. 6G). In the INs, no single synapse was able to depolarize the soma by more than ∼2.5 mV (on average in individual cells). In contrast, in the TCs, the upper limit for how large an EPSP_soma could be generated from a single synapse was much higher (∼8.0 mV).

It has been suggested that the transmission of synaptic signals from dendrites to soma is more limited by passive cable properties in INs than in TCs (Briska et al. 2003; Sherman, 2004). We suggest that this is the case only for particularly large synaptic conductances. Furthermore, we do not think that this is due to voltage attenuation but to voltage saturation in the IN dendrites which more easily approach the reversal potential for the synapses. Dendritic voltage saturation has been suggested to be important for the input–output function of other cell types as well (Cushing et al. 2005; Golding et al. 2005).

In theory, dendritic voltage saturation could be investigated experimentally with dendritic recordings; however, due to the low probability of encountering INs in the LGN this has proven to be difficult. Hopefully, the use of new transgenic animals where glutamic acid decarboxylase isoform 67 (GAD67)-expressing INs are selectively labelled (Tamamaki et al. 2003) will make this possible.