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* Abteilung Zellphysiologie, Max-Planck-Institut für medizinische Forschung, D-69120 Heidelberg, Germany; and Faculty of Life Sciences, Bar-Ilan University, Ramat-Gan 52900, Israel
Correspondence: Address reprint requests to Dr. Alon Korngreen, Faculty of Life Sciences, Bar-Ilan University, Ramat-Gan 52900, Israel. Tel.: +972-3-5318224; Fax: +972-3-5351824; E-mail: korngra{at}mail.biu.ac.il.
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXACKNOWLEDGEMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXACKNOWLEDGEMENTSREFERENCES |
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). However, voltage-clamp recordings can only be properly analyzed in isopotential structures. Problems that occur upon incomplete space clamp have been addressed extensively (Armstrong and Gilly, 1992; Augustine et al., 1985; Johnston et al., 1996; Koch, 1999; Larsson et al., 1997; Major, 1993; Major et al., 1993; Müller and Lux, 1993; Rall et al., 1992; Rall and Segev, 1985; Spruston et al., 1993; White et al., 1995). Typically, voltage-clamp recordings from neurons are severely distorted, which makes it impossible to directly determine properties of ionic conductances, even for small neurons. So far, only a few theoretical and experimental techniques enable true voltage-clamp recordings from specific branched structures either by analytical transformations (Cole and Curtis, 1941), or by recording conditions (Adrian et al., 1970; Augustine et al., 1985). Currently available reduced models only partially correct for space-clamp errors in somatic recordings of simple structures (Nadeau and Lester, 2000).
Recently, there have been advancements in the visualization and recording from dendrites (Johnston et al., 1996; Stuart et al., 1997). So far, information on dendritic channels has mainly been obtained from excised or cell-attached patch-clamp recordings (Bekkers, 2000; Bischofberger and Jonas, 1997; Hoffman et al., 1997; Korngreen and Sakmann, 2000; Magee, 1999; Magee and Johnston, 1995; Stuart and Sakmann, 1994) and from dendrosomes (Benardo et al., 1982; Kavalali et al., 1997; Takigawa and Alzheimer, 1999). However, cell-attached and outside-out recordings are laborious, and due to small currents and high variability in patch area, a large number of recordings is necessary to obtain an accurate description of dendritic channel density in such systems.
Here we present analytical, numerical, and experimental evidence, demonstrating that in general membrane conductances can be calculated from clamp-current recordings of nonregenerative conductances in a nonisopotential structure. We developed a numerical algorithm that performs this calculation and tested it on simplified potassium-channel models with systematically varied model parameters, and on a variety of potassium-channel models taken from the literature. Distortions due to the lack of space clamp were clearly seen and corrected in simulations carried out with an unbranched cylindrical cable, as well as in the soma and apical dendrite of a computer-reconstructed, layer 5 (L5) neocortical pyramidal neuron. This required i), the knowledge of morphology and passive parameters, and ii), an ionic current with hyperpolarized reversal potential. To explore the locality of the method, the cylinder was equipped with channel-density steps and gradients of various slopes were inserted into the apical dendrite of a L5 pyramidal neuron. Limitations of the correction algorithm were assessed by testing its sensitivity to incomplete knowledge of passive parameters, incorrect morphological reconstruction, and experimental noise. For voltage regions accessible to the activation voltage-clamp protocol, all channel parameters could be extracted faithfully under realistic conditions. In addition, the method was shown to be applicable to two-electrode voltage-clamp recordings from the apical dendrite of L5 neocortical pyramidal neurons.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXACKNOWLEDGEMENTSREFERENCES |
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) on a Silicon Graphics Origin 2000 or on a Pentium III, 650 MHz. The integration time step used was 25 µs; changing the time step did not alter the algorithm performance provided it stayed smaller than the channel activation time constants. Potassium-channel models were taken from the literature (for a full description see Appendix): delayed rectifier Kv, (Mainen et al., 1995), muscarinic potassium channel Km, (Mainen et al., 1995), A-type potassium channel Ka (Hoffman et al., 1997), and two types of K+ channels (Kslow and Kfast), which we recently described in neocortical pyramidal neurons (Korngreen and Sakmann, 2000). Simulation data were evaluated on a Macintosh Power PC G4 using custom-written routines in IgorPro 4 (Wavemetrics; Lake Oswego, OR). The unbranched cable (cylinder) had a diameter (d) of 3 µm and a length (l) of 2000 µm (Figs. 1, 4, 5 C and D, 6, 7, and 8). When comparison with analytical calculations in an infinite cylinder was attempted during simulations, a much longer cylinder was used to approximate this situation (d = 2 µm and l = 50,000 µm, Figs. 2, 5 A and B, and 11). For determination of morphological parameters, a L5 neocortical pyramidal neuron of a 48-day-old Wistar rat was biocytin-filled, computer-reconstructed (except for parts of the axonal arborization), and converted to the format of NEURON as previously described (Stuart and Spruston, 1998). The passive parameters were Ri = 250 
cm; Rm = 20,000 cm2, and Cm = 0.75 µF cm-2 with a passive reversal potential of Eleak = -65 mV and a potassium reversal potential of Ek = -80 mV, unless otherwise noted. All voltage-clamp simulations used the built-in MOD file SEClmp.mod. Dendritic two-electrode voltage-clamp configuration was simulated using the linear circuit builder in NEURON 5.0.0. Simulated voltage protocols consisted of an 8–15 s prepulse of -110 mV to relieve channels from inactivation, followed by a voltage step to the test potential. Series resistance was set to 10-5 to simulate an ideal voltage clamp. The total time necessary for correcting a 100-ms voltage-clamp experiment in a full model of a layer 5 pyramidal neuron sampled at 10 kHz was 6 h on a single processor of the SGI Origin 2000 and 8 h on the Pentium III PC.
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). The code of the correction algorithm can be downloaded from http://sun0.mpimf-heidelberg.mpg.de/~alon.
Slice preparation
Slices (sagittal, 300 µm thick) were prepared from the somatosensory cortex of 40–45 days old Wistar rats using previously described techniques (Stuart et al., 1993). Slices were perfused throughout the experiment with an oxygenated artificial cerebrospinal solution containing: (mM) 125 NaCl, 15 NaCO3, 2.5 KCl, 1.25 NaH2PO4, 1 MgCl2, 2 CaCl2, 25 Glucose (pH 7.4 with 5% CO2, 310 mosmol kg-1) at room temperature (20–22°C). Pyramidal neurons from L5 in the somatosensory cortex were visually identified using infrared differential interference contrast videomicroscopy (Stuart et al., 1993).
Solutions and drugs
The standard pipette solution contained (mM): 125 K-gluconate, 20 KCl, 10 HEPES, 4 MgATP, 10 Na-phosphocreatin, 0.5 EGTA, 0.3 GTP, and 0.2% biocytin (pH 7.2 with KOH, 312 mosmol kg-1). The bath solution for two-electrode voltage-clamp experiments contained (mM) 125 NaCl, 15 NaCO3, 2.5 KCl, 1 MgCl2, 2 CoCl2, 25 Glucose, 0.03 ZD2788, 0.001 tetrodotoxin (TTX pH 7.4 with 5% CO2, 308 mosmol kg-1). TTX (Tocris, Bristol, UK) was stored at -20°C as stock solutions in doubly distilled water and added directly to the bath solution. ZD2788 (Tocris) was stored at 4°C as stock solutions in doubly distilled water and added directly to the bath solution.
Two-electrode voltage clamp
Dendritic two-electrode voltage-clamp recordings were made with an Axoclamp-2B amplifier (Axon Instruments, Foster City, CA). Two HS-2Ax0.1M headstages were used. Whole-cell recordings were performed with two patch pipettes at a distance of 30–40 µm apart. Simulations have shown that the accuracy and stability of the correction algorithm were not affected by an interelectrode distance that was smaller than 50 µm (data not shown). No series resistance compensation was used. Capacitive coupling between the electrodes was minimized by placing a grounded shield that extended almost to the bath fluid level between the electrodes. To increase clamp gain, the feedback current injected via the current injecting electrode was low pass filtered by a built-in filter of the Axoclamp-2B (phase-lag control) with a time constant of 1–5 ms. Voltage and current were filtered at 10 kHz and sampled at 50 or 20 kHz using the program "Pulse" (Version 8.1, Heka Electronic, Lambrecht, Germany), digitized by an ITC-16 interface (Instrutech, Greatneck, NY, USA), and stored on the hard disk of a Macintosh computer. Capacitive and leak currents were subtracted off-line by scaling of pulses taken at hyperpolarized potentials. Patch pipettes (5–10 M) were pulled from thick-walled borosilicate glass capillaries (2.0 mm outer diameter, 0.5 mm wall thickness, Hilgenberg, Malsfeld, Germany) and were coated with Sylgard before the experiment. The distance of the dendritic recording from the soma and the interelectrode distance was measured from video pictures taken by a frame grabber.
Histology and morphology
At the end of the experiments, the slices were fixed in cold 100 mM phosphate buffer (PBS, pH = 7.4) containing 4% paraformaldehyde. After fixation, the slices were incubated for 2 h in avidin-biotinilated horseradish peroxidase (ABC Elite, Vector Laboratories, Peterborough, UK) and the stain was developed using 0.015% diaminobenzidine. The slices were mounted in Mowiol (Hoechst, Frankfurt, Germany) and stored at 4°C. The stained neurons were digitally traced using Neurolucida (MicroBrightField, Colchester, VT, USA).
Passive membrane parameters
The passive membrane parameters (Rm, Ri, and Cm) were determined as previously described (Roth and Häusser, 2001; Stuart and Spruston, 1998). Briefly, before engaging the two-electrode voltage clamp, both electrodes were in bridge mode of the Axoclamp-2B. In this configuration, a 0.5 ms pulse of 0.5 nA of current was injected via one of the electrodes and the voltage deflection was monitored by both electrodes. The passive membrane properties were determined simultaneously by direct fitting of the average of 30 voltage traces measured in the same cell (Clements and Redman, 1989; Roth and Häusser, 2001; Stuart and Spruston, 1998). The fitting was carried out using NEURON by routines kindly provided by A. Roth (for details see Roth and Häusser, 2001).
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXACKNOWLEDGEMENTSREFERENCES |
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The reverse problem in cable theory
Cable theory enables calculation of membrane currents in an arbitrary voltage-clamp experiment when the geometry, conductance-density distribution, and conductance kinetics are known (Rall, 1959). However, when obtaining clamp currents from voltage-clamp recordings, we are faced with the reverse problem, namely to determine the conductance kinetics, distribution, and density from the recorded currents. With a space-clamped structure, all the missing kinetic parameters can usually be extracted directly from the recorded currents, thus automatically solving the reverse problem, resulting in a full description of channel kinetics.
However, with nonisopotential structures, solution of the reverse problem is far from obvious. Our line of argument for the solution of this problem is presented in Figs. 2–4 for time-independent conductances of increasingly complex voltage dependence. In general, the reverse problem is equivalent to extracting multiple parameters (i.e., several gn(Vn) sufficient to accurately describe the conductance g(V)) from multiple measurements (i.e., the clamp currents In measured in various voltage-clamp experiments). As shown below, this multiparameter problem can be split into successive one-parameter problems such that only one conductance gn at a time has to be extracted from an appropriately chosen clamp-current recording. As argued above, with the conductance gn known, the clamp current I(gn) can be calculated by cable theory. If this function is unique, it can be inverted, yielding gn(In), and thus solving the reverse problem. In the one-parameter case, monotonicity of the clamp current-conductance relation is sufficient for the function I(gn) to be invertible.
Solving the reverse problem for simple conductances
A conductance with a hyperpolarized reversal potential, such as potassium, was distributed homogenously into a cylinder. A voltage clamp was applied (Fig. 2, A and D) and the steady-state situation was examined. First, a voltage-independent conductance was studied (Fig. 2, A and B). The relation between the measured clamp current Iclamp and the underlying ohmic conductance g can be determined analytically by cable theory as 
(diameter d, intracellular resistivity Ri, and reversal potential Ek (see Koch, 1999, chapter 2, adapted from Eq. 2.17)), which is indeed monotonic for a given clamp voltage V (Fig. 2 C).
A slightly more complex conductance was examined in Fig. 2 E: the conductance had a step dependence on voltage, i.e., being constant up to a voltage V0 and constant again for voltages larger than V0. Voltage clamping to voltages smaller than V0 yields the same situation as in Fig. 2, A–C: The entire cable was exposed to voltages more negative than V0 (Fig. 2 D), due to the assumption of a reversal potential at hyperpolarized voltages. Since g(V) is constant throughout the entire voltage range V < V0, g is therefore in this experiment constant throughout the entire cable (cf. Fig. 2 E). Thus, g0 = g(V, V 
V0) can be uniquely determined from a voltage-clamp experiment with a voltage step to V0 as described above for a nonvoltage-dependent conductance. In a second step, g1 = g(V, V > V0) can be extracted from a voltage-clamp experiment to V1: Since g(V, V V0) is known from the first step, again only one parameter, g1, remains to be determined. The relation between the clamp current and the conductance g1 is again monotonic as shown in the Appendix (Eqs. 1, 3, and 4, Fig. 2 F). Therefore, also for a one-step voltage-dependent conductance as in Fig. 2 E, g(V) can be calculated from the recorded clamp currents, solving the reverse problem.
This argument can be extended to an arbitrary conductance g(V) as follows: First, consider a g(V) that is a multistep function ("staircase") with identical voltage stepwidth 
V with N points (e.g., black line in the lower panel of Fig. 3 A). Then we can argue the following regarding the correction of one of the conductance steps (k).
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V will be faced with the same situation as in Fig. 2, D–F: Only one parameter has to be determined (that is the conductance g1 = g(V, V0 < V V1), as no voltages above V1 occur and the conductance for voltages less than V0 is known (i.e., assumed to be negligible). Since, as in Fig. 2, D–F, the relationship between clamp current and g1 is monotonic (see also Appendix), g1 can be determined unambiguously, thus yielding g(V, V < V1).
At the second stage of the correction (k = 2), performing a voltage-clamp experiment to V2 = V1 + V results in a similar situation again: Only one parameter has to be determined as V is more negative than V2 in the entire structure (due to the assumption of a hyperpolarized reversal potential) and g(V) is known for V < V1. This missing parameter is g2 = g(V, V1 < V V2). Since the clamp current-g2 relationship I(g2) is again monotonic (as in Fig. 2, cf. Appendix), g2 can be determined from the clamp current, resulting in the knowledge of g(V) for all voltages up to V2: g = 0 for V V0, g = g1 for V0 < V V1; g = g2 for V1 < V V2. At this step, g is still unknown for voltages above V2.
At the rest of the correction stages (2 < k < N), continuing this procedure in steps of V results in a stepwise calculation of g(V) for all voltages of interest. The conductance g(V, Vn-1 < V Vn) can be calculated from a voltage step to a voltage Vn = Vn-1 + V, relying on the knowledge of g(V, V Vn-1) from previous calculation steps (see Fig. 3). Essentially every smooth function g(V) can be approximated by a multistep "staircase" function with sufficiently small stepwidth V. Thus, in approximation, the reverse problem in a cylinder can be solved for any voltage-dependent conductance (with little activation at hyperpolarized potentials and a hyperpolarized reversal potential) by applying successive voltage-clamp experiments and using the data obtained from previous experiments to process current recordings for higher voltages.
The correction algorithm
To implement the approximation technique described above, we relied on i), accurate knowledge of passive properties and morphology (discussed in detail below); ii), the ionic current being nonregenerative (i.e., the reversal potential being below the region of activation for this channel—for simplicity, a potassium current with a hyperpolarized reversal potential was assumed); and iii), the conductance being negligible at resting membrane conditions, so that g(V, V < -80 mV) = 0 could be assumed as a starting condition. In the first step of the correction algorithm, the conductance between -80 mV and -70 mV (that was assumed to be constant throughout this voltage range) was determined from the clamp current I1 obtained from a voltage-clamp protocol stepping from -110 mV to -70 mV (see Fig. 1 C). This was achieved by simulating the voltage-clamp experiment in the given morphology with a time-independent potassium conductance that was different from 0 (but constant g1) only for voltages above -80 mV. The magnitude g1 of the potassium conductance was varied systematically to minimize the squared difference (Iclamp,simulated - Iexp,1)2 (using the golden section algorithm (Press et al., 1992)).
In the next step, the conductance was now known between -80 mV and -70 mV from the first correction step and assumed constant again but yet unknown for the next voltage region from -70 mV to -60 mV, in analogy to Fig. 2, D–F. The conductance for this more depolarized voltage region could again be obtained from the voltage-clamp data, this time using the recorded clamp current I2 for a voltage step from -110 mV to -60 mV. Again, simulations of the voltage-clamp experiment were performed in the given morphology with the potassium conductance g = 0 (for V -80 mV); g = g1 (for -80 mV < V -70 mV), and g = variable (for -70 mV < V -60 mV; higher voltages did not occur due to the assumption of a hyperpolarized reversal potential). To obtain g2 = g(V, -70 mV < V -60 mV), it was varied to minimize the difference (Iclamp,simulated - Iexp,2)2.
This process was continued in 10 mV steps until the entire activation curve g(V) of the conductance was retrieved by this approximation technique (Fig. 3 A). In the following, the accuracy was increased by replacing the stepwise constant approximation with a stepwise linear approximation of the conductance (see also legend to Fig. 3 B and Appendix). As the voltage profile narrowed with increasing voltage, the step size was increased to 20 mV for (V - Ek) > 80 mV, i.e., V > 0 mV.
In general, rather than correcting distorted clamp currents, then converting the corrected current to a conductance and finally to a conductance density, the algorithm is used to directly estimate the conductance density to fit the resulting clamp currents to the "experimentally recorded" ones.
Testing the algorithm
To test the algorithm, we used known conductances to obtain simulated voltage-clamp currents, distorted by a lack of space clamp. Here, the term "simulated currents/conductances" refers to these currents, and the term "corrected currents/conductances" describes currents/conductances generated by the correction algorithm. It has to be emphasized that the simulated currents were analyzed by the correction algorithm without any prior knowledge of the kinetics or distribution of the conductance that had been used to generate them.
First, a simple time-independent potassium conductance was implemented in an unbranched cable (Fig. 4 A). Its Boltzmann activation curve is depicted in Fig. 4 B (dashed line; gmax = 30 pS/µm2, V1/2 = -20 mV, inverse slope = 8 mV). Direct estimation of the activation parameters from the distorted clamp-current recording (solid line in Fig. 4 B) yielded a shallow slope and no saturation at high voltages. However, the correction algorithm accurately retrieved the activation parameters: gmax = 30.3 ± 0.1 pS/µm2, V1/2 = -19.1 ± 0.1 mV, inverse slope = 8.2 ± 0.1 mV. The range of parameters that can be accurately retrieved was assessed systematically by varying the slope and V1/2 of the activation curve (Fig. 4, C and D, respectively). Steep activation curves yielded increasingly worse fits for inverse slopes smaller than 2 mV (i.e., 25–75% activation within 4 mV). For shallow activation curves with inverse slopes >32 mV, a decrease of the accuracy of the fit was observed as a result of significant activation at the starting potential (V = -80 mV). When V1/2 was varied (Fig. 4 D), accurate values for all Boltzmann parameters were retrieved for V1/2 > -40 mV. For smaller values of V1/2, once again significant activation at -80 mV resulted in deviations from the original Boltzmann activation curve. Similar results were obtained with a nonmonotonic conductance-voltage relation 
(data not shown).
These findings indicate that our correction algorithm is capable of extracting steady-state activation parameters in a non-space-clamped situation for all but extremely steeply activating channels or those with a substantial activation at rest. Thus, the algorithm can in principle solve the reverse problem.
Spatial resolution of the algorithm
In realistic situations, the membrane properties are neither known completely nor necessarily homogenous. Thus, we investigated whether i), distal regions of unknown structure affect the precision of the algorithm, and whether ii), the obtained activation curve reflects local membrane properties or rather some average across parts of the structure.
These concerns were assessed by first studying the influence of changes in distal membrane properties on clamp-current recordings (see Fig. 5, A and B, and Appendix). A cylinder was divided into a middle region and two distal ones (Fig. 5 A). For simplicity, these cylindrical regions were modeled with constant membrane conductances gM and gD, respectively. Then, gD was varied and the relative change in Iclamp was noted for different gM/gD (or Rside/R
,middle, cf. Appendix) ratios. Even when the distal region was as close as 
, 50% changes of the distal membrane conductance gD did not alter Iclamp by more than 12% (Fig. 5 B, middle trace). The alteration of Iclamp was most prominent for situations where gD approximately equaled gM (Fig. 5 B).
In addition, a spatial conductance step was implemented in a cylinder. The simulated clamp currents at various positions (Fig. 5 C) were corrected without any a priori assumptions about the channel distribution. Since channel distributions are in general not known in an experimental situation, a homogenous channel distribution was assumed for correction of the individual recordings at different positions. The corrected conductance parameters for a conductance step from 3 to 30 pS/µm2 at various positions are depicted in Fig. 5 C: Both V1/2 and the inverse slope of the activation curve were retrieved accurately throughout the cylinder. Deviations from the actual values (dashes) were small and occurred primarily close to the conductance step. The conductance density was retrieved as a sigmoid, whose inverse slope indicated the spatial resolution of the correction algorithm. Repetition of this simulation and correction procedure for various conductance densities and passive parameters showed the dependence of the resolution on the "active space constant", 
active. This term refers to the apparent space constant when voltage-gated channels are open, 
. For Rm = 30,000 cm2 (gpas = 0.33 pS/µm2) and gact = 30 pS/µm2, 
which is only 10% of the passive space constant. Thus, information obtained was averaged over a small region that extended only on the order of active around the recording electrode (Fig. 5 D).
Temporal resolution of the algorithm
The applicability of the algorithm to non-steady-state situations was tested by studying the sensitivity of clamp-current recordings to temporal changes in channel activation, and in addition by correcting clamp-current recordings from potassium channels with various activation times. For the non-steady-state case, we have to assume a kinetic model for the time course of g(V,t), this model being as simple as possible. Therefore, clamp currents recorded during activation of voltage-gated channels were corrected independently for every time point t1: For each t1, we assumed a simple kinetic "model" g = const for all times less than t1 and approximated the voltage-dependent conductance g(V) to fit the corrected clamp current at time t1 to the simulated current, thus obtaining the activation curve at time t1: g(V,t1). This was repeated independently for each time point to yield the complete activation kinetics g(V,t) as outlined in Fig. 3.
To assess the errors made by this "pseudo-steady-state" approach, the dependence of the clamp current on the recent history of the membrane conductance was examined. A voltage clamp with a voltage step from -80 mV to 0 mV at t = 0 was performed, and the clamp current was measured at t1 = 100 ms. Then a constant membrane conductance was implemented in the cylinder and switched on at varying times ton, 0 < ton < t1 (Fig. 6 A). This conductance made the clamp current change as a function of t = t1 - ton. The half maximum of a log-sigmoid fit to Iclamp(t) yielded a measure of the temporal resolution ("memory") of the clamp-current recording. The same procedure was applied to a conductance that was initially on, switched off at varying times toff, and on again at t1 = 100 ms (Fig. 6 B), and to a conductance that increased linearly from 0 at time ton to a maximum value at t1 = 100 ms (data not shown). The dependence of the temporal resolution of the system on the apparent membrane time constant upon activation of all conductances (active membrane time constant: 
active, cf. legend to Fig. 6) is shown in Fig. 6 C.
Active time constants are usually small (active 
300 µs for a conductance with a density of 30 pS/µm2), which encouraged us to take the approach described above, i.e., not to assume a special kinetic model for the underlying conductances, but rather to fit every time point independently with a time-independent conductance. The fitted conductance is expected to reflect the time course of the actual underlying conductance with an accuracy of about active.
The validity of this "pseudo-steady-state" approach was further assessed by implementing a simplified potassium-channel model in a cable (Fig. 6 D). This channel model consisted of one activation particle with a voltage-independent activation time (channel = 8 ms) and a first-order steady-state activation model as in Fig. 4 (gmax = 10 pS/µm2, V1/2 = -20 mV, inverse slope = 8 mV). Channel kinetics directly estimated from the simulated distorted clamp currents deviated from the actual values (V1/2 = -12.8 ± 0.8 mV, inverse slope = 15.6 ± 0.7 mV, channel = 10.4 ± 0.1 ms; channel was determined for the voltage step to -10 mV). In contrast, by using the correction algorithm, the channel parameters could be retrieved accurately (gmax = 10.10 ± 0.04 pS/µm2, V1/2 = –18.7 ± 0.2 mV; inverse slope = 8.9 ± 0.2 mV and channel = 8.8 ± 0.4 ms). Variation of channel indicated that, for all activation time constants, the steady-state parameters were determined correctly (Fig. 6 E). However, estimation of the channel activation time was less precise for fast channels (channel < 300 µs), which confirmed the prediction based on Fig. 6 C. Higher or lower channel densities (from 1 to 1000 pS/µm2) or membrane resistances (from 2000 to 128,000 cm2) did not influence algorithm performance (not shown).
Sensitivity and numerical stability
The algorithm, due to its structure, cannot be tested analytically for numerical stability. Therefore, stability was tested by adding various white noise levels to a simulated current recording from a cylinder with a simple underlying conductance (Fig. 7 A; gmax = 10 pS/µm2, V1/2 = -20 mV, inverse slope = 8 mV, channel = 7 ms). The corrected conductance density for a noise level with root mean square (rms) of 10 pA (
1% of steady-state current at 0 mV) is depicted in Fig. 7 A. The original kinetics were easily retrieved. The activation curve was markedly distorted only for noise levels above 100 pA rms. (Fig. 7 B). Thus, for experimentally observed noise levels numerical stability of the algorithm can be obtained.
The sensitivity of the correction algorithm to morphological and electrical parameters was also examined. Recordings from a cylinder, equipped with the same conductance as in Fig. 7 A, were made with the same passive parameters as described above. However, for the correction algorithm, passive parameters were modified, thus simulating an incorrect estimate of the true passive parameters. Global increases or decreases in the diameter of the cylinder were used to simulate incorrect morphological reconstruction. Changes in the corrected conductance parameters occurred mainly with incomplete knowledge of the morphology (Fig. 7 C, compare to 7, D–F). Overall, neither noise nor variations in passive parameters or morphology to an extent expected from experimental accuracy resulted in significant deviations of the correction results from the actual underlying channel parameters.
Full channel models
The correction algorithm was tested on more complex channel kinetics by implementing several published models of potassium channels in a 2-mm long cable. Simulations of voltage-clamp experiments produced clamp currents with distortions due to lack of space clamp (black in Fig. 8 A). As expected, apparent activation and inactivation time constants increased and the voltage dependence of the steady-state conductance flattened. The correction algorithm yielded estimates for the conductance kinetics (blue trace in Fig. 8 A) that accurately reflected the underlying channel properties (red trace in Fig. 8 A). The kinetics of fast inactivating channels (Kfast and Ka) and channels with a steep conductance-voltage relation (Kv) were not corrected as properly as slower channels (Km and Kslow). This deviation corresponded to the predictions depicted in Figs. 4 and 6. The contributing membrane areas varied significantly, thus no a priori prediction could be made for the conductance densities. Therefore, the distorted currents are displayed as conductances rather than conductance densities and with different scales to facilitate comparison. The results of applying the algorithm to several conductance densities of each of the five realistic conductances (Kfast: 3, 10, and 30 pS/µm2; Km: 1, 3, and 10 pS/µm2; Kslow: 3, 10, and 30 pS/µm2; Kv: 10, 30, and 100 pS/µm2; and Ka: 10, 30, and 100 pS/µm2) are shown in Fig. 8 B. The maximal conductance obtained at each voltage was normalized to the conductance at the maximal voltage (V = 60 mV). The results obtained by applying the algorithm to these realistic channels demonstrated that correcting every time point independently is sufficient to accurately obtain the kinetics of a wide range of potassium channels.
Correcting in neuronal morphologies
The major goal in designing the correction algorithm was to devise a method that could overcome space-clamp problems in elongated and branched structures, such as large cortical neurons. Therefore, the algorithm was tested on a fully reconstructed L5 neocortical pyramidal neuron (Fig. 9 A), into which was inserted the fast inactivating potassium channel Kfast (Korngreen and Sakmann, 2000). This type of neuron was selected for the simulations because it provided a worst case of problems that arise from improper space clamp. As expected, in somatic recordings the uncorrected conductance calculated from currents that were obtained from a simulated somatic voltage clamp did not properly reflect the underlying kinetics (Fig. 9 A, bottom panel, black versus red traces). In this morphology, as with the cylinder, application of the correction algorithm considerably reduced the error (Fig. 9 A, bottom panel, blue traces). The same was true for simulated currents from the apical dendrite (Fig. 9 A, upper panels). Note that the uncorrected dendritic currents obtained at the indicated positions not only display distorted kinetics, but also incorrect values of the relative channel densities (Fig. 9 A, black traces).
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) and neocortical (Korngreen and Sakmann, 2000) pyramidal neurons, potassium-channel densities change along the apical dendrite. Therefore, we tested whether the correction algorithm could retrieve correct local conductance densities without a priori knowledge of a conductance density gradient. The same reconstructed pyramidal neuron was used with a Kfast conductance density that was constant across soma, axon, and basal dendrites, but increased linearly along the apical dendrite. Again, uncorrected conductances from several recording sites as in Fig. 9 A displayed distorted channel kinetics (not shown). For the correction algorithm, a homogenous density of potassium channels was assumed, since a priori knowledge about the existence of an inhomogeneous distribution is usually not available. Nevertheless, the correction resulted in a correct estimate of the local channel density (Fig. 9 B) and retrieval of the correct kinetics as predicted by the spatial (Fig. 5) and temporal (Fig. 6) resolutions of the algorithm. The correction was successful for all tested locations; the conductance density slope along the apical dendrite was accurately retrieved (Fig. 9 B, inverted triangles). Results of this and of the correction of both a steeper (solid circles) and shallower (triangles) upwards gradients are summarized in Fig. 9 B. A decreasing gradient was also accurately extracted (Fig. 9 C).
Dendritic recordings
All our simulations were performed under the assumption of a negligible series resistance. However, experimentally reached series resistances are at least several M, especially for dendritic recordings that can easily be above 20 M (Stuart and Spruston, 1995; Stuart and Sakmann, 1994). The current injected by the voltage-clamp amplifier results in a voltage drop across the series resistance generated in the whole-cell configuration. Therefore, the voltage reported by most single-electrode voltage-clamp amplifiers is not correct (Armstrong and Gilly, 1992). The bigger the current injected by the amplifier, the bigger will be the error due to the series resistance. Conventional series resistance compensation, a part of any modern patch-clamp amplifier, can reasonably compensate resistances of up to 20 M if the series resistance is measured accurately. When recording from neurons, the large currents and the high series resistance will greatly increase the error, thus rendering the use of patch-clamp amplifiers impossible. In principle, series resistance problems could be overcome by discontinuous voltage-clamp amplifiers. In these amplifiers, the current is injected for short durations followed by periods during which the voltage is recorded. Since the voltage is recorded during a time in which there is no current flowing across the series resistance, it is a true recording of the membrane potential. When this cycle is repeated at a high enough frequency, it is possible to voltage-clamp a cell even in the presence of a relatively high series resistance. However, application of such amplifiers to dendrites is not trivial because of the many membrane time constants inherent to neurons (Major, 1993; Wilson and Park, 1989). Furthermore, the existing discontinuous voltage-clamp amplifiers cannot cycle fast enough to accurately follow fast and large conductance changes. Therefore, the current method of choice for implementing our correction algorithm is the two-electrode voltage clamp. In this method, one electrode injects current while the other records the voltage. Both electrodes have series resistance due to the whole-cell configuration. However, since no current is injected via the voltage-sensing electrode, it is reporting the correct membrane potential. Furthermore, this technique can be tuned to follow very fast and large changes in membrane current.
To comply with the assumptions made while developing the correction algorithm, the morphology of the neuron and the passive parameters were determined as detailed in the methods. To isolate only K+ currents, we blocked voltage-gated Na+ channels with 1 µM TTX, voltage-gated Ca2+ channels by complete substitution of the Ca2+ in the bath with Co2+, and Ih with 30 µM of ZD2788. Two simultaneous whole-cell dendritic recordings were established at an interelectrode distance of 30–40 µm. Under these conditions it was possible to record K+ currents from the dendrites of L5 neocortical pyramidal neurons. Fig. 10 A displays the schematic representation of the two-electrode voltage-clamp circuit with the current-injecting electrode at 220 µm and the voltage-sensing electrode at 180 µm measured from the center of the soma. The distorted currents, recorded voltage, and corrected conductance density are shown alongside the neuron (Fig. 10 B). Due to the initial overshoot of the potential (Fig. 10 B), the activation time course of the current was distorted and therefore not analyzed. The decay of the corrected conductance density was faster than the distorted currents as expected from our simulations. The correction clearly revealed a biexponential decay of the conductance density that was obscured in the distorted currents. This biexponential decay is probably the manifestation of the fast inactivating and slow inactivating K+ conductances that are known to be present in the dendrites of layer 5 neocortical pyramidal neurons (Bekkers, 2000; Korngreen and Sakmann, 2000). The steady-state activation curve of the distorted conductances, calculated by dividing the distorted currents by the driving force, could be fitted to a Boltzmann function to obtain a V1/2 = 27 mV and k = 22 mV (average of 21 ± 3 mV and 17 ± 2 mV, n = 3). After correction, these parameters had values of V1/2 = 18 mV and k = 16 mV (average of 17 ± 2 mV and 14 ± 1 mV, n = 3). The corrected slope factor was not different (P > 0.1, ANOVA) from that obtained using somatic nucleated patches (12 ± 1 mV, n = 6) (Korngreen and Sakmann, 2000). The V1/2, after correction for a liquid junction potential of 12 mV, was significantly different (P < 0.05, ANOVA) from the published value (-1 ± 2 mV, n = 6) (Korngreen and Sakmann, 2000).
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXACKNOWLEDGEMENTSREFERENCES |
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). Such measures essentially flatten the voltage attenuation from the point of the clamp, thus reducing the error in the voltage. However, this only applies to structures with relatively short dendritic trees. Here we demonstrate that our algorithm performs best in the almost complete absence of space clamp. Since voltage attenuation upon channel activation is much narrower than in the well-studied passive case, the region that significantly influences the clamp current is limited to a narrow region around the electrode ("poor space clamp"). Thus, any method for estimation of membrane conductances will retrieve an averaged conductance only over this region, enabling high spatial resolution for poorly space-clamped structures (a small active ; Fig. 5). The same principle applies to the temporal domain as follows. Many channel kinetics are slow compared to the membrane time constant for activated conductances (that is again much smaller than the well-known passive membrane time constant), therefore facilitating adaptation of our correction algorithm for the study of channel kinetics (Fig. 6).
With simplified channel models, which are described by only four parameters (three parameters for the activation curve in steady state and one voltage-invariant activation time constant), the correction algorithm accurately retrieved the entire channel model. Limitations of the algorithm were addressed by varying the four parameters (Figs. 4 and 6). Channels with a steep steady-state activation curve (inverse slope of the Boltzmann curve, k 4 mV) could not be fully corrected due to the finite step size of the linear interpolation. Similarly, with very fast channel activation times (channel << active) a poorer correction was achieved, since essentially no kinetic model was assumed. Although these two difficulties are not a major concern in realistic channel models, they could be accounted for by including a priori assumptions about the activation curve or underlying channel kinetics. However, such assumptions substantially reduce the generality of the correction approach. Other problems, such as decreases in correction accuracy with decreasing V1/2 of the Boltzmann activation curve, can be overcome by minor modifications of the algorithm, such as omission of the assumption of no channel activation at rest.
Correct implementation of the algorithm requires determination of both the morphology and passive parameters of the structure being studied. Given enough reconstructions, implementation might be simplified by building a canonical view of the specific neuron to obtain at least a first-order correction at a much reduced computation time, especially since the correction is sensitive primarily to local morphological features (Fig. 5). The effects of changes in distal morphology are in general minor (as partially discussed in the Appendix). Simplifying distant parts of the dendritic tree could thus substantially both ease and speed up the reconstruction process and enhance computational efficacy. This becomes important not mainly for the demonstration of the basic principles (that are assessed in a cylinder model) but for the application to various complex dendritic trees. We observed significant errors in the correction only if the reconstruction became poor in regions closer than 200 µm to the recording site on the dendrite (data not shown). For experimental implementability it could therefore be useful to extract the local dendritic morphology during the experiment (e.g., by imaging methods) and combine it with a simple generic cell morphological model to result in a quick semi-online correction. In a post hoc analysis, a slightly more detailed morphological measurement might already be sufficient. This can always be tested by further simplifying the morphology and testing whether any significant changes in the corrected conductance values occur. If this is not the case, it is safe to assume that the morphological reconstruction is sufficient.
Of concern also is the accuracy of the passive membrane parameters (Rm, Ri, and Cm). A 10–20% error in the passive parameters did not significantly affect the final outcome (Fig. 7). The passive parameters of a given neuron can be estimated with reasonable accuracy once its morphology is known (Clements and Redman, 1989; Major et al., 1994; Roth and Häusser 2001; Stuart and Spruston, 1998). This requires an additional pulse protocol in current-clamp mode, before the voltage-clamp experiment, and a full reconstruction of the neuron afterward (Clements and Redman, 1989; Major et al., 1994; Roth and Häusser 2001; Stuart and Spruston, 1998). As the neuron usually is to be reconstructed for use in the correction algorithm, only additional pulse protocols in current clamp need to be performed. Previously, many recorded sweeps were claimed to be needed to obtain a sufficiently accurate estimate of the passive parameters (Major et al., 1994). However, using two electrodes to record such sweeps (Roth and Häusser, 2001
; V1/2 = -20 mV, gmax = 30 pS/µm2, inverse slope k = 8 mV (dashed line). The solid line indicates the direct estimate of the activation curve from the distorted clamp currents (clamp currents divided by the driving force). Note the shallow slope, lack of saturation at higher voltages, and different scales (nS versus pS/µm2) since normalization to the membrane area was not possible. The parameter estimates were gmax = 48 ± 1 nS, V1/2 = -14 ± 0.8 mV, and inverse slope k = 15.8 ± 0.8 mV. The corrected conductance densities are depicted as triangles (gmax = 30.3 ± 0.1 pS/µm2, V1/2 = -19.1 ± 0.1 mV, inverse slope = 8.2 ± 0.1 mV). The voltage-clamp experiment was simulated for different slopes (C) and different V1/2 (D) of the Boltzmann activation curve. The three conductance parameters that fully describe the conductance were calculated from a Boltzmann fit to the corrected conductance-voltage relation (triangles in B). Error bars indicate the goodness of the Boltzmann fit depicted as standard deviation in fit parameters estimated from the residuals. The actual parameters of the underlying conductances are shown as dashed lines.
; 
. (C) A voltage-clamp experiment was simulated in a cable equipped with a Boltzmann-like conductance as in 
1000 µm. For each position, the simulated clamp currents were corrected assuming a homogenous conductance distribution. The corrected g(V) obtained was analyzed as in 
, with the membrane conductance 
. Error bars indicate the goodness of the sigmoidal fit depicted as standard deviation in fit parameters.
, with 
. (D) A voltage-clamp experiment was simulated as in 
with 
, V1/2 = -20 mV, gmax = 10 pS/µm2, inverse slope k = 8 mV, and 
where g/gmax is the conductance normalized to its maximal value, V is membrane potential, V1/2 is the voltage at which half of the "gates" of all channels are in the open configuration, and k is the slope factor. gmax was normalized to the true somatic value for display purposes.