Abstract
The precise shape of action potentials in cortical neurons is a key determinant of action potentialdependent Ca^{2+} influx, as well as of neuronal signaling, on a millisecond scale. In cortical neurons, Ca^{2+}sensitive K^{+} channels, or BK channels (BKChs), are crucial for action potential termination, but the precise functional interplay between Ca^{2+} channels and BKChs has remained unclear. In this study, we investigate the mechanisms allowing for rapid and reliable activation of BKChs by single action potentials in hippocampal granule cells and the impact of endogenous Ca^{2+} buffers. We find that BKChs are operated by nanodomains of single Ca^{2+} channels. Using a novel approach based on a linear approximation of buffered Ca^{2+} diffusion in microdomains, we quantitatively analyze the prolongation of action potentials by the Ca^{2+} chelator BAPTA. This analysis allowed us to estimate that the mean diffusional distance for Ca^{2+} ions from a Ca^{2+} channel to a BKCh is ∼13 nm. This surprisingly short diffusional distance cannot be explained by a random distribution of Ca^{2+} channels and renders the activation of BKChs insensitive to the relatively high concentrations of endogenous Ca^{2+} buffers in hippocampal neurons. These data suggest that tight colocalization of the two types of channels permits hippocampal neurons to regulate global Ca^{2+} signals by a high cytoplasmic Ca^{2+} buffer capacity without affecting the fast and brief activation of BKChs required for proper repolarization of action potentials.
 microdomains
 calbindin
 calcium buffer
 action potential
 BK channel/Slo
 calcium
 diffusion
 hippocampus
 patch clamp
Introduction
Ca^{2+} is a versatile intracellular messenger that translates neuronal activity into a variety of short and longterm cellular actions such as transmitter release, gene transcription, or neuronal cell death. Action potentials represent one of the most potent Ca^{2+} influx pathways because they depolarize large parts of the neuronal membrane and thereby open many voltagegated Ca^{2+} channels. The amount of Ca^{2+} entering per action potential is crucially determined by the shape of the action potential, because most influx occurs during the repolarization of the action potential (Llinas et al., 1982). For this reason, the precise regulation of the duration of action potentials is important to many neuronal functions, and neurons must possess effective mechanisms to limit Ca^{2+} influx during action potentials and to avoid an overload with Ca^{2+} during repetitive spiking activity. Many cortical neurons use largeconductance Ca^{2+}activated potassium channels (BKChs) for that purpose. These channels have the unique property that they are activated by both the depolarizing voltage as well as by the rise in the intracellular free Ca^{2+} concentration during an action potential, which therefore provide ideal means for a negative feedback mechanism limiting the amount of Ca^{2+} entry (Vergara et al., 1998; Sah and Faber, 2002; Lu et al., 2006).
Although the biophysical properties of BKChs make them ideally suited to keep action potentials narrow and limit Ca^{2+} entry, it is unclear how neurons manage to use them effectively. For example, single action potentials potently activate BKChs but primarily avoid recruitment of other Ca^{2+}activated channels, such as Ca^{2+}activated, nonselective cation channels, which would depolarize the membrane and counteract the effect of BKChs (Partridge et al., 1994). Furthermore, BKChs remain open only for a brief period of time (a few milliseconds) (Shao et al., 1999) and therefore do not interfere with repetitive spiking, which is necessary for highfrequency encoding of neuronal information. It is therefore a fundamental question as to how neurons are able to assure the selective, rapid, brief, and reliable activation of BKChs during action potentials.
Neurons typically contain some kind of Ca^{2+}binding proteins that allow them a more complex signal integration by shaping the spatiotemporal spread of Ca^{2+} signals (Maeda et al., 1999; Jackson and Redman, 2003). Ca^{2+} buffers can potentially impose severe limitations on the speed, reliability, and duration of the activation of BKChs. Indeed, it is known that the Ca^{2+}binding protein calbindinD_{28k} strongly limits the activation of BKChs in certain sensory neurons (Roberts, 1994). Several cortical neurons, such as hippocampal granule cells, also contain large amounts of calbindinD_{28k} (Muller et al., 2005). However, it is as yet unknown whether endogenous Ca^{2+} buffers have any impact on the action potentialinduced activation of BKChs in cortical neurons.
In this study, we investigate the impact of endogenous and exogenous Ca^{2+} buffers on the repolarizing efficacy of BKChs and report that cortical neurons use a precise strategic placement of BKChs to optimize their recruitment by single action potentials under most physiological circumstances.
Materials and Methods
Preparation of slices and recording configuration in current clamp and hybrid clamp.
Wistar rats (20–40 d old) or adult mice (mice were used for experiments shown in Fig. 2, E and F, calbindin−/− mice) were decapitated under ether anesthesia. The brains were rapidly dissected and chilled in 4°C cutting solution consisting of the following (in mm): 87 NaCl, 2.5 KCl, 1.25 NaH_{2}PO_{4}, 7 MgSO_{4}, 0.5 CaCl_{2}, 25 NaHCO_{3}, 25 glucose, 75 sucrose, gassed with a 95% O_{2}/5% CO_{2} mixture, pH 7.4. Horizontal hippocampal slices (400 μm) were prepared with a Vibratome (Leica, Nussloch, Germany). Until recording, slices were stored at room temperature in artificial CSF (ACSF) containing the following (in mm): 124 NaCl, 3 KCl, 1.25 NaH_{2}PO_{4}, 2 MgSO_{4}, 2 CaCl_{2}, 26 NaHCO_{3}, 10 glucose, 300 mOsm osmolarity, pH 7.4, gassed with a 95% O_{2}/5% CO_{2} mixture. For recording, slices were transferred to an interface chamber perfused with ACSF (35 ± 0.5°C) that contained the following (in mm): 125 NaCl, 3 KCl, 2 CaCl_{2,} 2 MgCl_{2,} 1.25 NaH_{2}PO_{4,} 20.3 NaHCO_{3}, 12 dglucose, pH 7.4, gassed with 95% O_{2}/5% CO_{2}. Patch pipettes had resistances between 4 and 5 MΩ. The standard pipette solution was free of any Ca^{2+} chelators and contained the following (in mm): 140 K^{+}gluconate, 0.5 MgCl_{2}, 4 NaCl, 10 HEPES, 5 KCl. A small amount of EGTA (20 μm) was added to the solution to balance Ca^{2+} contamination of salts. When Ca^{2+} chelators [1,2bis(2aminophenoxy)ethaneN,N,N′,N′tetraacetic acid (BAPTA) or EGTA] were present, we supplemented CaCl_{2} to maintain a free [Ca^{2+}] of ∼38 nm (calculated with “webmaxc” available at http://www.stanford.edu/∼cpatton/maxc.html) (Patton et al., 2004). Four different pipette solutions were used (3, 6, and 10 mm BAPTA; 10 mm EGTA). Solution 1 contained the following (in mm): 120 K^{+}gluconate, 4 NaCl, 0.52 MgCl_{2}, 3 K_{4}BAPTA, 0.02 EGTA, 10 HEPES, 0.25 CaCl_{2}, 5 KCl. Solution 2 contained the following (in mm): 115 K^{+}gluconate, 4 NaCl, 0.54 MgCl_{2}, 6 K_{4}BAPTA, 0.02 EGTA, 10 HEPES, 0.49 CaCl_{2}, 5 KCl. Solution 3 contained the following (in mm): 80 K^{+}gluconate, 4 NaCl, 0.57 MgCl_{2}, 10 K_{4}BAPTA, 0.02 EGTA, 10 HEPES, 0.81 CaCl_{2}, 5 KCl. Solution 4 contained the following (in mm): 10 EGTA, 140 K^{+}gluconate, 4 NaCl, 0.84 MgCl_{2}, 10 HEPES, 2.56 CaCl_{2}, 5 KCl, pH 7.3, KOH, ∼295 mOsm. If necessary, the osmolarity was adjusted by adding sucrose. Currentclamp recordings made with potassium gluconatebased solutions were corrected for a liquid junction potential of −10 mV by setting the reference voltage to −10 mV before seal formation.
Wholecell recordings were obtained from granule cells in hippocampal slices using the “blind technique” and an SEC05LX amplifier (standard headstage; NPI, Tamm, Germany) as described previously (Muller et al., 2005). Voltage and current signals were filtered at 8 kHz and 700 Hz, respectively, and digitized at 30 kHz with an ITC16 interface (InstruTECH, Port Washington, NY) operated by TIDA (HEKA Elektronik, Lambrecht, Germany). Series resistance of the cells ranged between 20 and 30 MΩ and was estimated by bridge balancing after careful compensation of the pipette capacitance.
Hybridclamp recordings of the current underlying the fast afterhyperpolarization (I_{fAHP}) were performed as described previously (Dietrich et al., 2002). The switching frequency was set to 20–25 kHz. For a hybridclamp recording, the SEC05LX amplifier operates in discontinuous singleelectrode current clamp until an action potential triggered a mode switch to discontinuous singleelectrode voltage clamp (dSEVC) for several seconds by a pulse generator (Master8; A.M.P.I., Jerusalem, Israel). The mode switch was delayed by an interval (0.1–0.5 ms) chosen individually for every cell such that the switch coincided with that time point at which the falling phase of the action potential crossed approximately −40 mV. Because the cells were subsequently held at −40 mV in dSEVC mode, there was usually a small voltage step (±5 mV) associated with the mode switch that gives rise to a contaminating transient capacitive current. To correct for this current, we proceeded as follows. The amplitude of the voltage step during the switch was measured in the voltage trace. Then, an appropriately scaled current response to a −10 mV jump was aligned and subtracted from the hybrid trace.
Quantification of action potential duration.
We induced action potentials by the injection of a depolarizing current such that the granule cells fired at <10 Hz. For an exact and reproducible quantification of the action potential duration, we proceeded as follows. We calculated two “reference” average action potentials of the control and of the triple toxin group in each case of five action potentials. We superimposed both reference action potentials and projected a vertical line from the time point of the peak of the control fAHP (see Fig. 1B, lower dot) to the tripletoxin action potential (see Fig. 1B, upper dot). All action potentials to be measured were aligned to the fast rising phase of both reference action potentials (as shown in Fig. 1B). The spike duration of the aligned action potentials was then measured at the voltage level defined by the latter dot.
Incubation with BKCh blockers and in low [Ca^{2+}].
Slices were incubated at room temperature in a small Petri dish and with bubbled 95% O_{2}/5% CO_{2} ACSF containing albumin (0.1 mg/ml) and 10 μm paxilline, 1 μm charybdotoxin, or 1 μm iberiotoxin, a combination of all three blockers (triple toxin), or no blocker for ∼1–4 h. After incubation, slices were transferred to an interface chamber perfused with ACSF (35 ± 0.5°C) (as described above) that, in addition, contained albumin (0.1 mg/ml) and 10 μm paxilline (10 μm), 45 nm charybdotoxin, or 60 nm iberiotoxin, all three blockers, or no blockers. There were no differences regarding action potential configuration or duration between nonincubated slices and slices incubated without blockers. For experiments in which action potentials were analyzed in low extracellular Ca^{2+} concentrations, slices were stored immediately after preparation in ACSF with low Ca^{2+} (1.5, 0.7, 0.5, and 0.2 mm). Reductions in extracellular Ca^{2+} were equimolarly compensated for by Mg^{2+}.
Conventional voltageclamp recordings of Ca^{2+} currents
Wholecell recordings of voltagegated Ca^{2+} currents were either obtained from acutely isolated granule cells or from granule cells in slices. Because the results were not different between the two techniques, data were pooled.
For recording granule cells in situ, slices were prepared as described above, transferred to the stage of an upright microscope, and visualized using infrared video microscopy and differential interference contrast optics (E600 FN; Nikon Tokyo, Japan). Slices were perfused with ACSF at room temperature, which contained the following (in mm): 125 NaCl, 3 KCl, 2 CaCl_{2}, 2 MgCl_{2}, 20.3 NaHCO_{3}, 12 dglucose, 0.0005 TTX, pH 7.4, gassed with 95% O_{2}/5% CO_{2}. Patch electrodes (3–5 MΩ) were filled with the following solution (in mm): 120 Cs^{+}gluconate, 4 MgCl_{2}, 4 Na_{2}ATP, 2 EGTA, 10 HEPES, 20 TEACl, 15 CsCl, pH adjusted to 7.3 (CsOH), osmolarity adjusted to ∼295 mOsm. Currents were recorded with an EPC7 amplifier (HEKA Elektronik) operated by Igor Pro (WaveMetrics, Lake Oswego, OR). Series resistance ranged between 10 and 20 MΩ and was compensated electronically by 60–70%.
Isolated granule cells were prepared as described previously (Beck et al., 1997) and superfused with solution containing the following (in mm): 90 Na^{+}methanesulfonate, 20 TEA, 5 4aminopyridine, 2 CaCl_{2}, 2 MgCl_{2}, 3 KCl, 10 glucose, 26 NaHCO_{3}^{−}, 0.0005 TTX, pH 7.4, 315 mOsm osmolarity, gassed with 95% O_{2}/5% CO_{2}. Patch pipettes (4–5 MΩ) were filled with an intracellular solution containing the following (in mm): 87.5 Cs^{+}methanesulfonate, 20 TEA, 0.5 CaCl_{2}, 5 MgCl_{2}, 5 BAPTA, 10 HEPES, 10 glucose, 10 Na_{2}ATP, 0.5 Li_{2}GTP, pH 7.2, 300 mOsm (adjusted with sucrose). Wholecell recordings were obtained at room temperature, and currents were recorded using an EPC9 (HEKA Elektronik). Series resistance ranged between 8 and 15 MΩ and was compensated by 50–70%.
Cells were held at −75 mV, and Ca^{2+} currents were evoked every 10–30 s by an action potentiallike command potential. The potential was instantaneously (or within 1 ms) stepped to +25 mV and returned to −75 mV using a linear ramp waveform with a duration of 5 ms. This protocol evoked a tail current that peaked approximately after half of the voltage ramp. All currents were leak corrected using either a +P/10 or a +P/5 strategy. To test the Ca^{2+} sensitivity of the peak of the tail currents, extracellular Ca^{2+} was lowered from 2 mm to 1.5, 0.7, 0.5, and 0.2 mm. Reductions in extracellular Ca^{2+} were equimolarly compensated for by Mg^{2+}.
CalbindinD_{28k} knockout mice.
The calbindinD_{28k} null mutant mice used in this study were described previously (Airaksinen et al., 1997). We obtained mice from The Jackson Laboratory (Bar Harbor, ME), and homozygous knockout mice were achieved by heterozygous breeding. All animals were genotyped by PCR using DNA samples extracted from tail tissue. The wildtype primers used were as follows: 5′GCA AGT AAC TAA TGG CAT CG 3′ and 5′TGC AGC GGC TAG TTT GAG AGT G3′. The null mutant primers used were as follows: 5′ACA TCG CAT CGA GCG AGC AC3′ and 5′AAG GCG ATG CGC TGC GAA TC3′.
Numerical simulation of buffered Ca^{2+} diffusion around a single Ca^{2+} channel.
Numerical simulations were performed to validate the linear approximation of microdomains as described previously (Muller et al., 2005). We were interested in the spatial and temporal development of free Ca^{2+} around the channel after its opening in the presence and absence of the buffer BAPTA. We consider a single Ca^{2+} channel that opens for 0.8 ms and is embedded in a planar membrane that limits a pseudoinfinitely large hemisphere (radius, 30 μm) around the channel that was divided into 30,000 shells. Within the first 600 nm from the origin, the shell size was constant at 0.06 nm, and the size of the following shells increased by 5% successively. In most preparations, a poorly mobile or even fixed Ca^{2+} buffer that will not wash out during the recording is found (Neher, 1995). For this reason, we incorporated a low amount of fixed Ca^{2+} buffer capacity (κ_{fB} ∼ 10; Eq. 5) with Ca^{2+}binding properties similar to those assumed in previous simulation studies (Nowycky and Pinter, 1993) (Table 1). The fixed Ca^{2+} buffer was homogeneously distributed in the diffusion volume. Diffusion coefficients and Ca^{2+} binding properties are summarized in Table 1. The maximum singlechannel Ca^{2+} current (at physiological levels of divalent cations) was assumed to be 0.2 pA (Gollasch et al., 1992). Because we are interested in Ca^{2+} microdomains that develop during action potentials, we took into account the varying driving force of the singlechannel current during the repolarizing phase of the action potential. Thus, the singlechannel current is not square like; the channel opens and closes in a steplike manner, but the current steeply increases during the open time and reaches a value of 0.2 pA at the end of the opening. The time course of singlechannel current amplitude was therefore empirically approximated with the following function (in pA): 0.2 × (1 + tanh (5 × (t − 0.2)))/2, where t (in milliseconds) is the time (see Fig. 6A, inset). Simulation starts at t = 0 ms and ends at 1.6 ms, and the channel closes abruptly at t = 0.8 ms. Before the simulation starts, buffers and Ca^{2+} are homogeneously distributed and have reached chemical equilibrium at resting Ca^{2+} concentration (50 nm). The differential equations were integrated using “Calcium Calculator (CalC)” software [version 4.9.8, freeware by V. Matveev (Matveev et al., 2004)]. This scriptoperated program uses the Crank–Nicholson differences scheme (onedimensional) and an adaptive timestep method to maintain a given level of accuracy (<0.1%). Because CalC only offers full spherical geometry, we increased the singlechannel current by a factor of 2, which yields identical results as simulating a hemisphere because of the complete symmetry. The correctness of the program was thoroughly checked against the transient analytical solutions of unbuffered Ca^{2+} diffusion and of buffered Ca^{2+} diffusion in case of a very mobile buffer (Crank, 1975; Pape et al., 1998). The output of the program was imported into Igor Pro for additional analysis and graphing.
Derivation of Equation 13, the formula used to estimate the distance r_{BK} between Ca^{2+} channel and BKCh.
Ca^{2+} microdomains build up very rapidly after opening of a Ca^{2+} channel and reach values in the high micromolar range (Roberts, 1994). In the absence of any mobile Ca^{2+} buffer, a Ca^{2+} microdomain is described as the steadystate increase in Ca^{2+} above the resting concentration (Δ[Ca^{2+}]), which is given by an inverse function of the distance (r) from a single Ca^{2+} channel (embedded in a planar membrane) according to the following equation (Crank, 1975): In this equation, i_{Ca} is the singlechannel current amplitude, F is the Faraday constant, and D_{Ca} is the diffusion coefficient of Ca^{2+}. Under some simplifying assumptions (see Results for details), the spatial extent of Ca^{2+} microdomains in the presence of BAPTA can be described well by a linear approximation yielding the following: with (Neher, 1986, 1998), where k_{on} is the association rate constant of Ca^{2+} binding to BAPTA and [BAPTA]_{free} is the concentration of BAPTA not bound to Ca^{2+} (Eq. 4). λ is a buffer length constant (in meters) that describes how the inhibitory effect of BAPTA on free Ca^{2+} increases with distance from the Ca^{2+} channel.
Now we consider a BKCh that is placed at a distance r_{BK} from the Ca^{2+} channel. Then, with r = r_{BK}, Equation 1 describes the increase in free Ca^{2+} seen by the BKCh in the absence of BAPTA (Δ[Ca^{2+}]_{BK}). With r = r_{BK}, Equation 2 describes the free Ca^{2+} concentration at the BKCh in the presence of BAPTA (Δ[Ca^{2+}]_{BK}^{BAPTA}).
Dividing Equation 2 by Equation 1 yields the following expression for the relative Ca^{2+} increase at the BKCh remaining in the presence of BAPTA:
Other formulas and relationships
Ca^{2+}free buffer concentration was calculated according to the law of mass action: The Ca^{2+}binding ratio was calculated as follows (Neher, 1995): The diffusion time t_{90%} required to reach 90% of the steadystate Ca^{2+} concentration at a distance r from a permanently open single Ca^{2+} channel in the absence of buffers can be obtained from the following squareroot relationship of diffusion (Crank, 1975): With an error function (erf) being ∼0.1 at 0.0889 and D_{Ca} = 2.2 × 10^{−8} m^{2}/s (diffusion coefficient of free Ca^{2+} ions), it follows that: The reduction in wholecell Ca^{2+} currents evoked by mock action potentials caused by lowered extracellular Ca^{2+} concentration mainly reflects the reduction in singlechannel current amplitude The wholecell Ca^{2+} current is related to the singlechannel current by the following equation: Here, N is the number of Ca^{2+} channels contributing to the wholecell current, i_{Ca} is the singlechannel current, and P_{o} is the open probability of channel i. The index, k, indicates that each of the N Ca^{2+} channels is allowed to have a different i_{Ca} and P_{o} (e.g., different subtypes of channels). i_{Ca} is time dependent because the driving force changes during the mock action potential. P_{o} is time, voltage, and Ca^{2+} dependent because gating properties of Ca^{2+} channels are time, voltage, and Ca^{2+} dependent. However, mock action potentials are short (∼5 ms), and Ca^{2+}dependent modulation of channel gating in granule cells develops with time constants >20 ms (Kohr and Mody, 1991). Thus, it seems reasonable to assume P_{o} primarily reflects voltage and time dependent gating and should primarily remain unaffected when the extracellular Ca^{2+} concentration is lowered. On the other hand, it can be expected that i_{Ca} will be reduced by lowering extracellular Ca^{2+}, i_{Ca}*, because of a diminished occupancy of the channel pore by Ca^{2+} (Hille, 1992). Because the different Ca^{2+} channels bind Ca^{2+} with a similar affinity (Carbone and Lux, 1987; Church and Stanley, 1996), we assume that the fraction of i_{Ca} remaining in low Ca^{2+} solution, represented by a ∈ [0,1], is similar between all of the N channels according to the following equation: For the ratio “norm. I_{Ca},” under lowered extracellular Ca^{2+} concentrations (see Fig. 4D), it follows from Equations 8 and 9 that: In conclusion, under the abovementioned assumptions, the reduction in norm.I_{Ca} equals the fraction of i_{Ca} that remains in low Ca^{2+} solution.
The percentage of reduction in wholecell Ca^{2+} currents is a measure of the relative reduction in the increase in free Ca^{2+} concentration remaining at the BKChs in low Ca^{2+}, as follows: with Δ*[Ca^{2+}]_{BK} as the Ca^{2+} increase at the BKChs in low Ca^{2+}.
The following sigmoid function was used to interpolate the values in Figure 5A: with the following coefficients: base = 0.0016447; max = −0.00069526; xhalf = 0.30296; rate = 0.13025.
Equations were developed and checked in Derive for Windows 5.05 (Texas Instruments, Dallas, TX) and in Mathematica 5.1 (Wolfram Research, Champaign, IL). Fitting was performed in Igor Pro 5 for Windows (WaveMetrics)
Monte Carlo simulation of randomly distributed Ca^{2+} channels.
The peak amplitudes of Ca^{2+} tail currents measured in granule cells (compare Fig. 4) ranged ∼600 pA. Assuming a singlechannel current of ∼0.2 pA and an open probability of 0.15, this amplitude translates into ∼20,000 voltagegated Ca^{2+} channels on the granule cell surface. For one run of the simulations, we generated two series of 10,000 random numbers (Igor Pro) representing the spherical coordinates of Ca^{2+} channels on a hemisphere with a 5 μm radius. One series of numbers was equally distributed and ranged between 0 and 2π (ϕ), and the other equally distributed series of numbers (ϑ) ranged between 0 and 1 was converted to angles between 0 and π by the arccosine function to account for the variation of the perimeter of the sphere with ϑ. The BKCh was placed at angle (0, 0), and we determined the four smallest distances of the 10,000 distances from the Ca^{2+} channels to this BKCh. To simulate the collection of 27 granule cells (as used for calculating the data shown in Fig. 5B), we performed 30 runs of the simulation. For each run, we calculated the corresponding −ln(norm.Δ[Ca^{2+}]) curve for the four closest channels according to Equation 20 (with n = 4). We then averaged the 30 curves and calculated the SEM (see Fig. 5B). We also performed an analogous calculation with 10,000 runs to obtain the asymptotic curve. This asymptotic curve fell well within the SEM of the curve shown for 30 runs.
All of the drugs were from Sigma (Taufkirchen, Germany). Data are given as mean ± SEM. The level of statistical significance was set to α = 0.05.
Results
BKChs are necessary for rapid repolarization of action potentials in granule cells
In hippocampal neurons, BKChs are activated by single action potentials, accelerate repolarization, and lead to a pronounced fAHP (Storm, 1987a; Shao et al., 1999). We tested the contribution of BKChs to action potential repolarization in hippocampal granule cells using established blockers of BKChs and wholecell recordings. Paxilline (10 μm), charybdotoxin (1 μm), and iberiotoxin (1 μm) reduced the fAHP amplitude and significantly prolonged action potential duration in granule cells (Fig.1). The combined application of the three blockers increased action potential duration to a similar degree as each blocker alone, suggesting that all inhibitors block the same type of channels. Together, and in agreement with previous findings in the CA1 region (Storm, 1987a; Shao et al., 1999), charybdotoxin and iberiotoxinsensitive BKChs are also critical for proper action potential repolarization in granule cells.
Endogenous Ca^{2+} buffers do not interfere with action potential repolarization
In the next series of experiments, we tested whether endogenous Ca^{2+} buffers, such as calbindinD_{28k}, normally inhibit the activation of BKChs by action potentials. We tracked the activation of BKChs using hybridclamp recordings in wholecell mode. In this way, it is possible to activate BKChs by action potentials and to simultaneously quantify the associated conductance change as a tail current amplitude [Dietrich et al. (2002), their Fig. 5]. To determine the effect of endogenous Ca^{2+} buffers on the I_{fAHP}, we took advantage of the fact that these buffers are strongly diluted during prolonged wholecell recordings (>30 min) (Muller et al., 2005). If calbindinD_{28k} or any other similarly mobile cytoplasmic Ca^{2+} buffer interferes with the activation of BKChs, we should observe an increase in I_{fAHP} caused by washout of Ca^{2+} buffers. However, I_{fAHP} remained constant during a recording period of >30 min, which dilutes calbindinD_{28k} to <10% (Fig. 2D). As an additional test, we compared the action potential duration of granule cells from wildtype mice to those from calbindinD_{28k}deficient mice (Airaksinen et al., 1997). The properties of action potentials were not different between the two groups (Fig. 2E,F). Together, the data suggest that neither calbindinD_{28k} nor any other mobile endogenous Ca^{2+} buffer influence the activation of BKChs by action potentials in hippocampal neurons.
Our previous work implied that calbindinD_{28k} strongly reduces global Ca^{2+} signals [signals that spread over several micrometers (Augustine et al., 2003)] and localized Ca^{2+} signals down to a distance of ∼200 nm from the source of entry (Muller et al., 2005). Thus, the lack of inhibition of BKCh activation by calbindinD_{28k} indicates that Ca^{2+} ions diffuse <200 nm before they reach a BKCh. Ca^{2+} signals on such a restricted spatial scale are called Ca^{2+} microdomains. Ca^{2+} microdomains are steep submicroscopic Ca^{2+} concentration gradients around a Ca^{2+} channel or a cluster of channels that very rapidly equal steady state after opening of the channel and reach values in the high micromolar range (Roberts, 1994).
Millimolar BAPTA is required to inhibit BKCh activation by action potentials
To test the involvement of Ca^{2+} microdomains in the activation of BKChs, we compared the effects of the Ca^{2+} chelators BAPTA and EGTA on the repolarization of action potentials. If microdomain signaling is involved, Ca^{2+} ions diffuse only a brief period of time before they reach BKChs. Within this brief period, only a chelator with fast Ca^{2+} binding rates such as BAPTA (Tsien, 1980) is able to capture a significant fraction of Ca^{2+} ions (Roberts, 1993; Naraghi and Neher, 1997). Consistent with the involvement of microdomains, addition of a high concentration of the slow (Tsien, 1980) Ca^{2+} chelator EGTA (10 mm) to the solution in the patch pipette did not prolong action potentials (Fig. 3A,C). In contrast, as illustrated in Figure 3, A and B, spike duration clearly increased in a concentrationdependent manner when cells were loaded with BAPTA. A concentration of 10 mm BAPTA prolonged action potentials slightly less than pharmacological blockade of BKChs by toxins (compare Fig. 1). The insensitivity of BKCh activation to EGTA is in line with previous findings in different types of cells and qualitatively indicates a close association between a BK and a Ca^{2+} channel (Robitaille et al., 1993; Prakriya and Lingle, 2000; Protti and Uchitel, 1997; Storm, 1987b).
Linear approximation of Ca^{2+} diffusion in microdomains allows the estimation of the diffusional distance between BK and Ca^{2+} channels
The concentration dependence of the effect of BAPTA on the action potential repolarization can be used to estimate the mean diffusional distance between Ca^{2+} and BKChs. The linear approximation of Ca^{2+} diffusion within Ca^{2+} microdomains describes the relative increase in Ca^{2+} above resting concentration at a certain distance from the source of entry in the presence of a Ca^{2+} buffer such as BAPTA (Naraghi and Neher, 1997; Neher, 1998). According to this approximation, the local Ca^{2+} concentration at a BKCh placed at the distance r_{BK} from a Ca^{2+} channel is given by the following equation (see Materials and Methods for details): where Δ[Ca^{2+}]_{BK}^{BAPTA} and Δ[Ca^{2+}]_{BK} represent the differential increase in free Ca^{2+} at the BKCh in the presence and absence of BAPTA and λ is a bufferlength constant that describes how the inhibitory effect of BAPTA on free Ca^{2+} increases with the distance from the Ca^{2+} channel (Neher, 1986; Naraghi and Neher, 1997).
Principally, from Equation 13, the mean distance, r_{BK}, which Ca^{2+} ions diffuse from Ca^{2+} to BKChs, can be calculated provided that λ as well as norm.^{BAPTA}Δ[Ca^{2+}]_{BK} are known. λ can easily be computed (Eqs. 3, 4) from the well known biophysical properties of BAPTA and Ca^{2+} (Tsien, 1980). It is less straightforward to assess norm.^{BAPTA}Δ[Ca^{2+}]_{BK} because it is not possible to directly measure Ca^{2+} concentrations at the required spatial and temporal resolution (tens of nanometers, submillisecond range). Here, we use an alternative calibration procedure to obtain information on this local Ca^{2+} concentration in four steps, explained as follows, in brief: (1) we measured how the action potential duration increases when decreasing the local Ca^{2+} concentration at the BKCh (norm.^{low_Ca}Δ[Ca^{2+}]_{BK}) by perfusion of low extracellular Ca^{2+} concentration; (2) we estimated how the singlechannel Ca^{2+} current (i_{Ca}) reduces when perfusing low extracellular Ca^{2+} concentration; (3) we used the reduction in i_{Ca} as a measure of norm.^{low_Ca}Δ[Ca^{2+}]_{BK} and constructed a calibration curve (from the data obtained in steps 1 and 2) that provides the corresponding value of norm.Δ[Ca^{2+}]_{BK} for any given AP duration; (4) with the help of this curve, we translate the AP durations measured in the presence of BAPTA (see Fig. 5A) into the corresponding values of norm.^{BAPTA}Δ[Ca^{2+}]_{BK}.
Calibrating the prolongation of action potentials by BAPTA on the prolongation induced by low extracellular Ca^{2+}
In detail, we performed the following experiments. (1) We lowered the extracellular Ca^{2+} concentrations and kept the totaldivalent cation concentration constant by substitution of equimolar Mg^{2+} to avoid divalent screening effects. Figure 4, A and B, depicts the increase in spike duration when extracellular Ca^{2+} was lowered from 2 mm to 1.5, 0.7, 0.5, and 0.2 mm. (2) I_{Ca} was measured in wholecell voltageclamp mode and evoked by mock action potentials as command potential (Fig. 4C). Figure 4D illustrates the concentrationdependent decrease in I_{Ca} when lowering the extracellular Ca^{2+} concentration. As outlined in Materials and Methods, under the conditions used here, the fraction of I_{Ca} remaining in low extracellular Ca^{2+} (norm.I_{Ca}) is a measure of the fraction of the singlechannel current amplitude remaining in the same concentration. (3) Because microdomain Ca^{2+} scales proportionally to i_{Ca} (Eqs. 1, 2), norm.I_{Ca} is also a measure of the relative action potentialtriggered increase in free Ca^{2+} concentration remaining at the BKChs in low [Ca^{2+}]. Therefore, the following equation is satisfied: and we can construct a calibration curve by replacing the scaling of the abscissa in Figure 4B with the ordinate of Figure 4D (i.e., we plot the action potential duration measured at various extracellular Ca^{2+} concentrations versus the corresponding norm.^{low_Ca}Δ[Ca^{2+}]_{BK} values (Fig. 5A, data pairs taken from Fig. 4B,D). The data points were interpolated with a sigmoid function (dashed line; see Materials and Methods). For any given increment in spike duration, the corresponding relative decrease in the local Ca^{2+} concentration at the BKCh can be deduced from this curve. (4) In particular, this plot yields the values of norm.^{BAPTA}Δ[Ca^{2+}]_{BK} that gave rise to the increased action potential durations in the presence of different concentrations of intracellular BAPTA (Fig. 3). We used the inverse of the sigmoid function to compute the values of norm.^{BAPTA}Δ[Ca^{2+}]_{BK} corresponding to the spike durations recorded with 0, 3, 6, and 10 mm BAPTA (Fig. 5A, see example arrow).
Graphical approach to infer the diffusional distance of Ca^{2+}
Taking advantage of Equation 13, we developed a graphical approach to estimate the value of r_{BK} based on the concentrationdependent inhibitory effect of BAPTA. Rearranging Equation 13 yields a linear function of the reciprocal buffering length constant (1/λ) with a slope of r_{BK} as follows: Thus, when BKChs are only activated by Ca^{2+} channels that are placed at a distance of r_{BK}, the plot of −ln(norm.^{BAPTA}Δ[Ca^{2+}]_{BK}) versus 1/λ should be a straight line through the origin. The slope of that line represents r_{BK}. The values of norm.^{BAPTA}Δ[Ca^{2+}]_{BK} were obtained from the calibration plot in Figure 5A as described above, and λ was calculated according to Equation 3, Equation 4, and using published parameters summarized in Table 1 (yielding λ = 15, 10.6, and 8.2 nm for 3, 6, and 10 mm BAPTA respectively). As illustrated in Figure 5B, the data could be well fitted by a line crossing the origin with a slope of r_{BK} = 12.7 nm. This value is also obtained if we fit our data with a formula derived from the full model (from Eq. 20, supplemental Fig. 1, available at www.jneurosci.org as supplemental material). Even an unconstrained linear fit yields a very similar estimate of 13.5 nm (data not shown). Thus, there must be a very close spatial association between BKChs and voltagegated Ca^{2+} channels such that Ca^{2+} ions diffuse, on average, only ∼13 nm.
It is important to note that plotting −ln(norm.^{BAPTA}Δ[Ca^{2+}]_{BK}) versus 1/λ generates a line only when BKChs exclusively sense Ca^{2+} from a source (or sources) located at r_{BK} and not from additional Ca^{2+} sources at other distances. If additional remote channels contributed to the activation of the BKCh, then the resulting graph would be a curve that starts from the origin with a steep slope and that converges into a more shallow line (constant slope) as 1/λ increases (see Appendix). This case is illustrated by the dashed line that signifies how the plot would look if BKChs were activated by four Ca^{2+} channels placed at 12.7, 50, 60, and 70 nm (according to Eq. 20) (Fig. 5B,C). The initial slope of this curve is steeper because a low BAPTA concentration (low 1/λ values) can already cause a significant reduction in Δ[Ca^{2+}]_{BK} by capturing Ca^{2+} from remote sources. When BAPTA concentration (and 1/λ) is increased further, the contribution of more remote sources is shut off successively, and the slope of the curve flattens. Finally, when the BAPTA concentration is sufficiently high such that BKChs sense only Ca^{2+} from the closest source at r_{BK}, the curve continues as a line with the slope r_{BK}.
It can be shown that even when several channels contribute to the local Ca^{2+} concentration at the BKCh, the graphical analysis of a plot, as shown in Figure 5B, yields estimates of the distance of the closest Ca^{2+} channel, of the fraction of the local Ca^{2+} that this closest channel contributes, and of the total number of Ca^{2+} channels that participate in activating the BKCh (see Appendix).
Random placement of channels cannot explain the colocalization
We next asked whether such a tight colocalization may also result from a random placement of Ca^{2+} channels on the granule cell surface. Based on the amplitudes of Ca^{2+} currents measured above, we estimated the mean number of Ca^{2+} channels in granule cells (see Materials and Methods for details) and performed a simple Monte Carlo simulation of the distribution of channels. Based on the random distribution of the four closest Ca^{2+} channels, we calculated the corresponding −ln(norm.Δ[Ca^{2+}]) plot according to Equation 20. These simulations (Fig. 5) clearly show that the observed increase in action potential duration by BAPTA can hardly be the result of random channel distributions.
Validation of the linear approximation of Ca^{2+} diffusion by numerical simulations
In the above analysis, we rely on the accuracy of the linear approximation of Ca^{2+} diffusion in microdomains, which makes several simplifying assumptions (Naraghi and Neher, 1997; Neher, 1998), such as immediate steady state of free Ca^{2+} concentration during a squarelike singlechannel current and negligible buffer depletion. We used numerical simulations of buffered diffusion of Ca^{2+} to verify that this model also holds in a more physiological scenario incorporating a timedependent singlechannel Ca^{2+} current and the presence of fixed Ca^{2+} buffers that compete with BAPTA (see Materials and Methods for details).
In agreement with previous studies (Simon and Llinas, 1985; Roberts, 1994) and with analytical approaches (Naraghi and Neher, 1997; Pape et al., 1998), the free Ca^{2+} at a distance of 15 nm, and to a slightly lesser degree at 50 nm, closely follows the time course of the taillike singlechannel current and nearly approaches the steadystate concentration before the channel closes (Fig. 6A, arrowheads). Figure 6, A and B, shows that BAPTA effectively reduces the free Ca^{2+} concentration at 15 and 50 nm to ∼40 and ∼5% of the value reached without BAPTA, respectively. Because of the very fast Ca^{2+}binding rate of BAPTA, the free Ca^{2+} concentration in the presence of the buffer still closely follows the time course of unbuffered Ca^{2+} concentration (Fig. 6A), and the inhibition produced by BAPTA commences nearly instantaneously (Fig. 6B). The approximation of buffered Ca^{2+} microdomain diffusion linearizes around the concentration of Ca^{2+}bound buffer. It was therefore important to verify that the deviations in this parameter are insignificant (Fig. 6C). Together, the model of linear approximation of buffered Ca^{2+} diffusion used for estimating the distance between BK and Ca^{2+} channels describes the inhibition of free Ca^{2+} concentration in the immediate vicinity of a Ca^{2+} channel during an action potential very well (Fig. 6D).
Discussion
The main results of the present study is that BKChs are tightly colocalized with voltagegated Ca^{2+} channels such that Ca^{2+} ions have to diffuse, on average, only ∼13 nm after entering the cell before they bind to BKChs. The operation of BKChs by nanodomains of single Ca^{2+} channels provides several unique advantages, as discussed below. This functional colocalization was determined by a novel approach based on a linear approximation of buffered Ca^{2+} diffusion in microdomains (Neher, 1986; Stern, 1992; Naraghi, 1997).
This approach involved the quantitative analysis of the effect of the Ca^{2+} chelator BAPTA on the repolarization phase of action potentials, which we have shown to be potently shaped by BKChs. When we plotted the lntransformed normalized local Ca^{2+} versus the inverse buffer length constant (1/λ), the points could be well fitted with a line through the origin and a slope of 12.7 nm. Because our approach involves a calibration of the Ca^{2+}dependent response (action potential duration) on defined relative reductions of the local Ca^{2+} increases (obtained by lowering extracellular Ca^{2+}), it does not require estimates of the absolute local Ca^{2+} concentration at the target. Therefore, it should prove useful for Ca^{2+}dependent processes other than activation of BKChs as well.
How precise is this estimate of 13 nm diffusional distance? The data shown in Figure 5B potentially result from the summation of responses on two levels. First, the local Ca^{2+} at a single BKCh could be the sum of two Ca^{2+} channels very close to each other. If, for example, two Ca^{2+} channels are placed at 11 and 15 nm distances from the BKCh, then a plot would result that is indistinguishable from our results within the range of experimental accuracy. However, when the closest channel of such a cluster is located at a distance ≤8 nm, then the reduction in the Ca^{2+} elevation caused by 10 mm BAPTA would be detectably smaller than what we observed. Likewise, when the most remote channel is placed at a range ≥18 nm, the curve would clearly bend and pass into the asymptotic linear course in the presence of ≥3 mm BAPTA. Thus, several Ca^{2+} channels could contribute to the activation of a BKCh if they were placed in the range of ∼8–18 nm away from the BKCh. However, given that in hippocampal neurons BKCh openings seemed to follow sparse openings of Ca^{2+} channels (Marrion and Tavalin, 1998), it appears more likely that they are operated by single Ca^{2+} channels.
Second, BKCh activation was monitored in wholecell recordings that represent the summed response of hundreds of BKChs. If each BKCh is operated by a single Ca^{2+} channel, then the respective −ln plots of the fractional local Ca^{2+} will be linear functions of 1/λ for all participating channels. The slopes of those individual plots do not have to be identical but could vary around the arithmetical average of 13 nm. However, because the physical extent of an ion channel protein is ∼8 nm (Hille, 1992), the range of possible distances between individual BK and Ca^{2+} channels that yields an average of 13 nm has to be ∼8–18 nm (assuming a symmetrical distribution). Such a constant spacing is suggestive of a physical link between the two channels that could potentially be mediated by βcatenin (Lesage et al., 2004).
Together, our results clearly indicate a tight association between BK and Ca^{2+} channels, such that the mean diffusional distance for Ca^{2+} ions is as short as ∼8–18 nm. The most parsimonious explanation for this scenario is that each BKCh is operated by a Ca^{2+} nanodomain, a microdomain of a single Ca^{2+} channel (Augustine et al., 2003).
The nanodomain operation of BKChs imparts several unique functional advantages that are all connected to the spatial and temporal spread of this very local Ca^{2+} signaling. First of all, the short mean diffusional distance of Ca^{2+} ions assures that BKChs are activated very rapidly and are able to effectively contribute to the repolarization of an action potential. This is in striking contrast to other Ca^{2+} activated K^{+} channels, such as small conductance Ca^{2+}activated K^{+} (sK) channels, which open as rapid as BKChs (Xia et al., 1998; Ha et al., 2004); however, they do not contribute to repolarization, but instead produce a slower AHP that peaks with a delay of ∼5 ms (Sah, 1996). The reason for the different AHP kinetics may be that Ca^{2+} ions need only ∼0.03 ms to reach ∼90% of the steadystate concentration at a distance of 13 nm (nanodomain), whereas at a distance of 200 nm (microdomain), the time to reach this level is strongly prolonged to ∼5 ms (Eq. 7).
Second, at a distance of 13 nm from a Ca^{2+} channel, the free Ca^{2+} concentration also drops back to resting levels within microseconds after closure of the Ca^{2+} channel (Fig. 6) (Roberts, 1994; Naraghi and Neher, 1997). Thus, the period of activation of BKChs is given by the very brief open time of Ca^{2+} channels during an action potential [<1 ms (Lee and Elmslie, 1999)] and prevents BKChs from suppressing repetitive firing. In contrast, the kinetics of Ca^{2+}activated currents operated by a more remote regimen will primarily dependent on the removal of Ca^{2+} by sequestering mechanisms that typically requires several hundred milliseconds (Helmchen et al., 1996).
Third, there is a multitude of intracellular processes that are triggered by elevations in intracellular Ca^{2+}. Still, single action potentials do not usually trigger many of these processes, but they consistently lead to activation of BKChs. This selectivity of BKCh activation partially results from the steep spatial gradient of Ca^{2+} nanodomains. Ca^{2+} nanodomains reach high micromolar concentrations only very close to the source (the Ca^{2+} channel), and at a few hundred nanometers off of the source, the free Ca^{2+} concentrations only reach the low or submicromolar range. Therefore, Ca^{2+}binding partners placed far away from the source have much less chances to bind sufficient Ca^{2+} ions to be activated after single action potentials.
Fourth, the restricted spatial extent of Ca^{2+} nanodomains also confers reliability on BKCh activation. The actual concentration of Ca^{2+}binding partners competing with BKChs can change drastically. For example, in response to ischemia and neuronal activity, certain neurons strongly increase the expression of Ca^{2+}binding proteins such as calbindinD_{28k} (Lowenstein et al., 1991; Batini et al., 1993; Arnold and Heintz, 1997). If competing Ca^{2+}binding partners strongly increase in concentration, activation of BKChs could decrease concomitantly. However, considering that BAPTA binds Ca^{2+} two to four times faster than endogenous Ca^{2+}binding proteins (Tsien, 1980; Nagerl et al., 2000), the concentration of an endogenous Ca^{2+}binding protein would have to increase such that it provides ≥8 mm of free Ca^{2+}binding sites, which is unlikely to occur in cortical neurons (Muller et al., 2005).
Together, the speed and the spatial restriction of Ca^{2+} nanodomains helps to activate BKChs fast and transiently and allows for their selective and reliable recruitment by action potentials.
Although BKChs are most probably operated by nanodomains when recruited during action potentials, this does not mean that a BKCh cannot be activated by more than one channel under different conditions. For example, during sustained depolarizations, global (cytoplasmic) Ca^{2+} could rise into the micromolar range (>2 μm) and BKChs would then be activated by Ca^{2+} from several sources pooled in the cytoplasm. Thus, the question arises as to what factors establish the nanodomain operation mode of BKChs? First of all, this obviously requires a special geometry of channels in which one Ca^{2+} channel is situated particularly close to a BKCh such that it dominates the local Ca^{2+} sensed by the BKCh. Second, opening of Ca^{2+} channels by action potentials facilitates nanodomain signaling in two ways. Because of their brief duration, action potentials keep the open time of Ca^{2+} channels short. It has been estimated that Ntype Ca^{2+} channels, which couple to hippocampal BKChs (Marrion and Tavalin, 1998), very rapidly close (<0.1ms) as soon as the membrane is repolarized after an action potential (Lee and Elmslie, 1999). This brief open time counteracts the spatial spread of nanodomains. An additional effect of action potentials is that they induce a taillike singlechannel current (Fig. 6). This sharpens nanodomains in time and space and works against an overlap of individual nanodomains. Third, the cooperative action of Ca^{2+} on the open probability of BKChs (Magleby, 2003) facilitates nanodomain operation. This sigmoid dependence makes the BKCh relatively insensitive to small local Ca^{2+} elevation caused by remote channels during that time when the closest channel is shut. In light of the complex interaction of these numerous factors, it can be expected that the tightness of the coupling between BK and Ca^{2+} channels vary among different cell types.
In conclusion, the unique functional role of BKChs in the firing pattern of cortical neurons results from a strategic placement of BKChs in Ca^{2+} nanodomains. This placement is a key to selective Ca^{2+} signaling, rendering BKChs buffer insensitive, and allowing neurons to regulate global Ca^{2+} signals by a high cytoplasmic Ca^{2+}buffer capacity without affecting the crucial fast repolarization of action potentials.
Appendix
Superposition of multiple microdomains
The aim of this section is to show that the plot of the lntransformed fractional local Ca^{2+} versus the inverse bufferlength constant will not be a line through the origin when more than one channel contributes to the local Ca^{2+} concentration. Also, it will be illustrated that the following parameters can be graphically estimated from such a plot: the distance of the closest channel, the relative contribution to local Ca^{2+} by the closest channel, and the total number of Ca^{2+} channels contributing to local Ca^{2+}.
We consider a BKCh placed at the origin that is surrounded by n + 1 Ca^{2+} channels, which are located at the distances r_{j}, j = 0, …, n. Let r_{0} denote the distance of the closest channel such that the following equation is satisfied: We assume that the n + 1 Ca^{2+} channels are similar with respect to their singlechannel conductance and their gating properties and that all channels experience the same membrane potential. We also assume that free Ca^{2+} reaches a near steadystate value fast compared with the mean open time of the channels (see Discussion). In analogy to Equation 1, we can write for the increment in free Ca^{2+} at the origin in the presence of multiple Ca^{2+} channels and in the absence of mobile fibers: Because we are interested in the average [Ca^{2+}] over many BKChs, the gating of single channels is summarized in the timedependent open probability P_{o}(t) (cf. Neher, 1998). The equation describing the free [Ca^{2+}] in the presence of BAPTA can be obtained by superimposing the contributions of the different Ca^{2+} channels (Neher, 1998), as follows: For the fraction of [Ca^{2+}] remaining under BAPTA, we attain the following analogy to Equation 13: The logarithmic transformation yields the following: If the left side of Equation 20 is plotted versus 1/λ, a graph different from a line results because its initial slope (for 1/λ → 0) is larger than its asymptotic slope (for 1/λ → ∞). To determine the initial and the asymptotic slope, we substitute ν = 1/λ (λ > 0) and calculate the first derivative of Equation 20 with respect to ν, as follows: The asymptotic slope of Equation 20 can be obtained by calculating the limit of Equation 21 toward infinity (high BAPTA concentrations), as follows: because, by Equation 16, (r_{0} − r_{j}) < 0 for all j.
Thus, the asymptotic slope yields the distance of the closest Ca^{2+} channel, even in the presence of multiple channels. The initial slope of Equation 20 amounts to the following: by Equation 16, QED.
In the presence of high BAPTA concentrations, when λ is sufficiently small (such that 3 λ < r_{0} − r_{j}, for all j), Equation 20 can be approximated by the following linear equation of the reciprocal buffering length constant: Compared with Equation 17, it can be seen that the inner logarithmic term in Equation 24 represents the fraction of the local Ca^{2+} at the BKCh contributed by the closest channel located at r_{0}. Thus, backextrapolation of the linear portion of the plot of the lntransformed fractional Ca^{2+} versus 1/λ to the ordinate supplies the negative logarithm of the relative contribution of the closest channel to the total Ca^{2+} at the origin. Moreover, division of the initial slope (Eq. 23) by the inner logarithmic term (expand by r_{0} for clarity) and by the asymptotic slope (Eq. 22) yields an estimate of the total number of Ca^{2+} channels that contribute to the Ca^{2+} increase seen by the BKCh at the origin (n + 1).
It should be noted that this analysis provides reasonable estimates only when most channels are located within ∼100 nm around the BKCh. If channels are located beyond that distance, then the condition of near steadystate Ca^{2+} concentrations would be too strongly violated. Furthermore, the correctness of the estimate of the total number of channels requires that the closest channel at r_{0} can be well isolated from the others by a buffer concentration that would not result in a toostrong suppression of the Ca^{2+} from the channel at r_{0}. This is the case when there is a λ* (i.e., some buffer concentration), with 3 l* < r_{0} − r_{j}, for all j and with 2 l* ≥ r_{0}.
Footnotes

This work was supported by the Deutsche Forschungsgemeinschaft (SFB TR3 C5, GK 246), University Clinic Bonn grants (BONFOR), and the GertrudReemtsmaStiftung of the MaxPlanckGesellschaft. We thank Roland Krüppel for advice on the Monte Carlo simulation and for critically reading this manuscript. We are grateful to Pia Stausberg for excellent technical assistance.
 Correspondence should be addressed to Dr. Dirk Dietrich, Department of Neurosurgery, NCH U1 R035, Experimental Neurophysiology, University Clinic Bonn, SigmundFreud Street 25, D53105 Bonn, Germany. dirk.dietrich{at}ukb.unibonn.de