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1Department of Biological Sciences, Rutgers University; and 2Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey
Submitted 12 January 2005; accepted in final form 21 February 2005
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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) and theoretical studies (Poznanski 1999; Veenstra 2001) analyzing the contribution of electrical current spread between neurons and other cell types. However, because of the scarcity of specific pharmacological gap-junction blockers (Bennett and Zukin 2004) and of appropriate biological preparations, estimating the effects of electrical coupling on measurements of ionic currents in neurons presents a difficult problem that needs to be rigorously studied. This issue has acquired further importance as a result of recent studies that show the presence of electrical coupling in an increasing number of cell types (Bennett and Zukin 2004; Pfeuty et al. 2003; Vandecasteele et al. 2005).
Electrical coupling to other cells can, in principle, produce an increase in either the passive leak or active voltage-clamp currents measured from a neuron. The relative contribution in passive or active currents from one cell to its gap junctional neighbor also depends on the leakiness of both, including the effects of electrode impalement on input resistance because this will affect their electrotonic compactness. It is possible to isolate neurons from their gap-junctional neighbors using pharmacology (Bou-Flores and Berger 2001; Yamamoto et al. 1998) or by photo-ablating the coupled neurons (Eisen and Marder 1982; Miller and Selverston 1979). However, these techniques are problematic; pharmacological agents that block gap junctions are well known for their frequent lack of specificity (Bennett and Zukin 2004) and photoinactivation of the coupled neurons causes the release of cellular contents and free radicals that can damage or affect neighboring cells (Dahle et al. 2001). Therefore when doing voltage-clamp measurements from gap-junctionally coupled neurons, it would be useful to have appropriate criteria to determine when to choose between cell isolation versus accepting measurement errors produced by electrical coupling.
In this study, we used electrophysiology and conductance-based models to examine the effect of electrical coupling on the measurements of passive neuronal properties, voltage-gated potassium currents as well as elicited synaptic outputs. As a model system, we used somatic recordings of the 2 pyloric dilator (PD) neurons and the postsynaptic lateral pyloric (LP) neuron in the lobster stomatogastric ganglion (STG), as well as computational models of this network. The 2 PD neurons are strongly electrically coupled and are believed to be structurally and functionally similar. The PD neurons exhibit similar intrinsic properties; make and receive similar synapses; and, as members of the pyloric pacemaker group, produce synchronous membrane potential oscillations (Ayali and Harris-Warrick 1999; Eisen and Marder 1984; Hooper 1997; Miller and Selverston 1982). We compared measurements of passive and voltage-gated currents when the 2 PD neurons were simultaneously voltage clamped to the same membrane potential (and thus functionally uncoupled) and when only one PD neuron was voltage clamped. We then used a computational model of the PD neurons to examine the effects of current density and the location of the gap junction on these measurements. We also examined the role of electrical coupling on measurements of postsynaptic potentials and used the model to study the effect of the relative positions of gap junctions and chemical synapses on synaptic output. These comparisons provide a measure of the distortion that a gap-junctionally coupled neuron produces in voltage-clamp measurements of ionic currents and synaptic output.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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Experiments were conducted on adult male spiny lobsters (Panulirus interruptus), weighing 400–800 g. Animals were purchased from Don Tomlinson Fisheries (San Diego, CA) and were kept in artificial seawater tanks at 12–15°C until use. The animals were anesthetized by packing on ice for 30 min before the dissection. The stomatogastric nervous system (STNS; including the STG, the esophageal, and the paired commissural ganglia) was isolated using standard methods (Harris-Warrick 1992; Selverston et al. 1976). The STNS was pinned down in a Sylgard-coated Petri dish and the STG was desheathed to allow penetration of the cell bodies and effective superfusion of the neurons. The preparations were superfused with normal saline at 18°C, pH 7.35, containing (in mM): 12.8 KCl, 479 NaCl, 13.7 CaCl2, 10.0 MgSO4, 3.9 NaSO4, 11.2 Trizma base, and 5.1 maleic acid. For neuron impalement, microelectrodes were pulled using a Flaming–Brown micropipette puller (P87, Sutter Instruments, Novato, CA) and filled with 3 M KCl to give resistances of 8–13 M
. Identification of the neurons was accomplished by matching intracellular action potential recordings to their corresponding extracellular recordings on motor nerves (Harris-Warrick 1992; Selverston et al. 1976). After identification, the neurons were impaled with 2 electrodes each and the preparation was superfused with 10–7 M tetrodotoxin (TTX; Biotium, Hayward, CA) to block sodium inward currents and thus action potentials. Slow rhythmic activity is also blocked in TTX (Raper 1979).
Using the 2-electrode voltage-clamp method, either one (1VC) or both (2VC) PD neurons were voltage clamped using Axoclamp 2B amplifiers (Axon Instruments, Union City, CA). For the 1VC protocol, one PD neuron was voltage clamped to a holding potential of –40 mV to remove the contribution of the large transient A current (Harris-Warrick et al. 1995), whereas the other PD neuron was recorded in current clamp. A 2-s, 10-mV hyperpolarizing pulse followed by 2-s depolarizing pulses ranging in amplitude from 5 to 60 mV were applied to the voltage-clamped PD neuron. The initial hyperpolarizing pulse was used to determine the leak current and, when necessary, to leak subtract the depolarization-induced currents after appropriately scaling it. For the 2VC protocol, both PD neurons were always voltage clamped to the same membrane voltage. Assuming identical electrotonic structure of the 2 electrically coupled neurons, this protocol would effectively uncouple the neurons because the gap-junctional current is proportional to the difference between the membrane potentials of the 2 neurons. All depolarizing pulses elicited a total current Itotal that consisted largely of potassium outward currents IK and passive leak current Ileak. An inward Ca2+ current is also included in this total current but its amplitude is negligible (<1%) compared with IK + Ileak (Golowasch and Marder 1992; Graubard and Hartline 1991; Johnson et al. 2003).
Both PD neurons also establish inhibitory cholinergic synaptic connections to multiple follower neurons, including the LP neuron (Eisen and Marder 1982). These synapses show voltage-dependent (graded) release of neurotransmitter. To measure the graded component of the PD to LP synapse, each PD neuron was impaled with 2 electrodes and the LP neuron was impaled with a single electrode in saline containing 10–7 M TTX to block rhythmic activity and action potential–mediated synaptic transmission (Raper 1979). To study the effect of electrical coupling on PD to LP synaptic release, either PD1 (1VC protocol) or both PD1 and PD2 (2VC protocol) were voltage clamped at –60 mV, whereas LP (and PD2 in 1VC only) was recorded in current clamp. A series of 2-s depolarizing pulses to voltages ranging from –50 to –10 mV was applied to the voltage-clamped PD neuron(s) to activate the PD to LP synapse. The LP resting membrane potential was set to –50 mV by current injection so that the inhibitory postsynaptic potentials (IPSPs), which had typical amplitudes of 1–7 mV (depending on the presynaptic membrane potential), did not saturate the membrane potential at the synaptic reversal potential of –85 to –80 mV (Eisen and Marder 1982).
An NI PCI-6070-E board (National Instruments, Austin, TX) was used for data acquisition and for current injection with the data-acquisition software Scope (available for download at http://stg.rutgers.edu/software) developed in the LabWindows/CVI software environment (National Instruments, Austin, TX) on a Windows operating system. The acquired data were then saved as individual binary and ASCII files and were analyzed on a Linux platform. Statistica (Statsoft, Tulsa, OK) and Origin (OriginLab, Natick, MA) software packages were used for statistical and graphical analysis. Reported statistical significance indicates that the achieved significance level P was below the critical significance level 
= 0.05. All error bars shown denote SD.
Model
Integrations of membrane and cable equations were performed using Network, a home-developed software running on the Linux platform, using a 4th-order Runge–Kutta method with a time step of 5 µs. Two somata, 180 µm in diameter, connected to respective multicompartment cylindrical cables 600 µm long and 20 µm in diameter were coupled with a gap junction of conductance gC = 10–6 S between the last segment of each cable. These values were chosen to fit the passive properties of the PD neurons as shown in Fig. 2A. The model neuron is representative of the biological PD neuron, which is a monopolar neuron with a large soma coupled to a relatively large primary neurite (King 1976a,b). In this study we ignored the effects of the secondary and tertiary neurites. Each cable was divided into five 100-µm-long compartments and both have geometry-independent specific membrane resistivity Rm = 40,000 · cm2 (corresponding to a leak conductance of Gm = 0.025 mS/cm2), specific axial resistivity Ri = 60 · cm, and specific membrane capacitance Cm = 10–6 F/cm2 (Hartline and Castelfranco 2003). Each neuron was modeled as a "Rall neuron" by combining the soma with the first segment of the cable (Johnston and Wu 1995; Rall et al. 1995).
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KS = 5 mS/cm2, KF = 1.4 mS/cm2. Gap-junctional coupling between the 2 PD neurons was modeled by adding a symmetric coupling current term to the coupled compartments of each cell
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The LP neuron received identical synaptic input from each PD neuron (except when stated) and the synaptic current was described by the sum of a transient (T) and a persistent (P) component as in Manor et al. (1997)
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· cm2. The 1VC protocol was modeled by clamping the voltage of the PD1 neuron soma and recording the total current. The 2VC protocol was modeled by clamping the soma of PD1 and PD2 to the same voltage and recording the total current from PD1 (identical results are obtained if gC is set to zero and the 1VC protocol is applied). Leak subtraction was performed exactly as described for the biological current measurements.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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All depolarizing pulses elicited a total current Itotal that consisted mainly of potassium outward currents IK and passive leak current Ileak (see METHODS). When only one PD neuron was voltage clamped (1VC protocol) and stepped to different voltage levels, the membrane potential of the second PD neuron was also affected (see, e.g., thick VPD2 trace in Fig. 2A). Because VPD1 and VPD2 were not the same in the 1VC protocol, PD2 contributed an additional current to the measurement of the total current from the PD1 soma. This was the current that flowed through the gap junction between the 2 neurons. However, when both PD neurons were clamped to the same voltage (assuming they had similar electrotonic characteristics), this gap junction current was eliminated. A greater Ileak for 1VC (coupled) than that for 2VC (uncoupled) was therefore expected. Figure 2A shows a 10-mV hyperpolarizing step in the voltage-clamped PD neuron(s) and the (leak) current recorded. Ileak for the coupled neurons (1VC, thick traces) was greater than Ileak for the uncoupled neurons (2VC, thin traces). When only one neuron was voltage clamped (1VC), the second neuron (Fig. 2A, middle panel, thick trace) showed an attenuated change in voltage. The apparent coupling ratio (
Vpost/Vpre) determined using hyperpolarizing pulses was 0.232 ± 0.062 (n = 9). The leak conductance, calculated as the ratio Ileak/Vhyp, was significantly larger in PD1 for 1VC, measuring 0.238 ± 0.144 µS, than in 2VC, which measured 0.193 ± 0.109 µS (Fig. 2B; Student’s t-test, P = 0.00846; n = 12). This corresponds to a nearly 23% overestimate of the leak conductance of the isolated (uncoupled) PD neuron.
Effect of electrical coupling on voltage-gated current measurements
The flow of current across the gap junction when only one PD neuron is voltage clamped could, in principle, activate voltage-gated currents in the unclamped partner as well as in the clamped neuron. As in the case of the leak current measurement, we measured a greater total current with the 1VC protocol (coupled neurons) than when both cells were uncoupled with the 2VC protocol (Fig. 3A). This difference was present in both the leak-subtracted (Fig. 3, A and B, top panels) and the total (not shown) currents, indicating that the voltage step in PD1 with the 1VC protocol resulted in activation of voltage-gated currents in PD2 and not simply a recruitment of leak current from the unclamped neuron. The difference between leak-subtracted currents measured in the 1VC and 2VC conditions was observed at all voltage-clamp steps above –25 mV (2-way ANOVA P = 0.0179 for peak, P = 0.001 for steady state, n = 12). The 1VC total currents were also significantly larger than the 2VC total currents at all voltages above –25 mV (2-way ANOVA, P = 0.0107 for peak currents and P = 0.0083 for steady-state currents, n = 12; not shown). High-threshold K+ currents are known to activate at –25 mV and higher in these neurons (Golowasch and Marder 1992; Graubard and Hartline 1991; Harris-Warrick et al. 1995; Johnson et al. 2003). The peak current was measured between 50 and 100 ms and the steady-state current at the end of the 2-s-long voltage step.
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; Prinz et al. 2004). The model I–V curves are shown in Fig. 3B. The 2 PD model neurons were identical and consisted of a soma coupled to a single 600-µm-long neurite with the gap junction located between the tips of the neurites (Fig. 1, inset). To quantify the error in the voltage-clamp measurements stemming from electrical coupling, we calculated the difference between the leak-subtracted steady-state currents measured in the coupled and uncoupled neurons (data set same as that used in Fig. 3B, Biological) as a percentage of the current in the uncoupled neuron (Fig. 4A). For voltage steps to –30 mV the average difference current was close to 50% of the current measured in the uncoupled neuron. However, with larger voltage steps (and thus with larger currents activated), the difference current stemming from the electrical coupling was a smaller percentage of the total current. (For the voltage step to –35 mV, the currents were too small to accurately measure the difference.) We show only the normalized difference for the steady-state currents (Fig. 4A). However, these data are not significantly different from the differences observed for the peak currents (not shown).
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] to the steady-state (or peak, not shown) conductance versus membrane potential curves showed little difference between the 1VC and 2VC measurements in either the midpoint (1VC: V1/2 = –0.4 mV, 2VC: V1/2 = –0.6 mV) or slope values (1VC: k = 0.100 mV, 2VC: k = 0.107 mV) (Fig. 4B). The more prominent, but still small (<10%), difference was in the estimated maximal conductance (1VC: Gmax = 1.79 µS, 2VC: Gmax = 1.63 µS) (Fig. 4B).
It has been shown that averaged data do not necessarily adequately characterize the properties of individual neurons (Golowasch et al. 2002; Prinz et al. 2004). Our model data were fitted to the measurements of a single experiment (rather than to the average data) representative of the average trend that showed larger current values using the 1VC protocol than the 2VC protocol. However, in a small number of experiments (3 out of 15) the 2VC protocol elicited larger total currents than the 1VC protocol. We attempted to adjust the parameters of our model to obtain a good fit to these data. However, even with a thorough search of the parameter space (10-fold increase and decrease in Rm, Ri, KS, and KF, and changes of up to ± 25 mV of the activation/inactivation midpoints), we did not succeed. Our lack of success may indicate that the simple geometry of our model or the assumption of uniform distribution of channels does not capture all cases. However, it seems to correctly reproduce the global effects of uncoupling neurons connected by gap junctions.
Effect of gap junction position along the neurite
Using the model 2-neuron network, we examined the effect of the position of the gap junction on the total current measured. We varied the position of the gap junction along the neurite of the model neurons in a symmetric fashion and measured the total current in 1VC and 2VC conditions. To elucidate the effect of the distance of the gap junction clearly, we chose a 1,000-µm-long neurite for these measurements. We observed that when the gap junction was located close to the clamped soma, the total current was larger. With the 1VC protocol (red traces, neurons coupled), the closer the gap junction was to the soma, the closer the voltage across the gap junction was to the clamped voltage (not shown) and thus the total current measured was larger (Fig. 5A, i–iii and Fig. 5B, open red circles). In contrast, with the 2VC protocol (black traces) the total current was independent of the gap junction position because the 2 neurons were effectively uncoupled. As the gap junction was moved further away from the soma, the total currents under 1VC and 2VC conditions approach each other. However, the total current under 1VC was higher at all positions because the neurites were not infinitely long cables. Consequently, the second neuron contributed a fixed, nonzero current even when the gap junction was positioned at the tip of the neurites (Fig. 5B, open circles). Figure 5C (open symbols) shows the percentage of additional current measured in 1VC compared with 2VC (the current difference between 1VC and 2VC normalized to 2VC). The largest difference (about 18%) was observed for the gap junction positions closest to the soma, and the difference dropped rapidly as the gap junction was moved away from it (Fig. 5C, open symbols).
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Current measurements as a function of leak and K+ conductances
The electrotonic structure has a crucial influence on how well the voltage of a neuron can be controlled across a gap junction, as demonstrated by the effects of the position of a gap junction along the neurite (and the length of the neurite) described in the previous section. To understand the effect of changes in electrotonic structure attributed to membrane resistance on measured currents, we varied both the leak conductance and the maximal conductance of the K+ current in our standard model. Figure 6A shows the expected increase in total current measured as the leak conductance (Gm) is varied. With increasing leak conductance, the total measured current increased approximately linearly, whether the 2 neurons are coupled (1VC) or not coupled (2VC; note that the apparent exponential increase is explained by the fact that the abscissa has a logarithmic scale). We observed a small difference in total current between coupled and uncoupled conditions (Fig. 6A, inset), with the total current of the coupled neurons always higher. This difference was mostly a consequence of the activation of the K+ currents in the second neuron (in 1VC, as also seen in Fig. 3). Figure 6B shows the difference between coupled and uncoupled total and leak-subtracted currents as a percentage of the uncoupled current. The difference in the total current (black symbols) was constant at low leak conductance values but decreased monotonically to zero as the leak conductance increased and the gap junction became electrotonically distant from the soma, effectively uncoupling the 2 neurons.
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In contrast with the total current, the leak-subtracted K+ currents showed a different and somewhat nonintuitive dependency on the conductance of the cells (Fig. 6, B and D, red circles). For small values of Gm, the normalized leak-subtracted K+ current difference was positive, i.e., the leak-subtracted current was larger when the 2 neurons were coupled than when they were uncoupled (Fig. 6B, red circles). However, for larger Gm values, the K+ current became smaller when the cells were coupled than when they were uncoupled and the leak-subtracted current difference became negative. As Gm was increased even further, the difference reversed toward zero. This odd behavior can be explained as follows. When the soma of cell 1 is voltage clamped and a positive-voltage pulse is applied, the gap-junctionally coupled cell 2 contributes to the current measured from cell 1 soma in 2 ways. First, the gap junction allows some K+ channels to be opened in cell 2, thereby adding to the (leak-subtracted) K+ current measured from the soma of cell 1. Second, cell 2 puts a passive load on cell 1 and dampens the depolarization of the compartments of cell 1 that are near the gap junction. This latter effect is reflected in the fact that the membrane potential of the cell 1 compartment adjacent to the gap junction is lower when cell 1 is coupled to cell 2 than in an isolated cell 1 (not shown). This in turn results in less K+ current activation in cell 1, thereby reducing the K+ current measured in cell 1, whereas the leak current measured is still higher because the neurons are electrotonically more compact (less membrane conductance) during the hyperpolarizing pulse. Thus the 2 contributions of cell 2 to the K+ current measured from cell 1 work in opposition to one another. If the leak conductance is small, the first effect dominates and there is a net increase in K+ current as measured from the soma of cell 1. In contrast, if the leak conductance is large, the second effect dominates and there is a net decrease in the amount of activated K+ current and an increase in the measured leak current. The leak subtraction procedure then reduces the net current in the coupled case proportionally more than in the uncoupled case, thus leading to a negative current difference and strong nonlinearity of the (normalized) current difference (Fig. 6B, red symbols). The same effect on the electrotonic structure of the neurons occurred when the K+ conductance was changed and a similar nonlinear phenomenon was observed at high K+ conductance values, albeit much less pronounced than when the leak conductance was modified (Fig. 6D, red symbols; magnified in the inset).
Effect of gap junction on measuring the synaptic response
The 2 PD neurons synapse onto the LP neuron and, on depolarization, produce a graded inhibitory postsynaptic potential in the LP neuron (Fig. 7A, inset and schematic inset of Fig. 1). To analyze how gap-junctional coupling can affect the operation of circuits involving both electrical and chemical synapses, we measured the PD to LP neuron synaptic potential and expanded our standard model to include these graded inhibitory synapses (see METHODS). Both PD1 and PD2 were voltage clamped and held at –60 mV while the LP neuron was recorded in current clamp. Figure 7A (inset) shows the IPSP elicited in the LP neuron when either PD1 (1VC) or both PD1 and PD2 (2VC) were stepped to –20 mV. The larger IPSP amplitude evoked when the 2 PD neurons were depolarized simultaneously (and consequently uncoupled) is the result of transmitter release from both neurons compared with when a single PD neuron was depolarized. Note that the IPSP was not twice as large in 2VC compared with 1VC, indicating that, when only PD1 was voltage clamped, there was transmitter release from PD2 as well.
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Gap-junction effects on synaptic transmission
To analyze the effects of gap-junction coupling on the behavior of a network that includes chemical synapses, we used our model to identify the influence of the relative positions of the gap junction and the chemical synapse on synaptic release. Figure 8 shows the effect of the position of the gap junction and of the chemical synapse along the neuritic cable on synaptic efficacy. Three positions of the gap junction (0, 500, and 1,000 µm) were examined along a 1,000-µm cable, with 0 indicating the position of the soma. For each gap junction position, the position of the synapse was varied from 0 to 1,000 µm along the length of the cable, the soma of neuron 1 or both neurons 1 and 2 of the model was voltage clamped with a voltage pulse from –60 to –20 mV, and the postsynaptic potential was measured (neuron 3). Consistent with the observations shown in Fig. 7, the 2VC protocol produced a larger IPSP than the 1VC protocol for any position of the chemical synapse (Fig. 8A). This was the result of a better voltage control of the entire second neuron as shown in Fig. 8B, where we compare the voltage values along both neurites between 1VC and 2VC. In addition to producing better voltage control of neuron 2, the 2VC protocol effectively uncoupled both synapses, thus making the IPSP completely independent of the position (and conductance, not shown) of the gap junction. Consequently, the 2VC curve was identical for all gap-junction positions (Fig. 8A, black curve). Furthermore, Fig. 8A shows that moving the gap junction closer to the site of voltage clamp (i.e., the soma) increased the IPSP amplitude with the 1VC protocol. This was a consequence of less attenuation of the voltage along both neurites as the gap junction was moved closer to the soma (Fig. 8B; compare voltage values along both neurites in the 1VC cases; gap junction position indicated in each schematic diagram). Notice also that when the gap junction is at an intermediate position between the ends (here at 500 µm), there is a slight kink (discontinuity of the first derivative) in the IPSP amplitude curve at the value for which the position of the chemical synapse matches the position of the gap junction (Fig. 8A, blue trace and black arrowhead). This kink is also evident in the voltage values along the neurite of neuron 1, which show a considerably larger drop between soma and the center of the neurite (500 µm) when the gap junction is placed at the center than when it is placed either at the soma (0 µm) or at the tip (1,000 µm) (Fig. 8B, 1VC).
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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; Dermietzel and Spray 1993). Recently a number of studies have sought to determine algorithms to minimize errors in the measurement of gap-junction conductances (Poznanski 1999; Veenstra 2001). However, the complex morphology of a neuron affects the voltage-clamp measurement of ionic currents and the presence of gap junctions in a neuron that is electrically coupled to one or several other neurons can also influence these measurements. To our knowledge, the effect of gap junctions on voltage-clamp measurements of ionic currents has not been carefully studied.
We performed voltage-clamp recordings from the 2 electrically coupled PD neurons of the crustacean stomatogastric ganglion. To estimate the effect of electrical coupling on the measurements of ionic currents and synaptic outputs we used 2 voltage-clamp protocols. In one protocol, we voltage clamped only one PD neuron (1VC) and recorded the second (coupled) PD neuron in current clamp, whereas, in the second protocol, both PD neurons were voltage clamped (2VC) and held at exactly the same membrane potential. Assuming the 2 PD neurons are anatomically and physiologically similar (Ayali and Harris-Warrick 1999; Eisen and Marder 1984; Hooper 1997; King 1976a,b; Miller and Selverston 1982), the 2VC protocol would functionally uncouple the 2 PD neurons by drastically reducing current transfer through the gap junctions. We also used ball-and-stick model neurons with uniform active and passive properties to examine the effect of gap junctions on ionic current measurements.
Although, in this study we focused on voltage-clamp measurements of outward currents, electrical coupling also affects measurements of inward currents. We predict that the effect of gap junctions on nonregenerative inward currents, such as the hyperpolarization activated Ih, are similar to those discussed here for the outward currents. Measurements of regenerative inward currents, however, could lead to a different set of complexities, depending on the possible loss of voltage control in the gap-junctionally coupled neurons stemming from the activation of the inward current.
Gap-junction coupling and ionic current measurement errors
Accurate estimation of input resistance, membrane capacitance, and voltage-dependent ionic currents requires that these measurements be done in isolated neurons. Otherwise, these measurements would be contaminated by the extra membrane that would effectively be "recruited" through the electrical junctions to the recording configuration. At first glance, our measurements of the voltage-dependent outward currents and the model seem to suggest that the error arising from gap-junctional coupling is minimal because there is only a small contribution to the total current by the coupled cell (see Figs. 3 and 4B). However, when the current difference arising from the gap junctional coupling is expressed as a percentage of the current measured from the uncoupled neuron, it becomes evident that the error is extremely dependent on the size of the measured conductance, with the highest errors obtained for the smallest conductances (Fig. 4A). The largest errors in the biological measurement were at low-voltage steps, where the outward current was small. The model confirmed this result when the maximum conductance of the voltage-dependent current was varied: smaller conductances always resulted in larger errors, which can reach values of 
90% (Fig. 6D). This error in fact stems from the quality of space clamp in the voltage-clamped neuron (1VC protocol), but with the unusual consequence that better space clamp caused a larger error. When the amount of conductance activated is relatively low and therefore the neuron is electrotonically compact, the coupled neuron can be strongly influenced by the clamped neuron. As the conductance in neuron 1 is increased, it becomes less electrotonically compact, thus reducing the influence on the coupled neuron and functionally uncoupling the neurons. As a consequence, the error decreases toward zero. However, the disappearance of the error when conductances are high comes at a price: the distant portions of the neuron contribute little or nothing to the currents measured with the somatic electrodes. This was, for example, apparent in the study of Hartline et al. (1993) who showed that there is very little difference in parameter estimation for currents measured in intact PD neurons compared with currents measured in ligated somata of PD neurons.
The small apparent error in estimating voltage-dependent parameters that characterize the outward currents would be considerably larger if the gap-junction location was closer to the point of impalement (and current injection). Indeed, as suggested by the model (Fig. 5), the closer the gap junction is to the voltage-clamped region (i.e., the soma), the larger is the amount of current recruited from the coupled neuron. Because this current is the result of poorly controlled voltage across the gap junction, paradoxically, for a given gap-junctional conductance the error grows larger with improved space clamp (Fig. 5C). Therefore although errors may not appear large, placing the electrodes in a different position along a neurite that may result in better space clamp may actually further contaminate current measurements by increasing the current flow through gap junctions.
We also examined the effect of changing the strength of the gap junction on the outward current measurements. As expected, smaller or larger values of coupling strength simply reduced or increased the relative discrepancy between the 1VC and 2VC measurements, respectively (data not shown).
The relatively small errors in our experimental measurements [about 23% of the leak conductance (Fig. 2B), and about 10% of the maximal voltage-gated K+ conductance (Fig. 4B)] suggest that the gap junctions between PD neurons in the lobster STG are probably located far away from the soma and close to their dendritic endings, as suggested by the work of Cabirol-Pol et al. (2000). The experimental measurements also show a minimum error of around 5% (Fig. 4A), which remains approximately constant for all membrane potentials >0 mV. This is probably an indication that at these membrane potentials, the voltage-dependent current in the PD neurons is maximally activated and that the neuronal input conductance remains constant. Assuming a uniform distribution of channels, from this minimum error we can estimate that the maximum conductance density is close to 5 x 10–3 mS/cm2 (corresponding to 5% in Fig. 6D), a value very close to the maximal conductance we used in our model.
Errors attributed to leak subtraction
Our modeling results suggest that between strongly electrically coupled neurons leak subtraction can lead to an additional error than those discussed in the preceding section. This error originates from the subtraction of currents generated during a leak-current measurement protocol, when neurons are relatively electrotonically compact, from total current measurements, when neurons are significantly less compact because of the opening of new channels. The difference in actual membrane area under voltage-clamp control (higher during the leak current measurement protocol) accounts for an error that can change polarity as a function of the total conductance activated (Fig. 6B, and inset in Fig. 6D). This error, however, is relatively small (not exceeding about 8%) compared with the errors discussed in the preceding section, and is mainly observed at relatively high leak conductance levels (>0.1 mS/cm2). However, these high leak conductance levels are often encountered (Buchholtz et al. 1992; Hodgkin and Huxley 1952; Miller et al. 1985; Svirskis et al. 2001) and, in these cases, leak subtraction should be performed with caution. It is important to remember, however, that this leak-subtraction–associated error occurs when the leak conductance is homogeneously high along the neuronal structure. During an experiment, the leak conductance may increase as a consequence of pharmacological or photoinactivation manipulations. Because this increase in conductance would likely not be localized near the site of impalement, it may lead to errors during the subsequent application of a leak-subtraction protocol. In contrast, high measured leak conductances arising from microelectrode penetration damage will not have the same effect. Thus it is useful to determine the leak-conductance levels before deciding whether standard leak-subtraction protocols should be applied.
Effect of electrical coupling on synaptic properties
Clearly, 2 presynaptic neurons that are simultaneously active would release more neurotransmitter than a single neuron. However, the efficacy of voltage-dependent (graded) synapses, measured as the amplitude of the postsynaptic potential, depends on many factors. When the 2 presynaptic neurons are electrically coupled and both synapse onto a third neuron, depolarizing only one of them causes the second neuron to be recruited into releasing neurotransmitter and the combined effect will be larger than if a single, uncoupled neuron is depolarized, but smaller than if both neurons are depolarized. The release efficacy is nonlinear with respect to the position of the gap junction, with maximal release from the coupled neuron when the gap junction is closest to the soma. Release from both the stimulated and the coupled neurons also depends on the position of the synapse, again with maximal release when the synapse is closest to the soma.
However, if only one of the 2 coupled neurons is presynaptic, the position of the gap junction plays a more complex role on synaptic release: when the chemical synapse is located between the gap junction and the soma, the farther away the gap junction, the smaller the voltage attenuation and thus the stronger the synaptic release (compare blue and green curves in the range between 0 and 500 µm in Fig. 8C). Interestingly, in our ball-and-stick model, there is a point along the neurite at which the gap junction has the strongest attenuating effect on the membrane potential because the input resistance of the second neurite is lower at this point than that at either end. An efferent chemical synapse located between the soma and this point will be significantly more attenuated compared with the cases where the gap junction is located either closer to the soma or farther away toward the end of the neurite. A neuron coupled with a gap junction of sufficiently large conductance at this central position of the neurite effectively acts to almost clamp the voltage of its partner for all positions distal from that point. As a consequence of this "voltage-clamping" effect, for very distal locations along the neurite, voltage attenuation may turn out to be less significant if the gap junction is located close to the electrotonic center of the neurite than if it is located at the end of it (compare blue and green curves in the range between 700 and 1,000 µm in Fig. 8C).
Gap junctions are known to be more abundant in the developing than in the adult nervous system (Kandler and Katz 1995; Peinado et al. 1993a,b; Vandecasteele et al. 2005) and appear to be critical in the specification of axonal tree structure during development (Wolszon et al. 1994a,b). Gap junctional "pruning" has also been suggested to be a potential mechanism for the remodeling of embryonic rhythmic networks into the adult central pattern-generating networks (Bem et al. 2002). Gap junctions in the nervous system seem to occur between the tips of either dendrites or axon terminals as has been shown for retinal ganglion cells (Deboer and Vaney 2005) and leech Retzius neurons (Garcia-Perez et al. 2004). In spite of the possible effects of interacting chemical and electrical synapses, except for a few estimates (Fukuda and Kosaka 2003; Garcia-Perez et al. 2004), hardly any information exists about their co-localization. Our observations indicate potential interactions of the flowing currents between coupled neurons that may be important for synaptic efficacy. For instance, it seems that to preserve a maximal chemical synaptic output while maintaining electrical coupling by gap junctions, it would be best if the chemical synapses are closer to the locus of electrical activity and, in that case, the gap junctions should be distributed either near the tips of the neurites or as close to this locus as possible. These considerations about gap junction and chemical synapse co-localization may be important when using network models involving both types of connections as is the case of the GABAergic interneuron networks of the hippocampus (Fukuda and Kosaka 2000a,b; Skinner et al. 1999; Traub 1995).
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Address for reprint requests and other correspondence: J. Golowasch, Dept. of Biological Sciences, 101 Warren St., Newark, NJ 07102 (E-mail: Jorge.P.Golowasch{at}njit.edu)
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