INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Electrical coupling occurs in many vertebrate and invertebrate neural circuits (Bennett 1997; Kiehn and Tresch 2002). Experimental and modeling studies suggest that electrical synapses can shape the output of neural networks in ways that go beyond their classical role in synchronization of neural systems and rapid signal transmission (Marder 1998) and allow them to perform such complex tasks as temporally precise coincidence detection (Edwards et al. 1998). In several systems where electrical synapses contribute to complex functions, the junctional conductance of the synapses has been identified as an important parameter for the performance of the circuit. An example for this is found in the mammalian inferior olive, where electrical synapses-if they fall in the appropriate conductance rangecan theoretically contribute to the rhythmogenesis in a population of not intrinsically oscillating neurons and where the strength of the electrical connections influences the period and amplitude of the emerging rhythm (Manor et al. 1997). Similarly, gap junctions between interneuron dendrites in the hippocampus can enhance the synchrony of gamma oscillations in spatially extended interneuron networks, but their ability to do so strongly depends on the junctional conductance of the electrical synapses (Traub et al. 2001). Another example are models of electrically coupled neuron pairs where the phase relationship between coupled oscillators, the cycle period and transitions between spiking and bursting behavior all depend on the junctional conductance (Sherman and Rinzel 1992). Knowledge of the conductances of electrical synapses in a neural circuit can therefore be crucial for the understanding of circuit function.

The common method of computing the conductance of a synapse from dual voltage recordings at the two cell bodies is based on the assumption of isopotential neurons (Bennett 1966) as illustrated in Fig. 1A. In intact circuits, however, the coupling site can be electrotonically remote from the sites of current injection and recording at the cell bodies of the coupled neurons. In such cases, the approach will necessarily underestimate the actual synaptic conductance by an amount that depends on the geometry and electrical properties of the neurites of both cells between the cell bodies and the coupling site.

View larger version (32K): [in this window] [in a new window]   Fig. 1. Neuron pair with electrical synapse. A: conventional model for electrically coupled pair. The neurons are represented by single isopotential compartments with resting potentials Vrest, membrane capacitances c, ohmic input conductances g, and an effective synaptic conductance gsyn. B: equivalent circuit for nonisopotential neurons with unbranched neurites and a synaptic connection at the frontal contact of 2 neurites. Parameters are the membrane capacitances c, the membrane conductances g and the cytoplasmic resistances r per unit length of the neurites (differential length element dx), the synaptic conductance syn and the capacitances csoma and conductances gsoma of the somatic membranes. Battery symbols stand for resting potentials. C: cultured pair of neurons from Lymnaea stagnalis with straight neurons on a lane of growth promoting factors. The white arrow indicates the position of the electrical synapse at the frontal contact of the 2 neurites. Scale bar: 100 µm.

In this paper, we address the problem of electrical coupling in nonisopotential neurons choosing a simple geometry of the neurons with an electrical synapse located at the tips of two unbranched neurites. Using cable theory as illustrated in Fig. 1B, we derive an analytical expression that relates the synaptic conductance to the passive cable properties and somatic recordings and to the apparent conductance given by the isopotential theory. The equation implies that conductance estimates obtained with the conventional method significantly differ from the local junctional conductance if the length of the two neurites is comparable to the electrotonic length constant and if their resistance is comparable to the resistance of the synapse.

For an experimental test, we used neurons from the snail Lymnaea stagnalis in culture. Their outgrowth in vitro can be guided by chemical patterns to achieve straight and unbranched neuritic cables. When two neurons grow on the same lane, their colliding growth cones readily formed nonrectifying electrical synapses (Prinz 2000; Prinz and Fromherz 2000) as shown in Fig. 1C. The technique allows the design of pairs of neurons that share a single point of contact with an electrical synapse at the terminal of two straight neuritesa geometry that corresponds to the one considered in our theoretical approach. Isolating the neurons from their surrounding tissue and restricting their neuritic tree to a straight cable has the additional advantage of facilitating the computation of the neuritic cable properties from high-resolution optical-imaging data obtained with voltage-sensitive dyes (Fromherz and Müller 1994). This alternative way of obtaining cable properties uses voltage transients observed throughout the neuritic cable rather than transients measured at the cell body alone. The imaging technique thus avoids the ill-posed parameter identification problem (White et al. 1992) associated with applications of the somatic shunt cable model to somatic voltage and current transients (Durand 1984; Kawato 1984).

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Substrate for guided growth

Linearly patterned substrates for neurite guidance were prepared as previously described by UV lithography of a layer of brain-derived growth-promoting factors (Prinz and Fromherz 2000). Briefly, glass coverslips with attached silicone chambers (flexiPERM, In Vitro Systems and Services, Osterode, Germany) were incubated overnight with 1 mg/ml poly-L-lysine (Sigma, P6516) in Tris buffer (0.15 M, pH 8.4) (Wong et al. 1983). After several washes with water, the chambers were filled with defined medium (PAN Systems, Aidenbach, Germany) containing (in mM) 40.0 NaCl, 1.7 KCl, 4.1 CaCl₂, 1.5 MgCl₂, 1.0 glutamine, and 10.0 HEPES (pH 7.9), all other ingredients of Leibovitz L-15 medium at half of the standard concentration, and 20 µg/ml gentamycin (Sigma, G3632) (Ridgway et al. 1991). Central ganglionic rings were isolated from L. stagnalis that were raised in tap water and fed on lettuce. Prior to dissection, the snails were deshelled and soaked in 25% Listerine in normal saline consisting of (in mM) 51.3 NaCl, 1.7 KCl, 4.1 CaCl₂, 1.5 MgCl₂, and 5.0 HEPES 5.0 at pH 7.9 (all from Sigma), then pinned to silicone elastomer (Sylgard) in antibiotic saline, which is normal saline with 150 µg/ml gentamycin (Ridgway et al. 1991). The dissected brains were extensively washed in antibiotic saline and added to the defined medium in the chambers at 3 brains/ml. After 3-5 days at 20°C, the conditioned supernatant medium and the brains were removed. The cover slips were dried and illuminated for 20 min with the full spectrum of a mercury lamp (Osram HBO200) through a mask consisting of aluminum strips (14 µm wide and 33 µm apart) on a silica plate (Fromherz and Schaden 1994). Then the chamber was filled with defined medium or with defined medium mixed with the conditioned medium at a ratio of 1:1.

Cell culture

Neurons were isolated from the A-clusters in the pedal ganglia of L. stagnalis and cultured following standard procedures (Ridgway et al. 1991) as follows: after removing the outer connective tissue, snail brains were incubated in antibiotic saline for 15 min and in defined medium with 1.33 mg/ml collagenase/dispase (Boehringer Mannheim) and 0.67 mg/ml trypsin (Sigma, T8253) for 30-35 min, washed several times with defined medium, treated with 0.67 mg/ml trypsin inhibitor (Sigma, T9003) in defined medium for 15 min, and again washed several times. Then a high-osmolarity medium was applied, consisting of defined medium with 30 mM glucose added. The pedal ganglia were opened using a micro-needle. Neurons from the A-clusters were removed by aspiration through a fire-polished and siliconized (Sigmacote, Sigma) micropipette, immediately placed on the patterned substrate, and incubated for 1 day at 20°C (Ridgway et al. 1991). The cells were positioned in pairs at distances of several hundred µm on the same lane for the formation of synapses or positioned in isolation for the imaging experiments.

Electrophysiology

Both for the cell pair experiments and for the optical imaging experiments, neurons were impaled with sharp microelectrodes pulled from glass capillaries, filled with a saturated solution of K₂SO₄ and contacted with a chlorinated silver wire. Electrode resistances were 10-25 M. An electrode amplifier operating in discontinuous mode (Ba-1S, npi Advanced Electronic Systems, Germany) was used to inject current and monitor voltage. We applied hyperpolarizing currents of up to 0.5 nA amplitude and 900 ms duration in the experiments with cell pairs and hyperpolarizing current pulses of 10-ms duration and up to 2 nA amplitude in the imaging experiments.

Optical recording

The voltage-sensitive dye dibutyl-naphtalene-butylsulfonato-isoquinolinium (BNBIQ) (Ephardt and Fromherz 1993; Fromherz and Müller 1993) was dissolved at a concentration of 4.3 mM in 15 mM cholic acid (Sigma) (Meyer et al. 1997) in Lymnaea cell culture medium (Ridgway et al. 1991). To stain cultured snail neurons, this stock solution was diluted 50-fold with medium and immediately added to the culture chamber, which was then incubated for 10 min in the dark and subsequently washed with medium. To achieve satisfactory staining of the neural membrane, the staining procedure was repeated up to 10 times.

The straight and unbranched neurites of the stained cell were projected onto a linear array of 100 photodiodes (Müller 1994) (LD50-5, Centronic, England) through an inverted microscope (Axiovert 135, Zeiss, Germany), and the cell body was impaled with microelectrodes. For dye excitation, the light from a Xenon lamp was passed through a 450 to 190 nm band-pass onto the cell. The fluorescent light emitted by the dye in the somatic and neuritic membrane was long-pass filtered at 560 nm and recorded at a spatial resolution of 4-5 µm as previously described (Fromherz and Müller 1994; Meyer et al. 1997) while hyperpolarizing current pulses were injected into the cell.

When illuminated, the voltage-sensitive dye bleaches and the fluorescence intensity therefore decays over the exposure time. To separate this effect from the fluorescence change due to neuronal activity, we corrected the recorded fluorescence transient for each diode. This was achieved by fitting the parts of the fluorescence transient during which the cell was at resting potential to a function of the form f(t) = a + b*e^t/c and subtracting this baseline from the recorded transient (Fromherz and Vetter 1992). The difference to baseline was normalized to an amplitude of -1 for each diode individually to account for the unknown proportionality factor between the local membrane potential change and the recorded fluorescence change. The normalized transients obtained in this manner thus contain information about shape and timing of the local voltage transient but not about its amplitude.

Estimation of cable properties from imaging data

To estimate the electrical properties of cultured snail neurons with straight neurites, we used a model consisting of homogeneous cylindrical cables joined at one end to a spherical soma and sealed at the other end. The length l and diameter d of each cable were matched to the dimensions of the grown neurite. The resistance per unit length r is related to the cytoplasmic resistivity R as r = 4R/(d²), the membrane conductance per unit length g to the membrane conductance G per unit area as g = Gd and the capacitance per unit length c to the membrane capacitance C per unit area as c = Cd. The total capacitance c_soma and conductance g_soma of the cell body depend on the diameter d_soma and the capacitance per unit area C_soma and conductance per unit area G_soma as c_soma = C_somad and g_soma = G_somad. The model neuron was stimulated by injecting a hyperpolarizing current pulse into the soma. The dynamics of the membrane potential at the soma and along the neurites were obtained by numerical solution of the cable equation using an Euler forward algorithm (Press et al. 1997).

The voltage transient in each compartment of the model neuron in response to the hyperpolarizing somatic current pulse was normalized to an amplitude of -1. This was necessary to allow a comparison of the simulated voltage transients and the recorded fluorescence transients, which were normalized due to the lack of amplitude information inherent to the imaging technique. Thus the fit relied solely on the delay and shape of voltage transients at different locations along the neurites. A fit of only the timing and shape of the transients is sufficient because both of these features are influenced by the cable parameters and therefore contain information about them.

We adjusted the electrical parameters r, g, and c of the model neurites and the parameters g_soma and c_soma of the model cell body to achieve an optimal fit between the normalized voltage transients in the model and the normalized optical records. To reduce the number of parameters, we assigned the standard capacitance C = 1 µF/cm² to the neurite membrane. An initial estimate of the somatic membrane conductance G_soma was obtained from the experimental input conductance of a soma with sealed neurites. We used the relation   G_soma(d + dl) with length l and diameter d of the neurites, implying that the neuron is electrotonically compact and that the specific membrane conductances of cell body and neurites are similar. For given diameter d_soma of the soma and known diameters d and lengths l of the neurites, we chose a value for C_soma and varied G and R to minimize the deviation of the computed normalized voltage transients from the normalized experimental fluorescence transients. Let S and S be the experimental and simulated normalized signals for data point d of diode n, and let _n be the SD of the transient of diode n prior to the onset of current injection. With N fluorescence transients and D data points per transient, we used ²/(ND) as a measure for the deviation where ² = S  S)/_n]². The fit procedure was repeated for different values of C_soma to find a set C_soma, G, and R that best reproduced the recorded somatic voltage transient and the normalized optical recordings along the neurites. Because the somatic membrane conductance G_soma used as an input parameter for this fit was only a rough estimate (see preceding text), we repeated the fit procedure with other values of G_soma in a physiologically meaningful range to evaluate to what extent the fit results for C_soma, G, and R depend on the particular choice of G_soma.

Synaptic coupling coefficient

The strength of an electrical synapse is usually described in terms of the stationary voltage changes in the postsynaptic and presynaptic cell caused by a hyperpolarizing current injection into the presynaptic neuron. Given the stationary voltages without (denoted by 0) and with (denoted by ) an injection current i_inj, the coupling coefficient k_pre-post is defined by Eq. 1 with the voltage changes V_pre = V  V and V_post = V  V (Bennett 1966)

(1)

Synaptic conductance assuming isopotential neurons

The conventional model to describe two electrically coupled passive neurons is shown in Fig. 1A (Bennett 1966), although Bennett implicitly assumed the special case of V_pre,rest = V_post,rest. The parameters are an effective synaptic conductance g_syn, the input conductances g_pre and g_post of the pre- and postsynaptic neuron and the resting potentials V_pre,rest and V_post,rest. An injection current i_inj in the presynaptic cell gives rise to a change of the stationary currents through the presynaptic membrane, the synapse and the postsynaptic membrane. Equations 2 and 3 are the steady-state current balances for both neurons

(2)

(3)

We apply these relations to the steady states before and after a current step from 0 to i_inj in the presynaptic cell. When we subtract the two versions of Eqs. 2 and 3 before and after the step from each other, we obtain Eqs. 4 and 5, where V_pre and V_post are the changes in the steady-state voltages induced by the current step

(4)

(5)

From Eq. 5 follows Eq. 6 for the ratio of the post- and the presynaptic voltage change, which corresponds to the coupling coefficient k_pre-post according to Eq. 1

(6)

Equation 6 shows that the coupling coefficient does not reflect the synaptic conductance g_syn alone, but that it depends on the input conductance g_post of the postsynaptic neuron (Bennett 1966).

Using Eq. 4, we eliminate the presynaptic voltage change V_pre from Eq. 6 and solve for V_post to obtain Eq. 7

(7)

According to Eq. 7, the postsynaptic response V_post is symmetric in the input conductances g_pre and g_post of the two neurons. This means that the postsynaptic voltage change is the same no matter which of the two neurons is postsynaptic.

Finally, we can express the synaptic conductance g_syn in terms of experimentally accessible voltage changes V_pre and V_post. To do so, we first stimulate neuron A in a cell pair with a current step from 0 to i_inj (such that A is pre and B is post), and then apply the same current step to neuron B (such that B is pre and A is post). From the two versions of Eqs. 4 and 5 for the two directions of signal transfer we can derive Eq. 8

(8)

Here, V_pre,A is the voltage change in neuron A when it is presynaptic (and analogously for V_post,A, V_pre,B, and V_post,B and V_post = V_post,A = V_post,B according to Eq. 7.

Synaptic conductance assuming nonisopotential neurons

Figure 1B shows the circuit for two electrically coupled neurons that takes the passive cable properties of the neurites between the cell bodies and the synapse at their tips into account. In steady state, the current balances for the two neurons are given by Eqs. 9 and 10

(9)

(10)

In these equations, g_pre,soma is the membrane conductance of the presynaptic soma, g_pre,dist is the total input conductance of its distal neurite or neurites, i is the membrane current of its proximal neurite, and V_pre and V are its membrane potential at the soma and at the tip of the proximal neurite, respectively. The presynaptic resting potential _pre,rest and the junctional conductance _syn of the synapse have cable-like overbars to distinguish them from the corresponding parameters in the isopotential model, and all postsynaptic parameters are named in complete analogy to the presynaptic ones.

Equations 9 and 10 contain the experimentally not accessible membrane currents i and i and tip potentials V and V that don't occur in the current balances Eqs. 2 and 3 of the isopotential model. To allow a comparison between the two models, we use cable theory to express these currents and potentials in terms of electrical properties of the neurons and somatic potentials (see APPENDIX). After these substitutions, the current balances can be brought into the same mathematical form as the balances in the isopotential model, resulting in the (slightly unwieldy) Eqs. 11 and 12

(11)

(12)

Here

(13)

stand for the input resistances that the two neurons would have if their proximal neurites were sealed. L_pre and L_post are the electrotonic lengths of the proximal neurites, _pre and _post are their electrotonic length constants, r_pre and r_post are their axial resistances per unit length, and we have introduced the abbreviations, Eqs. 14 and 15

(14)

(15)

Comparison of the coeffients in the current balances Eqs. 11 and 12 with the balances Eqs. 2 and 3 in the isopotential model immediately suggests relations between the input conductances g_pre and g_post, the resting potentials V_pre,rest and V_post,rest and the synaptic conductance g_syn in the isopotential model and the corresponding parameters _pre, _post, _pre,rest, _post,rest, and _syn in the nonisopotential model. In particular, that comparison suggests the identification g_syn = .

The equation 1 g_syn =  is confirmed by using the same transformations that led from the current balances Eqs. 2 and 3 to expression Eq. 8 in the isopotential model. When we apply these transformations to the current balances Eqs. 11 and 12 we obtain Eq. 16

(16)

Comparison of Eqs. 16 and 8 shows that g_syn and  are indeed identical. Together with the definition (Eq. 15) of , we obtain the crucial relation Eq. 17 between the synaptic conductance _syn and the estimate g_syn given by the isopotential model

(17)

A complete neglect of the neurites leads to g_syn = _syn. For electrotonically short neurites with L_pre 1 and L_post 1, we obtain Eq. 18, a relation that reflects the serial resistances of the synapse and the proximal neurites

(18)

Assuming r_pre = r_post = r with a total neurite length l = l_pre + l_post, we obtain the relation _syn = g_syn/(1  g_synrl) for g_synrl < l. In this parameter range, the ratio of the apparent conductance from the isopotential theory and the synaptic conductance from the cable theory drops for increasing neurite lengths, for increasing neurite resistances, and for increasing synaptic conductances.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

We describe current-clamp experiments with neuron pairs from L. stagnalis joined by unbranched neurites with an electrical synapse at their tips. We first evaluate the experimental data in terms of the conventional theory that assumes isopotential neurons. Then, as a basis for the evaluation with the nonisopotential model, the cable parameters of the neurites are determined from optical mapping of voltage transients in the neurites. Using these results, we compute the junctional conductances of the electrical synapses and compare them to the estimates from the isopotential approach.

Synaptic coupling coefficient of Lymnaea neuron pairs

Neurons isolated from the A clusters in the pedal ganglia of pond snails were cultured on linearly patterned substrates as previously described (Prinz and Fromherz 2000). Over the course of one day in culture, most cells extended unbranched neurites along the straight lanes of growth-promoting substrate. When two cell bodies were positioned on the same lane at a distance of several hundred micrometers, their growth cones could make a contact as illustrated in Fig. 1C. Figure 2 shows recordings from such a pair of cultured neurons with simple neurite geometry (Fig. 2A). The voltage transients caused by depolarizing and hyperpolarizing current injections in either cell were accompanied by postsynaptic potential changes in the other cell (Fig. 2B). The transfer of hyperpolarization demonstrated that the coupling at the contact site was electrical. Figure 2C shows families of hyperpolarizing injection currents of increasing amplitude and the pre- and postsynaptic voltage transients they caused.

View larger version (19K): [in this window] [in a new window]   Fig. 2. Electrical coupling between pair of geometrically simple neurons. A: micrograph of 2 Lymnaea neurons whose growth cones touch after 1 day of culture on a linearly patterned substrate. The arrow indicates the location of the electrical synapse between the 2 growth cones. Scale bar: 100 µm. B: comparison of depolarizing and hyperpolarizing current steps injected into the cells shown in A. Current and voltage traces for both cells in response to injection into neuron A (top) and neuron B (bottom). The corresponding injection currents are shown above each set of voltage traces. Horizontal lines in each panel indicate 0 injection current and resting potential. Black arrows show the direction of signal transfer. Hyperpolarizing and depolarizing voltages changes can spread through the synapse in either direction. C: comparison of different hyperpolarizing current steps. Current and voltage traces for current injected into cell A (top) and cell B (bottom). Injection currents are plotted above each set of voltage traces. Families of traces like these were used to estimate the conductance of the electrical synapse as described in the text.

From the pre- and postsynaptic steady-state voltages in each cell before and after the current step we computed the coupling coefficient k_pre-post of the electrical synapse according to Eq. 1. We found no voltage dependence of the coupling between these cultured neurons (Prinz 2000), so we averaged the coupling coefficients obtained with different injection currents for each cell pair and each direction of signal transfer. From the data shown in Fig. 2C we obtained k_pre-post = 0.18 from cell A to cell B and k_pre-post = 0.08 from cell B to cell A. The difference is due to different input conductances of both neurons according to Eq. 6. Figure 3A shows the cumulative distribution function of the coupling coefficients k_pre-post of 11 cell pairs. We found a wide range of coupling strengths between extreme values k_pre-post = 0.01 and k_pre-post = 0.27 with an average of k_pre-post = 0.09. 

View larger version (11K): [in this window] [in a new window]   Fig. 3. Coupling coefficient and synaptic conductance with isopotential theory. A: cumulative distribution function of the coupling coefficients kpre-post obtained from current step injections in 11 geometrically simple pairs of snail neurons with a single point of contact. Note that there are 2 coupling coefficients for every cell pair1 for each direction of signal transfer. B: cumulative distribution function of synapse conductances gsyn computed for the same cell pairs under the assumption of isopotential cells. C: coupling coefficients kpre-post plotted vs. gsyn. The two measures of synapse strength are not well correlated.

Synaptic conductance under the assumption of isopotential cells

To estimate the synaptic conductance g_syn under the assumption of isopotential cells, we used relation Eq. 8 for the 11 cell pairs considered in the preceding paragraph and again averaged the results for different injection currents in each cell pair. Figure 3B shows the cumulative distribution function of g_syn. The synapse conductances are rather small, with an average of g_syn = 0.33 nS and extreme values of g_syn = 0.04 nS and g_syn = 0.98 nS.

In Fig. 3C, the coupling coefficients k_pre-post are plotted versus the conductances g_syn for the 11 neuron pairs. The figure shows that k_pre-post is not well correlated to g_syn and therefore conveys little information about the strength of the underlying synapse. The discrepancy must be assigned to the variability of the input conductance g_post of the postsynaptic neurons that affects the coupling coefficient for isopotential neurons according to Eq. 6.

Cable parameters from optical imaging

To determine the cable parameters of the neurites, individual neurons from the A clusters were cultured on linearly patterned substrates, stained with the voltage-sensitive dye BNBIQ and impaled with a microelectrode. A brief hyperpolarizing current pulse was injected into the soma, and the change of fluorescence intensity along the neurites was recorded with a photodiode array and processed as described in METHODS.

The outline drawing of a neuron with two straight neurites is shown in Fig. 4A. The somatic voltage response to a 10 ms current pulse is depicted in Fig. 4B. The spatiotemporal fluorescence pattern in the neurites is displayed as a color-coded map in Fig. 4C with the bleach-corrected fluorescence transients normalized to a peak value of -1 for each diode. Obviously, the hyperpolarization propagates rather fast through the entire cell without significant broadening. The neuron seems to be electrotonically fairly compact. Nonetheless, a careful inspection shows that the signal exhibits a delay along the neurite.

View larger version (55K): [in this window] [in a new window]   Fig. 4. Cable properties of cultured Lymnaea neurites. A: outline drawing of a snail neuron cultured on a linear pattern. The cell was stained with a voltage-sensitive dye and impaled with a microelectrode. B: somatic voltage transient (black) of the neuron in response to a 10-ms hyperpolarizing current pulse of 2 nA injected into the cell body and simulated response (red) of the corresponding model neuron to the same current injection for a best fit of the electrical parameters. C: color coded spatiotemporal map of the normalized fluorescence transients in the neurites and contour map of normalized simulated voltage transient for a best fit of the electrical parameters. The color code ranges from -1 (blue) to 0 (red) and the contour plot has solid lines at -0.1, 0.3, 0.5, and -0.7, and a dashed line at -0.9. The hyperpolarization propagates fast throughout the neuron within a few milliseconds.

We determined the soma and neurite dimensions of the neuron from a photomicrograph of the cell and constructed a model neuron of the same dimensions; in the case of the neuron in Fig. 4, the soma diameter was 57 µm, the neurite lengths were 311 and 154 µm and the neurite diameter was 7 µm, a diameter typical for Lymnaea neurites grown on patterned substrates. With this model neuron, we simulated voltage transients in response to the same hyperpolarizing current pulse used in the experiment and fitted them to the fluorescence transients as described in METHODS. The simulated somatic voltage transient was superimposed on the experimental transient in Fig. 4B, and the simulated spatiotemporal voltage in the neurite is displayed as a contour map in Fig. 4C. The electrical parameters obtained from the fit were C_soma = 1.5 µF/cm² and G_soma = 0.038 mS/cm² for the capacitance and the conductance of the somatic membrane and G = 0.043 mS/cm² and R = 469 cm for the membrane conductance and the cytoplasmic resistivity of the neurites. We repeated the measurement and fit for two other neurons and found values of 0.9 µF/cm² and 0.8 µF/cm² for C_soma, 0.005 mS/cm² and 0.016 mS/cm² for G_soma, 0.03 mS/cm² and 0.033 mS/cm² for G, and 476 and 236 cm for R. With ²/(ND) < 2 for the best fit in all three neurons, the experimental and simulated transient shapes agreed rather well (Press et al. 1997).

The value of G_soma used for the model neuron was the least well defined input to the fit and can only be a rough estimate of the actual somatic membrane conductance. For that reason, we tested whether other values of G_soma would also lead to adequate fits with different results for the other parameters. In fact, varying G_soma in the physiological range could lead to fits with similarly good agreement with the experimental data as the best fit reported above. This variation of G_soma left the values for C_soma and R virtually unchanged but could result in a significantly changed value obtained for the neuritic membrane conductance G, indicating that there is a trade-off between G_soma and G.

Averaged over all three experiments, the neurite parameters obtained from the best fits are G = 0.035 mS/cm² and R = cm, and the average neurite diameter is d = 5 µm. With these values, the electrotonic length constant  =  of the neurites is in the range of   1,000 µm. Given the typical length of cultured Lymnaea neurites of several hundred micrometers, their electrotonic length L = l/ is small but cannot be neglected. For such relatively short electrotonic lengths, Eq. 18 shows that the dominant cable parameter of influence for the synaptic conductance estimate is the cytoplasmic resistivity R. Fortunately, R is the most reliable parameter obtained from the optical recordings.

Theory for remote synapses

Figure 1, A and B, shows the equivalent circuits for a pair of electrically coupled neurons without and with neurites between the cell bodies and an electrical synapse. In the isopotential model in Fig. 1A, the effective synaptic conductance g_syn can be computed from somatic recordings in response to current injection according to Eq. 8. With cable theory we derived

(17)

which expresses the true synaptic conductance _syn between the neurite terminals in terms of g_syn and the experimentally determined cable properties of the presynaptic and postsynaptic neurites. These are the resistances per unit length r_pre and r_post, the geometric lengths l_pre and l_post, the electrotonic length constants _pre and _post, and the electrotonic lengths L_pre = l_pre/_pre and L_post = l_post/_post. Relation 18 is the approximation of Eq. 17 for electrotonically short neurites with L_pre 1 and L_post 1 

(18)

The synaptic resistance 1/_syn is smaller than the apparent resistance 1/g_syn because 1/g_syn includes the axial resistances r_prel_pre and r_postl_post of the two neurites.

Conductances of remote synapses

We used the average values of G = 0.035 mS/cm² and R = 394 cm and the neurite lengths and diameters measured from photomicrographs of the coupled neuron pairs to compute with Eq. 17 the actual junctional conductance _syn from the estimated value g_syn obtained with the isopotential model as described in the preceding text. The resulting values of _syn are shown in Fig. 5A in a cumulative distribution function and fall between 0.1 and 1.2 nS.

View larger version (14K): [in this window] [in a new window]   Fig. 5. Synaptic conductances. A: cumulative distribution function of synapse conductances syn computed for 11 cultured snail cell pairs evaluated with the cable theory. B: synapse conductances syn computed with the cable theory plotted vs. gsyn with the isopotential theory the same cells. All data are above the diagonal marking syn = gsyn.

For a more direct comparison, we plotted the synapse conductances _syn obtained with cable theory versus the conductances g_syn computed with the isopotential theory in Fig. 5B. The isopotential theory obviously underestimates the synaptic conductance with _syn, although the difference is small. The actual synapse conductances of the 11 cell pairs used here are on average 13% larger than would be expected from the isopotential model.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Our theoretical and experimental results indicate that neuritic cables between the (typically somatic) recording sites and the electrical synapse between two neurons can distort the conductance estimate for that synapse. This is particularly relevant in cases where electrical synapses are located far from the cell bodies and yet significantly shape the signaling between the coupled cells. Examples for this can be found in the Mauthner system of the goldfish and in rat hippocampus: In the goldfish, electrical synapses between the club endings of auditory afferent fibers and the distal part of the Mauthner cell's dendrite synchronize active fibers and promote the recruitment of new fibers via back-firing (Pereda et al. 1995). And in rat hippocampal slices, axo-axonal gap junctions allow for very fast signal transmission between principal neurons and are thought to enable highly precise signaling and high-frequency oscillations in networks of neurons (Schmitz et al. 2001).

Coupling coefficient and g_syn

A direct measure of the coupling by electrical synapses is the coupling coefficient k_pre-post, which is the ratio of the post- and presynaptic voltage change caused by a hyperpolarizing current injected into the presynaptic soma. For our geometrically simple cell pairs, the coupling coefficients mostly ranged between 0.02 and 0.12 (Fig. 3A), which puts them near the lower limit of the coefficients reported for neurons from different molluscan species both in vivo and in vitro (Benjamin and Pilkington 1986, Bodmer et al. 1984, Hadley et al. 1985, Magoski and Bulloch 1998, Magoski et al. 1994). Our cell pairs were grown on guiding substrates and formed synapses restricted to a single point of physical contact, whereas pairs grown under identical culture conditions on homogeneous substrates routinely established several points of physical contact and had about fourfold stronger synapses (Prinz and Fromherz 2000). So the small coupling coefficients reported here are most likely due to the restricted number of contacts between the two cells. Using the isopotential theory, we evaluated the synaptic conductances g_syn. With g_syn < 1 nS, we found them to be rather low compared with the values of typically several tens of nanoSiemens reported for molluscan nerve cells in vitro (Bodmer et al. 1988; Carrow and Levitan 1989) and in situ (Benjamin and Pilkington 1986; Bodmer et al. 1988; Kiss 1979; Wildering and Janse 1992). Although the values computed for k_pre-post and G_syn both indicate weak synaptic coupling, the correlation between the two measures is weak (Fig. 3C), presumably due to the variation of the input conductance of the neurons.

Cable properties from optical imaging

Cable properties of neurons in a number of different systems have been computed from somatic voltage transients in response to current injections at the cell body (Coleman and Miller 1989; Fleshman et al. 1988; Jackson 1992; Stafstrom et al. 1984). In cells whose somatic membrane properties differ from those of the neurites (Durand 1984, Kawato 1984), this parameter identification problem has, however, been demonstrated to be ill-posed in that very small errors in measured data can lead to large and unpredictable errors in parameter estimates (White et al. 1992). To avoid that problem, we took advantage of the simple geometry and good optical accessibility of cultured Lymnaea nerve cells and determined their cable properties from voltage-sensitive dye measurements of voltage transients in the entire cell rather than at the cell body alone (Fromherz and Müller 1994). We estimated the electrical parameters by fitting voltage transients simulated with a passive multi-compartment model to voltage transients reconstructed from fluorescence measurements throughout the cell and transients recorded directly at the cell body (Fig. 4). The propagation of voltage changes along the neurites was very fast, resulting in rather feature-less optical maps (Fig. 4C). For that reason, the evaluation of the cable parameters was less reliable than with neurites where larger propagation delays in the distal parts were observed (Meyer et al. 1997).

We found an average cytoplasmic resistivity R of about 400 cm, a value far higher than the classical 35 cm for the squid giant axon (Hodgkin and Huxley 1952) but similar to values reported for other narrower neurites such as the 250 cm assigned to cultured leech neurons (Fromherz and Müller 1994; Fromherz and Vetter 1992) and the 340 and 300 cm found for rat hippocampal neurons (Major et al. 1994; Meyer et al. 1997). The average neuritic membrane conductance G = 0.035 mS/cm² that led to the best fits in three experiments is similar to values found with imaging techniques in cultured leech neurons (G = 0.05 mS/cm²) (Fromherz and Müller 1994) and cultured rat hippocampal pyramidal neurons (G = 0.07 mS/cm²) (Meyer et al. 1997). The high propagation velocity evident in Fig. 4C is therefore largely due to the unusually large neurite diameters of Lymnaea neurites cultured on patterned substrates.

Impact of neuritic cables on synaptic conductance estimates

We consider Eq. 17 to be the main result of this paper. It relates the junctional conductance of a remote electrical synapse to experimentally accessible currents and voltages and neuritic parameters and will allow researchers to obtain more refined estimates of electrical synapse strength that they could previously achieve by applying the isopotential cell pair model of Bennett (1966). To illustrate the extent to which neuritic cables between the cell bodies and remote electrical synapses can influence the synaptic conductance estimate, the relationship between the apparent synaptic conductance g_syn obtained with Bennett's model and the actual junctional conductance _syn is illustrated in Fig. 6, A and B, for two sets of cable parameters. In Fig. 6A, the parameters were _pre = _post = 1,083 µm, r_pre = r_post = 1.3 × 10⁹ /cm, and l_pre = l_post = 30  1,000 µm. These values for the length constant  and the resistance per unit length r follow from d_pre = d_post = 6 µm, R_pre = R_post = 394 cm, and G_pre = G_post = 0.035 mS/cm², which are the values we obtained for snail neurites cultured on linear lanes (see RESULTS). The figure shows that the discrepancy between _syn and g_syn increases for long neurites and large synaptic conductances. For _syn  0.5 nS and l  300 µm, typical for the neurons used here, the devations are, however, modest.

View larger version (17K): [in this window] [in a new window]   Fig. 6. Effect of cable properties on synaptic conductance estimates. A: synapse conductances syn between the terminals of 2 neurites plotted vs. gsyn computed under the assumption of isopotential cells. The 4 curves show the values computed for neurite lengths lpre = lpost of 30, 100, 300, and 1,000 µm using the cable parameters of the cultured Lymnaea neurites. B: same as in A but using different cable parameters: the specific membrane conductance in this case was G = 0.1 mS/cm3, the cytoplasmic resistivity was R = 200 , and the neurite diameter was d = 1 µm, as reported for various mammalian neurons.

In Fig. 6B, we chose _pre = _post = 354 µm, r_pre = r_post = 25 × 10⁹ /cm, and l_pre = l_post = 30  1,000 µm. These values for  and r follow from d_pre = d_post = 1 µm, R_pre = R_post = 200 cm, and G_pre = G_post = 0.1 mS/cm², which are typical values for various mammalian neurons (Clements and Redman 1989; Koch 1999; Meyer et al. 1997; Stafstrom et al. 1984). Here, significant discrepancies between _syn and g_syn appear for much shorter neurites and much smaller synaptic conductances.

Figure 6 and Eq. 17 indicate that the rather modest differences of only about 13% between the actual junctional conductance g_syn between snail neurite tips and the estimate g_syn obtained with the isopotential model are due to the small electrotonic lengths and weak electrical synapses of the cultured pairs of snail neuronsboth are factors that favor good agreement between the isopotential and the nonisopotential model. For less compact and more strongly coupled pairs of neurons, the error in the synaptic conductance estimate due to the assumption of isopotential cells will, however, be much more dramatic.

Applicability of the nonisopotential theory in vivo

In electrically coupled cell pairs in vivo, the effect of the neuritic cable on the discrepancy of g_syn and _syn is likely to be more severe than in cultured snail neurons. The synapse conductance estimates g_syn can be as low as a few percent of the actual value _syn if the synapse is strong and far away from the cell bodies. Of course, to use Eq. 17 for a determination of the junctional conductance from somatic voltage recordings, one needs to know the length l of the neurites that connect the two cell bodies to the synapse, their resistance r per unit length, and their electrotonic length constant . For coupled cell pairs in vivo, only rough estimates of these values may be available.

The applicability of the model shown in Fig. 1B to electrically coupled neurons with extensive neuritic trees is further compromised by the fact that multiple coupling sites at different electrotonic distances from the cell body might exist and that neurites can extend beyond the coupling site. However, simple cable models have been successfully applied to complex neuritic trees before (Jackson 1992; Rose and Dagum 1988; Stafstrom et al. 1984; Whitehead and Rosenberg 1993), and even the isopotential cell pair model (Fig. 1A) has been used for obviously nonisopotential pairs of neurons (Benjamin and Pilkington 1986; Bodmer et al. 1988; Kiss 1979). We believe that in such situations the application of relation (Eq. 17) can lead to better estimates of the junctional conductances of electrical synapses than could previously be obtained by assuming isopotentiality, and can do so without the need for detailed compartmental models of the neurons involved.