Electrotonic Structure of Motoneurons in the Spinal Cord of the Turtle: Inferences for the Mechanisms of Bistability

Gytis Svirskis,1,2 Aron Gutman,1 and Jørn Hounsgaard2

 1Laboratory of Neurophysiology, Biomedical Research Institute, Kaunas Medical Academy, 3000 Kaunas, Lithuania; and  2Department of Medical Physiology, The Panum Institute, Copenhagen University, Copenhagen DK-2200, Denmark


    ABSTRACT
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

Svirskis, Gytis, Aron Gutman, and Jørn Hounsgaard. Electrotonic Structure of Motoneurons in the Spinal Cord of the Turtle: Inferences for the Mechanisms of Bistability. J. Neurophysiol. 85: 391-398, 2001. Understanding how voltage-regulated channels and synaptic membrane conductances contribute to response properties of neurons requires reliable knowledge of the electrotonic structure of dendritic trees. A novel method based on weak DC field stimulation and the classical method based on current injection were used to obtain two independent estimates of the electrotonic structure of motoneurons in an in vitro preparation of the turtle spinal cord. DC field stimulation was also used to ensure that the passive membrane properties near the resting membrane potential were homogeneous. In two cells, the difference in electrotonic lengths estimated with the two methods in the same cell was 6 and 9%. The majority of dendritic branches terminated at a distance of 1 electrotonic unit from the recording site. The longest branches reached 2lambda . In the third cell, the difference was 36%, demonstrating the need to use both methods, field stimulation and current injection, for reliable measurements of the electrotonical structure. Models of the reconstructed cells endowed with voltage-dependent conductances were used to explore generation mechanisms for the experimentally observed hysteresis in input current-voltage relation of bistable motoneurons. The results of modeling suggest that only some dendrites need to possess L-type calcium current to explain the hysteresis observed experimentally and that dendritic branches with different electrotonical lengths can be bistable. Independent bistable behavior in individual dendritic branches can make motoneurons complex processing units.


    INTRODUCTION
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

This paper deals with the well-known problem of defining the electrotonic image of a nerve cell. Electrotonic measurements provide necessary parameters for complex models of neurons and serve as a basis for understanding the functional role of linear and voltage-dependent membrane properties. For this reason, it is necessary to provide reliable estimates of the electrotonical parameters. Electrotonic measurements have been performed in morphologically reconstructed neurons of different types (Clements and Redman 1989; Fleshman et al. 1988; Major et al. 1994; Rapp et al. 1994; Thurbon et al. 1994, 1998). Unfortunately with the methods used, it was not possible to check for homogeneity of the passive membrane properties. Thus it was necessary either to assume homogeneity or include a fourth unknown parameter to account for the difference between somatic and dendritic membrane properties. However, recordings with two electrodes in a single cell have suggested that dendritic membrane properties can also be intrinsically heterogeneous even near the resting potential (Stuart and Spruston 1998).

Recently methods based on stimulation with weak electrical DC fields were proposed for detecting membrane heterogeneity and injury shunts and for estimating the electrotonic structure of neurons (Svirskis et al. 1997a,b). If the passive membrane resistance of a neuron is homogeneous, then the transient in response to an applied DC field has no characteristic shape peculiarities and develops faster than the response to current injected through the recording electrode (Svirskis et al. 1997b). This transient depends only on membrane time constant, tau , and electrotonical length constant, lambda  (Svirskis et al. 1997a), allowing for a more reliable estimation of the electrotonical structure of neurons. Since the check for homogeneity of the passive properties can be done during experiments, only cells fulfilling these criteria were chosen for electrotonical measurements. The combination of DC field method with the classical current injection method can increase the reliability of electrotonical measurements significantly.

In the present study, the electrotonic parameters were found for three motoneurons in slices of the turtle spinal cord. Two independent estimates of the electrotonic structure were obtained for each cell from responses to weak DC field stimulation and current pulse injection. An acceptable difference between parameters estimated with different methods in two cells suggests that errors introduced by staining procedures and reconstruction were inessential.

Because numerous potential-dependent currents are present in the neuronal dendrites, electrotonical structure alone can provide only a basis for further exploration of how synaptic input is processed. Models of reconstructed motoneurons were used to explore the generation of nonlinear properties due to potential-dependent inward current observed previously (Hounsgaard and Mintz 1988). It is known that part of this inward current is generated in the dendrites of motoneurons in turtles and cats (Hounsgaard and Kiehn 1993; Lee and Heckman 1996, 1998a). In voltage clamp, turtle and cat motoneurons show hysteresis in their input current-voltage relation (I-V) (Lee and Heckman 1998b; Schwindt and Crill 1980; Svirskis and Hounsgaard 1998). The presence of the hysteresis during very long voltage ramp stimulation (Lee and Heckman 1998b) suggests that dendrites are bistable (Gutman 1984; Jack et al. 1983) as theoretical studies had anticipated (Butrimas and Gutman 1978, 1981) from early experimental findings (Schwindt and Crill 1977, 1980). Qualitatively, the hysteresis can be explained by the relatively weak electrical coupling between soma and distal dendrites that allows distal inward current to be activated even when the proximal dendrite is relatively hyperpolarized. Here we explored how strong and where the inward current should be to generate hysteresis as observed experimentally.

Our models suggest that dendrites with different electrotonic length can be bistable and that only a fraction of the dendrites have to be nonlinear to generate the hysteresis observed experimentally. The results also show that a simple equation derived previously for nonlinear cables can be used to predict the behavior of more realistic models.


    METHODS
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

Methods for the experimental procedures

Transverse sections of the lumbar spinal cord were obtained from turtles (Pseudemys scripta elegans) deeply anesthetized with pentobarbitone (100 mg/kg) (Hounsgaard et al. 1988). The medium contained (in mM) 120 NaCl, 5 KCl, 15 NaHCO3, 20 glucose, 2 MgCl2, and 3 CaCl2. 6-Cyano-7-nitroquinoxaline-2,3-dione (CNQX; 40 µM; Tocris Cookson, Bristol, UK) was applied to block excitatory synaptic potentials. For experiments a section of the cord, 0.5-mm thick, was placed in the recording chamber between two silver-chloride electrodes (see Fig. 2A in Svirskis et al. 1997b) used to establish an extracellular DC field.

For recordings, patch electrodes were pulled from borosilicate glass tubes with an outer diameter of 1.5 mm and an inner diameter of 0.86 mm. Electrodes were filled with 125 mM potassium gluconate, 9 mM HEPES, and 1% biocytine; pH was adjusted to 7.4 with KOH. During whole cell recording, voltage transients were generated by injecting a current pulse of 0.3-1.0 nA for 1 ms through the recording electrode or by applying an extracellular current pulse of 1 µA between field electrodes for 150 ms (Svirskis et al. 1997b). The response to the DC field stimulation depends on the direction of the field. Since the transmembrane potential induced by the electric field is largest in direction of the applied field, the shape of the evoked transient reflects electrotonic structure of the dendrites oriented mainly in direction of the field. The DC field was applied in the lateral direction because turtle spinal cord motoneurons have their physically longest dendrites oriented laterally (Ruigrok et al. 1985). To reduce noise, 256 sweeps were averaged on a HIOKI digital oscilloscope (Hioki E. E. Corp., Nagano, Japan) and fed to a computer for later analysis.

After all measurements were accomplished, the electrode was withdrawn from the cell, and the extracellular potential, induced by the same field step stimulus, was recorded, averaged and subtracted from the intracellular potential to get the transmembrane potential. The extracellular potential gradient was measured in nine points at the depth of the recording performed. Extracellular potential values were obtained in three rows each of three points with a 0.5-mm distance between them. The field strength was homogeneous in the range of 10-15% and was 3-4 mV/mm 50-100 µm below the surface of the tissue. Weak electric field stimulation assured that membrane potential changes were less than a few millivolts and did not activate potential-dependent conductances. Because the field strength used for calculations is a proportionality coefficient (see following text), it was allowed to change within the limits of measured strengths. For this reason, only the transient part of the response to the field was used for electrotonical measurements. The dimensions of the slice were measured to assess and correct for shrinkage due to histological procedures. The position of the electrode in relation to the borders of the slice was obtained.

Electrotonic estimates were obtained from three motoneurons recorded with patch electrodes. All responses to the field step had no characteristic shape peculiarities and decayed faster than the response to current injection (Fig. 2), indicating homogeneity of passive membrane resistance (Svirskis et al. 1997a,b). The responses to current pulses of opposite sign were anti-symmetric, showing that membrane properties were linear in the range of the response amplitudes.

Staining and reconstruction

Standard procedures were applied to stain biocytin-injected neurons (Horikawa and Armstrong 1988). In brief, slices were kept in fixative overnight, then washed in phosphate buffer and treated with H2O2. After removal of hydroxy peroxide, slices were incubated with ABC complex and 1% Triton overnight. After wash, DAB treatment was used to visualize the stained neurons. Nickel salt was used to get almost black colored staining. To dehydrate slices, the incubation solution was gradually changed to pure ethanol. Then slices were cleared in xylene and mounted in Permount.

For reconstruction, semi-automatic Neurolucida software and hardware were used with Zeiss microscope. Water-immersed objective allowed to define the diameter with a precision of 0.3 µm in stained motoneurons. Soma was reconstructed as a part of the dendritic tree. The slice contour was outlined to define the shrinkage, which was 1.3-1.6 times, and to set the field direction in relation to reconstructed neuron. The position of the recording electrode was estimated after the coordinates were corrected for the shrinkage. In two motoneurons, this position hit the soma region, whereas in one, it coincided with the proximal dendrite (Fig. 3). Diameters of the dendritic branches were not corrected for the shrinkage, which could cause either the reduction or enlargement of the diameters depending on the intracellular contents of the dendrites (see Major et al. 1994) (see also DISCUSSION).

Calculations for electrotonic measurements

Homogeneous passive membrane properties were assured experimentally by using DC field stimulation as described in the preceding text. For the description of the linear response of the neurons with homogeneous membrane properties, we therefore used a set of electrotonic parameters: membrane time constant, tau  = RmCm, electrotonic length constant, lambda  = (Rma/RiPi )1/2, and characteristic resistance (resistance of semi-infinite cable), Rinfinity = (RmRi/aPi )1/2, where Rm is the specific membrane resistance, Cm is the specific capacitance, and Ri is the specific intracellular resistance. Here, a is the area of a cross-section of the cable and Pi  is the perimeter. The apparent diameter, D, is a morphologically measured quantity, Pi  ~ D and a ~ D2 with constant proportionality coefficients throughout the dendritic tree on a macroscopic scale lambda (Alaburda and Gutman 1996). Thus lambda  and Rinfinity are constant over the same scale. Because lambda  ~ D1/2 and Rinfinity  ~ D-3/2, it is necessary only to know values of parameters for the single particular diameter to define the response to the current and DC field stimulation of the complete dendritic tree. For this purpose, we used constants lambda 1 and R1infinity , which are defined as a electrotonical length and characteristic resistance for a homogeneous dendritic segment with an apparent diameter, D = 1 µm (Svirskis et al. 1997a). For any dendritic segment with a diameter of D µm, lambda  and Rinfinity were found by multiplying lambda 1 and R1infinity by dimensionless number equal to D1/2 and D-3/2, respectively.

The electrotonic parameters, tau , lambda 1, and R1infinity , were estimated by comparing experimental and simulated transients after current pulse injection and stimulation with DC field. Calculations in models of the reconstructed motoneurons were accomplished using Fourier transformation. The results were checked by solving the system of ordinary differential equations directly for the compartmental model of reconstructed motoneurons using the method of Cranck-Nicholson (Press et al. 1992). The same system of equations was also solved as a matrix equation for the eigenvalues, tau n, and eigencoefficients, Cn (Holmes et al. 1992; Perkel et al. 1981), with Matlab (Fig. 1).



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Fig. 1. Eigenvalues and eigencoefficients for the reconstructed motoneuron m2206. Time constants of the response to the 2 stimuli coincide, but their weights are very different. Note that the weight of the membrane time constant, tau  = 29 ms, is 0 in case of DC field stimulation. Electrotonic length constant lambda 1 = 800 µm for both cases. The nonproportional spectra of the eigencoefficients of responses to current and field demonstrates linear independence of the responses.

Here we only outline the method using Fourier transformation (Svirskis et al. 1997a). This gives us insight in the dependence of transients on the parameters. For the current injection the harmonic component of the response at the recording site
<IT>V</IT>(<IT>&thgr;</IT>)<IT>=</IT><FR><NU><IT>&OHgr;</IT></NU><DE><IT>&Lgr;</IT></DE></FR> <IT>I</IT>(<IT>&thgr;</IT>)<FENCE><FENCE><LIM><OP>∑</OP><LL><IT>n</IT></LL></LIM> <FR><NU><IT>D</IT><SUP><IT>2</IT></SUP><SUB><IT>n</IT></SUB></NU><DE><IT>A<SUB>n</SUB></IT>(<IT>&Lgr;</IT>)</DE></FR></FENCE></FENCE> (1)
Here I(theta ) is the harmonic component of the current injected through the microelectrode; theta  is cyclic frequency; Lambda  is the length constant for harmonic potentials, Lambda  = lambda 1/(1 + jtheta tau )1/2, where j is imaginary unit; Omega  is the characteristic impedance, Omega  = R1infinity /(1 + jtheta tau )1/2 for the segment with D = 1 µm, and Dn is the dimensionless number equal to the diameter, expressed in micrometers, of the nth proximal segment. If the transmembrane potential change is induced only by the field then the harmonic component
<IT>W</IT>(<IT>&thgr;</IT>)<IT>=</IT><IT>E</IT><IT>·</IT><FENCE><LIM><OP>∑</OP><LL><IT>n</IT></LL></LIM> <IT>D</IT><SUP><IT>2</IT></SUP><SUB><IT>n</IT></SUB> <FR><NU><IT>B<SUB>n</SUB></IT>(<IT>&Lgr;</IT>)</NU><DE><IT>A<SUB>n</SUB></IT>(<IT>&Lgr;</IT>)</DE></FR></FENCE><FENCE><FENCE><LIM><OP>∑</OP><LL><IT>n</IT></LL></LIM> <IT>D</IT><SUP><IT>2</IT></SUP><SUB><IT>n</IT></SUB> <FR><NU><IT>1</IT></NU><DE><IT>A<SUB>n</SUB></IT>(<IT>&Lgr;</IT>)</DE></FR></FENCE></FENCE> (2)
Here, E is the strength of the DC field. Coefficients A and B were calculated using recursive equations
<IT>A</IT><IT>=&Lgr;</IT><RAD><RCD><IT>D</IT></RCD></RAD> <FENCE><FENCE><IT>cosh </IT><FENCE><FR><NU><IT>X</IT></NU><DE><IT>&Lgr;</IT><RAD><RCD><IT>D</IT></RCD></RAD></DE></FR></FENCE><IT>+&Lgr;</IT><IT>G</IT><RAD><RCD><IT>D</IT></RCD></RAD><IT> sinh </IT><FENCE><FR><NU><IT>X</IT></NU><DE><IT>&Lgr;</IT><RAD><RCD><IT>D</IT></RCD></RAD></DE></FR></FENCE></FENCE></FENCE> (3)

<FENCE><FENCE><FENCE><IT>sinh </IT><FENCE><FR><NU><IT>X</IT></NU><DE><IT>&Lgr;</IT><RAD><RCD><IT>D</IT></RCD></RAD></DE></FR></FENCE></FENCE><IT>+&Lgr;</IT><IT>G</IT><RAD><RCD><IT>D</IT></RCD></RAD><IT> cosh </IT><FENCE><FR><NU><IT>X</IT></NU><DE><IT>&Lgr;</IT><RAD><RCD><IT>D</IT></RCD></RAD></DE></FR></FENCE></FENCE></FENCE>

<IT>B</IT><IT>=&Lgr;</IT><RAD><RCD><IT>D</IT></RCD></RAD> <FENCE><IT>F</IT><FENCE><FENCE><IT>sinh </IT><FENCE><FR><NU><IT>X</IT></NU><DE><IT>&Lgr;</IT><RAD><RCD><IT>D</IT></RCD></RAD></DE></FR></FENCE></FENCE><IT>+&Lgr;</IT><IT>G</IT><RAD><RCD><IT>D</IT></RCD></RAD><IT> cosh </IT><FENCE><FR><NU><IT>X</IT></NU><DE><IT>&Lgr;</IT><RAD><RCD><IT>D</IT></RCD></RAD></DE></FR></FENCE></FENCE><IT>−</IT><IT>A</IT><IT> cos &agr;</IT></FENCE> (4)
where X is the length of the segment, alpha  is the angle between the segment and the field. For terminal segments, G = 0, F = cos alpha . For all other segments, <IT>G</IT><IT>=</IT><LIM><OP><IT>&Sgr;</IT></OP><LL><IT>n</IT></LL></LIM> <IT>D</IT><SUP><IT>2</IT></SUP><SUB><IT>n</IT></SUB><IT>/</IT><IT>A<SUB>n</SUB>D</IT><SUP><IT>2</IT></SUP> and <IT>F</IT><IT>=cos &agr;+</IT><LIM><OP><IT>&Sgr;</IT></OP><LL><IT>n</IT></LL></LIM> <IT>D</IT><SUP><IT>2</IT></SUP><SUB><IT>n</IT></SUB><IT>B<SUB>n</SUB></IT><IT>/</IT><IT>A<SUB>n</SUB>D</IT><SUP>2</SUP>. The sum is carried out for segments adjacent to the distal end of the considered segment, Dn are dimensionless numbers equal to the diameters, expressed in micrometers, of adjacent segments, D- of the more proximal segment considered. Starting the calculation from terminal segments, coefficients A and B were found for the proximal segment of every dendrite emerging from the recording point. In our reconstruction of the neurons, the soma was represented as a part of the dendritic tree.

From Eqs. 1 and 2 and after inverse Fourier transformation, it is evident that in the case of current injection the response of transmembrane potential in any point of the neuron depends on lambda 1, tau , and linearly on R1infinity ; in the case of field stimulation, the response depends only on lambda 1 and tau . Another advantage of using the electrotonic set of parameters is that the solution for the complete dendritic tree does not require specifying the shape of the dendritic cross-section. Solution dependence on the shape is hidden in the electrotonic parameters.

Calculations for nonlinear models

To analyze the mechanisms responsible for the experimentally observed nonlinear properties of motoneurons, we added L-type calcium channel conductance and potassium delayed rectifier conductance in the membrane of models of the reconstructed neurons. Parameters and equations for these conductances were similar to whose used previously in models of bistable motoneurons (Booth et al. 1997). Calcium current Ica = Gca[m(V - Vca- minfinity (-65 - Vca)]; calcium channel activation variable, m, was governed by the equation dm/dt = (minfinity  - m)/tau ca, where minfinity  = 1/{1 + exp[(V - theta L)/kL]}, Vca = 40 mV, tau ca = 20 ms, theta L = -35 mV, kL = -5 mV. The conductance parameter, Gca, was used as a variable parameter. Since the activation kinetics of the K+ delayed rectifier channel is very fast compared with the stimulus protocol used (see following text), the activation was modeled as instantaneous. Thus potassium current IK = GKn4 (V - VK), where n = 1/{1 + exp[(V - theta K)/kK]}, VK = -90 mV, theta K -30 mV, kK = -12 mV, and GK =3/Rm. Input I-Vs for the reconstructed neurons were computed by simulating voltage clamp as triangular ramps of the soma voltage. The speed of the ramp was made 0.005 mV/ms to get almost stationary input I-V. Calculations were performed by directly solving the system of ordinary differential equations for the compartmental models of reconstructed motoneurons.


    RESULTS
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

First of all, we illustrate that the responses to intracellular current injection and field stimulation are independent, i.e., are not linearly transformable. For this purpose, we calculated the eigenvalues and eigencoefficients of the system of equations describing reconstructed motoneurons in both cases of stimulation (see METHODS). As seen in Fig. 1 the eigenvalues are the same for both types of responses except for the absence of the component with the membrane time constant in response to DC field stimulation (Svirskis et al. 1997b). However, the coefficients are very different implying that transients after current injection and field stimulation are independent. This shows that the two methods provide independent measures of the electrotonic structure of neurons and that they therefore also provide a mutual validation.

The transients in neurons with homogeneous passive properties depend on the electrotonic parameters in a way that allows the parameters to be estimated one by one. First, by using the response to injected current pulses, the membrane time constant, tau , was estimated from the exponentially decaying part of the transient (Fig. 2A). Values of tau  varied from 10 to 29 ms for the three motoneurons (see Table 1). Since the transient after a current pulse is proportional to R1infinity (see METHODS), the value of this parameter was made equal to 1 MOmega when calculating the response of the model of reconstructed motoneurons. Only lambda 1 value was changed to achieve the best correspondence between experimental and calculated transients. The fitting was done by dividing the experimentally obtained transient by the calculated transient; this gave the R1infinity value for each point of the transient (Fig. 2B, bottom). Because the calculated transient should be proportional to the R1infinity sought, large variance of R1infinity values at the points of the transient indicates a poor fit to the experimental data. Usually large variance was due not to the scatter of the values but to the global change with time in the transient. Thus the value of lambda 1 was assumed to be the best estimate if it produced minimal variance of R1infinity without a global change in the transient. The lambda 1 values obtained in this way were different for all three neurons and ranged from 550 to 850 µm (Table 1). The procedure described above simultaneously provided an estimate for the characteristic resistance, R1infinity (see Table 1), which was calculated as an average of R1infinity values for all points in the transient. To validate the procedure used, we also calculated the average of squared difference between the experimental and the theoretical transients. In this case the calculated transients were multiplied by the estimated R1infinity value. The estimates of lambda 1 provided the minimal average of the squared difference (Fig. 2B, inset). The electrical parameters were then calculated assuming that the dendritic cross-section is circular. The specific membrane capacitance, Cm, ranged from 0.6 to 1.1 µF/cm2, specific membrane resistance, Rm, had values from 14 to 26 kOmega · cm2, and cytoplasmic resistance, Ri, ranged from 100 to 160 Omega  · cm (Table 1).



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Fig. 2. Electrotonical measurements for motoneuron m2206. A: the membrane time constant, tau  = 29 ms, was found by fitting the decaying exponent to the tail of the transient after current impulse injection. Potential is shown in logarithmic scale. B: the transients were calculated with characteristic resistance, R1infinity , = 1 MOmega . The electrotonic length constant, lambda 1 = 800 µm, defined for the diameter of 1 µm, minimized the variance of R1infinity between the points in the transient (bottom). In this case, the average R1infinity value was 1,030 MOmega when obtained after division of experimental transient by the calculated transient (bottom). These parameter values also provided for the minimal average of squared difference, SD (inset), between the experimental (, top) and the scaled calculated (---) transient during the 1st 40 ms. C: an independent lambda 1 = 850 µm estimate was obtained, which minimized SD (inset) of the difference between the calculated response to the field step and the experimental response. · · · , standard deviation in the experimental traces.


                              
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Table 1. Ohmic parameters of spinal cord motoneurons

To check the validity of measurements, we also estimated lambda 1 by simulating the response of the models of reconstructed motoneurons to the field stimulation. This response depends only on tau  and lambda 1 (see METHODS). Since the membrane time constant, tau , could be reliably estimated from the response to current injection (Fig. 2A), we used it in these simulations as a known parameter. We varied the value of lambda 1 and found the estimate which produced the minimum of the average of the squared difference between experimental and calculated transients (Fig. 2C, inset). In two cells the difference between estimates of lambda 1 obtained by the two methods in the same cell was 6 and 9%, showing that possible errors due to histological procedures were inessential. In the third cell, the difference was 36%, demonstrating the necessity to use current and field stimulations to ensure the reliability of the measurements.

Knowing the electrotonical length constant, lambda 1, defined for the diameter of 1 µm allows calculating electrotonical length constant for any dendritic segment with a diameter of D µm as lambda  = lambda 1 · D1/2 (see METHODS). Surprisingly, despite very different membrane time constants between neurons the electrotonical structure of the dendritic trees did not vary much. The majority of the branches terminated at the electrotonical distance around 1 lambda  from the recording site (Fig. 3) for all three neurons. However, some branches reached as far as 2 lambda  (Fig. 3B).



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Fig. 3. Electrotonic structure of 3 turtle motoneurons. A: motoneuron m2206. Left: diagram showing electrotonic length of the branches in electrotonic units. Upper axis is for the length constant estimated from response to current injection, lower axis from the response to DC field stimulation. Right: the graphical view of the motoneuron in the transverse plane of the spinal cord. The crossing of the straight lines indicates the position of the recording electrode, while the horizontal lines are parallel to the direction of applied DC field. B: motoneuron m0607. C: motoneuron m0207. Note that the longest branch reaches as far as 2 lambda in B.

Since dendrites also possess voltage-dependent conductances, the electrotonic structure provides just a starting point for the exploration of synaptic integration. In motoneurons, dendritic inward current mediated by L-type calcium channels is responsible for plateaus and hysteresis in the input I-V (Hounsgaard and Kiehn 1993; Lee and Heckman 1998b; Schwindt and Crill 1980; Svirskis and Hounsgaard 1998). To investigate the generation of hysteresis in input I-V, reconstructed neurons were used to create nonlinear models with L-type calcium and potassium delayed rectifier conductances (see METHODS).

Input I-Vs were calculated using the complete model of reconstructed motoneuron m2206 (Fig. 4A). To mimic experiments, a slow triangular voltage-clamp ramp at the soma was used as a stimulus for these calculations. Hysteresis appeared in the I-V plot when the conductance of calcium current, Gca, was increased to 1.55/Rm. With the further increase of calcium conductance, Gca = 2.95/Rm, the hysteresis became broader and deeper (not shown). However, in experiments, the I-V hysteresis observed is not very deep. Possibly, only some dendrites have potential-dependent inward currents. As a check, we endowed only the third dendrite of the motoneuron (Fig. 3) with the voltage-dependent conductances while the other two dendrites were left only with the passive leakage current. In this case, the simulated I-V plot (Fig. 4A) became more similar to the experimentally observed plot (Svirskis and Hounsgaard 1997, 1998).



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Fig. 4. Hysteresis in input I-V of nonlinear model of motoneuron m2206. A: input I-Vs for different calcium conductance, Gca, values indicated in the legend. Note the increase of the number of jumps indicating the larger number of bistable dendritic branches as the conductance of the calcium current was increased. Arrows indicate the direction of the change of clamped soma voltage. Only the 3rd dendrite had potential dependent conductances when Gca = 2.95/Rm. B: distribution of membrane voltage in the case of Gca = 1.55/Rm when soma voltage was decreasing (lower branch of dotted hysteresis in A). Two longer branches keep their ends depolarized causing the hysteresis in input I-V. Activation of persistent inward current in the same two branches is responsible for the last jump in hysteresis when Gca = 2.95/Rm (solid hysteresis in A). C: stationary membrane I-V for the calcium conductance Gca = 1.55/Rm. Conductance was normalized to the passive membrane conductance. Dotted line indicates negative slope equal to -2.5.

The mechanisms of hysteresis could be understood from the distribution of membrane potential in the dendritic tree (Fig. 4B) when hysteresis is narrow, i.e., Gca = 1.55/Rm. When the soma voltage was decreasing during the falling phase of the triangular ramp, in some interval of the soma voltage the longest branches were more depolarized than the cell body due to activation of the persistent inward current and weak coupling to the soma (Fig. 4B). The depolarization of the terminal dendrites was absent in the same interval of clamped soma voltage when voltage ramp was rising because the inward current was not yet activated. Thus this bistability of branches is reflected as hysteresis in input I-V. The depolarization of the same two branches also caused the last jumps in input I-V for the larger value of Gca = 2.95/Rm (Fig. 4A), which made hysteresis broader and induced dendritic bistability (Fig. 4B) in shorter branches. Thus in case of broad hysteresis observed under experimental conditions the dendrites with different electrotonic length could be bistable. As the number of bistable branches increases, the variety of responses of motoneuron to synaptic input also increases (Gutman 1991).

Interestingly, the value of calcium conductance, Gca = 1.55/Rm, which induced hysteresis in the model of reconstructed motoneuron could be estimated by using a simple equation. For cables, a mathematical relation defines whether hysteresis is present in the stationary input I-V: the stationary negative slope-conductance, SN, in membrane I-V (Fig. 4C) should be SN > pi 2/4RmLcable2, where Lcable is cable electrotonic length (Baginskas et al. 1999; Gutman 1991; Jack et al. 1983). For the cable with the electrotonical length 1lambda , SN = 2.47. To achieve such a slope in membrane I-V, the calcium conductance, Gca, should be equal to 1.5/Rm, according to the equations in METHODS. In agreement with this estimate only the dendritic branches longer than 1lambda were bistable in the case of Gca = 1.55/Rm (Fig. 4B). This example shows that mathematical relations derived for the cable can be used to get a qualitative estimate of the behavior of nonlinear, branched dendrites. It also proves that hysteresis in the input I-V is a very general phenomenon, which does not depend on particular membrane channels generating the inward current in the dendrites.


    DISCUSSION
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

Electrotonic parameters were found for three spinal motoneurons in the turtle. The electrotonic structure was obtained by two independent methods after checking for homogeneity of the passive membrane properties. Comparison of electrotonic length constants obtained by two independent methods in the same cell suggests that anatomical procedures and reconstruction did not induce substantial errors in two out of three cells.

Electrotonic measurements are prone to errors. Tissue dimensions are changed by the histochemical procedures employed for fixation and staining. In our experiments, shrinkage in tissue dimensions was 1.3-1.6 times. While the lengths of dendrites were corrected, we have chosen not to change the diameter values measured because the shrinkage could cause either reduction or increase of the dendritic diameters (see Major et al. 1994). The change in the diameters is important for estimates of the electrical parameters Rm, Cm, Ri. However the electrotonical structure is less sensitive to global changes in diameters. For example, increase of all diameters by r times causes the reduction of the electrotonical length by the factor r0.5 if lambda 1 is kept constant. But this results in faster transients, which require a decrease of lambda 1 to fit the experimental transient. We estimated lambda 1 values for all three neurons after diameters were corrected for shrinkage. The reduction of lambda 1 was 500/600 = 1/(1.44)0.5, 675/800 = 1/(1.4)0.5, and 500/550 = 1/(1.2)0.5 for the shrinkage of 1.6, 1.3, and 1.3 respectively in three cells. Thus the electrotonical structure would not change significantly if diameters were corrected for shrinkage.

The other source of possible error is the morphological reconstruction. We evaluated the electrotonical structure after inducing a significant reduction in diameter of a compartment near the recording site. In this case, the estimates of lambda  obtained by injection of current pulses and by DC field stimulation differed several times. Since the experimental estimates we obtained differ by only 6 and 9% in two cells, we can be sure that the morphological reconstruction was without major errors.

The only modest correspondence between the estimates of the electrotonic structure in cell m0207 could be explained by the heterogeneity in field strength, which equaled 15% and was the largest among the three cells, distortion of the response to the current impulse by pipette capacitance, and/or heterogeneous shrinkage. At this stage, we cannot distinguish between these explanations. Note, however, that for this cell the value of Cm is smaller and the value of Ri considerably larger than for the other two cells.

The calculated values of eigencoefficiences (Fig. 1) show a striking difference between responses to the field stimulation and current injection. The eigencoefficiences of the fast components are very small in the response to DC field stimulation. Hence the decay rate of the whole transient is a satisfactory reflection of the electrotonic lengths of dendritic branches, which are oriented in the direction of the field. This result validates the DC field stimulation as a method for the estimation of the equivalent electrotonic length of the dendritic tree (Svirskis et al. 1997a). Cell m2206 is particularly good for illustration because the evolution of the response evoked by the DC field depends mainly on two exponents (Fig. 1). The slowest time constant of the transient, tau F = 5.6 ms, and the membrane time constant, tau  = 29 ms, allow us to calculate an estimate of the tip-to-tip electrotonic length L in the field direction. According to the classical Rall equation: L = pi / (tau /tau F -1)1/2 approx  1.5 and is much shorter than L for the two other cells. The estimation fits quite well to the electrotonical structure if we notice that cell m2206 has rather short laterally oriented dendrites and long lambda  (Fig. 3A).

As shown here and elsewhere (Svirskis et al. 1997a,b), DC field stimulation offers several advantages for electrotonic measurements. The essence of the method is that during weak DC field stimulation the total current passing the membrane of a neuron equals to zero (Svirskis et al. 1997b)---there are no net current sinks or sources. Although we used the extracellular current to create a potential gradient for polarizing neurons, other methods could be used as well. Recordings from the same neuron with two independent electrodes have proved feasible (Stuart and Spruston 1998; Stuart et al. 1993). Electrotonic measurements can be obtained with this configuration keeping the total current flowing through the neuronal membrane zero by injecting current of equal amplitude but opposite polarity through the two electrodes. Also the check for homogeneity of the specific membrane resistance is the same as with the field stimulation.

Neglecting the uncertainties in correcting diameters and assuming circular cross-sections of dendritic branches, we have obtained electrical parameters for dendritic cables. The specific membrane resistance, Rm, was in the range from 14 to 26 kOmega  · cm2; specific membrane capacitance was from 0.6 to 1.1 µF/cm2; and the specific cytoplasmic resistance was from 100 to 160 Omega  · cm (see Table 1). The values of these parameters are in the range observed in other studies. Specific membrane capacitance was estimated to be from less than 1 µF/cm2 (Major et al. 1994) to more than 2 µF/cm2 (Rapp et al. 1994; Thurbon et al. 1998). Estimates of other parameters varied even more. Specific membrane resistance was estimated to be from tens of kOmega  · cm2 (Larkman et al. 1992; Meyer et al. 1997; Thurbon et al. 1994, 1998) to hundreds of kOmega  · cm2 (Major et al. 1994; Rapp et al. 1994). Specific cytoplasmic resistance had values from less than 100 Omega  · cm (Thurbon et al. 1994, 1998) to several hundred Omega  · cm (Larkman et al. 1992; Major et al. 1994; Meyer et al. 1997; Rapp et al. 1994). The huge variation of the estimates may be attributed to the reasons outlined in the preceding text and/or to biological differences between cell types. The electrotonic set of parameters, tau , lambda 1, and R1infinity , has several advantages over the electrical set, Cm, Rm, and Ri. First, each parameter has its own functional meaning for the response of dendritic cables. Second, the estimates are less vulnerable to unverified assumptions, i.e., circular cross-section of the dendrites. Third, the parameters can be estimated one by one, which improves the reliability of estimates.

In motoneurons, voltage-dependent currents together with passive conductance shape the response to synaptic input. Persistent dendritic inward current mediated by L-type calcium channels is responsible for generation of plateau potentials in current-clamp mode (Hounsgaard and Kiehn 1993; Lee and Heckman 1996, 1998a) and for hysteresis in voltage-clamp mode (Lee and Heckman 1998b; Schwindt and Crill 1980; Svirskis and Hounsgaard 1997, 1998). In this study, we used a model with L-type calcium current to show that dendrites with different electrotonic length could be bistable in case of the experimentally observed broad hysteresis in input I-V. Although only three motoneurons were reconstructed in the present study, previous morphological investigations (Ruigrok et al. 1984, 1985) showed that short terminal dendritic branches in abundance is a general characteristic for turtle motoneurons.

In our model, membrane conductances were homogeneously distributed in the dendrites. This may not be true in real dendrites. Because the dendrites are not very long electrotonically, membrane potential changes in space are smooth (Fig. 4B) in our case of slow currents and slow somatic voltage-clamp ramps. Thus effects of any possible heterogeneities of potential dependent conductances would be smoothed over entire dendritic branches. In this case, negative slope in membrane I-V, which defines when dendritic bistability could occur, would represent an average of the membrane conductances over the dendritic branches.

As demonstrated here, even in branching dendrites, hysteresis in input I-V depends only on electrotonic length of dendrites and the maximal negative slope in membrane I-V (Fig. 4C). Consequently the inferences made do not depend on the detailed nature of persistent inward current and should be readily applicable to other types of motoneurons with observed bistability (Hsiao et al. 1998; Lee and Heckman 1998b; Rekling and Feldman 1997). Because of the independence of hysteresis on particular membrane mechanisms, we did not include other channels, like N-type calcium channel, calcium-sensitive potassium channel etc., in the model, although these channels are known to be present in turtle motoneurons. These currents could influence the temporal phenomenology, but they would not change our qualitative findings regarding hysteresis during very slow voltage ramps.

Very slow kinetic properties of inward currents could, however, have profound influence. A slow depolarization induced facilitation of inward current, possibly by changing voltage sensitivity of this current, was observed in motoneurons (Bennett et al. 1998; Svirskis and Hounsgaard 1997). An inward current with shifting voltage sensitivity due to slow facilitation has steep potential dependence and, accordingly, could increase the negative slope in membrane I-V and cause hysteresis in input I-V of electrotonically short cables (Baginskas et al. 1999). The slowness of inward current kinetics also explains why jumps in hysteresis were not observed experimentally. Thus it is very probable that dendritic branches with different electrotonic lengths can be bistable in turtle motoneurons.

In turtle motoneurons the persistent inward current is also modulated by neurotransmitters via metabotropic pathways (Svirskis and Hounsgaard 1998). Facilitation of persistent inward current by bath-applied agonist leads to bistability in current clamp and broad hysteresis in voltage clamp (Svirskis and Hounsgaard 1998). In contrast and in agreement with modeling results, focal synaptic facilitation of dendritic inward current merely increases excitability in current clamp mode and only induces narrow hysteresis in voltage clamp (Delgado-Lezama et al. 1997, 1999).

In conclusion, potential-dependent inward current causing bistability independently in numerous dendritic branches increases the richness of synaptic processing and makes motoneurons complex processing units (Gutman 1991).


    ACKNOWLEDGMENTS

We thank H. Markram for providing facilities for morphological reconstruction of neurons.

We acknowledge the support of the Lithuanian Department of Science and Education. The experimental work was supported by the Lundbeck Foundation, the Novo Foundation, and the Danish Medical Research Council.


    FOOTNOTES

Present address and address for reprint requests: G. Svirskis, Center for Neural Science, 4 Washington Place, Rm. 809, New York University, New York, NY 10003 (E-mail: gytis{at}cns.nyu.edu).

Received 7 March 2000; accepted in final form 12 September 2000.


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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
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0022-3077/01 $5.00 Copyright © 2001 The American Physiological Society