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Firing Properties and Electrotonic Structure ofXenopusLarvalSpinal Neurons

BENOIT SAINT MLEUX AND L. E. MOORELaboratoire de Neurobiologie des Reseaux Sensorimoteurs, Centre National de la Recherche Scientifique-Unite Propre deRecherche de l’Enseignement Superieur Associee-7060, 75270 Paris Cedex 06, France

Saint Mleux, Benoit and L. E. Moore. Firing properties and elec-trotonic structure ofXenopuslarval spinal neurons.J. Neurophysiol.83: 1366–1380, 2000. Whole cell voltage- and current-clamp mea-surements were done on intactXenopus laevislarval spinal neurons atdevelopmental stages 42–47. Firing patterns and electrotonic proper-ties of putative interneurons from the dorsal and ventral medialregions of the spinal cord at myotome levels 4–6 were measured inisolated spinal cord preparations. Passive electrotonic parameterswere determined with internal cesium sulfate solutions as well as inthe presence of active potassium conductances. Step-clamp stimuliwere combined with white-noise frequency domain measurements todetermine both linear and nonlinear responses at different membranepotential levels. Comparison of analytic and compartmental dendriticmodels provided a way to determine the number of compartmentsneeded to describe the dendritic structure. The electrotonic structureof putative interneurons was correlated with their firing behavior suchthat highly accommodating neurons (Type B) had relatively largerdendritic areas and lower electrotonic lengths compared with neuronsthat showed sustained action potential firing in response to a constantcurrent (Type A). Type A neurons had a wide range of dendritic areasand potassium conductances that were activated at membrane poten-tials more negative than observed in Type B neurons. The differencesin the potassium conductances were in part responsible for a muchgreater rectification in the steady-state current voltage (I-V curve) ofthe strongly accommodating neurons compared with repetitively fir-ing cells. The average values of the passive electrotonic parametersfound for Rall Type A and B neurons werecsoma5 3.3 and 2.6 pF,gsoma5 187 and 38 pS,L 5 0.36 and 0.21, andA 5 3.3 and 6.5 forsoma capacitance, soma conductance, electrotonic length, and theratio of the dendritic to somatic areas, respectively. Thus these ex-periments suggest that there is a correlation between the electrotonicstructure and the excitability properties elicited from the somaticregion.

I N T R O D U C T I O N

The Xenopusembryo and larvae have become extremelyuseful preparations for investigations of locomotor neural net-works. The pioneering studies of Roberts and his coworkers(Roberts 1989) have not only identified the few principalneurons of the network with their synaptic connections but alsointroduced minimal neural network models that simulate animpressive amount of behavior observed in this preparation(Dale 1995b; Roberts and Tunstall 1990; Roberts et al. 1995).Furthermore whole cell voltage-clamp measurements havebeen done on cultured (O’Dowd et al. 1988) or isolated cells

(Dale 1991, 1995a) as well as intact surface neurons (Desar-menien et al. 1993; Prime et al. 1998, 1999). Quantitativemodels describing these measurements have been restricted toa single somatic compartment (Dale 1995a; Lockery andSpitzer 1988) because dendritic structures are minimal at earlydevelopmental stages.

Although the dendritic structure at stage 37/38 is minimal, itclearly exists as shown by both morphology and electrophys-iological estimates of the electrotonic structure (Soffe 1990;Van Mier et al. 1985). This is in keeping with the originalempiric model of Roberts and Tunstall (1990) that has threecompartments representing the dendrite, axon, and the soma.Thus it would appear useful to understand the dendritic prop-erties as they begin to develop especially with regard to theappearance of NMDA receptors (Prime et al. 1999) becausethey appear to be correlated directly with the presence ofdendritic trees (Prime 1994; Prime et al. 1998).

The whole cell clamp experiments reported here were doneat larval stages 42–47 where the dendritic structure is clearlymore developed (Van Mier et al. 1985) and likely to play adefinitive role in locomotor behavior. The major advantages ofour approach are the normal electrical activity of functionalneurons can be measured as demonstrated by patterned net-work behavior (Fig. 1), the neurons are not isolated from theirnormal milieu, and thus minimal distortions in structure arelikely to have occurred because of measurement procedures asis demonstrated by the maintenance of synaptic events duringthe experiments.

In addition, both real-time and frequency domain measure-ments were used to determine the electrotonic behavior andtake into account inevitable electrode properties (Moore andChristensen 1985; Wright et al. 1996). These experimentssuggest that there is a correlation between the electrotonicstructure and the excitability properties elicited from the so-matic region. The action potentials of the neurons that showedstrong accommodation also had different electrotonic parame-ters compared with nonaccommodating neurons. Thus in ad-dition to the effects of the voltage-dependent conductances onfiring behavior (Dale and Kuenzi 1997), the structure in whichthese conductances are expressed is correlated with the rhyth-mic behavior of the neuron, perhaps as some function ofdevelopment from the embryo to later larval stages.

M E T H O D S

Experimental preparation

The developmental stages 42–47 ofXenopuslarvae were obtainedafter hormonally induced fertilization. Embryos and larvae were de-

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veloped at room temperature and the stages were morphologicallyselected as described by NieuwenKoop and Faber (1994) and VanMier et al. (1985). In accordance with the European CommunitiesCouncil directive of November 24, 1986 and following the proceduresissued by the French Ministere de l’Agriculture, the larvae wereanesthetized in tricaıne methanesulfate (MS222 Sigma) and placed inRinger solution [composed of (in mM) 110 NaCl, 3 KCl, 1.0 MgCl2,1.0 CaCl2, and 10 HEPES; pH5 7.4] containing 0.5 mg/ml dispase(Boehringer, Mannheim). The notochord, spinal cord, and overlyingmusculature were dissected, and the preparation was agitated for30–35 min at room temperature, after which the spinal cord wasremoved easily from all surrounding tissues. Figure 1 illustratesrhythmic locomotor or bursting behavior (Fig. 1A) of an isolated,stage 37–38, spinal cord similar to that observed in the intact prepa-ration (Roberts 1989). Similarly, isolated larval spinal cords (Fig. 1,Cand D) show network patterns like those observed in the intactpreparation (Fig. 1B). Figure 1Dalso illustrates bursting activityinduced by 50mM NMDA recorded with suction electrodes.

At the larval stage of development, the connective tissues even-tually making up the meninges had to be removed before a patchelectrode seal was possible. Although some recordings were made

from neurons on the external surface (Desarmenien et al. 1993), thesuccess rate was improved considerably by further dissection toreach the inner regions of the spinal cord and to visualize theneurons. The cells of the spinal cord cut between the otic capsuleand the 10th myotome were exposed for intracellular recording bycarefully splitting at the midline and mounting the half cord withthe inner face up (Fig. 2), revealing groups of neurons: dorsalsensory neurons, presumed medial interneurons, and presumedventral motoneurons. The positioning of the half cord was donewith two micromanipulators attached with suction electrodes orsharp glass rods. All recordings were done from neurons betweenthe 4th and 6th myotome levels. The Rohon-Beard sensory cells(Spitzer 1982) were identified easily on the dorsal surface. Longi-tudinally lined up large motoneurons in the ventral part of the cordcould be distinguished; however, other less visible motoneuronscould be confused with presumed interneurons. Intermediate neu-rons near the inner surface of the half cord in dorsal and ventralmedial positions were provisionally identified as interneurons andselected for analysis from a total of 200 recorded neurons of allgroups. Twenty presumed interneurons, which showed stable re-cordings for a minimum of 30 min, were fully analyzed.

FIG. 2. Photograph of isolated half spinal cord. Half spinal cord wasmaintained in a open position with 2 sharp glass electrodes. Top and bottomwhite lines indicate the borders of the dorsal and ventral regions of the cord.Indicated neurons represent presumed interneurons and the motoneurons.Large round cells on the dorsal surface were identified as Rohan Beard sensoryneurons. Calibration bar is 50mm.

FIG. 1. Rhythmic activity of isolatedXenopusspinal cords.A: extracellularrecording of bilateral activity from an isolatedXenopusembryo spinal cord(stage 37/38) using suction electrodes at the rostral and caudal ends. Rhythmicactivity was evoked by a 1-ms pulse of current. Combined frequency is;40Hz; however, an alternation of large and small spikes at 20 Hz can be seen inpart of the record, suggesting a fictive locomotor pattern typical of the intactpreparation.B: simultaneous extracellular recordings of rhythmic burstingactivity from both sides of an intact stage 46Xenopuslarvae. Alternating burstsof ;1 Hz occurred spontaneously.C: extracellular recording of bilateralactivity from an isolatedXenopuslarvae whole spinal cord (stage 46) usingsuction electrodes. Rhythmic activity was evoked by 50mM N-methyl-D-aspartate (NMDA) in the perfusion fluid. Burst frequency was;1 Hz. D:extracellular recording of bilateral activity from an isolatedXenopuslarvaehalf spinal cord (stage 46) that was obtained by cutting the spinal cord ofC atthe midline. Bursting frequency evoked by 50mM NMDA was nearly the sameas observed from the whole spinal cord; however the burst duration decreasedtwo- to threefold. One-second calibration bar applies toB–D.

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Electrophysiological recording

Figure 3 illustrates the combination current or voltage clamp andsum of sines (white noise) method that was used to obtain real-time(Fig. 3, B andC) and frequency (Fig. 3,D andE) domain responses(Moore and Christensen 1985; Moore et al. 1993). The experimentalprocedures were designed to measure both nonlinear responsesevoked by a constant current,I(t), or voltage clamp in real time, andsteady-state linear behavior in the frequency domain for both clampmodes. The stimulus protocol used for the two domains is illustratedin Fig. 3Aby a command step representing either a current or voltage,which is followed by a superimposed steady-state small white noisesignal that evokes a linear response (Wright et al. 1996). The corre-sponding transient voltage (Fig. 3B) or current (Fig. 3C) responses areschematic representations of the influence of active conductances.

The data were obtained with an Axoclamp 2B (Axon Instruments,Foster City, CA), filtered at 500 Hz with an 115 db/octave ellipticalfilter (Krohn-Hite, Model 3900, Avon, MA) and digitized at 12 bits.The values of both the current and the voltage were measured duringvoltage- and current-clamp measurements. This point is of someimportance because the value of the current measured in currentclamp was not identical to the command input. The linear responseswere analyzed in the frequency domain to obtain magnitude and phasefunctions of either the impedance or admittance corresponding tocurrent or voltage clamp, respectively. A fast Fourier transform (FFT)of 1,024 points (0.8 s) of the voltage,V(t), or current,I(t), responseprovided the corresponding output functions of frequency, namelyV(f ) and I( f ), as well as the stimulus or command input,I( f )ref orV(f )ref, for current and voltage clamp, respectively. The correspond-ing impedance,Z(f ) and admittance functions,Y(f ), then were com-puted asZ(f ) 5 V(f )/I( f )ref andY(f ) 5 I( f )/V(f )ref.

A step-by-step description of experimental procedure is as follows:1) 400-ms step constant currents are injected into the soma, immedi-ately followed by superimposed low amplitude white noise for 1 s.The steady-state responses for both the current and the voltage duringthe last 0.8 s are analyzed in the frequency domain to obtain imped-ance magnitude and phase functions, namely the output voltage withrespect to the input current.2) Similar voltage-clamp steps then aredone for a range of membrane potentials to obtain the admittancemagnitude and phase functions, i.e., output current with respect toinput voltage. 3) The resulting real-time transient responses andfrequency domain functions then are fitted using parameter estimationtechniques with an electrotonic model having voltage-dependent con-ductances to obtain quantitative descriptions of each neuron, i.e., acomplete neuronal model with its electrotonic structure and kineticbehavior at all membrane potentials for both constant current- andvoltage-clamp conditions.

In principle, small-signal current- and voltage-clamp measurementsshould lead to reciprocal frequency domain functions,Z(f ) 5 1/Y(f ).This property is a striking example of one of the advantages oftransform functions, namely two types of data can be comparedindependently of a theoretical model. The equivalence of the twofunctions provides a test of the current- and voltage-clamp instrumen-tation (Magistretti et al. 1996), where the latter requires stable elec-tronic circuitry to control the membrane potential at all measuredfrequencies. Current-clamp responses have the advantage that errorsfrom a control amplifier are minimal compared with the voltage clampand in general tend to be more reliable. For stable conditions, smallsignal measurements of the two modes should be consistent, indepen-dently of any particular model. Thus the linear frequency domainmeasurements in voltage- and current-clamp modes provide a mini-mal test of the voltage clamp and should be compared before anycomparisons of real-time voltage- and current-clamp responses aremeaningful. Because of this equivalence the data always was plottedas impedance functions, however, all voltage-clamp data are actuallyadmittance measurements. A difference in the impedance functionsmay occur due to changes in the leakage conductance or electrode

FIG. 3. Combination of current or voltage clamp with white noise analysis.Similarity of the current- and voltage-clamp protocols is indicated by diagramof a command stimulus that consists of a step function on which is superim-posed a small signal, sum of sines, called white noise.A: fast Fourier transform(FFT) of the white noise stimulus was done during the last second of thecommand stimulus. This transform provided current or voltage references,Iref( f ) or Vref( f ), respectively, which are magnitude and phase functions thatthen were used in the computation of the point impedance functions for the 2experimental conditions.B: real-time damped oscillatory voltage responsefollowed by a filtered white noise signal shows the typical constant or com-mand current behavior. FFT of the filtered response to the white noise currentprovides voltage magnitude and phase functions.C: similarly, in a separateexperiment, the current response to a somatic voltage-clamp command pro-vides real-time transient currents followed by small signal sum of sine currentsthat are responses to the white noise command voltage inputs. FFT of thesteady-state current gives the corresponding magnitude and phase functionsassociated with the voltage reference signals. In a linear system, the voltage-and current-clamp data are equivalent and satisfy the relation,Z(f ) 5 1/Y(f ).This was verified experimentally to show that the voltage clamp instrumenta-tion is adequate.D: impedance,Z(f ), magnitude plots are shown for 2 passiveneuronal models, a soma and 1 with an electrotonic structure (Rall neuron).Low-frequency impedance of the Rall neuron is less than the soma alonebecause the total surface area and size are correspondingly larger. Electrotonicstructure shows a small inflection in the mid frequency range.E: correspondingphase functions show that the Rall neuron has marked inflections that are easilydistinguished from the asymptotic behavior of a simple soma.

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properties; however, the electrotonic parameters,A and L (see defi-nitions in the following text), were required to be constant.

Because the impedance or its reciprocal, the admittance, is a ratioof response and stimulus, the final form of the data were corrected forthe effect of the antialiasing filter. However, the real-time responsesare not ratios and consequently contain a damped antialiasing filterresponse. The 500-Hz band was chosen because it provides the bestcompromise for the determination of electrotonic and voltage-depen-dent conductance properties. Because the sampling rate was 0.78125ms, the determination of transient real-time kinetics is limited to a fewmilliseconds with some rapid ringing because of the sharp antialiasingfilters. This limitation is less severe for the frequency domain becausea 1-ms relaxation time has a corner frequency of 160 Hz that is in themiddle of the frequency band measured. It is for these reasons that thefrequency domain is extremely useful for the estimation of the kineticbehavior.

Figure 3,D andE, illustrates typical magnitude and phase functionsfor passive neurons similar to those obtained in these experiments.The superimposed plots show that the dendrites impose marked in-flections on the phase function that are not present with isolatedsomatic structures. Magnitude functions are less sensitive; however,they do exhibit small inflections as well. The frequency domainfunctions thus provide an accurate method to determine the electro-tonic structure that is essential for the subsequent steps in the analysisof the active membrane properties.

Electrode properties and compensation methods

Electrodes were pulled with a laser heated puller (Sutter P2000,Sutter Instruments, Novato, CA) from 1.5 mm glass (GC150F, Elec-tromedical Insutruments, Pangbourne, UK). The electrodes were filledwith either (in mM) 90 K-gluconate, 20 KCl, 2 MgCl2, 10 HEPES, 10EGTA, 3 ATP, and 0.05 GTP or 55 Cs2SO4, 55 sucrose, 2 MgCl2, 10HEPES, 10 EGTA, 3 ATP, and 0.05 GTP. In both solutions, the pHwas adjusted to 7.4 with 10 mM KOH. Thus the internal perfusionfluids contained either potassium or cesium as the principle internalcation. Because different internal solutions were used no correctionfor liquid junction potentials was made. This correction is likely to be;10 mV based on the considerations of previous measurements(Neher 1992).

The electrode impedance was measured at the depth in the solutionof the selected neuron and just before making the gigaseal of thepatch. Because the electrode impedance is low and not well matched

to the Axoclamp headstage, it was measured in series with a parallelRC electronic model that was used to provide a calibration of themethod. Measurements and analyses then were done on the combi-nation electrode and electronic model just as with an actual neuron.The electronic model impedance is comparable with measured neu-rons and provides a method to simulate the contribution of theelectrode in the typical recording situation. Figure 4 illustrates that achange of solution levels of 1 mm led to changes in the electrodecapacitance of 2 pF. The data and superimposed model fits indicatethat real-time data are relatively insensitive to these small changes;however, the differences in the phase function are quite clear, asindicated byCe1 andCe2 for the phase impedance of Fig. 4C.Underthese conditions, the use of the Axoclamp electronic compensationcapabilities for the electrode capacitance and series resistance led tophase functions that could not be reliably estimated (Wilson and Park1989). Capacitance compensation alone was partially effective. Usingquartz glass or coating the electrodes with silicone elastomer (Syl-gard) reduced the value of the capacitance; however, it was still notpossible to achieve adequate compensation. Because these proceduresdistort the measured data in an uncalibrated manner, no electroniccompensation was done, and the electrode was modeled as part of themeasurement system. This had the additional advantage that anychanges in the series electrode impedance always would be taken intoaccount because each voltage-clamp record contained high-frequencydata that are sensitive to electrode properties. Thus fitting recordeddata over a reasonably wide frequency range requires accurate elec-trode parameters. If the electrode increases its resistance, this willbecome apparent at high frequencies in contrast to most voltage-dependent conductance effects that are more sensitive to lower fre-quency ranges. This approach is also more exact than traditional seriesresistance compensation methods, which are always partial because ofstability problems in the voltage clamp.

The mean parameter values of the electrode impedance in thesolution just before making the seal wereRe 5 10.5 MV andCe 55.69 pF (Table 1). All measurements were done with the electrodewithin a few micrometers of the recorded neuron. This procedureavoided errors in the electrode parameters that were due to differentsolution levels. The range ofCe corresponding to the lowest and thehighest levels of solution, 0.1 and 1 mm, was 2 to 6 pF, respectively.

After making the seal and establishing the whole cell clamp record-ing, the electrode was refitted using a neuronal model in series withthe electrode. The new fit included both the electrode and neuron;however, the electrode properties were estimated over the entire

FIG. 4. Effect of solution level on electrode capaci-tance. A whole cell electrode tip was placed in the record-ing chamber with its connection to ground through anelectronic model consisting of a parallel resistance,R,andcapacitance,C. Electrode was modeled as a resistance,Re,and capacitance,Ce, as discussed in theAPPENDIX. A:superimposed real-time responses to210 pA of currentwere measured and fitted for 2 solution levels. Four curvescannot be distinguished showing that the real-time re-sponse is insensitive to the solution level.B: impedancemagnitude plots for the same conditions asA. C: phasefunctions as inB in which differences in the responses andfitted curves can be seen at high frequencies. Smooth linesare model (D) fits for the 2 measurements as follows:C 580 pF,R 5 526 MV, Re 5 13.5 MV, Ce1 5 3.5 pF, andCe2 5 5.6 pF, for the low and high solution levels (bottomand top curves), respectively.D: schematic diagram ofelectronic and electrode model.


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frequency range. The effect of the electrode on the frequency domainfunctions is shown in Fig. 7,B and D (see following text). Asmentioned in the following, the passive electrotonic parameters weredetermined with one-half the frequency range and fixed electrodeparameters. The electrode parameter values during the whole cellrecording are given in Table 1. The average increase ofRe from 10 to17 MV is likely due to plugging of the electrode by membranefragments and cytoplasmic material during the breakthrough of themembrane. The decrease in the electrode capacitance may be relatedpartially to different factors, namely, the attachment of membranefragments on the inner cell wall at the tip, variations in solution levels,or a need for a distributed capacitance to accurately model the elec-trode. Because parameter estimation done with more complicatedelectrode models (Major et al. 1994) did not significantly alter theelectrotonic parameters, all of our analyses were done with an elec-trode modeled as a single resistance,Re, and a nondistributed capac-itance,Ce (seeAPPENDIX). It is worth noting that the determination ofthe electrode capacitance is a critical part of an accurate estimation ofthe soma capacitance. In this regard, electronic compensation methodsare especially subject to error because overcompensation can reducepart of the observed dendritic capacitance, which is seen from thesoma through a series resistance like the electrode itself.

Thus once the electrode properties are known, it is possible toseparate the properties of theattachedelectrode from those of theneuron and evaluate the electrotonic structure. This itself requires avalid theoretical formalism for the dendritic tree to obtain the excit-ability properties of both the soma and dendrites.

Data analysis and rationale

The goal of these experiments is to analyze dendritic membraneexcitability. The rationale of the analysis is to use different types ofmeasurements that allow the determination of the physiological elec-trotonic structure, both active and passive. The basic steps in theanalysis consist of the following:1) determination of passive elect-rotonic structure at hyperpolarized membrane potentials with a linearanalytic model, i.e., having perfect spatial resolution.2) Evaluation ofcompartmental models at different membrane potentials to evaluatethe required number of compartments needed to achieve adequatespatial resolution during the activation of ionic conductances.3) Theanalysis of real-time kinetic responses with compartmental modelsthat are constrained by the passive electrotonic structure.

THEORETICAL CONSIDERATIONS—ANALYTIC VERSUS COMPART-MENTAL MODELS.

Linear-analytical models—resting neurons.The experimental pro-tocol described in the preceding text involves an analysis of both largestep nonlinear data and small-signal linearized responses. Our goal isto obtain a minimal model that accurately describes the soma anddendritic behavior observed in the measured neurons. It was foundthat the dendritic tree could be well described by a single Rallequivalent cylinder, thus avoiding the need to use more complexmodels based on detailed morphology. Models with two dendriticcables connected to the soma did not significantly improve the error ofthe fit. This result supports the hypothesis that the branching dendriticstructure ofXenopusneurons follows the impedance matching criteriaat branch points developed by Rall (1960). More complex morpho-

logical models are only needed when the branch point matchingcriteria are not met. Although anatomic measurements on fixed tissuecan indicate such a discrepancy, it is not clear if these estimatesalways apply to living tissue. Electrophysiological measurements, inboth the time and frequency domains from neurons in their normalphysiological state, can provide data to evaluate the accuracy ofcollapsed dendritic models and as such are likely to be more suitablefor determining the adequacy of simplified dendritic models to de-scribe data than anatomic measurements from fixed tissues. Theability of a single uncoupled model to adequately describe our dataalso demonstrates that the neurons are not likely to be electrotonicallycoupled (Perrins and Roberts 1995a). Thus because interneurons donot show electrototonic coupling (Perrins and Roberts 1995b), ouranalysis further supports the presumption that the selected cells arefrom this class of neurons.

The implementation of the collapsed dendritic cylindrical modelhas been done with analytic (Major 1993a,b) and compartmentalmodels (Rapp et al. 1994). However, linear analytic models (seeAPPENDIX) are advantageous because they have perfect spatial resolu-tion and are computationally efficient. The applicability of thesemodels in different neurons was determined by comparing the analyticand compartmental formulations as a function of number of compart-ments. Near the resting potential there was good agreement usingfrom three to five compartments.

The passive electrotonic structure always was determined near theresting potential with both passive and activelinearized responses(Borst and Haag 1996; Major et al. 1994; Surkis et al. 1998). Underthese conditions, an analytic model equivalent to an infinite number ofcompartments was used (seeAPPENDIX). Although the linear analyticmodel avoids errors due to spatial resolution, it is only valid near theresting potential or under experimental conditions such that the neu-ron is entirely passive. It should be emphasized that linearized modelsare not necessarily passive and are capable of describing the kineticbehavior of the voltage-dependent conductances (Moore andBuchanan 1993) over a limited potential range if the steady-statepotentials throughout the cable are relatively constant (Murphey et al.1995).

LINEAR COMPARTMENTAL AND ANALYTIC MODELS—DEPOLARIZEDNEURONS IN STEADY STATE. At depolarized potentials the steady-state potential profile inherent in the dendritic structure leads tovariable activation of the ionic conductances. This requires a com-partmental model to correctly determine voltage-dependent admit-tance functions at each dendritic location (seeAPPENDIX). Neverthelessthe analytic model is a good approximation at moderate depolariza-tions and can provide an excellent initial estimate of the final imped-ance. This point is of computational importance because calculation ofthe analytic impedance can be orders of magnitude faster than com-partmental estimates.

The number of compartments necessary to assure adequate spatialresolution is clearly a function of both the electrotonic structure andthe nature of the ionic conductances (Bush and Sejnowski 1993). Atrest or with a passive neuron, the best test of the compartmental modelis a comparison with the analytic solution. However, active neurons atdepolarized potentials require increasing the number of compartmentsuntil no significant change occurs. Under these depolarized condi-tions, the compartmental model then can be used to evaluate theadequacy of the analytic model. For many neurons, the electrotonicstructure was sufficiently compact that very little error occurred withthe analytic model at depolarized potentials. Significantly larger errorsoccurred with compartmental models having too few segments thanever observed due to a steady-state potential profile error.

NONLINEAR COMPARTMENTAL MODELS—LARGE SIGNAL KINETICANALYSIS OF IONIC CURRENTS. The analysis of the constant currentresponses clearly requires nonlinear kinetic equations; however, this isalso true for a somatic voltage clamp because of the effects of thedendritic membrane potential transients. These kinetic models have

TABLE 1. Electrode parameters before and after seal formation

Electrode Parameters Parameter Values

Remod, MV 10.466 3.20

Cemod, pF 5.696 0.87

Recell, MV 17.056 6.39

Cecell, pF 3.166 0.80

n 5 9; values are means6 SD.


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been described extensively by numerous authors (Koch and Segev1998), and our specific implementation is given in theAPPENDIX.

Our initial analysis was done with the minimum number of threedendritic compartments in which the passive electrotonic parameters(ratio of dendritic to somatic surface areas,A, and electrotonic length,L) were identical to those of the analytic model (seeAPPENDIX). Thespatial resolution of the compartmental model then was evaluated byincreasing the number of compartments until minimal differenceswere obtained. In general, the resting neuron was described ade-quately by a three-compartmental dendritic model; however,$10compartments were essential to describe voltage-clamp currents atdepolarized potentials when the ionic conductances are significantlyactivated (Bush and Sejnowski 1993; D’Aguano et al. 1989). Adendritic model with 30 compartments was usually indistinguishablefrom the analytic model.

In brief, the parameter estimation methods, as previously described(Murphey et al. 1995, 1996), were used in the following order: Thepassive electrotonic and electrode parameters initially were estimatedat resting or hyperpolarized potentials with alinear analytic model(seeAPPENDIX) over the entire frequency range. Parameter estimationwas done by an iterative gradient descent method. Fits in a particularminimum were considered adequate if the minimal error change was,0.1%. In addition local minima were avoided by the use of both thefrequency and time domain data under voltage and current clamp.Afterward, the electrode parameters were fixed and the electrotonicparameters were estimated over one-half the frequency range. Thevalidity of a three-compartmental dendritic model also was confirmedfor the passive membrane by demonstrating an adequate fit of thesmall signal real-time data. The data then were fitted simultaneouslyin the frequency and time domains with the linear analytic model anda three-dendriticcompartmentalmodel, respectively. The adequacy ofboth the frequency and time domain fits were compared directly witha 30-compartmental model. If the fitting criteria were not met, the datawere fitted again with the higher resolution compartmental model thatwas implemented using the FindMinimum and FindRoot proceduresof Mathematica (Wolfram Research, Champaign, IL). These proce-dures then were applied to the active conductances after fixing thepassive parameters. Multiple records at different membrane potentialswere analyzed simultaneously in both the time and frequency do-mains. If necessary, the electrode parameters,Re andCe, were refittedduring the course of the experiment; however, the electrotonic param-eters remained at their original estimated values. In this case, con-straining limits based on measured electrode parameters (Fig. 4) wereplaced onCe andRe, as follows:Ce (2–6 pF) and a minimum forRe

of 6–10 MV, which was measured in the bath before the patch wasmade.

In summary, the approach developed in this paper is to analyze bothwhole cell voltage- and current-clamp data with a real-time three-dendritic compartment nonlinear model that is constrained simulta-neously by the linear analytic frequency domain form of the samemodel. These two forms of the neuronal model are computationallyefficient because of the small number of compartments in the formerand the analytic representation of the latter. The number of compart-ments then is increased to evaluate the errors due to inadequate spatialresolution for the real-time transients and the dendritic potentialprofile in the steady state.

R E S U L T S

Whole cell action potentials

All neurons perfused with the potassium electrode solution hadresting potentials more negative than255 mV and action poten-tial magnitudes that reached overshooting positive values. Theaction potential behavior varied between a Type I repetitive firingbehavior to a Type II single action potential in response to a

maintained constant current. Figure 5A shows a single actionpotential from a Type I neuron responding to a just thresholdstimulus. Increasing the stimulus strength increased the number ofaction potentials (Fig. 5B) and the average firing rate. In general,the cells responded repetitively with minimal accommodation to amaintained constant current that was twofold or greater thanthreshold (Fig. 5C). Figure 5Dillustrates superimposed constantcurrent steady-state current voltage (I-V) curves for nine neuronsshowing this type of action potential behavior. A marked rectifi-cation is observed where the maximal slope conductance isreached at about240 mV.

About 1/4 of the neurons, referred to as Type II neurons,showed marked accommodation in their firing responses. Fig-ure 6,A–C, illustrates two examples of this type of behavior.The most extreme form is seen in Fig. 6A in which a singleaction potential is evoked by constant current stimulation#10times threshold. An alternative type of adaptation is seen inFig. 6,B andC, in which a constant current evokes a series ofaction potentials that decrease in size and finally cease. Theseneurons show increased frequencies and numbers of actionpotentials as the current increases (Fig. 6C); however, theresponses are not maintained. The constant current voltagecurves of Type II neurons show a significant rectification overa wide potential range (Fig. 6D) leading to a maximum slopeconductance at a more positive potential than seen in Type Ineurons.

Passive electrotonic or structural parameters

The analysis of electrotonic behavior provides informationabout the passive structure of a neuron on which active prop-

FIG. 5. Current voltage (I-V) plots and action potential behavior of Type Ineurons. Current-clamp responses for different constant current levels for aType I neuron, namely,A: 210 and110 pA, B: 120 pA, andC: 130 pA. D:constant current-voltage curves from 9 neurons that show repetitive firing andsignificant rectification. Each neuron is represented by a different symbol.Neuron 97H05Aof A–C is represented byŒ in D. I-V curves were made fromdifference in the average values of the voltage 100 ms before the step and thelast 100 ms of the constant current step, just before the beginning of the whitenoise signal.


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erties are built. The excitable properties of neurons are inex-tricably linked to their electrotonic structure. Although mor-phological measurements provide a good indication ofdendritic cable properties, these must be measured in the intactcell to ascertain the actual electrotonic behavior. Ideally mor-phology could provide the basis of the model used for inter-preting electrophysiological measurements; however, suchmodels can require large numbers of compartments and arecomputationally cumbersome. Our approach is to obtain aminimal model that is consistent with the morphology andrigorously satisfies the spatial resolution requirements imposedby the electrophysiologically measured cable properties.

To measure passive neuronal properties, data at hyperpolar-ized membrane potentials were obtained in the presence ofTTX during internal perfusion of potassium gluconate or ce-sium sulfate. The electrode properties were analyzed in con-junction with those of essentially passive neurons in a sequen-tial manner such that fitted electrode parameters were fixedbefore finally evaluating the passive electrotonic structure (seeMETHODS). Neurons perfused with cesium sulfate had unstableresting potentials and survived better if voltage clamped at260 mV as rapidly as possible after establishing the whole cellrecording. Removal of the potassium conductance was presum-ably responsible for the instability of the resting potential andoften led to maintained depolarized potentials that damaged theneurons. The d.c. impedance of these neurons often exceeded10 GV.

Figure 7Aillustrates the potential time course in response toa hyperpolarizing current for a virtually passive cesium per-fused neuron. A multiexponential analysis of this response toobtain the membrane time constant and associated electrotonicproperties of the dendritic tree is notoriously difficult (Sprustonand Johnston 1992). In addition to problems associated with

separating exponential functions, the properties of the elec-trode are difficult to assess. The corresponding measurementsdone in the frequency domain (Fig. 7B) provide more sensi-tivity than real-time measurements for the estimation of elec-trode and electrotonic parameters. The fits shown in Fig. 7indicate good agreement for both the real-time and frequencydomain current-clamp data. Figure 7,C andD, shows voltage-clamp data and model fits for the same neuron using the modelparameters that describe the current-clamp data (Fig. 7,A andB) with the notable exception ofgsoma, which was reduced toone-half its value. This difference is likely due to the incom-plete exchange of cesium and potassium ions during the con-stant current measurement that was done at the beginning ofthe experiment. Otherwise the parameters are identical andshow that the two measurements are equivalent as is demon-strated in subsequent figures from other neurons. The effect ofthe electrode on the impedance functions also is illustrated forboth the current- (Fig. 7B at 270 mV) and voltage-clamp (Fig.7D at 263 mV) frequency domain experiments. The onlysignificant changes were in the phase functions showing adeviation in the phase at high frequencies when the electrodewas removed.

The mean values (Table 2) of the passive electrotonic pa-rameters of cesium perfused neurons show a soma capacitanceof ;3 pF and a dendritic structure that was approximated by asingle equivalent cylinder having an electrotonic length of 0.25and a dendritic to soma area ratio of 3.6. Assuming that theneuron is a perfect sphere with a specific capacitance of 0.5–1mF/cm2, then its diameter would be 10–14mm, respectively.These estimates are consistent with visual observations of theseneurons during the experiments (see Fig. 2). The majority ofexperiments were done at stage 47; however, some measure-ments at stages 42–46 showed that the electrotonic parametersbetween these stages are not radically different (Table 4). Theresults of this study should apply to all stages between 42 and47; however, it is clear that dendritic structures are in differentstates of development during these various stages.

Voltage-gated potassium conductances

Because part of the resting conductance is due to the potas-sium conductance, it is often difficult to determine the passiveelectrotonic structure with real-time hyperpolarizing pulses(Spruston and Johnston 1992; Surkis et al. 1998). Thus theeffects of voltage-dependent conductances must be generallytaken into account in resting neurons. We have been able todemonstrate that hyperpolarized interneurons perfused withcesium or potassium ions are essentially passive; however atthe resting potential most voltage-dependent conductances cancontribute to the linear properties. Therefore passive electro-tonic parameters of a normal resting neuron must be estimatedin the presence of active conductances. Thus it is necessary toassess the distribution of ionic channels between the soma anddendritic tree. Although morphological labeling methods canindicate receptor distributions, it is important to functionallyestimate these distributions. This is not generally possible bydirect measurements in intact tissues; however, in our analysis,a homogenous distribution of ionic channels was shown to beadequate for the experimental conditions explored. This resultis of some significance because it has been demonstratedpreviously that the frequency domain method is capable of

FIG. 6. I-V curves and firing behavior of Type II neurons.A: single or noaction potential response from a Type II neuron during210-, 110-, 120-,130-,140-, and1200-pA current injections,neuron 97F05C. B: multiple, butnot sustained, action potential responses from other Type II neuron during asustained 10-pA current and, inC, 20-pA stimulation,neuron 97J30A. D:constant current-voltage curves from 8 neurons show marked rectificationapproaching a 0 slope. Neurons ofA andB are identified by the same symbolsin D.


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detecting differences in the spatial distributions of activatedreceptors (Murphey et al. 1995); however, extremely accuratefrequency domain measurements with averaging would berequired before spatial effects can be seen easily. This was notdone in these experiments.

Figure 8 illustrates data obtained in the presence of TTX forboth current- and voltage-clamp modes and provides a strin-gent test of the full nonlinear model containing a single volt-age-dependent potassium conductance throughout the dendritictree. The time course of the hyperpolarizing potential responseduring K-gluconate perfusion is more rapid than observed inthe corresponding cesium perfused neuron of Fig. 7. Theseresults support the hypothesis that part of the conductance atrest is due to the voltage-dependent potassium conductance.The depolarizing potential response was lower in magnitudeand also faster due to the increased activation of the potassiumconductance (Table 3). Thus the normal resting conductancehas both voltage-dependent and -independent (leakage) com-ponents (Surkis et al. 1998).

The corresponding frequency domain data and fittedcurves (Fig. 8.B and D) show that the model parametersprovide a good description of both the passive and active

parameters. As with the passive case of Fig. 7, the voltage-clamp responses (Fig. 8,C and D) are shown to be wellpredicted by the current-clamp model of Fig. 8,A and B.These results confirm that the voltage clamp is adequatelycontrolling the soma potential and provides verification thatthe dendritic model used for both the current- and voltage-clamp experiments is an accurate description. The electro-tonic length of this neuron was;0.25 and was sufficientlysmall to allow the three-compartment dendritic model toaccurately describe the voltage-clamp currents. The com-pactness of the dendritic tree also allowed observation ofputative excitatory inward synaptic currents that decrease inamplitude during depolarizing voltage-clamp steps.

The steady-state electrotonic structure of the dendrite isgiven by a minimum of three parameters, the soma conduc-tance (gsoma), electrotonic length (L), and the ratio of thedendritic to somatic areas (A). Differences among neurons thatmight occur due to the dendritic structure should be revealedby a grouping of these neurons according to these variables(see Table 4). The three dimensional plot of these variables inFig. 9 shows that there is a clustering of neurons (stars) towardthe front corner of theA versusL surface defined byA . 4 and

FIG. 7. Passive electrotonic structure of single neurons.Four types of data were used to test the ability of a Rallmodel neuron to quantitatively describe the passive elect-rotonic behavior in a cesium perfused neuron. The dashedlines represent the model without an electrode, which is justvisible in A and completely superimposed inC. A: hyper-polarizing constant current response to20.01 nA. B: im-pedance magnitudes in megohms (MV) and phase re-sponses in radians (rad) at270 and2130 mV. Effect of theelectrode is only illustrated at270 mV. C: voltage-clampresponses to a 10-mV depolarization and hyperpolarizationfrom a holding level of263 mV. D: impedance magnitudeand phase plots of admittance data obtained during the samevoltage-clamp steps. Effect of the electrode is only illus-trated at263 mV. Same model is used for all the superim-posed smooth curves and all 4 data sets. Parameter valuesfor the passive neuronal model were as follows:Csoma 52.39 pF; L 5 0.133; A 5 6.03; gsoma 5 0.013 nS; andVleak 5 225.6 mV. Electrode model parameters wereCe 52.85 pF andRe 5 17 MV. A small bias current inC wasrequired because of the depolarization caused by cesiumperfusion.Cell 97K17C,Type B.


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L , 0.3. This Type B group (stars) has a high-input impedanceand agsoma , 50 pS. The remaining Type A (circles) has abroader range of all three parameters; however, no neuronswere found that hadA . 4 whenL . 0.4. The Type A neuronshave a higher averagegsomaand electrotonic length in potas-sium gluconate compared with cesium sulfate perfused cells(Table 2 and encircled symbols in Fig. 9). This difference isconsistent with the assumption that removal of potassium ionsdecreases the passive dendritic conductance and correspond-ingly decreases the electrotonic length. Thus part of the vari-ation observed in the Type A neurons is probably due to thecesium perfusion, which suggests the average value ofL withnormal internal potassium ions is more physiological. Type Bneurons have averagegsoma and electrotonic lengths that areboth lower than Type A neurons and independent of the inter-nal potassium concentration (Table 2).

The excitability properties of these two classes, Type A andType B, appear to be correlated with the Type I and II classi-fication of Figs. 5 and 6. The closed symbols in Fig. 9 indicateneurons in which the correlation was confirmed. The firingproperties of the remaining neurons were not measured. Themeans of these two types are given in Table 2 show that theType A neurons have an average electrotonic length of 0.36and a dendritic to soma area ratio of 3.3; however, with normalinternal potassium ions,L 5 0.44 andA 5 3.64. The Type Bneurons have a larger dendritic to soma area ratio of 6.5;however, a smaller electrotonic length of 0.21. These latterneurons represent;30% of the presumed interneurons thatwere measured. Because the correlation of electrotonic struc-ture with excitability properties required consideration of bothL andA, it would be unlikely that only the morphological areaof the dendritic tree would distinguish neurons with differentfiring properties.

Spatial resolution and dendritic potential profile

The potassium conductance near rest is well described bya three-compartmental dendritic model in real time and theanalytic model for the frequency domain; however, majordiscrepancies can occur at large depolarizations. As dis-cussed inMETHODS, as the effective electrotonic length in-creases because of the activation of voltage-dependent con-ductances, the number of compartments must be increaseduntil there are no changes in the linear and nonlinear re-sponses of the model equations. Under these conditions, thecompartmental models can be used to obtain an increasedspatial resolution for the transient response and the steady-state dendritic potential profile for the frequency domain.Figure 10Aillustrates a neuron having a passive electrotoniclength of 0.5 where the time domain fits for a three-dendriticcompartment model (—) are progressively worse with de-polarization. At these potentials, the effective electrotoniclength is considerably.0.5, which means that the level ofdepolarization of the peripheral dendrites is considerablyless than the soma. Therefore the spatial resolution of three-compartmental model for this data are not adequate andcannot quantitatively describe the voltage-clamp results.Curiously, the predictions of a three-compartmental modelshow less current than measured, probably because the lackof spatial resolution leads to an abnormally low level ofdepolarization in the end compartments that produces lesscurrent than observed with more compartments.

Figure 10A, - - -, shows a marked improvement of themodel, using the same passive and active parameters, when thenumber of compartments for the real-time response was in-creased to 30. The frequency domain fits (Fig. 10B) show thatthe analytic (—) and 30 compartmental (- - -) models showbetter agreement with the data near the resonant peak; how-ever, the low-frequency impedance of the analytic modelmatches the data better. The decreased magnitude at low fre-quencies of the analytic model compared with the compart-mental model is not due to spatial resolution errors but occursbecause the analytic model assumes a uniform potentialthroughout the neuron. The increased low-frequency imped-ance magnitude of the 30-compartmental model is a conse-quence of the reduced activation of the potassium conductancein the peripheral compartments as would be expected becauseof the increased electrotonic length. The analytic and 30-compartment models superimpose at the resting potential;however, a frequency domain model with only three dendriticcompartments is completely inadequate at all membrane po-tentials. The relatively good agreement between the analyticand 30-compartmental model frequency domain fits suggeststhat spatial resolution errors are more significant than those dueto the potential profile. Although we used 30 compartments, 10compartments were generally sufficient for most neuronalstructures (Bush and Sejnowski 1993; D’Aguano et al. 1989).Nevertheless, we emphasize that the preceding procedure is arelatively simple way to evaluate the correct number of com-partments for a particular set of conductances and consequentlyis preferred to assuming a fixed number.

The neuron of Fig. 10 also shows a marked linear impedanceresonance that is determined by the interaction of the relaxationtime of the potassium conductance and the passive electrotonicproperties. The half-activation potassium conductance time

TABLE 2. Passive electrotonic parameters of Type Aand B neurons

Stade 42/7 K1 Cs1 Mean

Type An 7 6 13Csoma, pF 3.566 1.70 3.046 0.76 3.326 1.33Elength 0.446 0.13 0.286 0.12 0.366 0.15Aratio 3.646 1.85 2.986 1.51 3.346 1.67gsoma, pS 2626 323 98.76 46.3 1876 245vleak, mV 259.116 1.00 258.146 10.94 258.666 7.11Rin, MV 15856 1216 29166 1463 22006 1452

Type Bn 5 2 7Csoma, pF 2.676 1.07 2.376 0.02 2.596 0.89Elength 0.236 0.05 0.176 0.05 0.216 0.05Aratio 6.916 1.67 5.466 0.80 6.506 1.57gsoma, pS 37.86 16.6 36.66 33.2 37.56 19.1vleak, mV 258.486 3.79 245.306 27.90 254.716 13.44Rin, MV 26506 1318 40006 1414 30366 1388

Meann 12 8 20Csoma, pF 3.246 1.47 2.876 0.71 3.096 1.21Elength 0.356 0.15 0.256 0.12 0.316 0.14Aratio 4.946 2.36 3.606 1.74 4.406 2.19gsoma, pS 1676 266 83.26 50.2 1346 209vleak, mV 258.006 4.95 254.936 15.23 256.776 10.10Rin, MV 20296 1319 31876 1437 24936 1451

Values are means6 SD.


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constant,tn, is of the order of milliseconds, which leads totransients that cannot be resolved in the real-time measure-ments; however, because a 1-ms time constant has the cornerfrequency of 160 Hz, it is possible to estimate this parameterfrom the impedance data. Thus the activation time constantprincipally was determined by the frequency domain reso-nance, which is sensitive to its value. Both Type A and Bmodels do show impedance resonances; however, the Type Bmodel shows a resonance at more depolarized potentials. Thissimulation result is consistent with our observation that, over a

limited range of membrane depolarizations, resonance wasmore frequently observed in Type A than B neurons.

The slow decay seen in the voltage-clamp currents was notmodeled; however, we have obtained essentially identical ac-tivation time constants with an inactivating potassium conduc-tance model. An additional cause of the slow decay in thecurrent could be a change in the internal concentration ofpotassium ions because these neurons are relatively smallcompared with other preparations. In general, the potassiumconductance has a positive slope conductance; however, aninactivating potassium conductance could in principle show anegative slope conductance. Tables 3 and 5 show the results forthe fast voltage-dependent potassium conductance for the twogroups of neurons, Type A and Type B.

D I S C U S S I O N

The passive membrane properties of the larval Type A andB neurons showed input resistances (Rin) of 1–3 GV that wasmeasured at270 mV (Table 2). These values are slightlyhigher than the measured resistances of embryonic neurons,

TABLE 3. Potassium conductance parameters of Type Aand B neurons

Type A Type B

gK, nS 5.906 9.36 2.436 4.27vn, mV 210.606 11.61 215.366 4.45sn, mV21 0.0376 0.008 0.0906 0.062tn, ms 1.76 0.7 15.06 18.0rn, mV21 0 6 0.042 20.0306 0.058n 6 5

FIG. 8. Voltage-dependent potassium conductance.A–Dshow the responses of the potassium conductance for 4 dif-ferent stimulus conditions measured in 1mM TTX. A: hyper-and depolarizing currents show a decreasing time constant asthe conductance is activated. Current levels were20.01, 0.01,0.02, and 0.03 nA.B: corresponding impedance plots aregiven for each current-clamp step indicating a decrease in thelow-frequency impedance with depolarization.C: voltage-clamp currents for 4 step potentials show rapid activationkinetics that are too fast to be resolved in the real-timerecords. Inward synaptic currents are shown decreasing inamplitude with depolarization.D: impedance plots of theadmittance data are shown for 3 of the voltage-clamp steps.Superimposed smooth fits of both frequency and time domaindata show fits for a 3-compartment dendritic model. A nearlytwofold increase in the conductance occurred at230 mV.Electrotonic parameters areCsoma5 3.67 pF;L 5 0.247;A 51.77; gsoma 5 0.13 nS; andVleak 5 259.8 mV. Electrodeparameters areCe 5 2 pF and Re 5 12.5 MV. Activepotassium conductance parameters are:gK 5 0.36 nS;vn 524.2 mV;sn 5 0.047 mV21; tn 5 2.4 ms;rn 5 20.001; andVK 5 290 mV (thisvK was used for all figures and tables).Neuron 97K03D,Type A.


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which were generally 1 GV(Dale 1991). Embryonic soma aresomewhat larger than those of the larva; however, the dendriticstructure would tend to reduce the effect on the input resistanceof the smaller soma. Estimates of the embryo soma capacitancevary from 10 to 120 pF (Dale 1995b; Prime et al. 1998; Robertsand Tunstall 1990; Soffe 1990), possibly reflecting a largedistribution of sizes due to different developmental states andmeasurement conditions. A value of 9.5 pF was used by Dale(1995b) in model simulations that were based on measure-ments of isolated embryonic neurons. Furthermore isolatedlarval neurons have been reported to have a lower cell capac-itance of 3.2 pF (Sun and Dale 1998). This value is quitecomparable to our soma capacitances of 3.3 and 2.6 pF forType A and B neurons, respectively (Table 2). In our measure-ments of intact neurons, there is a significant additional den-dritic capacitance that was 4–7 times that of the soma. Thus thetotal capacitance of the intact larval neuron is considerablylarger than the isolated neuron.

The choice of a correct dendritic model is dependent onthe electrotonic structure and how it might change dynam-ically. If the assumption of a single equivalent cylinder isreasonable, then the analytic model provides the best linear

description, which can be passive or active. We have shownthat the resting neurons in theXenopuslarva can be de-scribed by an analytic model and that a three-dendriticcompartmental model is usually adequate for real-time re-sponse. This small number of compartments is not adequatefor depolarized neurons because the dynamic electrotoniclength increases and the voltage-clamp currents cannot bedescribed correctly. We have found that 10 –30 compart-ments is sufficient for describing the real-time behavior ofvoltage-clampedXenopuslarval neurons.

The relative contribution of the dendritic cable to the small-signal passive conductance often is referred to asr, namely,gdendrite/gsoma where r 5 (A/L) tanh L in our terminology(APPENDIX). For the range ofL values found in these experi-ments, r is nearly equal toA. We were not able to findcorrelations between active properties and individual electro-tonic parameters, such asA or L; however, the two together doseem to allow the separation a particular group, Type B, fromthe remaining Type A neurons. The mean parameter values ofthese groups suggest that the larger dendritic area of the TypeB neuron is associated with a lower electrotonic length com-pared with Type A neurons. This relationship enhances inte-

TABLE 4. Electrotonic passive parameters for individual neurons

Neurons* Solution re, MV ce, pF Csoma, pF Elength Aratio gsoma, pS Vleak, mV

A. Passive parameters of Type A cells

4797J01A K1 25 2.9 3.95 0.479 2.89 152 257.4597J31J K1 20 4.21 1.82 0.354 6.85 130 25897K03D K1 12.5 2 3.67 0.247 1.77 135 259.8197K05C K1 10.62 2.2 6.45 0.660 1.80 987 260.1297K06B K1 23.56 5.28 4.84 0.399 5.29 202 259.33

4697K04A K1 8.99 2.5 2.18 0.492 3.62 169 259.66

4397K21D K1 13.8 3.27 2.04 0.447 3.27 57.8 259.41

B. Passive parameters of Type A cells with a Cs1 intracellular solution

4797K12A Cs1 9 4 2.95 0.194 2.53 59.2 259.9397K13A Cs1 6.5 2.5 4.03 0.504 2.02 168 270.6797K13C Cs1 12.99 2.96 3.40 0.21 1.97 45.2 237.89

4597K18A Cs1 10 3.3 2.27 0.329 5.86 100 260.6

4297K19C Cs1 10.71 4.5 2.07 0.2 3.39 84.7 262.797K19D Cs1 12.9 3.1 3.52 0.223 2.13 135 257.08

C. Passive parameters of Type B cells

4597J08B K1 12.48 3.15 1.75 0.278 8.00 48.5 262.34

4797J20B K1 18.43 2.85 3.01 0.193 5.74 43.8 243.397K06D K1 16.26 2.00 2.10 0.234 8.54 15.4 255.5

4697J28B K1 16.14 3.33 2.57 0.148 4.257 28.2 26097J30A K1 23.07 2.93 4.51 0.211 7.224 39.7 261.04

4597K17C Cs1 17 2.85 2.39 0.133 6.029 13.2 225.57

4397K20A Cs1 47.8 3.76 2.36 0.207 4.89 60.1 265.02

* Numbers (i.e., 47, 46, 45, etc.) represent stages of development.


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grative mechanisms because the larger dendritic region is elec-trotonically closer to the soma. Even Type A neurons do notappear to have electrotonic lengths.0.4 for dendritic arearatios.4.

The membrane conductance of neurons at rest appears to bedifferent for Type A and B neurons. In contrast to Type Bneurons, Type A neurons have a component of the restingpassive conductance that is dependent on potassium ions inaddition to the voltage-dependent potassium conductance. Ourelectrotonic analysis separates active and passive propertiesand should show a different value ofL for neurons havingsimilar dendritic areas and different passivegsoma’s. Table 2indicates thatL is lower for Type A neurons whengsoma isreduced by cesium compared with control values with normalinternal potassium ions. Furthermore Fig. 9 shows thatgsoma

increases withL and decreases withA, as is suggested by therelationships betweengsoma, A and L given in theAPPENDIX.Correspondingly, Type B neurons do not show a change ingsoma or L. The demonstration that Type A neurons show alower L in the presence of cesium ions supports the hypothesisthat the membrane resistivity of the dendritic membrane issimilar to that of the soma. If the cesium perfused neurons wereremoved from Fig. 9, the Type A neurons would be morehomogenous in their properties and would show a greaterdifference in electrotonic length compared with Type B neu-rons.

In the Type B group, action potentials showed markedadaptation and occasionally remained depolarized withsmall levels of injected current. The action potential behav-ior was measured in 8 of the 19 neurons of Fig. 9, as

FIG. 9. Distribution of neuronal types based onelectrotonic structure. Coordinates of the 3-dimen-sional plot were the d.c. electrotonic parameters:soma conductance (gsoma), electrotonic length (L),and the ratio of the dendritic to soma areas (A).Clustering of the star symbols provides a way todefine Type A (circles) and Type B (stars) neurons.All neurons in which the firing behavior was mea-sured are indicated by filled (8 of 19) symbols. OneType I, A neuron was not shown because one of itscoordinates (gsoma), is out of the range (Table 4,97K05C) of the graph. In each instance, there is astrict correlation between Type A and Type I mul-tiple firing neurons or Type B and Type II accom-modating neurons. Circled symbols represent neu-rons perfused with cesium sulfate, which in the caseof Type A neurons usually show a lower value ofL.

FIG. 10. Influence of electrotonic structure on potassiumcurrents.A: voltage-clamp currents at 3 levels of depolar-ization.B: corresponding impedance plots for the each volt-age-clamp step fitted with an analytic model. A significantdiscrepancy is seen between the fit of the 3-compartmentalmodel inA, suggesting a need for increased spatial resolu-tion. Dashed lines show an improved model fit for theincrease in number of compartments from 3 to 30. Otherparameters of the model were identical to those used in the3-compartmental model. Parameters wereCsoma5 3.95 pF;L 5 0.479;A 5 2.89;gsoma5 0.15 nS; andVleak 5 257.5mV. Electrode model parameters areCe 5 2.9 pF andRe 525 MV. Active parameters weregK 5 4.34 nS;vK is 290mV; vn 5 230.2 mV;sn 5 0.031 mV21; tn 5 2.4 ms;rn 50.02 mV21. Neuron 97J01A,Type A.


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indicated by the filled symbols. In each instance, there wasa match between Types A and B and Types I and II,respectively (Figs. 5 and 6). The Type II accommodatingneurons also have been observed at stage 37/38 (Roberts andSillar 1990) for dorsolateral commissural (dlc) neurons incontrast to single impulse responses from ventral interneu-rons. Because this behavior appears to be correlated withdendritic structure in our measurements of larvae, it istempting to speculate that embryonic dlc interneurons mayhave a significant dendritic structure.

The differences between Type I and II neurons also arereflected clearly in theI-V curves. Similar behavior for centralcochlear neurons has been observed in brain slices of theguinea pig cochlear nucleus (White et al. 1994). Both the highpassive resistance and steeper potassium conductance activa-tion curve give Type B neurons more steady-state rectification.The lower gsoma would allow steady-state negative conduc-tances due to sodium and calcium ions to influence theI-Vcurve, especially for the condition that the potassium current isnot activated strongly at moderate depolarizations. AlthoughType A and B neurons have similar half activation potentials,the steeper slope of the Type B activation curve delays acti-vation of the potassium conductance. This would be mani-fested in anI-V curve by an abrupt increase in current near thehalf activation potential, as is observed in theI-V plots forType II neurons.

In summary, the analysis of these experiments has provided aquantitative description of the passive electrotonic properties ofputative spinal interneurons ofXenopuslarvae that consists of asoma with one equivalent dendritic cable. The models with alimited number of compartments are remarkably accurate for thedifferent types of neurons; however, active conductances arelikely to require increasing numbers of compartments because ofdynamical variations in the space constant. The use of an analyticmodel provides electrotonic parameters where spatial resolution isperfect. This procedure then allows a systematic and accurate wayto determine the number of compartments needed for describingthe active properties of any given neuron.

A P P E N D I X

Model equations

As described previously (Borg-Graham 1991; Moore and Buchanan1993; Moore et al. 1999; Murphey et al. 1995), the kinetic formulationusingx as a generalized kinetic variable is

I i 5 I p 1 I core 1 I l (A1)

I l 5 gsoma~Vi 2 Vl! (A2)

I p 5 Op

gpx~Vi 2 Vp! (A3)

I core 5 gcore~Vi 2 Vi11! (A4)

Vi

t5

2N

Acsoma

I i (A5)

x

t5 ax~1 2 x! 2 bx 5 ~x` 2 x!/tx (A6)

tx 5 1/~ax 1 bx! (A7)

x` 5 ax/~ax 1 bx! (A8)

ax 5 ~1/tx! exp~V 2 vx!~2sx 2 r x! (A9)

bx 5 ~1/tx! exp~V 2 vx!~2sx 1 r x! (A10)

wherecsomais the capacitance of the soma,Ii is the current in theithcompartment,Icore is the current between compartments,Vi is themembrane potential in theith compartment,gsomaand I l represent anonspecific leakage conductance and current havingVl as a reversalpotential,A is the total area of the dendritic compartments to the soma,N is the number of compartments,gp is a generic voltage-dependentionic conductance with a reversal potential ofVp and whose kineticsis governed by the unitless variable,x, which has a steady-state valueof x`. Thus at the half-activation (x5 1/2) voltage,vx, sx is the slopeof x`, tx is the time constant,tx, andrx is the normalized slope oftx.In this papergp represents potassium (gK) where the variable,x, is n.The analytic linearized admittance for one variable is given as

Ysoma~V, f ! 5 j2pfcsoma1 gsoma1 Op

gp$x`~V! 1 ~V 2 Vp!dx~V, f !% (A11)

dx~V, f! 5 txHax

V2 x`Sax

V1

bx

VDJY ~2jpftx 1 1! (A12)

or

Ysoma~V, f ! 5 j2pfcsoma1 gsoma

1 Op

gp$x`~V! 1 ~V 2 Vp!~dx`/dV!/~1 1 j2ptx! (A13)

where j 5 =21 and f 5 frequency in Hertz. The compartmentalmodel was recursively computed from the end compartment, either ina simple loop or symbolic notation (see Murphey et al. 1995) usingMathematica (Wolfram Research, Champaign, IL). Alternatively, an

TABLE 5. Potassium conductance parameters of Type A and Type B neurons

Neurons gK, nS vn, mV sn, mV21 tn, ms rn, mV21

Type A97J01A 4.34 230.19 0.031 2.0 0.02097J31J 1.84 21.62 0.042 2.0 0.00897K03D 3.66 24.16 0.047 2.4 20.00197K04D 0.296 210.84 0.041 1.0 20.07797K06B 0.556 217.32 0.035 1.0 0.05097K21D 0.176 227.16 0.043 37.0 20.133

Type B97J08B 1.50 214.07 0.050 1.8 20.07097J20B 0.132 221.92 0.056 5.9 20.07597J28B 0.203 217.64 0.056 30.0 20.08297J30A 10.00 212.15 0.198 38.6 097K06D 0.327 211.00 0.090 1.0 0.050

K1 intracellular solution was used.


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analytic model was used (see Moore et al. 1999; Rall 1960), whichleads to an active cable with an admittance,Ya, as follows

Ya 5 Ysoma1A* gsomaÎYsoma/gsoma

Ltanh ~LÎYsoma/gsoma! (A14)

wherel is the space constant,L is the electrotonic length,l/gcore 5L/(A*gsoma), N is the number of compartments andl 5 N/L, is in unitsof N. Note that the units of the hyperbolic tangent term are given bygsoma. In a completely passive neuron at d.c., the dendritic conduc-tance,gdendrite 5 (A*gsoma/L)*tanh L. Finally, the total admittancewith the electrode properties is

Yt 5 2pfCe 1 Ya/~1 1 ReYa! (A15)

whereCe is the electrode capacitance andRe is the electrode resis-tance.

This work was supported in part by the Centre National de la RechercheScientifique, France.

Address for reprint requests: L. E. Moore, Laboratoire de Neurobiologie desReseaux Sensorimoteurs, Centre National de la Recherche Scientifique-Unite Propre de Recherche de l’Enseignement Superieur Associee-7060, 45Rue des Saints-Peres, 75270 Paris Cedex 06, France.

Received 7 July 1999; accepted in final form 28 October 1999.

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