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Division of Neurobiology, Barrow Neurological Institute, St. Joseph's Hospital and Medical Center, Phoenix, Arizona 85013
Submitted 8 September 2003; accepted in final form 13 April 2004
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSACKNOWLEDGMENTSREFERENCES |
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), electrotonic length (L), and dendritic-to-somatic conductance ratio (
) was examined, showing that the relation between impedance phase and differed markedly between models with uniform and nonuniform membrane resistivity. Dendritic synaptic conductances decreased impedance magnitude at low frequencies; at higher frequencies, impedance magnitude increased. The frequency at which the change in impedance magnitude reversed from a decrease to an increase—the reversal frequency, Fr—was a good estimator of electrotonic synaptic location. A measure of the average normalized impedance change at frequencies less than Fr, cu
Z, estimated relative synaptic conductance. Fr and cuZ provided useful estimates of synaptic location and conductance in models with nonuniform (step, sigmoidal) and uniform membrane resistivity. Fr also provided good estimates of spatial synaptic location on the equivalent cable in both step and sigmoidal models. Variability in relations between Fr, cuZ, and conductance location and magnitude between neurons was reduced by normalization with and . The effects on Fr and cuZ of noise in experimental recordings, different synaptic distributions, and voltage-dependent conductances were also assessed. This study indicates that location and conductance of tonic dendritic conductances can be estimated from Fr, cuZ, and basic electrotonic motoneuron parameters with the exercise of suitable precautions. |
INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSACKNOWLEDGMENTSREFERENCES |
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). Within the dendrites of spinal motoneurons, synaptic locations have been identified through arduous reconstructions of single motoneurons and presynaptic neurons after intracellular staining (e.g., Brown and Fyffe 1981; Burke and Glenn 1996; Burke et al. 1979; Fyffe 1991; Redman and Walmsley 1983). The electrotonic positions of synapses have also been identified using the shapes of the postsynaptic potentials produced by single afferents (Iansek and Redman 1973; Jack et al. 1971; Mendell and Henneman 1971; Rall et al. 1967; Redman and Walmsley 1983). These determinations depend on the satisfaction of several conditions, such as having reasonable estimates of the duration of synaptic current and knowledge of the electrotonic characteristics of the motoneuron (Jack et al. 1975), and, of course, the ability to obtain suitable recordings from pairs of neurons for the electrophysiological measurements.
Considering the inherent difficulties and limitations of these methods, it is not surprising that our knowledge of the synaptic organization in the dendrites of motoneurons is still limited. An alternative method for determining synaptic location through electrophysiological measurements was proposed by Fox (1985). Fox demonstrated through compartmental models that the impedance function Z(f) of a neuron, the ratio of the amplitude of voltage response to the amplitude of injected sinusoidal current of frequency f, varies in a systematic way with the location of a synapse during its tonic activation. Fox and Chan (1985) confirmed the utility of this method in recordings from cultured neurons, but no further application has been made in experimental investigations.
The goal of this study was to develop the impedance measurement for application in studies of spinal motoneurons. We sought to use impedance functions to determine both the location and the magnitude of the conductance change produced by a steady synaptic input. The magnitude of this conductance change is critical for assessing the contribution of the synapse to dendritic integration, but is not readily obtained in most morphological studies.
We examined how the electrotonic parameters of a motoneuron, different models of nonuniform membrane resistivity, and voltage-dependent conductances affect both a motoneuron's impedance function and the changes in impedance created by synaptic inputs. In a companion paper (Maltenfort et al. 2004), we describe the application of this method to the synaptic conductances produced by recurrent inhibition. Preliminary accounts of this work have been published (Hamm and McCurdy 1992; Hamm et al. 1993).
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSACKNOWLEDGMENTSREFERENCES |
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, which were based on electrophysiological measurements and anatomical reconstructions of 6 type-identified triceps surae 
-motoneurons (Cullheim et al. 1987). The following will describe how each model was constructed and used to determine the sensitivity of impedances and synaptic effects to changes in the electrotonic parameters of the neuron. Motoneuron models
Fleshman et al. (1988) found that 2 models with nonuniform membrane resistivity provided equally good fits to their results, obtained using sharp electrodes. In the somatic shunt (or "step") model, the dendritic resistivity was uniform, but larger than somatic resistivity. In the "sigmoidal" model, the dendritic resistivity increased monotonically with distance from the soma, proportionately with the percentage of dendritic membrane area. Previous investigations have used either the somatic shunt model (e.g., Clements and Redman 1989; Fu et al. 1989; Jones and Bawa 1997; Rapp et al. 1994; Segev et al. 1990) or the sigmoidal model (Powers and Binder 1996) to represent the electrotonic properties of motoneurons.
This paper and its companions (Maltenfort and Hamm 2004; Maltenfort et al. 2004) focus on the somatic shunt model, which facilitates comparison to other studies of motoneuron properties and synaptic location (cf. Burke and Glenn 1996; Clements and Redman 1989; Fyffe 1991). Some simulations were performed using the sigmoidal model or a model with uniform membrane resistivity (i.e., without a somatic shunt) for purposes of comparison. Motoneuron models were uniquely defined by Rms and Rmd, the somatic and dendritic specific resistivities; As, somatic area, determined from , the dendritic-to-somatic conductance ratio, and the conductance of the dendritic cable; Deq, the initial diameter of the equivalent cable, at the first dendritic compartment; Ri, cytoplasmic resistivity; and Cm, specific membrane capacity. Values of 70 
-cm and 1 µF/cm2 were used for Ri and Cm, respectively. The other parameters for each neuron were determined according to values provided by Fleshman et al. (1988; Table 5 and Fig. 9 therein).
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for underlying theory). The electrotonic length of each compartment Lc was 0.05 in the step model and 0.01 in the sigmoidal model. The shorter compartment length in the sigmoidal model was used to reduce the variation in membrane resistivity within compartments. We used a "standard profile" for the equivalent cable of each motoneuron based on Fig. 9 in Fleshman et al. (1988; see also Clements and Redman 1989); the cable diameter Dc was constant for 0.25 cm from the soma, and then decreased linearly over the next 0.4 cm. The diameter of the proximal 0.25 cm segment in each model Deq was taken from the Fig. 9 legend of Fleshman et al. (1988). Starting at the soma, the cable length of each dendritic compartment lc (in cm) was determined using the equation lc = Lc x 
= Lc x (RmdDc/4Ri)0.5, where is the cable length constant. Once the cumulative cable length l exceeded 0.25 cm, Dc was determined by Dc = Deq – Deq(l – 0.25)/0.4. Rmd was a constant in the step model, but varied by compartment in the sigmoidal model and was defined as Rms + cuA(lT) x (Rmax – Rms), where cuA(lT) is the cumulative fraction of total dendritic membrane area between the soma and the current compartment, and Rmax is the maximum value for Rmd.
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Once the parameters were determined for all dendritic compartments, impedance functions were determined using equations from Fox (1985). These equations for impedance can be derived from the solution to the cable equation (e.g., Rall 1977). A succinct presentation of this derivation is given by Yang and Chapman (1983).
The "characteristic impedance" at the end of a semi-infinite cable is defined as
 | (1a) |
) = (1 + jD)1/2; is angular frequency; and D is the time constant of the dendritic membrane, the product of Rmd and the membrane capacitance per unit area.
Using characteristic impedance, the input impedance of a finite cable of length Lc under different boundary conditions can be expressed as (Fox 1985; Rall 1977)
 | (1b) |
 | (1c) |
) is the impedance at one end of the finite cable, when the other end is insulated so that no current flows out of the cable at its end (i.e., a sealed end termination); Zk,clp() is the impedance at one end of a finite cable when the other end is voltage clamped at resting potential, sometimes described as an "open-circuit" termination. Note that at = 0, Eq. 1b is equivalent to Rall's equation for the steady-state input resistance of a finite-length dendritic cable with sealed end, Rd = RmdLc/Ad tanh (Lc), where Ad is the membrane area of the equivalent cable. This equivalence can be demonstrated by noting that 2(Ri/RmdDeq)0.5 = Lc/l, 
Deql = Ad, and substituting into Eqs. 1a and 1b.
The impedance of the entire equivalent dendrite arises from the nonlinear sum of the individual compartment impedances. The admittances Yk,ins and Yk,clp are the reciprocals of the impedances defined in Eq. 1 above. Define Y0...k() as the combined admittance of contiguous compartments 0 to k, 0 being the index of the most distal compartment. Then the admittance of the entire equivalent dendrite, as seen from the soma, arises from iterative application of the following equation
 | (2) |
) = 1/Y0...N–1().
The impedance of the entire neuron observed from the soma [Z()] is the combination of the somatic and dendritic impedance, ZSZD/(ZS + ZD). The impedance of the soma (ZS) is calculated as ZS() = (Rms/AS)/(1 + jS), where S is the time constant of the somatic membrane.
Simulations with voltage-dependent conductances
Simulations of neurons with voltage-dependent conductances were also performed in which each compartment was modeled as a parallel resistor–capacitor combination in parallel with an inductive term representing the voltage-dependent conductance. The admittance Yc() = 1/Zc() of each compartment was modeled as
 | (3) |
m are the membrane conductance and time constant, respectively, and GV and V are the voltage-dependent conductance and time constant, respectively. The term on the right in Eq. 3 is a linear approximation to a voltage-dependent conductance conventionally described by a Hodgkin–Huxley-type equation. This approximation is valid for small-voltage excursions around a set membrane potential (Koch 1984) and with this restriction could be used to represent a current such as Ih, for example. Small hyperpolarizations increase activation of Ih at most resting membrane potentials, producing an inward current. With voltage changes that are slow relative to the Ih time constant at resting membrane potential (V), the magnitude of this inward current would be the product of GV and the change in membrane potential. With rapid changes in membrane potential ( > 1/V), Ih would change less and the right-hand term in Eq. 3 would contribute correspondingly less to the compartmental admittance.
Gm was determined by dividing the membrane area of the compartment by the specific membrane resistivity. When the voltage-dependent conductance was restricted to the soma, impedance functions were calculated as described above. For dendritic locations, the cable equations were replaced with equivalent circuit models, using Eq. 3 to represent the membrane properties of each compartment. Compartments were interconnected with axial resistances, given by the product of Ri and lc, divided by cross-sectional cable area. The impedance function of the dendritic cable was computed by starting at the last compartment, adding axial resistance to 1/Yc(), then taking the inverse to determine the combined admittance. This process was continued until all dendritic compartments had been incorporated. Impedance functions, based on cable equations and equivalent circuits, were compared to ensure that the different methods yielded the same results.
Estimation of time constant and electrotonic length
The system time constant () and electrotonic length of each modeled neuron were determined to examine the dependency of the impedance functions on electrotonic parameters. To determine in neuron models with nonuniform membrane resistivities, effective values of were estimated from the product of specific membrane capacity and an effective membrane resistivity. This resistivity was the inverse of the sum of the somatic and dendritic conductances, weighted by the relative surface areas of the somatic and dendritic compartments. The resulting estimate eff was then divided by an empirical correction factor K, which compensates for a systematic underestimation that is tightly correlated (r2 = 0.98) with (Fleshman et al. 1988). For the somatic shunt model, K = 1.0–0.01 x [11.2 – 22.0 x ln ()]. Electrotonic length (L) was defined as the length (normalized by ) at which the cumulative surface area of the dendritic cable reached 97% of its total area, a convention based on the finding of Fleshman et al. (1988) that the measured value of L corresponded to the length at which 96–98% of dendritic surface area was attained.
Comparison of model properties and experimental observations
Because the dendritic tree of each motoneuron was simplified to an equivalent cable and the dimensions of our standard cable differed from the equivalent cables of Fleshman et al. (1988) to some degree, we compared the calculated properties of the 6 models to those observed experimentally. Excellent matches were seen for both input resistance and , as indicated by regressions (Rexpt = 1.08 x Rmodel – 0.17 M, r2 = 0.98; exp = 0.95 x model + 0.04 ms, r2 = 0.99). As an additional check on , the transient response to a 1-ms pulse was calculated by multiplying the impedance function times the fast Fourier transform (FFT) of the pulse, and performing the inverse FFT on the product. A model transient computed in this way is shown in Fig. 1C with the experimental transient, plotted as open circles. The agreement between experimental time constants and those determined from model transients (fit between 30 and 35 ms of the transient tail) was also good (exp = 0.95 x model + 0.18 ms, r2 = 0.98).
In summary, errors introduced as a result of the simplifications inherent in the model are not large. The equivalent cable models described herein should be adequate.
Simulation of changes in the impedance function during synaptic activation
To model the effects of synaptic input, the conductivity of the affected compartments was increased by dividing the membrane resistivity by the factor 1 + x, where x represents the fractional increase in conductance attributed to the synapse; for example, to produce a 25% increase in membrane conductance, the membrane resistivity was divided by 1.25. In most simulations, each active synaptic input was represented by an increased conductance in 3 adjacent compartments, a span of 0.15. This number of compartments was chosen because 0.15 is an approximate mean for the range of locations spanned by the synapses of individual Renshaw cells on the dendrites of motoneurons (Fyffe 1991). In some cases, broader distributions of synaptic input were simulated, as described in RESULTS. Typically, the middle compartment containing active synapses was placed at one of 5 dendritic locations on the model neurons (0.15, 0.30, 0.45, 0.60, and 0.75 from the soma), and the membrane conductance was increased by 10, 25, 50, or 100%. This approach implicitly assumes that the probability of synaptic contacts is proportional to the membrane area. In the step model, the total conductance corresponding to a 100% conductance increase ranged from 25.8 to 71.5 nS (mean of 47.3).
In the sigmoidal model, in which resistivity varies by compartment, an alternative approach was used. First, the effective mean Rmd of the dendritic cable was determined as the inverse of the sum of all compartment conductivities, weighted by the fractional area of each compartment relative to total dendritic membrane area. Then the membrane area of a 0.15-long cable segment having this mean Rmd was determined. The conductance increase of this cable segment produced by the specified fractional change in conductance (e.g., 25%) was determined, using the same procedure as for the step model. The conductivity of the affected compartments was then increased by the amount needed to match this conductance increase. This approach added the same synaptic conductance at each synaptic site independent of synaptic location, as done implicitly in the step model. The conductance corresponding to a 100% conductance increase ranged from 32.2 to 106.9 nS (mean of 66.6).
The impedance function for the neuron was reevaluated with each change in synaptic input, using the altered resistivities, electrotonic lengths, and time constants of the compartments with synapses.
Simulations of impedance functions obtained from noisy recordings
The effects of noise on estimates of synaptic location and conductance magnitude were assessed using impedance functions of the model neurons to which random Gaussian noise was added. In each trial of the simulations, one of 3 motoneuron models (41/2, 43/5, and 42/4) and one of 3 dendritic locations (0.15, 0.3, and 0.45) were selected randomly using a second random-number generator. A conductance increase of 50% was added to the selected synaptic compartments, spanning 0.15. The level of added noise was based on the normalized random error for an impedance estimate EZ(f):
 | (4) |
2(f) is the coherence between injected current and the voltage response, and n is the number of samples used to determine the impedance function (Bendat and Piersol 1986). The coherence estimates the fraction of power in the voltage signal produced by the injected current acting on the impedance; that is, a 2(f) of 0.99 means that at frequency f, 99% of the power of the power spectrum of the membrane voltage is linearly related to the injected current (Bendat and Piersol 1986). Simulations used typical values from the accompanying study of Maltenfort et al. (2004). Coherence was 0.89 at the lowest frequency, 4.88 Hz, increased linearly to 0.98 at 20 Hz, and was constant at higher frequencies. Fifty experimental trials were often taken in this study, and each trial was divided into 5 segments of data (each overlapping the next by 50%) for computation of power and cross spectra and impedance functions. For example, if segment 1 is composed of points 1–1,024, segment 2 would constitute points 513–1,536, segment 3 would constitute points 1,025–2,048, and so forth. This procedure minimized the variance in the spectral estimates, providing effectively 4.1 samples per record (Press et al. 1992), and a total n of 4.1 x 50 = 205. In the simulations, values produced by a Gaussian random-number generator, set to have a SD determined by the preceding equation and the cited values of 2(f) and n, were added to each point in the impedance functions, with and without synaptic activation. The difference between the 2 impedance functions in each run of the simulation was computed, the reversal frequency was selected, and the change in impedance magnitude was determined. This process was performed without the operator's knowledge of the model neuron or synaptic location used in the trial. From 11 to 20 trials were collected for each combination of model and synaptic location. Computations were performed using either programs written in C code or using MATLAB (MathWorks, Natick, MA).
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSACKNOWLEDGMENTSREFERENCES |
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MAGNITUDE OF THE IMPEDANCE FUNCTION.
Impedance functions calculated for 6 motoneurons modeled with the somatic shunt model using values from the data of Fleshman et al. (1988) are shown in Fig. 1B. Representation of motoneurons modeled with the sigmoidal model and based on the same data of Fleshman et al. (1988) yielded impedance functions that did not differ considerably from results with the somatic shunt model (Fig. 1B). Transient responses calculated from both impedance functions matched experimental responses well (see METHODS). For example, Fig. 1C shows the transient response of a neuron modeled with the somatic shunt model to a 1-ms current pulse (parameters based on cell 42/4; Fleshman et al. 1988). An electrophysiologically recorded transient from cell 42/4 is plotted on the model response and shows good agreement (Fig. 1C).
The effect of electrotonic parameters on the form of these impedance functions was explored. Figure 2A shows the effect of system time constant on the normalized impedance function of the model of motoneuron 43/5. Increasing or decreasing by 50% shifts the impedance function proportionally to the left or right, respectively, along the frequency axis. Multiplying frequencies by compensates for variations in time constant, producing impedance functions for neurons with different values that are superimposable (figures not shown). Figure 2B illustrates the dependency of the impedance function on the dendritic-to-somatic conductance ratio . Increasing by 50% (by decreasing soma area) shifts the impedance function to the left, whereas decreasing shifts the curve to the right, so that the roll off in the impedance function occurs at higher frequencies. However, the effect of is not a simple proportional shift, given that the curvature in the roll off increases at larger values of . The electrotonic length (L) of the dendritic cable also affected the impedance function, although to a lesser degree. Decreasing L produced a slightly faster roll off of impedance with frequency, whereas increasing L had the opposite effect (not shown).
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for each neuron. The differences between these impedance functions are best correlated with differences in . The 3 models with = 0.14–0.32 show a later and more linear fall-off of impedance with frequency than do the 3 with = 1.1–1.2 (whose lines overlie each other in this figure). The 2 impedance functions, which have similar values of (0.3 vs. 0.32), are close together and situated between the impedance curves representing cells with larger and smaller values of .
The influence of on the impedance function of neurons with low somatic resistivity can be attributed to the difference in the somatic and dendritic impedance functions and the difference in membrane time constants. The somatic and dendritic contributions to the motoneuron's impedance function depends on their relative conductance, as indicated by Z = ZSZS/(ZS + ZD) = 1/(YD + YS). The shapes of somatic and dendritic impedances differ, as shown for one of the model neurons in Fig. 2D, in which the somatic impedance behaves as a simple one-pole low-pass filter, whereas the curvature in dendritic impedance reflects its distributed, cable structure. Because Rms and Rmd and their associated time constants may differ by 2 orders of magnitude, somatic and dendritic impedance functions roll off at very different frequencies, with dendritic impedance decreasing over a broad frequency range where somatic impedance remains nearly constant. Within this range, dendritic conductance increases and dendritic characteristics dominate the impedance function, as indicated by the relation. This effect is greater with larger values of .
PHASE OF THE IMPEDANCE FUNCTION.
Phases of the impedance function for all 6 model motoneurons are shown in Fig. 3A. Similar to impedance magnitude, the impedance phase of the sigmoidal and step models did not differ significantly. Some of the variability between cells is accounted for by differences in . Changes in affected phase in the same manner as they affected the magnitude of the impedance function. Phase curves of models that differed only in their values of could be superimposed after normalization by (not illustrated). The effects of L and were more complex. Changing L produced changes in curvature of the phase relation, such that greater curvature was observed as L decreased, and less was observed as L increased (Fig. 3B).
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demonstrated the sensitivity of phase to , based on modeling work of Rall (1960). In our simulations, we found that as was varied, the curvature of the phase plot changed, but the frequency at which the phase lag was 45° was constant, equal to 1/(2s), where s was the somatic time constant. Because the dendritic membrane has a much longer time constant than s (Clements and Redman 1989; Fleshman et al. 1988), the dendritic impedance phase reaches a maximum of 45° at relatively low frequencies, and s determines the frequency at which the neuron impedance phase is 45°. Rall (1960) showed that phase shifted as a function of without the crossover shown in Fig. 3C, assuming uniform membrane resistivity. When the phase plots were recalculated making that same assumption (Fig. 3D), we obtained a similar result. Changes in impedance function with tonic synaptic input
DEPENDENCY OF THE CHANGE IN THE IMPEDANCE FUNCTION ON LOCATION AND MAGNITUDE OF THE SYNAPTIC CONDUCTANCE.
Figure 4A compares impedance functions of a neuron with synaptic conductances of 100% at one of 3 dendritic locations (0.15, 0.30, and 0.55). This change in conductance totals 34.6 nS in the 3 dendritic compartments. The impedance functions with dendritic synaptic conductances (broken lines) lie between impedance functions of neurons without any synaptic conductance (top solid line) and with a somatic synaptic conductance of the same magnitude (34.6 nS; bottom solid line). The normalized impedance changes are shown in Fig. 4B. Dendritic conductances produce changes in the magnitude of the impedance function that decrease rapidly with frequency. At greater frequencies, an increase in impedance magnitude is actually observed, in agreement with Fox (1985).
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R), as shown in Fig. 5B. %R increases linearly with the magnitude of the conductance change at each synaptic location.
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R can provide estimates of synaptic location and conductance magnitude, respectively. However, it is obvious from Fig. 4A that dendritic conductances of this magnitude produce small impedance changes measured at the soma (cf. Carlen and Durand 1981; Rall 1967). To explore the sensitivity of Fr and %R estimates to noise in the impedance records, a set of simulations was performed in which noise was added to the impedance functions of 3 of the model motoneurons with and without a synaptic conductance of 50%. The added noise was comparable to the level observed in the accompanying study by Maltenfort et al. (2004). Several alternative methods of measuring reversal frequency and measures of the impedance change at low frequencies were assessed in these simulations. Fr could be identified most reliably by visual identification after passing the impedance magnitude function through a median filter: each point was replaced by the median value of points in a window (ranging from ±3 to 10 points) around the original point.
The variability in %R was deemed unacceptable, and an alternative measure of the low-frequency impedance change was adopted. This measure, cuZ, is a normalized, frequency-weighted cumulative sum, which approximates the average value of Z in a semilogarithmic plot
 | (5) |
Z is the normalized change in impedance magnitude, 100 x (|Z(f)| – |Zsyn(f)|)/Z(0); f/f is the frequency interval divided by the frequency of each term in the summation [approximating ln (f)]; and the summation is taken from the lowest frequency (4.88 Hz in these studies) in the spectrum to Fr. Division by 
f/f reduces the dependency of cuZ on Fr. CuZ is plotted as a function of synaptic conductance for several dendritic locations in Fig. 5C, showing that this measure is proportional to the synaptic conductance change.
The means and SDs of cuZ and Fr for sets of measurements made with noisy impedance functions from 3 motoneuron models are shown in Fig. 6, A and B, respectively. Figure 6A indicates the level of uncertainty to be expected in estimates of cuZ and its variation between cells. The least variability in relation to the expected cuZ value occurs with model 43/5, in which is largest and the relative change in impedance greatest. The greatest variability occurs with model 41/2, which has the lowest value of . The corresponding variability in Fr is shown in Fig. 6B, in which Fr estimates are plotted against the ratio of the mean to SD of the cuZ estimate. At the larger values of this ratio, above 1–1.5, reasonably accurate Fr estimates can be obtained. These observations suggest a strategy for experimental implementation: determine the expected SD of cuZ using Eq. 4 and reject Fr estimates if the expected SD is too large in relation to the cuZ estimate. This approach is applied in the accompanying paper (Maltenfort et al. 2004).
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Z on the magnitude and position of conductance changes for 2 of the model neurons. In these grids, specification of the reversal frequency provides an estimate of synaptic location that is scarcely influenced by conductance magnitude. Once synaptic location is specified, the value of cuZ estimates the relative change in conductance in the dendritic compartments with active synapses. The differences in these 2 diagrams also show that the relationships between Fr, cuZ, and synaptic location and conductance magnitude vary between cells, evidently depending on the electrotonic parameters of each motoneuron.
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Z values attenuate less at distal locations than in the step-model grids because of the different electrotonic profiles of the 2 models.
DEPENDENCY OF CHANGES IN IMPEDANCE ON AND .
Fox (1985) reported that impedance functions were invariant with when plotted as a function of normalized frequency (i.e., frequency x ). Considering the importance of relative somatic and dendritic conductances in determining impedance (see above), should be an important determinant of cuZ. That is, dendritic conductance changes will have a greater effect on impedance in neurons with larger relative dendritic conductance (larger ). We examined the dependency of Fr and cuZ on and in the limited set of 6 model motoneurons by comparing parameters in pairs of neurons. When reversal frequencies of each motoneuron for a set of synaptic locations were compared with the corresponding values of Fr for the other 5 motoneurons, strong linear relations were found (r2 > 0.99). A similar finding was made when values of cuZ for each motoneuron, obtained for multiple synaptic locations and conductance magnitudes, were compared with the corresponding cuZ values of other motoneurons (r2 > 0.99). Regression slopes varied between cell pairs.
The dependency of Fr on was examined by comparing the slopes of the regressions between Fr values with the ratios of the system time constants of each pair of cells. If Fr is linearly proportional to 1/, as suggested by Fox (1985), then the regression slope should equal the inverse ratio of time constants. Regression lines were calculated between reversal frequencies for each cell pair. The slopes of these regressions matched the inverse ratios of time constants of the 2 neurons (i.e., Fr1/Fr2 = 2/1; r2 = 0.96), indicating that reversal frequency is proportional to 1/. A similar analysis was performed to examine the relationship between cuZ and . In this case, cuZ was found to depend on both and . This relation (r2 = 0.96) was described by: cuZ1/cuZ2 = (1/2)0.46(2/1)0.33.
The use of and to adjust the estimates of synaptic conductance magnitude and location is illustrated in Fig. 8. In Fig. 8A, the grids have been normalized by multiplying Fr values by and multiplying cuZ values by 0.33/0.46. This rescaling causes a substantial, if not exact, overlap. Normalizations were also applied to sigmoidal and no-shunt models, illustrated in Fig. 8, B and C, based on analyses such as those described for the step model. CuZ in the sigmoidal model was proportional to 0.20/0.27 (r2 = 0.91), and normalization by this factor removed most of the variance in cuZ values between neurons. However, the correlation between Fr and 1/ was weaker (r2 = 0.56), as evident in the difference in Fr values of the normalized grids in Fig. 8B. The failure of to normalize Fr in this model can be attributed to the nonuniformity of Rmd and membrane time constant. Differences in Fr values for conductances at 0.15, for example, were associated with differences in membrane within this section of dendritic cable.
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is not a significant factor. The relatively small variance in cuZ values (slopes of cuZ regressions ranged from 0.85 to 1.17) depended on dendritic geometry, specifically Deq, and on , with cuZ proportional to Deq0.23/0.45 (r2 = 0.95). The cuZ differences between normalized grids seen in Fig. 8C are among the worst among the normalized no-shunt grids. Fr values of the grids represented in Fig. 7, E and F (shown normalized in Fig. 8C) are similar, but normalized Fr for other grids with dissimilar values, corresponding to the observation that Fr was proportional to 1/ (r2 = 0.97) in this model. Comparisons were also made between Fr values in relation to mean spatial synaptic location (rather than electrotonic location) on the equivalent dendritic cable in the 3 models. Fr values of the step and sigmoidal models were essentially the same for corresponding cable positions (Fig. 9). Fr values for models without a somatic shunt were also similar but somewhat lower for a given cable position. Consequently, without adjustment for electrotonic parameters, Fr can be used to estimate the spatial location of synapses along an equivalent dendritic cable, with little dependency on the underlying model assumptions.
DEPENDENCY ON DISTRIBUTION OF SYNAPSES.
All simulations described to this point were based on a conductance distributed equally over 3 dendritic compartments (0.15). A specific synaptic input may produce conductance changes over a range of electrotonic distances (Burke and Glenn 1996; Fyffe 1991), so we examined the effect of changing the width of the synaptic distribution.
Figure 10A shows that increasing the width of the distribution to cover longer dendritic segments moves the relation between Fr and mean synaptic location upward and to the right. This effect seems to be produced by the more proximal synapses in the wider distribution, which have a greater effect on Fr. Despite this proximal bias, plots of Fr versus electrotonic location of the most proximal synapse in the distributions demonstrated a greater dependency on distribution width (not illustrated), indicating that Fr is a better estimator of mean synaptic location.
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). Half the total conductance change occurs within the proximal 29.3% of the proximally weighted distribution, accounting for the smaller influence of width on Fr and the larger values of Fr for a given mean synaptic location (compare Fig. 10A and 10B). Fr was even less dependent on the width of triangular distributions (not shown), in which synaptic density is maximum at the center of the distribution and falls linearly to zero at either end. Allowing for variations in the width and shape of synaptic distributions, Fr estimates electrotonic location within 0.2 length constants of its true value.
Simulations were also performed to examine the use of Fr to determine the location of synapses confined to part of a dendritic arbor. Models were constructed with 2 dendritic cables. The ratios of the initial diameters of the 2 cables ranged from 0.25 to 1, and the ratios of their electrotonic lengths (at 97% dendritic area) ranged from 0.66 to 1. The profiles of the 2 cables were adjusted so that the electrotonic profile of the combined cables (using the 3/2-power law) matched that of the corresponding one-cable model. Fr values for a tonic conductance at a given electrotonic distance from the soma varied as the conductance was placed on either or both cables of these 2-cable models. The SD of Fr values varied from <1 to 6% of mean Fr at different locations. This variability was sufficient to produce some overlap in Fr values of distal locations (>0.45), but was generally small. Values of cuZ scaled to the total synaptic conductance in each model, taking into account differences produced by different synaptic locations.
EFFECT OF VOLTAGE-DEPENDENT CONDUCTANCE.
Impedance functions obtained experimentally may show characteristics at low frequencies, indicating the presence of a voltage-dependent conductance (Maltenfort and Hamm 2004; Moore and Christensen 1985; Weckström et al. 1992). This conductance changes both the magnitude and phase of the impedance function, potentially affecting the relationships between synaptic location, conductance magnitude, cuZ, and reversal frequency. An additional set of simulations was performed to examine the effect of a voltage-dependent conductance on these relationships. Voltage-dependent conductances were included in somatic or dendritic compartments, or both, by adding an inductive term, GV/(1 + jV), to compartmental admittance (see METHODS). The range of values used in these simulations for GV and V, the magnitude and time constant of the voltage-dependent conductance, covered the values found by Maltenfort and Hamm (2004).
Figure 11 shows the influence of a voltage-dependent conductance, distributed uniformly throughout the neuron, on the impedance function of one model motoneuron. Addition of a voltage-dependent conductance (GV = 100 µS/cm2, V = 20 ms) altered the form of the impedance change produced by a synaptic conductance, as shown in Fig. 11A. The voltage-dependent conductance increased neuron conductance and lowered impedance over a low-frequency range determined by V (see Maltenfort and Hamm 2004). This conductance increase had opposing effects on the normalized impedance change induced by synaptic activity, shunting the synaptic conductance at low frequencies, but also reducing the low-frequency impedance value in the denominator of the normalization. Provided that V was short enough that the frequency range of the voltage-dependent conductance extended to the reversal frequency, Fr was also affected. These effects are evident in the 2 grids shown in Fig. 11B. This voltage-dependent conductance increases cuZ values and increases Fr progressively with more distal synaptic locations.
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V values for a uniformly distributed voltage-dependent conductance. Fr increases as V decreases, the effect of which was greater at more distal synaptic locations. Except for V < 20 ms and GV 
100 µS/cm2, these increases are relatively small. A voltage-dependent conductance decreases the apparent input resistance, tending to increase cuZ, but the added conductance also reduces the change in impedance produced by a synaptic conductance, tending to decrease cuZ. Consequently cuZ is decreased by voltage-dependent conductances with short time constants, but is increased when the time constant is long. These changes may be substantial. Large voltage-dependent conductances that produce this effect should be evident as a low-frequency dip in a neuron's impedance function (Maltenfort and Hamm 2004).
The effects of voltage-dependent conductance restricted to dendritic locations were similar to those with uniform distribution (not shown). Voltage-dependent conductances restricted to the soma produced qualitatively similar but smaller effects (not shown). (Somatic GV values were increased by a factor of 10 in these simulations to produce impedance functions similar to those produced by uniform GV.) Changes in Fr and cuZ with somatic GV were 10 to 25% and 30 to 75%, respectively, of those with uniform GV. Changes with the 2 GV distributions were more similar in cells with larger .
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSACKNOWLEDGMENTSREFERENCES |
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. The impedance functions can also be used to estimate the relative magnitude of the conductance change. In the following discussion, we address the relation of this study to previous work that concerned the determination of synaptic conductance changes in neurons, the basis of the impedance changes, and the experimental applicability of the impedance method. Comparison to previous studies
The technique described in this report complements several approaches for determining synaptic location and/or conductance. Analysis of postsynaptic potential shape has been particularly effective in estimating the location of synapses producing individual postsynaptic potentials (PSPs; e.g., Iansek and Redman 1973; Jack et al. 1971; Rall et al. 1967; Redman and Walmsley 1983). Unlike the present method, analysis of PSP shape requires a brief conductance change, or an estimate of conductance time course. Estimates of relative location for inhibitory synapses can be obtained by comparing the sensitivity of inhibitory PSPs (IPSPs) to reversal by chloride injection and hyperpolarization (e.g., Burke et al. 1971). Smith et al. (1967) applied impedance methods to determine the time course and magnitude of conductance changes produced by several synaptic systems in spinal motoneurons. However, interpretation of their results was limited by use of a single frequency, which can yield seemingly paradoxical results, as discussed by Fox (1985). More recently, Häusser and Ro
