Passive Normalization of Synaptic Integration Influenced by Dendritic Architecture

David B. Jaffe1 and Nicholas T. Carnevale2

 1Division of Life Sciences, University of Texas at San Antonio, San Antonio, Texas 78249; and  2Department of Psychology, Yale University, New Haven, Connecticut 06520


    ABSTRACT
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ABSTRACT
INTRODUCTION
METHODS
THEORY
RESULTS
DISCUSSION
REFERENCES

Jaffe, David B. and Nicholas T. Carnevale. Passive Normalization of Synaptic Integration Influenced by Dendritic Architecture. J. Neurophysiol. 82: 3268-3285, 1999. We examined how biophysical properties and neuronal morphology affect the propagation of individual postsynaptic potentials (PSPs) from synaptic inputs to the soma. This analysis is based on evidence that individual synaptic activations do not reduce local driving force significantly in most central neurons, so each synapse acts approximately as a current source. Therefore the spread of PSPs throughout a dendritic tree can be described in terms of transfer impedance (Zc), which reflects how a current applied at one location affects membrane potential at other locations. We addressed this topic through four lines of study and uncovered new implications of neuronal morphology for synaptic integration. First, Zc was considered in terms of two-port theory and contrasted with dendrosomatic voltage transfer. Second, equivalent cylinder models were used to compare the spatial profiles of Zc and dendrosomatic voltage transfer. These simulations showed that Zc is less affected by dendritic location than voltage transfer is. Third, compartmental models based on morphological reconstructions of five different neuron types were used to calculate Zc, input impedance (ZN), and voltage transfer throughout the dendritic tree. For all neurons, there was no significant variation of Zc with location within higher-order dendrites. Furthermore, Zc was relatively independent of synaptic location throughout the entire cell in three of the five neuron types (CA3 interneurons, CA3 pyramidal neurons, and dentate granule cells). This was quite unlike ZN, which increased with distance from the soma and was responsible for a parallel decrease of voltage transfer. Fourth, simulations of fast excitatory PSPs (EPSPs) were consistent with the analysis of Zc; peak EPSP amplitude varied <20% in the same three neuron types, a phenomenon that we call "passive synaptic normalization" to underscore the fact that it does not require active currents. We conclude that the presence of a long primary dendrite, as in CA1 or neocortical pyramidal cells, favors substantial location-dependent variability of somatic PSP amplitude. In neurons that lack long primary dendrites, however, PSP amplitude at the soma will be much less dependent on synaptic location.


    INTRODUCTION
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ABSTRACT
INTRODUCTION
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THEORY
RESULTS
DISCUSSION
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The computations of single neurons involve the integration of synaptic inputs, thousands of which may be distributed over the cell surface. The seminal work of Rall (1962, 1970, 1977) provided a theoretical basis for understanding how location affected the amplitude and shape of synaptic signals. One prediction of cable theory is that the ability of a synapse to alter somatic membrane potential (Vm) declines as its distance from the soma increases. Theoretical and experimental studies of dendritic electrotonus have generally been presented in terms of the voltage transfer ratio from synapse to soma, i.e., Vsoma/Vsyn, where Vsyn is the postsynaptic potential (PSP) amplitude at the synaptic location and Vsoma is its amplitude observed at the soma. In this paper we will represent this ratio by ksynright-arrow soma (read as "k from synapse to soma"), after the notation introduced by Carnevale and Johnston (1982).

Models of neurons, whether based on equivalent cylinders or multicompartmental representations, have generally demonstrated large variations in ksynright-arrow soma with synaptic location (Jack et al. 1975; Rall 1962, 1970, 1977). More recent studies have emphasized the direction-dependence of voltage transfer, which is less efficient for signals spreading toward the soma than for somatic voltage spreading into the dendrites (Carnevale et al. 1997; Cauller and Connors 1992; Mainen et al. 1996; Nitzan et al. 1990; Tsai et al. 1994; Zador et al. 1995): ksynright-arrow soma ksomaright-arrow syn.

It is essential to know whether ksynright-arrow soma is the most reliable index of the relationship between synaptic location and synaptic efficacy, because this has a strong bearing on our understanding of the integrative properties of neurons. If there is a large variability in voltage transfer from synapse to soma, and action potentials are generally initiated near the soma (Colbert and Johnston 1996; Mainen et al. 1995; Stuart and Sakmann 1994), then how do distant synapses trigger action potentials? All else being equal, proximal excitatory inputs would seem to exert a much greater depolarizing action on the soma than do distal inputs. Anderson et al. (1987) suggested that active conductances in dendrites might "boost" excitatory PSPs (EPSPs) as they propagate toward the soma. We now know that the dendrites of many classes of neurons contain active conductances that could enhance EPSP amplitude (Gillessen and Alzheimer 1997; Lipowsky et al. 1996; Magee et al. 1995; Magee and Johnston 1995a, b; Schwindt and Crill 1995, 1997; Stuart and Sakmann 1995), but assessing their contribution to synaptic efficacy demands that we first have a clear grasp of the baseline electrotonic properties of these cells.

Recent results from this laboratory have compelled us to reevaluate the role of voltage transfer in synaptic integration. In a study of the electrotonic properties of hippocampal CA3 interneurons, Chitwood et al. (1999) modeled the effects of unitary synaptic conductances placed at various distances from the soma. They found that the range of relative EPSP amplitudes measured at the soma was strikingly narrower than the range observed at the synaptic sites. The relative variation of the somatically recorded EPSP was also much smaller than the variation of ksynright-arrow soma. This unexpected disparity suggested that ksynright-arrow soma might not always be the best predictor of synaptic integration.

In this report we examine the basis for this intriguing result, which leads to the conclusion that it is necessary to consider local input impedance (ZN), transfer impedance (Zc), and the magnitude and time course of the synaptic conductance change (Delta gs) itself when examining synaptic integration. We demonstrate how some neuron morphologies favor relative uniformity of Zc throughout the cell in the face of substantial variation of both ksynright-arrow soma and ZN with synaptic location. In these cells there is a large class of synaptic inputs for which location has only a small effect on the amplitude of a PSP observed at the soma, a phenomenon that we call "passive synaptic normalization" to emphasize the fact that it does not depend on contributions from active currents. We illustrate these conclusions by examining a number of cell types that reveal what is required for dendritic geometry to be a major determinant of synaptic variability.


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Neuronal reconstructions

Two sets of neuron morphology were used in this study. In the first set, input resistance (RN) and the slowest membrane time constant (tau 0, which was used to calculate specific membrane resistivity, Rm) were measured experimentally for each reconstructed cell with the use of standard whole cell recording methods (Chitwood et al. 1997). This set consisted of 15 CA3 nonpyramidal neurons from s. radiatum and s. lacunosum-moleculare, 3 CA3 pyramidal neurons, and 1 layer V pyramidal neuron from the medial prefrontal cortex (PFC). All of these cells were obtained from 17- to 30-day-old Sprague-Dawley rats (see Chitwood et al. 1999). A computer-assisted cell reconstruction system was used to determine three-dimensional morphology (Claiborne 1992).

Dr. Brenda Claiborne (University of Texas at San Antonio) generously provided the second data set. These included four CA1 pyramidal neurons and four CA3 pyramidal neurons, labeled using sharp electrodes filled with horseradish peroxidase, and reconstructed with morphometric techniques that were nearly identical to those for the first data set. These cells have been subjected to other analyses in prior studies (Carnevale et al. 1997; Claiborne 1992; Mainen et al. 1996).

Simulations

All simulations were performed with NEURON (Hines 1989; Hines and Carnevale 1997) on Silicon Graphics R4400 workstations. For all models, specific membrane capacitance Cm was assumed to be 1 µF/cm2. For the morphological models (described below) intracellular resistance (Ri) was set to 200 Omega  cm (Carnevale et al. 1997), although recent experimental measurements suggest that this value may be 70-100 Omega  cm (Stuart and Spruston 1998). Excitatory non-N-methyl-D-aspartate (NMDA) receptor-mediated synaptic conductances were modeled using the reaction scheme proposed by Holmes and Levy (1990)
<IT>A</IT><IT>+</IT><IT>R</IT> <LIM><OP><ARROW>←</ARROW></OP><UL><IT>k</IT></UL></LIM> <IT>AR</IT> <LIM><OP><ARROW>⇄</ARROW></OP><LL><IT>&bgr;</IT></LL><UL><IT>&agr;</IT></UL></LIM> <IT>AR</IT><IT>°</IT>
where the agonist (A) is assumed to bind instantaneously to the receptor (R), such that AR = 1, and AR° is the open form of the channel. The rate constants were k = 1.0 ms-1, alpha  = 1.0 ms-1, and beta  = 0.5 ms-1, and the synaptic current (Isyn) was given by
<IT>I</IT><SUB><IT>syn</IT></SUB><IT>=</IT><IT>g</IT><SUB><IT>max</IT></SUB><IT>·1.42·</IT><IT>AR</IT><IT>°</IT>(<IT>V</IT><SUB><IT>m</IT></SUB><IT>−</IT><IT>E</IT><SUB><IT>rev</IT></SUB>)
where gmax is the maximum conductance and Erev is the synaptic reversal potential (0 mV). The scale factor 1.42 ensured that the peak synaptic conductance equaled gmax.

Electrotonic analysis tools that are part of the standard distribution of NEURON (Carnevale et al. 1997) were used to determine Zc, local ZN, and ksynright-arrow soma. In a number of simulations, we normalized Zc at any dendritic location to ZN at the soma (&Zcirc;c). This facilitated within-cell comparisons at different signal frequencies and between-cell comparisons.

In a previous study (Chitwood et al. 1999) we found that the location-dependent distributions of depolarization at the soma, as well as synapse to soma voltage attenuation, for the non-NMDA receptor-mediated synapse model were best fit by 20-Hz signals. Therefore most calculations of impedance and voltage transfer in the present study were for 20 Hz.

EQUIVALENT CYLINDER MODEL. An equivalent cylinder model, modified from Spruston et al. (1993), was used to simulate generalized electrotonus. The model emulates the electrotonic properties of a CA3 pyramidal neuron by having both an apical and basal equivalent cylinder. The somatic compartment was a 50-µm-long cylinder with a diameter of 20 µm. The apical dendrite was 720 µm long (3 µm diam), and the basal dendrite was 310 µm long (3.8 µm diam). Both dendritic cylinders for this model were subdivided into 50 compartments each. Apical tufts were added to this model as 10 additional dendrites (5 compartments each) 100 µm long and 3 µm diam. Rm for these models was 50,000 Omega  cm2 and Ri was 100 Omega  cm.

MORPHOLOGICAL MODELS. Morphometric data were converted to a format compatible with NEURON using custom software. The electrical effects of spines on pyramidal neurons were emulated in some simulations by doubling Cm and halving Rm (Holmes 1989).

For the first data set, Rm was determined from tau 0 measured from CA3 pyramidal and nonpyramidal cells (Table 1), given that estimated Cm in these cells is close to 1 µF/cm2 (Chitwood et al. 1999; Major et al. 1994). The time constant for the layer V pyramidal neuron was not determined experimentally. The range of tau 0 in these neurons is ~20-80 ms (Kawaguchi 1993; A. T. Gulledge and D. B. Jaffe, unpublished observations), and a value of 40 ms was chosen for this study. In the second data set of four CA1 pyramidal neurons, four CA3 pyramidal neurons, and one dentate granule cell, Rm was taken from the experimental measurements of Spruston and Johnston (1992). All measurements were performed at either room temperature (24-26°C) or 31-32°C.


                              
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Table 1. Membrane time constants for neuron models


    THEORY
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ABSTRACT
INTRODUCTION
METHODS
THEORY
RESULTS
DISCUSSION
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Two-port theory

Two-port electrotonic analysis, which was introduced by Carnevale and Johnston (1982), is a good starting point for understanding the origins of the disparity between the variability of Vsyn and Vsoma (see also Koch 1999). Although it has been used to describe both current and voltage attenuation in dendrites (Carnevale and Johnston 1982; Carnevale et al. 1997; Tsai et al. 1994), the functional consequences of the relationship between input impedance ZN and transfer impedance Zc have not been elaborated.

Two-port electrotonic analysis draws on the basic principle that the electrical coupling between any two points in a linear system can be described by an equivalent circuit that consists of three impedances. The top of Fig. 1 shows a cartoon of a cell with a recording electrode attached to the soma and an activated synapse located somewhere on its dendritic tree. The membrane potentials at the soma and synapse are Vsoma and Vsyn. The current injected into the cell through the electrode at the soma is Isoma, and the synaptic current is Isyn. The bottom of this figure shows the electrical equivalent for this experimental arrangement. Here we have used an equivalent T circuit (Carnevale and Johnston 1982) to represent the coupling between the somatic recording electrode and the synaptic location.



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Fig. 1. T-circuit representation of the electrotonic interaction between synapse and soma. Top: schematic diagram of a neuron with current-clamp electrode at the soma (Isoma). The current (Isynapse) produced by synaptic activation causes membrane potential changes at the synapse (Vsynapse) and at the soma (Vsoma). Bottom: equivalent T-circuit of the situation shown above. This circuit consists of 2 axial impedances (Za and Zb) and a mutual or transfer impedance (Zc). The potentials Vsynapse and Vsoma are relative to resting potential.

Despite the suggestive appearance of this diagram, the transfer impedance Zc is not a direct counterpart of the cell membrane between the synapse and the soma, nor do the axial impedances Za and Zb correspond to the resistance of the intervening cytoplasm. These impedances are complex functions of frequency that depend on the anatomic and biophysical properties of the entire cell and the locations of the synapse and the soma (e.g., Eqs. 9 and 10 in Carnevale et al. 1997). For any finite structure, it should be noted that generally Za not equal  Zb (Carnevale and Johnston 1982; Carnevale et al. 1997) except for two special cases: the trivial instance where the synapse is located at the soma and the special case of a uniform cylinder in which the "soma" and the synapse are equidistant from the geometric midpoint of the cylinder.

According to Kirchhoff's voltage law, the membrane potentials at the soma and synapse are
<IT>V</IT><SUB><IT>soma</IT></SUB><IT>=</IT><IT>I</IT><SUB><IT>soma</IT></SUB>(<IT>Z<SUB>a</SUB></IT><IT>+</IT><IT>Z<SUB>c</SUB></IT>)<IT>+</IT><IT>I</IT><SUB><IT>syn</IT></SUB><IT>Z<SUB>c</SUB></IT> (1)
and
<IT>V</IT><SUB><IT>syn</IT></SUB><IT>=</IT><IT>I</IT><SUB><IT>soma</IT></SUB><IT>Z<SUB>c</SUB></IT><IT>+</IT><IT>I</IT><SUB><IT>syn</IT></SUB>(<IT>Z<SUB>b</SUB></IT><IT>+</IT><IT>Z<SUB>c</SUB></IT>) (2)

SOMATIC AND SYNAPTIC INPUT IMPEDANCES. The input impedance (ZN) at any point in a cell is proportional to the local change of Vm produced by injecting a current at that point, so we have the input impedances of the cell at the soma
<IT>Z</IT><SUB><IT>N</IT><SUB><IT>soma</IT></SUB></SUB><IT>=</IT>(<IT>V</IT><SUB><IT>soma</IT></SUB><IT>/</IT><IT>I</IT><SUB><IT>soma</IT></SUB>)<IT>‖</IT><SUB><IT>I</IT><SUB><IT>syn</IT></SUB><IT>=0</IT></SUB><IT>=</IT><IT>Z<SUB>a</SUB></IT><IT>+</IT><IT>Z<SUB>c</SUB></IT> (3)
and at the synapse
<IT>Z</IT><SUB><IT>N</IT><SUB><IT>syn</IT></SUB></SUB><IT>=</IT>(<IT>V</IT><SUB><IT>syn</IT></SUB><IT>/</IT><IT>I</IT><SUB><IT>syn</IT></SUB>)<IT>‖</IT><SUB><IT>I</IT><SUB><IT>soma</IT></SUB><IT>=0</IT></SUB><IT>=</IT><IT>Z<SUB>b</SUB></IT><IT>+</IT><IT>Z<SUB>c</SUB></IT> (4)
Equations 3 and 4 underscore the inequality of Za and Zb because input impedance at the soma is generally different from the input impedance elsewhere in a neuron. In fact, because the input impedance of most mammalian neurons is thought to be larger at nonsomatic locations than at the soma, these equations imply that Zb > Za in general (but see Magee 1998; Stuart and Spruston 1998).

TRANSFER IMPEDANCE. The transfer impedance (Zc) between two points is the change of Vm produced at one point by applying a current at the other. Referring to Fig. 1, it is immediately clear that transfer impedance is symmetric, i.e., it does not depend on which point is "upstream" and which is "downstream."
<IT>Z<SUB>c</SUB></IT><IT>=</IT>(<IT>V</IT><SUB><IT>soma</IT></SUB><IT>/</IT><IT>I</IT><SUB><IT>syn</IT></SUB>)<IT>‖</IT><SUB><IT>I</IT><SUB><IT>soma</IT></SUB><IT>=0</IT></SUB><IT>=</IT>(<IT>V</IT><SUB><IT>syn</IT></SUB><IT>/</IT><IT>I</IT><SUB><IT>soma</IT></SUB>)<IT>‖</IT><SUB><IT>I</IT><SUB><IT>syn</IT></SUB><IT>=0</IT></SUB> (5)

  VOLTAGE TRANSFER. From Eqs. 1-4, the voltage transfer ratio from the soma to the synapse is
<IT>k</IT><SUB><IT>soma→syn</IT></SUB><IT>=</IT>(<IT>V</IT><SUB><IT>syn</IT></SUB><IT>/</IT><IT>V</IT><SUB><IT>soma</IT></SUB>)<IT>‖</IT><SUB><IT>I</IT><SUB><IT>syn</IT></SUB><IT>=0</IT></SUB><IT>=</IT><IT>Z<SUB>c</SUB></IT><IT>/</IT>(<IT>Z<SUB>a</SUB></IT><IT>+</IT><IT>Z<SUB>c</SUB></IT>)<IT>=</IT><IT>Z<SUB>c</SUB></IT><IT>/</IT><IT>Z</IT><SUB><IT>N</IT><SUB><IT>soma</IT></SUB></SUB> (6)
and voltage transfer from the synapse to the soma is
<IT>k</IT><SUB><IT>syn→soma</IT></SUB><IT>=</IT>(<IT>V</IT><SUB><IT>soma</IT></SUB><IT>/</IT><IT>V</IT><SUB><IT>syn</IT></SUB>)<IT>‖</IT><SUB><IT>I</IT><SUB><IT>soma</IT></SUB><IT>=0</IT></SUB><IT>=</IT><IT>Z<SUB>c</SUB></IT><IT>/</IT>(<IT>Z<SUB>b</SUB></IT><IT>+</IT><IT>Z<SUB>c</SUB></IT>)<IT>=</IT><IT>Z<SUB>c</SUB></IT><IT>/</IT><IT>Z</IT><SUB><IT>N</IT><SUB><IT>syn</IT></SUB></SUB> (7)
The inequality of ZNsoma and ZNsyn guarantees that ksomaright-arrow syn not equal  ksynright-arrow soma. That is, unlike transfer impedance, voltage transfer is not symmetrical but instead depends on the direction of signal propagation. Furthermore, for most mammalian neurons one would expect that ksomaright-arrow syn > ksynright-arrow soma because ZNsyn > ZNsoma.

INTEGRATION OF SYNAPTIC INPUTS. The effect of synaptic location on the somatically observed PSP depends on whether the synapse acts more like a voltage source or more like a current source. If a synapse acts like a voltage source, the change of Vm that it produces in its immediate vicinity (Vsyn) is almost independent of synaptic location. According to Eq. 7, the PSP at the soma is given by
<IT>V</IT><SUB><IT>soma</IT></SUB><IT>=</IT><IT>k</IT><SUB><IT>syn→soma</IT></SUB><IT>V</IT><SUB><IT>syn</IT></SUB> (8)
Because Vsyn is relatively constant, most of the variation of the PSP observed at the soma with synaptic location must reflect regional variation of ksynright-arrow soma.

However, if the synapse acts more like a current source, then the time course and amplitude of the current it delivers to the cell will be relatively independent of synaptic location. From Eq. 5, the somatic PSP generated by a synaptic current is
<IT>V</IT><SUB><IT>soma</IT></SUB><IT>=</IT><IT>Z<SUB>c</SUB>I</IT><SUB><IT>syn</IT></SUB> (9)
This shows that the transfer impedance between the synapse and the soma is the principal determinant of the efficacy of a current source synapse. Therefore variability of somatic PSP amplitude with synaptic location is best described by the variation of Zc throughout the cell.

To gain a different perspective on this result, we use Eq. 7 to express Zc in terms of input impedance and voltage transfer, which have received more attention in experimental and theoretical publications
<IT>Z<SUB>c</SUB></IT><IT>=</IT><IT>k</IT><SUB><IT>syn→soma</IT></SUB><IT>Z</IT><SUB><IT>N</IT><SUB><IT>syn</IT></SUB></SUB> (10)
so
<IT>V</IT><SUB><IT>soma</IT></SUB><IT>=</IT><IT>k</IT><SUB><IT>syn→soma</IT></SUB><IT>Z</IT><SUB><IT>N</IT><SUB><IT>syn</IT></SUB></SUB><IT>I</IT><SUB><IT>syn</IT></SUB> (11)
Equation 11 tells us that the PSP observed at the soma when a current source synapse is activated will be a function of both the voltage transfer ksynright-arrow soma and ZNsyn, the input impedance of the cell at the synapse. If ZNsyn is relatively uniform throughout a cell, then Eqs. 10 and 11 imply that regional variations in ksynright-arrow soma will cause proportional fluctuations of both the transfer impedance Zc and the somatically observed PSP. However, if ZNsyn increases with distance from the soma, ksynright-arrow soma may decrease while both Zc and the somatic PSP remain relatively unaffected by synaptic location.


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Empirical dissociation of Zc and ksynright-arrow soma

To probe the relationship between Zc and dendritic geometry, we employed two simplified neuron models. The soma of the first model was bracketed by two cables that correspond to the apical and basal dendrites of a CA3 pyramidal neuron (Spruston et al. 1993). The second model was a modification of the first, with a tuft of 10 short daughter branches attached to the distal end of the apical cable. For convenience we will call these the "plain" and "tufted" models.

The spatial profile of ksynright-arrow soma for 20-Hz signals along the apical cable was identical in both models (Fig. 2A, top). However, ZN increased steadily with distance from the soma in the plain model, whereas it changed little along the apical cable of the tufted model (Fig. 2A, middle). Consistent with the relationship of ZN and ksynright-arrow soma to Zc (Eq. 10 in THEORY), Zc was relatively uniform across the apical dendrite of the plain model (Fig. 2A, bottom). This relative uniformity of Zc implies that a current applied to any dendritic location will produce nearly the same change in Vsoma, i.e., the somatic response to synaptic inputs is approximately normalized (Fig. 2B, top). We call this phenomenon passive normalization because it happens even though active currents are not present. In contrast to the plain model, the tufted model shows a significant decrease of Zc with distance so that passive normalization did not occur (Fig. 2B, bottom). Thus a dendritic geometry that leads to an increase of ZN with distance from the soma may prevent a steep decline of Zc and result in passive normalization.



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Fig. 2. Neuronal geometry affects synaptic integration. Simulations were performed on 2 models: a "plain" model of a CA3 pyramidal neuron adapted from Spruston et al. (1993), and a "tufted" model that included 10 short branches attached to the distal end of the apical dendrite. These models were analyzed for 20-Hz signals placed progressively farther from the soma on the apical dendrite. A, top: voltage transfer (ksynright-arrow soma) from any apical dendritic location to the soma was not affected by the presence of the tuft. Middle: in contrast, the profile of input impedance (ZN) varied little along the apical dendrite in the tufted model, whereas in the plain model there was a progressive increase in ZN with distance from the soma. Bottom: transfer impedance (Zc = ksynright-arrow somaZN) showed very little variation with location in the plain model, but when the apical tufts were present, Zc decreased with distance from the soma. Although the apical dendrites had the same electrotonic lengths in the Vin direction, the presence of the apical tufts affected the profile of Zc. B: the profile of Zc nearly predicts the amplitude of postsynaptic potentials (PSPs) at the soma. An excitatory conductance-change synapse (1 nS) was placed progressively along the apical dendrite of the plain (top) and tufted (bottom) models. Both Zc and Vsoma [peak excitatory PSP (EPSP) amplitude at the soma] are plotted normalized to an input onto the soma (&Zcirc;c and &Vcirc;soma, respectively). Insets show EPSPs from the most distal and proximal inputs.

The spatial profiles of Zc were spatially symmetric: somatodendritic Zc was equal to dendrosomatic Zc (simulations not shown). However, unlike Zc, voltage transfer was spatially asymmetric, with ksynright-arrow soma falling off more quickly with distance than ksomaright-arrow syn (simulations not shown). These symmetry properties confirm the predictions of two-port linear circuit theory (see prior section) (also see Cauller and Connors 1992; Koch 1999).

Does Zc actually predict the location-dependence of EPSPs produced by conductance-change synapses? To answer this question, a 1-nS peak conductance synapse (peak conductance latency ~2 ms) was sequentially placed along the apical dendrite of both models (Fig. 2B). In both cases, the profile of Zc normalized to an input onto the soma (&Zcirc;c) for 20-Hz signals closely followed the peak amplitude of the somatic PSP (&Vcirc;soma, again normalized to an input onto the soma) produced by the dendritic synapse. As predicted by the different profiles of Zc, the model without an apical tuft exhibited passive normalization.

Because of membrane capacitance, the attenuation of electrical signals in a neuron increases with frequency (Jack et al. 1975; Johnston and Brown 1983; Rall 1977; Spruston et al. 1993, 1994). Therefore we examined the profile of Zc in the apical dendrite of the plain model at several frequencies (Fig. 3A). As frequency increased, the absolute magnitude of Zc became smaller (Fig. 3A1), and at frequencies above 10-25 Hz its location-dependence grew progressively steeper (best seen in the plots of &Zcirc;c, Fig. 3A2). By 100 Hz, &Zcirc;c at the distal end of the apical dendrite was only ~60% of its peak value, a much greater reduction than the ~7% seen at 0 Hz.



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Fig. 3. Effects of varying frequency and specific membrane resistivity (Rm) on the location-dependence of transfer impedance. Simulations were performed on the 2-dendrite model of a CA3 pyramidal neuron without an apical tuft (see Fig. 2). A: the magnitude of Zc from any location to the soma decreased with signal frequency (A1). Signal frequency affected the location-dependence of Zc, as measured by &Zcirc;c (A2). Above 30 Hz there was more than a 20% difference in &Zcirc;c between the most proximal and distal input locations. B: decreasing Rm reduced the magnitude of Zc across the entire dendrite. Note, however, that reducing Rm from its baseline (50 kOmega cm2) by a factor of 5 resulted in less than a 25% decrease in &Zcirc;c at the most distal end of the apical dendrite.

The spatial profile of Zc was also sensitive to differences in Rm. Uniformly decreasing Rm from 50 to 1 kOmega cm2 reduced the magnitude of transfer impedance (Fig. 3B1) and accelerated its decay with distance from the soma (Fig. 3B2).

Because Zc governs the somatic response to synaptic current flow, these simulations imply that, when signals are of relatively low frequency (<25 Hz) and Rm is high (25-50 kOmega cm2), the amplitude of the PSP observed at the soma will vary significantly less with location than would be expected from ksynright-arrow soma.

Transfer impedance and voltage transfer in neurons

The simulations presented above suggest that the geometry of a neuron can have quantitatively different effects on the spatial profiles of ksynright-arrow soma and Zc, and in turn on the peak somatic response to a dendritic synaptic input (Vsoma). To further define the role of geometry, we compared the profiles of these electrotonic indices across five morphologically distinct cell classes, starting with CA1 and CA3 pyramidal neurons.

Figure 4 shows ksynright-arrow soma and Zc normalized to the soma (&Zcirc;c) for 20-Hz signals in a CA1 pyramidal cell model as functions of path distance to the soma. Points along the basal dendrites are shown at negative distances to distinguish them from apical locations.



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Fig. 4. Location-dependence of ksynright-arrow soma, Zc, and somatically observed EPSPs in a model CA1 pyramidal neuron. Values shown here and in Figs. 5 and 8-12 are for 20-Hz signals; qualitatively similar results were obtained at DC. A: reconstruction of a CA1 pyramidal neuron. Scale bar = 100 µm here and in Figs. 5 and 8-11. B: ksynright-arrow soma decreased rapidly with distance from the soma. This decrease continues out to the distal dendritic terminations, a fact that is somewhat obscured by the scale of the graph when ksynright-arrow soma < 0.1. C: &Zcirc;c vs. distance from the soma illustrating a >90% change in the apical dendrite, whereas there was less than a 40% difference for the basal dendrite. Note that the slope of terminal branches is nearly zero while most of the location-dependent decrease in &Zcirc;c occurs in the apical trunk. D: peak EPSP amplitude at the soma (Vsoma) resulting from a conductance change synapse (500 pS) sequentially placed at all dendritic locations. Similar results were obtained with 1-2 nS conductance synapses.

The voltage transfer ratio (Fig. 4B) fell rapidly below 0.2 within 200 µm of the soma and continued to decrease noticeably all the way to the distal dendritic terminations. Transfer impedance also decreased with distance from the soma (Fig. 4C). However, little additional decline was seen along terminal branches, which appear as nearly horizontal chains of points (the derivative of &Zcirc;c with respect to path distance is ~0). This means that a synaptic current that enters anywhere along a terminal branch will produce almost the same PSP at the soma regardless of its exact location on that branch. Most of the variation of &Zcirc;c in this cell occurred along the bifurcated primary apical dendrites, which can be discerned as two diagonal chains of points. Similar results were observed for three other CA1 pyramidal neuron reconstructions. This suggests that the primary apical dendrite sets the electrotonic coupling of synapses to the soma by virtue of anatomic distance along its length, whereas the side branches serve some other function, e.g., providing the surface area needed to receive multiple converging inputs that will all have the same relative effect on the soma.

As with the simplified models, an important question is how these properties condition the efficacy of conductance change synapses. To answer this question, we sequentially placed a 500-pS non-NMDA synapse (see METHODS) on different dendritic locations of the CA1 pyramidal neuron. The peak EPSP amplitude at the soma (Vsoma) produced by these dendritic inputs is illustrated in Fig. 4D. Like &Zcirc;c, Vsoma decreased as synapse distance increased, and there was little further change along terminal branches. The profile of EPSP amplitudes observed at the soma paralleled the profile of transfer impedance Zc. This means that the synaptic conductance amplitude and time course were such that these synapses acted more like current sources than voltage sources. Further evidence for this conclusion can be drawn from the peak EPSP amplitudes at the synaptic locations. For example, in one CA1 pyramidal neuron the mean peak EPSP amplitude at 200 randomly chosen dendritic synaptic locations was 5.2 ± 2.5 (SE) mV for 500-pS unitary inputs. Because resting potential was -65 mV in these simulations, this depolarization represents only ~8% loss of driving force. Finally, increasing the synaptic conductance from 500 pS to 2 nS to enhance the loss of driving force (mean peak EPSP amplitude at the same dendritic locations was 9.4 ± 4.2 mV) had no qualitative effect on the location-dependence of depolarization at the soma (simulations not shown).

The basal dendrites were an interesting feature of this cell type. Although the range of voltage transfer ratios from the basal dendrites to the soma was comparable with what was seen in the apical dendrites, there was much less variability of &Zcirc;c and Vsoma. Therefore these simulations predict that, from the standpoint of impact on Vsoma, synaptic locations in the basal dendritic field are significantly more isoefficient than inputs onto the apical dendrites.

For comparison, we analyzed seven CA3 pyramidal neuron models obtained from morphometric reconstructions. These cells also exhibited steep profiles of ksynright-arrow soma with distance from the soma (Fig. 5B). Another similarity was that high-order branches showed almost no change of &Zcirc;c or Vsoma along their length (Fig. 5, C and D). In contrast to CA1 pyramidal neurons, however, the location-dependence of &Zcirc;c and Vsoma throughout the cell was quite small (Fig. 5, C and D): they never fell below 80% of their maximum values. Thus, if a unitary synaptic current was placed at any point in the cell, there would be very little change in the amplitude (but not the shape) of the PSP detected at the soma. In addition, like the basal dendrites of CA1 pyramidal neurons, the basal dendrites of CA3 pyramidal neurons also exhibited very little variation in &Zcirc;c and Vsoma.



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Fig. 5. Location-independence of Zc in a CA3 pyramidal neuron. A: reconstruction of the CA3 pyramidal neuron. B: ksynright-arrow soma declined rapidly with distance from the soma to <0.2, indicating a >80% loss of signal from the most distal dendrites. C: in contrast to ksynright-arrow soma, &Zcirc;c varied by <20% over the entire dendritic tree. This result predicts that the depolarization at the soma produced by unitary synapses placed anywhere in the dendrites will vary by <20%. D: peak EPSP amplitude at the soma (Vsoma) elicited by a conductance change synapse (500 pS) sequentially placed at all dendritic locations. As predicted from &Zcirc;c, Vsoma varied by <20% across the dendritic tree.

Effects of varying frequency and Rm

We next examined the frequency dependence of &Zcirc;c between CA1 and CA3 pyramidal neuron geometries. As the preceding simulations suggested, the mean &Zcirc;c was higher for the CA3 neuron at all frequencies we examined (up to 100 Hz; Fig. 6A). However, the variance of &Zcirc;c was significantly lower for CA3 than for CA1 pyramidal neurons (Fig. 6B). This combination of larger mean (approaching 1) and a smaller variance of &Zcirc;c distinguishes the electrotonic architecture of CA3 pyramidal neurons from CA1 pyramidal neurons. Such reduced variability of synaptic amplitude, which emerges from the anatomic and basic passive properties of a cell without requiring the participation of active currents, is a hallmark of passive synaptic normalization. At frequencies below 10 Hz, the CA1 variance was almost four times greater. With increasing frequency, the CA3 variance also grew larger, so that the two cell classes showed practically identical variance at 100 Hz.



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Fig. 6. Effect of frequency on the spatial-distribution of Zc. A: mean &Zcirc;c from the CA1 and CA3 pyramidal neurons illustrated in Figs. 4 and 5 for frequencies from DC to 100 Hz. Mean &Zcirc;c for the CA3 pyramidal neuron was larger than for the CA3 pyramidal neuron at all frequencies tested. B: the standard deviation (sigma ) of &Zcirc;c in these same cells was used as measure of location-dependent variability. The variance of &Zcirc;c in the CA3 pyramidal neuron was smaller than in the CA1 pyramidal neuron for frequencies <= 50 Hz.

Assuming that Cm is similar in CA1 and CA3 pyramidal neurons, the apparent twofold difference in tau 0 between these cells implies a twofold difference in Rm (Spruston and Johnston 1992). To see how differences in Rm might contribute to the location-dependence of transfer impedance in these cells, we determined the minimum &Zcirc;c from three data sets (Fig. 7A). The first set was four CA1 pyramidal neurons reconstructed from sharp-electrode impaled cells filled with horseradish peroxidase (HRP). In the second set, models of CA3 pyramidal neurons (n = 4), also obtained from sharp-electrode/HRP fills, had significantly larger values of &Zcirc;c at the most distal locations. Varying Rm between 30 and 60 kOmega cm2 had no significant effect on &Zcirc;c for either CA1 or CA3 pyramidal neuron models (Fig. 7C). The third data set was from CA3 pyramidal neurons filled with biocytin via whole cell pipettes. Minimum values of &Zcirc;c for these cells were also significantly larger than for the CA1 pyramidal neurons, but not significantly different from CA3 pyramidal neuron reconstructions obtained from sharp-electrode fills. The results from these comparisons indicate that dendritic morphology, rather than Rm, is the major determinant of Zc between CA1 and CA3 pyramidal neurons.



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Fig. 7. Varying Rm and RN had little effect on the location-independence of Zc in CA3 pyramidal neurons. A: analysis of CA1 and CA3 pyramidal neuron models provided by Dr. Brenda Claiborne and CA3 pyramidal neuron models obtained by R. Chitwood and A. Hubbard in this laboratory (CH&J). The minimum &Zcirc;c for 20-Hz signals was used as a measure of location-dependent variability (low &Zcirc;c = high variability). The time constant (tau 0) of each model was set by changing Rm to either 30 or 60 kOmega cm2. Note that the minimum &Zcirc;c was significantly higher for CA3 pyramidal neurons (significantly less location-dependent variability) than for CA1 pyramidal neurons, regardless of the choice of Rm. B1: the anatomic features of the CA3 pyramidal neuron in Fig. 5 made it electrotonically quite compact for the wide range of Rm that we examined. Consequently, RN at the soma was very nearly a linear function of Rm. This linearity allowed &Zcirc;c to be plotted as a function of Rm and RN in a single graph (B2 of this figure). B2: the mean and standard deviation of &Zcirc;c for this cell, plotted here vs. Rm and the corresponding RN, are indicators of location-dependent variability: an increase in location-dependent variability is signified by a decrease in the mean and an increase in the standard deviation of &Zcirc;c. The experimentally observed RN for this cell at the soma was ~200 MOmega , which corresponded to a mean &Zcirc;c of ~0.9 and implies little location-dependent variability. Note that mean &Zcirc;c was >0.7 even if RN was reduced to 50 MOmega (25% of the measured value).

There is evidence to suggest that RN in vivo is smaller than the values observed in vitro because of differences in synaptic activity (Paré et al. 1998; Raastad et al. 1998). The effect of synaptically induced conductance changes on apparent Rm can be inferred from the magnitude of the change of RN, depending on the electrotonic architecture of the cell. Apparent Rm is nearly proportional to RN for an isopotential cell and proportional to the square of RN for an infinite cylinder. For real neurons the relationship between Rm and RN will lie between these two extremes. Thus a 50% decrease of RN implies a reduction of Rm by ~50-75% (Fig. 7B).

If Rm is smaller in vivo, then electrical signals should decay more rapidly with distance (Jack et al. 1975; Rall 1969; Spruston et al. 1993). Reducing Rm might also reasonably be expected to reduce the magnitude and alter the spatial profile of &Zcirc;c (see Fig. 3). To look for such an effect, we calculated the mean and variance of &Zcirc;c in a CA3 pyramidal neuron model for Rm ranging from 1 to 100 kOmega cm2. In Fig. 7B2, mean &Zcirc;c is plotted against Rm. Location-independence of &Zcirc;c was consistent over wide ranges of Rm, and therefore RN. Only when Rm (and in turn RN) was reduced by ~70% from its original value (66 kOmega cm2) were the mean and variance of &Zcirc;c decreased and increased more than 10%, respectively. Such an extreme reduction of RN is much larger than the synaptic effects observed in neonatal rat spinal cord by Raastad et al. (1998) and is at the upper limit of the findings reported by Paré et al. (1998) for pyramidal neurons in cat neocortex. It therefore seems unlikely that the reasonable differences between in vitro and in vivo empirical observations of RN will have significant effects on the location-independence of Zc in CA3 pyramidal neurons, or on the spatial profile of &Zcirc;c in other morphological cell types.

These simulations assume that the passive membrane properties are uniform throughout the dendritic tree. The possibility that the apparent Rm of a neuron is nonuniform was examined recently by Stuart and Spruston (1998) and Magee (1998). They found that RN in both CA1 and neocortical pyramidal neurons decreases progressively with distance from the soma due to a progressively higher density of hyperpolarization-activated channels (Ih) in the distal dendrites. Therefore we examined the profile of &Zcirc;c for models in which the decrease of Rm followed a sigmoidal function, whereas somatic RN and tau 0 were approximately the same as when Rm was uniform. This nonuniformity of Rm had no qualitative effect on the location-dependence of &Zcirc;c in CA1 pyramidal neurons or the relative location-independence of &Zcirc;c in CA3 pyramidal or nonpyramidal neurons. Figure 8 illustrates that profoundly different spatial distributions of Rm produced only slight variations in the spatial profile of Zc in a CA3 pyramidal neuron model.



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Fig. 8. Location-independence of Zc was unaffected by nonuniformity of Rm. The spatial profiles of Zc in a CA3 pyramidal neuron were computed for Rm that was uniform (left) or governed by the sigmoid function Rm(x) = {(Rmax - Rmin)/<RAD><RCD>1 + exp[(x − 134)/0.1]</RCD></RAD>} + Rmin (middle and right). The values of Rmax and Rmin at the proximal and distal ends were adjusted so that somatic RN and tau 0 were approximately the same as when Rm was uniform.

Other neuronal geometries

We next examined 15 CA3 nonpyramidal neurons. These cells have significantly fewer branches and higher RN than CA3 pyramidal neurons (Chitwood et al. 1999). They also have very different configurations of dendritic arbors compared to CA3 pyramidal neurons, although their electrotonic architectures display certain parallels, such as small variation of &Zcirc;c and Vsoma across the dendritic tree (Fig. 9, C and D) and similar decay of ksynright-arrow soma with distance (Fig. 9B; cf. Fig. 5B).



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Fig. 9. Location-independence of Zc in a CA3 nonpyramidal neuron. A: reconstruction of a CA3 interneuron. B: ksynright-arrow soma decreased >90% with distance from the soma. C: &Zcirc;c vs. distance from the soma. In this case, the least efficient synaptic inputs had 75% of the impact of the most proximal inputs. When Ri was reduced to 100 Omega  cm, the effect of the least efficient synaptic inputs on the soma was ~90% of the most proximal inputs (simulations not shown). D: peak EPSP amplitude at the soma (Vsoma) resulting from a conductance change synapse (500 pS) sequentially placed at all dendritic locations.

The morphology of deep layer cortical neurons differs greatly from hippocampal pyramidal neurons, being dominated by a very long (~1 mm) primary apical dendrite that ends in a distal tuft of higher order dendrites. From a reconstruction of a layer V pyramidal neuron in rat PFC, we analyzed ksynright-arrow soma, &Zcirc;c, and Vsoma. As in CA1 pyramidal neurons, ksynright-arrow soma decayed rapidly with distance (Fig. 10B). The smallest voltage transfer ratios were seen in the branches of the apical tuft. The spatial profiles of &Zcirc;c and Vsoma in these cells were steeper than for CA3 pyramidal and nonpyramidal neurons (Fig. 10, C and D). Like CA1 pyramidal neurons, the primary apical dendrite stood out distinctly from all other branches because of the steady decay of &Zcirc;c and Vsoma with distance along it. Secondary and higher order dendrites showed very little change in &Zcirc;c or Vsoma with location (slope ~0), a feature that was common to all neurons that we examined. The basal and oblique dendrites of these cells, like those of CA1 and CA3 pyramidal neurons, also showed very little variation in &Zcirc;c and Vsoma with location.



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Fig. 10. Location-dependence of Zc in a layer V pyramidal neuron from rat prefrontal cortex. A: the quantitative cell morphology was obtained by A. Gulledge in this laboratory. B: as in all neuron models in this paper, ksynright-arrow soma showed a severe decline with distance from the soma. C: &Zcirc;c vs. distance from the soma showing location-dependence of Zc in this neocortical neuron. The plot indicates that the primary apical dendrite is responsible for most of the location-dependence of Zc, as happened in CA1 pyramidal cells (Fig. 4). Like their counterparts in all the other neuron models, the terminal dendrites had slopes near zero indicative of isoefficiency along their length. D: peak EPSP amplitude at the soma (Vsoma) resulting from a conductance change synapse (500 pS) sequentially placed at all dendritic locations.

The last neuron we studied was a dentate gyrus granule cell (Fig. 11). This cell resembled the basal dendrites of hippocampal and neocortical pyramidal neurons in that ksynright-arrow soma decayed rapidly with distance, whereas &Zcirc;c and Vsoma showed considerably less variation (compare Fig. 11 with the basal dendritic fields of Figs. 4 and 10). Other parallels between the anatomic and electrotonic architectures of these cells and the basal dendrites of hippocampal pyramidal neurons have been noted by Carnevale et al. (1997).



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Fig. 11. Location-independence of Zc in a granule cell from the dentate gyrus. A: this cell was provided by Dr. Brenda Claiborne. B: the decline of ksynright-arrow soma with distance was almost as severe as in cells that were anatomically far more extensive. C: the profile of &Zcirc;c in this cell was similar to the profile of &Zcirc;c in the CA3 pyramidal and nonpyramidal neurons (Figs. 5 and 9). D: peak EPSP amplitude at the soma (Vsoma) resulting from a conductance change synapse (500 pS) sequentially placed at all dendritic locations.

Spatial variation of input impedance

ZN increases with distance from the soma and is maximal at the distal dendritic terminations (simulations not shown). This has been observed in lamprey spinal neurons (Buchanan et al. 1992) and suggested for a number of mammalian neuron types (Cauller and Connors 1992; Nitzan et al. 1990; Rapp et al. 1994; Segev et al. 1995). For every compartment of the five types of model neurons, we plotted the 20-Hz input impedance of that location versus the 20-Hz transfer impedance (Zc) between it and the soma (Fig. 12). In all of these cells, the greatest variation of ZN occurred along the terminal branches, which are easily discerned in these graphs. This is consistent with the previous simulations that showed Zc to be nearly constant along secondary and higher order branches. Most of the variation of Zc tended to occur along branches that were more proximal, especially in pyramidal cells. The existence of a primary apical dendrite is marked by the presence of a long region where ZN shows the least variation (CA1 and PFC layer V). This agrees with the hypothesis that a cable with uniform ZN will exhibit the most dramatic spatial variation of Zc (see INTEGRATION OF SYNAPTIC INPUTS in THEORY and see How ZN and Zc can normalize synaptic responses in DISCUSSION). In contrast, CA3 pyramidal neurons, CA3 nonpyramidal neurons, and granule cells do not have apical dendrites with relatively flat ZN profiles. Dendrites in these cells all had significant changes in ZN with Zc. Therefore it appears that the presence of a large primary dendrite is a prerequisite for significant location-dependent variability of Zc.



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Fig. 12. Relationship between input impedance (ZN) and Zc. ZN was plotted against Zc for 20-Hz signals from all compartments of a CA3 nonpyramidal, CA3 pyramidal, dentate gyrus granule, CA1 pyramidal, and layer V neocortical neuron. Terminal dendrites appear as almost vertical traces in these plots, which means that ksynright-arrow soma will be almost exactly inversely proportional to ZN along their lengths (see Eqs. 7 and 10). Note that the primary dendrites of the CA1 pyramidal neuron and the layer V neocortical pyramidal neuron are the only segments that exhibit relatively little change in ZN, and that these are also the ones that show the most variation of Zc.


    DISCUSSION
TOP
ABSTRACT
INTRODUCTION
METHODS
THEORY
RESULTS
DISCUSSION
REFERENCES

In general, proximal synaptic inputs tend to produce larger somatic PSPs than synapses that are distal but otherwise identical (Spruston et al. 1994). Proximal excitatory synapses are therefore more likely to trigger spikes, assuming a spike-generating zone in the vicinity of the soma (Colbert and Johnston 1996; Stuart and Sakmann 1994). The results of the simulations presented here are consistent with this rule of thumb in each of the cell classes that we studied. For example, a synapse onto a proximal dendrite of a hippocampal CA1 or PFC pyramidal neuron could elicit a PSP at the soma ~2.5-fold greater than an identical synaptic input located on one of the most distal branches. This estimate is based on unitary inputs; synapses with slower kinetics or trains of synaptic input would show somewhat less location-dependent variability.

It might therefore seem surprising that somatic depolarizations evoked by synaptic inputs onto proximal dendrites were only slightly larger than distally generated signals in CA3 pyramidal neurons, CA3 nonpyramidal neurons, and dentate granule cells. This unexpected finding is explained by the spatial profile of transfer impedance Zc. For example, in CA3 pyramidal neurons Zc for 20-Hz signals in the most proximal dendrites was only 1.1 times larger than for distal inputs. This suggests that a synapse onto a proximal dendrite would produce a somatic PSP that is only 10% larger than the PSP that would result from an identical synapse placed in the most distal locations. Simulations using a fast non-NMDA receptor-mediated synaptic conductance were consistent with this prediction. In other words, the anatomic and "passive" biophysical properties of the cell combine to remove most of the dependence of somatic PSP amplitude on synaptic location. The outcome is an approximate normalization of the impact of synaptic inputs on somatic Vm.

This is in contrast to the voltage transfer ratio ksynright-arrow soma, which could be more than fourfold larger for proximal synapses than for the most distal inputs. These simulations point out that analysis of voltage transfer alone does not reveal how unitary postsynaptic currents would affect somatic potential. When the amplitude and time course of a synaptic conductance are such that the synapse acts more like a current source than a voltage source, the transfer impedance Zc is a better indicator of the relative efficacy of synapses at different locations. The decision whether to use Zc or ksynright-arrow soma depends on both the anatomic and biophysical properties of the cell and the magnitude and time course of the Delta gs. A conductance change synapse will act like a current source when Delta gs is relatively small or of brief duration, so that Delta Vm in the subsynaptic region is only a small fraction of the driving force for current flow. Under this condition, the amplitude and time course of synaptic current will be relatively independent of synaptic location, and Zc is the better descriptor. However, if Delta gs is large and lasts long enough, the driving force for synaptic current will dissipate. This limits the peak amplitude of the PSP in the vicinity of the synapse, i.e., the synapse starts to behave more like a voltage source. In this case, ksynright-arrow soma is more appropriate.

To the best of our knowledge, this is the first systematic investigation of a mechanism for normalization of synaptic inputs that does not invoke active currents. Our study has the further distinction of documenting this phenomenon, which we call "passive synaptic normalization," in several different types of neurons through simulations of morphometrically detailed models. These simulations also demonstrate that there is very little effect of location on a given secondary or higher order dendrite, for any of the neurons we studied. A synapse placed anywhere along the length of a higher order dendrite will have comparable influence on somatic Vm.

The literature contains isolated reports of insensitivity of peak EPSP amplitude to synaptic location in previous studies of other cell types. For example, in an arbitrary fourth-order binary tree model extrapolated from a ball-and-stick representation of Cs+-filled retinal ganglion cells, Taylor et al. (1996) observed that somatic EPSP peak amplitudes were "essentially independent" of synaptic position. Segev et al. (1995), who were primarily concerned with the effects of spines in an anatomically detailed model of spiny stellate neurons, noted that EPSP amplitude at the soma was relatively independent of synaptic location in the dendritic tree. Models of presumed motoneurons in rat spinal cord slice cultures by Larkum et al. (1998) found that EPSP peak amplitudes were nearly uniform throughout the cell regardless of synaptic location, except for the immediate vicinity of the synapse where Vm showed a brief high-amplitude depolarization. Taken as a whole, these prior studies and our present findings offer good reason to suppose that synaptic normalization without the participation of active currents may be an important principle of synaptic integration that is as common as temporal summation.

This is also the first description of systematic differences in Zc between different classes of neurons. Other measures have been used to characterize and compare different classes of neurons, such as voltage attenuation (1/ksynright-arrow soma) and its logarithm (Carnevale et al. 1997). However, the importance of Zc has largely gone unnoticed (but see Cauller and Connors 1992; Koch 1999). This is most likely due to the fact that convenient tools for determining Zc and ZN for large-scale models (~3,000 compartments) have only recently been developed (Carnevale et al. 1997; Tsai et al. 1994).

How ZN and Zc can normalize synaptic responses

These observations demonstrate that, in some neurons, there is a broad class of synaptic inputs that can produce somatic PSPs whose amplitude is practically independent of synaptic location. How does this normalizing effect on somatically observed PSPs arise, and how do differences in neuronal geometry account for it?

The degree of synaptic normalization is a reflection of the spatial profile of the transfer impedance Zc, which in turn is related by the two-port theory of electrotonus to the spatial profiles of input impedance ZNsyn and voltage transfer ratio ksynright-arrow soma (Eq. 7). Along any dendritic branch, the voltage transfer ratio ksynright-arrow soma falls off with distance from the soma (Fig. 13, top). If ZNsyn is roughly constant along a dendritic branch, then in order to account for the decline of ksynright-arrow soma there must be a similar drop of Zc with distance (Fig. 13, middle), and synaptic normalization will not occur. This is what happens in a long cylindrical dendrite with no side branches (Fig. 2A) and in the primary apical branches of CA1 and PFC pyramidal cells (Figs. 4B and 10B). If instead ZNsyn increases rapidly enough with distance, the spatial profile of Zc will be much more shallow (Fig. 13, bottom), creating the conditions that allow passive synaptic normalization to happen.



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Fig. 13. Relationships between ksynright-arrow soma, ZN, and Zc. A: for any cell morphology, ksynright-arrow soma will decrease with distance from the soma. B: for a long, cylindrical cable, the profile of ZN will be relatively uniform. Therefore Eqs. 10 and 11 predict that Zc (and the depolarization produced at the soma) will vary substantially with synaptic location. This model is consistent with our simulations of the large apical trunks of CA1 pyramidal and neocortical pyramidal neurons. C: in higher-order dendrites of pyramidal neurons, the increase of ZN with distance from the soma can account for the decline of ksynright-arrow soma, whereas Zc remains relatively location-independent. The same effect occurs in CA3 pyramidal neurons, CA3 nonpyramidal neurons, and dentate granule cells, which do not have primary apical dendrites.

What dendritic properties affect the somatodentritic profile of ZN to permit passive normalization? One possibility is that the tapering of dendrites from proximal to distal terminations could lead to an increasing ZN in some cells (Holmes 1989) and account for passive normalization. This is in contrast to the relatively constant diameter of primary apical dendrites of CA1 and neocortical pyramidal neurons where there is significant location-dependence of synaptic responses observed at the soma. However, dendritic branching pattern is also an important determinant (see Fig. 2). The dendrites of the dentate granule cells, CA3 interneurons, and CA3 pyramidal neurons modeled here taper very quickly once they emerge from the soma and are of relatively constant diameter thereafter. For these neurons, the combination of their electronically short dendrites and specific branching pattern leads to the location-independence of Zc.

The similarity between granule cell dendrites and the basal dendrites of pyramidal cells provides a clue as to why the profiles of ksynright-arrow soma and Zc could be so different among the different types of neurons: these dendrites do not have extended primary branches. The apical dendritic fields of the PFC neuron and the CA1 pyramidal neuron are characterized by long, large-diameter primary branches that extend from the soma. Input impedance (ZN) showed only slight variation with distance along these primary apical dendrites (Fig. 12). Hence for a given synaptic current there was less variation in local PSP amplitude but greater variability in the EPSP observed at the soma (Fig. 13, middle). The proximal dendrites of granule cell, CA3 pyramidal and nonpyramidal neurons, and the basal dendrites of CA1 and PFC pyramidal neurons are much less prominent, and ZN increased much more rapidly with distance along them, so Zc was more uniform (Fig. 12). Consequently local EPSP amplitude would increase rapidly with distance from the soma, whereas the EPSP at the soma would show little change (Fig. 13, bottom). Finally, in all neurons, secondary and higher order branches showed significant increases in ZN (Fig. 12), whereas Zc in these dendrites was generally uniform with distance. Therefore a synaptic input placed at any location on such a dendrite would have roughly equal potency at the soma.

Physiological relevance

At this point it is useful to reflect on several questions that bear on the relevance of passive synaptic normalization to neuronal function. How might this phenomenon be affected by alterations of Rm produced by generalized synaptic bombardment? How does it vary with the frequency content of the synaptic signal? What about possible contributions from inward or outward voltage-gated currents, and how might it be influenced by nonuniform distributions of passive or active channels? Finally, what implications does synaptic normalization have for information processing in dendrites?

Paré et al. (1998) recently demonstrated that synaptic bombardment has significant effects on somatic RN in neocortical neurons. They concluded that in vitro measurements of RN may be up to ~70% higher than in vivo. However, when we decreased Rm so that RN was reduced by ~70%, the mean Zc fell by only 18% (Fig. 7C). Therefore synaptic bombardment that globally reduces ZN is likely to have only a nominal effect on Zc and the location-dependence of synaptic integration.

Signal frequency does affect Zc, and our findings (e.g., Fig. 3) lead us to expect that slow PSPs (i.e., NMDA or GABAB mediated) or the summed baseline of a burst of fast PSPs will show the greatest degree of uniformity throughout the cell. Single, fast PSPs should display greater variability with respect to dendritic location, with Vm showing a prominent peak in the near neighborhood of the synapse itself. However, outside of this narrow spatial zone, the voltage transient evoked by a fast PSP will have been slowed to the point where its time course approaches that of a slow PSP, and consequently its amplitude too will be relatively independent of location. This is supported by the observations of Larkum et al. (1998), who studied responses to synaptic currents that were much faster than the membrane time constant.

Voltage-gated inward current has been proposed as a mechanism for reducing synaptic variability due to location (Andersen et al. 1987; Cook and Johnston 1997, 1999). It is becoming apparent that many, if not all, mammalian CNS neurons have dendritic voltage-gated Na+ and Ca2+ channels (Christie et al. 1995; Jaffe et al. 1992, 1994; Magee et al. 1995; Magee and Johnston 1995a, b).

The simulations presented here show that extensive normalization may result from passive membrane properties alone. They also suggest that the most efficient way to supplement passive synaptic normalization would be for active Na+ and Ca2+ conductances to be concentrated in the primary dendrites of cortical or CA1 pyramidal neurons, where the largest changes in Zc occur, instead of throughout the dendritic tree. For neocortical neurons, it has been suggested that a high density or "hot spot" of these channels in the primary apical dendrite may amplify EPSPs (Schwindt and Crill 1995; Yuste et al. 1994). Otherwise, these signals would be significantly attenuated by passive electrotonus. In contrast, higher order dendritic branches and the dendrites of CA3 pyramidal neurons, nonpyramidal neurons, and granule cells do not need to have a high density of inward current channels to ensure that synapses at all dendritic locations have equal strengths at the soma.

Recent articles have highlighted possible functional roles of hyperpolarization-activated currents (Ih) in the dendrites of neocortical and hippocampal CA1 pyramidal neurons (Magee 1998; Stuart and Spruston 1998), and in particular how Ih may cause significant reduction of apparent Rm and dendritic RN. Although Ih may be quite prominent in these and other cells, and is unquestionably important for neuronal function, there are several reasons why it is unlikely to confound the synaptic normalization that we describe here.

First, its voltage-dependence and slow time course imply that Ih will have little effect on low- to moderate-amplitude EPSPs. Indeed, the results presented by Stuart and Spruston (1998) and Magee (1998) indicate that the principal action of Ih on somatically observed EPSPs is not to alter peak amplitude but instead to produce a temporal sharpening of the waveform, which compensates for much of the broadening caused by electrotonic filtering. So in a sense Ih improves the fidelity of the somatic response to the dendritic stimulation. This is particularly noteworthy because the underlying conductance (gh) has been estimated to increase from the soma to distal dendrites by anywhere from fivefold (Magee 1998) to three orders of magnitude (see Fig. 5 in Stuart and Spruston 1998). Here we should also point out that, despite the large estimated variation of gh, Magee's own data display a remarkable symmetry: the somatic response to a long dendritic current pulse was nearly identical to the dendritic response when the same pulse was applied at the soma, both in the absence and presence of bath-applied Cs+ (Fig. 9, A and B, of Magee 1998). This is exactly as predicted by two-port linear electrotonic theory, and it suggests that the notion of transfer impedance Zc may have practical value even when active currents make obvious contributions to the time course of Vm (Fig. 9A of Magee 1998).

Second, passive synaptic normalization is robust in the face of major reductions of distal dendritic Rm (Figs. 7C and 8). Like other rapid fluctuations of Vm, synaptic potentials are attenuated by ohmic loss via axial resistance (Ra) and the escape of signal currents through membrane capacitance (Cm). Because Ra and Cm are the principal determinants of signal attenuation, it is not surprising that nonuniformity of Rm has little effect on passive normalization. Furthermore, even if a distal Ih current was activated by a large, prolonged hyperpolarization, it would have to cause a quite profound increase of membrane conductance before its effect on normalization would be felt.

It should be noted that passive or active nonuniformities of apparent Rm would change the response of a cell to DC and slow inputs, but the synaptic normalization we describe involves transient signals, and so it is governed primarily by cytoplasmic resistivity and specific membrane capacitance. Our findings indicate that passive synaptic normalization will be altered only if local membrane time constant varies by at least an order of magnitude. In this connection, the studies of Stuart and Spruston (1998) and Magee (1998) suggest only a sevenfold and twofold reduction in Rm, respectively, between the soma and distal dendrites of neocortical and hippocampal pyramidal neurons.

The simulations presented here should not be taken to imply that dendrites are not important for local computations. To the contrary, they predict significant differences in local PSP amplitudes in the dendrites by virtue of regional variations of ZN, as Segev et al. (1995) also found in a model of a spiny stellate neuron. Therefore synapses onto distal dendrites are more likely to activate voltage-gated conductances than proximal inputs. This may explain why the density of certain K+ channels increases with distance from the soma, particularly those for A-type K+ currents (Hoffman et al. 1997). Such channels might compensate for differences in ZN, but the resulting compensation may be use-dependent; sustained depolarization, as may occur during bursts of EPSPs, might transiently inactivate these channels, briefly opening a window in which subsequent synaptic inputs are relatively boosted.

Passive normalization might at first glance seem to have limited importance for synaptic integration. After all, there is a substantial body of evidence that active currents can enhance synaptic efficacy and, under the proper conditions, may be responsible for synaptically triggered dendritic spikes (Golding and Spruston 1998; Schwindt and Crill 1997). We propose that passive normalization may actually play a much more widespread and important role in synaptic integration. This suggestion is based on the observation that low-amplitude fluctuations of Vm, presumably of synaptic origin, are commonly seen in many types of neurons, and action potentials generally appear to be triggered by these noiselike fluctuations. Mainen and Sejnowski (1995) have shown how such apparent "noise" can trigger action potentials reliably and with high temporal precision. Passive normalization, perhaps in combination with Ih to achieve some temporal sharpening of the somatic response as we suggested above, is an ideal mechanism for transforming locally high-amplitude postsynaptic signals, which are scattered widely over a cell but have limited range, into low-amplitude fluctuations of Vm at the soma. It ensures that all synapses have a nearly equal "vote" at the soma, regardless of where they are located in the dendritic tree.


    ACKNOWLEDGMENTS

We thank R. A. Chitwood, A. T. Gulledge, and A. Hubbard for the electrophysiology, histology, and 3D reconstructions that constituted a major portion of the neuron morphology used in this study. We also thank Dr. Brenda J. Claiborne for providing additional neuron morphology data and for assistance with 3D reconstruction.

This work was supported by National Science Foundation Grants IBN-9511309 and HRD-9628514 and National Institute of General Medical Sciences Grant GM08194-17S1.


    FOOTNOTES

Address for reprint requests: D. B. Jaffe, Division of Life Sciences, University of Texas at San Antonio, 6900 North Loop 1604 West, San Antonio, TX 78249.

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

Received 23 December 1998; accepted in final form 26 August 1999.


    REFERENCES
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ABSTRACT
INTRODUCTION
METHODS
THEORY
RESULTS
DISCUSSION
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0022-3077/99 $5.00 Copyright © 1999 The American Physiological Society