The Journal of Neurophysiology Vol. 78 No. 2 August 1997, pp. 703-720
Copyright ©1997 by the American Physiological Society

Comparative Electrotonic Analysis of Three Classes of Rat Hippocampal Neurons

Nicholas T. Carnevale^{1, 2}, Kenneth Y. Tsai^{1, 2}, Brenda J. Claiborne⁴, and Thomas H. Brown^{1, 2, 3}

¹ Center for Theoretical and Applied Neuroscience, ² Department of Psychology, and ³ Department of Cellular and Molecular Physiology, Yale University, New Haven, Connecticut 06520; and ⁴ Division of Life Sciences, University of Texas at San Antonio, San Antonio, Texas 78249

	ABSTRACT

Abstract Introduction Methods Results Discussion References

Carnevale, Nicholas T., Kenneth Y. Tsai, Brenda J. Claiborne, and Thomas H. Brown. Comparative electrotonic analysis of three classes of rat hippocampal neurons. J. Neurophysiol. 78: 703-720, 1997. We present a comparative analysis of electrotonus in the three classes of principal neurons in rat hippocampus: pyramidal cells of the CA1 and CA3c fields of the hippocampus proper, and granule cells of the dentate gyrus. This analysis used the electrotonic transform, which combines anatomic and biophysical data to map neuronal anatomy into electrotonic space, where physical distance between points is replaced by the logarithm of voltage attenuation (log A). The transforms were rendered as "neuromorphic figures" by redrawing the cell with branch lengths proportional to log A along each branch. We also used plots of log A versus anatomic distance from the soma; these reveal features that are otherwise less apparent and facilitate comparisons between dendritic fields of different cells. Transforms were always larger for voltage spreading toward the soma (V_in) than away from it (V_out). Most of the electrotonic length in V_out transforms was along proximal large diameter branches where signal loss for somatofugal voltage spread is greatest. In V_in transforms, more of the length was in thin distal branches, indicating a steep voltage gradient for signals propagating toward the soma. All transforms lengthened substantially with increasing frequency. CA1 neurons were electrotonically significantly larger than CA3c neurons. Their V_out transforms displayed one primary apical dendrite, which bifurcated in some cases, whereas CA3c cell transforms exhibited multiple apical branches. In both cell classes, basilar dendrite V_out transforms were small, indicating that somatic potentials reached their distal ends with little attenuation. However, for somatopetal voltage spread, attenuation along the basilar and apical dendrites was comparable, so the V_in transforms of these dendritic fields were nearly equal in extent. Granule cells were physically and electrotonically most compact. Their V_out transforms at 0 Hz were very small, indicating near isopotentiality at DC and low frequencies. These transforms resembled those of the basilar dendrites of CA1 and CA3c pyramidal cells. This raises the possibility of similar functional or computational roles for these dendritic fields. Interpreting the anatomic distribution of thorny excrescences on CA3 pyramidal neurons with this approach indicates that synaptic currents generated by some mossy fiber inputs may be recorded accurately by a somatic patch clamp, providing that strict criteria on their time course are satisfied. Similar accuracy may not be achievable in somatic recordings of Schaffer collateral synapses onto CA1 pyramidal cells in light of the anatomic and biophysical properties of these neurons and the spatial distribution of synapses.

	INTRODUCTION

Abstract Introduction Methods Results Discussion References

By governing the spread of electrical signals, electrotonic structure establishes the context for information processing in neurons. It conditions the global integration of synaptic inputs to drive spiking (Bekkers and Stevens 1990; Claiborne et al. 1992; Edwards et al. 1994; Holmes and Rall 1992; Jack et al. 1983; Rall 1977), it sets the extent of local interactions between synaptic inputs (Mainen et al. 1996; Shepherd and Koch 1990; Shepherd et al. 1989), and it is relevant to the voltage-dependent synaptic modifications that are thought to underlie certain types of learning (Brown et al. 1988, 1990-1992; Fisher et al. 1993; Kairiss et al. 1992; Kelso et al. 1986; Mainen et al. 1990, 1991; Tsai et al. 1994a). Any understanding of the consequences of active currents for neuronal function must take electrotonic structure into account, because it provides the framework within which the signals generated by active currents spread and interact (Gillessen and Alzheimer 1997; Jaffe et al. 1992; Lipowsky et al. 1996; Magee and Johnston 1995; Schwindt and Crill 1995; Stuart and Sakmann 1995); recent modeling (Mainen and Sejnowski 1966) suggests that electrotonic structure may be a critical determinant of neuronal firing patterns. Furthermore, electrotonic structure is important to the design and interpretation of experimental studies of synaptically mediated and voltage-gated currents and potentials (Barrionuevo et al. 1986; Carnevale et al. 1994; Cauller and Connors 1992; Claiborne et al. 1993; Jaffe and Brown 1994; Jaffe and Johnston 1990; Jaffe et al. 1994; Johnston and Brown 1983; Johnston et al. 1992; Kairiss et al. 1992; Mainen et al. 1996; Siegel et al. 1992; Spruston et al. 1993, 1994).

Although mammalian neurons can be classified on the basis of morphological differences, understanding the relevance of these differences to electrotonus requires that anatomy be interpreted in the context of biophysics. To this end, we have explored various new approaches to the problem of defining and analyzing the consequences of cellular anatomy and biophysics for electrical signaling in neurons (Brown et al. 1992; Carnevale et al. 1995a,b; O'Boyle et al. 1993, 1996; Tsai et al. 1993; 1994b; Zador et al. 1995). We have developed a method that combines these properties, mapping the branched architecture of a neuron into "electrotonic space" through a transformation that lends itself to graphic displays that provide a quick and intuitive grasp of the spread of current and voltage (Carnevale et al. 1995a; O'Boyle et al. 1996; Tsai et al. 1993, 1994b).

We implemented the transformation with a powerful, efficient algorithm that makes it practical to study a large number of cells with unprecedented resolution in frequency and space. Using this approach, we have analyzed electrical signaling in the dendritic trees of three classes of rat hippocampal neurons: CA1 pyramidal neurons, CA3c pyramidal neurons, and granule cells of the dentate gyrus. This comparative analysis disclosed striking contrasts and unexpected similarities between these cells that may have important implications for hippocampal operation. These findings also suggest new strategies for neuronal classification. By virtue of its close relationship to function, the electrotonic transformation may reveal useful insights to the organization of the brain that would remain undetected by methods based solely on morphological criteria.

	METHODS

Abstract Introduction Methods Results Discussion References

Anatomic reconstructions

Adult Sprague-Dawley rats were anesthetized deeply with pentobarbital sodium (Nembutal; 60 mg/kg body wt) and decapitated. The brain was removed quickly, and 400 µm-thick slices of the middle third of the hippocampal formation were prepared and maintained at 32°C in a recording chamber (Claiborne et al. 1986). Horseradish peroxidase (HRP) was injected into pyramidal neurons of the CA1 and CA3c fields of the hippocampus and granule cells of the dentate gyrus using previously described techniques (Claiborne 1992; Claiborne et al. 1986, 1990; Rihn and Claiborne 1990; Seay-Lowe and Claiborne 1992). Cells were impaled with sharp electrodes filled with 2-3% HRP in KCl/tris(hydroxymethyl)aminomethane buffer (pH 7.6). To decrease the chance of labeling neurons whose dendrites had been severed during the slicing process, only cells located in the middle of the slice were impaled. Neurons with a resting potential of at least -60 mV were injected with HRP using 3-5 nA positive current pulses with a 250-ms duration at a rate of 2 Hz for 20-25 min.

The slices were left intact during tissue processing. After an interval of 2-3 h for HRP diffusion, they were fixed and processed with diaminobenzidine for visualization (Claiborne et al. 1990). To minimize shrinkage, they were cleared in ascending concentrations of glycerol and mounted on slides in 100% glycerol. Slices prepared in this manner shrink by <5% in linear dimension, and the dendrites are not distorted (Claiborne 1992). Therefore no corrections were needed for shrinkage or "wiggle".

HRP was the label of choice for two reasons. First, it has been shown to fill hippocampal neurons in their entirety, including the finest dendritic branches (Claiborne 1992; Claiborne et al. 1990; Ishizuka et al. 1995). Second, the histochemical process required to visualize HRP can be done with intact thick slices so there is no need for resectioning. Thus the anatomic structure of an entire neuron can be analyzed directly from a slice whole-mount. Thinner slices are required for the histochemical reactions used to visualize biocytin (reaction with avidin coupled to HRP) or to produce a dense product from the fluorescent dye Lucifer yellow (reaction with antibodies coupled to HRP).

This study included seven pyramidal cells from the CA1 field, four pyramidal cells from the CA3c field, and six granule cells from the dentate gyrus. These cells were selected because they were well filled with HRP and had a minimum number of cut branches. Staining was uniformly dense throughout, with no fading toward the dendritic tips. The numbers of cut branches in each dendritic field of the pyramidal neurons were CA1: apical, 0-6 (average 3); basilar, 0-4 (average 1.8); CA3c: apical, 0-4 (average 1.6); basilar, 1 each. None of these cuts affected a proximal or "primary" branch. Granule cells were rejected if a dendrite in the proximal half of the molecular layer was cut or if two or more branches were severed in the distal two-thirds of the layer. Further confirmation of the anatomic integrity of these cells was provided by comparing their total dendritic lengths with values that have been reported previously for granule (Claiborne et al. 1990; Rihn and Claiborne 1990) and pyramidal (Ishizuka et al. 1995) neurons of rat hippocampus in studies using the same techniques; in all cases these lengths were well within the corresponding range.

The CA3 neurons we examined were from the CA3c field, the portion of CA3 that lies closest to the hilus of the dentate gyrus. Part of the mossy fiber projection from the dentate runs through the basilar region of this field, and the transition from CA3c to CA3b is approximately at the distal end of this infrapyramidal bundle (Lorente de No 1934). The morphology of pyramidal cells in CA3c is reportedly more heterogeneous than in CA3a or CA3b (Scharfman 1993). We selected CA3c because these cells are frequently the target of physiological investigations. They are of particular interest to the study of synaptic transmission in mammalian brain (Xiang et al. 1994) because their relative proximity to the dentate may favor the experimental isolation of a pure, monosynaptic mossy fiber input (Claiborne et al. 1993).

Camera lucida drawings of filled neurons were made using a ×63 objective (Zeiss Neofluar oil immersion, working distance 0.5 mm, NA 1.25) attached to a Nikon Optiphot microscope. Three-dimensional reconstructions of cells were obtained directly from the thick slices using a computer-microscope system designed by John Miller (University of California, Berkeley), with software written by Rocky H.W. Nevin (Claiborne 1992; Jacobs and Nevin 1991; Nevin 1989; Rihn and Claiborne 1990). The system consisted of a Nikon Optiphot microscope interfaced to an IBM AT computer that controlled motors mounted on both the microscope stage and the focus-control knob. Accurate positioning of the stage was ensured by optical encoders capable of 0.2 µm resolution. Labeled neurons were digitized in three dimensions by an operator using a computer mouse. A video camera was mounted on the microscope and the dendrites were viewed on a monitor. Diameter measurements were taken from a reference cursor superimposed over the dendrite on the monitor (Rihn and Claiborne 1990). Each datum included XYZ coordinates and a diameter measurement. Further details are provided elsewhere (Claiborne 1992; Claiborne et al. 1990).

The effect of dendritic spines on cell electrical properties is often compensated by adjusting surface area or membrane properties based on spine dimensions and density (Cauller and Connors 1992; Claiborne et al. 1992; Stratford et al. 1989). However, a significant variation of spine density with dendritic diameter recently has been reported in CA1 pyramidal neurons (Bannister and Larkman 1995), and significant if less striking variation long has been recognized in granule cells (Desmond and Levy 1985). Furthermore, even within a single cell class there can be a wide range of spine dimensions (Chicurel and Harris 1992; Desmond and Levy 1985; Harris et al. 1992), so it is unclear how large this compensation should be. In addition, our laboratories recently have been exploring the use of confocal scanning laser microscopy to improve the accuracy of diameter measurements (O'Boyle et al. 1993, 1996). Diameters tend to be overestimated by as much as 0.5-1.0 µm when standard light microscopic techniques are applied to thick slices (O'Boyle et al. 1993). The resulting increase of apparent surface area amounts to ~1.6-3.1 µm²/µm length, which brackets the weighted estimate of 2.85 µm²/µm that we previously derived (Mainen et al. 1996) from measurements in CA1 pyramidal neurons reported by Harris et al. (1992). Therefore in this study, we made no alterations in membrane properties or measured diameters and thereby accomplished a partial compensation for the effect of dendritic spines. This seemed preferable to compounding the uncertainties of spine density and diameter measurement by applying estimated correction factors that are themselves uncertain.

Electrotonic analysis

Because of the central importance of R_i and R_m to the construction of the transforms, we based the values we used on the results of Spruston and Johnston (1992), who exercised great care to obtain measurements that were as physiological and as accurate as possible. The passive electrical properties were R_i = 200  cm, C_m = 1 µF/cm² for all three cell classes, and R_m = 30 k cm² for CA1 pyramidal cells, 70 k cm² for CA3 pyramidal cells, and 40 k cm² for granule cells. As we have noted elsewhere (Mainen et al. 1996), these should be regarded as "linearized" rather than "passive" parameters, because the defining experiments did not employ channel blockers or attempt to inactivate currents and membrane potential changes were kept within the linear range of the cells' current-voltage relationships (Spruston and Johnston 1992). Therefore these parameter estimates are a linearized approximation of all active and passive mechanisms that contributed to the total clamp current for potential fluctuations within ~5 mV of resting potential.

Transforms were performed at several frequencies because of the frequency dependence of attenuation. For the purpose of illustration, we show the transforms at DC (0 Hz) and 40 Hz.

The cells were mapped into electrotonic space by computing the attenuation of voltage for signal spread away from (V_out) or toward (V_in) the soma. This mapping or transformation is presented from a more theoretical standpoint elsewhere (Tsai et al. 1994b). The following sections briefly review the transformation and previously undescribed computational strategies that enable its practical application.

Signal attenuation in neurons

The principles that underlie our analytic strategy derive from the application of two-port network theory to linear electrotonus by Carnevale and Johnston (1982). Their use of two-port theory was motivated by the fact that the spread of electrical signals in a neuron is best described in terms of the efficacy of signal transfer. Three principal conclusions of their work have a major bearing on our new approach: the direction-dependence of signal transfer, the identity of current and charge transfer, and the directional reciprocity between the transfer of voltage and the transfer of current and charge.

SIGNAL TRANSFER IS DIRECTION-DEPENDENT. Carnevale and Johnston (1982) described the loss of amplitude suffered by a signal that propagates through a neuron with a factor k. This factor was always 1 because it was the ratio of the "downstream" (output) amplitude to the "upstream" (input) amplitude. The electrotonic transform uses the inverse of this ratio because it leads to a natural definition of electrotonic distance.

For any two points i and j in a cell, if a voltage V_i applied at upstream location i produces a voltage V_j measured at location j, we define the voltage attenuation to be A^V_ij = V_i/V_j. If the direction of propagation is reversed, so that j is upstream relative to i, the voltage attenuation is A^V_ji = V_j/V_i. Because of the direction dependence of signal transfer (Carnevale and Johnston 1982), these attenuations will generally not be equal

(1)

The degree of inequality depends on factors such as anatomic asymmetry, regional variation of biophysical properties, and the locations of i and j.

Current and charge attenuation are also direction dependent. Suppose a current I_i enters the cell at i, and a voltage clamp is attached to the cell at j. The current attenuation A^I_ij is the ratio of I_i to the current I_j measured by the clamp (A^I_ij = I_i/I_j). If the sites of current entry and clamp attachment are exchanged, the current attenuation is A^I_ji = I_j/I_i. As with voltage attenuations (Eq. 1), the direction-dependence of signal transfer implies that

(2)

CURRENT AND CHARGE TRANSFER ARE IDENTICAL. Carnevale and Johnston (1982) showed that the propagation of charge Q and current are equally efficient

(3)

DIRECTIONAL RECIPROCITY OF VOLTAGE AND CURRENT/CHARGE TRANSFER. The third and final conclusion of two-port theory that we need here is the fact that voltage transfer in one direction is identical to the transfer of current and charge in the opposite direction

(4)

Mapping from anatomic to electrotonic space

Equations 1 and 2 show that a complete description of electrotonus in a neuron requires knowledge of the attenuation of electrical signals along each branch of the cell in two directions. The identity of current and charge transfer (Eq. 3) and the directional reciprocity between voltage and current transfer (Eq. 4) imply that electrotonus could be specified equally well in terms of the attenuation of voltage or current. However, voltage attenuation is the most pragmatic choice because of the central importance of membrane potential to neuronal function.

Starting from these premises, we have shown that the electrotonic structure of a neuron is defined completely by two sets of voltage attenuations: the attenuation of voltage as it propagates away from and toward a reference location (Brown et al. 1992; Tsai et al. 1994b). We then advanced a new definition of electrotonic distance, the logarithm of attenuation, which is a metric for mapping the architecture of the cell from anatomic to electrotonic space (Brown et al. 1992; Tsai et al. 1994b).

We combine detailed, accurate morphometric data with the best available estimates of the biophysical properties of membrane and cytoplasm to calculate these attenuations at DC and several frequencies of interest along each of the branches of a cell. This accomplishes a partial mapping of physical space into electrotonic space.

The next step is to organize these attenuations around a reference location. For each point of interest in the cell, we must find the total attenuation for voltage signals propagating away from and toward the reference location. Figure 1 illustrates how this is done. The endpoints of two adjacent branches are labeled as i, j, and k, where j is the junction between the two branches. From the anatomy and biophysics of this cell, we already have computed the attenuations along these two branches for the two directions of signal flow: A_ij and A_jk (Fig. 1, left), and A_kj and A_ji (Fig. 1, right).

View larger version (9K): [in this window] [in a new window]   FIG. 1. The total attenuation over path ik is the product of the attenuations along the branches ij and jk. See text for details.

Suppose i is the reference location. Then the total attenuation for voltage spreading from i to k is the product of the attenuations along each branch that lies on the direct path starting at i and ending at k. Because there are only two branches along this path, we have

(5)

We say that A_ik is part of the V_out transform with respect to reference location i. The product of the attenuations along this same path, but in the opposite direction, gives the total voltage attenuation from k to i

(6)

where A_ki is part of the V_in transform with respect to i.

The extension to cases where the direct path from the location of interest k to the reference location i involves more than two branches is straightforward. For voltage propagating in one direction along path ik, the total attenuation equals the product of the attenuations of voltage propagating along the intervening branches in the same direction.

To summarize, the transformation from anatomic to electrotonic space is started by computing the attenuations in both directions along each branch of the cell and completed by multiplying these attenuations in proper combination and order so as to find the total attenuation between the reference location and each point of interest in the cell. Any point in the cell could be used as the reference, but the soma is generally a good choice. With a somatic reference, the V_out transform reveals the influence of somatic potentials on voltage-dependent mechanisms of synaptic plasticity in the dendrites, and the V_in transform suggests the ability of dendritic synaptic inputs to drive spiking at the cell body (Brown et al. 1990-1992; Fisher et al. 1993; Kairiss et al. 1992; Mainen et al. 1990, 1991; Tsai et al. 1994a,b). A nonsomatic reference may be more useful for studies of interactions among dendritic synaptic inputs (Carnevale et al. 1995a).

A NEW DEFINITION OF ELECTROTONIC DISTANCE. Long lists of numbers, such as tables of morphometric data or signal attenuations, are ill-suited for human use. A graphic representation is a better vehicle for communicating a large body of information in a way that fosters the rapid development of qualitative insights. For example, morphometric data can be rendered as two-dimensional projections that portray the anatomy of the cell, emulating the traditional microscopic images obtained by camera lucida drawings or photography. The length of each branch in such a figure is related directly to the physical distance between corresponding branch points in the cell.

The key to developing an intuitive graphic depiction of the electrotonic architecture of a neuron is to define a measure of "electrical distance" that expresses the signal attenuation between locations in the cell in a consistent manner. Then points widely separated in electrotonic space would correspond to anatomic locations that are poorly coupled to each other (large attenuations), whereas points that are adjacent would correspond to sites that are nearly isopotential (attenuations close to 1).

To this end we have advanced a new definition of electrotonic distance as the natural logarithm of voltage attenuation (Brown et al. 1992; Tsai et al. 1994b). Like attenuation, this electrotonic distance L is direction dependent. That is, each pair of anatomic locations i and j is associated with two different electrotonic distances: L_ij = ln A_ij for signal spread from i to j and L_ji = ln A_ji for the opposite direction. At every frequency of interest, each branch of the cell has two representations with different lengths in electrotonic space.

Our definition of L has the special property that a cascade of attenuations translates into a sum of distances in electrotonic space. In other words, electrotonic distances are additive over a path that has a consistent direction of signal propagation. Thus if location j is on the direct path between locations i and k, as in Fig. 1, then L_ik = L_ij + L_jk and L_ki = L_kj + L_ji. This unique property is a consequence of Eqs. 5 and 6 and the definition of L as the logarithm of attenuation (Carnevale et al. 1995a; Tsai et al. 1994b). As we note in the next section, it allows L to be used as the metric for graphic representations of mappings from anatomic to electrotonic space.

GRAPHIC RENDERINGS OF THE ELECTROTONIC TRANSFORMATION. For the electrotonic transform to be useful, its results must be presented in a form that is functionally relevant and easily understood. This requires rendering the electrotonic distances in convenient graphic forms.

The most intuitive graphic representations are the "neuromorphic figures" (Brown et al. 1992; Carnevale et al. 1995a; Tsai et al. 1993, 1994b), in which the branching pattern of the cell and the relative orientations of the branches are preserved but the physical branch lengths are replaced by segments that are proportional to their electrotonic lengths. These are generated in pairs, one image using the electrotonic lengths of the branches for voltage spread away from the reference location (V_out) and the other using the electrotonic lengths for voltage spread toward the reference location (V_in). Because attenuation also depends on frequency, we generate a pair of these graphs at each frequency of interest. Because of the directional reciprocity of voltage and current or charge attenuation, the renderings of V_out and I_in transforms are identical, as are the renderings of V_in and I_out transforms.

An alternative rendering plots the electrotonic distance L = ln A as a function of physical distance x from the reference location (O'Boyle et al. 1996). This enables convenient evaluation of synaptic inputs that have a laminar organization and reveals the spatial voltage gradient along neurites clearly. As with the neuromorphic figures, these "log A versus x" plots are generated in pairs, one for voltage propagation away from (V_out) and the other for voltage propagation toward (V_in) the reference location.

The voltage attenuation between any two points in the cell can be found by combining the appropriate segments of the somatocentric V_in and V_out transforms (Tsai et al. 1994b). Regardless of what reference location s we initially select for the V_in and V_out transforms, the additive property of L makes it easy to generate the transforms for any other reference location w. The only difference between using s or w as a reference is in the direction of signal propagation in the branches along the direct path between these two points, where V_in relative to s is the same as V_out relative to w and vice versa. Changing the reference location does not affect the direction of signal flow in the remainder of the cell, so the attenuations along all other branches and their corresponding representations in electrotonic space are unaltered. The additive property of L is responsible for this simple relationship. Without it, generating the transforms for a new reference location would require a laborious recalculation of all the mappings from anatomic to electrotonic space.

Our approach to electrotonic analysis differs in several important ways from the traditional equivalent cylinder method. First, our definition of electrotonic distance L as the logarithm of attenuation contrasts strongly with the conventional definition of electrotonic length X as the ratio of the physical distance x to the space constant  (Jack et al. 1983; Rall 1977). The classical X lacks the additive property that makes L so useful for graphic representations of attenuation over a chain of dendritic branches. Furthermore, attenuation is a simple exponential function of L, whereas its variation with X is much more complicated and depends on boundary conditions (Jack et al. 1983; Rall 1977). Attenuation is an exponential function of X only in the case of an infinitely long cylindrical cable with uniform biophysical properties. Finally and, perhaps most importantly, the electrotonic transform encodes both anatomic and electrophysiological data, so it does not require collapsing the cell into an equivalent cylinder and hence does not destroy the anatomic relationships among synaptic inputs distributed throughout the dendritic tree (see Mainen et al. 1996). Like our definition of electrotonic length L, the transform is directly applicable to any architecture.

THE TRANSFORM ALGORITHM. In principle, voltage attenuations can be determined by computing the distribution of potential in response to an applied signal using a simulator such as NEURON (Hines 1984, 1989, 1993, 1994), and this is what we did initially. However, this approach is feasible only for the DC V_out transform. Simulation run times for non-0 frequencies were excessively long because many cycles had to pass before the response settled: a single run to find the V_out attenuations at 40 Hz took >20 h on a SUN Sparc 10-40 (Tsai et al. 1994b) compared with a few seconds for DC. Computing a full set of V_in attenuations would require a separate simulation using a signal applied to each terminal dendritic branch in turn. This was out of the question because the pyramidal cells in this study have ~100 terminations each; so finding the V_in transform at one frequency for a single cell would have taken ~2000 h (almost 3 mo).

For this reason, we developed a new program that computes the V_in and V_out attenuations in 2 s per frequency of interest (Tsai et al. 1994b). This program achieves its speed by operating in the frequency domain rather than the time domain, exploiting the branched architecture of a neuron to compute attenuations by a multipass recursive strategy.

The user specifies the file that contains the morphometric data, the frequencies at which the attenuations are to be calculated, and the biophysical properties of the membrane and cytoplasm. The program then reads the morphometric data and builds a model of the cell that consists of a branched tree of cylindrical segments. The internal representation of the architecture of the cell and the anatomic and electrical properties of each segment is in the form of a doubly linked binary tree (Sobelman and Krekelberg 1985; Wirth 1976).

The fundamental operation of the program is the repeated application of Kirchhoff's laws (Kuo 1966) to the equivalent circuit of the cell. This relies on the fact that each cylindrical segment can be represented by an equivalent T circuit (Fig. 2) with transverse impedance Z_m and axial impedances Z_a (Carnevale and Johnston 1982).

View larger version (12K): [in this window] [in a new window]   FIG. 2. Top and middle: each unbranched dendritic segment can be described by an equivalent T circuit consisting of one transverse and two axial impedances (Carnevale and Johnston 1982). If the diameter is constant and the electrical properties of the membrane and cytoplasm are uniform over the segment, then the axial impedances (Za) are symmetric as shown here. However, the conclusions from two-port theory still apply even if the axial impedances are not symmetric. Bottom: voltage attenuation V0/V1 over a cylindrical segment depends on the axial (Za) and transverse (Zm) impedances and any load impedance (Zload) at the downstream end of the segment. For voltage spread away from the cell body, Zload includes the somatofugal input impedances of any daughter branches. For voltage spread toward the cell body, Zload includes the somatopetal input impedance of the parent and the somatofugal input impedance of any sibling branches.

The program performs a series of recursive passes through the binary tree. Some of these passes could be combined to maximize computational efficiency, but for the sake of clarity, we present them separately. On the first pass, the morphometric and biophysical data are used to compute Z_a and Z_m for each segment at the frequency of interest. The values of these impedances may be approximated by simply lumping the properties of the membrane and cytoplasm

(7)

(8)

where d and l are the segment diameter and length, _m = R_mC_m is the membrane time constant, j =  and  is the frequency in radians/second. These formulas are adequate for DC and low frequencies. At frequencies where the physical length of the segment exceeds 5-10% of the AC length constant  = /2, accuracy requires using these functions which we derived from cable theory (Tsai et al. 1994b)

(9)

(10)

where r = 2/d^3/2 is the DC input resistance of a semi-infinite cylindrical cable of diameter d, x is the physical length of the branch, and  = /2 is the DC length constant.

The next four passes are illustrated through reference to the bottom of Fig. 2. In pass 2, the somatofugal input impedance at the proximal end of each segment is calculated (i.e., looking away from the soma). If the segment is a terminal branch, this is simply Z_out = Z_a + Z_m. Otherwise, the load imposed by distal segments must be included so Z_out = Z_a + Z_p, where Z_p = Z_m(Z_a + Z_load)/(Z_m + Z_a + Z_load), and Z_load is the parallel combination of the Z_out impedances of the daughter branches. Pass 3 calculates the V_out attenuation along each branch as V₀/V₁ = (1 + Z_a/Z_p)(1 +Z_a/Z_load).

The fourth pass is similar to the second, but it starts at the soma working outward to compute the somatopetal input impedance Z_in at the distal end of each segment (looking toward the soma). For signals propagating in this direction, the load is the (somatopetal) Z_in of the parent in parallel with the (somatofugal) Z_out of any sibling branch. The same equation that was used in pass 3 is applied in the fifth pass to find the V_in attenuation.

In the final pass through the tree, the attenuations are written to the output. Afterward, a new set of attenuations is computed at the next frequency of interest.

As noted above, this program evaluates attenuations several orders of magnitude faster than is possible with time-domain simulations using software such as NEURON, GENESIS, or SPICE. Computation time for our algorithm is O(N) where N is the number of compartments in the neuron model, i.e., it scales linearly with anatomic complexity. Both run time and accuracy are independent of frequency.

The power and efficiency of our program arise from two factors. The first is our use of a recursive algorithm that exploits the branched topology of the cell. Superficially, matrix methods using an upper-diagonalization-backsubstitution scheme (Carnevale and Lebeda 1987) adapted for sparse, nearly tridiagonal systems of equations might seem conceptually different, but they are computationally equivalent. A more general approach that resorts to formal matrix inversion would be inferior because the inverse of a sparse matrix is typically highly nonsparse (Jennings 1977).

The second factor that enhances computational efficiency is our strategy for circumventing the effect of increasing frequency on the effective length constant. As we pointed out elsewhere (Tsai et al. 1994b), the length constant shortens considerably starting at frequencies near f_m = 1/2_m. This frequency is quite low for hippocampal principal neurons (~5.3 Hz for CA1 cells, ~2.3 Hz for CA3 cells, and ~4 Hz for granule cells). Approaches that lump the properties of membrane and cytoplasm into simple RC equivalents must resort to smaller compartmental size to preserve accuracy at higher frequencies (Oran and Boris 1987). Using complex impedance functions derived from cable theory eliminates the need to reduce compartmental size as frequency increases.

We have evaluated the accuracy of this program by comparing its predicted attenuations with simulations performed with NEURON (Hines 1984, 1989, 1993, 1994) using an anatomically and biophysically realistic model of a CA1 neuron. In all cases, the results agreed within 0.02%.

AVAILABILITY OF ANALYTIC TOOLS. The neural simulation environment NEURON (Hines 1993, 1994) now includes the Electrotonic Workbench (Carnevale et al. 1996), a set of analytic tools that provide a convenient way to perform the electrotonic transformation. The Electrotonic Workbench is fast and efficient because it operates in the frequency domain. Because NEURON is freely available and runs under the three leading operating systems (UNIX and its variants, all varieties of MS Windows, and the Mac OS), the transformation is maximally accessible to interested members of the neuroscience community. It can be obtained via the WWW at http://www.neuron.yale.edu and http://neuron.duke.edu

Statistical analysis

Anatomic distances and electrotonic lengths are reported as sample means ± SD (Table 1). Three between-class comparisons were carried out for each measure of anatomic or electrotonic extent, testing the null hypothesis that population means were equal. We used the protected t-test (Howell 1995), which is also known as Fisher's least significant difference (LSD) test, to avoid the increased risk of Type I errors that can occur when multiple comparisons are performed with the ordinary t-test. Before performing the protected t-test, we first calculated the overall F statistic for each measure.

	RESULTS

Abstract Introduction Methods Results Discussion References

The anatomic and electrotonic architectures of representative hippocampal neurons are shown in five pairs of figures: two pairs for each of two CA1 pyramidal cells (Figs. 3-6), one pair for a CA3c cell (Figs. 7 and 8), and two more pairs for two granule cells (Figs. 9-12). The first figure of each pair presents the "raw" anatomy of a cell on the left, with the neuromorphic renderings of its DC V_out (somatofugal) and V_in (somatopetal) transforms on the right (Figs. 3, 5, 7, 9, 11). The second of each pair shows the log A versus x plots of the transforms (Figs. 4, 6, 8, 10, and 12). The basilar dendrites of the pyramidal cells are plotted in Figs. 4, 6, and 8 at "negative" anatomic distances from the soma. The last figure (Fig. 15) compares log A versus x plots of a granule cell and the basilar dendrites of a CA1 pyramidal cell.

View larger version (19K): [in this window] [in a new window]   FIG. 3. Left: two-dimensional projection of the anatomy of a CA1 pyramidal neuron (cell 524). Right: neuromorphic renderings of the electrotonic transforms of this cell computed at DC (0 Hz) and 40 Hz for somatofugal (Vout, top) and somatopetal (Vin, bottom) signal flow. The primary apical dendrite dominates the Vout transforms, while the basilar and terminal branches appear much smaller. In the Vin transforms the basilars and terminal branches are the most prominent features.

View larger version (18K): [in this window] [in a new window]   FIG. 4. Plots of the logarithm of attenuation at DC as a function of physical distance from the soma (log A vs. x) for the Vout (left) and Vin (right) transforms of the CA1 neuron of Fig. 3. Positive distances along the x axis correspond to the apical dendrites, and the basilars are shown at negative distances. For Vout, the primary apical dendrite stands out as a diagonal that gives rise to many tributaries that are almost horizontal (the nearly isopotential terminal branches). In the Vin transform these branches are much steeper than the primary apical because of the rapid attenuation of voltage along their length.

View larger version (22K): [in this window] [in a new window]   FIG. 5. Left: two-dimensional projection of the anatomy of a CA1 pyramidal neuron (cell 503). Right: Vout (top) and Vin (bottom) neuromorphic figures at DC and 40 Hz. The primary apical dendrite bifurcates close to the cell body, giving rise to a pair of branches that dominate the Vout transforms. The basilar and terminal branches, nearly inconspicuous in the Vout transforms, are the most noticeable aspects of the Vin transforms.

View larger version (20K): [in this window] [in a new window]   FIG. 7. Left: two-dimensional anatomic projection of a CA3c pyramidal neuron (cell 701). Right: Vout and Vin neuromorphic figures at DC and 40 Hz. This cell has several roughly equivalent apical dendritic branches instead of a single primary apical dendrite. As in the CA1 cells of Figs. 3 and 5, however, the basilar and terminal branches of this cell are very small in the Vout transforms yet quite prominent in the Vin transforms.

View larger version (19K): [in this window] [in a new window]   FIG. 8. Log A vs. x plots at DC for the Vout and Vin transforms of the CA3c neuron in Fig. 7. The four steep diagonals correspond to the cluster of proximal apical branches in Fig. 7. The slopes of the terminal branches and basilar dendrites depend on the direction of signal propagation, as in Figs. 4 and 6.

View larger version (16K): [in this window] [in a new window]   FIG. 9. Left: two-dimensional anatomic projection of a granule cell (cell 964). Right: Vout and Vin neuromorphic figures at DC and 40 Hz. Granule cells are electrotonically more compact than either CA1 or CA3 neurons. The dendritic branches have nearly identical electrotonic lengths.

View larger version (17K): [in this window] [in a new window]   FIG. 11. Left: two-dimensional anatomic projection of a granule cell (cell 950). The soma of this neuron lies in a deeper layer of the dentate gyrus than the cell displayed in Figs. 9 and 10. It has a short apical branch that gives rise to the remainder of the dendritic tree. Right: Vout and Vin neuromorphic figures at DC and 40 Hz. The short initial apical segment accounts for about one-third of the electrotonic extent of the cell for somatofugal voltage transfer.

View larger version (18K): [in this window] [in a new window]   FIG. 6. Log A vs. x plots at DC for the Vout and Vin transforms of the CA1 neuron in Fig. 5. For Vout, the steep diagonals of the twinned primary apical dendrites are quite distinct from their nearly horizontal daughter branches. As in Fig. 4, the terminal branches are much steeper in the Vin plot because of the more rapid decline of voltage with distance for voltage spread toward the soma.

View larger version (19K): [in this window] [in a new window]   FIG. 10. Log A vs. x plots at DC for the Vout and Vin transforms of the granule cell in Fig. 9. As in pyramidal cells, terminal branches are nearly horizontal in the Vout figure because of their sealed-end terminations. The Vin figure displays much greater attenuation for potential spread from the dendrites toward the soma.

View larger version (18K): [in this window] [in a new window]   FIG. 12. Log A vs. x plots at DC for the Vout and Vin transforms of the granule cell in Fig. 11. The initial apical segment is nearly vertical in the Vout plot because of the steep spatial gradient of voltage spreading from the soma to the dendrites. This segment is nearly horizontal in the Vin plot because it is almost isopotential along its length for voltage spreading from the dendrites to the soma.

View larger version (22K): [in this window] [in a new window]   FIG. 15. Side by side comparison of the log A vs. x plots at DC for Vout in a CA1 neuron (cell 503) and a granule cell (cell 964). The range of attenuations and their variation with distance are very similar. This suggests possible functional parallels between dendritic fields in these cell classes.

For each dendritic field of each neuron in this study, we also found the anatomic distance x^max from the soma to the most remote dendritic termination, and computed the L_out and L_in along this path at DC and 40 Hz. In addition, we determined the anatomic and electrotonic distances of the termination(s) that were electrotonically most remote from the soma at DC and 40 Hz, i.e., the terminals that were associated with L^max_out and/or L^max_in. Because CA1 and CA3 neurons have two dendritic fields (apical and basilar), whereas granule cells have only one, there were 34 dendritic fields (22 for the 11 pyramidal neurons, plus 12 for the 12 granule cells). Thus there were 34 values of x^max (one for each dendritic field), 68 L_out values (one for each field at 0 Hz and another for 40 Hz), and 68 L_in values. These measures are plotted in Figs. 13 and 14, and their sample means ± SE are summarized in Table 1.

View larger version (24K): [in this window] [in a new window]   FIG. 13. Maximum anatomic and electrotonic measures of the dendrites of granule cells and the apical dendrites of CA1 and CA3c pyramidal cells. A: values of xmax. B and C: Lmaxout for DC and 40 Hz. D and E: Lmaxin for DC and 40 Hz. Obvious differences between cell classes in these plots turned out to be statistically significant (Table 1). See text for details.

View larger version (26K): [in this window] [in a new window]   FIG. 14. Maximum anatomic and electrotonic measures of the dendrites of granule cells and the basilar dendrites of CA1 and CA3c pyramidal cells. A: values of xmax. B and C: Lmaxout for DC and 40 Hz. D and E: Lmaxin for DC and 40 Hz. Obvious differences between cell classes in these plots turned out to be statistically significant (Table 1). See text for details.

General observations

DC V_out transforms of all three classes of neurons were relatively compact (top right of Figs. 3, 5, 7, 9, and 11), which indicates only slight to moderate attenuation of voltage spread away from the soma in the steady state. Most of the voltage drop occurred in proximal branches. Branches that were more distal are nearly invisible in these figures because they were almost isopotential from their origins to their distal terminations.

The V_in transforms were considerably larger (bottom right of these figures) because voltage suffered more attenuation as it spreads toward the soma. Distal small-diameter branches accounted for a large fraction of attenuation in this direction, illustrating the general principle that electrotonic architecture is direction dependent (Carnevale and Johnston 1982).

In all branches of all cells, attenuation worsened with increasing frequency as a consequence of membrane capacitance, and the transforms of all cells grew more extensive. This effect was first noticeable at frequencies in the range of 5-10 Hz (  32-16 ms), and it was quite prominent at frequencies 40 Hz (  4 ms).

Within each neuron, the physical path lengths from the soma to the dendritic terminations were scattered over a range of values in all three cell classes. Electrotonic path lengths between the soma and dendritic terminations were also nonuniform in both directions. This confirms and extends our recently reported observations in CA1 pyramidal neurons (Mainen et al. 1996). The log A versus x plots show this particularly well (Figs. 4, 6, 8, 10, and 12). This nonuniformity was most pronounced in the apical dendritic trees of CA1 and CA3c cells, but it also appeared in their basilar dendrites and in granule cells.

Although L tended to increase with physical distance from the soma, the dendritic branch termination that was anatomically most remote was also electrotonically most remote in only 28 of 68 cases. That is, the "x^max path" was associated with both the greatest L_out and L_in less than half the time. The x^max path had either L^max_out or L^max_in but not both in an additional 15 instances. In six cases, a physically shorter path had both L^max_out and L^max_in.

Although either or both of these electrotonic measures was occasionally >30% larger than the corresponding L of the x^max path, the difference was usually 10%. Out of a total of 136 individual values of L^max_out and L^max_in computed at 0 and 40 Hz in these three cell classes, the largest was 10-20% greater than the corresponding L of the x^max path in 17 comparisons and larger yet in 14 cases.

The anatomic lengths of the L^max_out and L^max_in paths were usually no less than 90% of the x^max path. They were 10-20% shorter than the corresponding x^max in just 13 of 136 comparisons, and only two were >20% shorter. Even so, some disparities between the paths with greatest L and those with x^max were rather striking: in one CA3c cell a basilar dendrite was anatomically 20% shorter than the basilar x^max path, yet it had an L_out that was 19% greater at 40 Hz. At the same frequency, another CA3c cell had a basilar dendrite that was 4.4% shorter than the x^max path, whereas its L_out was 70% greater.

CA1 and CA3c pyramidal cells

The apical dendrites were the major component of the V_out transforms of the pyramidal cells (Figs. 3, 5, and 7). The dominance of the V_out transforms by the apical dendrites was preserved through the physiologically interesting range of frequencies (0-10 kHz).

The CA1 pyramidal cells generally had a single primary apical dendrite (Fig. 3, cell 524), but in some neurons this dendrite bifurcated (Fig. 5, cell 503). It was the most prominent feature of the V_out transforms of CA1 cells (top right in Figs. 3 and 5), whereas its side branches were nearly invisible. This indicates that most of the attenuation for voltage propagation away from the soma occurred along its length. This is particularly clear in the log A versus x plots for V_out (Figs. 4 and 6), which contrast the steep longitudinal voltage gradient in the primary apicals (the long diagonal rows of points) with the nearly flat spatial profile of voltage in the side branches (the almost horizontal rows of points).

The apical dendritic trees of the CA3c pyramidal neurons were not organized around a primary stem. Instead they consisted of multiple proximal branches of similar electrotonic extent (top right of Fig. 7, cell 701) which accounted for much of the attenuation. This is especially clear in the log A versus x plots (Fig. 8), which disclose no single or bifurcating primary apical dendrite.

In the V_out transforms of CA1 and CA3c pyramidal cells, the basilar dendrites were very short for DC and frequencies lower than _m = 1/_m (top right in Figs. 3, 5, and 7). This indicates that they were virtually isopotential with the soma at low frequencies. For the particular CA1 pyramidal neuron of Fig. 5 (cell 503), the V_out transforms of the basilar dendrites seem to be grouped into two different electrotonic paths, but this was not a consistent feature of pyramidal cell basilar dendrites.

The relative extent of the basilar dendrites was substantially larger in the V_in transforms (bottom right in Figs. 3, 5, and 7). This means that voltage attenuation toward the soma along the basilar dendrites was roughly comparable with attenuation in the anatomically longer apical dendrites. This is due to the loading effect of downstream membrane on these narrow processes. The proximal end of each basilar dendrite is attached to a low impedance load: the soma and all the other dendrites that arise from it. If a synaptic input on a basilar dendrite is to evoke a voltage transient at the soma, it not only has to supply current to the membrane capacitance and conductance of the soma, but it also must supply current to the proximal ends of the remainder of the dendritic tree. Therefore producing a small potential change at the soma requires a large axial current in the basilar dendrite, which results in a steep longitudinal voltage gradient.

Granule cells

Granule cells of the dentate gyrus have a simpler branching pattern than pyramidal cells, and most dendritic terminations appear to be physically nearly equidistant from the cell body. However, the electrotonic path lengths of the dendrites were surprisingly nonuniform. This nonuniformity could affect either the V_in or the V_out transform.

We present two examples of granule cells. The first shows the usual pattern of multiple branches arising close to the soma (Fig. 9, cell 964). Log A versus x plots at DC for the V_out and V_in transforms for this cell are shown in Fig. 10. The soma of the second neuron lay relatively deep in the granule cell layer (Fig. 11, cell 950) and gave rise to a single unbranched process that traveled ~30 µm before its first division.

Both cells were nearly isopotential for DC voltage spread from the soma (top right of Figs. 9 and 11). At higher frequencies, unexpected differences in attenuation emerged along their various dendritic branches (e.g., at 40 Hz in Fig. 9).

As in the pyramidal cells, there was greater attenuation of voltage propagating toward the cell body. The electrotonic path lengths from the dendritic terminations to the soma were nearly identical in some cells (Fig. 9) but nonuniform in others (Fig. 11).

The neuron with an initial solitary apical branch (cell 950) showed the loading effects of downstream membrane quite clearly. The short proximal apical branch occupied only a small fraction of the total anatomic length of dendritic tree (Fig. 11). However, voltage drop along it accounted for about one-third of the extent of the V_out transform of this cell (top right in Fig. 11), and it was prominent in the corresponding log A versus x profile (Fig. 12). This is because almost the entire dendritic tree of this cell arose from its distal end, i.e., it had a very low impedance load. In contrast, the load for voltage spread in the opposite direction along this branch was quite small (just the soma). Consequently there was little signal loss, and the branch was nearly undetectable in the V_in transforms. The difference between the L_out and L_in of this branch was due to the difference in the loads attached to its proximal and distal ends.

Comparisons between classes

Although CA1 and CA3c pyramidal neurons bear some overall resemblance to each other, there are important anatomic and electrotonic differences and similarities between these two cell classes. A rough indication of these differences and similarities is provided by comparison of the maximum anatomic (x^max) and electrotonic lengths (L^max_out and L^max_in; Table 1, Figs. 13 and 14). The apical field of CA1 cells was anatomically and electrotonically longer than that of CA3c cells (Fig. 13). The basilar dendrites of these two cell classes were anatomically comparable (Fig. 14A), but at DC and low frequencies, they were electrotonically more extensive in CA1 cells (Fig. 14, B and D). This difference disappeared with increasing frequency (Fig. 14, C and E).

Figure