THE JOURNAL OF COMPARATIVE NEUROLOGY 323:137-152 (1992)
Dendritic Morphology of Pyramidal Neurones of the Visual Cortex of the Rat. IV: Electrical Geometry A.U. LARKMAN, G. MAJOR, K.J. STRATFORD, AND J.J.B. JACK University Laboratory of Physiology, Oxford University, Oxford OX1 3PT, United Kingdom
ABSTRACT Features of the dendritic morphology of pyramidal neurones of the visual cortex of the rat that are relevant to the development of models of their passive electrical geometry were investigated. The sample of 39 neurones that was used came from layers 213 and 5. They had been recorded from and injected intracellularly with horseradish peroxidase (HRP) in vitro as part of a previous study (Larkman and Mason, J. Neurosci 10:1407,1990). These cells had been reconstructed and measured previously by light microscopy. The relationship between the diameters of parent and daughter dendrites during branching was examined. It was found that most dendrites did not closely obey the ‘‘% branch power relationship” required for representation of the dendrites as single equivalent cylinders. Estimates of total neuronal membrane area ranged from 27,100 ? 7,900 pm2 for layer 213 cells to 52,200 t 11,800 km2 for thick layer 5 cells. Dendritic spines contributed approximately half the total membrane area. Both neuronal input resistance and the ratio of membrane time constant to input resistance were correlated with neuronal membrane area as measured anatomically . The relative electrical lengths of the different dendrites of individual neurones were investigated, by using simple transformations to take account of the differences in diameter and spine density between dendritic segments. A novel “morphotonic” transformation is described that represents the purely morphological component of electrotonic length. Morphotonic lengths can be converted into electrotonic lengths by division by a “morphoelectric factor” ( LR,/Ri]l’Z). This procedure has the advantage of separating the steps involving anatomical and electrical parameters. These transformations indicated that the dendrites of the apical terminal arbor were much longer electrically than the basal or apical oblique dendrites. In relative electrical terms, most apical oblique trees arose extremely close to the soma, and terminated at similar distances to the basals. These results indicate that the dendrites of these pyramidal cells cannot be represented as single equivalent cylinders. The electrotonic lengths of the dendrites were calculated by using the electrical parameters specific membrane capacitance (C,), intracellular resistivity (81, and specific membrane resistivity (Rm).Conventional values were assumed for C, (1.0 pFcm-2) and R, (100 Rcm), but three different R, values were used for each cell. Two of these were within the conventionally accepted range (10,000-20,000 Rcmz), while the third value was an order of magnitude higher, in line with some recent evidence from modeling and whole-cell recording studies. The high R, value yielded dendrites that were electrotonically very short, some 0.15 space constants for basal and oblique dendrites and 0.3-0.6 space constants for terminal arbor dendrites. The distribution of dendritic spines, as markers for the location of excitatory synaptic inputs, with electrotonic distance from the soma was estimated for each R,value. With conventional R,values, 50% of each neurone’s spines were within 0.23-0.33 space constants from the soma, depending on cell class. With high R, values, 50% of spines were within 0.08-0.10 space constants. Thus the majority of excitatory inputs to these cells appear to be located close to the soma in electrical terms. However, there is some evidence that Ri may be higher than conventionally
Accepted April 29, 1992
o 1992 WILEY-LISS, INC.
A.U. LARKMAN ET AL.
138
assumed, which would increase the electrotonic length of dendrites for a given R,. Further data are therefore required before adequate electrical models for these cells can be developed. C>
1992 Wiley-Liss, Inc.
Key words: pyramidal cell, passive modelling, electrotonic length, dendritic spines, membrane area
One of the most influential models of the passive electrical properties of neuronal dendrites has been the "soma and equivalent cylinder" model developed by Rall ('771, originally for the spinal motoneurone. In this model, all the dendrites of the neurone are represented as a single, unbranched cylindrical cable, and sophisticated mathematical methods for describing electrical current flow within such structures have been developed (Jack et al., '75; Rall, '77). For the classes of neurone for which this type of model is appropriate, important properties of the dendrites can be determined experimentally by analysis of relatively simple electrophysiological records, such as the time course of the transient following a voltage- or current-clamp step applied and recorded at the soma. This approach has been adopted in numerous studies of motoneurones (Rall et al., '67; Burke and ten Bruggencate, '71; Iansek and Redman, '73; see Rall, '77), and a demonstration of at least the internal consistency of the approach has been provided by a combination of electrophysiological recording and anatomical reconstruction of the same cells and afferent fibres (Redman and Walmsley, '83). In the simplest and most widely used form of the model, the major requirements for the representation of the dendrites of a neurone as a single equivalent cylinder are: 1. The electrical properties of the neuronal membrane must be uniform over the entire surface of the soma and dendrites. 2. At dendritic branch points, the diameters of the parent and daughter dendrites must follow the ''% power relationship," such that the diameter of the parent dendrite raised to the power of % must equal the sum of the diameters of the daughters each raised to the same power. 3. All the dendrites must terminate at the same electrotonic distance from the soma. This approach has also been applied to a number of cell types other than motoneurones, including relay cells of the lateral geniculate nucleus (Bloomfield et al., '87) and hippocampal pyramidal cells (Brown et al., '81; Johnston, '81). A careful study of the passive properties of neocortical pyramidal cells, using electrophysiological data and assuming that the equivalent cylinder model was appropriate, has also been performed (Stafstrom et al., '84). In recent years, there have been a number of detailed, quantitative anatomical studies of spinal motoneurones to determine morphological features relevant to passive model development, and in particular to assess whether the dendrites of these neurones meet the anatomical requirements for representation as single equivalent cylinders. Extensive use has been made of the intracellular horseradish peroxidase (HRP) injection technique, which permits detailed anatomical and electrophysiological information to be obtained from the same neurone (e.g., Ulfhake and Cullheim, '81, '88; Ulfhake and Kellerth, '81, '82, '83; Burke et al., '82; Cameron et al., '83, '85; Rose et al., '85;
Russell-Mergenthal et al., '86; Cullheim et al., '87; Clements and Redman, '89; Kernel1 and Zwaagstra, '89). These studies have indicated that at least some classes of motoneurone deviate significantly from the anatomical requirements for equivalent cylinder representation, particularly as a result of the premature termination of some dendrites. Doubt has also been expressed concerning the uniformity of membrane properties, particularly over the possibility that the membrane resistivity at or near the soma is lower than in the dendrites during conventional sharp electrode recording (Iansek and Redman, '73; Barrett and Crill, '74; Jack, '79; Fleshman et al., '88; Rose and Vanner, '88; Clements and Redman, '89). Prior to impalement, or during wholecell recording, the membrane may have a more uniform and very high resistivity (Blanton et al., '89; Coleman and Miller, '89; Edwards et al., '89; Stratford et al., '89; Pongracz et al., '91; Spruston and Johnston, '92). With these concerns in mind, we have undertaken a combined electrophysiological and morphological study of some classes of pyramidal neurone from the visual cortex of the rat. Slices were maintained in vitro, and a sample of 39 cells from layers 213 and 5 were recorded intracellularly and injected with HRP. Some electrophysiological and morphological features of these cells have been described in previous papers (Larkman and Mason, '90; Mason and Larkman, '90; Larkman, '91a-c). In this paper we present anatomical data relevant to the passive electrical properties of the dendrites of these cells and the particular issue of whether the soma and equivalent cylinder model can be applied to them (Stafstrom et al., '84). We also illustrate the effects of simple electrical transformations of the dendritic geometry, to take account of differences in segment diameter and the additional membrane area contributed by dendritic spines. We also explore the distribution of the dendritic spines, as indicators of the distribution of excitatory synaptic inputs, as a function of electrotonic distance from the soma. Rather than attempting to derive a definitive set of electrical membrane parameters for each cell class, we consider three illustrative example sets, including one in which the membrane resistivity in the dendrites is much higher than at the soma. This analysis provides insights into the electrical geometry of these cells, but falls short of providing a model that permits accurate prediction of the attenuation and time course of synaptic potentials arising from different parts of the neurone.
MATERIALS AND METHODS The data for this study were taken from the same sample of 39 pyramidal neurones from layers 213 and 5 of the visual cortex of the rat that was used in previous studies in this series (Larkman and Mason, '90; Larkman, '91a), in which details of the experimental procedures have been given. Briefly, 400 pm slices were prepared from young adult albino rats (Wistar; 130-160 g) and maintained in vitro, by conventional techniques. Neurones were impaled
PYRAMIDAL NEURONE ELECTRICAL GEOMETRY using fine-tipped glass microelectrodes and intracellular recordings were made. After recording, the neurones were injected with HRP. The slices were then fixed, resectioned a t 60 pm, and reacted histochemically to visualize the injected neurones. The cells were reconstructed from camera lucida drawings and measured with the light microscope. The length, diameter, and spine density of each dendritic segment were entered into a computer program that allowed a range of statistics to be calculated. On the basis of their laminar location and dendritic morphology, these cells were divided into three classes. The layer 2/3 cells formed one class, while the layer 5 cells were divided into "slender" and "thick" types (Larkman and Mason, '90). The terminology used to describe their dendrites is the same as in Larkman ('91a). Electrophysiological recordings were obtained, averaged, and analyzed as described in Mason and Larkman ('90). Neuronal membrane time constants and input resistances were obtained from the voltage responses following brief (0.2-0.6 ms) hyperpolarizing pulses of injected current (Durand et al., '83).
Neuronal membrane area estimation The membrane areas of the somata and dendritic shafts were estimated for each cell in previous studies (Larkman and Mason, '90). The total numbers and distribution of dendritic spines have also been estimated previously (Larkman, '91c). The determination of overall membrane area therefore additionally required the estimation of the membrane area contributed by the dendritic spines. Spines show considerable local variation in size, and we did not attempt to address this in detail. The dimensions we used were taken from our own light microscope measurements of spines from our HRP-filled neurones (Larkman, '91c) and from published electron microscope measurements of Golgiimpregnated cells (Peters and Kaiserman-Abramof, '70). The diameter of the spine head was taken to be 0.65 Fm for all spines, and the length of the spine neck varied depending on the diameter of the parent dendritic shaft, as described in Larkman ( ' 9 1 ~ )The . diameter of the spine neck could not be measured even approximately with the light microscope, and we used a value of 0.15 pm for all spines. This figure is in the middle of the range quoted by Peters and KaisermanAbramof ('70) and is close to the mean value reported for hippocampal CA1 spines using serial electron microscopy (Harris and Stevens, '89). These dimensions gave membrane areas for individual spines of 1.7 pm2 for those on shafts less than 1.5 pm in diameter, 1.5 pm2 for those on shafts between 1.5 and 2.0 pm, and 1.4 pm2 for those on shafts thicker than 2.0 p,m in diameter.
139 TABLE 1. Branch Power Ratios' Branch power ratio (n = 312) Basal dendrite branch points Layer 213 1.08 f 0.13 (n = 424) (0.80-1.56) Slender layer 5 1.05 f 0.13 (n = 193) (0.59-1.49) Thicklayer5 1.13f 0.17 (0.31-1.84) (n = 313) Oblique dendrite branch paints Layer 213 1.05 2 0.09 (n = 92) (0.86-1.35) Slender layer 5 1.08 f 0.10 ( n = 46) (0.86-1.31) Thicklayer5 1.15 f 0.15 (n = 139) (0.861.91) Terminal arbor dendrite branch points Layer 2/3 0.97 i 0.16 (n = 111) (0.46-1.52) Slender layer 5 Thick layer 5 0.97 f 0.13 (n = 37) (0.57-1.21) Apical trunk branch points Layer 213 1.17 ? 0.19 (n = 93) (0.65-1.79) Slender layer 5 1.10 i 0.20 (n = 88) (0.59-1.52) Thick layer 5 1.18 f 0.20 (n = 132) (0.74-1.84)
Branch power ratio (n = 2)
Branching ratio (n = 312)
Branching ratio (n = 2)
0.89 i 0.15 (0.59-1.43) 0.86 2 0.14 (0.40-1.35) 0.95 f 0 19 (0.16-1.83)
0 52 2 0.03
(0.44-0.61) 0.51 i 0.03 (0.37-0.60) 0.53 t 0.04 (0.23-0.65)
0.47 2 0.04 10.37-0.59) 0.46 i 0.04 (0.28-0.58) 0 48 -t 0.05 (0.14-0.65)
0.85 t 0.10 (0.65-1.20) 0.89 ? 0.11 (0.65-1.14) 0.97 f 0.18 (0.65-1.89)
0.51 0.02 (0.46-0.58) 0.52 f 0.02 (0.460.57) 0.53 f 0.03 (0.46-0.66)
0.46 f 0.03 (0.40-0.55) 0.47 f 0.03 (0.40-0.53) 0.49 f 0.04 (0.40-0.65)
0.78 f 0.17 (0.28-1.39)
0.49 i 0.04 (0.31-0.60)
0.43 i 0.06 (0.22-0.58)
-
*
-
-
0.76 f 0.14 (0.38-1.06)
0.49 ? 0.04 (0.36-0.55)
0.43 ? 0.05 (0.27-0.52)
1.04 f 0.21 (0.45-1.74) 0.97 f 0.23 (0.41-1.44) 1.08 i 0.22 (0.54-1.82)
0.54 2 0.04 (0.39-0 64) 0.52 f 0.05 10.37-0.60) 0.54 2 0.04 10.43-0.65)
0.51 f 0.05 (0.31-0.64) 0.48 ? 0.06 ( 0 29-0.59) 0.51 2 0.05 (0.35-0.65)
'In this and subsequent tables values are expressed as mean parentheses.
?
SD, with range in
The direct calculation of n from measured dendritic segment diameters is highly sensitive to measurement error. It is therefore common to calculate branch power ratios (BP,) for a given value of n , defined as
BPn = (dp + d g ) / q .
(2)
Measured branch power ratios close to unity indicate that the value of n used was appropriate for the branching of the tree considered. Rose et al. ('85) used a slightly different method to quantify the branch power of motoneurone dendrites, which they claimed reduced the difficulties associated with averaging ratios. They calculated branching ratios (BR,) for a given value of n, according to (3)
Measured branching ratios close to 0.5 indicate that the value of n was appropriate for the branching of the tree considered. We calculated both BP, and BRn for each dendritic branch point in the sample for the two cases, n = 7 2 and n = 2. These cases are of interest because n = 3/2 is a requirement for the representation of branched dendritic trees as RESULTS single unbranched cylinders, while n = 2 conserves the Branch power ratios total cross-sectional area of the dendrites of the tree during The branch power (n) describes the relationship between branching. It has previously been reported that n = 2 for the diameters of parent and daughter dendritic segments at neocortical pyramidal neurones (Hillman, '79). All the branch points and is important in determining the electri- branch points in our sample were recorded as bifurcations, cal properties of a dendritic tree (Rall, '77; Hillman, '79). although some of the intermediate segments were very For a bifurcating tree, it may be defined as the exponent short. Dendritic segments that showed obvious taper along their length were omitted from this analysis. that satisfies The branch point and branching ratios obtained varied d i = dr + dE (1) widely between branch points, even within a single cell class and dendritic type. Mean values are given in Table 1,and where dp is the diameter of the parent dendrite and dl and show that branch points within basal and apical oblique dz are the diameters of the daughter dendritic branches. dendritic trees appeared to obey a relationship with n
A.U. LARKMAN ET AL.
140 somewhere between YL and 2. Within terminal arbors, the n = s/z relationship was more closely followed. For the apical trunks of all three cell classes, n = 2 seemed more appropriate than n = 3/2. Indeed, perhaps the clearest violations of the d3J2branch power law were found on the apical trunks of particularly the layer 2/3 and thick L5 cells, where the trunk commonly showed no measurable reduction in diameter following branch points, that gave rise to oblique trees of considerable size.
Dendritic trunk parameter The effect of deviations from the d312branch power law was explored further by calculating the dendritic trunk parameter at 1 km steps along the path length of the dendrites from the soma. The dendritic trunk parameter was introduced by Rall(’59)and may be defined as Dendritic trunk parameter
=
X d;”
nated visual cortical neurones. The areas of single spines used here ranged between 1.4and 1.7 pm2, depending on the diameter of the parent dendrite (see Materials and Methods). From these values, we calculated the total area contributed by spines for each cell, and hence the total neuronal membrane area, the dendritic membrane area, and the fractions contributed by the spines and the soma. A summary of these measures for the three classes of cell is given in Table 2. It can be seen that the inclusion of the dendritic spines results in an approximate doubling of the dendritic membrane area. It is also striking that the soma represents only some 3% of the total membrane area of these cells.
Electrophysiological correlations An advantage of the intracellular HRP injection tech-
(4) nique, particularly when applied in vitro, is that anatomical
where d, is the diameter of the ith dendritic segment at that path length from the soma. The basal and apical dendrites were considered separately, and within the apicals, the obliques were separated from the apical trunk and the terminal arbor. In most cases, the basal dendrites showed an initial slight increase in the dendritic trunk parameter with distance from the soma (“flare”), followed by a plateau and then an abrupt decline (“taper”) as the basal dendrites all terminated over a fairly narrow range of path lengths (Fig. 1). The effects of deviation from the d312 law were more apparent in the apical dendrites. After some 50 Fm from the soma, there was in most cases a rapid and dramatic increase in dendritic trunk parameter, as the tapering of the apical trunk was insufficient to compensate for the oblique trees arising from it. This was most pronounced for thick layer 5 cells, in which the apical trunk shows relatively little taper for most of its length, in spite of giving rise to numerous oblique branches. When cells within each class were pooled and all types of dendrite considered together, all cell classes showed an initial flare followed by dramatic taper as the basal and oblique dendrites terminated at similar path lengths (Fig. 2A-C). The initial flare was much less dramatic for slender layer 5 cells than for the other types. The ratio of peak to initial dendritic trunk parameter was 1.33c 0.25 for layer 2/3cells, 1.46 t 0.24 for thick layer 5 cells, and only 1.12? 0.13 for slender layer 5 cells.
Membrane area and dendritic spines Most dendritic segments of pyramidal iieurones in the neocortex and the hippocampus bear large numbers of dendritic spines. The estimated numbers of spines for the cells used in this study ranged between some 3,500 and 18,000,with clear differences between the cell classes (Larkman, ’91~).It is likely that a significant proportion of the total membrane area of a pyramidal cell is contributed by spines, which therefore must be taken into account in any consideration of their electrical geometry. Estimates of the soma1 areas, dendritic shaft areas, and total numbers of spines for each neurone have been given in previous papers (Larkman and Mason, ’90;Larkman, ’91c), but the area contributed by spines was not estimated. We therefore derived estimates of the membrane area of single, typical spines, based on our own HRP-injected neurones and on previous ultrastructural studies of Golgi-impreg-
and intracellular electrophysiological data can be obtained from the same neurones. We were able to obtain estimates of the neuronal input resistance and membrane time constant for 32 of the 39 neurones whose dendritic morphology was examined, and the following analysis is based on these cells. For a neurone whose membrane properties are uniform over its entire surface and whose dendrites can be represented as an equivalent cylinder, it can be shown that its input resistance, membrane resistivity, and electrotonic length are related to its membrane area according to
AN = R,L/(R,,tanh L)
(5)
where AN = total neuronal membrane area, R, = specific membrane resistivity, R,, = neuronal input resistance, and L = electrotonic length (Rall, ’77). Making the same assumptions, an estimate of R, can be obtained from the measured membrane time constant (T,), using T,
=
RmCm
(6)
where C, is the specific membrane capacitance. The precise value for C, for neocortical pyramidal neurones is not known, but it is thought to be close to 1.0pFcm-2 for a wide range of cell types. Thus AN (Y T,L/(R,,tanh
L)
(7)
Unfortunately, pyramidal cells may not meet the requirements for representation as equivalent cylinders, and we have been unable to determine their electrotonic lengths with any certainty (Stratford et al., ’89). There are also difficulties associated with the measurement Of T,, particularly for layer 5 cells that show time-dependent rectification or “sag” (Mason and Larkman, ’90). Nevertheless, it is interesting to explore whether there is some relationship between the anatomically derived value for AN and the ratio Tm/R,n. Anatomical neuronal membrane area was found not to be correlated with membrane time constant (Fig. 3A).There was a weak, but significant, negative correlation with input resistance (Fig. 3B). Membrane area was, however, positively correlated with the T,,,/R,~ ratio (Fig. 3C;r2 = 0.714; P < 0.001).It is interesting to note that the correlation between T,/R,, and combined dendritic length (Larkman and Mason, ’901, another anatomical measure of neuronal size, was less close (r2= 0.558; data not shown). This
PYRAMIDAL NEURONE ELECTRICAL GEOMETRY
141
L
a,
zE (d
a Y
C
2
t-
0
-
100
200
300
400
500
0
100
200
300
400
500
0
200
400
600
800
1000
200
400
600
800
1000
-
2c
N
N
; i
-5
E,
Y
20
L
0
. d
0
E
f
10
a Y C
0
200
400
.= 600
0
200
400
600
04
.
I
'
I
2
+ o 800
1000
800
1000
Path Distance from Soma (pm) Fig. 1. Graphs showing how dendritic trunk parameter varies with path length from the soma for individual cells from each of the cell classes. Filled squares, total for all dendrites; hollow squares, apical dendrites (apical trunk and obliques); hollow circles, apical trunk only. Insets show basal dendrites only; note different scale in A. A: Layer 213 cell. B: Slender layer 5 cell. C: Thick layer 5 cell. In each case the reduction in dendritic trunk parameter shown by the apical trunk over the proximal 200 km or so does not compensate for the formation of the apical oblique branches. Thus the apical dendrites as a whole show a flaring in dendritic trunk parameter, especially for layer 213 and thick layer 5 cells. The basal dendrites also show a slight flare over the first 100 pm. After the termination of the majority of the basal and oblique dendrites, at approximately 200 pm from the soma, there is an abrupt decline in dendritic trunk parameter.
suggests that the inclusion of additional measurements such as segment diameter, spine density, and spine area, which are used for membrane area estimation but not for dendritic lengths, has improved the correlation. This is a pleasing finding in view of the difficulties and likely errors associated with making these measurements.
0
Path Distance from Soma (pm) Fig. 2. Graphs showing mean dendritic trunk parameter for all dendrites against path length from the soma. Points show mean values for each cell class (errorbars SD). Horizontal dotted line indicates the initial value. A: Layer 213 (n = 18).B: Slender layer 5 (n = 10).C: Thick layer 5 (n = 11).All classes show some initial flare (only minor for slender layer 5 cells) followed by dramatic taper after 150-200 bm from the soma. TABLE 2. Neuronal Membrane Areas
Layer 213 (n = 18) Dendritic membrane area including spines (rm2) (Dendritic area + spinesil (dendritic area - spines) Total neuronal area (with soma; &m2) (Dendriticarea + spines)/ somal area
26,200 5 7,900 112,800-47,800) 2.04t 0.12 (1.92-2.331 27,100 2 7,900 i13,400-48,700) 31.5 t 9.8 118.2-50.6)
Slender layer 5 ( n = 10) 30.800 2 7,400 i15,60044,100) 1.902 0.29 (151-2.36) 32,100 t_ 7,500 (16,70046,200) 26.8 i 8.1 (14.5-37.0)
Relative electrical lengths
Thick layer 5 (n = 11) 50.800 2 11,400 (31,400-67,700) 1.96 i 0.16 (1.75-2.21) 52,200 i 11,800 (32,ion-70.000~ 37.42 8.1 (27.9-49.9)
Equivalent electrical length transformations. If the passive electrical properties of the membrane are uniform
A.U. LARKMAN ET AL.
142
40000 13
E
B
0
30000
n 0
20000
0
0
10000
5
.
'
'
" '
'
'
"
' .
'
' .
'
20 Membrane Time Constant (mS) 10
15
. +
'
5
Fig. 4. Dendrograms of a single basal dendritic tree with eight terminal segments (from a layer 213 cell, but typical of all classesl, showing the effect of the electrical transformations on relative segment lengths. Segment lengths are drawn to scale, but line thickness is only an approximate guide to segment diameter. A: Dendrogram showing physical path lengths of segments. B: Electrical lengths following transformation according t.0 segment diameters. C: Electrical lengths following transformations for both segment diameters and dendritic spines. The effect of these transformations is to increase the lengths of distal and terminal segments that are of low diameter and densely covered in spines relative to the proximal segments, which are thick and nonspiny or only sparsely spiny. Within a given tree, the equivalent length and morphotonic length transformations have exactly the same effect on relative segment lengths, although the units are different in the two cases. Scale bar: A = 100 pm; B,C = 100 "electrical pm" for equivalent lengths and 1.27 for morphotonic lengths.
70000]B
.
ionoo-l ---- . , . 0 20 40
.
I
60
.
I
80
.
I
.
100
.
.
120 1 4 0
Input Resistance (MQ) 70000
4:
1c
the stem segment of the tree. The various dendritic trees arising from a given neuronal soma may be compared (assuming their membrane properties to be the same) by transforming their segment physical lengths according to
b
40000-
and
0.1
0.2
0.3
0.4
0.5
Time Constant / Input Resistance (ms/MQ) Fig. 3. Relationships between total neuronal membrane area (including spines), as determined anatomically, and electrophysiological measures. Hollow circles, layer 213 cells; filled circles, thick layer 5 cells; hollow squares, slender layer 5 cells. Correlations calculated for cells of all classes taken together In = 32). A With membrane time constant-no significant correlation. B: With input resistance-negative correlation (r2 = -0.178; P < 0.05). C: With membrane time constant1 input resistance-positive correlation (rZ= 0.714; P < 0.001).
over the dendritic surface, the electrical length of a given dendritic segment will be inversely proportional to the square root of its diameter (Rall, '59). If we ignore for now the complication posed by the dendritic spines, a simple method of comparing the equivalent electrical lengths (1*d) of the various segments comprising a single dendritic tree is to transform the physical lengths (1) of all the segments according to
e,* = ~(d,,,m/d,eg)"2
(8)
where dsegis the segment diameter and dstemthe diameter of
where dstem,,,is the diameter of the stem segment of the ith tree. Dendritic segments showing obvious tapering were subdivided and the transformation applied to the portions separately. Figure 4A and B shows the effect of these segment diameter transformations on a single basal dendritic tree. Small diameter, distal segments are elongated more than thicker, proximal segments, exaggerating the tendency for pyramidal cell terminal segments to be physically longer than intermediate segments (Larkman, '91a). This equivalent electrical length transformation procedure follows naturally from the method of collapsing dendritic trees into single equivalent cylinders devised by Ra11 (e.g., Rall, '59, '62). The equivalent electrical length of a segment (l*d) can be used to calculate its electrotonic length (L) using measured or assumed values for specifir membrane resistivity and intracellular resistivity (Ft,) according to
where h is the space constant (Rall, '77), defined as
PYRAMIDAL NEURONE ELECTRICAL GEOMETRY Without reference to the space constant, however, the equivalent electrical length cannot be used to make comparisons between different cells, unless they have identical Dee], values. The space constant also involves both anatomical (Dcell)and electrical (Rm,RJ parameters. In order to more clearly separate anatomical and electrical factors and facilitate between-cell comparisons, we have devised an alternative procedure, which we have termed the ‘morphotonic length’ transformation. Morphotonic length transformations. The two-stage procedure for calculating the electrotonic length of a segment described above can be summarized by combining Equations 9-12:
143 current passing from the spine into the dendritic shaft, as a result of, for example, a conductance increase in the spine head membrane as during a synaptic current. In these cases, the spine morphology would have to be represented explicitly). The simplest way of incorporating the additional membrane is to calculate the “folding factor” (F) for each dendritic segment, defined as segment area including spines F = segment area ignoring spines
(17)
The segment length is then transformed according to = eFF”2
(18)
where 1*, is the equivalent electrical segment length adjusted for spines. The folding factor is a dimensionless quantity, and this transformation may be performed either before or after the equivalent or morphotonic length transformations described above. Variations in spine density along a given segment will not affect its overall equivalent (14) electrical length. Thus the dendritic spines have the effect of increasing the This form of the equation offers two advantages. First, Dcell electrical length of the segment. Since most proximal is not required, and second, the numerator now contains intermediate segments of pyramidal cells are spine-free or only anatomical parameters and the denominator only only sparsely spiny, and all terminal and distal intermedielectrical parameters. The estimation of the electrotonic ate segments are densely covered in spines, the effect on length of a segment can proceed in two stages as before, the entire dendritic trees is to increase the electrical length of first stage being to calculate its “morphotonic length” ( 1 ~ ) distal segments relative to proximal ones (Fig. 4C). from [A more rigorous method for incorporating spines (Jack et al., ’89) involves transforming segment diameters as well (15) as lengths, according to Note that while Dcell is used in the calculation of both equivalent electrical length and the space constant, it is unnecessary for the final calculation of L. Equation 13 can be rearranged to give
Morphotonic length encapsulates the relevant morphological information, and, unlike equivalent electrical length, can be used to make comparisons between cells. The second stage is to calculate electrotonic length from
L
= tM/(R,/RJ1’2.
(16)
The quantity (R,/$)1‘2 contains only the relevant electrical information and we suggest it could be termed the “morphoelectric factor.” If R, and Ri are expressed in their conventional units of acm2 and Ocm, respectively, then the morphotonic length must be calculated using segment lengths and diameters expressed in centimeters. Both morphotonic length and morphoelectric factor will then have units of cm1/2; when divided they yield the dimensionless electrotonic length. Within a given tree or cell, the morphotonic length transformation will have the same relative effect on segment lengths as the equivalent length transformation; only the units will be different. Thus Figure 4A and B, with different scale bars, also illustrates the effect of this transformation on a sample basal dendritic tree. Dendritic spines. The presence of the dendritic spines will have a significant influence on the electrical lengths of segments. For all purposes involving the flow of electrical current outwards through the spine membrane, the resistance to current flow of the spine head membrane is likely to be orders of magnitude greater than the longitudinal resistance of the spine neck, which can therefore be ignored (Jack et al., ’89). The spines can then be regarded as simple folds in the dendritic membrane. (This simplification of the geometry is unlikely to be appropriate for the case of
but the overall effect on relative segment lengths will be the same as the simpler procedure used here.] Combined transformations. The effect of the electrical transformations for segment diameters and spine densities is perhaps most dramatic when considering apical dendritic trees. The terminal segments of the terminal arbor are of low diameter, and so, in electrical terms, are considerably lengthened relative to the much thicker segments of the apical trunk. The most proximal apical trunk segments are both thick and only sparsely spiny, and so become relatively extremely short. This is illustrated by the dendrograms in Figures 5 and 6. Figure 5 shows the apical tree and one representative basal tree for a cell from layer 3 , before and after the electrical transformations for both diameters and spines. The dendrograms have been scaled so the overall lengths of the apical trees are similar. It can be seen that the electrical transformations reduced the relative length of the apical trunk to less than that of most basal and oblique dendrites. Also the points of origin of most of the apical oblique trees become relatively very close to the soma, with the result that the oblique dendrites terminate at much the same electrical distance from the soma as the basal dendrites. Similar changes are seen for thick layer 5 cells (Fig. 6). These cells are characterized by a long, thick, apical trunk, giving rise to numerous apical oblique branches (Fig. 6A,C). After the electrical transformations, most oblique trees arise relatively very close to the soma. For the thick layer 5 cells as a whole, the median relative electrical distance to the branch points giving rise to the oblique trees was only 11.6 4.7% of the length of the apical trunk. In
*
A.U. LARKMAN ET AL.
144
A
PHYSICAL
B
ELECTRICAL
‘ I
Terminal Arbor
Apical Obliques
I
’7
’i I
Apical
Apical
Basal
Fig. 5. Dendrograms showing the effect of the electrical transformations for segment diameters and spines on the apical dendrite and a single basal tree from a layer 213 cell. A Dendrograrn of physical path lengths. Scale bar = 100 pm. B: Dendrogram of electrical lengths, (either equivalent electrical or morphotonic), rescaled so that the length to the terminal arbor tips is drawn similar to A. Scale bar = 400
“equivalent Fm” for equivalent lengths or 3.4 cm1’2for morphotonie lengths. C: Camera lucida reconstruction of cell. Scale bar = 100 Fm, The most dramatic effect of the electrical transformations is the relative shortening of the proximal segments of the apical trunk, so the apical oblique trees are seen to arise relatively very close to the origin of the tree at the soma.
other words, on average, half the oblique trees arose within the first 11.68 of the electrical length of the apical trunk, (or within only 7.7% of the overall electrical length of the whole apical tree). Since the branching patterns and segment lengths of the oblique trees tend to be similar to the basals, the equivalent electrical lengths of the oblique dendrites are little greater than those of the basals. The morphotonic distance from the soma to the tip of every dendrite in the sample was calculated, and the lengths of the basal, apical oblique, and terminal arbor dendrites were compared. Figure 7 shows examples from each cell class, and the results for all cells are summarized in Table 3 . In every case, the morphotonic lengths of the basal and oblique dendrites show a high degree of overlap, and both are much shorter than the terminal arbor. The ratios of the morphotonic lengths of the oblique to the basal dendrites were only slightly above unity (ranging from 1.05
for layer 2/3 cells to 1.27for slender layer 5 cells). The ratios of terminal arbor to basal morphotonic lengths were much higher, ranging from 2.37 for layer 2 / 3 cells to 4.53 for thick layer 5 cells.
Electrotonic lengths The electrotonic lengths of dendritic segments can be calculated from their morphotonic lengths by dividing by the morphoelectric factor (Eq. 16) if values for R,,, and R, can be estimated or assumed. R, is often assumed to be approximately 100 Ocm, although there is evidence that higher values might be more appropriate in some cases (Shelton, ’85; Stratford et al., ’89; Major et al., ’90). The value for R, appropriate for a given neurone, however, is highly uncertain, and values between 10,000 and 200,000 Ocm2 were found to be consistent with available data for the present sample of neurones (Stratford et al., ’89).
PYRAMIDAL NEURONE ELECTRICAL GEOMETRY
A
PHYSICAL
C
145
B
ELECTRICAL
-3
1
I l-
4
-Apical
Basal
Trunk
Apical Obliques
I
Apical
Terminal Arbor
Apical
Basal
Fig. 6 . As Figure 5, but for a thick layer 5 cell. These cells are characterized by a long, thick apical trunk giving rise to a terminal arbor in layer 1. The electrical transformations result in a relative shortening of the trunk, especially the more proximal segments. This
again results in the apical oblique trees arising very close to the soma in relative electrical terms. Scale bars: A, C = 200 pm; B = 500 “equivalent pm” or 3.5 cm1:2.
Therefore, the morphoelectric factors were calculated using three different R, values for each cell, with Ri always assumed to be 100 Ocm. Results for each cell class are summarized in Table 3. R, = 10,000 0 cm2. In the first case, R, was taken to be 10,000 Qcm2for all cells, which, with an R.,of 100 Rcm, gave a morphoelectric factor of 10 cm1I2. The median electrotonic lengths of the basal, apical oblique, and terminal arbor dendrites were calculated separately. For all three cell classes, the basal dendrites were approximately 0.5 A, with the basals of the slender layer 5 cells slightly longer than the other classes. In each case the oblique dendrites were slightly longer than the basals. The terminal arbors of layer 213 cells extended for approximately 1.0 A, and those of thick layer 5 cells and the apical dendrites of slender layer 5 cells extended for some 2.0 A. The similarity between the two classes of layer 5 cell is because although the slender layer 5 cells have shorter apical trunks and no terminal arbors, the distal parts of their apical trunks are of very low diameter compared with the thick layer 5 cells.
R, based on 7,. If it is assumed that there is no appreciable leak around the recording microelectrode, and that the passive membrane properties of the cell are uniform over its entire surface, then R, can be calculated from the cell’s membrane time constant, using Equation 6. If C, is assumed to be 1.0 pFcm-Z, an approximate estimate of R, can thus be obtained electrophysiologically. Membrane time constants were measured from the voltage decay following brief pulses of current injected through the microelectrode, and were found to be significantly different between the three cell classes (Mason and Larkman, ’90). The mean 7, values for each cell class were used to calculate R,, giving values of 12,000 Rcm2 for layer 213 cells, 19,500 Rcm2for slender layer 5 cells, and 10,400 Rcm2for the thick layer 5 cells. These gave morphoelectric factors of 10.95, 13.96,and 10.2 cm1I2,respectively, which were then used to calculate the electrotonic length of each dendrite. These values now take some account of the membrane electrical properties as well as the characteristic dendritic morphology shown by each cell class. Median values for the basal,
A.U. LARKMAN ET AL.
146
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Fig. 7. Frequency histograms showing the morphotonic lengths from the soma at which dendrites of different types terminate, for individual neurones from the three classes. Clear bars, basal dendrites; hatched bars, apical oblique dendrites; filled bars, terminal arbor dendrites (tip of apical trunk for slender layer 5 cell). A Layer 213 cell
with soma in layer 2. B: Layer 213 cell with soma in layer 3. C: Slender layer 5 cell. D: Thick layer 5 cell. Note the high degree of overlap between the basal and oblique dendrites in each case, with the terminal arbor dendrites always terminating at much greater morphotonic distances.
oblique, and terminal arbor dendrites are summarized for each class in Table 3. The longer time constants and hence higher Rm values of the slender layer 5 cells have shortened their dendrites relative to the other classes, so their apical dendrites terminate at about 1.4 A compared with 2.0 A for thick layer 5 cells. R,,,based on T,,, x 10. There is growing evidence that the assumptions made above to estimate Rm from the membrane time constant are not justified. In particular, it seems likely that microelectrode impalement produces a substantial leak o r region of decreased membrane resistance at the soma, often referred to as a "somatic shunt" (see Discussion).The effect of this shunt is to reduce the apparent time constant of the cell, and lead to an underestimation of Rm.It has been suggested that, in the more natural state prior to impalement, Rmvalues dramatically higher than the conventional values considered above might be more appropriate. We have therefore repeated the electrotonic length calculations using R, values that are tenfold higher than the values derived directly from 7,. These give higher morphoelectric factors and point to dendrites that are electrically extremely short. Basal and oblique dendrites are now only about 0.15 A, and even the terminal arbors of thick layer 5 cells are only about 0.6 A.
TABLE 3. Morphotonic and Electrotonic Lengths'
Spine distributions A major goal in any study of dendritic electrical geometry is to describe the locations of synapses on the dendrites in electrical terms. For neocortical pyramidal cells an indicator of the location of the excitatory synaptic inputs is available in the form of the dendritic spines. The ovenvhelm-
Murphotonic lengths (cm"'1 Basal Oblique Terminal arbor Obliqueibasal Terminal arburibasal
Layer 213
Slender layer 5
Thick layer 5
4.72 i- 0.81 (2 68-6.0fiJ 4.92 2 0 76 (2.85-5 93) 11.01 i- 2 16 (7.80-14.69) 105 2 0.09 (0.81-1.211 2.37 0.51 (1.71-:1.971
5 45 F 1.21 13.55-7.85) 6.85 i- 2.50 (4 69-11.48) 20.02 ? 6 20 (13.85-32.75) 1.27 f 0.39 (0.90-2.021 3.73 2 1.00 (2.71-5.751
4.52 i 0.62 (3.66-5.791 5 30 i 0 . m (3.98-6 47) 20.22 -c 3.77 (12.0;!-26.47, 1.17 2 0.13 ( 1 041.451 4.53 i 1 0 3 (2.9,1-6.62J
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Electrotonic lengths (space constants) R,, (flcm") 10,000 Morphoelectric factor ( c ~ i i ' ! ~ ) 10.00 Basal 0.47 i 0 08 (0.27-0.611 Oblique 0.49 5 0.08 Terminal arbor R, (based on 7,; Ilcm2) Morphoelectric factor (cm'") Basal Oblique Terminal arbor R,(based on T, x 1 0 lkm21 Morphoelectric Factor (crnlsZJ Basal Obliques Terminal arbor
(0.29-0.591 1.10 i: 0.22 (0.78-1 471 12,000 10 95 0 4 3 ~t 0 0 7 (0.25--0.56) 0 45 f 0.07 (0.26-0.54) 1 0 1 f 0.20 (0.71-1.341 120,000 34.64 0.14 5 0.2 (0.084 181 0.14 L 0.02 (0.08-0.17) 0.3" ? 0 06 10.23-0.42)
(0.36-0.79)
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(0 47-1.151 2.00 ? 0.62 (1.39-3.28) 19,500 13.96 0.39 ? 0 09 (0 26-0.57) 0.49 0.18 (0.34-0.82) 1.43 i 0 44 (0.99-2.35) 195,000 44.16 0.12 i- 0.03 (0 08-0 18) 0.16 2 0.06 (0.11-0.261 0.45 5 0.14 (0.31-0.74)
(1.:!0-2.651 10,400 10 20 0.44 t 0.06 (0.36-0.57) 0.52 ? 0.08 (0 39-0.631 I.R8 2 0.37 (1 18-2.60) 104,000 32.25 0.14 r 0 0 2 (0.11-0.18) 0.16 r 0.03 ((1 12-0.20) 0 63 i. 0 12
(0.37-0.H2)
~_____
'Within a given cell, the morphotonic and elertrotonic lengths of the apical oblique dendrites form a skewed rather than a normal distribution The median rather than mean values for each cell have therefore been used to calculate the c h s s mran and SD Fbr all types of dendrite
PYRAMIDAL NEURONE ELECTRICAL GEOMETRY
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Fig. 8. Graphs illustrating the distribution of dendritic spines with electrotonic distance from the soma for individual neurones, using morphoelectric factors based on R, = 10,000 Cicm2, Ri = 100 Rcm. Ordinate shows the number of spines located in each 0.1 A width zone. Filled squares, all dendrites; hollow squares, apical dendrites (trunk and obliques). Insets show the cumulative percentage distribution of
spines with electrotonic distance for each cell. A Layer 213 cell with soma in layer 2. B: Layer 213 cell with soma in layer 3. C: “Short” slender layer 5 cell, with apical dendrite terminating near the top of layer 4. D: “Long” slender layer 5 cell, with apical dendrite terminating in upper layer 2. E,F: Thick layer 5 cells.
ing majority of excitatory synapses on pyramidal cells are located on dendritic spines, which are clearly visible with the light microscope. The number and distribution of spines on these cells were described in a previous paper in this series (Larkman, ’91c). Based on these estimates, the distribution of spines with respect to electrotonic distance from the soma will now be considered. For each cell, the dendritic diameter and spine area transformations were performed to convert the physical length of each dendritic segment into morphotonic length, and hence derive morphotonic dendrograms like those shown in Figures 5 and 6. These morphotonic lengths were then converted into electrotonic lengths using the same three morphoelectric factors as before. A series of probe lines, 0.1 space constants apart, was then applied across each dendrogram, and a computer program used to calculate the number of spines contained in each zone between adjacent probe lines. The number of spines was then plotted against electrotonic distance from the soma. The contributions from the basal and apical dendrites were calculated separately. First, a morphoelectric factor of 10.0 cm1’2was applied to all cells; examples are shown in Figure 8. All the cells had few spines within the first 0.1 h, but the number rapidly increased to a maximum at 0.3-0.5 h. There was then a dramatic decline, corresponding to the terminations of the basal and oblique dendrites at 0.6-0.9 A. The apical trunk and terminal arbor then contribute a much smaller number of spines per zone until they terminate at a distance that varied greatly from cell to cell. Within layer 2/3, the length of the apical dendrites was related to the depth of the soma in the layer, with deeper cells having longer dendrites in electrical as well as physical terms (Fig. 8A,B). Slender layer 5 cells showed even greater variation, mainly because
of variations in physical lengths. Some had apical dendrites that terminated in layer 4 (Fig. 8C), while others extended to the layer 1border (Fig. 8D). All the distal apical dendrites in this class were of low diameter and bore relatively few spines. Thick layer 5 cells also showed some variation in the electrotonic length of the apical dendrites, but this was mainly the result of variations in diameter rather than physical length (Fig. 8E,F). Thick layer 5 cells had many more spines at the relatively greater electrotonic distances from the soma than the other classes. This is perhaps best illustrated in the cumulative percentage spine distribution plots, shown as insets in Figure 8 for individual cells, and pooled by cell class in Figure 9. These show clearly that a high proportion of spines was located in the electrically proximal part of the dendritic systems for all cell classes. The different shape of the plot for the thick layer 5 cells reflects the greater proportion of more remote spines in this class. Figure 10 shows cumulative percentage plots, pooled by cell class, for the other two morphoelectric factors considered. These were based on the ,T value and 10 times the T, value for R, for each class, respectively. Table 4 summarizes this data, showing the electrotonic distances within which 50% and 75% of the spines were located, and the percentage of spines situated closer than three example electrotonic distances, for each of the three different morphoelectric factors. It is very striking that with morphoelectric factors based on conventional R, values, although both classes of layer 5 cells had dendrites extending to some 2 A, half the spines were closer than about 0.3 A and 75% within 0.5 h. Using the high R, values, all the cells were electrically very compact, and, remarkably, half the spines were within 0.1 h of the soma. If higher Ri values had been used, however, the morphoelectric factors would have been lower
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Fig. 10. Graphs showing the mean cumulative percentage spine distribution with electrotonic distance from the soma for each cell class (all dendrites taken together), using values of R, based on T,, values (hollow circles) and 10 x T, (filled circles), with other electrical parameters as Figure 8. In the latter case, with high R,, 50% of all spines are located within 0.1 h in each cell class.
for each R, value, the dendrites would have been electrotonically longer, and hence more spines would have been electrically remote. The appropriate R, value remains to be determined.
TABLE 4. Electrotonic Distribution of Suines 50% Spines
75% Spines
closer than&)
closer than (A)
'Z Spines located within 0.5 A
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0.24 I 0.03 (0.17-0.30) 0.22 2 0.04 10.17-0.291 0.32 I 0.04 (0.25-0.361
0 36 I 0.05 10.26-0.44) 0.35 i 0 07 10.26-0.471 0.53 i 0.09 (0.38-0 66)
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Layer 213
DISCUSSION % Branch power law The dendritic branch points in this sample of cells did not closely follow the ''VLbranch power law" requirement for representation as single equivalent cylinders, except for the case of the terminal arbor dendrites. Branch points along the apical trunk appeared to be better described by square power relationship, as had been reported earlier for neocortical pyramidal neurones (Hillman, '79). The branch points in the basal and oblique dendrites of the thick layer 5 cells were also better described by a square power law, while a relationship intermediate between the square and the $4 laws seemed more appropriate for those of the other cell classes. This led t o a slight initial flare in the dendritic trunk parameter (Rall, '59) for basal dendrites, and a more dramatic flare for the apicals, particularly for the layer 213
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PYRAMIDAL NEURONE ELECTRICAL GEOMETRY and thick layer 5 cells. The flare would have been more dramatic if the membrane area in the dendritic spines had been taken into account. A rigorous incorporation of spine membrane area into an electrical representation requires increasing the diameter as well as the length of spiny segments (Jack et al., '89). This would increase the diameter of densely spiny distal segments relative to the usually less spiny proximal ones. The dendritic trunk parameter always showed an abrupt and dramatic decline or taper as the basal and oblique dendrites terminated approximately together, much sooner than the apical terminal arbor dendrites. This dendritic trunk parameter profile resembles a more extreme version of that shown by spinal motoneurones (e.g., Rose et al., '85; Cullheim et al., '87). Motoneurones often show some initial flare followed by taper, but the taper is more gradual, and is due to segment tapering and dendritic termination over a wide range of path lengths, rather than in two distinct phases as in these pyramidal neurones.
Neuronal membrane area The present estimates of total neuronal membrane area ranged from a mean of 27,000 pm2 for layer 2 / 3 cells to 52,000 km2 for thick layer 5 cells. These may be compared with published estimates of 24,400 pm2 for CA1 and 47,000 km2 for CA3 pyramidal cells, respectively (Turner, '84). Thus our layer 213 and slender layer 5 cells were similar in area to CA1 pyramids, and our thick layer 5 cells close to CA3 cells. Estimates for dentate granule cells range from 11,000 pm2 (Turner, '84) to 19,000 pm2 (Desmond and Levy, '84). Turner ('84) also reported that the inclusion of dendritic spines in the estimation resulted in an approximate doubling of the membrane area, consistent with our findings for visual cortex. Desmond and Levy ('84) reported a rather smaller increase (by a factor of about 1.6) for dentate granule cells. All these types of spiny neurone are small compared with spinal motoneurones, whose total membrane area may exceed 500,000 pm2 (Cullheim et al., '87; Ulfhake and Cullheim, '88; Segev et al., '90). The ratio of dendritic to somal membrane area for the pyramidal cells in this study was approximately 32, which is close to the values obtained previously for pyramidal cells of the cat somatic sensory cortex (Mungai, '67). Similar high values of 45 have been reported for dentate granule cells (Desmond and Levy, '84) and values as high as 50-60 have been found for spinal motoneurones (Cullheim et al., '87; Ulfhake and Cullheim, '88; Segev et al., '90). It is clear that the soma represents only a small proportion of the total membrane area in all these cell types. These anatomical findings may be contrasted with the dendritic to somal conductance ratios of between 2 and 4 derived from electrophysiological data for large pyramidal neurones from cat neocortex (Stafstrom et al., '84). These estimates relied on making the assumption that the dendrites could be represented as single equivalent cylinders, and on the method of exponential peeling of transients. The validity of this approach for pyramidal neurones will be discussed below.
Relative electrical lengths A further requirement for the representation of neuronal dendritic systems as single, unbranched equivalent cylinders is that all dendrites must terminate at the same electrical distance from the soma. Simple inspection of pyramidal neurone morphology suggests that the physical lengths of basal and oblique dendrites are usually much less
149
than those of apical terminal arbor dendrites, and this has been demonstrated quantitatively in an earlier paper in this series (Larkman, '91a). The question remains whether this is also the case for their electrical lengths. The apical trunks are usually of much greater diameter than most basal or oblique segments, and this would tend to offset, in electrical terms, their greater physical length. Making the assumption of uniform membrane properties between dendrites, we performed simple electrical transformations of the physical lengths to account for differing segment diameters and spine densities. The results were clear; the terminal arbor dendrites were electrically much longer than the basals or obliques. Thus it is not appropriate to represent the dendrites of these neurones as single equivalent cylinders. This means that attempts to derive model parameters from purely electrophysiological data, by exponential peeling of the voltage decay following an injected current pulse or the response to a voltage clamp step (Stafstrom et al., '841, may be unreliable. Not only are the formulae used to derive the model parameters inappropriate (Glenn, '88; Rose and Dagum, '88) but the peeling procedure itself is unreliable (Stratford et al., '89). Another interesting finding from the electrical transformations was that the most proximal segments of the apical trunk were extremely short electrically, because of their large diameter and low spine density. Thus the origins of the majority of apical oblique dendritic trees were electrically extremely close to the soma. In earlier papers in this series, attention was drawn to the high degree of similarity in many respects between the more proximal oblique and basal trees (Larkman, '91a,b). The electrical proximity of the origins of most oblique trees to the soma further emphasizes this point. We performed the electrical transformations in two slightly different ways. The first method, derived directly from the procedure for combining dendrites into equivalent cylinders developed by Rall (e.g., Rall, '59, '62), involved calculating the equivalent electrical lengths of the dendrites, which can be converted into electrotonic lengths by dividing by the space constant. The second, novel, method was to calculate "morphotonic lengths" which could be converted to electrotonic lengths by dividing by the "morphoelectric factor" ([Rm/Ft,]1'2).1This procedure offers the advantage of clearly separating the steps involving morphological and electrical parameters. Morphotonic length represents the purely morphological component of electrotonic length and can be used to make comparisons of dendritic architecture between neurones without reference to their passive electrical properties. We suggest that this is a useful property, given the difficulty of establishing the electrical parameters with certainty (Stratford et al., '89). The morphotonic length transformation also offers a flexible starting point for modeling studies into the effects of variations in electrical parameters within a given cell. The morphoelectric factor can simply be varied between, for example, basal and apical trees, or as a function of distance from the soma (Fleshman et al., '88). The methods we describe for incorporating the membrane area of dendritic spines are not appropriate in all circumstances. For studies concerned with the local effects of synaptic inputs on spines, those spines receiving input must be represented explicitly. If the effects of active 'This transformation was discussed at a workshop held at the Neurosciences Institute, New York City, in August, 1987 (see Rall et al., '92).
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conductances located on spines are to be investigated, the spines can be represented either by the continuum method of Baer and Rinzel ('91) or explicitly. However, for passive spines, not receiving input, our methods are adequate and simple, and must be preferable to ignoring the presence of spines (e.g., Koch et al., '90). In our procedures, the spines are accounted for by adjustments to the morphological representation of the dendrite, which may be preferable to adjusting the electrical parameters C, and R,, as in a recent study (Holmes, '89). This is consistent with the approach underlying the morphotonic length transformation, in trying to keep anatomical and electrical parameters separate wherever possible.
Dendritic electrotonic lengths The observed deviation from the requirements for equivalent cylinder representation does not mean that accurate models of dendritic electrical geometry cannot be developed. If detailed anatomical data are available, then compartmental and other modeling techniques exist which can be applied to neurones of arbitrary geometry. In addition to anatomical measurements, these models commonly make use of the parameters &, R,, and C, to describe the electrical properties of the neuronal membrane. R, and C , might not be expected to vary beyond fairly narrow limits, so reasonably reliable assumptions can be made, based on measurements in other systems. The model then hinges on the value for R,,,, which is likely to vary over a wide range in different cells. If R, is uniform over the entire soma1 and dendritic surface, and if there is a perfect electrical seal between the membrane and the recording microelectrode, then R, can be estimated from the membrane time constant (7,) as described above. A technical difficulty with this approach is that the voltage decay used to measure T , may be contaminated by voltage-dependent conductances, such as the time-dependent inward rectification or "sag" commonly shown by motoneurones (Fleshman et a]., '88; Rose and Vanner, '88) and pyramidal cells (Spain et al., '87; Stratford et al., '89; Mason and Larkman, '90). We tried to minimise this problem by using the decay following very brief current impulses, which are less likely to activate nonlinear membrane behaviour than longer pulses (Iansek and Redman, '73; Durand et al., '83), but sag was still a problem for some layer 5 cells. A much more serious difficulty is that R, may not be uniform. R, could vary from one dendritic region to the next in an infinity of ways, and there is little experimental evidence to allow the alternatives to be distinguished. One particular case, for which there is some evidence and which has been explored in some detail, occurs when the dendritic R, is uniform but higher than the somatic R,. The lower somatic R, could be the result of tonic inhibitory action close to the soma, an imperfect electrical seal with the recording microelectrode, or a higher density of ionic channels open at rest near the soma. Some of these channels could be activated by impalement damage or poor seal permitting the ingress of Ca2+ or Na+ ions. The likelihood of a "somatic shunt" has been suggested by a number of studies (Iansek and Redman, '73; Barrett and Crill, '74; Jack, '79; Durand and Carlen, '85; Fleshman et al., '88; Rose and Vanner, '88; Clements and Redman, '89; Stratford et al., '89). The possibility that much of the shunt is due to impalement by the recording microelectrode has received strong support recently from results obtained using the tight-seal, whole-cell recording technique. It has
been observed that neurones recorded in this way show much higher input resistances and longer membrane time constants than when impaled conventionally (Blanton et al., '89; Coleman and Miller, '89; Edwards et al., '89; Pongracz et al., '91; Spruston and Johnston, '92). Coleman and Miiler ('89) proposed that R, might be approximately a n order of magnitude higher than suggested by conventional impalement techniques. Values of R, as high i l s this were found to be consistent with the anatomical and electrophysiological data obtained for the present sample of pyramidal neurones using conventional sharp-electrode recording, if the possibility of a somatic shunt was included in the model (Stratford et al., '89). However, given likely anatomical measurement errors and a degree of uncertainty over appropriate values for C, and R, in these cells, it was impossible to choose between a wide range of possible R, values. Nevertheless, it seems likely that R, values much higher than conventionally used may be appropriate for the dendritic membranes of pyramidal cells. In view of these complexities, in this paper we have not attempted to derive a definitive electrical model for these cells. Instead, we have looked at the electrotonic lengths of dendrites and the distribution of dendritic spines with respect to electrotonic distance from the soma for three example values of' R, (10,000 &m2; T,/C,; 10 x T J C ~ , ) , assuming it to be uniform over the dendrites, and taking conventional values for R, and C,. The two lower, ''conventional,'' It, values gave electrotonic lengths of about 0.5 A for basal and oblique dendrites, 1A for the apical dendrites of layer 213 cells, and 2 A for those of layer 5 cells. These are similar to the lengths reported for hippocampal pyramidal cells (Turner and Schwartzkroin, '83; Turner, '84). In the high R, case, the dendrites would be much shorter electrically. For all cell classes, the basals and obliques were only some 0.15 A, and even the terminal arbors only 0.3-0.6 A.
Spine distributions The existence of dendrites of very different electrical lengths on the same cell means that quoting a single value for electrotonic length, even for a given set of membrane parameters, is virtually meaningless for these cells. What would be more useful would be an indication of the distribution of synaptic inputs along the electrotonic length of the cell. For spinal motoneurones, attempts have been made to determine the distribution of dendritic surface area with respect to electrotonic distance from the soma (Clements and Redman, '89). Information on synapse density from electron microscope studies might then be used to estimate the distribution of synapses (Conradi, '69; Conradi and Ronnevi, '75; Ulfhake and Cullheim, '88), although we are not aware of any such attempts to date. For at least the large majority of excitatory synapses on visual cortical pyramidal cells, a more direct indication of synapse distribution can be obtained from the dendritic spines. For the conventional R, cases, essentially all the spines on the basal dendrites were situated closer than 0.5 h from the soma. Perhaps more surprisingly, a high proportion of apical spines were also electrically close. For layer 2/3 and slender layer 5 cells, approximately 75% of apical spines were closer than 0.5 A. This reflects the compactness of the majority of the apical obliques. For thick layer 5 cells, with thick apical trunks and well-developed terminal arbors, only just over half of the apical spines were closer than 0.5 A. These electrical distributions of spines reinforce the
PYRAMIDAL NEURONE ELECTRICAL GEOMETRY concept expressed in earlier papers in this series (Larkman, '91a-c) that the dendrites of neocortical pyramidal cells can be considered in two parts. A proximal part, consisting of the basal and proximal oblique dendrites and the proximal part of the apical trunk, is highly compact electrically and receives the majority of the excitatory synaptic inputs (some 80-90%, depending on cell type). The other more distal part, consisting of the distal apical trunk, a small number of distal obliques, and the terminal arbor (if present), receives only a small minority of excitatory inputs which may be relatively remote from the soma in electrical terms (depending on the electrical parameters). For the cell types considered in this study, most of the inputs to these dendrites will be located in layer 1 and the upper part of layer 213. For the conventional R, cases, some 5% of all spines on thick layer 5 cells (representing some 500-1,000 spines) were further than 1.5 A from the soma, which is as remote as the estimate for the most distal synapses on spinal motoneurones (Rall, '77). In the high R, case, all the spines may be considered to be situated electrically close to the soma. The electrotonic spine distributions for all the R,values considered here suggest that if, during electrophysiological experiments, afferent fibres are stimulated essentially at random (as during extracellular electrode stimulation in brain slice studies of synaptic action), in the absence of knowledge of well-defined afferent pathways, the first assumption should be that the synapses stimulated will be located proximally on the postsynaptic cell. Although there is increasing evidence that the high R, case may be the most plausible, some modeling studies have suggestedthat Ri might be substantially higher than conventionally assumed (Shelton, '85; Stratford et al., '89; Spruston and Johnston, '92). This would have the effect of increasing the electrotonic length of a dendrite for a given R, value. There is also some doubt over the precise value for C, for these cells. Thus there are still a number of challenges to be met in developing adequate models of passive dendritic behaviour for these types of cells. Consideration of how such models should then be extended to include the active processes that undoubtedly play a role in shaping synaptic potentials in neocortical cells (Thomson et al., '88; Deisz et al., '91) is only in its infancy.
ACKNOWLEDGMENTS The authors wish to thank Mr. L.F. Waters and Miss A.L. Morgan for photographic assistance. This work was supported by the Wellcome Trust and the Royal Society.
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152 Larkman, A.U. (1991b) Dendritic morphology of pyramidal neurones in the visual cortex of the rat: 11. Parameter correlations. J. Comp. Neurol. 306:320-331. Larkman, A.U. ( 1 9 9 1 ~Dendritic ) morphology of pyramidal neurones in the visual cortex of the rat: 111. Spine distributions. J. Comp. Neurol. 306:332-343. Larkman, A.U., and A. Mason (1990) Correlations between morphology and electrophysiology of pyramidal neurones in slices of rat visual cortex. I. Establishment of cell classes. J. Neurosci. 10:1407-1414. Major, G., A.U. Larkman, and J.J.B. Jack (1990) Constraining nonuniqueness in passive electrical models of cortical pyramidal neurones. J. Physiol. 430t61P. Mason, A., and A.U. Larkman (1990) Correlations between morphology and electrophysiology of pyramidal neurones in slices of rat visual cortex. 11. Electrophysiology. J. Neurosci. IOr1415-1428. Mungai, J.M. (1967) Dendritic patterns in the somatic sensory cortex of the cat. J. Anat. 101:403-418. Peters, A., and I.R. Kaiserman-Abramof (1970) The small pyramidal neuron of the rat cerebral cortex. The perikaiyon, dendrites and spines. Am. J. Anat. 127t321-356. Pongracz, F., S. Firestein, and G.M. Shepherd (1991) Electrotonic structure of olfactory sensory neurons analyzed by intracellular and whole cell patch techniques. J. Neurophysiol. 65747-758. Rall, W. (1959) Branching dendritic trees and motoneuron membrane resistivity. Exp. Neurol. 1:491-527. Rall, W. (1962) Electrophysiology of a dendritic neuron model. Biophys. J. 2145-167. Rall, W. (1977) Core conductor theory and cable properties of neurons. In E.R. Kandel (ed): Handbook of Physiology. The Nervous System, Section 1,Vol. 1.Bethesda: American Physiological Society, pp. 39-97. Rall, W., R.E. Burke, T.G. Smith, P.G. Nelson, and K. Frank (1967) Dendritic location of synapses and possible mechanisms for the monosynaptic EPSP in motoneurons. J. Neurophysiol. 30r1169-1193. Rall, W., K.E. Burke, W.R. Holmes, J.J.B. Jack, S.J. Redman, and 1. Segev (1992) Matching neuron model complexity to experimental data. Physiol. Rev. (in press). Redman, S.J.,and B. Walmsley (1983) The time course of synaptic potentials evoked in cat spinal motoneurones at identified group l a synapses. J. Physiol. 343:117-133. Rose, P.K., and A. Dagum (1988) Nonequivalent cylinder models of neurons: Interpretation of voltage transients generated by somatic current injection. J. Neurophysiol. 6Ot125-148. Rose, P.K., and S.J. Vanner (1988) Differences in somatic and dendritic specific membrane resistivity of spinal motoneurons: An electrophysiological study of neck and shoulder motoneurons in the cat. J. Neurophysiol. 60;149-166.
Rose, P.K., S.A. Keirstead, and S.J.Vanner (1985) A quantitative analysis of the geometry of cat motoneurons innervating neck and shoulder muscles. J. Comp. Neurol. 239;89-107. Russell-Mergenthal, H., J.R. McClung, and S.J. Goldberg (1986) The determination of dendrite morphology on lateral rectus motoneurons in cat. J. Comp. Neurol. 245:11(3-122. Segev, I., J.W. Fleshman, and R E . Burke (1990) Computer simulation of group l a EPSPs using morphologically realistic models of cat a-motnneurons. J. Neurophysiol. 64:648-660. Shelton, D.P. (1985) Membrane resistivity estimated for the Purkinje neuron by means of a passive computer model. Neuroscience 14t111131. Spain, W.J., P.C. Schwindt, and W.E. Crill(1987) Anomalous rectification in neurons from cat sensorimotor cortex in vitro. J. Neurophysiol. 57;15.551576. Spruston, N., and D. Johnson (1992) Perforated patch-clamp analysis of the passive membrane properties of three classes of hippocampal neurons. J. Neurophysiol. 67:508-529. Stafstrom, C.E., P.C. Schwindt., and W.E. Crill (1984) Cable properties of layer V neurons from cat sensorimotor cortex in vitro. J. Neurophysiol. 52:278-289.
Stratford, K., A. Mason, A. Larkman, G. Major, and J. Jack (1989) The modelling of pyramidal neurones in the visual cortex. In R. Durbin, C. Miall, and G. Mitchison (eds): The Computing Neuron. Wokingham: Addison-Wesley, pp. 296-321. Thomson, A.M., D. Girdlestone, and D.C. West (19881 Voltage-dependent currents prolong single-axon postsynaptic potentials in layer I11 pyramidal neurons in rat neocortical slices. J. Neurophysiol. 60;1896-1907. Turner, D.A. (1984) Segmental cable evaluation of somatic transients in hippocampal neurons (CA1, CA3, and dentate). Biophys. J. 46:73-84. Turner, D.A., and P.A. Schwartzkroin (1983) Electrical characteristics of dendrites and dendritic spines in intracellularly stained CA3 and dentate hippocampal neurons. J. Neurosci. 3:2381-2394. Ulfhake, B., and S. Cullheim (1981)A quantitative light microscopic study of the dendrites of cat spinal y-motoneurons after intracellular staining with horseradish peroxidase. J. Comp. Neurol. 202:585-596. Ulfhake, B., and J.-0.Kellerth (1981) A quantitative light microscopic study of the dendrites of cat spinal a-motoneurons after intracellular staining with horseradish peroxidase. J. Comp. Neurol. 202571-583. Ulfhake, B., and J.-0. Kellerth (1982) Does a-motoneurone size correlate with motor unit type in cat, triceps surae? Brain Res. 251:201-209. Ulfhake, B., and J.-0.Kellerth (1983) A quantitative morpholoL6cal study of HRP-labelled cat a-motoncurones supplying different hindlimb musc1e:i. Brain Res. Z64:l-19. Ulfhake, B., and S. Cullheim (1988) Postnatal development of cat hind limb motoneurons. 111.:Changes in size of motoneurons supplying the triceps surae muscle. J. Comp. Neurol. 278;103-120.
Dendritic Morphology of Pyramidal Neurones of the Visual Cortex of the Rat. IV: Electrical Geometry A.U. LARKMAN, G. MAJOR, K.J. STRATFORD, AND J.J.B. JACK University Laboratory of Physiology, Oxford University, Oxford OX1 3PT, United Kingdom
ABSTRACT Features of the dendritic morphology of pyramidal neurones of the visual cortex of the rat that are relevant to the development of models of their passive electrical geometry were investigated. The sample of 39 neurones that was used came from layers 213 and 5. They had been recorded from and injected intracellularly with horseradish peroxidase (HRP) in vitro as part of a previous study (Larkman and Mason, J. Neurosci 10:1407,1990). These cells had been reconstructed and measured previously by light microscopy. The relationship between the diameters of parent and daughter dendrites during branching was examined. It was found that most dendrites did not closely obey the ‘‘% branch power relationship” required for representation of the dendrites as single equivalent cylinders. Estimates of total neuronal membrane area ranged from 27,100 ? 7,900 pm2 for layer 213 cells to 52,200 t 11,800 km2 for thick layer 5 cells. Dendritic spines contributed approximately half the total membrane area. Both neuronal input resistance and the ratio of membrane time constant to input resistance were correlated with neuronal membrane area as measured anatomically . The relative electrical lengths of the different dendrites of individual neurones were investigated, by using simple transformations to take account of the differences in diameter and spine density between dendritic segments. A novel “morphotonic” transformation is described that represents the purely morphological component of electrotonic length. Morphotonic lengths can be converted into electrotonic lengths by division by a “morphoelectric factor” ( LR,/Ri]l’Z). This procedure has the advantage of separating the steps involving anatomical and electrical parameters. These transformations indicated that the dendrites of the apical terminal arbor were much longer electrically than the basal or apical oblique dendrites. In relative electrical terms, most apical oblique trees arose extremely close to the soma, and terminated at similar distances to the basals. These results indicate that the dendrites of these pyramidal cells cannot be represented as single equivalent cylinders. The electrotonic lengths of the dendrites were calculated by using the electrical parameters specific membrane capacitance (C,), intracellular resistivity (81, and specific membrane resistivity (Rm).Conventional values were assumed for C, (1.0 pFcm-2) and R, (100 Rcm), but three different R, values were used for each cell. Two of these were within the conventionally accepted range (10,000-20,000 Rcmz), while the third value was an order of magnitude higher, in line with some recent evidence from modeling and whole-cell recording studies. The high R, value yielded dendrites that were electrotonically very short, some 0.15 space constants for basal and oblique dendrites and 0.3-0.6 space constants for terminal arbor dendrites. The distribution of dendritic spines, as markers for the location of excitatory synaptic inputs, with electrotonic distance from the soma was estimated for each R,value. With conventional R,values, 50% of each neurone’s spines were within 0.23-0.33 space constants from the soma, depending on cell class. With high R, values, 50% of spines were within 0.08-0.10 space constants. Thus the majority of excitatory inputs to these cells appear to be located close to the soma in electrical terms. However, there is some evidence that Ri may be higher than conventionally
Accepted April 29, 1992
o 1992 WILEY-LISS, INC.
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assumed, which would increase the electrotonic length of dendrites for a given R,. Further data are therefore required before adequate electrical models for these cells can be developed. C>
1992 Wiley-Liss, Inc.
Key words: pyramidal cell, passive modelling, electrotonic length, dendritic spines, membrane area
One of the most influential models of the passive electrical properties of neuronal dendrites has been the "soma and equivalent cylinder" model developed by Rall ('771, originally for the spinal motoneurone. In this model, all the dendrites of the neurone are represented as a single, unbranched cylindrical cable, and sophisticated mathematical methods for describing electrical current flow within such structures have been developed (Jack et al., '75; Rall, '77). For the classes of neurone for which this type of model is appropriate, important properties of the dendrites can be determined experimentally by analysis of relatively simple electrophysiological records, such as the time course of the transient following a voltage- or current-clamp step applied and recorded at the soma. This approach has been adopted in numerous studies of motoneurones (Rall et al., '67; Burke and ten Bruggencate, '71; Iansek and Redman, '73; see Rall, '77), and a demonstration of at least the internal consistency of the approach has been provided by a combination of electrophysiological recording and anatomical reconstruction of the same cells and afferent fibres (Redman and Walmsley, '83). In the simplest and most widely used form of the model, the major requirements for the representation of the dendrites of a neurone as a single equivalent cylinder are: 1. The electrical properties of the neuronal membrane must be uniform over the entire surface of the soma and dendrites. 2. At dendritic branch points, the diameters of the parent and daughter dendrites must follow the ''% power relationship," such that the diameter of the parent dendrite raised to the power of % must equal the sum of the diameters of the daughters each raised to the same power. 3. All the dendrites must terminate at the same electrotonic distance from the soma. This approach has also been applied to a number of cell types other than motoneurones, including relay cells of the lateral geniculate nucleus (Bloomfield et al., '87) and hippocampal pyramidal cells (Brown et al., '81; Johnston, '81). A careful study of the passive properties of neocortical pyramidal cells, using electrophysiological data and assuming that the equivalent cylinder model was appropriate, has also been performed (Stafstrom et al., '84). In recent years, there have been a number of detailed, quantitative anatomical studies of spinal motoneurones to determine morphological features relevant to passive model development, and in particular to assess whether the dendrites of these neurones meet the anatomical requirements for representation as single equivalent cylinders. Extensive use has been made of the intracellular horseradish peroxidase (HRP) injection technique, which permits detailed anatomical and electrophysiological information to be obtained from the same neurone (e.g., Ulfhake and Cullheim, '81, '88; Ulfhake and Kellerth, '81, '82, '83; Burke et al., '82; Cameron et al., '83, '85; Rose et al., '85;
Russell-Mergenthal et al., '86; Cullheim et al., '87; Clements and Redman, '89; Kernel1 and Zwaagstra, '89). These studies have indicated that at least some classes of motoneurone deviate significantly from the anatomical requirements for equivalent cylinder representation, particularly as a result of the premature termination of some dendrites. Doubt has also been expressed concerning the uniformity of membrane properties, particularly over the possibility that the membrane resistivity at or near the soma is lower than in the dendrites during conventional sharp electrode recording (Iansek and Redman, '73; Barrett and Crill, '74; Jack, '79; Fleshman et al., '88; Rose and Vanner, '88; Clements and Redman, '89). Prior to impalement, or during wholecell recording, the membrane may have a more uniform and very high resistivity (Blanton et al., '89; Coleman and Miller, '89; Edwards et al., '89; Stratford et al., '89; Pongracz et al., '91; Spruston and Johnston, '92). With these concerns in mind, we have undertaken a combined electrophysiological and morphological study of some classes of pyramidal neurone from the visual cortex of the rat. Slices were maintained in vitro, and a sample of 39 cells from layers 213 and 5 were recorded intracellularly and injected with HRP. Some electrophysiological and morphological features of these cells have been described in previous papers (Larkman and Mason, '90; Mason and Larkman, '90; Larkman, '91a-c). In this paper we present anatomical data relevant to the passive electrical properties of the dendrites of these cells and the particular issue of whether the soma and equivalent cylinder model can be applied to them (Stafstrom et al., '84). We also illustrate the effects of simple electrical transformations of the dendritic geometry, to take account of differences in segment diameter and the additional membrane area contributed by dendritic spines. We also explore the distribution of the dendritic spines, as indicators of the distribution of excitatory synaptic inputs, as a function of electrotonic distance from the soma. Rather than attempting to derive a definitive set of electrical membrane parameters for each cell class, we consider three illustrative example sets, including one in which the membrane resistivity in the dendrites is much higher than at the soma. This analysis provides insights into the electrical geometry of these cells, but falls short of providing a model that permits accurate prediction of the attenuation and time course of synaptic potentials arising from different parts of the neurone.
MATERIALS AND METHODS The data for this study were taken from the same sample of 39 pyramidal neurones from layers 213 and 5 of the visual cortex of the rat that was used in previous studies in this series (Larkman and Mason, '90; Larkman, '91a), in which details of the experimental procedures have been given. Briefly, 400 pm slices were prepared from young adult albino rats (Wistar; 130-160 g) and maintained in vitro, by conventional techniques. Neurones were impaled
PYRAMIDAL NEURONE ELECTRICAL GEOMETRY using fine-tipped glass microelectrodes and intracellular recordings were made. After recording, the neurones were injected with HRP. The slices were then fixed, resectioned a t 60 pm, and reacted histochemically to visualize the injected neurones. The cells were reconstructed from camera lucida drawings and measured with the light microscope. The length, diameter, and spine density of each dendritic segment were entered into a computer program that allowed a range of statistics to be calculated. On the basis of their laminar location and dendritic morphology, these cells were divided into three classes. The layer 2/3 cells formed one class, while the layer 5 cells were divided into "slender" and "thick" types (Larkman and Mason, '90). The terminology used to describe their dendrites is the same as in Larkman ('91a). Electrophysiological recordings were obtained, averaged, and analyzed as described in Mason and Larkman ('90). Neuronal membrane time constants and input resistances were obtained from the voltage responses following brief (0.2-0.6 ms) hyperpolarizing pulses of injected current (Durand et al., '83).
Neuronal membrane area estimation The membrane areas of the somata and dendritic shafts were estimated for each cell in previous studies (Larkman and Mason, '90). The total numbers and distribution of dendritic spines have also been estimated previously (Larkman, '91c). The determination of overall membrane area therefore additionally required the estimation of the membrane area contributed by the dendritic spines. Spines show considerable local variation in size, and we did not attempt to address this in detail. The dimensions we used were taken from our own light microscope measurements of spines from our HRP-filled neurones (Larkman, '91c) and from published electron microscope measurements of Golgiimpregnated cells (Peters and Kaiserman-Abramof, '70). The diameter of the spine head was taken to be 0.65 Fm for all spines, and the length of the spine neck varied depending on the diameter of the parent dendritic shaft, as described in Larkman ( ' 9 1 ~ )The . diameter of the spine neck could not be measured even approximately with the light microscope, and we used a value of 0.15 pm for all spines. This figure is in the middle of the range quoted by Peters and KaisermanAbramof ('70) and is close to the mean value reported for hippocampal CA1 spines using serial electron microscopy (Harris and Stevens, '89). These dimensions gave membrane areas for individual spines of 1.7 pm2 for those on shafts less than 1.5 pm in diameter, 1.5 pm2 for those on shafts between 1.5 and 2.0 pm, and 1.4 pm2 for those on shafts thicker than 2.0 p,m in diameter.
139 TABLE 1. Branch Power Ratios' Branch power ratio (n = 312) Basal dendrite branch points Layer 213 1.08 f 0.13 (n = 424) (0.80-1.56) Slender layer 5 1.05 f 0.13 (n = 193) (0.59-1.49) Thicklayer5 1.13f 0.17 (0.31-1.84) (n = 313) Oblique dendrite branch paints Layer 213 1.05 2 0.09 (n = 92) (0.86-1.35) Slender layer 5 1.08 f 0.10 ( n = 46) (0.86-1.31) Thicklayer5 1.15 f 0.15 (n = 139) (0.861.91) Terminal arbor dendrite branch points Layer 2/3 0.97 i 0.16 (n = 111) (0.46-1.52) Slender layer 5 Thick layer 5 0.97 f 0.13 (n = 37) (0.57-1.21) Apical trunk branch points Layer 213 1.17 ? 0.19 (n = 93) (0.65-1.79) Slender layer 5 1.10 i 0.20 (n = 88) (0.59-1.52) Thick layer 5 1.18 f 0.20 (n = 132) (0.74-1.84)
Branch power ratio (n = 2)
Branching ratio (n = 312)
Branching ratio (n = 2)
0.89 i 0.15 (0.59-1.43) 0.86 2 0.14 (0.40-1.35) 0.95 f 0 19 (0.16-1.83)
0 52 2 0.03
(0.44-0.61) 0.51 i 0.03 (0.37-0.60) 0.53 t 0.04 (0.23-0.65)
0.47 2 0.04 10.37-0.59) 0.46 i 0.04 (0.28-0.58) 0 48 -t 0.05 (0.14-0.65)
0.85 t 0.10 (0.65-1.20) 0.89 ? 0.11 (0.65-1.14) 0.97 f 0.18 (0.65-1.89)
0.51 0.02 (0.46-0.58) 0.52 f 0.02 (0.460.57) 0.53 f 0.03 (0.46-0.66)
0.46 f 0.03 (0.40-0.55) 0.47 f 0.03 (0.40-0.53) 0.49 f 0.04 (0.40-0.65)
0.78 f 0.17 (0.28-1.39)
0.49 i 0.04 (0.31-0.60)
0.43 i 0.06 (0.22-0.58)
-
*
-
-
0.76 f 0.14 (0.38-1.06)
0.49 ? 0.04 (0.36-0.55)
0.43 ? 0.05 (0.27-0.52)
1.04 f 0.21 (0.45-1.74) 0.97 f 0.23 (0.41-1.44) 1.08 i 0.22 (0.54-1.82)
0.54 2 0.04 (0.39-0 64) 0.52 f 0.05 10.37-0.60) 0.54 2 0.04 10.43-0.65)
0.51 f 0.05 (0.31-0.64) 0.48 ? 0.06 ( 0 29-0.59) 0.51 2 0.05 (0.35-0.65)
'In this and subsequent tables values are expressed as mean parentheses.
?
SD, with range in
The direct calculation of n from measured dendritic segment diameters is highly sensitive to measurement error. It is therefore common to calculate branch power ratios (BP,) for a given value of n , defined as
BPn = (dp + d g ) / q .
(2)
Measured branch power ratios close to unity indicate that the value of n used was appropriate for the branching of the tree considered. Rose et al. ('85) used a slightly different method to quantify the branch power of motoneurone dendrites, which they claimed reduced the difficulties associated with averaging ratios. They calculated branching ratios (BR,) for a given value of n, according to (3)
Measured branching ratios close to 0.5 indicate that the value of n was appropriate for the branching of the tree considered. We calculated both BP, and BRn for each dendritic branch point in the sample for the two cases, n = 7 2 and n = 2. These cases are of interest because n = 3/2 is a requirement for the representation of branched dendritic trees as RESULTS single unbranched cylinders, while n = 2 conserves the Branch power ratios total cross-sectional area of the dendrites of the tree during The branch power (n) describes the relationship between branching. It has previously been reported that n = 2 for the diameters of parent and daughter dendritic segments at neocortical pyramidal neurones (Hillman, '79). All the branch points and is important in determining the electri- branch points in our sample were recorded as bifurcations, cal properties of a dendritic tree (Rall, '77; Hillman, '79). although some of the intermediate segments were very For a bifurcating tree, it may be defined as the exponent short. Dendritic segments that showed obvious taper along their length were omitted from this analysis. that satisfies The branch point and branching ratios obtained varied d i = dr + dE (1) widely between branch points, even within a single cell class and dendritic type. Mean values are given in Table 1,and where dp is the diameter of the parent dendrite and dl and show that branch points within basal and apical oblique dz are the diameters of the daughter dendritic branches. dendritic trees appeared to obey a relationship with n
A.U. LARKMAN ET AL.
140 somewhere between YL and 2. Within terminal arbors, the n = s/z relationship was more closely followed. For the apical trunks of all three cell classes, n = 2 seemed more appropriate than n = 3/2. Indeed, perhaps the clearest violations of the d3J2branch power law were found on the apical trunks of particularly the layer 2/3 and thick L5 cells, where the trunk commonly showed no measurable reduction in diameter following branch points, that gave rise to oblique trees of considerable size.
Dendritic trunk parameter The effect of deviations from the d312branch power law was explored further by calculating the dendritic trunk parameter at 1 km steps along the path length of the dendrites from the soma. The dendritic trunk parameter was introduced by Rall(’59)and may be defined as Dendritic trunk parameter
=
X d;”
nated visual cortical neurones. The areas of single spines used here ranged between 1.4and 1.7 pm2, depending on the diameter of the parent dendrite (see Materials and Methods). From these values, we calculated the total area contributed by spines for each cell, and hence the total neuronal membrane area, the dendritic membrane area, and the fractions contributed by the spines and the soma. A summary of these measures for the three classes of cell is given in Table 2. It can be seen that the inclusion of the dendritic spines results in an approximate doubling of the dendritic membrane area. It is also striking that the soma represents only some 3% of the total membrane area of these cells.
Electrophysiological correlations An advantage of the intracellular HRP injection tech-
(4) nique, particularly when applied in vitro, is that anatomical
where d, is the diameter of the ith dendritic segment at that path length from the soma. The basal and apical dendrites were considered separately, and within the apicals, the obliques were separated from the apical trunk and the terminal arbor. In most cases, the basal dendrites showed an initial slight increase in the dendritic trunk parameter with distance from the soma (“flare”), followed by a plateau and then an abrupt decline (“taper”) as the basal dendrites all terminated over a fairly narrow range of path lengths (Fig. 1). The effects of deviation from the d312 law were more apparent in the apical dendrites. After some 50 Fm from the soma, there was in most cases a rapid and dramatic increase in dendritic trunk parameter, as the tapering of the apical trunk was insufficient to compensate for the oblique trees arising from it. This was most pronounced for thick layer 5 cells, in which the apical trunk shows relatively little taper for most of its length, in spite of giving rise to numerous oblique branches. When cells within each class were pooled and all types of dendrite considered together, all cell classes showed an initial flare followed by dramatic taper as the basal and oblique dendrites terminated at similar path lengths (Fig. 2A-C). The initial flare was much less dramatic for slender layer 5 cells than for the other types. The ratio of peak to initial dendritic trunk parameter was 1.33c 0.25 for layer 2/3cells, 1.46 t 0.24 for thick layer 5 cells, and only 1.12? 0.13 for slender layer 5 cells.
Membrane area and dendritic spines Most dendritic segments of pyramidal iieurones in the neocortex and the hippocampus bear large numbers of dendritic spines. The estimated numbers of spines for the cells used in this study ranged between some 3,500 and 18,000,with clear differences between the cell classes (Larkman, ’91~).It is likely that a significant proportion of the total membrane area of a pyramidal cell is contributed by spines, which therefore must be taken into account in any consideration of their electrical geometry. Estimates of the soma1 areas, dendritic shaft areas, and total numbers of spines for each neurone have been given in previous papers (Larkman and Mason, ’90;Larkman, ’91c), but the area contributed by spines was not estimated. We therefore derived estimates of the membrane area of single, typical spines, based on our own HRP-injected neurones and on previous ultrastructural studies of Golgi-impreg-
and intracellular electrophysiological data can be obtained from the same neurones. We were able to obtain estimates of the neuronal input resistance and membrane time constant for 32 of the 39 neurones whose dendritic morphology was examined, and the following analysis is based on these cells. For a neurone whose membrane properties are uniform over its entire surface and whose dendrites can be represented as an equivalent cylinder, it can be shown that its input resistance, membrane resistivity, and electrotonic length are related to its membrane area according to
AN = R,L/(R,,tanh L)
(5)
where AN = total neuronal membrane area, R, = specific membrane resistivity, R,, = neuronal input resistance, and L = electrotonic length (Rall, ’77). Making the same assumptions, an estimate of R, can be obtained from the measured membrane time constant (T,), using T,
=
RmCm
(6)
where C, is the specific membrane capacitance. The precise value for C, for neocortical pyramidal neurones is not known, but it is thought to be close to 1.0pFcm-2 for a wide range of cell types. Thus AN (Y T,L/(R,,tanh
L)
(7)
Unfortunately, pyramidal cells may not meet the requirements for representation as equivalent cylinders, and we have been unable to determine their electrotonic lengths with any certainty (Stratford et al., ’89). There are also difficulties associated with the measurement Of T,, particularly for layer 5 cells that show time-dependent rectification or “sag” (Mason and Larkman, ’90). Nevertheless, it is interesting to explore whether there is some relationship between the anatomically derived value for AN and the ratio Tm/R,n. Anatomical neuronal membrane area was found not to be correlated with membrane time constant (Fig. 3A).There was a weak, but significant, negative correlation with input resistance (Fig. 3B). Membrane area was, however, positively correlated with the T,,,/R,~ ratio (Fig. 3C;r2 = 0.714; P < 0.001).It is interesting to note that the correlation between T,/R,, and combined dendritic length (Larkman and Mason, ’901, another anatomical measure of neuronal size, was less close (r2= 0.558; data not shown). This
PYRAMIDAL NEURONE ELECTRICAL GEOMETRY
141
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0
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100
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Path Distance from Soma (pm) Fig. 1. Graphs showing how dendritic trunk parameter varies with path length from the soma for individual cells from each of the cell classes. Filled squares, total for all dendrites; hollow squares, apical dendrites (apical trunk and obliques); hollow circles, apical trunk only. Insets show basal dendrites only; note different scale in A. A: Layer 213 cell. B: Slender layer 5 cell. C: Thick layer 5 cell. In each case the reduction in dendritic trunk parameter shown by the apical trunk over the proximal 200 km or so does not compensate for the formation of the apical oblique branches. Thus the apical dendrites as a whole show a flaring in dendritic trunk parameter, especially for layer 213 and thick layer 5 cells. The basal dendrites also show a slight flare over the first 100 pm. After the termination of the majority of the basal and oblique dendrites, at approximately 200 pm from the soma, there is an abrupt decline in dendritic trunk parameter.
suggests that the inclusion of additional measurements such as segment diameter, spine density, and spine area, which are used for membrane area estimation but not for dendritic lengths, has improved the correlation. This is a pleasing finding in view of the difficulties and likely errors associated with making these measurements.
0
Path Distance from Soma (pm) Fig. 2. Graphs showing mean dendritic trunk parameter for all dendrites against path length from the soma. Points show mean values for each cell class (errorbars SD). Horizontal dotted line indicates the initial value. A: Layer 213 (n = 18).B: Slender layer 5 (n = 10).C: Thick layer 5 (n = 11).All classes show some initial flare (only minor for slender layer 5 cells) followed by dramatic taper after 150-200 bm from the soma. TABLE 2. Neuronal Membrane Areas
Layer 213 (n = 18) Dendritic membrane area including spines (rm2) (Dendritic area + spinesil (dendritic area - spines) Total neuronal area (with soma; &m2) (Dendriticarea + spines)/ somal area
26,200 5 7,900 112,800-47,800) 2.04t 0.12 (1.92-2.331 27,100 2 7,900 i13,400-48,700) 31.5 t 9.8 118.2-50.6)
Slender layer 5 ( n = 10) 30.800 2 7,400 i15,60044,100) 1.902 0.29 (151-2.36) 32,100 t_ 7,500 (16,70046,200) 26.8 i 8.1 (14.5-37.0)
Relative electrical lengths
Thick layer 5 (n = 11) 50.800 2 11,400 (31,400-67,700) 1.96 i 0.16 (1.75-2.21) 52,200 i 11,800 (32,ion-70.000~ 37.42 8.1 (27.9-49.9)
Equivalent electrical length transformations. If the passive electrical properties of the membrane are uniform
A.U. LARKMAN ET AL.
142
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30000
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20000
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10000
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.
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20 Membrane Time Constant (mS) 10
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Fig. 4. Dendrograms of a single basal dendritic tree with eight terminal segments (from a layer 213 cell, but typical of all classesl, showing the effect of the electrical transformations on relative segment lengths. Segment lengths are drawn to scale, but line thickness is only an approximate guide to segment diameter. A: Dendrogram showing physical path lengths of segments. B: Electrical lengths following transformation according t.0 segment diameters. C: Electrical lengths following transformations for both segment diameters and dendritic spines. The effect of these transformations is to increase the lengths of distal and terminal segments that are of low diameter and densely covered in spines relative to the proximal segments, which are thick and nonspiny or only sparsely spiny. Within a given tree, the equivalent length and morphotonic length transformations have exactly the same effect on relative segment lengths, although the units are different in the two cases. Scale bar: A = 100 pm; B,C = 100 "electrical pm" for equivalent lengths and 1.27 for morphotonic lengths.
70000]B
.
ionoo-l ---- . , . 0 20 40
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60
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80
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the stem segment of the tree. The various dendritic trees arising from a given neuronal soma may be compared (assuming their membrane properties to be the same) by transforming their segment physical lengths according to
b
40000-
and
0.1
0.2
0.3
0.4
0.5
Time Constant / Input Resistance (ms/MQ) Fig. 3. Relationships between total neuronal membrane area (including spines), as determined anatomically, and electrophysiological measures. Hollow circles, layer 213 cells; filled circles, thick layer 5 cells; hollow squares, slender layer 5 cells. Correlations calculated for cells of all classes taken together In = 32). A With membrane time constant-no significant correlation. B: With input resistance-negative correlation (r2 = -0.178; P < 0.05). C: With membrane time constant1 input resistance-positive correlation (rZ= 0.714; P < 0.001).
over the dendritic surface, the electrical length of a given dendritic segment will be inversely proportional to the square root of its diameter (Rall, '59). If we ignore for now the complication posed by the dendritic spines, a simple method of comparing the equivalent electrical lengths (1*d) of the various segments comprising a single dendritic tree is to transform the physical lengths (1) of all the segments according to
e,* = ~(d,,,m/d,eg)"2
(8)
where dsegis the segment diameter and dstemthe diameter of
where dstem,,,is the diameter of the stem segment of the ith tree. Dendritic segments showing obvious tapering were subdivided and the transformation applied to the portions separately. Figure 4A and B shows the effect of these segment diameter transformations on a single basal dendritic tree. Small diameter, distal segments are elongated more than thicker, proximal segments, exaggerating the tendency for pyramidal cell terminal segments to be physically longer than intermediate segments (Larkman, '91a). This equivalent electrical length transformation procedure follows naturally from the method of collapsing dendritic trees into single equivalent cylinders devised by Ra11 (e.g., Rall, '59, '62). The equivalent electrical length of a segment (l*d) can be used to calculate its electrotonic length (L) using measured or assumed values for specifir membrane resistivity and intracellular resistivity (Ft,) according to
where h is the space constant (Rall, '77), defined as
PYRAMIDAL NEURONE ELECTRICAL GEOMETRY Without reference to the space constant, however, the equivalent electrical length cannot be used to make comparisons between different cells, unless they have identical Dee], values. The space constant also involves both anatomical (Dcell)and electrical (Rm,RJ parameters. In order to more clearly separate anatomical and electrical factors and facilitate between-cell comparisons, we have devised an alternative procedure, which we have termed the ‘morphotonic length’ transformation. Morphotonic length transformations. The two-stage procedure for calculating the electrotonic length of a segment described above can be summarized by combining Equations 9-12:
143 current passing from the spine into the dendritic shaft, as a result of, for example, a conductance increase in the spine head membrane as during a synaptic current. In these cases, the spine morphology would have to be represented explicitly). The simplest way of incorporating the additional membrane is to calculate the “folding factor” (F) for each dendritic segment, defined as segment area including spines F = segment area ignoring spines
(17)
The segment length is then transformed according to = eFF”2
(18)
where 1*, is the equivalent electrical segment length adjusted for spines. The folding factor is a dimensionless quantity, and this transformation may be performed either before or after the equivalent or morphotonic length transformations described above. Variations in spine density along a given segment will not affect its overall equivalent (14) electrical length. Thus the dendritic spines have the effect of increasing the This form of the equation offers two advantages. First, Dcell electrical length of the segment. Since most proximal is not required, and second, the numerator now contains intermediate segments of pyramidal cells are spine-free or only anatomical parameters and the denominator only only sparsely spiny, and all terminal and distal intermedielectrical parameters. The estimation of the electrotonic ate segments are densely covered in spines, the effect on length of a segment can proceed in two stages as before, the entire dendritic trees is to increase the electrical length of first stage being to calculate its “morphotonic length” ( 1 ~ ) distal segments relative to proximal ones (Fig. 4C). from [A more rigorous method for incorporating spines (Jack et al., ’89) involves transforming segment diameters as well (15) as lengths, according to Note that while Dcell is used in the calculation of both equivalent electrical length and the space constant, it is unnecessary for the final calculation of L. Equation 13 can be rearranged to give
Morphotonic length encapsulates the relevant morphological information, and, unlike equivalent electrical length, can be used to make comparisons between cells. The second stage is to calculate electrotonic length from
L
= tM/(R,/RJ1’2.
(16)
The quantity (R,/$)1‘2 contains only the relevant electrical information and we suggest it could be termed the “morphoelectric factor.” If R, and Ri are expressed in their conventional units of acm2 and Ocm, respectively, then the morphotonic length must be calculated using segment lengths and diameters expressed in centimeters. Both morphotonic length and morphoelectric factor will then have units of cm1/2; when divided they yield the dimensionless electrotonic length. Within a given tree or cell, the morphotonic length transformation will have the same relative effect on segment lengths as the equivalent length transformation; only the units will be different. Thus Figure 4A and B, with different scale bars, also illustrates the effect of this transformation on a sample basal dendritic tree. Dendritic spines. The presence of the dendritic spines will have a significant influence on the electrical lengths of segments. For all purposes involving the flow of electrical current outwards through the spine membrane, the resistance to current flow of the spine head membrane is likely to be orders of magnitude greater than the longitudinal resistance of the spine neck, which can therefore be ignored (Jack et al., ’89). The spines can then be regarded as simple folds in the dendritic membrane. (This simplification of the geometry is unlikely to be appropriate for the case of
but the overall effect on relative segment lengths will be the same as the simpler procedure used here.] Combined transformations. The effect of the electrical transformations for segment diameters and spine densities is perhaps most dramatic when considering apical dendritic trees. The terminal segments of the terminal arbor are of low diameter, and so, in electrical terms, are considerably lengthened relative to the much thicker segments of the apical trunk. The most proximal apical trunk segments are both thick and only sparsely spiny, and so become relatively extremely short. This is illustrated by the dendrograms in Figures 5 and 6. Figure 5 shows the apical tree and one representative basal tree for a cell from layer 3 , before and after the electrical transformations for both diameters and spines. The dendrograms have been scaled so the overall lengths of the apical trees are similar. It can be seen that the electrical transformations reduced the relative length of the apical trunk to less than that of most basal and oblique dendrites. Also the points of origin of most of the apical oblique trees become relatively very close to the soma, with the result that the oblique dendrites terminate at much the same electrical distance from the soma as the basal dendrites. Similar changes are seen for thick layer 5 cells (Fig. 6). These cells are characterized by a long, thick, apical trunk, giving rise to numerous apical oblique branches (Fig. 6A,C). After the electrical transformations, most oblique trees arise relatively very close to the soma. For the thick layer 5 cells as a whole, the median relative electrical distance to the branch points giving rise to the oblique trees was only 11.6 4.7% of the length of the apical trunk. In
*
A.U. LARKMAN ET AL.
144
A
PHYSICAL
B
ELECTRICAL
‘ I
Terminal Arbor
Apical Obliques
I
’7
’i I
Apical
Apical
Basal
Fig. 5. Dendrograms showing the effect of the electrical transformations for segment diameters and spines on the apical dendrite and a single basal tree from a layer 213 cell. A Dendrograrn of physical path lengths. Scale bar = 100 pm. B: Dendrogram of electrical lengths, (either equivalent electrical or morphotonic), rescaled so that the length to the terminal arbor tips is drawn similar to A. Scale bar = 400
“equivalent Fm” for equivalent lengths or 3.4 cm1’2for morphotonie lengths. C: Camera lucida reconstruction of cell. Scale bar = 100 Fm, The most dramatic effect of the electrical transformations is the relative shortening of the proximal segments of the apical trunk, so the apical oblique trees are seen to arise relatively very close to the origin of the tree at the soma.
other words, on average, half the oblique trees arose within the first 11.68 of the electrical length of the apical trunk, (or within only 7.7% of the overall electrical length of the whole apical tree). Since the branching patterns and segment lengths of the oblique trees tend to be similar to the basals, the equivalent electrical lengths of the oblique dendrites are little greater than those of the basals. The morphotonic distance from the soma to the tip of every dendrite in the sample was calculated, and the lengths of the basal, apical oblique, and terminal arbor dendrites were compared. Figure 7 shows examples from each cell class, and the results for all cells are summarized in Table 3 . In every case, the morphotonic lengths of the basal and oblique dendrites show a high degree of overlap, and both are much shorter than the terminal arbor. The ratios of the morphotonic lengths of the oblique to the basal dendrites were only slightly above unity (ranging from 1.05
for layer 2/3 cells to 1.27for slender layer 5 cells). The ratios of terminal arbor to basal morphotonic lengths were much higher, ranging from 2.37 for layer 2 / 3 cells to 4.53 for thick layer 5 cells.
Electrotonic lengths The electrotonic lengths of dendritic segments can be calculated from their morphotonic lengths by dividing by the morphoelectric factor (Eq. 16) if values for R,,, and R, can be estimated or assumed. R, is often assumed to be approximately 100 Ocm, although there is evidence that higher values might be more appropriate in some cases (Shelton, ’85; Stratford et al., ’89; Major et al., ’90). The value for R, appropriate for a given neurone, however, is highly uncertain, and values between 10,000 and 200,000 Ocm2 were found to be consistent with available data for the present sample of neurones (Stratford et al., ’89).
PYRAMIDAL NEURONE ELECTRICAL GEOMETRY
A
PHYSICAL
C
145
B
ELECTRICAL
-3
1
I l-
4
-Apical
Basal
Trunk
Apical Obliques
I
Apical
Terminal Arbor
Apical
Basal
Fig. 6 . As Figure 5, but for a thick layer 5 cell. These cells are characterized by a long, thick apical trunk giving rise to a terminal arbor in layer 1. The electrical transformations result in a relative shortening of the trunk, especially the more proximal segments. This
again results in the apical oblique trees arising very close to the soma in relative electrical terms. Scale bars: A, C = 200 pm; B = 500 “equivalent pm” or 3.5 cm1:2.
Therefore, the morphoelectric factors were calculated using three different R, values for each cell, with Ri always assumed to be 100 Ocm. Results for each cell class are summarized in Table 3. R, = 10,000 0 cm2. In the first case, R, was taken to be 10,000 Qcm2for all cells, which, with an R.,of 100 Rcm, gave a morphoelectric factor of 10 cm1I2. The median electrotonic lengths of the basal, apical oblique, and terminal arbor dendrites were calculated separately. For all three cell classes, the basal dendrites were approximately 0.5 A, with the basals of the slender layer 5 cells slightly longer than the other classes. In each case the oblique dendrites were slightly longer than the basals. The terminal arbors of layer 213 cells extended for approximately 1.0 A, and those of thick layer 5 cells and the apical dendrites of slender layer 5 cells extended for some 2.0 A. The similarity between the two classes of layer 5 cell is because although the slender layer 5 cells have shorter apical trunks and no terminal arbors, the distal parts of their apical trunks are of very low diameter compared with the thick layer 5 cells.
R, based on 7,. If it is assumed that there is no appreciable leak around the recording microelectrode, and that the passive membrane properties of the cell are uniform over its entire surface, then R, can be calculated from the cell’s membrane time constant, using Equation 6. If C, is assumed to be 1.0 pFcm-Z, an approximate estimate of R, can thus be obtained electrophysiologically. Membrane time constants were measured from the voltage decay following brief pulses of current injected through the microelectrode, and were found to be significantly different between the three cell classes (Mason and Larkman, ’90). The mean 7, values for each cell class were used to calculate R,, giving values of 12,000 Rcm2 for layer 213 cells, 19,500 Rcm2for slender layer 5 cells, and 10,400 Rcm2for the thick layer 5 cells. These gave morphoelectric factors of 10.95, 13.96,and 10.2 cm1I2,respectively, which were then used to calculate the electrotonic length of each dendrite. These values now take some account of the membrane electrical properties as well as the characteristic dendritic morphology shown by each cell class. Median values for the basal,
A.U. LARKMAN ET AL.
146
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2
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3 4 5 6 7 8 9 10111213141516 Morphotonic Distance from Soma (crnl'*)
2
6 8 1 0 1 2 1 4 1 6 18 2 0 2 2 2 4 2 6 Morphotonic Distance from Soma ( c m l '2)
4
Fig. 7. Frequency histograms showing the morphotonic lengths from the soma at which dendrites of different types terminate, for individual neurones from the three classes. Clear bars, basal dendrites; hatched bars, apical oblique dendrites; filled bars, terminal arbor dendrites (tip of apical trunk for slender layer 5 cell). A Layer 213 cell
with soma in layer 2. B: Layer 213 cell with soma in layer 3. C: Slender layer 5 cell. D: Thick layer 5 cell. Note the high degree of overlap between the basal and oblique dendrites in each case, with the terminal arbor dendrites always terminating at much greater morphotonic distances.
oblique, and terminal arbor dendrites are summarized for each class in Table 3. The longer time constants and hence higher Rm values of the slender layer 5 cells have shortened their dendrites relative to the other classes, so their apical dendrites terminate at about 1.4 A compared with 2.0 A for thick layer 5 cells. R,,,based on T,,, x 10. There is growing evidence that the assumptions made above to estimate Rm from the membrane time constant are not justified. In particular, it seems likely that microelectrode impalement produces a substantial leak o r region of decreased membrane resistance at the soma, often referred to as a "somatic shunt" (see Discussion).The effect of this shunt is to reduce the apparent time constant of the cell, and lead to an underestimation of Rm.It has been suggested that, in the more natural state prior to impalement, Rmvalues dramatically higher than the conventional values considered above might be more appropriate. We have therefore repeated the electrotonic length calculations using R, values that are tenfold higher than the values derived directly from 7,. These give higher morphoelectric factors and point to dendrites that are electrically extremely short. Basal and oblique dendrites are now only about 0.15 A, and even the terminal arbors of thick layer 5 cells are only about 0.6 A.
TABLE 3. Morphotonic and Electrotonic Lengths'
Spine distributions A major goal in any study of dendritic electrical geometry is to describe the locations of synapses on the dendrites in electrical terms. For neocortical pyramidal cells an indicator of the location of the excitatory synaptic inputs is available in the form of the dendritic spines. The ovenvhelm-
Murphotonic lengths (cm"'1 Basal Oblique Terminal arbor Obliqueibasal Terminal arburibasal
Layer 213
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Thick layer 5
4.72 i- 0.81 (2 68-6.0fiJ 4.92 2 0 76 (2.85-5 93) 11.01 i- 2 16 (7.80-14.69) 105 2 0.09 (0.81-1.211 2.37 0.51 (1.71-:1.971
5 45 F 1.21 13.55-7.85) 6.85 i- 2.50 (4 69-11.48) 20.02 ? 6 20 (13.85-32.75) 1.27 f 0.39 (0.90-2.021 3.73 2 1.00 (2.71-5.751
4.52 i 0.62 (3.66-5.791 5 30 i 0 . m (3.98-6 47) 20.22 -c 3.77 (12.0;!-26.47, 1.17 2 0.13 ( 1 041.451 4.53 i 1 0 3 (2.9,1-6.62J
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Electrotonic lengths (space constants) R,, (flcm") 10,000 Morphoelectric factor ( c ~ i i ' ! ~ ) 10.00 Basal 0.47 i 0 08 (0.27-0.611 Oblique 0.49 5 0.08 Terminal arbor R, (based on 7,; Ilcm2) Morphoelectric factor (cm'") Basal Oblique Terminal arbor R,(based on T, x 1 0 lkm21 Morphoelectric Factor (crnlsZJ Basal Obliques Terminal arbor
(0.29-0.591 1.10 i: 0.22 (0.78-1 471 12,000 10 95 0 4 3 ~t 0 0 7 (0.25--0.56) 0 45 f 0.07 (0.26-0.54) 1 0 1 f 0.20 (0.71-1.341 120,000 34.64 0.14 5 0.2 (0.084 181 0.14 L 0.02 (0.08-0.17) 0.3" ? 0 06 10.23-0.42)
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(0 47-1.151 2.00 ? 0.62 (1.39-3.28) 19,500 13.96 0.39 ? 0 09 (0 26-0.57) 0.49 0.18 (0.34-0.82) 1.43 i 0 44 (0.99-2.35) 195,000 44.16 0.12 i- 0.03 (0 08-0 18) 0.16 2 0.06 (0.11-0.261 0.45 5 0.14 (0.31-0.74)
(1.:!0-2.651 10,400 10 20 0.44 t 0.06 (0.36-0.57) 0.52 ? 0.08 (0 39-0.631 I.R8 2 0.37 (1 18-2.60) 104,000 32.25 0.14 r 0 0 2 (0.11-0.18) 0.16 r 0.03 ((1 12-0.20) 0 63 i. 0 12
(0.37-0.H2)
~_____
'Within a given cell, the morphotonic and elertrotonic lengths of the apical oblique dendrites form a skewed rather than a normal distribution The median rather than mean values for each cell have therefore been used to calculate the c h s s mran and SD Fbr all types of dendrite
PYRAMIDAL NEURONE ELECTRICAL GEOMETRY
147
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Fig. 8. Graphs illustrating the distribution of dendritic spines with electrotonic distance from the soma for individual neurones, using morphoelectric factors based on R, = 10,000 Cicm2, Ri = 100 Rcm. Ordinate shows the number of spines located in each 0.1 A width zone. Filled squares, all dendrites; hollow squares, apical dendrites (trunk and obliques). Insets show the cumulative percentage distribution of
spines with electrotonic distance for each cell. A Layer 213 cell with soma in layer 2. B: Layer 213 cell with soma in layer 3. C: “Short” slender layer 5 cell, with apical dendrite terminating near the top of layer 4. D: “Long” slender layer 5 cell, with apical dendrite terminating in upper layer 2. E,F: Thick layer 5 cells.
ing majority of excitatory synapses on pyramidal cells are located on dendritic spines, which are clearly visible with the light microscope. The number and distribution of spines on these cells were described in a previous paper in this series (Larkman, ’91c). Based on these estimates, the distribution of spines with respect to electrotonic distance from the soma will now be considered. For each cell, the dendritic diameter and spine area transformations were performed to convert the physical length of each dendritic segment into morphotonic length, and hence derive morphotonic dendrograms like those shown in Figures 5 and 6. These morphotonic lengths were then converted into electrotonic lengths using the same three morphoelectric factors as before. A series of probe lines, 0.1 space constants apart, was then applied across each dendrogram, and a computer program used to calculate the number of spines contained in each zone between adjacent probe lines. The number of spines was then plotted against electrotonic distance from the soma. The contributions from the basal and apical dendrites were calculated separately. First, a morphoelectric factor of 10.0 cm1’2was applied to all cells; examples are shown in Figure 8. All the cells had few spines within the first 0.1 h, but the number rapidly increased to a maximum at 0.3-0.5 h. There was then a dramatic decline, corresponding to the terminations of the basal and oblique dendrites at 0.6-0.9 A. The apical trunk and terminal arbor then contribute a much smaller number of spines per zone until they terminate at a distance that varied greatly from cell to cell. Within layer 2/3, the length of the apical dendrites was related to the depth of the soma in the layer, with deeper cells having longer dendrites in electrical as well as physical terms (Fig. 8A,B). Slender layer 5 cells showed even greater variation, mainly because
of variations in physical lengths. Some had apical dendrites that terminated in layer 4 (Fig. 8C), while others extended to the layer 1border (Fig. 8D). All the distal apical dendrites in this class were of low diameter and bore relatively few spines. Thick layer 5 cells also showed some variation in the electrotonic length of the apical dendrites, but this was mainly the result of variations in diameter rather than physical length (Fig. 8E,F). Thick layer 5 cells had many more spines at the relatively greater electrotonic distances from the soma than the other classes. This is perhaps best illustrated in the cumulative percentage spine distribution plots, shown as insets in Figure 8 for individual cells, and pooled by cell class in Figure 9. These show clearly that a high proportion of spines was located in the electrically proximal part of the dendritic systems for all cell classes. The different shape of the plot for the thick layer 5 cells reflects the greater proportion of more remote spines in this class. Figure 10 shows cumulative percentage plots, pooled by cell class, for the other two morphoelectric factors considered. These were based on the ,T value and 10 times the T, value for R, for each class, respectively. Table 4 summarizes this data, showing the electrotonic distances within which 50% and 75% of the spines were located, and the percentage of spines situated closer than three example electrotonic distances, for each of the three different morphoelectric factors. It is very striking that with morphoelectric factors based on conventional R, values, although both classes of layer 5 cells had dendrites extending to some 2 A, half the spines were closer than about 0.3 A and 75% within 0.5 h. Using the high R, values, all the cells were electrically very compact, and, remarkably, half the spines were within 0.1 h of the soma. If higher Ri values had been used, however, the morphoelectric factors would have been lower
A.U. LARKMAN ET AL.
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Fig. 10. Graphs showing the mean cumulative percentage spine distribution with electrotonic distance from the soma for each cell class (all dendrites taken together), using values of R, based on T,, values (hollow circles) and 10 x T, (filled circles), with other electrical parameters as Figure 8. In the latter case, with high R,, 50% of all spines are located within 0.1 h in each cell class.
for each R, value, the dendrites would have been electrotonically longer, and hence more spines would have been electrically remote. The appropriate R, value remains to be determined.
TABLE 4. Electrotonic Distribution of Suines 50% Spines
75% Spines
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Layer 213
DISCUSSION % Branch power law The dendritic branch points in this sample of cells did not closely follow the ''VLbranch power law" requirement for representation as single equivalent cylinders, except for the case of the terminal arbor dendrites. Branch points along the apical trunk appeared to be better described by square power relationship, as had been reported earlier for neocortical pyramidal neurones (Hillman, '79). The branch points in the basal and oblique dendrites of the thick layer 5 cells were also better described by a square power law, while a relationship intermediate between the square and the $4 laws seemed more appropriate for those of the other cell classes. This led t o a slight initial flare in the dendritic trunk parameter (Rall, '59) for basal dendrites, and a more dramatic flare for the apicals, particularly for the layer 213
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PYRAMIDAL NEURONE ELECTRICAL GEOMETRY and thick layer 5 cells. The flare would have been more dramatic if the membrane area in the dendritic spines had been taken into account. A rigorous incorporation of spine membrane area into an electrical representation requires increasing the diameter as well as the length of spiny segments (Jack et al., '89). This would increase the diameter of densely spiny distal segments relative to the usually less spiny proximal ones. The dendritic trunk parameter always showed an abrupt and dramatic decline or taper as the basal and oblique dendrites terminated approximately together, much sooner than the apical terminal arbor dendrites. This dendritic trunk parameter profile resembles a more extreme version of that shown by spinal motoneurones (e.g., Rose et al., '85; Cullheim et al., '87). Motoneurones often show some initial flare followed by taper, but the taper is more gradual, and is due to segment tapering and dendritic termination over a wide range of path lengths, rather than in two distinct phases as in these pyramidal neurones.
Neuronal membrane area The present estimates of total neuronal membrane area ranged from a mean of 27,000 pm2 for layer 2 / 3 cells to 52,000 km2 for thick layer 5 cells. These may be compared with published estimates of 24,400 pm2 for CA1 and 47,000 km2 for CA3 pyramidal cells, respectively (Turner, '84). Thus our layer 213 and slender layer 5 cells were similar in area to CA1 pyramids, and our thick layer 5 cells close to CA3 cells. Estimates for dentate granule cells range from 11,000 pm2 (Turner, '84) to 19,000 pm2 (Desmond and Levy, '84). Turner ('84) also reported that the inclusion of dendritic spines in the estimation resulted in an approximate doubling of the membrane area, consistent with our findings for visual cortex. Desmond and Levy ('84) reported a rather smaller increase (by a factor of about 1.6) for dentate granule cells. All these types of spiny neurone are small compared with spinal motoneurones, whose total membrane area may exceed 500,000 pm2 (Cullheim et al., '87; Ulfhake and Cullheim, '88; Segev et al., '90). The ratio of dendritic to somal membrane area for the pyramidal cells in this study was approximately 32, which is close to the values obtained previously for pyramidal cells of the cat somatic sensory cortex (Mungai, '67). Similar high values of 45 have been reported for dentate granule cells (Desmond and Levy, '84) and values as high as 50-60 have been found for spinal motoneurones (Cullheim et al., '87; Ulfhake and Cullheim, '88; Segev et al., '90). It is clear that the soma represents only a small proportion of the total membrane area in all these cell types. These anatomical findings may be contrasted with the dendritic to somal conductance ratios of between 2 and 4 derived from electrophysiological data for large pyramidal neurones from cat neocortex (Stafstrom et al., '84). These estimates relied on making the assumption that the dendrites could be represented as single equivalent cylinders, and on the method of exponential peeling of transients. The validity of this approach for pyramidal neurones will be discussed below.
Relative electrical lengths A further requirement for the representation of neuronal dendritic systems as single, unbranched equivalent cylinders is that all dendrites must terminate at the same electrical distance from the soma. Simple inspection of pyramidal neurone morphology suggests that the physical lengths of basal and oblique dendrites are usually much less
149
than those of apical terminal arbor dendrites, and this has been demonstrated quantitatively in an earlier paper in this series (Larkman, '91a). The question remains whether this is also the case for their electrical lengths. The apical trunks are usually of much greater diameter than most basal or oblique segments, and this would tend to offset, in electrical terms, their greater physical length. Making the assumption of uniform membrane properties between dendrites, we performed simple electrical transformations of the physical lengths to account for differing segment diameters and spine densities. The results were clear; the terminal arbor dendrites were electrically much longer than the basals or obliques. Thus it is not appropriate to represent the dendrites of these neurones as single equivalent cylinders. This means that attempts to derive model parameters from purely electrophysiological data, by exponential peeling of the voltage decay following an injected current pulse or the response to a voltage clamp step (Stafstrom et al., '841, may be unreliable. Not only are the formulae used to derive the model parameters inappropriate (Glenn, '88; Rose and Dagum, '88) but the peeling procedure itself is unreliable (Stratford et al., '89). Another interesting finding from the electrical transformations was that the most proximal segments of the apical trunk were extremely short electrically, because of their large diameter and low spine density. Thus the origins of the majority of apical oblique dendritic trees were electrically extremely close to the soma. In earlier papers in this series, attention was drawn to the high degree of similarity in many respects between the more proximal oblique and basal trees (Larkman, '91a,b). The electrical proximity of the origins of most oblique trees to the soma further emphasizes this point. We performed the electrical transformations in two slightly different ways. The first method, derived directly from the procedure for combining dendrites into equivalent cylinders developed by Rall (e.g., Rall, '59, '62), involved calculating the equivalent electrical lengths of the dendrites, which can be converted into electrotonic lengths by dividing by the space constant. The second, novel, method was to calculate "morphotonic lengths" which could be converted to electrotonic lengths by dividing by the "morphoelectric factor" ([Rm/Ft,]1'2).1This procedure offers the advantage of clearly separating the steps involving morphological and electrical parameters. Morphotonic length represents the purely morphological component of electrotonic length and can be used to make comparisons of dendritic architecture between neurones without reference to their passive electrical properties. We suggest that this is a useful property, given the difficulty of establishing the electrical parameters with certainty (Stratford et al., '89). The morphotonic length transformation also offers a flexible starting point for modeling studies into the effects of variations in electrical parameters within a given cell. The morphoelectric factor can simply be varied between, for example, basal and apical trees, or as a function of distance from the soma (Fleshman et al., '88). The methods we describe for incorporating the membrane area of dendritic spines are not appropriate in all circumstances. For studies concerned with the local effects of synaptic inputs on spines, those spines receiving input must be represented explicitly. If the effects of active 'This transformation was discussed at a workshop held at the Neurosciences Institute, New York City, in August, 1987 (see Rall et al., '92).
A.U. LARKMAN ET AL.
150
conductances located on spines are to be investigated, the spines can be represented either by the continuum method of Baer and Rinzel ('91) or explicitly. However, for passive spines, not receiving input, our methods are adequate and simple, and must be preferable to ignoring the presence of spines (e.g., Koch et al., '90). In our procedures, the spines are accounted for by adjustments to the morphological representation of the dendrite, which may be preferable to adjusting the electrical parameters C, and R,, as in a recent study (Holmes, '89). This is consistent with the approach underlying the morphotonic length transformation, in trying to keep anatomical and electrical parameters separate wherever possible.
Dendritic electrotonic lengths The observed deviation from the requirements for equivalent cylinder representation does not mean that accurate models of dendritic electrical geometry cannot be developed. If detailed anatomical data are available, then compartmental and other modeling techniques exist which can be applied to neurones of arbitrary geometry. In addition to anatomical measurements, these models commonly make use of the parameters &, R,, and C, to describe the electrical properties of the neuronal membrane. R, and C , might not be expected to vary beyond fairly narrow limits, so reasonably reliable assumptions can be made, based on measurements in other systems. The model then hinges on the value for R,,,, which is likely to vary over a wide range in different cells. If R, is uniform over the entire soma1 and dendritic surface, and if there is a perfect electrical seal between the membrane and the recording microelectrode, then R, can be estimated from the membrane time constant (7,) as described above. A technical difficulty with this approach is that the voltage decay used to measure T , may be contaminated by voltage-dependent conductances, such as the time-dependent inward rectification or "sag" commonly shown by motoneurones (Fleshman et a]., '88; Rose and Vanner, '88) and pyramidal cells (Spain et al., '87; Stratford et al., '89; Mason and Larkman, '90). We tried to minimise this problem by using the decay following very brief current impulses, which are less likely to activate nonlinear membrane behaviour than longer pulses (Iansek and Redman, '73; Durand et al., '83), but sag was still a problem for some layer 5 cells. A much more serious difficulty is that R, may not be uniform. R, could vary from one dendritic region to the next in an infinity of ways, and there is little experimental evidence to allow the alternatives to be distinguished. One particular case, for which there is some evidence and which has been explored in some detail, occurs when the dendritic R, is uniform but higher than the somatic R,. The lower somatic R, could be the result of tonic inhibitory action close to the soma, an imperfect electrical seal with the recording microelectrode, or a higher density of ionic channels open at rest near the soma. Some of these channels could be activated by impalement damage or poor seal permitting the ingress of Ca2+ or Na+ ions. The likelihood of a "somatic shunt" has been suggested by a number of studies (Iansek and Redman, '73; Barrett and Crill, '74; Jack, '79; Durand and Carlen, '85; Fleshman et al., '88; Rose and Vanner, '88; Clements and Redman, '89; Stratford et al., '89). The possibility that much of the shunt is due to impalement by the recording microelectrode has received strong support recently from results obtained using the tight-seal, whole-cell recording technique. It has
been observed that neurones recorded in this way show much higher input resistances and longer membrane time constants than when impaled conventionally (Blanton et al., '89; Coleman and Miller, '89; Edwards et al., '89; Pongracz et al., '91; Spruston and Johnston, '92). Coleman and Miiler ('89) proposed that R, might be approximately a n order of magnitude higher than suggested by conventional impalement techniques. Values of R, as high i l s this were found to be consistent with the anatomical and electrophysiological data obtained for the present sample of pyramidal neurones using conventional sharp-electrode recording, if the possibility of a somatic shunt was included in the model (Stratford et al., '89). However, given likely anatomical measurement errors and a degree of uncertainty over appropriate values for C, and R, in these cells, it was impossible to choose between a wide range of possible R, values. Nevertheless, it seems likely that R, values much higher than conventionally used may be appropriate for the dendritic membranes of pyramidal cells. In view of these complexities, in this paper we have not attempted to derive a definitive electrical model for these cells. Instead, we have looked at the electrotonic lengths of dendrites and the distribution of dendritic spines with respect to electrotonic distance from the soma for three example values of' R, (10,000 &m2; T,/C,; 10 x T J C ~ , ) , assuming it to be uniform over the dendrites, and taking conventional values for R, and C,. The two lower, ''conventional,'' It, values gave electrotonic lengths of about 0.5 A for basal and oblique dendrites, 1A for the apical dendrites of layer 213 cells, and 2 A for those of layer 5 cells. These are similar to the lengths reported for hippocampal pyramidal cells (Turner and Schwartzkroin, '83; Turner, '84). In the high R, case, the dendrites would be much shorter electrically. For all cell classes, the basals and obliques were only some 0.15 A, and even the terminal arbors only 0.3-0.6 A.
Spine distributions The existence of dendrites of very different electrical lengths on the same cell means that quoting a single value for electrotonic length, even for a given set of membrane parameters, is virtually meaningless for these cells. What would be more useful would be an indication of the distribution of synaptic inputs along the electrotonic length of the cell. For spinal motoneurones, attempts have been made to determine the distribution of dendritic surface area with respect to electrotonic distance from the soma (Clements and Redman, '89). Information on synapse density from electron microscope studies might then be used to estimate the distribution of synapses (Conradi, '69; Conradi and Ronnevi, '75; Ulfhake and Cullheim, '88), although we are not aware of any such attempts to date. For at least the large majority of excitatory synapses on visual cortical pyramidal cells, a more direct indication of synapse distribution can be obtained from the dendritic spines. For the conventional R, cases, essentially all the spines on the basal dendrites were situated closer than 0.5 h from the soma. Perhaps more surprisingly, a high proportion of apical spines were also electrically close. For layer 2/3 and slender layer 5 cells, approximately 75% of apical spines were closer than 0.5 A. This reflects the compactness of the majority of the apical obliques. For thick layer 5 cells, with thick apical trunks and well-developed terminal arbors, only just over half of the apical spines were closer than 0.5 A. These electrical distributions of spines reinforce the
PYRAMIDAL NEURONE ELECTRICAL GEOMETRY concept expressed in earlier papers in this series (Larkman, '91a-c) that the dendrites of neocortical pyramidal cells can be considered in two parts. A proximal part, consisting of the basal and proximal oblique dendrites and the proximal part of the apical trunk, is highly compact electrically and receives the majority of the excitatory synaptic inputs (some 80-90%, depending on cell type). The other more distal part, consisting of the distal apical trunk, a small number of distal obliques, and the terminal arbor (if present), receives only a small minority of excitatory inputs which may be relatively remote from the soma in electrical terms (depending on the electrical parameters). For the cell types considered in this study, most of the inputs to these dendrites will be located in layer 1 and the upper part of layer 213. For the conventional R, cases, some 5% of all spines on thick layer 5 cells (representing some 500-1,000 spines) were further than 1.5 A from the soma, which is as remote as the estimate for the most distal synapses on spinal motoneurones (Rall, '77). In the high R, case, all the spines may be considered to be situated electrically close to the soma. The electrotonic spine distributions for all the R,values considered here suggest that if, during electrophysiological experiments, afferent fibres are stimulated essentially at random (as during extracellular electrode stimulation in brain slice studies of synaptic action), in the absence of knowledge of well-defined afferent pathways, the first assumption should be that the synapses stimulated will be located proximally on the postsynaptic cell. Although there is increasing evidence that the high R, case may be the most plausible, some modeling studies have suggestedthat Ri might be substantially higher than conventionally assumed (Shelton, '85; Stratford et al., '89; Spruston and Johnston, '92). This would have the effect of increasing the electrotonic length of a dendrite for a given R, value. There is also some doubt over the precise value for C, for these cells. Thus there are still a number of challenges to be met in developing adequate models of passive dendritic behaviour for these types of cells. Consideration of how such models should then be extended to include the active processes that undoubtedly play a role in shaping synaptic potentials in neocortical cells (Thomson et al., '88; Deisz et al., '91) is only in its infancy.
ACKNOWLEDGMENTS The authors wish to thank Mr. L.F. Waters and Miss A.L. Morgan for photographic assistance. This work was supported by the Wellcome Trust and the Royal Society.
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