## Abstract

Interneurons in layer 2/3 (L2/3) of the somatosensory cortex show 4 *types* of *axonal projection* patterns with reference to the laminae and borders of columns in rat barrel cortex (Helmstaedter et al. 2008a). Here, we analyzed the dendritic geometry and electrical excitability of these interneurons. First, dendritic polarity, measured based on the insertion points of primary dendrites on the soma surface, yielded a continuous one-dimensional measure without a clustering of dendritic polarity types. Secondly, we analyzed polar and vertical distributions of dendritic length. A cluster analysis allowed the definition of 7 *types* of *dendritic arborization*. Thirdly, when dendritic polarity was related to the intrinsic electrical excitability we found that the ratio of frequency adaptation in trains of action potentials (APs) evoked by current injection was correlated with the number of primary dendrites. Numerical simulations of spiking patterns in L2/3 interneurons suggested that the number of primary dendrites could account for up to 50% of this correlation. Fourthly, *dendritic arborization* was not correlated with *axonal projection*, and *axonal projection types* could not be predicted by electrical excitability parameters. We conclude that 1) dendritic polarity is correlated to intrinsic electrical excitability, and 2) the axonal projection pattern represents an independent classifier of interneurons.

## Introduction

Approximately 10–20% of neocortical neurons are nonpyramidal, mostly gamma-aminobutyric acidergic (GABAergic) interneurons (Beaulieu 1993). The shape of the soma, dendrite and axon geometry of nonpyramidal neurons is highly diverse, and numerous classifications have been suggested based on this diversity (Ramón y Cajal 1904; Lorente de No 1938; Szentagothai 1973; Jones 1975; Feldman and Peters 1978; Gibson et al. 1999; Cauli et al. 2000; Gupta et al. 2000). In several studies electrical membrane properties of nonpyramidal neurons were characterized, and a matching of electrical and morphological parameters was attempted (McCormick et al. 1985; Connors and Gutnick 1990; Foehring et al. 1991; Gupta et al. 2000; Krimer et al. 2005; Zaitsev et al. 2005). In addition, immunohistochemical markers have been suggested to define classes of nonpyramidal neurons (Cauli et al. 1997; Gonchar and Burkhalter 1997; Kawaguchi and Kubota 1997; Miyoshi et al. 2007; Gonchar et al. 2008).

The concomitant measurement of all of the suggested classification parameters in a single experiment is a major methodological challenge. Consequently, subsets of parameters (typically shape of the soma or AP firing pattern) were used to preselect populations of cells at the beginning of an experiment (e.g., Gibson et al. 1999; Beierlein et al. 2000; Cauli et al. 2000; Wang et al. 2002; Karube et al. 2004; Toledo-Rodriguez et al. 2005). It is however not clear a priori, whether subsets of dendritic, electrical or axonal parameters are in fact valid surrogates for the detailed classifications that were originally based on a much larger set of classifying parameters (Ramón y Cajal 1904; Somogyi et al. 1983; Tamás et al. 1998).

Here, geometries of dendritic arbors, passive and active membrane properties, and axonal projection patterns (cf. Helmstaedter et al. 2008a) were evaluated in a sample of 64 interneurons in layer 2/3 of rat somatosensory cortex in the fourth postnatal week (P20–29). Neurons were not preselected based on DIC image or firing patterns (cf. Methods). The analysis was pursued as follows:

Dendritic geometry was quantified using (a) an analytical one-dimensional measure of dendritic polarity and (b) a categorical scale of dendritic types derived by a cluster analysis (CA) based on the dendritic length density distribution. The 2 measures of dendritic geometry were highly correlated.

Dendritic polarity was related to electrical excitability. Here, strong correlations were found. Numerical simulations of the AP firing patterns were used to separate any possible causal effect of dendrite geometry from the contribution of membrane conductances.

Canonical correlation analysis was then used to identify whether dendritic geometry or electrical excitability could predict the type of axonal projection. This was however not the case: thus axonal projections represent an independent classifier of interneurons.

## Methods

### Preparation, Solutions

All experimental procedures were performed according to the animal welfare guidelines of the Max-Planck Society. Wistar rats (20–29 days old) were anesthetized with isoflurane and decapitated, and slices of somatosensory cortex were cut in cold extracellular solution using a vibrating microslicer (DTK-1000; Dosaka, Kyoto, Japan). Slices were cut at 350-μm thickness in a thalamocortical plane (Agmon and Connors 1991) at an angle of 50° to the interhemispheric sulcus. Slices were incubated at room temperature (22–24°C) in an extracellular solution containing 4 mM MgCl_{2}/1 mM CaCl_{2} to reduce synaptic activity. Slices were continuously superfused with an extracellular solution containing (in mM): 125 NaCl, 2.5 KCl, 25 glucose, 25 NaHCO_{3}, 1.25 NaH_{2}PO_{4}, 2 CaCl_{2}, and 1 MgCl_{2} (bubbled with 95% O_{2} and 5% CO_{2}). The pipette (intracellular) solution contained (in mM): 105 K-gluconate, 30 KCl, 10 HEPES, 10 phosphocreatine, 4 ATPMg, and 0.3 GTP (adjusted to pH 7.3 with KOH); the osmolarity of the solution was 300 mOsm Biocytin (Sigma, Munich, Germany) at a concentration of 3–6 mg/mL was added to the pipette solution, and cells were filled during 1–2 h of recording, with the exception of neurons that were later used for immunohistochemical analysis (Supplementary Material).

### Identification of Barrels and Neurons

Slices were placed in the recording chamber and inspected with a 2.5×/0.075 numerical aperture (NA) Plan objective using bright-field illumination. Barrels were identified as evenly spaced, light “hollows” separated by narrow dark stripes (Agmon and Connors 1991; Feldmeyer et al. 1999) and photographed using a frame-grabber for later analysis. Interneurons were searched in layer 2/3 above barrels of 250 ± 70 μm width (*n* = 64) using a water immersion objective 40×/0.80 NA and infrared differential interference contrast microscopy (Dodt and Zieglgansberger 1990; Stuart et al. 1993). Somata of neurons in layer 2/3 were selected for recording if a single apical dendrite was not visible. This was the sole criterion for the a priori selection of interneurons. After recording, the pipettes were positioned above the slice at the position of the recorded cells, and the barrel pattern was again photographed using bright-field illumination.

### Histological Procedures

After recording, slices were fixed at 4°C for at least 24 h in 100 mM phosphate buffer, pH 7.4, containing either 4% paraformaldehyde or 1% paraformaldehyde and 2.5% glutaraldehyde. Slices containing biocytin-filled neurons were processed using a modified protocol described previously (Lübke et al. 2000). Slices were incubated in 0.1% Triton X-100 solution containing avidin-biotinylated horseradish peroxidase (ABC-Elite; Camon, Wiesbaden, Germany); subsequently, they were reacted using 3,3-diaminobenzidine as a chromogen under visual control until the dendritic and axonal arborizations were clearly visible. To enhance staining contrast, slices were occasionally postfixed in 0.5% OsO_{4} (30–45 min). Slices were then mounted on slides, embedded in Moviol (Clariant, Sulzbach, Germany), and enclosed with a thin coverslip (see below).

### Reconstruction of Neuronal Morphologies

Subsequently, neurons were reconstructed with the aid of Neurolucida software (MicroBright-Field, Colchester, VT) using an Olympus Optical (Hamburg, Germany) BX50 microscope at a final magnification of 1000× (using a 100×, 1.25 NA objective). The reconstructions provided the basis for the quantitative morphological analysis (see below). The soma was reconstructed by drawing a contour along its largest circumference. To do so, the soma circumference was first observed in the plane of the slice (*xy*-plane). Then, at each location along the largest soma circumference in the *xy*-plane, the focus was set such that the soma border appeared most clear. This focal depth was then recorded as the *z*-coordinate of this point. In the *Rembrandt II*-software, the resulting contour line was sampled in 1-μm steps along the *y*-axis. The soma midline was determined as average *x*- and *z*-coordinate of the circumference contour for each *y*-segment. From this data, the soma was converted to a sequence of cylindrical pieces with midpoints along the soma midline, radius of half the distance between the contour lines in each *y*-segment and of 1-μm thickness along the *y*-axis. This procedure yielded a volume approximation of the soma that was rotationally symmetric along the midline of the soma. Given the sampling precision and optical limitations, we estimate the spatial precision of the soma reconstruction to be on the order of 1–2 μm.

The pial surface of the slice was also reconstructed. The reconstruction was rotated in the plane of the slice such that the pial surface above the reconstructed neuron was horizontally aligned. Thus, the *x*-axis of the reconstruction was parallel to the pia in the plane of the slice, the *y*-axis perpendicular to the pia in the plane of the slice.

Reconstructions were used for quantitative analysis of axonal projections, if the axonal arbor had a path length of at least 4 mm (cf. Helmstaedter et al. 2008a). The average axonal path length however was much higher (20 ± 11 mm, Helmstaedter et al. 2008a). Due to the geometry of the in vitro slice preparation (slice thickness: 350 μm), longer-range (>1 mm) lateral axonal projections could potentially be lost during sectioning. We chose a quantitative criterion (total axonal path length) for judgment of slicing artifacts to prevent an observer bias introduced by qualitative exclusion criteria. Dendrites were analyzed if the dendritic arbor was not cut by the sectioning procedure.

### Tissue Shrinkage

To estimate shrinkage in the *xy*-plane, we compared the distance between the soma and the pia, or the 2 somata of paired recordings before fixation (using the bright-field image taken during the experiment) and after fixation (using the computer reconstruction). Shrinkage in the *xy*-plane was 11 ± 6% (range 0–25%). A recent study (Egger et al. 2008) found nonlinear shrinkage in the *z*-direction that was strongest at the slice surface (presumably due to adhesive forces between the tissue and the glass coverslip) and amounted to approximately 50% in the *z*-direction. To reduce shrinkage in the *z*-direction, we used very thin glass coverslips (0.04 - 0.06 mm thickness, Menzel Thermo Fisher Scientific, Braunschweig, Germany). For some samples, the coverslip was removed before reconstruction. Dendritic polarity is generally assumed to occur along an axis perpendicular to the cortical surface (here: the *y*-axis; for an overview, see e.g., Jones 1975). Thus, for measures of dendritic polarity, shrinkage along the *z*-axis is of less relevance. The systematic underestimation of axonal or dendritic path length by shrinkage was estimated to be up to 25% (assuming isotropic dendritic distributions and shrinkage of 15% in *xy*, and 50% in *z*). We did not use shrinkage correction for the following analyses.

### Dendritic Polarity

The reconstruction of the neuronal morphology (ASC file format as provided by the Neurolucida software) was then read-in by the custom-made software *Rembrandt II* (written in IGOR PRO, Lake Oswego, OR). The insertion points of the primary dendrites on the approximated surface of the soma were determined as the intersection of the somatic surface and the linear interpolation between the first reconstructed point of the dendrite and the soma center. The calculation of the polarity index is described in the Results and further detailed in the Supplementary Material (Supplementary Figs 1 and 2). We use the term “dendritic polarity” for reference to the polarity of primary dendrites, and the term “dendritic geometry” to summarize geometrical properties of the dendrites (including the dendritic polarity index, dendritic density distributions, length per primary dendrite, total dendritic length, and the number of primary dendrites).

### Dendritic Density Distributions

For the analysis of dendritic density distributions, dendritic reconstructions were converted to the NEURON format (.hoc) using the custom-made parser *Rembrandt II*. In NEURON (Hines and Carnevale 1997), 3d arc length of dendritic segments were measured and summed in either radial or Cartesian bins in the plane of the slice.

The final dendritic feature vector consisted of 1) the angular dendritic length distribution for the complete dendritic tree (36 parameters, bin size 10°), 2) the angular dendritic length distribution within a sphere of 30 μm radius around the soma (36 parameters, bin size 10°), 3) the angular dependence of the maximum dendritic extent from the soma (12 parameters, bin size 30°, weight factor 2), and 4) the distribution of dendritic length along the *y*-axis (17 parameters, bin size 50 μm). The angular length distributions were aligned such that the maximum was at 0° allowing a maximal rotation of ±20°. The cluster analyses were made on this 101-dimensional feature vector in MATLAB (Mathworks, Natick, MA) using Ward's method (Ward 1963) for linkage and Euclidean distances. For an independent criterion for cluster cutoff, 4 L2/3 pyramidal neurons were included in the analysis (reconstructions are shown in the Supplementary Material). The criterion for cluster cutoff distance was that no pyramidal neuron should be linked to an interneuron (this defined the upper boundary for cluster cutoff linkage distance), and that pyramidal neurons should be linked in one cluster (this defined the lower boundary for cluster cutoff linkage distance). The correlation analyses presented in this study did not depend on the choice of cluster cutoff distance within these boundaries.

### Canonical Correlations

To test the predictive power of parameter sets, multidimensional multivariate correlations were calculated using SPSS for Windows 14.0 (SPSS, Inc, Chicago, IL): The multivariate tests of significance for within cells regression of 4 dendritic parameters with 2 axonal parameters (Fig. 10*C*) were at significance levels >0.11 for all used test metrics (Pillais, Hotellings, Wilks Lambda). For the test of 5 electrophysiological parameters with 2 axonal parameters (Fig. 11*D*), significance levels were >0.14. For the test of the 9 combined electrophysiological and dendritic parameters with 2 axonal parameters (Fig. 12), significance levels were >0.11.

### Electrical Excitability Parameters

We recorded basic parameters describing the electrical excitability of the neurons that have been typically used in the literature. A sequence of depolarizing 500-ms-long rectangular current pulses of varying amplitudes was applied, and the action potential (AP) sequences were recorded. For analysis, the trace with an initial inter-AP-interval that was closest to 10 ms (100 Hz) was used (Beierlein et al. 2003). First, the intervals between subsequent APs were measured. The inverse (“AP frequencies”) was then fitted to a monoexponential function. The ratio of the offset of this function to its initial value was recorded as the “AP frequency adaptation ratio” at 100 Hz. Then, the minimum voltage of the AP-after hyperpolarization was measured as the minimum voltage between 2 AP peaks. A monoexponential function was fitted to the after hyperpolarization (AHP) minimum voltages within one AP train, and the ratio of the offset to the initial value was recorded as “AHP-adaptation ratio.” The half width of all APs in the 100-Hz trace were determined and averaged. Half width was measured as the AP width at half-maximal amplitude. Half-maximal amplitude was measured as half the difference between the median membrane voltage during the current pulse and the AP peak voltage. Finally, the apparent AP threshold was determined as the median voltage during the depolarizing current pulse, averaged for the last sweep with no AP firing and the first sweep with at least one AP firing.

### Nomenclature

The description of morphological and physiological features in this manuscript made—where applicable—use of the nomenclature recently suggested by the Petilla Convention (Ascoli et al. 2008).

### Sample Size

In total, 64 L2/3 interneurons were analyzed. Electrical properties were analyzed for all 64 interneurons with the exception of AP threshold potential (*n* = 56). The dendritic arbor of 7 of 64 interneurons did not show sufficient staining for quantitative analysis, thus dendritic and electrical properties were analyzed simultaneously for 57 interneurons (including AP threshold potential: *n* = 49 neurons). The axonal arbor of 51 interneurons met the criteria for completeness of axonal reconstruction as described above; thus axonal and electrical properties were analyzed simultaneously for 51 interneurons. These 51 interneurons were also analyzed in the previous paper (Helmstaedter et al. 2008a). Axonal, dendritic and electrical properties could be analyzed simultaneously for 39 interneurons. These 39 interneurons also were used for the CA presented in the following study (Helmstaedter et al. 2008b). The original reconstructions are shown in the Supplementary material and are also provided online in the .asc format: http://www.mpimf-heidelberg.mpg.de/∼mhelmsta/interneurons.

## Results

### Dendritic Polarity

We devised a quantitative measure of “dendritic polarity” as a function of the distribution of the primary dendrite insertion points on the surface of the soma. Figure 1 shows 2 examples of dendritic geometries, which are designated “multipolar” (Fig. 1*A*) and “bipolar” (Fig. 1*E*). Surface maps of the soma of the 2 interneurons are shown in Figure 1*B*,*F*. The origins of the primary dendrites are marked on the soma surface maps (Fig. 1*B*,*F*, crosses). Cluster analyses (CAs) were applied to the x, y and z-coordinates of the dendrite origins. Figure 1*A* shows a neuron with eleven primary dendrites. Their origins are almost homogeneously distributed on the soma surface (Fig. 1*B*). The CA on dendrite insertion points yielded the cluster linkage plot displayed in Figure 1*C*. The maximum linkage distance was normalized to the maximum of the soma diameters *d _{x}*,

*d*(Fig. 1

_{y}*B*). The smaller linkage distances were normalized to the minimum of the 2 soma diameters. Figure 1

*D*shows the resulting distribution of normalized cluster distances for the neuron in Figure 1

*A*. The largest linkage distance is only 66% of the larger soma diameter. The second largest linkage distance is 71% of the smaller soma diameter; the third and fourth linkage distances are 50% and 40% of the smaller soma diameter. This geometry contrasts with the geometry of the “bipolar” cell shown in Figure 1

*E*–

*H*. Figure 1

*G*shows the CA linkage plot for the dendrite origins of the neuron in Figure 1

*E*. Two dendrites (dend01, dend02) are clustered at a small linkage distance of 1.4 μm (linkage distances are defined as spatial Euclidean distances). This “cluster” of 2 dendrites is linked to the remaining third dendrite (dend03) at a much larger linkage distance of 10.3 μm. Figure 1

*H*shows the 2 normalized cluster distances for the neuron in Figure 1

*E*. The maximum linkage distance spans 80% of the maximum soma diameter, the second linkage distance is low (<20% of the minimum soma diameter).

The first 3 cluster distances (*d*_{1}, *d*_{2}, *d*_{3}) and the number of dendrites (*n*) were used as independent variables for a ratiometric scale of dendrite polarity. The highest linkage distance (*d*_{1}) was normalized to the larger soma diameter, and the second and third linkage distance (*d*_{2}, *d*_{3}) were normalized to the smaller soma diameter. Polarity was defined as a function of these 4 parameters (for details see Methods and Supplementary Material). The interneurons shown in Figure 1 had a dendrite polarity index of 6.1 (Fig. 1*A*) and 2.3 (Fig. 1*E*), respectively. This dendritic polarity index was then used to investigate possible functional consequences of “bipolarity” or “multipolarity” of interneurons. Figure 2 shows the relation between dendritic polarity and the number of primary dendrites (*n* = 57). Dendritic polarity was highly correlated to the number of primary dendrites (*r* = 0.91, *p* < 1e-6).

### Dendritic Density Distributions

We next investigated the spatial distribution of dendritic length of L2/3 interneurons in 2D and 3D. To detect the polarity of dendritic trees we computed polar histograms of dendritic path lengths both for the entire dendritic tree (Fig. 3*B*) and for the part of the dendritic tree that was located within a sphere of 30-μm radius from the soma center (Fig. 3*C*). Figure 3*D* shows the polar distribution of the maximal distance of dendrites from the soma center. To measure dendritic arborizations with tufts we also quantified the dendritic length distribution along an axis perpendicular to the brain surface in the plane of the slice (shown in Fig. 3*E*). These parameters were used to detect clusters of types of dendritic geometry. Figure 4*A* shows the linkage plot of a CA (Ward's Method; Ward 1963). Four L2/3 pyramidal neurons were included in the analysis, and the cutoff linkage distance was chosen such that pyramidal neurons were not linked to interneurons (dashed line in Fig. 4*A*, for a detailed account of the cutoff definition see Methods; reconstruction are shown in the Suppl. Material). This analysis yielded 7 types of dendrite geometries (“*dendritic types*” shown in Supplementary Figs 3–6). We then evaluated the relation between this high-dimensional measure of dendrite geometry and the one-dimensional dendrite polarity index derived above. Figure 4*B* shows the dendritic polarity index for the 7 dendritic types, which was significantly different between types (ANOVA, *P* < 0.0005; a post hoc Newman–Keuls test showed that clusters 1,2 were significantly different from clusters 6,7, *P* < 0.05).

### Dendritic Polarity and Electrical Membrane Properties

Subsequently, we investigated the relation between dendritic polarity and electrical membrane properties using measures of intrinsic electrical excitability that have been employed in many studies (e.g., Connors and Gutnick 1990). Figure 5*A* shows the AP discharge patterns of 2 inhibitory neurons in response to 500-ms depolarizing current pulses. The dendritic reconstructions and soma surface plots of the 2 cells are shown in Figure 5*B*. We analyzed 5 relevant characteristics of AP discharge patterns: 1) The AP frequency adaptation ratio at about 100-Hz initial spike frequency (Fig. 5*C*), 2) adaptation of the AP-AHP in the same 100-Hz trace (Fig. 5*D*), 3) AP half widths (Fig. 5*E*), 4) AP threshold, and 5) the apparent somatic input resistance measured as I/V ratio for assumedly passive membrane potentials around *V*_{rest} (not illustrated; for details of the analysis, cf. Methods).

We found a strong correlation between dendritic polarity and AP frequency adaptation ratio (Fig. 6*A*, *r* = 0.49, *P* < 0.005). Interneurons with positive AP frequency adaptation ratios (i.e., an increase in AP frequency during the train) had a high dendritic polarity index (≥4). The number of primary dendrites was also highly correlated with AP frequency adaptation ratio (*r* = 0.55, *P* < 0.0001, Fig. 6*B*).

This correlation between the number of primary dendrites and AP frequency adaptation ratio could be either purely correlative, or could represent a direct causal effect of the morphology on electrical properties. We therefore investigated by numerical simulations, to what degree the dendritic morphology alone can account for these correlations.

The interneuron shown in the left panels of Figure 7*A*,*B* had 7 primary dendrites (polarity index 4.3) and had a positively adapting AP pattern (AP frequency adaptation ratio = +0.7). The interneuron shown in the right panels of Figure 7*A*,*B* had 4 primary dendrites (polarity index = 2.7) and a negatively adapting firing pattern (AP frequency adaptation ratio = −1). The dendritic arbors were used in a multicompartment numerical model with a normalized set of conductance distributions (Mainen and Sejnowski 1996; Schaefer et al. 2003) assuming Na^{+}-, K^{+}-, K^{+}_{Ca}, Ca^{2+}-conductances with constant densities for the soma, dendrites and a prototypic axon, respectively. The absolute values of conductance densities were adjusted such that 4 control pyramidal neurons showed a strongly adapting firing pattern (not shown). Figure 7*C* shows the AP firing patterns that were simulated for the 2 interneurons shown in Figure 7*A*,*B* with this set of conductances (conductance set 1). All electrical parameters were similar between the 2 neuron simulations, except for the geometry of the soma and dendrites as shown in Figure 7*B*. The examples indicate that the dendritic geometry can partially account for differences in AP frequency adaptation ratios.

Then, these numerical simulations were made in all interneurons, and AP frequency adaptation ratio was measured in each of these simulations. Figure 8*A* shows the relation between the simulated AP frequency adaptation ratio and the number of primary dendrites, which were significantly correlated (*r* = 0.38, *r*^{2} = 0.14, *P* < 0.01, *n* = 53). When compared with the experimentally measured correlation between AP frequency adaptation ratio and the number of primary dendrites (Fig. 6*B*, *r*^{2} = 0.3), the simulations could thus account for up to 50% of the experimentally measured correlation between AP frequency adaptation and the number of primary dendrites. Figure 8*B* shows the relation between the simulated and the measured AP frequency adaptation ratios. Results for pyramidal neurons are indicated by open triangles for comparison. Results for different sets of conductances are shown in the Supplemental Material.

### Dendritic Geometry and Axonal Projection Types

Next, the relation between dendritic polarity and the axonal projection geometry was investigated. Figure 9*A*,*B* shows the axonal projection of the 2 interneurons illustrated in Figure 1. The neuron with a high dendritic polarity index (=6.1) had a local axonal projection in layer 2/3 (Fig. 9*A*), whereas the neuron with a bipolar dendritic polarity index (=2.3) projected vertically within the home column (Fig. 9*B*). Figure 9*C* shows the relation between dendritic polarity and *verticality* of axonal projection (*verticality* was quantified as the ratio of axonal path length extending below layer 2/3 within the home column, cf. Helmstaedter et al. 2008a). Dendritic polarity was only weakly correlated with *verticality* of axonal projection (*r* = −0.37, *P* < 0.02, Figure 9*C*, the correlation was outlier-dominated, rank correlation was not significant). Dendritic polarity was also not correlated with *laterality* of axonal projection (*r* = 0.07, Fig. 9*D*; *laterality* was quantified as the ratio of axonal path length extending to neighboring columns, cf. Helmstaedter et al. 2008a).

The weak parametric correlation between dendritic polarity and axonal *verticality* was mainly caused by 3 cells with vertical axon projections and a low polarity index <3 and a group of 4 cells with no vertical axon projection and a high polarity index >5 (without these cells: *r* = 0.04). It is important to note that for the majority of interneurons this correlation did not allow to predict axonal projection patterns. “Bipolar” or “tripolar” cells could both be highly vertical or entirely locally projecting (Fig. 9*C*). Dendritic polarity was thus not a predictor of *verticality* or *laterality* of axonal projection.

To validate this finding we also investigated whether the 7 *dendritic types* were related to the 3 main *axonal projection types* found in the previous study (Helmstaedter et al. 2008a). Figure 10*A* shows the axonal laterality and verticality of the 7 dendrite geometry types as determined above (color code in Fig. 10*A*). There was no significant difference in axonal laterality and verticality among the 7 dendritic types (ANOVA, *P* > 0.05). Figure 10*B* shows the complementary distribution of the axonal types in the dendritic parameter space (reduced to its first 3 principal components for display). As a fourth test we measured the canonical correlation between 4 dendritic parameters (number of primary dendrites, total dendritic length, dendritic polarity index, length per primary dendrite, cf. Table 1) and the 2 axonal parameters (laterality and verticality). No multidimensional correlation could be detected (Fig. 10*C*, *P* > 0.07, cf. Methods). We conclude that dendritic density is not a predictor of *axonal projection types*.

Dendritic polarity (a.u.) | n of primary dendrites | Length per primary dendrite (μm) | Total dendritic length (mm) | ||

Types | |||||

Local inhibitors | (n = 19) | 4.4 ± 28% | 6.1 ± 41% | 550 ± 52% | 2.8 ± 21% |

Lateral inhibitors | (n = 14) | 4 ± 21% | 5.7 ± 35% | 670 ± 50% | 3.5 ± 38% |

Translaminar inhibitors | (n = 10) | 3.4 ± 23% | 4 ± 31% | 700 ± 29% | 2.7 ± 26% |

L1 inhibitors | (n = 4) | 3.6 ± 18% | 4.2 ± 35% | 860 ± 52% | 3.2 ± 29% |

All | (n = 57) | 3.9 ± 27% | 5.1 ± 42% | 660 ± 48% | 2.9 ± 33% |

Dendritic polarity (a.u.) | n of primary dendrites | Length per primary dendrite (μm) | Total dendritic length (mm) | ||

Types | |||||

Local inhibitors | (n = 19) | 4.4 ± 28% | 6.1 ± 41% | 550 ± 52% | 2.8 ± 21% |

Lateral inhibitors | (n = 14) | 4 ± 21% | 5.7 ± 35% | 670 ± 50% | 3.5 ± 38% |

Translaminar inhibitors | (n = 10) | 3.4 ± 23% | 4 ± 31% | 700 ± 29% | 2.7 ± 26% |

L1 inhibitors | (n = 4) | 3.6 ± 18% | 4.2 ± 35% | 860 ± 52% | 3.2 ± 29% |

All | (n = 57) | 3.9 ± 27% | 5.1 ± 42% | 660 ± 48% | 2.9 ± 33% |

Note: Data are reported as mean ± SD pooled by the types of axonal projection (cf. Helmstaedter et al. 2008a). Briefly, L2/3 local inhibitors projected to L2/3 of the home column; L2/3 lateral inhibitors projected a significant part of the axon to L2/3 of the surround columns; L2/3 translaminar inhibitors projected a significant part of their axon to L4 and L5 of the home column. SD is reported in percent of mean to allow comparison of parameter homogeneity between types. For statistical analysis cf. Results and Figures 10–12.

### Clustering of Electrical Membrane Properties

We next investigated whether electrical membrane properties that were measured in many previous studies allowed a definition of interneuron classes in the unselected sample of L2/3 interneurons. Interneurons with positive AP frequency adaptation and small AP half width have been previously defined as “fast-spiking (FS)” cells (e.g., Kawaguchi 1993; Kawaguchi and Kubota 1993). Figure 11*A* shows the correlation between AP half width and AP frequency adaptation (*r* = −0.37, *P* < 0.005). Positive frequency adaptation ratios predict half widths of <0.6 ms. A CA suggested 2 clusters (e.g., FS vs. non-FS cells, cf. inset in Fig. 11*A*). Figure 11*B* shows the correlation between somatic input resistance and AP half width (*r* = 0.46, *P* < 0.0005) that was also suggested to define FS cells (Kawaguchi 1993; Kawaguchi and Kubota 1993). This distribution was, however, only weakly clustered.

### Electrical Membrane Properties and Axonal Projection Types

We then examined whether electrical membrane properties could be matched to *axonal projection types* (defined by the projection of L2/3 interneuron axons with reference to cortical columns; Helmstaedter et al. 2008a). Briefly, the axons of L2/3 *local inhibitors* projected to L2/3 of the home column; L2/3 *lateral inhibitors* projected to a significant degree to L2/3 of the surround columns; the axons of L2/3 *translaminar inhibitors* projected to a significant degree to L4 and L5 of the home column. Figure 11*A*,*B* shows the distribution of *local*, *lateral*, and *translaminar inhibitors* in the relation between AP half width, AP frequency adaptation ratio and input resistance. Figure 11*C* shows the distribution of *local*, *lateral*, and *translaminar inhibitors* in the space of electrical excitability parameters (depicted in their first 3 principal components, cf. Table 2). *Local*, *lateral* and *translaminar inhibitors* showed no significant differences in electrical excitability parameters (ANOVA, *P* > 0.29, Fig. 11*A*–*C*Supplementary Fig. 7, color code). Figure 11*D* shows the canonical correlation between the 5 major electrical excitability parameters (see above) and axonal projection patterns. As there was no significant correlation (*P* > 0.1, cf. Methods), we conclude that the suggested classifications of interneurons by electrical excitability parameters could not predict the type of axonal projections.

Input resistance (GΩ) | AP half width (ms) | AP frequency adaptation ratio | AHP adaptation (mV) | AP threshold potential (mV) | ||

Types | ||||||

Local inhibitors | (n = 19) | 0.24 ± 59% | 0.51 ± 32% | −0.48 ± 89% | 0.98 ± 350% | −44 ± 16% |

Lateral inhibitors | (n = 15) | 0.22 ± 56% | 0.55 ± 38% | −0.33 ± 158% | 2.4 ± 149% | −45 ± 16%, n = 13 |

Translaminar inhibitors | (n = 10) | 0.29 ± 39% | 0.48 ± 27% | −0.63 ± 63% | 1.6 ± 180% | −52 ± 7%, n = 9 |

L1 inhibitors | (n = 4) | 0.2 ± 24% | 0.76 ± 52% | −0.64 ± 18% | 7.9 ± 46% | −46 ± 21%, n = 2 |

All | (n = 64) | 0.26 ± 53% | 0.54 ± 41% | −0.5 ± 86% | 2.1 ± 171% | −46 ± 17%, n = 56 |

Input resistance (GΩ) | AP half width (ms) | AP frequency adaptation ratio | AHP adaptation (mV) | AP threshold potential (mV) | ||

Types | ||||||

Local inhibitors | (n = 19) | 0.24 ± 59% | 0.51 ± 32% | −0.48 ± 89% | 0.98 ± 350% | −44 ± 16% |

Lateral inhibitors | (n = 15) | 0.22 ± 56% | 0.55 ± 38% | −0.33 ± 158% | 2.4 ± 149% | −45 ± 16%, n = 13 |

Translaminar inhibitors | (n = 10) | 0.29 ± 39% | 0.48 ± 27% | −0.63 ± 63% | 1.6 ± 180% | −52 ± 7%, n = 9 |

L1 inhibitors | (n = 4) | 0.2 ± 24% | 0.76 ± 52% | −0.64 ± 18% | 7.9 ± 46% | −46 ± 21%, n = 2 |

All | (n = 64) | 0.26 ± 53% | 0.54 ± 41% | −0.5 ± 86% | 2.1 ± 171% | −46 ± 17%, n = 56 |

Note: Data are reported as mean ± SD pooled by the types of axonal projection (cf. Helmstaedter et al., titled “Neuronal correlates of lateral, local and translaminar inhibition with reference to cortical columns”). Standard deviation is reported in percent of mean to allow comparison of parameter homogeneity between types. For statistical analysis cf. Results and Figures 10, 11.

### Electrical Membrane Properties, Dendritic Geometry, and Axonal Projections

We finally tested whether the combination of electrical and dendritic parameters could predict the type of axonal projection. Figure 12*A* shows the distribution of *local*, *lateral* and *translaminar inhibitors* in the first 3 principal components of the combined dendritic and electrical excitability parameter space. The canonical correlation analysis (Fig. 12*B*) still showed no predictive capacity (*r*^{2} < 0.25, *P* > 0.1, cf. Methods).

## Discussion

### Experimental Constraints

The measurement of multiple morphological, electrical excitability, and antigen expression parameters in neocortical interneurons represents a major challenge: the concurrent measurement of many of these parameters is hardly feasible, neither in vitro nor in vivo. Therefore, in many studies only a subset of the available parameters (often pairs of parameters) could be used for the definition of neuron identity. Findings were related to the classification in this highly reduced parameter space (Beierlein et al. 2000, 2003; Cauli et al. 2000; Porter et al. 2001; Wang et al. 2002; Bacci et al. 2004; Toledo-Rodriguez et al. 2005; Dumitriu et al. 2007). The present study was aimed at an unbiased quantitative analysis of the low-dimensional relations between interneuron properties (i.e., between pairs, triplets or quadruplets of parameters). The goal was to find out whether small (2–5) sets of parameters could be used as valid predictors of higher-dimensional classifications. We found that dendritic geometry was partly predictive of intrinsic electrical excitability, as expected from previous simulation studies comparing electrical excitability in excitatory and inhibitory neurons with strongly differing morphologies (Mainen and Sejnowski 1996). Our results suggest, however, that axonal projection patterns are almost independent of dendritic geometry or electrical excitability parameters. These findings are consistent with a series of studies that used quantitative methods for the analysis of electrophysiological or dendritic classes of neurons (Nowak et al. 2003; Krimer et al. 2005; Zaitsev et al. 2005). A quantitative definition of electrophysiological types did only partially correlate with immunohistochemical marker expression. It did not allow a prediction of qualitative morphological properties in monkey dorsolateral prefrontal cortex (Zaitsev et al. 2005). In a second study from the same preparation, electrophysiologically defined classes of interneurons did not predict the qualitative description of axonal and dendritic geometry (with the exception of chandelier neurons, Krimer et al. 2005). This was also found in in vivo recordings from cat visual cortex (Nowak et al. 2003).

### Limitations of the In Vitro Slice Preparation

The morphological analysis of neurons stained intracellularly in vivo provides reconstructions of complete axonal arbors with isotropic distribution, including longer-range projections spanning several millimeters (e.g., Bruno and Sakmann 2006). Although in vivo staining of excitatory neurons is well feasible, there are only very few reports of reconstructions of interneurons stained in vivo (e.g., Hirsch et al. 2003). Therefore, a quantitative analysis of interneuron morphology on a sufficiently large sample (*n* > 30) had to be done in an in vitro preparation. Errors introduced by the in vitro slice preparation include 1) putative truncation of axons or dendrites; 2) anisotropic shrinkage during fixation (especially in the direction normal to the coverslip, cf. Egger et al. 2008); 3) introduction of a direction bias by the choice of the slice plane. We attempted to minimize these artifacts by 1) quantitative criteria for inclusion of reconstruction based on the total axonal path length (cf. Methods); 2) usage of very thin glass coverslips, and projection of the measured 3D path lengths into the plane of the slice for analysis; 3) choice of a slice plane and slice thickness that presumably contains one cortical column (*z*-direction) and at least 2 neighboring cortical columns (*x*-direction). In addition, the term dendritic polarity is usually employed for polarity along an axis perpendicular to the cortical surface (e.g., Jones 1975). Thus, for the measures analyzed in this study, the projection of dendritic path length into the plane of the slice could be expected to introduce only minor artifacts (for details, cf. Methods).

### Comparison to Previous Studies of Dendritic Geometry

We have presented approaches for quantifying dendritic geometry. A multidimensional approach included the radial and vertical distribution of dendritic densities around the soma. In qualitative descriptions of dendritic shape, typically, “bipolar” geometries were distinguished from “bitufted” geometries (Kawaguchi and Kubota 1997; Feldman and Peters 1978; Porter et al. 1998). Although both types were described as having bipolar dendritic origins, bitufted geometries were defined to show extensive branching at a particular distance from the soma. All of these features should be captured in the set of radial and vertical dendritic density distributions (cf. Fig. 3). However, the unsupervised clustering did not yield 2 distinct populations of bipolar versus bitufted dendritic geometries in our sample (cf. Supplementary Material Figs 3–6, types 1,2,4 for illustration). It may be speculated that the distinction of bitufted versus bipolar dendritic geometries is not a valid classifier of interneurons. Our quantitative classification of dendritic types also yielded types with similarities to large multipolar basket cell dendrites (types 5,7; Supplementary Figs 5 and 6; Fairén et al. 1984), to neurogliaform neurons (dendritic type 6, Supplementary Fig. 5; Jones 1984), and to tripolar neurons with a major descending dendrite (dendritic type 3, Supplementary Fig. 4).

### Correlation between Dendritic Geometry and Electrical Excitability

The electrical excitability of a neuron can be properly modeled by cable theory using the geometry of the soma, dendrites and axon, as well as the distribution of transmembrane conductances along the neuron's membrane (e.g., Rall 1959; Koch 1999). Both the geometry and the conductance distributions are thus causal for the electrical excitability measured at the soma. This causality was first quantified for the variability between cell types by Mainen and Sejnowski (1996). Schaefer et al. (2003) showed that specific morphological parameters (the number and position of oblique dendrites) could be predictive of specific functional properties of pyramidal cells (i.e., the likelihood of calcium-spikes in the distal apical dendrites). Here, we showed that the morphological variability within a cell class, L2/3 interneurons, could explain a large fraction of the variability in electrical excitability (i.e., in the AP discharge patterns measured at the soma). Therefore, electrical excitability and dendritic geometry cannot be considered as independent classifiers of interneurons. Instead we show that axonal projection types were such independent classifiers.

### Comparison to Classifications of Hippocampal Interneurons

The attempts of classification of hippocampal interneurons have been more successful in finding clear correlations between dendritic geometry and axonal projections (for a review, see e.g., Somogyi and Klausberger 2005). This may be due to a more clearly defined stratification into layers in the hippocampus. Layers in the hippocampus correspond to defined parts of pyramidal neurons in CA1 (soma, basal dendrites, apical dendrites). Axonal projections of hippocampal interneurons have been shown to be restricted to single layers of the hippocampus, resulting in subcellular specificity of axonal projections (for reviews: McBain and Fisahn 2001; Whittington and Traub 2003; Jonas et al. 2004). In contrast, the spatial precision of intracortical axonal projections is not yet fully understood (e.g., Di Cristo et al. 2004; Douglas and Martin 2007).

### Surrogate Parameters

Low-dimensional subsets of parameters can only be used as surrogates for higher-dimensional classifications if they have been shown to be predictors of this higher-dimensional classification in an unselected sample of neurons. Typically employed surrogate parameter sets (Beierlein et al. 2000, 2003; Cauli et al. 2000) could not be shown to be sufficient for this requirement in our data set. Thus, low-dimensional classifications for neocortical interneurons have to be questioned and a multidimensional analysis is needed to define interneuron *groups* (cf. the accompanying study; Helmstaedter et al. 2008b).

## Supplementary Material

Supplementary material can be found at: http://www.cercor.oxfordjournals.org/.

## Funding

This work was supported by the Max-Planck Society.

We would like to thank Drs Jochen Staiger, Arnd Roth, Andreas Schaefer and Idan Segev for fruitful discussions, and Drs Jochen Staiger and Andreas Schaefer for helpful comments on previous versions of the manuscript, Ms Marlies Kaiser for expert technical assistance and Jonas Tesarz for help with reconstructions. *Conflict of Interest*: None declared.