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## Abstract

The neocortical neuropil has a strong vertical (orthogonal to pia) orientation, constraining the intracortical flow of information and forming the basis for the functional parcellation of the cortex into semi-independent vertical columns or ‘modules’. Apical dendrites of excitatory pyramidal neurons are a major component of this vertical neuropil, but the extent to which inhibitory, GABAergic neurons conform to this structural and functional design is less well documented. We used a gene gun to transfect organotypic slice cultures of mouse and rat neocortex with the enhanced green fluorescent protein (eGFP) gene driven by the promoter for glutamic acid decarboxylase 67 (GAD67), an enzyme expressed exclusively in GABAergic cells. Many GAD67–GFP expressing cells were highly fluorescent, and their dendritic morphologies and axonal patterns, revealed in minute detail, were characteristic of GABAergic neurons. We traced 150 GFP-expressing neurons from confocal image stacks, and estimated the degree of vertical bias in their dendritic trees using a novel computational metric. Over 70% of the neurons in our sample had dendritic trees with a highly significant vertical bias. We conclude that GABAergic neurons make an important contribution to the vertical neocortical neuropil, and are likely to integrate synaptic inputs from axons terminating within their own module.

## Introduction

Neurons are inherently polarized, the result of having one process (the axon) structurally and functionally different from all their other processes (the dendrites) (Black and Baas, 1989; Mattson, 1999). While the morphological polarity between axon and dendrites is common to virtually all neurons, a significant subset of neurons exhibit morphological polarity also within their dendritic tree. For example, pyramidal neurons of the cerebral cortex extend a single, thick apical dendrite in one direction, and a multitude of smaller, basal dendrites in other directions. The apical dendrite is preferentially oriented upwards, towards the pial surface. The vertically oriented apical dendrites of pyramidal neurons (Escobar et al., 1986; Peters and Sethares, 1991; White and Peters, 1993), together with vertically oriented afferent, efferent and intrinsic axons (Lorente de Nó, 1949), are the major contributors to the predominantly vertical grain of the cortical neuropil and impart a strong vertical bias to the flow of information in the neocortex, forming the structural basis for the well-documented parcellation of the cerebral cortex into semi-autonomous anatomical and functional units, called ‘columns’ or ‘modules’ (Colonnier, 1966; Hubel and Wiesel, 1969; Szentagothai, 1975; Mountcastle, 1997).

Pyramidal neurons are the dominant cell type of the neocortex, and function to excite postsynaptic neurons, both locally and in other cortical areas and subcortical targets, by secreting glutamate as a neurotransmitter (Ottersen and Storm-Mathisen, 1984; Carder and Hendry, 1994). In contrast, ~15–25% of all neocortical neurons (Fitzpatrick et al., 1987; Ren et al., 1992; Micheva and Beaulieu, 1995; Gabbott et al., 1997) are inhibitory, non-pyramidal neurons, which utilize GABA as a neurotransmitter. Dendrites of GABAergic cortical neurons usually lack morphological polarity (i.e. all their dendrites appear morphologically equivalent), but may be spatially polarized. Indeed, some GABAergic neurons have bipolar or bitufted dendritic trees, in which dendrites emanate mainly from two opposite poles of the cell body. The majority of GABAergic neurons, however, have multipolar, or ‘stellate’, morphology with dendrites emanating in all directions (Houser et al., 1983; Kisvarday et al., 1990; Prieto et al., 1994). Although many multipolar neurons appear to have vertically elongated dendritic trees (Jones, 1975; Feldman and Peters, 1978), this vertical bias has rarely been studied quantitatively (McMullen et al., 1984).

All GABAergic neurons express GAD (glutamic acid decarboxylase), the major biosynthetic enzyme for GABA (Ribak, 1978; Houser et al., 1983; Martin and Rimvall, 1993). There are two known GAD isoforms, GAD65 and GAD67, which are the products of two different genes (Erlander et al., 1991; Kaufman et al., 1991; Esclapez et al., 1994). Neuron-specific and region-specific expression of the GAD67 gene is conferred by its complex 5′ flanking region, which includes several neuron-restrictive silencer sequences and putative region-specific enhancers (Szabo et al., 1996; Katarova et al., 1998). To visualize GABAergic neurons in living tissue, and to study their axonal and dendritic morphologies, we linked the cloned mouse GAD67 promoter region to an enhanced green fluorescent protein (eGFP) reporter gene, transfected neurons in organotypic cortical slices with the resulting construct, and used a novel metric to quantify the degree of vertical bias in dendrites of transfected neurons expressing GFP. Some of these results have been previously reported in abstract form (Jin et al., 1999).

## Materials and Methods

### GAD67–GFP DNA Preparation

A 9 kb segment of the GAD67 gene's 5′ regulatory region, including the first intron, drives expression of a lacZ reporter transgene in hippo-campus and neocortex of transgenic mice in a partially cell-type specific pattern (Katarova et al., 1998). To improve specificity, we used a slightly longer segment, and to allow visualization of the reporter gene product in living tissue, we fused it in-frame to eGFP. We cloned a 10.3 kb segment of the murine GAD67 genomic DNA, including the 5′ upstream regulatory region, the first (untranslated) exon and intron and a portion of exon 2, from the HindIII site (at the 5′ end) to the BamHI site (at the 3′ end) (Katarova et al., 1998), and inserted it in-frame between the HindIII and the BamHI insertion sites of the pEGFP-1 vector (Clontech, Palo Alto, CA). The resulting construct encodes a fusion protein consisting of the first 17 amino acids of GAD67, four amino acids of the vector and the full-length eGFP.

### Organotypic Culture Preparation

Organotypic cortical slice cultures were prepared using a previously described method (Stoppini et al., 1991). Postnatal day 2–7 mouse or rat pups (postnatal day 0 being the first 24 h after birth) were deeply anesthetized by Metofane inhalation, decapitated, and the brains rapidly removed and immersed in ice-cold artificial cerebrospinal fluid (ACSF, containing 126 mM NaCl, 3.0 mM KCl, 1.3 mM MgSO4, 2.5 mM CaCl2, 1.2 mM NaH2PO4, 26 mM NaHCO3, and 20 mM dextrose, saturated with 95%:5% O2:CO2). The following procedures were performed under sterile conditions. Coronal brain slices of parietal cortex, 350 μm thick, were cut with a Vibraslicer (WPI, Sarasota, FL) into ice-cold ACSF. Slices were then placed on transparent Millicell membrane inserts (Millipore, Bedford, MA), usually two slices/insert, in 30 mm Petri dishes containing 1 ml of culture medium, composed of (per 100 ml) 46 ml Eagle's basal medium, 25 ml Earle's balanced salt solution, 25 ml horse serum, 1 ml penicillin– streptomycin–glutamine (all from Life Technologies, Rockville, MD), and 3 ml of 20% glucose solution. The slices were kept in a humidified incubator at 37°C with a 5% CO2-enriched atmosphere, and the medium was changed twice a week.

### Gene Gun Mediated Transfection

Cultured slices were transfected using a Helios gene gun (Bio-Rad, Hercules, CA). Cartridges were prepared according to the manufacturer's manual, using manufacturer's supplies. The full, supercoiled plasmid DNA (total size 14.6 kb), prepared using an endotoxin-free kit (Qiagen, Valencia, CA), was used for transfection. Gold particles (1.6 μm diameter) were coated with plasmid DNA at a ratio of 1 mg gold to 2 μg plasmid DNA, precipitated onto the inner wall of Tefzel tubing, and the tubing cut into individual cartridges. Each cartridge contained ~1 μg plasmid DNA. Assuming (arbitrarily) a 50% coating efficiency, each gold particle carried on average ~2700 copies of the DNA construct. After 1–2 days in culture, the slices were bombarded with one cartridge per insert under 100 psi helium pressure. After 2–35 more days in culture, slices were removed for imaging.

### Confocal Microscopy

Explants were cut out together with the membrane and transferred to an imaging chamber filled with fresh ACSF. Well-separated neurons with bright GFP expression and no signs of truncation were selected for imaging. An inverted Zeiss LSM510 laser-scanning confocal microscope was used to capture images. Images were generally taken with a ×20, 0.7NA objective, using the 488 nm argon laser line and a 505–550 nm bandpass emission filter. Image stacks were collected at ~1.5–2 μm Z-axis steps to cover the full depth of the dendritic tree. Stacks were superimposed digitally, but in most cases the full 3-D dataset was saved for further analysis. At the end of each session, images were taken at a lower magnification (×10 objective) in fluorescence and transmission modes, to record the locations of the GFP-expressing neurons relative to the pia and white matter.

### Neuronal Tracing

Confocal image stacks of individual neurons were traced digitally using Neurolucida software (Microbrightfield, Colchester, VT); all subsequent analysis was done on the traced data. For each neuron, a line was drawn on a low-power image from the cell body to the nearest point on the pial surface, and the coordinates of the two endpoints measured; these were later used in the analysis to calculate angle of growth relative to the vertical axis.

### Data Analysis

The Neuroexplorer software (Microbrightfield) was used to calculate numerical parameters of dendritic morphology, including soma area, total dendritic length and number of dendritic end points, and to measure the maximal vertical and horizontal extent of the dendritic tree. Dendritic coverage area was defined as the area of the smallest convex polygon that could encompass all the dendritic branches. Dendritic tree aspect ratio was defined as the ratio of the maximal vertical extent to the maximal horizontal extent of the dendritic tree. To analyze dendritic growth directions, Neurolucida data in ASCII format was imported into HiQ (National Instruments, Austin, TX) and calculations were carried out using custom-programmed HiQ scripts. Since organotypic slices grown in serum-supplemented medium tend to become quite thin, coordinates along the depth (Z) dimension varied very little, compared with coordinates in the plane of the slice; we therefore ignored the depth dimension in this analysis and treated the neurons as planar projections. The program first converted all dendritic chords (i.e. the segments defined by two sequential data points along the dendrite) into vectors anchored at the origin of axes, then calculated the sum of the absolute values of the dot product of all vectors with a normalized vector representing the vertical direction (equivalent to the total length of the projections of the vectors on the vertical axis), and divided by the sum of all the vector norms (equivalent to the total dendritic length), resulting in the weighted average of the absolute value of the cosine of the angle of growth relative to the vertical axis (WACAV). The average WACAV of neurons with randomly oriented dendrites should be 2/π or ~0.64; this applies both to a two-dimensional neuron with dendrites oriented randomly in the plane, and to a three-dimensional neuron with dendrites oriented randomly in space, as long as the WACAV is calculated on the two-dimensional projection of the neuron on a vertical plane (see Appendix).

To allow for an easier comparison of polarizations, we normalized the WACAV values by calculating an Index of Polarization (IOP) equal to

$\mathit{IOP}\ {=}\ \frac{(\mathit{WACAV}\ {\mbox{--}}\ 2/{\pi})}{(1\ {\mbox{--}}\ 2/{\pi})}$
The IOP should give values ranging from ~(–1.75) for purely horizontal dendrites, to 0 for random dendrites, to 1 for purely vertical dendrites.

## Results

### Morphology of Neurons Expressing GAD67–GFP in Organotypic Cortical Slices

The data presented here were collected from 143 organotypic slice cultures prepared from 40 mice and 8 rats. To visualize GABAergic, non-pyramidal cortical neurons, we biolistically transfected organotypic cortical slices with the eGFP gene controlled by the mouse GAD67 promoter (see Materials and Methods). The number of GFP-expressing neurons observed in successfully transfected slices varied from ∼10 (or occasionally less) to more than 100, possibly reflecting fluctuations in the number of gold particles deposited on the wall of each gene gun cartridge. Invariably, any GFP-expressing neuron examined was found to contain a gold particle in its nucleus. Levels of GFP expression varied between neurons: in many, mostly small neurons, GFP fluorescence was low and mostly restricted to the cell body, but in many medium and large transfected neurons, fluorescence level was high and allowed clear visualization of the full dendritic and axonal morphology. GFP-expressing neurons occurred in all cortical layers, although in younger slices they had a tendency to be more concentrated in the deep layers.

With almost no exceptions, GAD67–GFP expressing neurons were non-pyramidal, as typical of GABAergic neurons. Their dendritic and axonal morphologies were varied; Figures 1 and 2 illustrate confocal images of neurons representative of the range of morphologies observed. The great majority of GFP-expressing neurons were either bitufted (Figs 1A, 2C, 3B) or multipolar (Figs 1B,D, 2A,D, 3CG). While most of the GFP-expressing neurons had non-spiny or sparsely spiny dendrites typical of mature inhibitory cells in the cortex, some were more densely spiny (Fig. 1D), possibly reflecting an immature stage of development (Lund et al., 1977; McMullen et al., 1988a; Seay-Lowe and Claiborne, 1992). The GFP fluorescence also revealed highly detailed axonal arbors, rivaling in complexity those visualized by the Golgi or the intracellular dye-filling methods (Figs 2 and 3). Some GFP-containing axons exhibited varicosities at regular intervals, which were potentially en passant synaptic boutons (Fig. 2B).

Many of the GFP-expressing neurons had dendritic and axonal morphologies comparable to previously described putative or confirmed GABAergic neurons of the cerebral cortex. For example, upper layers multipolar or bitufted neurons with large cell bodies (>20 μm in diameter) and with axons giving rise to horizontally directed branches (Fig. 1B, leftmost neuron; Fig. 2C,D), resembled Golgi-stained large basket cells in monkey somatosensory cortex [type 1 of Jones (Jones 1975)] and parvalbumin-containing, axosomatic ‘fast spiking’ cells in rat frontal cortex [Fig. 6A in Kawaguchi and Kubota (Kawaguchi and Kubota, 1998)]. Among neurons in our sample with less common morphologies were lower layer VI cells such as the fan-shaped neuron of Figure 1A, which was similar to Golgi-impregnated neurons described by Ferrer et al. (their fig. 3C) in the rat sensorimotor cortex (Ferrer et al. 1986), and the inverted pyramidals in Figure 1C (upper two neurons), which were comparable with the GAD-immunostained neurons described by Prieto et al. (their figs 13:4 and 14:4) in cat auditory cortex (Prieto et al., 1994), and to the somatostatin-immunopositive neurons in rabbit visual cortex described by Ramon y Cajal-Agueras et al. — see their figures 5 and 7C (Ramon y Cajal-Agueras et al., 1989). The multipolar GFP-expressing neuron in layer I (Fig. 2D), with a horizontal axonal spread in layers I and II/III, resembled some intracellularly stained layer I neurons in the rat (Zhou and Hablitz, 1996) (their fig. 13).

Computer-assisted reconstruction of neurons with well-visualized axonal arbors (Fig. 3) allowed comparison with additional recognized classes of non-pyramidal cortical neurons. Among upper-layers cells expressing GFP were small neurons with mostly ascending dendrites and many horizontal axonal branches (Fig. 3A), similar to small basket cells in monkey somatosensory cortex [type 6 of Jones (Jones, 1975); compare also with Prieto et al.'s fig. 8:3 (Prieto et al., 1994), and with Meyer's fig. 2C (Meyer, 1983)]. Other neurons in the upper layers had initially ascending axons with downward recurving ‘arcades’ giving rise to a loose bundle of parallel, vertically oriented descending axonal branches below the level of the cell body (Fig. 3BD), similar to type 2 of Jones (Jones, 1975) [compare also with the ‘regular spiking non-pyramidal’ neurons in Kawaguchi and Kubota's figs 3:B4 and 8:7,8 (Kawaguchi and Kubota, 1998)]. GFP-expressing neurons in the deep layers often had ascending axons (Fig. 3E,F,G), and of these some had axonal tufts ascending to and extending in layer I, typical of Martinotti cells [Fig. 3G; compare with Wahle's fig. 1 (Wahle, 1993), with Kawaguchi and Kubota's fig. 2 (Kawaguchi and Kubota, 1996) and with Klostermann and Wahle's fig. 11A (Klostermann and Wahle, 1999)].

Occasionally observed in our slices, but not included in the present report, were neurogliaform cells [type 5 of Jones (Jones, 1975)] and bipolar neurons, while apparently missing from our sample were axo-axonic (‘chandelier’) cells [type 4 of Jones (Jones, 1975)], possibly because of the relative immaturity of our cultures, since the characteristic chandelier-shaped terminals appear late in development (Meyer and Ferres-Torres, 1984). We were also not able to identify so-called double bouquet cells [(type 3 of Jones (Jones, 1975)], which to our knowledge have not been clearly documented in rodents. Very rarely (four cells, not shown), neurons with classical pyramidal morphology and spiny dendrites were observed to express GFP, possibly reflecting a low-level expression of GAD mRNA in pyramidal neurons (Cao et al., 1996). Finally, we consistently observed a small number per slice of GFP expressing cells with astroglia-like morphology; these were most often found near the pial or white matter boundaries (not shown).

### Quantitative Analysis of GFP-expressing Multipolar Cortical Neurons

For quantitative analysis, 150 multipolar neurons were selected, solely based on their being brightly fluorescent and well separated from other neurons in their vicinity. (We defined as ‘multipolar’ any non-pyramidal neuron with three or more primary dendrites, a definition that may include neurons traditionally classified as bitufted.) Initial analysis included computation of the following morphological features: soma cross-sectional area, number of primary dendrites, number of dendritic branch points, total dendritic length, vertical and horizontal extent of the dendritic tree and total area covered by the dendritic tree (see Materials and Methods for definitions). Figure 4 shows a matrix of pairwise scatterplots of the main four parameters — soma cross-sectional area, number of dendritic ends (primary dendrites + branch points), total dendritic length and area of dendritic coverage. All plots showed some degree of linear correlation, but the strongest correlations were found between total dendritic length and the three other parameters (Fig. 4, bottom row), with correlation coefficients between 0.57 and 0.70. Thus, total dendritic length increased nearly linearly with soma cross-sectional area, with number of dendritic end points and with dendritic coverage area.

## Discussion

We describe here a novel method for visualizing the detailed morphology of non-pyramidal, presumed GABAergic neurons in living neocortical slice cultures by GAD67–GFP transfection. Using a novel computational metric of dendritic polarization, we demonstrated that dendrites of ~70% of the medium and large GAD67–GFP expressing neurons are not randomly oriented, but instead show a highly significant tendency to extend along the vertical axis of the cortex.

### Novel Method for Visualizing GABAergic Neuronal Morphology in Living Slice Cultures

In this study we used particle-mediated transfection with the eGFP cDNA linked to the GAD67 gene's 5′ regulatory region, to visualize GABAergic neurons in living cortical cultures. A similar DNA construct, using a much shorter segment of the GAD67 promoter, was inserted as a transgene into transgenic mice, but GFP expression in these animals was reported to be restricted to a small subset of all GABAergic neurons (Oliva et al., 2000). Our construct appeared to label a wide variety of non-pyramidal neurons, but virtually never pyramidal neurons. With the exception of spiny stellate neurons in layer IV, which were also never observed in our cultures, all non-pyramidal neurons in the cortex are GABAergic (Houser et al., 1983; Prieto et al., 1994), so there is little doubt that our construct labeled specifically GABAergic neurons, but with no apparent subtype specificity (see Results and below in the Discussion for morphological identification of the neurons in our sample vis-à-vis accepted classifications of GABAergic cortical neurons).

In its level of morphological detail, GFP transfection appears to equal or rival the Golgi and intracellular staining methods, revealing even fine dendritic and axonal processes. Like the Golgi method, gene-gun mediated transfection labels neurons in a spatially random and scattered pattern, which in favorable cases allows one to follow dendrites and even axons of isolated neurons for their full length. Unlike the Golgi technique, however, our transfection method is cell-type specific, in that it labels selectively GABAergic neurons, and allows visualization of neuronal morphology in the living state. These features of the method have important potential applications. For example, this method could be used to follow changes in morphology of live neurons during development in vitro, and to test whether such development is dependent on neuronal activity or on various growth factors. Another application would be electrophysiological recordings of synaptic responses in postsynaptic neurons, upon minimal stimulation of GFP-tagged axons (Jin et al., 2000). Finally, future studies could take advantage of the well-characterized activity-dependent regulation of GAD expression in the cortex (Hendry and Jones, 1988; Akhtar and Land, 1991; Martin and Rimvall, 1993; Benson et al., 1994) and use dynamic changes in GAD67–GFP expression to identify activation patterns in the cortex at the single-cell level.

### Novel Metric for Dendritic Polarization

The observation that dendrites of multipolar cortical neurons have a vertical orientation bias was made by numerous previous studies (Jones, 1975; McMullen and Glaser, 1982; Winer, 1984a; DeFelipe et al., 1986a; Lund et al., 1988; McMullen et al., 1988a), but has rarely been examined quantitatively [a notable exception is a study by McMullen et al. (McMullen et al., 1984)]. Polarization of dendritic trees has been quantified previously using a variety of metrics [see Uylings et al.'s review (Uylings et al., 1986)], most of which are adequate for describing and comparing the spatial distribution of dendritic branches, but provide little insight into the mechanisms guiding dendritic growth. In this study, we followed a small number of previous studies (Glaser et al., 1979; McMullen et al., 1984) by measuring the local orientation of each dendritic chord in the brain coordinate system, rather than in a coordinate system centered on its parent cell body. Assuming that the trajectory of the mature dendrite closely resembles the trajectory taken by the dendritic growth cone during development, such analysis can provide important clues on the factors influencing dendritic growth. We expanded upon previous studies by employing a novel metric, the WACAV (or its normalized equivalent, the IOP), as a single quantity indicating the degree of directional bias in the dendritic tree of a given neuron. This novel approach allowed us to distinguish clearly between vertically biased and randomly oriented dendritic trees, the latter having IOP values scattered closely around zero. A major advantage of this method of analysis is that it applies equally well to the two-dimensional projection of neurons growing in three-dimensional space, and to neurons confined to the nearly planar organotypic slice (see Appendix).

### Potential Caveats

Three potential caveats need to be considered before accepting our interpretation. The first is that these data were acquired from neurons whose morphological development occurred mostly in cultured slices, since at the age of culturing (first postnatal week) non-pyramidal neurons are only poorly developed morphologically (Parnavelas and Uylings, 1980; Miller, 1986; McMullen et al., 1988a). This implies that the neurons in our sample developed, to a large degree, without sensory input, and may therefore have abnormal properties. Previous studies have documented similarities in many of the morphological and electrophysiological properties of neurons that developed in organotypic slice cultures to those of neurons that developed in vivo, suggesting that proper development of these properties does not require intact sensory input (Bolz et al., 1990, 1992; Behan et al., 1991; Annis et al., 1993; Novak and Bolz, 1993; Massengill et al., 1997; Sieg et al., 1998). Similar conclusions were reached in a recent study that focused specifically on morphological and physiological properties of inhibitory cortical interneurons (Klostermann and Wahle, 1999). On the other hand, dendritic orientation does depend on afferent input (Pinto Lord and Caviness, 1979). A classic example of this dependence is the morphology of both excitatory and inhibitory layer IV neurons in rodent somatosensory cortex, which display strong dendritic (and to a lesser degree axonal) polarization towards the center of their ‘barrel’ (Woolsey and Van der Loos, 1970; Lorente de Nó, 1992). Development of this polarity requires intact thalamocortical afferents during a ‘critical period’ in early development (Steffen and Van der Loos, 1980; Harris and Woolsey, 1981). Similar (though less dramatic) dependence of dendritic orientation on patterned sensory input has been described in the visual cortex (Borges and Berry, 1978; Coleman et al., 1981; Tieman and Hirsch, 1982; Kossel et al., 1995), and elimination of all auditory input during early development causes a shift of the dendritic orientation of non-pyramidal neurons in auditory cortex towards the horizontal (tangential) plane (McMullen et al., 1988b). Thus, vertical orientation in our cultures may have been reduced by lack of sensory input, compared with vertical orientation in vivo, and our estimate of vertical dendritic bias in vitro should be regarded as a lower estimate of vertical bias in the intact brain.

A second caveat is that biolistically transfecting neurons with GFP may have altered their normal development and shape. Preliminary studies (X. Jin, P.H. Mathers and A. Agmon, unpublished observations) have revealed no apparent differences in electrophysiological and morphological properties between GAD67–GFP expressing and non-expressing GABAergic neurons in cultured slices. In addition, previous studies of pyramidal cells biolistically transfected with GFP in vitro (Lo et al., 1994; McAllister et al., 1995) have revealed seemingly normal morphologies. We therefore believe that biolistic transfection and GAD67–GFP expression did not alter the morphology of non-pyramidal cells.

A third concern is that the dendritic orientation of the neurons in our sample was altered by the slicing procedure, since some dendrites would be truncated during the slicing. However the effect of truncation on our cultured slices was most likely negligible, since the cultures were prepared from animals less than 1 week old, in which the diameter (in the horizontal plane) of the dendritic tree of non-pyramidal cells is less than 100 μm (Parnavelas and Uylings, 1980; McMullen et al., 1988a). Therefore, only neurons with cell bodies 50 μm or less from either cut surface, or less than 30% of all neurons in a 350 μm thick slice, suffered any truncation at all, and these would have been truncated on one side only and most of them by a negligible amount. Moreover, it is quite likely that the more severely truncated neurons did not survive in the long-term culture, leaving only intact or mildly truncated neurons. Most importantly, truncation had absolutely no effect on the calculated IOP, since the truncation occurred in a plane parallel to the plane of the projection, and therefore both dendrites oriented vertically and dendrites oriented horizontally were truncated to the same extent.

### Classification Schemes of Non-pyramidal Neocortical Neurons and Dendritic Orientation

Neocortical non-pyramidal, GABAergic neurons have been classified in a multitude of ways, based on a variety of morphological, electrophysiological and immunocytochemical criteria. Although morphological cell types can be defined on the basis of cell body shape (Cobas et al., 1987) or dendritic arbors (Feldman and Peters, 1978; Peters and Regidor, 1981; Prieto et al., 1994), most modern classification schemes follow the lead of Cajal (DeFelipe and Jones, 1988) in classifying non-pyramidal neurons by relying heavily on their axonal arborization patterns (Jones, 1975; Szentagothai, 1978; Meyer, 1983). Axonal arborization patterns are predictive of the type of target favored by each inhibitory neuron (Somogyi et al., 1998), and most classifications now recognize at least three major types: basket cells, favoring somata and proximal dendritic shafts (Somogyi et al., 1983; DeFelipe et al., 1986a); double bouquet cells (described most often in primates and carnivores, rarely in rodents), favoring distal dendritic shafts and spines (Somogyi and Cowey, 1981; DeFelipe and Fairén, 1988); and chandelier cells, making synapses exclusively on initial segments of axons (Fairén and Valverde, 1980; Peters et al., 1982; Somogyi et al., 1982). These classes are also distinguishable by molecular markers: basket and chandelier cells most often contain parvalbumin, while double bouquet cells contain calbindin (DeFelipe et al., 1989; Hendry et al., 1989; DeFelipe et al., 1990; Williams et al., 1992; Peters and Sethares, 1997; Kawaguchi and Kubota, 1998). Falling outside this classification are several classes of neuropeptide-containing cortical interneurons, e.g. Martinotti cells, characterized by an ascending axon forming an extended arbor in layer I, and containing somatostatin, Substance P and/or Neuropeptide Y (Kuljis and Rakic, 1989; Wahle, 1993; Kawaguchi and Kubota, 1996), and bipolar cells, containing vasoactive intestinal peptide (Connor and Peters, 1984; Morrison et al., 1984; Bayraktar et al., 2000; Cauli et al., 2000). Finally, some morphological and/or immunocytochemical types also show characteristic firing patterns (Kawaguchi and Kubota, 1997). A definitive identification of a non-pyramidal neuron as belonging to one of these classes requires immunocytochemical identification, electron microscopical examination of its postsynaptic targets and/or electrophysiological characterization, in addition to knowledge of its dendritic and axonal morphologies, and was therefore outside the scope of the present study. Nevertheless, based on dendritic and axonal morphology, our sample of neurons expressing GAD67–GFP appeared to include representatives of many of the putative or confirmed GABAergic cell classes observed previously with the Golgi and intracellular staining methods, including some types with specific immunocytochemical identities. For example, our sample contained representatives of at least four of the six types of smooth non-pyramidal cells described in monkey somatosensory cortex by Jones (Jones, 1975) (see Results for specific comparisons with previous studies). Our characterization of GAD67–GFP expressing neurons according to the level of vertical bias in their dendritic tree is not intended to add yet another system of classification, and we have not attempted to equate a specific level of vertical orientation bias with specific classes of nonpyramidal cells. Indeed, it appears from our own and previous studies that a vertical dendritic orientation bias is a property superimposed, to varying degrees, on all classes of nonpyramidal GABAergic neurons.

### What Dendritic Measurements Tell Us About Dendritic Development

Our analysis shows that dendritic length of non-pyramidal, putative GABAergic neurons is strongly correlated with soma cross-sectional area, with number of dendritic ends and with total dendritic coverage area (Fig. 4), confirming and expanding previous findings (McMullen et al., 1984). The first of these correlations, the correlation between soma size and dendritic length, most likely reflects the need for a larger cell body to support a larger dendrite. The other correlations may be rooted in basic rules of dendritic development. For example, the linear relationship between number of end points and dendritic length suggests that the average length that a dendrite grows between branch points is similar between different neurons, implying a common underlying mechanism involved in determining dendritic bifurcation. The linear relationship between total dendritic length and dendritic coverage area suggests that dendrites branch so that a unit of cortical area (in the nearly planar organotypic slice) is always sampled by the same total dendritic length. The three-dimensional analogue of this last correlation remains to be investigated.

### Possible Mechanisms of Vertically Biased Dendritic Outgrowth

How dendrites acquire a vertical orientation bias is not known. An attractive hypothesis accounting for vertically biased growth is that developing neurons sense a pia-to-white matter gradient of a diffusible or surface-attached molecule. If a single extracellular gradient is to account for extension of processes both towards and away from the pia, one needs to assume that the receptive or signal transducing mechanism in the neuron is asymmetrically distributed, causing opposite poles of the neuron to extend neurites in opposite directions, in response to the same gradient. Evidence for precisely such a mechanism has recently been reported, indicating that the same diffusible morphogen, Semaphorin 3A, can simultaneously behave as an attractant for apical dendrites of cortical pyramidal neurons and as a repellent for their axons, by virtue of an asymmetric distribution of the enzyme guanylate cyclase (Polleux et al., 2000). The organotypic slice preparation could potentially be used to test such models. Interestingly, our analysis demonstrates that multipolar neurons did not form a continuum with respect to their Index of Polarization; rather, there was a well-separated population (NP cells, ~20% of neurons in our sample) with a statistical spread of IOP values around 0, indicating purely random dendritic orientations. An intriguing and testable possibility is that this population lacked the molecular machinery to detect or to transduce the postulated extracellular gradient.

### Functional Significance of Directionally Biased Dendritic Trees

Our results imply that a vertical orientation bias, which is characteristic of dendrites of pyramidal cortical cells, is also a feature of non-pyramidal neurons — not only of the bipolar and bitufted types, but also of those with ‘stellate’ or multipolar morphologies. What advantage(s) may be conferred on the information processing capabilities of the neocortex by vertically elongated dendritic trees? The major afferents to primary sensory and motor cortical areas are thalamocortical axons, which are organized along the horizontal (tangential) dimensions in a precise topological order (Hohl-Abrahao and Creutzfeldt, 1991; Agmon et al., 1995), and terminate in narrow vertical (radial) columns and in specific laminar patterns (LeVay and Gilbert, 1976; Jensen and Killackey, 1987; Landry et al., 1987; Agmon et al., 1993). A columnar pattern of terminations, although with a fractured topology and with a different laminar distribution, is also formed by long-range corticocortical axons (Jones et al., 1975; Goldman and Nauta, 1977; Code and Winer, 1986; DeFelipe et al., 1986b; Chapin et al., 1987; Ojima et al., 1991; Saleem et al., 1993; Pucak et al., 1996). In turn, both excitatory and inhibitory cortical neurons of all layers extend much of their local axonal projections along the vertical dimension (Lorente de Nó, 1949; Somogyi et al., 1981; Bernardo et al., 1990; Kritzer and Goldman-Rakic, 1995; Fujita and Fujita, 1996; Zhang and Deschenes, 1997; Lubke et al., 2000; Porter et al., 2001), again with specific laminar distributions (Lund and Boothe, 1975). These organizational principles ensure that afferent information pertaining to specific features of sensory space will be distributed and processed within relatively narrow radial modules, whether in primary (Hubel and Wiesel, 1977; Mountcastle, 1997) or in high-order (Fujita et al., 1992; Britten, 1998) cortical areas. By extending their dendritic trees vertically, pyramidal and non-pyramidal cortical neurons can potentially sample and integrate afferent and reentrant inputs terminating at different laminar positions and representing different stages of information processing, while retaining the specificity of location, modality and/or features unique to their own module. A similar organizational rule, manifested as dendritic arbors elongated in parallel to a topologically ordered system of afferent fibers and orthogonal to the plane of the mapping, is found also in other brain areas, such as the dorsal cochlear nucleus (Blackstad et al., 1984; Berrebi and Mugnaini, 1991), the inferior and superior colliculi (Rockel and Jones, 1973; Langer and Lund, 1974), the reticular thalamic nucleus (Montero et al., 1977; Liu et al., 1995; Crabtree, 1996) and the cerebellar cortex (Voogd and Bigare, 1980; Mulle et al., 1987; Wiklund et al., 1990).

## Notes

This study was supported by National Institutes of Health grants HD33463 (A.A.) and EY12152 (P.H.M.) and by OTKA grant T-029369 (G.S.). We thank Drs Albert Berrebi, Paul Brown and Ron Millecchia for critical comments on the manuscript, Dr Arie Gingold for helpful discussions on mathematical methods, and Cary Johnson, Colette Ramsburg and Jubin Ryu for excellent technical support.

Address correspondence to A. Agmon at Department of Neurobiology and Anatomy, West Virginia University, Morgantown, WV 26506, USA. Email: aagmon@wvu.edu.

## Appendix

As explained in Materials and Methods, to calculate the WACAV one takes the sum of the lengths of the projections of all dendritic segments on the vertical axis, and divides it by the sum of all dendritic lengths. The calculation can be done either in two dimensions (using two-dimensional coordinates) or, if a suitable dataset is available, in three dimensions. We show here that the two-dimensional and the three-dimensional WACAV values will be different, but that when calculated in two dimensions, the WACAV of neurons with randomly oriented dendrites is 0.64, regardless of whether the neurons have developed in a plane (e.g. in the organotypic slice) or in three-dimensional space (in vivo).

We start by calculating the average WACAV of a planar neuron with randomly oriented dendrites. We simulate the neuron by dividing it into a large number of small dendritic segments, and represent each segment as a unit vector forming a random angle ϑ with the vertical axis [this is equivalent to Glaser et al.'s ‘stick analysis’ (Glaser et al., 1979)]. Since the cosine is symmetric and since we consider the upward and downward directions equivalent, we can restrict the discussion to a quarter-circle between 0 and π/2. The normalized probability that a random angle will fall within the infinitesimal arc segment (ϑ, ϑ + dϑ) is the ratio of the length of the arc segment to the quarter circle, or (2/π)•dϑ. The average of the cosines of all angles (the WACAV of a randomly oriented dendritic tree) would therefore be 0.64, given by the integral

A1
$\frac{2}{{\pi}}{{\int}_{0}^{{\pi}/2}}cos\ {\vartheta}\ {\cdot}\ d\ {\vartheta}\ =\ \frac{2}{{\pi}}$
Next we calculate the three-dimensional WACAV of a neuron with dendrites randomly oriented in three-dimensional space. Calculations are simplified by using spherical coordinates, with θ being the angle from the vertical (Z) axis and φ being the angle of the projection of the vector on the horizontal (XY) plane relative to the positive X-axis. In space, the normalized probability of occurrence of a randomly oriented vector within the infinitesimal solid arc bounded by (ϑ, ϑ + dϑ) and (φ, φ + dφ) is the ratio of the area of the solid arc to the area of the 1/8 unit sphere, or (2/π)•sin ϑ•dϑ•dφ. The three-dimensional WACAV of a random neuron is therefore given by the integral
A2
$\frac{2}{{\pi}}{{\int}_{0}^{{\pi}/2}}{{\int}_{0}^{{\pi}/2}}cos\ {\vartheta}{\cdot}\ sin\ {\vartheta}{\cdot}d{\vartheta}{\cdot}d{\phi}\ =\ \frac{1}{2}$
Thus, if one has a true three-dimensional dataset describing the neuron, the WACAV should be calculated as described in Materials and Methods using three-dimensional dot products and norms, and any significant departure from 0.5 will indicate non-random dendritic growth.

Often, however, only a two-dimensional dataset is available, describing the projection of the neuron on the plane of the drawing (for example, when tracing a neuron using a camera lucida device). In this case, calculating the WACAV in two dimensions would give the same sum of projections as in the three-dimensional case, since the projection of a vector on the vertical axis is not altered by first projecting it on a vertical plane, but the (apparent) lengths of the projected dendrites would be smaller than their real three-dimensional lengths. The length of the projection of a unit vector on the vertical (XZ) plane is given by (cos2θ + sin2θ•cos2φ)1/2, so for a set of unit vectors randomly oriented in space, the average projected length of a vector is given by

A3
$\frac{2}{{\pi}}{{\int}_{0}^{{\pi}/2}}{{\int}_{0}^{{\pi}/2}}\sqrt{cos^{2}\ {\vartheta}\ +\ sin^{2}\ {\vartheta}\ {\cdot}\ cos^{2}\ {\phi}\ {\cdot}\ sin\ {\vartheta}\ {\cdot}\ d{\vartheta}\ {\cdot}\ d{\phi}\ =\ \frac{{\pi}}{4}}$

Since this quantity appears in the denominator when calculating the WACAV, one needs to divide the three-dimensional WACAV (which is 0.5) by π/4, to account for the distortion due to the projection. The WACAV of a neuron with randomly oriented dendrites would therefore be, on average, 2/π, the same as in the planar case.

Figure 1.

Confocal images of representative GAD67–GFP expressing neurons. Each panel is a projection of confocal images taken at multiple focal planes (Z-stacks) and superimposed. Dashed lines in (A,C,D) indicate white matter (WM) boundary. (A) A large fan-shaped neuron in lower layer VI; pia is towards the lower right. (B) A high-power montage of several microscope fields, including three large multipolar neurons in the upper layers; note the extended axonal arbor in layer I, emanating from the leftmost multipolar neuron. Pial surface is seen at the top of the image. The horizontal line at right represents the approximate border between the superficial (II/III) and deep (V/VI) layers; layer IV was not identifiable in this part of the cortex. (C) GFP-expressing neurons in layer VI, including two inverted pyramidal cells (top two neurons); pial surface is towards the left. (D) A multipolar neuron in lower layer VI; pia is upwards. Inset shows enlarged spiny dendritic segment. Scale bar (in D) represents 100 μm except for inset in (D), 15 μm.

Figure 1.

Confocal images of representative GAD67–GFP expressing neurons. Each panel is a projection of confocal images taken at multiple focal planes (Z-stacks) and superimposed. Dashed lines in (A,C,D) indicate white matter (WM) boundary. (A) A large fan-shaped neuron in lower layer VI; pia is towards the lower right. (B) A high-power montage of several microscope fields, including three large multipolar neurons in the upper layers; note the extended axonal arbor in layer I, emanating from the leftmost multipolar neuron. Pial surface is seen at the top of the image. The horizontal line at right represents the approximate border between the superficial (II/III) and deep (V/VI) layers; layer IV was not identifiable in this part of the cortex. (C) GFP-expressing neurons in layer VI, including two inverted pyramidal cells (top two neurons); pial surface is towards the left. (D) A multipolar neuron in lower layer VI; pia is upwards. Inset shows enlarged spiny dendritic segment. Scale bar (in D) represents 100 μm except for inset in (D), 15 μm.

Figure 2.

Confocal images (projected Z-stacks) of representative GAD67–GFP expressing neurons with extensive local axonal arborizations. (A) A multipolar neuron in the middle layers with an ascending axon. (B) A deep layer VI multipolar neuron with horizontally extended dendrites and a horizontally arborizing, varicose axon. (C) A bitufted neuron in the upper layers with several horizontally directed axonal branches. (D) A multipolar neuron in layer I with horizontal axonal branches in layers I and II/III. In all panels, arrows point to the initial segment of the axon, identified on the original stack of confocal images. Pial surface is indicated by the solid line in (C,D), and is upward and to the left in (A,B). Dashed line in (B) indicates the white matter boundary. The diagonal lines in (A) and (D) represent the estimated laminar borders and are drawn parallel to the pia; layer IV was not clearly identifiable in (A). Scale bar: 50 μm in (A,C); 12 μm in (B); 60 μm in (D).

Figure 2.

Confocal images (projected Z-stacks) of representative GAD67–GFP expressing neurons with extensive local axonal arborizations. (A) A multipolar neuron in the middle layers with an ascending axon. (B) A deep layer VI multipolar neuron with horizontally extended dendrites and a horizontally arborizing, varicose axon. (C) A bitufted neuron in the upper layers with several horizontally directed axonal branches. (D) A multipolar neuron in layer I with horizontal axonal branches in layers I and II/III. In all panels, arrows point to the initial segment of the axon, identified on the original stack of confocal images. Pial surface is indicated by the solid line in (C,D), and is upward and to the left in (A,B). Dashed line in (B) indicates the white matter boundary. The diagonal lines in (A) and (D) represent the estimated laminar borders and are drawn parallel to the pia; layer IV was not clearly identifiable in (A). Scale bar: 50 μm in (A,C); 12 μm in (B); 60 μm in (D).

Figure 3.

Computer-assisted tracings of GAD67–GFP expressing neurons with representative axonal morphologies. Dendrites and cell bodies are shown in blue, axons in red. For clarity, dendrites and axons are represented as lines of uniform width. Pia, white matter and laminar boundaries indicated as in Figs 1–2. See Results for identification of the different morphological cell types.

Computer-assisted tracings of GAD67–GFP expressing neurons with representative axonal morphologies. Dendrites and cell bodies are shown in blue, axons in red. For clarity, dendrites and axons are represented as lines of uniform width. Pia, white matter and laminar boundaries indicated as in Figs 1–2. See Results for identification of the different morphological cell types.

Figure 4.

Scatterplot matrix of four morphometric somatodendritic parameters. Lines represent best-fit (least squares) linear regressions, and were not forced to pass through the origin of axes. Five outlier neurons with soma area >500 μm2 were excluded from this analysis. The correlation coefficients for the six paired correlations are indicated on the plots. Probabilities that the regression slope was 0 were P < 0.0001, except for top left (P < 0.001) and middle right (P < 0.005).

Figure 4.

Scatterplot matrix of four morphometric somatodendritic parameters. Lines represent best-fit (least squares) linear regressions, and were not forced to pass through the origin of axes. Five outlier neurons with soma area >500 μm2 were excluded from this analysis. The correlation coefficients for the six paired correlations are indicated on the plots. Probabilities that the regression slope was 0 were P < 0.0001, except for top left (P < 0.001) and middle right (P < 0.005).

Figure 5.

Classification of GAD67–GFP expressing multipolar neurons according to the degree of vertical bias in their dendritic trees. (A) A smoothed histogram of the index of polarization (IOP, see Materials and Methods) of all neurons (solid line), with a superimposed histogram of the IOP of 10 000 simulated neurons (dashed line). Vertical lines above graph indicate the cut-off values for dividing the sample into tangentially polarized (TP, IOP < –0.05), non-polarized (NP, –0.05 < IOP < 0.05), moderately polarized (MP, 0.05 < IOP < 0.3) and highly polarized (HP, IOP > 0.3) neurons. (B) The fraction of neurons in our sample falling into each category.

Figure 5.

Classification of GAD67–GFP expressing multipolar neurons according to the degree of vertical bias in their dendritic trees. (A) A smoothed histogram of the index of polarization (IOP, see Materials and Methods) of all neurons (solid line), with a superimposed histogram of the IOP of 10 000 simulated neurons (dashed line). Vertical lines above graph indicate the cut-off values for dividing the sample into tangentially polarized (TP, IOP < –0.05), non-polarized (NP, –0.05 < IOP < 0.05), moderately polarized (MP, 0.05 < IOP < 0.3) and highly polarized (HP, IOP > 0.3) neurons. (B) The fraction of neurons in our sample falling into each category.

Figure 6.

Tracings of dendritic trees of representative GAD67–GFP expressing multipolar neurons from the four classes defined in Figure 4. Note that the computed degree of vertical polarization often, but not always, agrees with the visually perceived one. Pia is upwards in all panels.

Tracings of dendritic trees of representative GAD67–GFP expressing multipolar neurons from the four classes defined in Figure 4. Note that the computed degree of vertical polarization often, but not always, agrees with the visually perceived one. Pia is upwards in all panels.

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text:
Class 9 MOTION AND MEASURING THE RATE OF MOTION

### Topic covered

color{red} ♦ INTRODUCTION
color{red} ♦ DESCRIBING MOTION
color{red} ♦ MEASURING THE RATE OF MOTION

### INTRODUCTION

In everyday life, we see some objects at rest and others in motion.

Birds fly, fish swim, blood flows through veins and arteries and cars move. Atoms, molecules, planets, stars and galaxies are all in motion.

We often perceive an object to be in motion when its position changes with time. However, there are situations where the motion is inferred through indirect evidences.

For example, we infer the motion of air by observing the movement of dust and the movement of leaves and branches of trees.

What causes the phenomena of sunrise, sunset and changing of seasons? Is it due to the motion of the earth? If it is true, why don’t we directly perceive the motion of the earth?

An object may appear to be moving for one person and stationary for some other. For the passengers in a moving bus, the roadside trees appear to be moving backwards.

A person standing on the road-side perceives the bus along with the passengers as moving. However, a passenger inside the bus sees his fellow passengers to be at rest. What do these observations indicate?

Most motions are complex. Some objects may move in a straight line, others may take a circular path. Some may rotate and a few others may vibrate. There may be situations involving a combination of these.

In this chapter, we shall first learn to describe the motion of objects along a straight line. We shall also learn to express such motions through simple equations and graphs. Later, we shall discuss ways of describing circular motion.

### DESCRIBING MOTION

We describe the location of an object by specifying a reference point.

Let us assume that a school in a village is 2 km north of the railway station. We have specified the position of the school with respect to the railway station.

In this example, the railway station is the reference point. We could have also chosen other reference points according to our convenience. Therefore, to describe the position of an object we need to specify a reference point called the origin.

ul"MOTION ALONG A STRAIGHT LINE"

The simplest type of motion is the motion along a straight line. We shall first learn to describe this by an example.

Consider the motion of an object moving along a straight path. The object starts its journey from O which is treated as its reference point (Fig. 8.1).

Let A, B and C represent the position of the object at different instants. At first, the object moves through C and B and reaches A. Then it moves back along the same path and reaches C through B.

The total path length covered by the object is OA + AC, that is  60 km + 35 km = 95 km. This is the distance covered by the object.

To describe distance we need to specify only the numerical value and not the direction of motion. There are certain quantities which are described by specifying only their numerical values.

The numerical value of a physical quantity is its magnitude. From this example, can you find out the distance of the final position C of the object from the initial position O?

This difference will give you the numerical value of the displacement of the object from O to C through A. The shortest distance measured from the initial to the final position of an object is known as the displacement.

Can the magnitude of the displacement be equal to the distance travelled by an object? Consider the example given in (Fig. 8.1).

For motion of the object from O to A, the distance covered is 60 km and the magnitude of displacement is also 60 km.

During its motion from O to A and back to B, the distance covered = 60 km + 25 km = 85 km displacement, are used to describe the overall motion of an object and to locate its final position with reference to its initial position at a given time.

Activity ______________ 8.3

♦ Take a metre scale and a long rope.
♦ Walk from one corner of a basket-ball court to its oppposite corner along its sides.
♦ Measure the distance covered by you and magnitude of the displacement.
♦ What difference would you notice between the two in this case?

Activity ______________ 8.4

♦ Automobiles are fitted with a device that shows the distance travelled. Such a device is known as an odometer. A car is driven from Bhubaneshwar to New Delhi. The difference between the final reading and the initial reading of the odometer is 1850 km.
♦Find the magnitude of the displacement between Bhubaneshwar and New Delhi by using the Road Map of India.

ul"UNIFORM MOTION AND NONUNIFORM MOTION"

Consider an object moving along a straight line. Let it travel 50 km in the first hour, 50 km more in the second hour, 50 km in the third hour and 50 km in the fourth hour. In this case, the object covers 50 km in each hour.

As the object covers equal distances in equal intervals of time, it is said to be in uniform motion.

The time interval in this motion may be small or big. In our day-to-day life, we come across motions where objects cover unequal distances in equal intervals of time, for example, when a car is moving on a crowded street or a person is jogging in a park.
These are some instances of non-uniform motion.

Activity ______________ 8.5

♦ The data regarding the motion of two different objects A and B are given in Table 8.1.

♦ Examine them carefully and state whether the motion of the objects is uniform or non-uniform.

### MEASURING THE RATE OF MOTION

Look at the situations given in Fig. 8.2. If the bowling speed is 143 \ \ km \ \ h^(–1 ) in Fig. 8.2(a) what does it mean?
What do you understand from the signboard in Fig. 8.2(b)?

Different objects may take different amounts of time to cover a given distance. Some of them move fast and some move slowly.

The rate at which objects move can be different. Also, different objects can move at the same rate. One of the ways of measuring the rate of motion of an object is to find out the distance traveled by the object in unit time.

This quantity is referred to as speed. The SI unit of speed is metre per second. This is represented by the symbol m s^(–1) or m//s. The other units of speed include centimetre per second (cm \ \ s^(–1)) and kilometre per hour (km \ \ h^(–1)).

To specify the speed of an object, we require only its magnitude. The speed of an object need not be constant. In most cases, objects will be in non-uniform motion.

Therefore, we describe the rate of motion of such objects in terms of their average speed. The average speed of an object is obtained by dividing the total distance travelled by the total time taken. That is,

average speed  = text( Total distance travelled)/text( Total time taken)
If an object travels a distance s in time t then its speed v is,

 v = s/t ........(8.1)

Let us understand this by an example. A car travels a distance of 100 km in 2 h. Its average speed is 50 km \ \ h^(–1). The car might not have travelled at 50 km \ \ h^(–1) all the time. Sometimes it might have travelled faster and sometimes slower than this.

ul"SPEED WITH DIRECTION"

The rate of motion of an object can be more comprehensive if we specify its direction of motion along with its speed. The quantity that specifies both these aspects is called velocity.

Velocity is the speed of an object moving in a definite direction. The velocity of an object can be uniform or variable. It can be changed by changing the object’s speed, direction of motion or both.

When an object is moving along a straight line at a variable speed, we can express the magnitude of its rate of motion in terms of average velocity. It is calculated in the same way as we calculate average speed.

In case the velocity of the object is changing at a uniform rate, then average velocity is given by the arithmetic mean of initial velocity and final velocity for a given period of time. That is,

average velocity  = text( initial velocity + final velocity)/2

Mathematically, v_(av) = (u + v)/2 .........(8.2)

where v_(a v) is the average velocity, u is the initial velocity and v is the final velocity of the object.

Speed and velocity have the same units, that is, m \ \ s^(–1) or m//s.

Activity ______________ 8.6

♦ Measure the time it takes you to walk from your house to your bus stop or the school. If you consider that your average walking speed is 4 km \ \ h^(–1), estimate the distance of the bus stop or school from your house.

Activity ______________ 8.7

♦ At a time when it is cloudy, there may be frequent thunder and lightning. The sound of thunder takes some time to reach you after you see the lightning.
♦ Can you answer why this happens?
♦ Measure this time interval using a digital wrist watch or a stop watch.
♦ Calculate the distance of the nearest point of lightning. (Speed of sound in air = 346 m \ \ s^(-1) .)

Q 3224791651

An object travels 16 m in 4 s and then another 16 m in 2 s. What is the average speed of the object?
Class 9 Chapter 8 Example 1
Solution:

Total distance travelled by the object = 16 m + 16 m = 32 m

Total time taken = 4 s + 2 s = 6 s

average speed  = text( Total distance travelled)/text( Total time taken)

 = (32 m)/(6 s) = 5.33 m \ \ s^(–1)

Therefore, the average speed of the object is 5.33 m \ \ s^(–1).
Q 3234791652

The odometer of a car reads 2000 km at the start of a trip and 2400 km at the end of the trip. If the trip took 8 h, calculate the average speed of the car in km \ \ h^(–1) and m s^(–1).

Class 9 Chapter 8 Example 2
Solution:

Distance covered by the car,

s = 2400 km – 2000 km = 400 km

Time elapsed, t = 8 h

Average speed of the car is,

 v_(av) = s/t = (400 km)/(8 h)

= 50km \ \ h^(-1)

= 50 (k m)/h xx (1000 m)/(1 k h) xx ( 1 h)/(3600 s)

= 13.9 m \ \ s^(–1)
The average speed of the car is 50 km\ \ h^(–1) or 13.9 m s^(–1).
Q 3244791653

Usha swims in a 90 m long pool. She covers 180 m in one minute by swimming from one end to the other and back along the same straight path. Find the average speed and average velocity of Usha.
Class 9 Chapter 8 Example 3
Solution:

Total distance covered by Usha in 1 min is 180 m.
Displacement of Usha in 1 min = 0 m

Average speed  = text( Total distance covered)/text( Total time taken)

 = (180 m)/(1 min ) = (180 m)/(1 min ) xx (1 min )/(60 s)

 = 3 m \ \ s^(-1)

Average velocity  = text( Displacement)/text( Total time taken)

 = (0 m)/( 60 s)

= 0 m \ \ s^(–1)
The average speed of Usha is 3 m \ \ s^(–1)
and her average velocity is 0 m \ \ s^(–1).

format: plain


text:
# $T(1) = 1 , T(n) = 2T(n/2) + n^3$? Divide and conquer

$T(1) = 1 , T(n) = 2T(n/2) + n^3$? Divide and conquer, need help, I dont know how to solve it?

#### Solutions Collecting From Web of "$T(1) = 1 , T(n) = 2T(n/2) + n^3$? Divide and conquer"

Hmm, possibly another way of heuristics is instructive.
First write the undisputable elements of the sequence:
$$\begin{array} & a(1) &=&a(2^0) & = & 1 \\\ a(2)&=&a(2^1) &=&10 &= & 2^3 + 2*1 &=& 2*(4^1+1) \\\ a(4)&=&a(2^2) &=&84 &=& 4^3 + 2*10 &=& 4*(4^2+4^1+1) \\\ a(8)&=&a(2^3) &=&680 &=& 8^3 + 2*84 &=& 8^3+2*4^3+4*2^3+8*1^3\\\ & & & & &=& 8*(4^3+4^2+4^1+1) \\\ \ldots &=&a(2^k)&=& \ldots \end{array}$$
It is obvious how this can be continued, because at the exponent k we get always $8^k$ plus two times the previous, thus the weighted sum of all powers of 8 which can be expressed as consecutive powers of 4:
$$a(2^k) = 2^k*(4^k+4^{k-1} \ldots +4^0)= 2^k*\frac{4^{k+1}-1}{4-1}$$
Now the step “divide” can be taken: the above gives also a meaningful possibility for interpolation of the non-explicitely defined elements. If we allow base-2 logarithms for k we get for

$$\begin{array} & & a(2^k) &= & 2^k*\frac{4^{k+1}-1}{4-1} \\\ & &= & 2^k*\frac{4*(2^{k})^2-1}{3} \\\ \text{assuming }& k&=& \frac{\log(n)}{\log(2)} \\\ & a(n) &=& n*\frac{4*n^2-1}{3} \\\ & &=& n^3 + \frac{(n-1)n(n+1)}{3!} \\\ & &=& n^3 + 2*\binom{n+1}{3} \\\ \end{array}$$
where the expression in the fourth line is the same as Fabian’s result.

The homogeneous part of the equation $T(n) =2 T(n/2)$ has the general solution $$T_0(n) = C n.$$ So all what we have to do is find a particular solution of the inhomogeneous equation $$T(n) = 2 T(n/2) + n^{3} .$$ Quite often it is good to try an ansatz which has the same form as the inhomogeneous term. Therefore, we try $T_p(n) = c n^3$ which yields $$c n^3 = \frac{c}{4} n^3 + n^3.$$ Solving for $c$, we get the particular solution $T_p(n)= \frac{4}{3} n^3$.

The general solution therefore has the form $T(n) =T_0(n) + T_p(n)$. With the initial condition $T(1)=1$, we obtain $C= – \frac{1}{3}$. So the solution is given by
$$T(n) = \frac{n}{3} (4 n^2 -1).$$

Use Akra-Bazzi which is more useful than the Master Theorem.

Using Akra-Bazzi, I believe you get $$T(x) = \theta(x^3)$$

You can also use the Case 3 of Master theorem in the wiki link above. (Note: That also gives $\theta(x^3)$.)

There is another closely related recurrence that admits an exact
solution. Suppose we have $T(0)=0$ and for $n\ge 1$ (this gives
$T(1)=1$)
$$T(n) = 2 T(\lfloor n/2 \rfloor) + n^3.$$

Furthermore let the base two representation of $n$ be
$$n = \sum_{k=0}^{\lfloor \log_3 n \rfloor} d_k 2^k.$$

Then we can unroll the recurrence to obtain the following exact
formula for $n\ge 1$
$$T(n) = \sum_{j=0}^{\lfloor \log_2 n \rfloor} 2^j \left( \sum_{k=j}^{\lfloor \log_2 n \rfloor} d_k 2^{k-j} \right)^3.$$

Now to get an upper bound consider a string of one digits which yields
$$T(n) \le \sum_{j=0}^{\lfloor \log_2 n \rfloor} 2^j \left( \sum_{k=j}^{\lfloor \log_2 n \rfloor} 2^{k-j} \right)^3.$$

Note that this bound is attained and cannot be improved. It simplifies
to
$$\frac{32}{3} 8^{\lfloor \log_2 n \rfloor} – 24 \times 4^{\lfloor \log_2 n \rfloor} + \left(6 \lfloor \log_2 n \rfloor + \frac{40}{3}\right) \times 2^{\lfloor \log_2 n \rfloor} + 1.$$

The lower bound is for the case of a one digit followed by a string of
zeros and yields
$$T(n) \ge \sum_{j=0}^{\lfloor \log_2 n \rfloor} 2^j \left( 2^{\lfloor \log_2 n \rfloor-j} \right)^3.$$
It simplifies to
$$\frac{4}{3} 8^{\lfloor \log_2 n \rfloor} – \frac{1}{3} 2^{\lfloor \log_2 n \rfloor}$$
and this matches the results posted by the other contributors.

Joining the dominant terms of the upper and the lower bound we obtain
the asymptotics
$$2^{3 \lfloor \log_2 n \rfloor} \in \Theta\left(2^{3 \log_2 n}\right) = \Theta\left(n^3\right).$$

These are both in agreement with what the Master theorem would produce.

This MSE link has a series of similar calculations.


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# Tag Info

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Thanks to the commenters above. They suggest a grammar definition for parsing. Here's one in PyPEG2 (untested and incomplete), but shows how I'll handle variable and sub/superscripting. from term import Term from pypeg2 import * alphabet = RegEx( r'\alpha|\beta|\gamma|\delta|\epsilon|' + \ r'\zeta|\eta|\theta|\vartheta|\iota|' + \ ...

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Here is a new paper that covers a similar topic. The idea is that by doing algebra in enriched categories (2-categories are like categories enriched in categories), you can talk about more fine grained semantic structure on the algebra. (I haven't read through the whole paper myself, but I know enough to see some of the ideas behind it.) The way it relates ...

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Yes, in that context Hindley-Milner polymorphism is let-polymorphism, since such language uses $\sf let$ to introduce polymorphic functions. In the untyped lambda calculus, we can consider a (non recursive) ${\sf let}\ x = e \ {\sf in}\ t$ to be syntactic sugar for $(\lambda x.t)e$. In System F, where polymorphism is introduced by explicit $\Lambda \alpha$ ...

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There is just nothing that you can generalise. Every language is different. A class describes what instances of the class (objects) look like. In Java, a class is also an object in its own right. All classes are instances of a class named "class", which allows the programmer to ask the class for example "what is your name", "what are the instance variables ...

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I'm not confident enough about the later questions, but I hope I can give a partial answer, since this hasn't been given much love. Yes Yes There is nothing in particular, since first-order logic quantifies over non-logical objects, but a priori there are no non-logical objects we're reasoning about for the lambda calculus. There is a similar-sounding type ...

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Unfortunately I don't have a copy of TAPL with me, so I can't figure out exactly what the author intends. But there is a point we should make, that types are something which classifies terms or values, even if there are other pieces that are related to the type. As such, classes and interfaces are not literally types themselves in Java or C++. Rather, each ...

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In the Gaster and Jones paper, rows are encoded using type application of a few type constants, most notably row extension. Your implementation needs to have the following constructors in Typeconstant: data Typeconstant = Arrow | EmptyRow | Rec | Var | RowExtension Label Absence of labels is handled using predicates in qualified types: data QualifiedType =...

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### Abstract

Retailers in Southeast Queensland of supplies for saltwater aquariums were surveyed in August 1989 with a view to obtaining some indication of the likely demand in this region (and more broadly in Australia for giant clams as aquarium specimens. Information was gathered on the size of the market, the possible demand in Australia for giant clams from clam farms for the aquarium trade, prices, attitudes of aquarium retailers to clam farms and about economic prospects for supplies from clam farms, as well as other factors such as the desirability or otherwise of allowing imports of live clams for the aquarium trade. It emerged that the Australian market for giant clams is likely to be small. Possibly it could absorb about 5,000 clams per year but it seems that these will need to be sold to retailers at a price of $10 each or less (possibly$6 each or less) if a market of this size is to obtain. The market is so small that to supply it would be a sideline for a clam farm and indeed one farm could more than supply the whole market. At a turnout of say 1 00,000 clams per year by a farm, it would only require 5 per cent of its production to satisfy the whole Australian market for clams as aquarium specimens.


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# Eigenvalues of a symmetric matrix that every block is circulant

For a matrix like \begin{bmatrix} A & B \\ B & A \\ \end{bmatrix} which A and B are block matrix and are circulant, is there any simple way to find eigenvalues and eigenvectors? To be clear, following matrix is an example of my described matrix. \begin{bmatrix} 1 & 2 & 3 & 4 \\ 2 & 1 & 4 & 3 \\ 3 & 4 & 1 & 2 \\ 4 & 3 & 2 & 1 \\ \end{bmatrix}

Your matrix commutes with the matrices $$P_2 = P \oplus P$$ and $$X = \begin{pmatrix} 0 & I \\ I& 0\end{pmatrix}$$ where $P$ is the cyclic permutation matrix on one subblock. Moreover, also $P_2$ and $X$ commute. Thus the three matrices can be simultaneously diagonalized. The combined eigenspaces of $P_2$ and $X$ have the form $$E_{\lambda,\pm} = \mathop{\rm span}\begin{pmatrix}v_\lambda \\ \pm v_\lambda \end{pmatrix}$$ with the eigenvalues $\lambda_j =e^{2\pi i j/n}$ and $$v_\lambda =(1,\lambda, \lambda^2, ...)^t.$$
As the eigenspaces are 1 dimensional, the eigenvectors of your matrix are given by $$\begin{pmatrix}v_{\lambda_j} \\ \pm v_{\lambda_j} \end{pmatrix}$$ and the eigenvalues are easily found.
In particular, the unitary transformation $$U=\frac{1}{\sqrt{2}} \begin{pmatrix} U_\text{DFT} & U_\text{DFT} \\ U_\text{DFT} & -U_\text{DFT}\end{pmatrix}$$ with $U_\text{DFT}$ the discrete Fourier transform brings your matrix onto diagonal form.