Structure of cortical microcircuit theory
1 Department of Anatomy and Neurobiology, University of California, Irvine, CA, USA
Abstract
Recent experimental and theoretical investigations have made considerable advances in three major areas relating to the structural basis of quantitative cortical microcircuit theory. The first concerns the nature of the cellular units, encompassing the increasingly precise identification and progressively more complete listing of the individual cellular species that constitute the various cortical networks. The second element addresses the problem of heterogeneity, including the demonstration of the importance of cell to cell variability within defined interneuronal populations and the application of the Shannon-Wiener diversity index for the quantitative assessment of the number and relative abundance of interneuronal species. The third component relates to the discovery of basic topological principles underlying the circuit wiring, revealing a surprising order in the architectural design of networks. These new advances deepen our understanding of the computational principles embedded in cortical microcircuits, and they also provide novel opportunities for building realistic models of mammalian cortical microcircuits.
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Atoms of the cortical crystal: interneuronal species as fundamental units of microcircuits
It is well established that many more cellular subtypes can be defined among GABAergic interneurones than among glutamatergic cells. The general, albeit not yet rigorously tested, assumption is that the larger diversity of the numerically less dominant interneurones has its evolutionary origins in the many functional roles that interneurones perform in the network. An important clue regarding the diversity of interneuronal species comes from recent developmental studies that indicate that different interneuronal populations originate from distinct areas in the embryo. In general, glutamatergic cells are generated locally within the cortical areas, originating from the proliferative ventricular zone of the dorsal telencephalon (Sidman & Rakic, 1973). The newly born glutamatergic cells migrate radially from the ventricular zone to the cortical plate, where they form columns of neurones that originate at the same place (Rakic, 1972). In contrast, GABAergic interneurones derive from distant subcortical areas, and interneuronal migration is orthogonal to the direction of the radial migration, referred to as tangential migration. Evidence from several species, including humans, indicates that many, possibly all, cortical GABAergic interneurones derive from the ventral telencephalon in the primordium of the basal ganglia (Marin & Rubenstein, 2003). Importantly, there are multiple origins for interneurones within the subpallium. The medial ganglionic eminence appears to be the primary source of cortical interneurones, and several other embryonic brain areas have also been implicated, such as the lateral ganglionic eminence. Fate-mapping experiments in vivo, together with in vitro assays, indicated that distinct interneuronal sybtypes originate from different areas, e.g. parvalbumin- and somatostatin-expressing interneurones originate from the medial ganglionic eminence, whereas calretinin-positive interneurones originate from other areas (Wichterle et al. 2001; Anderson et al. 2002; Valcanis & Tan, 2003; Xu et al. 2004). Therefore, there is a strict segregation, at least for cortical parvalbumin and somatostatin cells versus calretinin cells, regarding the places of origin for the distinct interneuronal subtypes. Although much more needs to be determined about the exact sites of origin and mechanisms of migrations for the various cortical interneuronal species, these results show that the anatomically and histochemically identified interneurones not only have distinct functions in the adult circuit, but they also have distinct ontogenetic histories during development. In addition, these developmental data reinforce the notion that distinct interneuronal species exist in the network, in spite of the naturally occurring cell to cell variability (see below).
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The concept of interneuronal subtypes lies at the heart of cortical microcircuit research, and the identification and counting of interneuronal species are also central to determining the fate of interneuronal populations in pathological states (Santhakumar & Soltesz, 2004). In spite of its importance, there is no widely accepted, precise definition of interneuronal species (Maccaferri & Lacaille, 2003). The exact definition of a cellular species within an organ of a multicellular organism is obviously fraught with challenges. However, it is useful to remind ourselves that species definitions are notoriously difficult even in the case of animal or plant species (the common definition of species involves the potential for the successful production of fertile offspring via sexual reproduction, but this definition obviously cannot apply to species that reproduce only asexually, or to individuals that are beyond the reproductive age), yet the concept of species is still central to many quantitative branches of biology, e.g. population genetics and mathematical ecology (Wilson & Bossert, 1971). Interneuronal species definitions in the everyday practice often involve postsynaptic target specificity and the expression of species-characteristic markers (Freund & Buzsáki, 1996). Indeed, this practical approach works well in most situations, albeit it can run into problems in pathological states where certain interneurones may change their dendritic and axonal morphologies (Davenport et al. 1990; Deller et al. 1995; Mathern et al. 1997; Wittner et al. 2001; Wittner et al. 2002) or the species-defining marker may be down-regulated in an activity-dependent manner (Wittner et al. 2001). In spite of these difficulties, interneuronal species that were identified based on morphological, cytochemical and single cell physiological grounds (Buhl et al. 1994a,b; Freund & Buzsáki, 1996) subsequently have been shown to serve different functional roles in the circuit (Klausberger et al. 2003; Wang et al. 2004). For example, cholecystokinin (CCK)-positive neuronal subtypes were identified on anatomical grounds (Freund et al. 1986), and subsequent research resulted in the discovery that CCK+ neurones express cannabinoid receptor type 1 (CB1) (Katona et al. 1999), which endows these cells with the ability to regulate GABA release from their terminals in a manner that depends on both the presynaptic (Losonczy et al. 2004) and postsynaptic (Wilson et al. 2001; Wilson & Nicoll, 2001) activity levels. A recent, equally striking example of how an initially purely anatomical classification many years later led to the discovery of a cell-type specific functional property is the demonstration that neurogliaform cells are the sources of unitary GABAB responses in cortical networks (Tamas et al. 2003). Although a final answer to the question of how many species make up a given cortical area is not yet available in the form of a widely accepted, complete cataloguing of every abundant and rare interneuronal subtype, it appears that a reasonable estimate is that the number of identifiable distinct interneuronal species will be in the order of low to mid teens. For the more common interneuronal subtypes, there is already a substantial amount of precise, quantitative structural and functional information that allows the construction of realistic computational single cell models, e.g. for basket cells and certain hilar interneurones in the dentate gyrus (Santhakumar et al. 2004). However, a major challenge from this respect is that the rarer interneuronal subtypes are represented by only a handful of known specimens, severely limiting the extraction of reliable information for model construction.
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New approaches to handling interneuronal heterogeneity: separating cell to cell variability from species diversity
The most interesting, and, at the same time, the most challenging aspect of interneuronal research concerns the seemingly bewildering array of interneuronal forms and properties. An added problem is that electrophysiological single cell parameters often are not found to readily segregate along lines that correspond to currently known interneuronal subgroups (e.g. Pawelzik et al. 2002). While further precise, combined anatomical and electrophysiological experimental studies will be needed to clarify these issues, there have been new approaches concerning the thorny issue of interneuronal heterogeneity. As a starting point, more precise definitions of terms relating to interneuronal variability need to be established. Although in most texts the terms interneuronal heterogeneity, variability and diversity are often used interchangeably, the loose terminology often results in confusion. We propose to apply interneuronal heterogeneity as a broad term, referring to either cell to cell variability and/or species diversity (Aradi et al. 2004; Foldy et al. 2004; Santhakumar & Soltesz, 2004). In turn, we suggest that the term interneuronal variability is exclusively reserved to be used in connection with the variance of a particular parameter, reflecting the cell to cell variability within a population of a given interneuronal species. Parameter variance could be derived from any quantitatively measured property, e.g. the cell to cell variance of the peak sodium conductance within a population (note that our discussion here focuses on cellular variability, but these terms could also be applied to synaptic event populations as well, e.g. Aradi et al. 2004; Santhakumar & Soltesz, 2004). It must be emphasized that the notion of interneuronal intraspecies variability, i.e. the existence of certain measurable differences in some parameter between individual interneurones belonging to the same species, does not go against the notion that the concept of interneuronal species forms one of the major foundations of quantitative microcircuit theory. On the contrary, given the functional consequences of cell to cell variability within interneuronal subpopulations (see below), together with the biological reality that no two neurones are exactly the same, recognition of the extent and importance of the intraspecies variability clarifies and enriches the cellular species-based functional neuroanatomical approach that was originated by Ramon y Cajal. Finally, we propose to reserve the term diversity for the heterogeneity that is reflected in the number of subgroups, i.e. the number of interneuronal species (Foldy et al. 2004; Santhakumar & Soltesz, 2004). These simple definitions of interneuronal heterogeneity, variability and diversity should help to clarify the issues and aid in the construction of a much-needed framework, where experimental data on cell to cell variability and the existence of well-defined interneuronal species are not regarded as mutually exclusive and in perpetual conflict, but are treated as intrinsically and necessarily linked objective entities that can be studied in a quantitative, precise manner, as discussed below.
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Quantitative approaches to the measurement of interneuronal population variability and species diversity
In order for the above-defined terms to be useful and widely applied, they need to be based on solid quantitative foundations. Typically, the statistical tests used in interneuronal research are geared to detect differences in means or medians. Such statistical tests are often used to determine differences in particular parameters between two experimentally recorded and subsequently anatomically identified interneuronal species, and are also applied in interneuronal plasticity studies, where the change in a certain parameter is examined before or after a certain manipulation, as well as in pathophysiological investigations comparing the same interneuronal species in control and in diseased states. In order to arrive at a better understanding of cell to cell variability within interneuronal populations, it is important to experimentally determine the differences in variations which may or may not be associated with differences in means or medians. As we shall discuss below, it may be equally important to know not only what the average value of a given experimentally measured parameter (e.g. a particular sodium conductance) is within a specific interneuronal species, but also how typical the average or mean value is across the cell population. The assumption of normality plays an important role in testing for equality of variances. In the two independent sample situation when we can assume normality (e.g. based on the Kolmogorov-Smirnov normality test), the appropriate statistic to test equality of population variances, irrespective of whether the means are equal, is the variance ratio F test,
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where s1 and s2 are the standard deviations of the two samples, which has an F distribution under the hypothesis that the population variances are equal (Sprent & Smeeton, 2001). If we need to compare the variances of two groups without assumptions about normality (e.g. because the sample sizes are relatively small, which can lead to problems with normality tests), the non-parametric Conover squared-rank test can be used (Aradi & Soltesz, 2002). The Conover test for equality of variance is based on the squared ranks of absolute deviations from the means (where the data from both samples is combined before ranking) (Sprent & Smeeton, 2001). Certain statistical packages (e.g. StatXact from Cytel Software Corporation) provide a specific program for the Conover squared-rank test.
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Clearly, these measures of intraspecies parameter variance cannot be readily applied to the entirely different problem of quantification of interneuronal species diversity. The chosen measure should be able to reflect the diversity of interneuronal species in at least two fundamental ways. First, it should take into account how many species make up a particular network, since most researchers would agree that a network with more interneuronal species is more diverse. However, a second aspect is the relative abundance of the species. Intuitively, most of us would agree that a network where all species are found at exactly the same rate is more diverse than a network where certain species are abundant but others are exceedingly rare. These two related, but distinct aspects of diversity can be jointly taken into account with the Shannon-Wiener diversity index (Foldy et al. 2004). This particular measure of diversity is used in other disciplines, e.g. it is used to assess the number and relative abundance of animal and plant species in ecosystem studies (Wilson & Bossert, 1971), and it is also widely used in information theory and neuronal coding (Dayan & Abbott, 2001).
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The Shannon-Wiener diversity index is defined as
where k is the number of categories, and pi is the proportion of observations in category i (Zar, 1999). The measure D is formally equivalent to entropy, and it is a dimensionless number that describes how ‘interesting’ or ‘surprising’ a set of events (observations) is. For a single category (when there is only a single cellular species present in the network), the diversity index is zero (D = 0), and for two or more subgroups, D increases. D takes different values for different distributions (e.g. a uniform distribution where all events occur with the same probability versus a Gaussian distribution). The actual diversity index value for the various parameters of GABAergic systems in cortical microcircuits is only beginning to be assessed. Table 1 illustrates, in a step by step fashion, the details of its calculation for the interneuronal species that are present in the rat dentate gyrus (Santhakumar & Soltesz, 2004). The precise assessment of D-values in various cortical areas in control and pathological situations will be extremely useful for determining, in precise terms, the fundamental, quantitative features of interneuronal diversity.
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Functional consequences of cell to cell variability and changes in species diversity
It may be argued that the existence of small differences between two interneurones belonging to the same species may have little relevance for the functioning of the network. Recent modelling and experimental studies, however, have shown that even relatively small changes in a variety of cellular and synaptic parameters in interneurones can result in significant alterations in firing rates and network synchrony, even in the absence of changes in mean values (Aradi & Soltesz, 2002; Aradi et al. 2004). For example, increasing the cell to cell variance in expression of Ca2+ channels and K+ channels that influence action potential adaptation in an interneuronal population results in marked decreases in the synchrony of gamma frequency (40 Hz) interneuronal firing (Aradi & Soltesz, 2002) (see also Wang & Buzsaki, 1996; White et al. 1998; Tiesinga & Jose, 2000; Tiesinga et al. 2002), even if the changes in variance is introduced without altering the mean values. In addition, the changes in Ca2+ and K+ channel variance in interneurones can also modulate the amplitude of the degree of synchrony of the IPSPs (appearing as subthreshold oscillations) in the postsynaptic cells, which, in turn, can potently modulate the cells' responses to incoming excitatory inputs (Aradi & Soltesz, 2002). Given the strong effects of altered cell to cell variance in these modelling studies, it has been suggested that the cell to cell variance in interneuronal populations may be subject to modulation by certain regulatory processes, e.g. by the various ascending neuromodulatory pathways (Aradi & Soltesz, 2002). For example, if a given neuromodulator acts in a non-uniform manner on members of a given interneuronal species (e.g. depolarizing some to a greater degree than others), that neuromodulator will effectively regulate the cell to cell variance (in addition to possibly also changing the mean values). Interestingly, muscarinic receptor activation has been reported to induce a variety of effects within identified interneuronal populations (Parra et al. 1998; McQuiston & Madison, 1999), although further studies will be needed to determine the precise degree of cell to cell variability in the response of particular, well-defined interneuronal species to ascending modulatory substances.
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Recent studies also revealed the importance of interneuronal diversity (as defined above), by showing that altering the diversity index can significantly modulate neuronal firing and synchrony (Foldy et al. 2004). As illustrated in Fig. 1, increasing the number of subgroups (simulated by changing the number of different levels of excitatory current inputs that the cells received) in an interneuronal network resulted in a robust decrease in coherence (a measure of synchronous neuronal firing), even though the mean and variance of the depolarizing current input did not change across the population. These results indicate that both intragroup variance and intergroup diversity can have significant effects on the functional properties of cortical microcircuits.
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A, spike raster plots illustrating the effect changing diversity by altering the number of different levels of excitation (Idepol) that interneurones receive from three levels (upper panel; D = 0.6) to five (lower panel; D = 0.8) levels. B, summary plot of the network coherence (White et al. 1998) shown as a function of the normalized diversity index (D = 1 is for six subgroups) for a 120-cell interneuronal network. Note that the network coherence decreased with increasing diversity, even though the mean and CV of Idepol for the interneuronal population did not change (mean Idepol = 0.6 nA; CV = 0.115). Modified from Foldy et al. (2004) with permission from Blackwell Publishing.
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Graph theoretical approaches for determining the basic architecture of cortical microcircuits
What is the basic architectural property of all neuronal networks A recent insight into this difficult, but important question came from studies that applied graph theoretical techniques to real-world networks, ranging from the biochemical and social networks to the electrical grid of the United States and the internet (Watts & Strogatz, 1998; Albert et al. 1999; Jeong et al. 2000; Barabasi et al. 2002). In a landmark paper, Watts & Strogatz (1998) used two measures, the average shortest path length (L) and the clustering coefficient (C, describing the probability that two nodes connected to a common node are also connected to each other) to characterize the so-called small world network nature of many real networks (Fig. 2A). In highly ordered, regular networks (Fig. 2A, left panel), individual nodes are richly linked to their neighbours (high C), but it takes a lot of steps to go from a node to other nodes on average (high L). In random networks containing the same number of nodes and links (Fig. 2A, right panel), L is low, but the local connectivity is relatively poor (low C). Interestingly, if even a few links from the ordered network are reconnected in a completely random fashion, the resulting network will be unlike any of these other two networks, because they will have a low L (like random networks, due to the presence of long-range connections) and a high C (like regular networks). Such networks are called small world networks. Watts & Strogatz (1998) determined the graph structure of the nervous system of the neuronal network of the worm Caenorhabditis elegans (Fig. 2B). They found that the 308 cells in the nervous system of the worm were connected in a manner that precisely corresponded to the small world network graphs, characterized by a low L and a high C. The low L in the worm's nervous system indicated that the network was globally well-connected, since it took less than three steps to reach any other neurone from any neurone on average. In contrast, the high C indicated that the network was also well-connected locally, i.e. the neurones formed local clusters whose members were especially heavily interlinked. Theoretical studies, simulating network activity on artificial circuits, indicated that a high C may underlie a special propensity for strong local synchrony, while a low L allows activity to easily spread throughout the network, leading to global synchrony (Lago-Fernandez et al. 2000; Barahona & Pecora, 2002; Li & Chen, 2003; Masuda & Aihara, 2004). In addition, recent modelling studies also indicated that changes in L and C can strongly modulate the action potential discharge patterns (e.g. tonic firing versus bursting) in model CA1 and CA3 networks (Netoff et al. 2004).
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A, basic types of graph structure (regular, small world and random) are illustrated, with the characteristic L and C values indicated above the graphs. B, the graph structure of the nervous system of the C. elegans. Cells were arranged in a circle (as the nodes in panel A), and the synaptic connections were indicated by links (data generously supplied by Dr Watts, from the same data base used in Watts & Strogatz, 1998). At the bottom of the graph, the L and C values for the actual nervous system of the worm is shown, as well as the L and C values for the equivalent random graphs (containing the same number of nodes and links). Modified from Watts & Strogatz (1998); reproduced by permission from Nature393, 440–442, Macmillan Publishers Ltd (http://www.nature.com/).
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It is interesting that many real, non-neuronal networks also conform to the mathematical definitions of small world networks (Watts, 1999), indicating a strong advantage for the emergence of such networks in a variety of situations. In spite of the apparently widespread nature of small world topology, it is not known whether mammalian neuronal networks are, in fact, also small world networks. In order to determine the basic topology of cortical networks, precise information is needed about the number of neuronal species that reside in a network and the connectivity rules between each of these species. In addition, since real neuronal networks are topographic in nature (i.e. the probability of connectivity between two neurones depends not only on their respective subtypes but also on their relative spatial positions in the network), precise information is also required on the axonal distributions of each neuronal subtype in space. One of the best-studied brain area in the mammalian CNS is the dentate gyrus, and a recent study (Dyhrfjeld-Johnsen et al. 2004) assembled the necessary anatomical information, and, for the first time, determined the L and C values for a mammalian neuronal network. Surprisingly, the L value in the dentate gyrus was virtually identical to the L value determined for the nervous system of the worm, indicating that, in spite of over a million cells connected by over a billion links in the mammalian dentate gyrus, the average number of steps separating any two neurones in the network is still astonishingly low. The low L, together with the fact that the C value for the network was many times higher than the C value for an equivalent random graph (containing the same number of nodes and links as the real dentate graph), indicated that the neuronal network of the mammalian dentate gyrus is also a small world network (Dyhrfjeld-Johnsen et al. 2004). With the rapid expansion of our knowledge about neuronal connectivity in various areas of the mammalian CNS, it is likely that within a short time the L and C values can be determined for larger and larger parts of the mammalian nervous system. It is interesting to note that, given the number of long-distance connections between distinct brain areas (e.g. septum and hippocampus; Freund & Antal, 1988) and between distinct parts of the same general brain area (e.g. between the dentate gyrus and CA1 and the subiculum; Sik et al. 1994; Ceranik et al. 1997; see also Buzsaki et al. 2004), it is expected that the L value will increase only slightly when larger neuronal assemblies are considered. In addition to helping us understand the basic topological features of neuronal networks, a more precise knowledge of the L and C values will also be useful in scaling down the large biological networks for computational modelling studies (Traub et al. 1999; Bibbig et al. 2002).
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Closing thoughts and outlook
In conclusion, recent experimental and theoretical research efforts have made substantial advances in uncovering three structural foundations for cortical microcircuit theory. The increasingly exact description of the various interneuronal species that constitute cortical networks, together with quantitative data on their relative abundance, will help to determine how interneuronal diversity changes in development as well as in neurological diseases. It is also interesting to note that, with more data provided by future studies, we will also be able to differentiate between diversity (the number and abundance of interneuronal species, as defined above) and the so-called diversity (Duivenvoorden et al. 2002), which describes how species composition varies between one brain area to another. Precise measurements of the extent of cell to cell variability within identified interneuronal populations in various brain states (Klausberger et al. 2003; Klausberger et al. 2004) and in response to neuromodulatory substances (Aradi & Soltesz, 2002) are likely to provide new insights into the regulation of interneuronal synchrony through activity-dependent modulation of population variance. Finally, the new microcircuit connectivity data are likely to allow us to determine in the near future the exact graph architecture of hippocampal and neocortical neuronal networks.
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Footnotes
This report was presented at The Journal of Physiology Symposium in honour of the late Eberhard H. Buhl on Structure/Function Correlates in Neurons and Networks, Leeds, UK, 10 September 2004. It was commissioned by the Editorial Board and reflects the views of the authors.
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Xu Q, Cobos I, De La Cruz E, Rubenstein JL & Anderson SA (2004). Origins of cortical interneuron subtypes. J Neurosci 24, 2612–2622., http://www.100md.com(Csaba Fldy, Jonas Dyhrfje)
Abstract
Recent experimental and theoretical investigations have made considerable advances in three major areas relating to the structural basis of quantitative cortical microcircuit theory. The first concerns the nature of the cellular units, encompassing the increasingly precise identification and progressively more complete listing of the individual cellular species that constitute the various cortical networks. The second element addresses the problem of heterogeneity, including the demonstration of the importance of cell to cell variability within defined interneuronal populations and the application of the Shannon-Wiener diversity index for the quantitative assessment of the number and relative abundance of interneuronal species. The third component relates to the discovery of basic topological principles underlying the circuit wiring, revealing a surprising order in the architectural design of networks. These new advances deepen our understanding of the computational principles embedded in cortical microcircuits, and they also provide novel opportunities for building realistic models of mammalian cortical microcircuits.
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Atoms of the cortical crystal: interneuronal species as fundamental units of microcircuits
It is well established that many more cellular subtypes can be defined among GABAergic interneurones than among glutamatergic cells. The general, albeit not yet rigorously tested, assumption is that the larger diversity of the numerically less dominant interneurones has its evolutionary origins in the many functional roles that interneurones perform in the network. An important clue regarding the diversity of interneuronal species comes from recent developmental studies that indicate that different interneuronal populations originate from distinct areas in the embryo. In general, glutamatergic cells are generated locally within the cortical areas, originating from the proliferative ventricular zone of the dorsal telencephalon (Sidman & Rakic, 1973). The newly born glutamatergic cells migrate radially from the ventricular zone to the cortical plate, where they form columns of neurones that originate at the same place (Rakic, 1972). In contrast, GABAergic interneurones derive from distant subcortical areas, and interneuronal migration is orthogonal to the direction of the radial migration, referred to as tangential migration. Evidence from several species, including humans, indicates that many, possibly all, cortical GABAergic interneurones derive from the ventral telencephalon in the primordium of the basal ganglia (Marin & Rubenstein, 2003). Importantly, there are multiple origins for interneurones within the subpallium. The medial ganglionic eminence appears to be the primary source of cortical interneurones, and several other embryonic brain areas have also been implicated, such as the lateral ganglionic eminence. Fate-mapping experiments in vivo, together with in vitro assays, indicated that distinct interneuronal sybtypes originate from different areas, e.g. parvalbumin- and somatostatin-expressing interneurones originate from the medial ganglionic eminence, whereas calretinin-positive interneurones originate from other areas (Wichterle et al. 2001; Anderson et al. 2002; Valcanis & Tan, 2003; Xu et al. 2004). Therefore, there is a strict segregation, at least for cortical parvalbumin and somatostatin cells versus calretinin cells, regarding the places of origin for the distinct interneuronal subtypes. Although much more needs to be determined about the exact sites of origin and mechanisms of migrations for the various cortical interneuronal species, these results show that the anatomically and histochemically identified interneurones not only have distinct functions in the adult circuit, but they also have distinct ontogenetic histories during development. In addition, these developmental data reinforce the notion that distinct interneuronal species exist in the network, in spite of the naturally occurring cell to cell variability (see below).
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The concept of interneuronal subtypes lies at the heart of cortical microcircuit research, and the identification and counting of interneuronal species are also central to determining the fate of interneuronal populations in pathological states (Santhakumar & Soltesz, 2004). In spite of its importance, there is no widely accepted, precise definition of interneuronal species (Maccaferri & Lacaille, 2003). The exact definition of a cellular species within an organ of a multicellular organism is obviously fraught with challenges. However, it is useful to remind ourselves that species definitions are notoriously difficult even in the case of animal or plant species (the common definition of species involves the potential for the successful production of fertile offspring via sexual reproduction, but this definition obviously cannot apply to species that reproduce only asexually, or to individuals that are beyond the reproductive age), yet the concept of species is still central to many quantitative branches of biology, e.g. population genetics and mathematical ecology (Wilson & Bossert, 1971). Interneuronal species definitions in the everyday practice often involve postsynaptic target specificity and the expression of species-characteristic markers (Freund & Buzsáki, 1996). Indeed, this practical approach works well in most situations, albeit it can run into problems in pathological states where certain interneurones may change their dendritic and axonal morphologies (Davenport et al. 1990; Deller et al. 1995; Mathern et al. 1997; Wittner et al. 2001; Wittner et al. 2002) or the species-defining marker may be down-regulated in an activity-dependent manner (Wittner et al. 2001). In spite of these difficulties, interneuronal species that were identified based on morphological, cytochemical and single cell physiological grounds (Buhl et al. 1994a,b; Freund & Buzsáki, 1996) subsequently have been shown to serve different functional roles in the circuit (Klausberger et al. 2003; Wang et al. 2004). For example, cholecystokinin (CCK)-positive neuronal subtypes were identified on anatomical grounds (Freund et al. 1986), and subsequent research resulted in the discovery that CCK+ neurones express cannabinoid receptor type 1 (CB1) (Katona et al. 1999), which endows these cells with the ability to regulate GABA release from their terminals in a manner that depends on both the presynaptic (Losonczy et al. 2004) and postsynaptic (Wilson et al. 2001; Wilson & Nicoll, 2001) activity levels. A recent, equally striking example of how an initially purely anatomical classification many years later led to the discovery of a cell-type specific functional property is the demonstration that neurogliaform cells are the sources of unitary GABAB responses in cortical networks (Tamas et al. 2003). Although a final answer to the question of how many species make up a given cortical area is not yet available in the form of a widely accepted, complete cataloguing of every abundant and rare interneuronal subtype, it appears that a reasonable estimate is that the number of identifiable distinct interneuronal species will be in the order of low to mid teens. For the more common interneuronal subtypes, there is already a substantial amount of precise, quantitative structural and functional information that allows the construction of realistic computational single cell models, e.g. for basket cells and certain hilar interneurones in the dentate gyrus (Santhakumar et al. 2004). However, a major challenge from this respect is that the rarer interneuronal subtypes are represented by only a handful of known specimens, severely limiting the extraction of reliable information for model construction.
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New approaches to handling interneuronal heterogeneity: separating cell to cell variability from species diversity
The most interesting, and, at the same time, the most challenging aspect of interneuronal research concerns the seemingly bewildering array of interneuronal forms and properties. An added problem is that electrophysiological single cell parameters often are not found to readily segregate along lines that correspond to currently known interneuronal subgroups (e.g. Pawelzik et al. 2002). While further precise, combined anatomical and electrophysiological experimental studies will be needed to clarify these issues, there have been new approaches concerning the thorny issue of interneuronal heterogeneity. As a starting point, more precise definitions of terms relating to interneuronal variability need to be established. Although in most texts the terms interneuronal heterogeneity, variability and diversity are often used interchangeably, the loose terminology often results in confusion. We propose to apply interneuronal heterogeneity as a broad term, referring to either cell to cell variability and/or species diversity (Aradi et al. 2004; Foldy et al. 2004; Santhakumar & Soltesz, 2004). In turn, we suggest that the term interneuronal variability is exclusively reserved to be used in connection with the variance of a particular parameter, reflecting the cell to cell variability within a population of a given interneuronal species. Parameter variance could be derived from any quantitatively measured property, e.g. the cell to cell variance of the peak sodium conductance within a population (note that our discussion here focuses on cellular variability, but these terms could also be applied to synaptic event populations as well, e.g. Aradi et al. 2004; Santhakumar & Soltesz, 2004). It must be emphasized that the notion of interneuronal intraspecies variability, i.e. the existence of certain measurable differences in some parameter between individual interneurones belonging to the same species, does not go against the notion that the concept of interneuronal species forms one of the major foundations of quantitative microcircuit theory. On the contrary, given the functional consequences of cell to cell variability within interneuronal subpopulations (see below), together with the biological reality that no two neurones are exactly the same, recognition of the extent and importance of the intraspecies variability clarifies and enriches the cellular species-based functional neuroanatomical approach that was originated by Ramon y Cajal. Finally, we propose to reserve the term diversity for the heterogeneity that is reflected in the number of subgroups, i.e. the number of interneuronal species (Foldy et al. 2004; Santhakumar & Soltesz, 2004). These simple definitions of interneuronal heterogeneity, variability and diversity should help to clarify the issues and aid in the construction of a much-needed framework, where experimental data on cell to cell variability and the existence of well-defined interneuronal species are not regarded as mutually exclusive and in perpetual conflict, but are treated as intrinsically and necessarily linked objective entities that can be studied in a quantitative, precise manner, as discussed below.
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Quantitative approaches to the measurement of interneuronal population variability and species diversity
In order for the above-defined terms to be useful and widely applied, they need to be based on solid quantitative foundations. Typically, the statistical tests used in interneuronal research are geared to detect differences in means or medians. Such statistical tests are often used to determine differences in particular parameters between two experimentally recorded and subsequently anatomically identified interneuronal species, and are also applied in interneuronal plasticity studies, where the change in a certain parameter is examined before or after a certain manipulation, as well as in pathophysiological investigations comparing the same interneuronal species in control and in diseased states. In order to arrive at a better understanding of cell to cell variability within interneuronal populations, it is important to experimentally determine the differences in variations which may or may not be associated with differences in means or medians. As we shall discuss below, it may be equally important to know not only what the average value of a given experimentally measured parameter (e.g. a particular sodium conductance) is within a specific interneuronal species, but also how typical the average or mean value is across the cell population. The assumption of normality plays an important role in testing for equality of variances. In the two independent sample situation when we can assume normality (e.g. based on the Kolmogorov-Smirnov normality test), the appropriate statistic to test equality of population variances, irrespective of whether the means are equal, is the variance ratio F test,
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where s1 and s2 are the standard deviations of the two samples, which has an F distribution under the hypothesis that the population variances are equal (Sprent & Smeeton, 2001). If we need to compare the variances of two groups without assumptions about normality (e.g. because the sample sizes are relatively small, which can lead to problems with normality tests), the non-parametric Conover squared-rank test can be used (Aradi & Soltesz, 2002). The Conover test for equality of variance is based on the squared ranks of absolute deviations from the means (where the data from both samples is combined before ranking) (Sprent & Smeeton, 2001). Certain statistical packages (e.g. StatXact from Cytel Software Corporation) provide a specific program for the Conover squared-rank test.
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Clearly, these measures of intraspecies parameter variance cannot be readily applied to the entirely different problem of quantification of interneuronal species diversity. The chosen measure should be able to reflect the diversity of interneuronal species in at least two fundamental ways. First, it should take into account how many species make up a particular network, since most researchers would agree that a network with more interneuronal species is more diverse. However, a second aspect is the relative abundance of the species. Intuitively, most of us would agree that a network where all species are found at exactly the same rate is more diverse than a network where certain species are abundant but others are exceedingly rare. These two related, but distinct aspects of diversity can be jointly taken into account with the Shannon-Wiener diversity index (Foldy et al. 2004). This particular measure of diversity is used in other disciplines, e.g. it is used to assess the number and relative abundance of animal and plant species in ecosystem studies (Wilson & Bossert, 1971), and it is also widely used in information theory and neuronal coding (Dayan & Abbott, 2001).
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The Shannon-Wiener diversity index is defined as
where k is the number of categories, and pi is the proportion of observations in category i (Zar, 1999). The measure D is formally equivalent to entropy, and it is a dimensionless number that describes how ‘interesting’ or ‘surprising’ a set of events (observations) is. For a single category (when there is only a single cellular species present in the network), the diversity index is zero (D = 0), and for two or more subgroups, D increases. D takes different values for different distributions (e.g. a uniform distribution where all events occur with the same probability versus a Gaussian distribution). The actual diversity index value for the various parameters of GABAergic systems in cortical microcircuits is only beginning to be assessed. Table 1 illustrates, in a step by step fashion, the details of its calculation for the interneuronal species that are present in the rat dentate gyrus (Santhakumar & Soltesz, 2004). The precise assessment of D-values in various cortical areas in control and pathological situations will be extremely useful for determining, in precise terms, the fundamental, quantitative features of interneuronal diversity.
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Functional consequences of cell to cell variability and changes in species diversity
It may be argued that the existence of small differences between two interneurones belonging to the same species may have little relevance for the functioning of the network. Recent modelling and experimental studies, however, have shown that even relatively small changes in a variety of cellular and synaptic parameters in interneurones can result in significant alterations in firing rates and network synchrony, even in the absence of changes in mean values (Aradi & Soltesz, 2002; Aradi et al. 2004). For example, increasing the cell to cell variance in expression of Ca2+ channels and K+ channels that influence action potential adaptation in an interneuronal population results in marked decreases in the synchrony of gamma frequency (40 Hz) interneuronal firing (Aradi & Soltesz, 2002) (see also Wang & Buzsaki, 1996; White et al. 1998; Tiesinga & Jose, 2000; Tiesinga et al. 2002), even if the changes in variance is introduced without altering the mean values. In addition, the changes in Ca2+ and K+ channel variance in interneurones can also modulate the amplitude of the degree of synchrony of the IPSPs (appearing as subthreshold oscillations) in the postsynaptic cells, which, in turn, can potently modulate the cells' responses to incoming excitatory inputs (Aradi & Soltesz, 2002). Given the strong effects of altered cell to cell variance in these modelling studies, it has been suggested that the cell to cell variance in interneuronal populations may be subject to modulation by certain regulatory processes, e.g. by the various ascending neuromodulatory pathways (Aradi & Soltesz, 2002). For example, if a given neuromodulator acts in a non-uniform manner on members of a given interneuronal species (e.g. depolarizing some to a greater degree than others), that neuromodulator will effectively regulate the cell to cell variance (in addition to possibly also changing the mean values). Interestingly, muscarinic receptor activation has been reported to induce a variety of effects within identified interneuronal populations (Parra et al. 1998; McQuiston & Madison, 1999), although further studies will be needed to determine the precise degree of cell to cell variability in the response of particular, well-defined interneuronal species to ascending modulatory substances.
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Recent studies also revealed the importance of interneuronal diversity (as defined above), by showing that altering the diversity index can significantly modulate neuronal firing and synchrony (Foldy et al. 2004). As illustrated in Fig. 1, increasing the number of subgroups (simulated by changing the number of different levels of excitatory current inputs that the cells received) in an interneuronal network resulted in a robust decrease in coherence (a measure of synchronous neuronal firing), even though the mean and variance of the depolarizing current input did not change across the population. These results indicate that both intragroup variance and intergroup diversity can have significant effects on the functional properties of cortical microcircuits.
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A, spike raster plots illustrating the effect changing diversity by altering the number of different levels of excitation (Idepol) that interneurones receive from three levels (upper panel; D = 0.6) to five (lower panel; D = 0.8) levels. B, summary plot of the network coherence (White et al. 1998) shown as a function of the normalized diversity index (D = 1 is for six subgroups) for a 120-cell interneuronal network. Note that the network coherence decreased with increasing diversity, even though the mean and CV of Idepol for the interneuronal population did not change (mean Idepol = 0.6 nA; CV = 0.115). Modified from Foldy et al. (2004) with permission from Blackwell Publishing.
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Graph theoretical approaches for determining the basic architecture of cortical microcircuits
What is the basic architectural property of all neuronal networks A recent insight into this difficult, but important question came from studies that applied graph theoretical techniques to real-world networks, ranging from the biochemical and social networks to the electrical grid of the United States and the internet (Watts & Strogatz, 1998; Albert et al. 1999; Jeong et al. 2000; Barabasi et al. 2002). In a landmark paper, Watts & Strogatz (1998) used two measures, the average shortest path length (L) and the clustering coefficient (C, describing the probability that two nodes connected to a common node are also connected to each other) to characterize the so-called small world network nature of many real networks (Fig. 2A). In highly ordered, regular networks (Fig. 2A, left panel), individual nodes are richly linked to their neighbours (high C), but it takes a lot of steps to go from a node to other nodes on average (high L). In random networks containing the same number of nodes and links (Fig. 2A, right panel), L is low, but the local connectivity is relatively poor (low C). Interestingly, if even a few links from the ordered network are reconnected in a completely random fashion, the resulting network will be unlike any of these other two networks, because they will have a low L (like random networks, due to the presence of long-range connections) and a high C (like regular networks). Such networks are called small world networks. Watts & Strogatz (1998) determined the graph structure of the nervous system of the neuronal network of the worm Caenorhabditis elegans (Fig. 2B). They found that the 308 cells in the nervous system of the worm were connected in a manner that precisely corresponded to the small world network graphs, characterized by a low L and a high C. The low L in the worm's nervous system indicated that the network was globally well-connected, since it took less than three steps to reach any other neurone from any neurone on average. In contrast, the high C indicated that the network was also well-connected locally, i.e. the neurones formed local clusters whose members were especially heavily interlinked. Theoretical studies, simulating network activity on artificial circuits, indicated that a high C may underlie a special propensity for strong local synchrony, while a low L allows activity to easily spread throughout the network, leading to global synchrony (Lago-Fernandez et al. 2000; Barahona & Pecora, 2002; Li & Chen, 2003; Masuda & Aihara, 2004). In addition, recent modelling studies also indicated that changes in L and C can strongly modulate the action potential discharge patterns (e.g. tonic firing versus bursting) in model CA1 and CA3 networks (Netoff et al. 2004).
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A, basic types of graph structure (regular, small world and random) are illustrated, with the characteristic L and C values indicated above the graphs. B, the graph structure of the nervous system of the C. elegans. Cells were arranged in a circle (as the nodes in panel A), and the synaptic connections were indicated by links (data generously supplied by Dr Watts, from the same data base used in Watts & Strogatz, 1998). At the bottom of the graph, the L and C values for the actual nervous system of the worm is shown, as well as the L and C values for the equivalent random graphs (containing the same number of nodes and links). Modified from Watts & Strogatz (1998); reproduced by permission from Nature393, 440–442, Macmillan Publishers Ltd (http://www.nature.com/).
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It is interesting that many real, non-neuronal networks also conform to the mathematical definitions of small world networks (Watts, 1999), indicating a strong advantage for the emergence of such networks in a variety of situations. In spite of the apparently widespread nature of small world topology, it is not known whether mammalian neuronal networks are, in fact, also small world networks. In order to determine the basic topology of cortical networks, precise information is needed about the number of neuronal species that reside in a network and the connectivity rules between each of these species. In addition, since real neuronal networks are topographic in nature (i.e. the probability of connectivity between two neurones depends not only on their respective subtypes but also on their relative spatial positions in the network), precise information is also required on the axonal distributions of each neuronal subtype in space. One of the best-studied brain area in the mammalian CNS is the dentate gyrus, and a recent study (Dyhrfjeld-Johnsen et al. 2004) assembled the necessary anatomical information, and, for the first time, determined the L and C values for a mammalian neuronal network. Surprisingly, the L value in the dentate gyrus was virtually identical to the L value determined for the nervous system of the worm, indicating that, in spite of over a million cells connected by over a billion links in the mammalian dentate gyrus, the average number of steps separating any two neurones in the network is still astonishingly low. The low L, together with the fact that the C value for the network was many times higher than the C value for an equivalent random graph (containing the same number of nodes and links as the real dentate graph), indicated that the neuronal network of the mammalian dentate gyrus is also a small world network (Dyhrfjeld-Johnsen et al. 2004). With the rapid expansion of our knowledge about neuronal connectivity in various areas of the mammalian CNS, it is likely that within a short time the L and C values can be determined for larger and larger parts of the mammalian nervous system. It is interesting to note that, given the number of long-distance connections between distinct brain areas (e.g. septum and hippocampus; Freund & Antal, 1988) and between distinct parts of the same general brain area (e.g. between the dentate gyrus and CA1 and the subiculum; Sik et al. 1994; Ceranik et al. 1997; see also Buzsaki et al. 2004), it is expected that the L value will increase only slightly when larger neuronal assemblies are considered. In addition to helping us understand the basic topological features of neuronal networks, a more precise knowledge of the L and C values will also be useful in scaling down the large biological networks for computational modelling studies (Traub et al. 1999; Bibbig et al. 2002).
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Closing thoughts and outlook
In conclusion, recent experimental and theoretical research efforts have made substantial advances in uncovering three structural foundations for cortical microcircuit theory. The increasingly exact description of the various interneuronal species that constitute cortical networks, together with quantitative data on their relative abundance, will help to determine how interneuronal diversity changes in development as well as in neurological diseases. It is also interesting to note that, with more data provided by future studies, we will also be able to differentiate between diversity (the number and abundance of interneuronal species, as defined above) and the so-called diversity (Duivenvoorden et al. 2002), which describes how species composition varies between one brain area to another. Precise measurements of the extent of cell to cell variability within identified interneuronal populations in various brain states (Klausberger et al. 2003; Klausberger et al. 2004) and in response to neuromodulatory substances (Aradi & Soltesz, 2002) are likely to provide new insights into the regulation of interneuronal synchrony through activity-dependent modulation of population variance. Finally, the new microcircuit connectivity data are likely to allow us to determine in the near future the exact graph architecture of hippocampal and neocortical neuronal networks.
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Footnotes
This report was presented at The Journal of Physiology Symposium in honour of the late Eberhard H. Buhl on Structure/Function Correlates in Neurons and Networks, Leeds, UK, 10 September 2004. It was commissioned by the Editorial Board and reflects the views of the authors.
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