Review TRENDS in Neurosciences Vol.27 No.4 April 2004 Interneuron Diversity series: Circuit complexity and axon wiring economy of cortical interneurons Gyo¨ rgy Buzsa´ ki1, Caroline Geisler2, Darrell A. Henze3 and Xiao-Jing Wang2 1Center for Molecular and Behavioral Neuroscience, Rutgers, The State University of New Jersey, Newark, NJ 07102, USA 2Volen Center for Complex Systems and Physics Department, Brandeis University, Waltham, MA 02454-9110, USA 3Department of Neuroscience, Merck Research Laboratories, West Point, PA 19486, USA The performance of the brain is constrained by wiring sparsely connected (e.g. feedforward ‘synfire’ chains of length and maintenance costs. The apparently inverse pyramidal cells across many layers [9]), signals become too relationship between number of neurons in the various long to propagate across the network owing to synaptic interneuron classes and the spatial extent of their axon and conduction delays. However, if the network is densely trees suggests a mathematically definable organization, recurrent, the number of connections should scale with the reminiscent of ‘small-world’ or scale-free networks network size by some rule. Whereas all-to-all wiring is observed in other complex systems. The wiring-econ- possible in a tissue culture involving dozens of neurons, it omy-based classification of cortical inhibitory inter- becomes less and less feasible when millions of neurons neurons is supported by the distinct physiological are involved, owing to space and energy supply limitations patterns of class members in the intact brain. The com- [10,11]. The second approach for increasing network per- plex wiring of diverse interneuron classes could rep- formance is adding novel types of constituents (e.g. func- resent an economic solution for supporting global tionally different types of interneurons), whose activity synchrony and oscillations at multiple timescales with can exert qualitatively different effects on network functions minimum axon length. [12]. Combining distinct computational elements endows networks with the ability to carry out novel computations One of the main challenges of neuroscience is to under- (e.g. oscillations of different frequencies). Mathematical stand how complex behaviors of the brain emerge from its modeling indicates a power-law relationship between the cellular constituents. The mammalian cortex consists of number of computation types (complexity) and the number two basic neuron types: excitatory principal cells and of distinct constituents in physical networks (e.g. elec- inhibitory interneurons. In contrast to the more homo- tronic devices and the Internet) [12,13]. The economy of geneous principal cell population, interneurons are excep- wiring in physical systems has received special attention tionally diverse in their morphological appearance and recently (Box 1) and excellent reviews are available on functional properties [1–7]. To date, there is no univer- this topic [13–16]. sally accepted taxonomy of cortical interneurons. Classi- Brains have evolutionary goals but implementation fication schemes vary from a dozen or so defined classes of brain structures has physical constraints [11,17]. Brain [1–7] to views that regard interneurons as a single group systems with ‘simple’ computational demands evolved with virtually unlimited heterogeneity of its members [1]. only a few neuron types. The basal ganglia, thalamus and Interneurons differ from each other in intrinsic biophysi- the cerebellum possess a low degree of variability in their cal properties and in morphological and molecular bio- neuron types. By contrast, cortical structures have evolved logical features, as well as in connectivity [2]. This article in a manner that most closely resembles a relatively considers how wiring of interneurons affects their contri- sparsely connected network of few principal cell types and bution to network performance and suggests that connec- many classes of GABAergic interneurons. An important, tivity is a useful approach for examining how complex but hitherto unaddressed, issue is whether diversity of functions (e.g. oscillations) emerge from elementary interneurons increases with the evolution of the mammal- features (e.g. inhibition) [8]. ian cortex. Even if the same interneuron types are present in small and large brains, some unique wiring rules must Building networks for multiple functions be implemented so that functions have a preserved The repertoire and complexity of network performance can continuity in brains of various complexities. One hypothe- be augmented in two fundamentally different ways. The sis is that the diversity of interneurons in the mammalian first approach is to use relatively few constituents in large cortex [1–7] reflects a compromise between computational numbers. However, physical realization of this approach needs and wiring economy [10,18]. The diversity of inter- in growing networks is problematic. If the network is neurons in the hippocampus and neocortex might have evolved to meet the need for multiple functions, as will Corresponding author: Gyo¨rgy Buzsa´ki ([email protected]). be discussed in this review. To date, the many facets of www.sciencedirect.com 0166-2236/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.tins.2004.02.007 Review TRENDS in Neurosciences Vol.27 No.4 April 2004 187 already indicates a rich multiplicity of neocortical inter- Box 1. Small-world and scale-free network architecture neurons [4,5,7,19] and their basic similarity to hippo- How fast can a message propagate from one neuron (‘node’) to campal interneurons [5]. Examples of how interneurons distant neurons in large networks? If one defines a ‘characteristic implement novel functions in the brain include sculpting path length’ lpath as the average number of monosynaptic connec- stimulus selectivity of cortical neurons in sensory and tions in the shortest path between two neurons, how does it scale memory systems [20,21] or generating coherent oscil- with the network size N? In a completely random network with a lations at different frequencies [22]. Given the hypothesis sufficiently large number of links per node, lpath can be very short (and the network is a ‘small world’). But what if, more realistically, most that diversity reflects a compromise between compu- connections are local? This question was studied in a landmark paper tational needs and wiring economy, we need to discover by Watts and Strogatz [14]. They first prescribed a local architecture the rules that can describe how interneuron diversity and with neighboring connections, for which lpath increases linearly with connectivity result in economical computational complexity. N: Then they reconnected a fraction r of existing links to nodes that were chosen uniformly at random over the entire network (Figure Ia). Surprisingly, they found that even with a small number of shortcuts Scalable interneuronal clocks: connectivity is of the ðr < 0:05Þ; the dependence of lpath on N becomes , logðNÞ. The small essence number of shortcuts dramatically reduces the average path length Complex brains have developed specialized mechanisms 6 lpath of the network. For example, if N ¼ 10 ; with local connections 6 for keeping time: inhibitory interneuron networks [23]. lpath is also ,10 , whereas with a small world architecture lpath < log 106 ¼ 6: Hence, a ‘small-world’ architecture can be realized with Oscillatory timing can transform unconnected principal only a few shortcuts (long-range connections). cell groups into temporal coalitions, providing maximal In the Watts–Strogatz model, the reconnection was assumed to be flexibility and economic use of their spikes [24]. Various uniform across the network. What if the reconnection probability architectures of inhibitory and excitatory neurons can give p(i,j ) from node i to node j decreases with the distance d(i,j )? rise to oscillations [25–28]. The simplest one consists of Intuitively, if p(i,j ) is local, (e.g. a narrow Gaussian or exponential distribution), there would be no chance for long-range connections. interneurons of the same type [26,28–34]. Let us illustrate However, if p(i,j ) decreases with d(i,j ) as a power law, p(i,j ) , the importance of connectivity using this simplest network. d(i,j )2a, then there is a significant (although small) probability for Suppose that the goal of a interneuron network is to connections across long distances, especially if the exponent a is provide oscillatory timing for the principal cells and that small so that p(i,j ) falls off slowly with d(i,j ), and a small world this function should be preserved in different animals – becomes realizable. Power distributions lack a characteristic scale and, hence, are ‘scale-free’. Many recent studies have been devoted that is, independent of the brain size. How should the to ‘scale-free networks’ where the number k of links per node obeys a network be wired? Because synchronization requires a power law, P(k) , k2g. The skewed distribution with a heavy tail minimum connectedness, one possibility is to keep a given means that there are a few nodes (‘hubs’) with an exceptionally large fractional connectivity among other neurons, regardless of number of links [13,58,59]. Although it is unlikely that the concepts of small-world and scale-free networks directly apply to neural net- the network size. However, physical implementation of works in the brain, the demonstration of the effectiveness of a few but this strategy is often not feasible or economic,