As demonstrated by C. elegans example, a synaptic connectivity matrix (or a statistical description thereof) is not by itself sufficient to understand brain function.

In developing a theoretical description of neuronal circuits, it is important to choose an appropriate level of abstraction. Such choice must rely on our (rather rudimentary) understanding of the basics of circuit function. For example, how much of circuit functionality is described by the connectivity matrix? What is the significance of neuronal layout and shape? How important is the location of synapses on the neuron? Why are some connections implemented with chemical synapses, others - with gap junctions?

To help answer these questions we study neuronal circuits from the engineering point of view, which includes answering why questions. As biological systems have evolved over hundreds of millions of years, design tradeoffs are embedded in the blueprint of such systems. It is thus natural that a theoretical approach to biology must incorporate aspects of constrained optimization. Particular success has been enjoyed by the wiring economy principle, which is rooted in Cajal’s laws of wiring economy. By using this principle, we were able to explain many aspects of brain design, such as neuronal placement, dimensions of axons, dendrites and the existence of spines, as well as the segregation of the neocortex into the gray and white matter.

What determines the placement of neurons in the body? We pursued the hypothesis that neurons minimize the cost of wiring between them. By using a method borrowed from computer engineering we found the optimal layout for the given wiring diagram and compared it with actual, Figure 7. Although most neurons follow wiring optimization predictions, some are displaced significantly. Interestingly, we find that outlier neurons have additional developmental functions (they pioneer axonal growth along the ventral cord), which may explain their misplacement. This finding illustrates how comparing theoretical predictions of the optimization approach with experimental data leads to the detection of discrepancies, which point to functional constraints that were not included in the original formulation, Figure 1.

Figure 7. Theoretically predicted placement of C. elegans neurons vs. their actual locations (shown are neuron coordinates along the body axis in fractions of body length). Neurons shown by the same color belong to the same ganglion and are predicted to cluster together (B. Chen, D. Hall & D. B. Chklovskii, in preparation).

What is the role of axonal and dendritic arbors in information processing? Since C. elegans neurons do not have elaborate arbors, we addressed this question in the context of cortical neurons (Chklovskii 2004). We find that wiring up of a highly inter-connected network such as the cortical column is a difficult problem. Solution of such problem in the allotted volume requires all the major features of neuronal morphology, such as axons and spiny dendrites. Therefore, one needs not look further to find a reason for their existence. Our past research addressed the relationship between the sizes of dendritic and axonal arbors and the minimization of wiring volume according to the wiring economy principle. We plan to extend this work to understand branching and specific placement of synapses. Comparing optimization predictions with anatomy will help us understand the functional significance of neuronal shape.

A related pursuit in the realm of wiring optimization is the study of conduction delay, a primary factor among the costs of connectivity in extensive cortical neuron networks. The segregation of the brain into white and gray matter is one characteristic of brain design that may have resulted from an effort to minimize conduction delays. Our lab's own Quan Wen is shedding light on the brain's structure-function relationships with his research on this topic, which also helps to explain the shape and topology of axonal and dendritic arbors.

In addition to these applications of the wiring economy principle, we found that the specific ocular dominance patterns and orientation preference maps in mammalian visual cortex are accounted for by the principle.

Another difficulty with using the connectivity matrix, Figure 1, is that such representation of the neuronal circuit is too detailed for comprehensive understanding. In an attempt to simplify the description of the neuronal circuit, we search for multi-neuron modules (smaller than invertebrate ganglia or vertebrate nuclei and cortical columns). The advantage of modular description is illustrated by electrical engineering, where a description of a circuit in terms of operational amplifiers, logical gates and memory registers is often preferred to that showing each transistor, resistor and diode. However, unlike electrical engineers who designed these modules themselves, neurobiologists need to discover such modules.

We are trying to find modules by statistical analysis of the connectivity matrix (Milo et al. 2002). Specifically, we search for few-neuron connectivity patterns that are statistically over-represented in C. elegans relative to the expectations from the random matrix ensemble (Reigl et al. 2004). We performed similar analysis on the neocortical connectivity data, obtained by sampling (Song et al. 2005). Interestingly, we have found that similar patterns, which we call motifs, are over-represented both in C. elegans and rat neocortex. In particular, reciprocally connected neuronal pairs are four times more common than expected by chance. Several triplet connectivity patterns are over-represented both in C. elegans and cortex, Figure 8. A notable exception is the feedback loop (#11), which is over-represented only in neocortex. In the future, we plan to determine significance of these motifs for circuit function.

Section 2           Research Home