Morphological characterization of in vitro neuronal networks Orit Shefi, 1,2 Ido Golding, 1, * Ronen Segev, 1 Eshel Ben-Jacob, 1 and Amir Ayali 2,² 1 School of Physics and Astronomy, Raymond & Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel 2 Department of Zoology, Faculty of Life Sciences, Tel-Aviv University, Tel-Aviv69978, Israel Received 25 February 2002; revised manuscript received 20 May 2002; published 14 August 2002 We use in vitro neuronal networks as a model system for studying self-organization processes in the nervous system. We follow the neuronal growth process, from isolated neurons to fully connected two-dimensional networks. The mature networks are mapped into connected graphs and their morphological characteristics are measured. The distributions of segment lengths, node connectivity, and path length between nodes, and the clustering coefficient of the networks are used to characterize network morphology and to demonstrate that our networks fall into the category of small-world networks. DOI: 10.1103/PhysRevE.66.021905 PACS numbers: 87.10.+e, 87.17.-d I. INTRODUCTION One of the most profound questions in science is how a collection of elements self-organize to form new and ex- tremely complex systems 1 and references therein, 2,3. This question becomes far more challenging when talking about biological systems, where the building blocks them- selves are living entities 4. In the case of the nervous sys- tem this issue translates to the open question of how a func- tioning neuronal network a small circuit as well as a complex brain emerges from a collection of single entities, the individual neurons 5–9. As in networks in general, there is a strong relation be- tween the neuronal network structure, or ‘‘wiring diagram,’’ and its function, i.e. the form-function relation 10. This enables determination of the dynamics and activity of a net- work by analyzing its morphology and topology of connec- tivity. An attempt in this direction has been recently made by Watts and Strogatz in introducing their ‘‘small-world net- works’’ concept 10–13. A small world network is one that interpolates between the two extreme cases of a regular lat- tice, on the one hand, and a random graph, on the other. It is characterized by a local neighborhood, which is highly clus- tered as in regular lattices, and by a short path length be- tween vertices as in random networks. Watts and Strogatz state that small-world characteristics are a prevalent feature of real life biological networks. Yet, so far, only a few such systems have been examined experi- mentally. These include metabolic networks in various or- ganisms 14, as well as the large-scale organization of meta- bolic networks 15, and the nervous system of the worm Caenorhabditis elegans 11. We are presently studying two-dimensional in vitro neu- ronal networks. While these cultured networks lack some features of in vivo neuronal networks, they retain many oth- ers 16 and references therein. They develop organotopic synaptic connections and exhibit a rich variety of electrical properties similar to those observed in vivo. The two- dimensional system enables easy access for noninvasive op- tical observations, allowing us to follow the dynamics of neuronal growth and network organization. In addition, our use of invertebrate locust cells is advantageous due to the large size of the neurons and the ease with which they can be cultured under various conditions 2,8,10,17–21. All the above, together with recent progress in multielectrode array technology, optical imaging, and fluorescence microscopy, make invertebrate cultured neuronal networks a favorable model system for studies of neuronal networks and the ner- vous system. In our culture preparations, fully differentiated adult neu- rons, which lose their dendrites and axon during dissociation, regenerate neurites that interconnect to form an elaborate network. During the growth process, growth cones connect to nonself, as well as self previously extended neurites, with no clear evidence for self-avoidance see Fig. 1. It appears that the cultured neurons cannot be considered as simple el- ements; even the single isolated cell shows spontaneous elec- trical activity and forms a complex morphological structure. Neuronal systems can be modeled as networks or graphs of coupled systems, where the vertices represent the ele- ments of the system, and the edges represent the interactions between them. Once in the framework of a wired graph, one *Present address: Department of Molecular Biology, Princeton University, Princeton NJ 08544-1014. ² Corresponding author. Fax: 972-3-6409403. Email address: ayali@post.tau.ac.il FIG. 1. A single cultured neuron, two days after plating. The neurites outgrow from the round soma, branch and connect to other neurites extending from the same cell. Scale bar = 50 . PHYSICAL REVIEW E 66, 021905 2002 1063-651X/2002/662/0219055/$20.00 ©2002 The American Physical Society 66 021905-1