Morphological characterization of in vitro neuronal networks.
ABSTRACT 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.
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ABSTRACT: Networks of coupled dynamical systems have been used to model biological oscillators, Josephson junction arrays, excitable media, neural networks, spatial games, genetic control networks and many other self-organizing systems. Ordinarily, the connection topology is assumed to be either completely regular or completely random. But many biological, technological and social networks lie somewhere between these two extremes. Here we explore simple models of networks that can be tuned through this middle ground: regular networks 'rewired' to introduce increasing amounts of disorder. We find that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. We call them 'small-world' networks, by analogy with the small-world phenomenon (popularly known as six degrees of separation. The neural network of the worm Caenorhabditis elegans, the power grid of the western United States, and the collaboration graph of film actors are shown to be small-world networks. Models of dynamical systems with small-world coupling display enhanced signal-propagation speed, computational power, and synchronizability. In particular, infectious diseases spread more easily in small-world networks than in regular lattices.Nature 07/1998; 393(6684):440-2. · 38.60 Impact Factor
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ABSTRACT: In a cell or microorganism, the processes that generate mass, energy, information transfer and cell-fate specification are seamlessly integrated through a complex network of cellular constituents and reactions. However, despite the key role of these networks in sustaining cellular functions, their large-scale structure is essentially unknown. Here we present a systematic comparative mathematical analysis of the metabolic networks of 43 organisms representing all three domains of life. We show that, despite significant variation in their individual constituents and pathways, these metabolic networks have the same topological scaling properties and show striking similarities to the inherent organization of complex non-biological systems. This may indicate that metabolic organization is not only identical for all living organisms, but also complies with the design principles of robust and error-tolerant scale-free networks, and may represent a common blueprint for the large-scale organization of interactions among all cellular constituents.Nature 11/2000; 407(6804):651-4. · 38.60 Impact Factor
Article: Mathematical guidance for axons.[show abstract] [hide abstract]
ABSTRACT: Axon guidance by gradients plays an important role in wiring up the developing nervous system. Growth cones seem to sense a concentration difference across their spatial extent, and convert this into a signal to move up or down a gradient. In this article, a simple mathematical framework is developed to understand when and where gradient detection can occur as a function of gradient shape. This framework is applied to two examples:the guidance of axons by target-derived diffusible factors in vivo and in collagen gels, and guidance by substrate-bound gradients of optimal shape, as might be relevant in the retinotectal system.Two distinct spatial limits on guidance emerge: I mm for a target-derived diffusible gradient, and I cm for a substrate-bound gradient.Trends in Neurosciences 07/1998; 21(6):226-31. · 13.58 Impact Factor
Morphological characterization of in vitro neuronal networks
Orit Shefi,1,2Ido Golding,1,* Ronen Segev,1Eshel Ben-Jacob,1and Amir Ayali2,†
1School of Physics and Astronomy, Raymond & Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel
2Department of Zoology, Faculty of Life Sciences, Tel-Aviv University, Tel-Aviv 69978, 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.66.021905PACS number?s?: 87.10.?e, 87.17.?d
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-
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-
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.
†Corresponding author. Fax: 972-3-6409403. Email address:
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
can apply the mathematical tools of graph theory to analyze
the system under study and to look for universal, generic
features that are common to different kinds of networks
within as well as outside the nervous system.
As a first step we needed to define vertices and edges in
our system. According to the ‘‘neuron paradigm’’ the build-
ing blocks of the nervous system are the neurons ?vertices?
and synapses ?edges?. However, based on the branching and
growth process of the cultured networks, we chose the neu-
rons, the synapses, and synapselike connections between the
neurites of the same neuron, to be the vertices. Our main
working assumption was that these structures are essential
for information processing in the network.
In this work we describe the results for three different
neuronal networks grown in culture under controlled condi-
tions. These networks are characterized by a high clustering
coefficient compared to random graphs, and a path length
that is closer to random networks than to regular ones. We
thus classify the studied neuronal networks as small-world
II. GROWTH OF THE NEURONAL NETWORK
Cell cultures. Neurons were dissociated from the frontal
ganglion of adult locusts and maintained under controlled
conditions. Culturing method followed Shefi et al. ?2?. Plated
neurons varied in size from 10 ?m to 50 ?m. The number
of ganglia per dish determined the density of the culture, and
thus the average distance between cells.
A charge coupled device camera mounted onto a phase
contrast microscope was used to acquire images of the cul-
tured neurons and networks into a PC for image processing
Growth process. Time-lapse observations on the growth
process of cultured neurons revealed that the most intense
stage of development was between day 1 and day 5. After
this rapid growth stage there was a pronounced decrease in
growth rate ?2,3?. By day 6 in culture most of the neurons
had developed interconnections and were already a part of an
elaborate network ?Fig. 2?. Hence, we analyzed the network
at that point.
During the growth process, growth cones connected not
just to neighbor cells but also to neurites previously extended
from their own cell body, with no evidence for self-
avoidance. They thereby formed close loops. The junctions
or interconnection points acted as anchors that seemed to be
more firmly attached to the substrate than the neurites them-
selves. Tension was generated along the neurites as they
stretched between these anchors to form straight segments,
giving the close loops polygonal shapes ?Figs. 1 and 2?.
Connectivity statistics was found to change significantly
with the age of the single neuronal cell in culture and the
developmental stage of the network as a whole ?2?. After the
initial stage of intense neurite formation, the neuronal cell
bodies started to aggregate into packed clusters. The cluster-
ing of cells was accompanied by absorption of branches and
even whole neurites, together with rearrangements of neu-
rites and what appeared to be fusion of parallel ones. The
somata were observed to migrate along newly formed
bundles toward one another. Thus, relatively homogenous
cultures, in which single neurons were scattered, evolved
into cultures organized into a few centers comprised of clus-
ters of neurons connected by thick nervelike bundles ?2,22?.
FIG. 2. A part of a mature 6-day-old network. At this stage all
the neurons in the network are connected to each other. The neurites
are straight segments that show high tension along the processes
between junctions. The junction points appear to be more firmly
attached to the substrate than are the neurites that connect them.
Scale bar ? 50?.
FIG. 3. Illustration of the points considered as vertices in the
networks: numbers 1 and 8 are somata of neurons and numbers 2–7
are connection points between neurites. In the adjacency matrix
pairs of vertices that are connected as 1-2, 2-3, 3-4, 4-5, 5-6, 5-7,
etc. obtain the value 1 while the nonconnected pairs obtain the
SHEFI, GOLDING, SEGEV, BEN-JACOB, AND AYALIPHYSICAL REVIEW E 66, 021905 ?2002?
III. MORPHOLOGY ANALYSIS
Abstraction process. We took a static snapshot of the
evolving networks structure at a particular time point, day 6
in culture, on which the networks demonstrate maximum in-
terconnections between neurites. We mapped the neuronal
network into a simple graph using the following assumptions
?see Fig. 3?.
?1? All vertices are identical.
?2? All edges are identical. That is, we ignored edge
length, the possibility that different edges have different syn-
aptic efficacies and/or edges directionality.
?3? We ignored edge multiplicity, i.e., we considered only
whether vertices were adjacent or not.
Physical properties of the networks. Our main findings
obtained for three 6-day-old networks cultured and grown
under the same conditions are summarized in Table I. Before
taking the graph theory approach and describing the net-
works as simple labeled graphs ?which are devoid of any
physical properties?, we examine in further detail the statis-
tics obtained for one network, the largest of the three ?net-
work number 3 in Table I, with 240 nodes and 290 edges?.
Distribution of the measured lengths between all pairs of
nodes on a semilogarithmic scale is shown in Fig. 4. The
bimodal distribution, with the majority of the samples at the
lower end of the distribution ?short segments, r?10 ?m),
and the remaining samples distributed in a symmetrical man-
ner in the log space between ?101
(10–300 ?m), is typical for networks with spatially clus-
tered structures ?22?.
Mapping of the network into a graph. The neuronal net-
work is described as labeled graph G. Such a graph can be
described solely by its adjacency matrix A(G) ?23?. This
symmetric matrix is defined as follows: Let the nodes of G
be labeled v1,v2, . . . ,vn. The Adjacency Matrix A?ai,jof
G is a binary matrix of order n, with
1 if viand vjare joined by an edge
?i.e., adjacent, or neighbors?
For each of the networks under study, A(G) is constructed
by manually labeling all nodes and marking all connected
pairs ?see Fig. 3?. The properties of the adjacency matrix
facilitate calculation of all of the required characteristics of
the graph ?see below?, through simple algebraic manipula-
tions. In particular, we stress the following important prop-
erty ?23?: If G is a labeled graph with adjacency matrix A,
then the (i,j) element of Alis the number of walks of length
l from vito vj.
Node Connectivity. The degree k of a vertex is the number
of other vertices to which it is directly connected
??adjacent?. For each vertex vi,k is obtained by adding up
the elements in row ?or column? i in A(G). The distribution
of k values for network 3 is depicted in Fig. 5. The average
connectivity is k¯?2.38. It can be seen that the network has a
FIG. 4. Distribution of the physical lengths of connecting seg-
ments in network number 3 ?240 nodes, 290 segments?. Horizontal
axis is log10of segment length ?measured in micrometers?. Vertical
axis is the normalized frequency of occurrence. Note the bimodality
of the distribution in log space, and the apparent symmetry of the
FIG. 5. Distribution of the node connectivity in network number
3 ?240 nodes, 290 connections?. Horizontal axis is k, the number of
nodes to which each node is connected. Vertical axis is the normal-
ized frequency of occurrence. k¯?2.38.
TABLE I. Descriptive parameters measured for three 6-day-old
neuronal networks, cultured and grown under controlled conditions:
n, number of nodes; k¯, average node connectivity, l¯, characteristic
path length. Values for a regular ring graph (lreg?n/2k) and a
random graph are also given for comparison. crandand lrandare
calculated for ten numerically generated random graphs with the
same parameters as the corresponding studied network.
1104 2.3322.35/11.03/4.88?.28 0.092/.017?.010
2 1402.62 26.70/9.66/4.82?.120.129/.016?.010
MORPHOLOGICAL CHARACTERIZATION OF IN VITRO . . . PHYSICAL REVIEW E 66, 021905 ?2002?
pronounced scale of connectivity. That is, it is far from being
‘‘scale free,’’ a feature claimed to be common among ‘‘real-
world’’ networks ?24?. However, as Amaral et al. have
shown ?25?, the node connectivity of various networks ?real
as well as manmade? can exhibit either scale-free, broad
scale or single scale statistics.
Path length. The path length l between viand vjis de-
fined as the number of edges included in the shortest path
between viand vj. The characteristic path length l¯of the
graph G is l averaged over all pairs of vertices. The distribu-
tion of path length values in network 3 is shown in Fig. 6.
The characteristic path length of the corresponding graph is
Clustering coefficient. Another important parameter in the
context of small-world networks is the clustering coefficient
of the graph. The clustering coefficient of vertex v, C(v), is
defined as the number of edges among the kvneighbors of v
??adjacent vertices?, divided by the maximal number of
such edges, kv(kv?1)/2. Thus, C(v) ?which is in the range
?0–1?? measures the ‘‘cliquishness’’ of the neighborhood of
v, i.e., what fraction of the vertices adjacent to v are also
adjacent to each other. By extension, the clustering coeffi-
cient of the graph G, C¯, is the average of C(v) over all
vertices. For network 3 we obtain C¯?0.113.
Small-world test. Using the results presented in Table I,
we can now attempt to test whether our in vitro neuronal
networks fall into the category of small-world networks. For
this purpose, the table contains a comparison to two bench-
mark cases: a random graph and a regular graph ?11?, with
the same number of nodes n and average connectivity k¯as
the network under study. The formal definition of a small-
world network requires that such a network satisfies ?1?
C¯?Crandom (Crandom?k/n), that is, a small-world net-
work is much more highly clustered than the correspond-
the characteristic free path of a small-world network is
close to that of a random graph, and much smaller than
that of a regular graph. Specifically, l¯ should scale as
ln(n)/ln(k), rather than as n/2k.1
A comparison between the average path length and clus-
tering coefficient of our three networks and various real net-
works ?following Albert and Barabsi ?26??, with the theoret-
ical values for random and regular graphs, is presented in
Fig. 7. It can be seen that our three networks fall within the
1One should note that a random graph with the same n and k¯as
our networks will not, in general, be fully interconnected. The char-
acteristic path length for such a graph corresponds only to con-
nected subgraphs. The fact that our graphs are completely con-
nected at these parameter values is, of course, a feature that
distinguishes them from random graphs.
FIG. 6. Distribution of the path length values in network number
3 ?240 nodes, 290 connections?. Horizontal axis is the path length,
i.e., the number of edges in the shortest path between two nodes.
Vertical axis is the normalized frequency of occurrence. l¯?17.58.
FIG. 7. ?a? Average path length of the neuronal networks studied
?filled dots?, compared to the prediction of random graph theory
?lrand?ln(N)/ln(?k?, solid line? and regular graphs ?lreg?n/2k,
open triangles?. Data for our networks are also compared to other
real networks ?open dots, data taken from Albert and Barabasi ?26?,
Table I?. ?b? The network’s clustering coefficient compared to the
prediction of random graph theory (crand??k?/N), and other real
networks ?26? ?symbols as in ?a??.
SHEFI, GOLDING, SEGEV, BEN-JACOB, AND AYALIPHYSICAL REVIEW E 66, 021905 ?2002?
‘‘cloud’’ of real-world networks. As presented in Table I, the
clustering coefficient is indeed much higher ?5–13 times?
than that of the corresponding random graphs. The data do
not allow us to verify that l¯scales as ln(n)/ln(k) but the
characteristic path length is closer to lrandomthan to lregular,
as in small-world networks.
During development of the nervous system, two opposing
forces impose the morphology of the evolving neuronal net-
works. On the one hand, single neurons grow axons and
highly branched dendritic trees in order to achieve maximum
interconnected networks. This enables efficient information
flow, and adds to the strength of the networks as computa-
tional units. On the other hand, developing extended and
vastly branched neurites has a high energetic cost. Hence, the
final structure of the neuronal network is a consequence of
the interplay between these factors. One category of net-
works that could be the result of such competition is small-
world networks, combining fast information transmission
with maximal economy in wiring length ?energetic cost?.
We studied in vitro two-dimensional neuronal networks
generated by culturing neurons dissociated from locust gan-
glia. The in vitro networks were mapped onto graphs where
the vertices represent the elements of the system and the
edges represent the interactions between them. We examined
our networks at the stage where they were practically fully
connected. In order to determine whether the networks fall
within the small-world regime, we calculated the clustering
coefficient and path length of each network and compared
these parameters to random and regular graphs with the same
n and k¯.
For the three networks tested, the clustering coefficients
were indeed much higher than those of the corresponding
random graphs, and the characteristic path lengths were
closer to lrandomthan to lregular. According to this test, the
networks can be classified as small-world networks.
Distribution of the lengths of segments connecting the
nodes in our networks is typical to networks with a spatially
clustered structure. This becomes very apparent as the net-
works mature. The culture goes through a dynamical process,
starting with single entities, continues to a fully connected
network, and finally develops to cultures organized into a
few centers comprised of groups of neurons connected by
thick nervelike bundles. The latter can be characterized by
efficient information transmission together with tight and
thrifty structure, the features of a small-world network.
The growing process that was observed in our two-
dimensional cultures serves to demonstrate the self-
organization process that leads to the characteristic structure
of the nervous system in vivo: concentrations of neuronal cell
bodies, namely, ganglia, interconnected by nerve tracts.
We are grateful to Paul Meakin, Daniel Rothman, and
Peter Dodds for their useful advice.
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