# Who's with whom?

The mathematical analysis of regulatory networks is becoming
increasingly important in different fields of biology - since such
networks exist everywhere in nature. Species of animals and plants in
an ecosystem, genes and proteins in a cell or neurons in the brain
constitute networks of interacting units. Marc Timme, researcher at the
Bernstein Center for Computational Neuroscience and at the Max Planck
Institute for Dynamics and Self-Organization in Göttingen, has now
developed a mathematical method to infer the circuitry structure of a
regulatory network from its dynamical response properties. The
implementation of this theoretical method can make it possible to
determine the exact connections between the units of a network - for
example, the interaction between the molecules of a cell or the
connections in a neuronal network. The study will appear in June in the
scientific journal "Physical Review Letters".

As a precondition, Timme's method requires that the network is in a
stable, balanced equilibrium, similar to that of a mobile toy in a
balanced state. If a small weight is carefully attached to a figure of
the mobile, can one infer information about the structure of the mobile
from the altered up- and downwards positions of the other figures? And
if so, how many times would one have to place weights onto the mobile
in different variations before it is possible to reliably determine the
connections between all its figures? Timme pursued these questions -
not for mobiles but for balanced regulatory networks in general.

In nature, we find many regulatory network systems, for example, for
generating the recurring pattern of activity when breathing or when the
heart beats. If in such a network the dynamics of one component is
altered, the other components respond to it. If a neuronal network is
fed somewhere with an external signal, a new, slightly shifted balance
is restored - that is, some neurons now transmit signals later, some
earlier than previously. "The dynamical response of the network not
only depends on the type of signal applied but also characteristically
on the structure of the network", Timme explains. This fact was
successfully demonstrated in a previous work published in the journal
"Europhysics Letters" in November 2006. In his latest work, Timme
showed how the structure of a network can be determined by studying its
response and how the connectivity can be reconstructed on a
mathematical level. With the help of his method, it is not only
possible to determine which elements of a network are connected with
one another, but also how strong the connections are.

Precisely determining every connection within a network requires the
recording of a vast amount of data. "To reconstruct the entire network
we need to collect sufficient information", Timme explains, "in general
this means that the number of different experiments needs to equal the
number of all components of a network." Only few researchers have dared
to draw conclusions about the structure of a network from its dynamic
responses. Up to now such mathematical analysis methods have mainly
allowed for statistical assertions, e.g. about the percentage of
connected units or the expected strengths of the connections. Timme's
approach is different in that his mathematical model determines every
single connection of the network. He has tested the validity of this
theory with the help of computer simulations of different networks.

In reality, regulatory networks consist of thousands or millions of
components, making a recording of the responses of a network to a large
number of signals practically impossible. A further significant step in
Timme's method nonetheless allows to determine the structure of even
very large networks with just a moderate number of experiments. Nature
normally arranges its network units very efficiently such that a
desired function can be accomplished with a minimal number of
connections. In his method, Timme makes use of nature's principle of
sparseness as a feature beneficial for finding the structure of larger
networks. He shows how to reconstruct the structure of a sparsely
connected network with a number of experiments far fewer than the
number of components represented in a network. The novel method
provides researchers with the basic principles for a tool to
systematically investigate the interdependences between structure and
function of regulatory networks.

Publications:

Marc Timme. Revealing Network Connectivity From Response Dynamics. Physical Review Letter, 98:224101 (2007)

Marc Timme. Does dynamics reflect topology in directed networks? Europhysics Letters 76 (3), 367-373 (2006).