Our efforts in computational neuroscience have started only
recently with a project on networks of excitable neurons. In
this project our emphasis is not on a specific biological
system; instead we are investigating the dynamics of a whole
class of neuronal networks. Specifically, in this project we
have been studying how a heterogenous topology characterized by
a combination  of local and non-local connections somewhat
similar to a small-world  topology affects the ability of a
network to sustain persistent activity.  We have considered
neurons that are excitable rather than oscillatory and that
support propagating waves of excitation when coupled in a
network.  The central finding is that at small density non-local
connections can lead to persistent activity, which is not
possible  without short-cuts. A raster plot, i.e. a space-time
plot of the firing  times of the neurons is shown below. In this
regime the firing patterns and the temporal evolution of the
spatially integrated firing rate typically exhibit significant
oscillations. 

At higher densities the short-cuts quickly induce a large burst
in the activity of the population after which the activitiy dies
out completely. In this regime the activity cannot persist
because all neurons are simultaneously in the recovery period
during which they cannot be excited by the weak input provided
by the other neurons. 

The transition from persistent activity to failure is 
illustrated in the figure below, which gives the fraction of 
network configurations that fail as a function of the density of
short-cuts for different system sizes. 


Detailed results  can be found in a paper in Phys. Rev Lett. and
in a talk given at the  ICAM Workshop Frontiers in Biological
Physics III: Neurobiology Workshop.



In this first project the model for the neurons as well as for
their connections (synapses)  have been chosen to be minimal. To
some extent this is necessary to limit the computational effort
since a large number of different configurations have to be
simulated for  a sufficiently long time to determine the failure
rate. To some extent, however, it also reflects the goal of this
project, which is to condense the phenomenon to its minimal
ingredients and to identify the relevant mechanisms. 

Our long-term goal is to pursue a two-pronged approach to
computational neuroscience. Thus, while on the one hand we
intend to investigate general properties of relatively simple
neural network models, we are aiming on the other hand to
develop biophysically realistic models for parts of the
olfactory system. The latter requires the close collaboration
with an experimental group and is still in the planning stage.