In our first project we investigate the dynamics of 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, 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 a small density of 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 the short-cuts allow the activity to spread so fast through the network that all neurons simultaneously end up 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 N. The location of the transition and its dependence on the system size can be obtained quantitatively from a mean-field theory for small-world networks.
More details including the investigation of exceedingly long chaotic transients 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 (2004).
In this first project the models 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.