Alex Roxin, Hermann Riecke, Sara Solla

Phys. Rev. Lett. 92 (2004) 198101 Chaos 17 (2007) 026110 This research was funded by the National Science Foundation

In this project we have investigated the dynamics of networks of excitable neurons. In this project our emphasis was not on a specific biological system; instead we have investigated 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, while high densities lead to transient, eventually failing activity.

Without any short-cuts the activity stops when the two excitation waves that are triggered by the excitation of a single neuron annihilate.

With a small number of shprt-cuts, marked by grey arrows that become red when they are activated, the activity can become persistent, since short-cuts can re-activate previously excited neurons. This can only happen if these neurons have sufficiently recovered, i.e. if sufficient time has elapsed since the last firing of those neurons.

If the density of short-cuts is high, the activity spreads so rapidly across the whole system that all neurons are in the recovery period at the same time. Thus, no reactivation is possible. The latter two regimes are also shown in raster plots, i.e. in space-time plots of the firing times of the neurons. In the low-density regime with persistent activity 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 the regime of persistent activity these networks have an extraordinarily large number of attractors. Even if only those attracts are included that are reached with initial conditions in which only a single neuron is excited, the number of attractors is of the order of the number of neurons. Many of these attractors have the same period, however, and many of the attractors can be reached from only one of these initial condition ('1-attractors'). The figure below shows how the number of attractors grows with the size of the network.

While for fast wave propagation the population activity dies out after essentially a single population spike if the density of connections is high, the activity can persist for a very long time if the wave propagation is slow. These transients exhibit chaotic dynamics and their duration increases very strongly when the wave speed is decreased by increasing the delay between the excitation of successive neurons.

Dependence of the duration of the chaotic transient on the wave speed for the same initial condition.

Surprising - and not understood - is the fact that the fraction of network realizations for whom the activity fails by a fixed time depends non-monotonically on the wave speed and shows resonance-like structure.
a) b)
Fraction of networks with activity that fails within time T. Panel b) shows an enlarged detail of a).

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.