Criteria for robustness of heteroclinic cycles in neural microcircuits
1 Mathematics Research Institute, University of Exeter, Exeter, EX4 4QF, UK
2 Faculty of Electrical and Electronics Engineering, Electronics and Communication Department, Istanbul Technical University, TR-34469, Maslak-Istanbul, Turkey
3 Centre for Computational Neuroscience and Robotics, Informatics, University of Sussex, Falmer, Brighton, BN1 9QJ, UK
The Journal of Mathematical Neuroscience 2011, 1:13 doi:10.1186/2190-8567-1-13Published: 28 November 2011
We introduce a test for robustness of heteroclinic cycles that appear in neural microcircuits modeled as coupled dynamical cells. Robust heteroclinic cycles (RHCs) can appear as robust attractors in Lotka-Volterra-type winnerless competition (WLC) models as well as in more general coupled and/or symmetric systems. It has been previously suggested that RHCs may be relevant to a range of neural activities, from encoding and binding to spatio-temporal sequence generation.
The robustness or otherwise of such cycles depends both on the coupling structure and the internal structure of the neurons. We verify that robust heteroclinic cycles can appear in systems of three identical cells, but only if we require perturbations to preserve some invariant subspaces for the individual cells. On the other hand, heteroclinic attractors can appear robustly in systems of four or more identical cells for some symmetric coupling patterns, without restriction on the internal dynamics of the cells.