Heteroclinic network

In mathematics, a heteroclinic network is an invariant set in the phase space of a dynamical system. It can be thought of loosely as the union of more than one heteroclinic cycle. Heteroclinic networks arise naturally in a number of different types of applications, including fluid dynamics and populations dynamics.

The dynamics of trajectories near to heteroclinic networks is intermittent: trajectories spend a long time performing one type of behaviour (often, close to equilibrium), before switching rapidly to another type of behaviour. This type of intermittent switching behaviour has led to several different groups of researchers using them as ways to model and understand various type of neural dynamics.

References

• Kirk, V; Silber, M (1994-11-01). "A competition between heteroclinic cycles". Nonlinearity. IOP Publishing. 7 (6): 1605–1621. Bibcode:1994Nonli...7.1605K. doi:10.1088/0951-7715/7/6/005. ISSN 0951-7715.
• Rabinovich, M.; Volkovskii, A.; Lecanda, P.; Huerta, R.; Abarbanel, H. D. I.; Laurent, G. (2001-07-20). "Dynamical Encoding by Networks of Competing Neuron Groups: Winnerless Competition". Physical Review Letters. American Physical Society (APS). 87 (6): 068102. Bibcode:2001PhRvL..87f8102R. doi:10.1103/physrevlett.87.068102. ISSN 0031-9007. PMID 11497865.
• Rabinovich, Mikhail I.; Afraimovich, Valentin S.; Varona, Pablo (2010). "Heteroclinic binding". Dynamical Systems. Informa UK Limited. 25 (3): 433–442. doi:10.1080/14689367.2010.515396. ISSN 1468-9367.
• Ashwin, Peter; Lavric, Aureliu (2010). "A low-dimensional model of binocular rivalry using winnerless competition". Physica D: Nonlinear Phenomena. Elsevier BV. 239 (9): 529–536. Bibcode:2010PhyD..239..529A. doi:10.1016/j.physd.2009.06.018. hdl:10871/14537. ISSN 0167-2789.