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Heteroclinic Networks in Homogeneous and Heterogeneous Identical Cell Systems

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Abstract

We prove results that enable realization of heteroclinic networks in coupled homogeneous and heterogeneous systems of identical cells. We also consider various models for network dynamics, which allow variation in the number of inputs to identical cells.

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Notes

  1. Interest has continued since that time notwithstanding the authors remark that ‘Biologically, the behaviour illustrated in Figs. 4 and 5 is nonsense’.

  2. We allow either asymmetric or symmetric inputs.

  3. Strictly speaking, unique if \(j_i:\mathbf {k_i}\rightarrow \varvec{\ell }\) is injective, \(i \in \varvec{\ell }\).

  4. We do not discuss the issue of spike amplification that can occur in pyramidal neurons in the hypo-campus Fricker and Miles (2000)—this can be modelled using a non-autonomous version of additive input structure that depends on relative timings.

  5. We review the methods used in Aguiar et al. (2011) when we address the general case.

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Acknowledgments

We thank Peter Ashwin for helpful comments on preliminary drafts of the paper and the referees for their remarks and questions which have led to clarifications and improvements in the exposition.

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  1. M. J. Field

    Present address: Department of Mathematics, Imperial College, London, SW7 2AZ, UK

Authors and Affiliations

  1. Department of Mathematics, Rice University, Houston, TX, 77005-1892, USA

    M. J. Field

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Correspondence to M. J. Field.

Additional information

Communicated by Paul Newton.

Research supported in part by NSF Grant DMS-1265253 and Marie Curie IIF Fellowship (Project 627590).

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Field, M.J. Heteroclinic Networks in Homogeneous and Heterogeneous Identical Cell Systems. J Nonlinear Sci 25, 779–813 (2015). https://doi.org/10.1007/s00332-015-9241-1

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