### Abstract

Networks of neurons in some brain areas are flexible enough to encode new memories quickly. Using a standard firing rate model of recurrent networks, we develop a theory of flexible memory networks. Our main results characterize networks having the maximal number of flexible memory patterns, given a constraint graph on the network's connectivity matrix. Modulo a mild topological condition, we find a close connection between maximally flexible networks and rank 1 matrices. The topological condition is H _{1}(X;ℤ)=0, where X is the clique complex associated to the network's constraint graph; this condition is generically satisfied for large random networks that are not overly sparse. In order to prove our main results, we develop some matrix-theoretic tools and present them in a self-contained section independent of the neuroscience context.

Original language | English (US) |
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Pages (from-to) | 590-614 |

Number of pages | 25 |

Journal | Bulletin of Mathematical Biology |

Volume | 74 |

Issue number | 3 |

DOIs | |

State | Published - Mar 2012 |

### All Science Journal Classification (ASJC) codes

- Neuroscience(all)
- Immunology
- Mathematics(all)
- Biochemistry, Genetics and Molecular Biology(all)
- Environmental Science(all)
- Pharmacology
- Agricultural and Biological Sciences(all)
- Computational Theory and Mathematics

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## Cite this

*Bulletin of Mathematical Biology*,

*74*(3), 590-614. https://doi.org/10.1007/s11538-011-9678-9