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Stimulus-Response Reliability of Biological Networks

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Nonautonomous Dynamical Systems in the Life Sciences

Part of the book series: Lecture Notes in Mathematics ((LNMBIOS,volume 2102))

Abstract

If a network of cells is repeatedly driven by the same sustained, complex signal, will it give the same response each time? A system whose response is reproducible across repeated trials is said to be reliable. Reliability is of interest in, e.g., computational neuroscience because the degree to which a neuronal network is reliable constrains its ability to encode information via precise temporal patterns of spikes. This chapter reviews a body of work aimed at discovering network conditions and dynamical mechanisms that can affect the reliability of a network. A number of results are surveyed here, including a general condition for reliability and studies of specific mechanisms for reliable and unreliable behavior in concrete models. This work relies on qualitative arguments using random dynamical systems theory, in combination with systematic numerical simulations.

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Notes

  1. 1.

    Theta neurons can also model neurons operating in an excitable regime. The reliability of excitable theta neuron networks is studied in [22].

  2. 2.

    See, e.g., [5], but note that phase truncations can sometimes miss important dynamical effects [30], and their use in biological modeling should be carefully justified.

  3. 3.

    The fiber exponent can be defined exactly as in (4.6), but with the tangent vector v chosen to lie in the subspace tangent to each fiber; note these subspaces are invariant due to the skew product structure.

  4. 4.

    It is straightforward to show that there is always a unique modular decomposition connected by an acyclic graph that is “maximal” in the sense that it cannot be refined any further without introducing cycles into the quotient graph.

  5. 5.

    These results have been extended to certain nonlinear parabolic PDEs [32] and periodically-kicked homoclinic loops [37].

  6. 6.

    A rough estimate shows that when A = 2, each kick should be sufficient to drive the oscillator roughly 1/3 of the way around its cycle.

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Acknowledgements

The work described in this review were supported in part by the Burroughs-Wellcome Fund (Eric Shea-Brown) and the NSF (Lai-Sang Young and Kevin K Lin).

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  1. Department of Mathematics, University of Arizona, Tucson, AZ, USA

    Kevin K. Lin

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  1. Kevin K. Lin
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Correspondence to Kevin K. Lin .

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  1. Institut für Mathematik, Goethe-Universität Frankfurt, Frankfurt am Main, Germany

    Peter E. Kloeden

  2. Institut für Mathematik, Alpen-Adria Universität Klagenfurt, Klagenfurt am Wörthersee, Austria

    Christian Pötzsche

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Lin, K.K. (2013). Stimulus-Response Reliability of Biological Networks. In: Kloeden, P., Pötzsche, C. (eds) Nonautonomous Dynamical Systems in the Life Sciences. Lecture Notes in Mathematics(), vol 2102. Springer, Cham. https://doi.org/10.1007/978-3-319-03080-7_4

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