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A Combined Offline–Online Algorithm for Hodgkin–Huxley Neural Networks

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Abstract

Spiking neural networks are widely applied to simulate cortical dynamics in the brain and are regarded as the next generation of machine learning. The classical Hodgkin–Huxley (HH) neuron is the foundation of all spiking neural models. In numerical simulation, however, the stiffness of the nonlinear HH equations during an action potential (a spike) period prohibits the use of large time steps for numerical integration. Outside of this stiff period, the HH equations can be efficiently simulated with a relatively large time step. In this work, we present an efficient and accurate offline–online combined method that stops evolving the HH equations during an action potential period, uses a pre-computed (offline) high-resolution data set to determine the voltage value during the spike, and restarts the time evolution of the HH equations after the stiff period using reset values interpolated from the offline data set. Our method allows for time steps an order of magnitude larger than those used in the standard Runge–Kutta (RK) method, while accurately capturing dynamical properties of HH neurons. In addition, this offline–online method robustly achieves a maximum of a tenfold decrease in computation time as compared to RK methods, a result that is independent of network size.

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Acknowledgements

This work was supported by National Key R&D Program of China (2019YFA0709503), NSFC-11671259, NSFC-11722107, SJTU-UM Collaborative Research Program, and the Student Innovation Center at Shanghai Jiao Tong University (D.Z.); the NSF Mathematical Sciences PostDoctoral Research Fellowship (MSPRF) DMS-1703761 (J.C.). We dedicate this paper to our late professor David Cai.

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  1. School of Mathematical Sciences, MOE-LSC, Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai, China

    Zhong-qi Kyle Tian & Douglas Zhou

  2. Courant Institute of Mathematical Sciences, New York University, New York, NY, USA

    Jennifer Crodelle

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Tian, Zq.K., Crodelle, J. & Zhou, D. A Combined Offline–Online Algorithm for Hodgkin–Huxley Neural Networks. J Sci Comput 84, 10 (2020). https://doi.org/10.1007/s10915-020-01261-6

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