TY - JOUR
T1 - Structure-preserving numerical integrators for hodgkin-huxley-type systems
AU - Chen, Zhengdao
AU - Raman, Baranidharan
AU - Stern, Ari
N1 - Funding Information:
\ast Submitted to the journal's Computational Methods in Science and Engineering section December 18, 2018; accepted for publication (in revised form) November 18, 2019; published electronically February 18, 2020. https://doi.org/10.1137/18M123390X Funding: The work of the first author was partially supported by an ARTU research fellowship in the Department of Mathematics and Statistics at Washington University in St. Louis. The work of the second author was partially supported by NSF CAREER grant 1453022. The work of the third author was partially supported by NSF through grant DMS-1913272 and by the Simons Foundation through grant 279968. \dagger Mathematics, Courant Institute of Mathematical Sciences, New York University, New York, NY 10012-1110 (zc1216@nyu.edu). \ddagger Biomedical Engineering, Washington University in St. Louis, Saint Louis, MO 63130-4899 (barani@wustl.edu). \S Mathematics and Statistics, Washington University in St. Louis, Saint Louis, MO 63130-4899 (stern@wustl.edu). 1It is straightforward to generalize what follows to the case where xi and bi are vector-valued and ai is matrix-valued, so that xi satisfies a first-order linear system of ODEs with constant coefficients when xj is stationary for j
Publisher Copyright:
© 2020 Society for Industrial and Applied Mathematics.
PY - 2020
Y1 - 2020
N2 - Motivated by the Hodgkin-Huxley model of neuronal dynamics, we study explicit numerical integrators for "conditionally linear"" systems of ordinary differential equations. We show that splitting and composition methods, when applied to the Van der Pol oscillator and to the Hodgkin-Huxley model, do a better job of preserving limit cycles of these systems for large time steps, compared with the "Euler-type"" methods (including Euler's method, exponential Euler, and semi-implicit Euler) commonly used in computational neuroscience, with no increase in computational cost. These limit cycles are important to preserve, due to their role in neuronal spiking. Splitting methods even compare favorably to the explicit exponential midpoint method, which is twice as expensive per step. The second-order Strang splitting method is seen to perform especially well across a range of nonstiff and stiff dynamics.
AB - Motivated by the Hodgkin-Huxley model of neuronal dynamics, we study explicit numerical integrators for "conditionally linear"" systems of ordinary differential equations. We show that splitting and composition methods, when applied to the Van der Pol oscillator and to the Hodgkin-Huxley model, do a better job of preserving limit cycles of these systems for large time steps, compared with the "Euler-type"" methods (including Euler's method, exponential Euler, and semi-implicit Euler) commonly used in computational neuroscience, with no increase in computational cost. These limit cycles are important to preserve, due to their role in neuronal spiking. Splitting methods even compare favorably to the explicit exponential midpoint method, which is twice as expensive per step. The second-order Strang splitting method is seen to perform especially well across a range of nonstiff and stiff dynamics.
KW - Computational neuroscience
KW - Exponential integrators
KW - Hodgkin-Huxley
KW - Limit cycles
KW - Splitting methods
KW - Van der Pol oscillator
UR - http://www.scopus.com/inward/record.url?scp=85083758670&partnerID=8YFLogxK
U2 - 10.1137/18M123390X
DO - 10.1137/18M123390X
M3 - Article
AN - SCOPUS:85083758670
SN - 1064-8275
VL - 42
SP - B273-B298
JO - SIAM Journal on Scientific Computing
JF - SIAM Journal on Scientific Computing
IS - 1
ER -