Structure-Preserving Numerical Integrators for Hodgkin--Huxley-Type Systems

Abstract

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. The second-order Strang splitting method is seen to perform especially well across a range of non-stiff and stiff dynamics.2010 Mathematics Subject Classification. 65P30, 37M20, 92C20. 1 It 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 = i, but the scalar case covers the applications we are interested in.

Geometric Numerical Integration of Liénard Systems via a Contact Hamiltonian Approach

“…Our results show that the resulting geometric integrators are very stable, even when the system is stiff, and they preserve the qualitative features of the limit cycle even for large values of the time step, which permits sparing computational resources and is of primal importance in applications to, e.g., neuronal dynamics [7]. Moreover, from the use of the modified equations, we can prove analytical results on the preservation and the period of the limit cycle that show a very good agreement with the numerical simulations.…”

“…Both these approaches involve a four-dimensional phase-space, and in both the authors have focused on the perturbation theory and have not explored the consequences of the Hamiltonisation for the numerical integration. From yet another perspective, in [7] the authors have presented various splitting schemes for "conditionally linear systems" (these include Liénard systems) which, although not geometric, are based on the standard splitting schemes for symplectic Hamiltonian systems, and showed good qualitative and quantitative results.…”

Probabilistic solvers enable a straight-forward exploration of numerical uncertainty in neuroscience models

“…However, this method has the disadvantage that the ODE system at hand has to put into the proper form and therefore requires specific knowledge about the model and solver. In a more recent study, Chen et al [11] recommended to use splitting methods, such as second-order Strang splitting, instead of exponential integrators. The diversity in attempts to achieve the best accuracy vs. computational cost trade-off shows that the scientific debate regarding this topic is far from settled and the ideal solver choice is problem specific.…”

“…For some of the most common mechanistic models in neuroscience like the Hodgkin-Huxley or Izhikevich neuron model, errors in numerical integration have been studied for a range of solvers and different integration step-sizes [9][10][11]. These studies have shown that standard solvers are often not the best choice in terms of accuracy or the accuracy vs. run time trade-off.…”

“…Therefore, the authors of these studies proposed to use specific solvers for the analyzed models, e.g. the Parker-Sochacki method for the Hodgkin-Huxley and Izhikevich neuron [9], an exponential midpoint method [10] or second-order Strang splitting [11] for Hodgkin-Huxley-like models. While improving computations for the specific problems, applying these to other scenarios requires a detailed understanding of the kinetics of the neuron model of interest; and while choosing a "good" solver for a given model is important, it is typically not necessary to choose the "best" ODE solver.…”