Morphologically accurate reduced order modeling of spiking neurons

Abstract

Accurately simulating neurons with realistic morphological structure and synaptic inputs requires the solution of large systems of nonlinear ordinary differential equations. We apply model reduction techniques to recover the complete nonlinear voltage dynamics of a neuron using a system of much lower dimension. Using a proper orthogonal decomposition, we build a reduced-order system from salient snapshots of the full system output, thus reducing the number of state variables. A discrete empirical interpolation method is then used to reduce the complexity of the nonlinear term to be proportional to the number of reduced variables. Together these two techniques allow for up to two orders of magnitude dimension reduction without sacrificing the spatially-distributed input structure, with an associated order of magnitude speed-up in simulation time. We demonstrate that both nonlinear spiking behavior and subthreshold response of realistic cells are accurately captured by these low-dimensional models.

Appendix

The following table contains the information pertaining to the ion channels and gating variable kinetics that allow weakly excitable dendrites. The model is based on that of Migliore et al. (1999) for I _Na , $I_{K_{\text{DR}}}$, $I_{K_{\text{A}}}$, and $I_{\text{leak}}$, though there have been some modifications to the time constants for $l_{\text{prox}}$ and $l_{\text{dist}}$, which was done to obtain a good fit to the original functions without the need for defining them piecewise. Some of the gating variables in this model have time constants τ which depend on temperature T via a so-called Q ₁₀ factor. When we use this model, we set T = 35°C to match the value used in Migliore et al. (1999).

The complete HH and HHA channel models can be found in the appendix of Kellems et al. (2009).

Table 12 Channel model and kinetics that allow weakly excitable dendrites. For G(x), x is measured in μm from the soma. The model for $I_{K_{\text{A}}}$ is divided into two components, proximal and distal, which are denoted $I_{K_{\text{A(prox)}}}$ and $I_{K_{\text{A(dist)}}}$, respectively