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Abstract

Model Reduction of Large Spiking Neurons
by

Anthony Richard KeIlems

This thesis introduces and applies model reduction techniques to problems associ-
ated with simulation of realistic single neurons. Neurons have complicated dendritic
structures and spatially-distributed ionic kinetics that give rise to highly nonlinear
dynamics. However, existing model reduction methods compromise the geometry,
and thus sacrifice the original input-output relationship. I demonstrate that linear
and nonlinear model reduction techniques yield systems that capture the salient dy-
namics of morphologically accurate neuronal models and preserve the input-output
maps while using significantly fewer variables than the full systems. Two main dy-
namic regimes characterize the voltage response of a neuron, and I demonstrate that
different model reduction techniques are well-suited to each regime.

Small perturbations from the neuron’s rest state fall into the subthreshold regime,
which can be accurately described by a linear system. By applying Balanced Trunca-
tion (ВТ), a model reduction technique for general linear systems, I recover subthresh-
old voltage dynamics, and I provide an efficient Iterative Rational Krylov Algorithm
(IRKA), which makes large problems of interest tractable. However, these approxi-
mations are not valid once the input to the neuron is sufficient to drive the voltage
into the spiking regime, which is characterized by highly nonlinear behavior. To re-

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