The name is absent

21

• Any number of ion channels and gating variables can be allowed.

• Either current injection or synaptic conductance may be used as input.

With these assumptions in mind, we introduce and explain the nonlinear cable equa-
tion, which is the partial differential equation that will be the focus of the model
reduction techniques in this thesis.

The remaining subsections of this chapter come nearly verbatim from my first
paper with Roos, Xiao, and Cox (Kellems et al., 2009), which is used here, with some
modifications, with kind permission from Springer ScienceTBusiness Media: Journal
of Computational Neuroscience, Low-dimensional, morphologically accurate models
of subthreshold membrane potential, 27(2): 161-176, 2009, A. R. Kellems, D. Roos,
N. Xiao, and S. J. Cox, copyright Springer ScienceTBusiness Media, LLC 2009.

2.3.1 Nonlinear Cable Equation

We consider dendritic neurons (see, e.g., Figure 2.5) with D branched dendrites
meeting at the soma. The dth dendrite possesses Bd branches, and we denote by
£b the length of branch b and encode its radius, as a function of distance from its
distal end, by ʤ(ʃ). The transmembrane potential along branch b will be denoted by
vt,(x, t)∙ We assume that the axial resistivity, R_i (kΩ cm), and membrane capacitance,
C_m (μF∕cm²), are uniform throughout the cell. We suppose that branch b carries C
distinct currents, with associated densities, G⅛_c(τ) (mS∕cm²) and reversal potentials
E_c, c= 1,..., C. The kinetics of current c on branch b are governed by (powers of)

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