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TANH Spline Interpolation for Analytical Modelling of BK Ion Channels in Smooth Muscle

Ion channels are molecular entities that are responsible for electrical signal generation in nerve, muscle and other cells. The paper discusses the use of tangent hyperbolic functions (TANH) in the modelling of the Big Conductance Potassium (BK) ion channel, especially for the smooth muscle of the urinary bladder. Ion channels are often modelled using the Hodgkin-Huxley formalism, in which equations governing the dependencies of various state parameters on cellular variables are derived from experimental data. When there is sufficient experimental data, formulating equations that replicate the trend becomes relatively easy. However, where the data for these dependencies is sparse, usual approaches fail. Here, we describe a TANH spline interpolation technique as an efficient means of fitting sparse data. TANH is used to "braid" functions in two adjacent regions and is done, unlike with splines, in a way such that a single function operates over the whole region. The resultant function is whole range, continuous and differentiable to any arbitrary order. We have demonstrated the application of this technique in modelling the parameter "slope factor" for the BK channel of mammalian smooth muscle. We show that, by the use of this technique, BK channel current outputs thus generated, faithfully replicate physiologically recorded BK currents from the detrusor smooth muscles of guinea-pig bladder. We propose this method as one of the general application in modelling ion channels constrained by sparse experimental data.