### Why it is hard to fit tau1 and tau2 in k3.mod

Because tau1 and tau2 are not parameters but volatile assigned variables that serve as the voltage sensitive time constants on the last call to rates_khh(v). They are essentially functions of v, ta1_khh, tk1_khh, ta2_khh, tk2_hh, and vrest_khh and are defined by:

    vr = v - vrest : v = vrest means rates at 0
    tau1 = ta1*exp(tk1*vr)
    tau2 = ta2*exp(tk2*vr)

The simplest work around is to set tk1 to 0 and use ta1 as a proxy for tau1.

### Where did that functional form for the time constants come from?

The steady state Boltzmann distribution has a highly satisfying and simple physical interpretation. Voltage dependent time constants, on the other hand, seem to be based on more speculative mechanisms and I have been satisfied with merely capturing the phenomenology. In fact previous incarnations of the last exercise in this lesson used the FUNCTION_TABLE specification to directly use the measured time constants without imagining an underlying mechanism at all.

If you plot ntau_hh(v) you'll see that it is more complicated than my simple exponential above. Prior to the existence of the multiple run fitter, when the tau's were fit one at a time and plotted using GatherVector tool I noticed the downward trend at hyperpolarizing potentials and the apparent saturation at high depolarizing potentials and was not unpleased with the similarity to hh, not noticing that the hh peak occurred hyperpolarized from rest and the data clearly shows a slower (larger) time constant at 6mV than at 19mV.

Rewriting this exercise to take advantage of the multiple run fitter required that I come up with some kind of functional form for the voltage sensitive rates. Of the forms easy to imagine, only a constant would be simpler and that can't fit the data. The exponential has a simple interpretation in terms of a fixed size energy barrier a la Eyring rate theory and has only two parameters per state transition pair. The forms for alpha and beta when all this is put together have a not unpleasing mathematical symmetry. The final result of four parameters per state transition pair is probably a minimum. The chain of closed states suggests some simplified models analogous to diffusion that might be interesting to pursue that would naturally account for delay.

NEURON hands-on course
Copyright © 1998, 1999 by N.T. Carnevale and M.L. Hines, all rights reserved.