where gNao = 2 TmNa = 2/(am + Pm) mS, mNax = am/(am + Pm), ThNa = 2/(ah + Ph)

ms, hNax = ah/(ah + ph) ms, am = 0.36(v + 33)/{1 - exp[-(v + 33)/3]}, Pm = -0.4(v + 42)/{1 - exp[(v + 42)/20]}, ah = -0.1(v + 55)/{1 - exp[(v + 55)/6]}, and ph = 4.5/[1 + exp(-v/10)].

The bullfrog axon also has one type of fast, voltage-gated, calcium channel with dynamics given by:

lCa = gCaomCahCa VCa) nA 1.4-35

where gCao = 0.116 |S, [Ca++]o = 4 mM (fixed), hCa = 0.01/(0.01 + [Ca++]i) deactivation, mCax = 1/{1 + exp[-(v - 3)/8]}, and xmCa = 7.8/{exp[(v + 6)/16] + exp[-(v + 6)/16]} ms.

The leakage current is given by

where gLo = 0.02 |S, VL = -10 mV, T = 295 K, Cm = 0.15 nF. The very small capacitance and the small conductances listed above are because the bullfrog ion currents were determined using a test membrane area considerably less than the standard 1-cm2 area used in the HH equations as originally formulated. The author estimated that the axon area used was between 50 and 225 x 10-6 cm2. Note that as Ca ions enter the membrane through their channels, [Ca++]i will rise. The local concentration of [Ca++]i just inside the membrane must be calculated using the diffusion equation and ICa. Finite-difference calcium diffusion equations can be found in Yamada et al. (1989), and are not given here.

So why is such a detailed model for the bullfrog nerve axon voltage- and ion-concentration-dependent behavior needed? The extra detail allows one to examine and understand the effects of long-term changes in Vm by voltage clamp, and the effects of changing ion concentrations and replacing ions (e.g., Na+, K+, Ca++, Cl-) with nonpermeable equivalents. Channel blockers can also be emulated (e.g., tetro-dotoxin to block Na+ channels) to isolate components of total membrane current under voltage clamp conditions.

Apparently, nature is far more complex in its design of certain nerve axon membranes than originally described by Hodgkin and Huxley in 1952 for the squid. Still, their model has withstood the test of time and has provided the basis for subsequent dynamic models for spike generation in a variety of neurons.

1.4.4 Discussion

This section has been devoted to a detailed description of the HH model for action potential generation in active nerve membrane. Most modern neural modeling software packages offer the option of a HH-type model for spike generation. Often, the user can modify the HH auxiliary equation structure to reflect simplifications, or to include other voltage-dependent ionic conductances (other than for K+ and N+). The HH model was stimulated using Simnon, the nonlinear ODE solver.

The section has shown how the original HH model acts as a nonlinear, current-to-voltage converter, and how the HH equations can be simply modified to produce a voltage clamp in which the properties of the ionic conductances can be examined.

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