480 Chapter 12. Nerve impulses[[Student version, January 17, 2003]]
The big picture
Returning to the Focus Question, this chapter has developed a picture of the unmyelinated nerve
axon as an excitable medium, capable of transmitting nonlinear waves of excitation over long
distances without loss of signal strength or definition. Hodgkin and Huxley’s insight had an immense
impact even on applied mathematics, helping launch the theory of such waves. In biology, too, the
notion of nonlinearwaves in anexcitable medium has led to an understanding of systems as diverse
as the cooperative behavior of individual slime mold cells (as they spontaneously coalesce into the
multicellular fruiting body) and the cells in your heart (as they contract synchronously).
Welocated the source of the axon’s excitability in a class of allosteric ion channels. Channels
of the voltage-gated superfamily are the target of drugs widely used against pain, epilepsy, cardiac
arrhythmias, cardiac failure, hypertension, and hyperglycemia. These advances are all rooted in
Hodgkin and Huxley’s biophysical measurements—which contained no direct evidence for individual
ion channels!
Key formulas
- Capacitors: The potential across a capacitor isV =q/C.Sothe current flowing into a
capacitor isI=dq/dt=C(dV/dt)(Equation 12.5).
Forcapacitors in parallel, the total capacitance isC=C 1 +C 2 since both share the sameV;
this is why the capacitance of a membrane patch is proportional to its area. - Membrane conductance: The symboljqalways denotes net electric charge per time per
area from inside the cell to outside (also called charge flux). The charge flux through the
membrane due to ion speciesiisjq,i;thusjq=
∑
ijq,i.Inthis chapterVdenotes the electric
potential inside the membrane (in our model the potential is zero everywhere outside). V^0
denotes the quasi-steady value ofV,andv=V−V^0 .Sov=0is the quasi-steady state. Also
ViNernst=−kzBieTln(ci, 2 /ci, 1 )isthe Nernst potential for ion speciesi.
Westudied three increasingly realistic models of membrane conductance:- Ohmic: The fluxes arejq,i=(V−ViNernst)gi,wheregiare positive constants (Equa-
tion 11.8). Thus the current of ion typeiflows in the direction tending to bringV
back to that ion’s equilibrium value,ViNernst.Since equilibrium is the state of maximum
disorder, this is a dissipative process converting free energy to heat, like a resistor. - Voltage-Gating: One or more of the conductances are not constant, but rather depend
on the instantaneous value ofv.Weexplored a simplified model withjq,r=vg^0 tot+
B(v−H)v^2 (Equation 12.20). HereBandHare positive constants. - Hodgkin–Huxley:Some conductances depend not only on the value ofv,but also on its
recent past history via relaxation-type relations. - Chord:∑ Neglecting ion pumping, the Ohmic hypothesis yields the chord formula: V^0 =
i
gi
gtotV
Nernst
i ,wheregtot=∑
igi(Equation 12.3). V(^0) describs a quasisteady potential ap-
proximating the true resting potential. The formula shows that this value is a compromise
between the various Nernst potentials, dominated by the ion species with the greatest con-
ductance.