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**Hodgkin-Huxley Nerve Model**

Original source of the model: A. L. Hodgkin, A. F. Huxley: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. **117** (1952) 500-544.
The experiments reported in Figures 1-5 concentrate on the importance of *threshold values*. Here it is essential whether the initial voltage *U*_{0} is below or above a threshold: the cell will fire for *U*_{0}=-6, and will basically be quiet for *U*_{0}=-5; compare Figures 1 and 2. Figures 4 and 5 depict in addition the dynamics of some intermediate values.

Figure 3 shows a simulation that approaches a limit cycle. Taking the applied current as bifurcation parameter,
,
we find a Hopf bifurcation for
with an interval of bistability of approximately
.

Figure 1

(*U*,*V*_{1})-phase diagram of the Hodgkin-Huxley equation; two trajectories: *U*(0)=-5 and -6, with
*V*_{1}(0)=0.31,
*V*_{2}(0)=0.05,
*V*_{3}(0)=0.6 and *I*_{a}=0.

Figure 2

*U*(*t*) for
;
Hodgkin-Huxley equation; two trajectories: *U*(0)=-5 and -6, with
*V*_{1}(0)=0.31,
*V*_{2}(0)=0.05,
*V*_{3}(0)=0.6 and *I*_{a}=0.

Figure 3

(*U*,*V*_{1})-phase diagram of the Hodgkin-Huxley equation; trajectory approaching a limit cycle. *U*(0)=-5,
*V*_{1}(0)=0.31,
*V*_{2}(0)=0.05,
*V*_{3}(0)=0.6 and *I*_{a}=-10.

Figure 4

(*U*,*V*_{1})-phase diagram of the Hodgkin-Huxley equation; six trajectories *U*(0)=*U*_{0}, with
*V*_{1}(0)=0.31,
*V*_{2}(0)=0.05,
*V*_{3}(0)=0.6 and *I*_{a}=0;
*U*_{0}=-5.0/-5.2/-5.4/-5.6/-5.8/-6.0

Figure 5

*U*(*t*) for
;
membrane potential of the Hodgkin-Huxley equation; six trajectories *U*(0)=*U*_{0}, with
*V*_{1}(0)=0.31,
*V*_{2}(0)=0.05,
*V*_{3}(0)=0.6 and *I*_{a}=0;
*U*_{0}=-5.0/-5.2/-5.4/-5.6/-5.8/-6.0

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Figure 1

Figure 2

Figure 3

Figure 4

Figure 5