Further Developments

All of the foregoing models for cardiac cell electrical activity represent various approximations and simplifications. Consequently, there has been considerable activity to improve these models to better represent the correct physiology. Here we mention a few of these models.

Pacemaker activity

Both SA node and Purkinje fiber cells exhibit autonomous oscillatory behavior. Examples of improved ionic models that reconstruct this behavior are given by DiFrancesco and Noble (1985) for Purkinje fiber cells and by Noble and Noble (1984) for the SA node.

The fast sodium current

At the time the Beeler-Reuter equations were published, it was not possible to measure accurately the fast sodium inward current, because it activates so rapidly. As a result, all the early models (Noble, MNT, BR) used the Hodgkin-Huxley formulation of the sodium current. However, it is known that this does not give a sufficiently rapid upstroke for the action potential. This has little effect on the space-clamped action potential, but it has an important effect on the propagation speed for propagated action potentials.

Once appropriate data became available, it was possible to suggest an improved description of the sodium current. Thus, a modification of the sodium current was proposed by Ebihara and Johnson (1980) (EJ), which has since become the standard for most myocardial simulations. For the EJ model, the sodium current is given by

exp(-0.09(V + 10.66)) + 1 3.56exp(0.079V ) + 3.1 x 105exp(0.35V)

Calcium

The final form for a myocardial ionic model has not yet been determined, as there are continual suggestions for improvements and modifications. A major difficulty with the Beeler-Reuter model is with the calcium current and internal calcium concentration. This is not unexpected, as at the time the model was formulated, little was known about the mechanisms of calcium release and uptake.

While the inclusion of proper calcium kinetics into a myocardial ionic model is a topic of active research, one recent model deserves mention. This model is known as the Luo-Rudy (LR) model (Luo and Rudy, 1994a,b). An earlier model (Luo and Rudy, 1991) was a generalization of the Beeler-Reuter model. There is still significant debate about many of the details of the LR models, so it is unlikely that the LR models are the final word.

4.4 Exercises_

1. Show that, if k > 1, then (1 - e-x)k has an inflection point, but (e-x)k does not.

2. Use the independence principle of Chapter 3 to derive an expression for K in (4.6). Show that K is independent of time, as assumed by Hodgkin and Huxley. How can this expression be used to check the accuracy of the assumption that all the initial current in response to a voltage step is carried by Na+ ions? Derive another expression for K by assuming that the Na+ channel has a linear I-V curve (which, as we discussed in Chapter 3, does not obey the independence principle).

3. Explain why replacing the extracellular sodium with choline has little effect on the resting potential of an axon. Calculate the new resting potential with 90% of the extracellular sodium removed. Why is the same not true if potassium is replaced?

4. Plot the nullclines of the Hodgkin-Huxley fast subsystem. Show that vr and ve in the Hodgkin-Huxley fast subsystem are stable steady states, while vs is a saddle point. Compute the stable manifold of the saddle point and compute sample trajectories in the fast phase-plane, demonstrating the threshold effect.

5. Show how the Hodgkin-Huxley fast subsystem depends on the slow variables; i.e., show how the v nullcline moves as n and h are changed, and demonstrate the saddle-node bifurcation in which ve and vs disappear.

6. Plot the nullclines of the fast-slow Hodgkin-Huxley phase-plane and compute a complete action potential. How does the fast-slow phase-plane behave in the presence of an applied current? How much applied current is needed to generate oscillations?

7. Suppose that in the Hodgkin-Huxley fast-slow phase-plane, v is slowly decreased to v* < v0 (where v0 is the steady state), held there for a considerable time, and then released. Describe what happens in qualitative terms, i.e., without actually computing the solution. This is called anode break excitation (Hodgkin and Huxley, 1952d. Also see Peskin, 1991). What happens if v is instantaneously decreased to v* and then released immediately? Why do these two solutions differ?

8. Solve the full Hodgkin-Huxley system numerically with a variety of constant current inputs. For what range of inputs are there self-sustained oscillations? Why should one expect self-sustained oscillations for some current inputs?

9. The Hodgkin-Huxley equations are for the squid axon at 6.3°C. Using that the absolute temperature enters the equations through the Nernst equation, determine how changes in temperature affect the behavior of the equations. In particular, simulate the equations at 0° C and 30° C to determine whether the equations become more or less excitable with an increase in temperature.

10. Show that a Hopf bifurcation occurs in the generalized FitzHugh-Nagumo model when fv(v*,w*) = -egw(v* ,w*), assuming that fv(v*,w*)gw(v*,w*) - gv(v*,w*)fw(v*,w*) > 0.

11. Morris and Lecar (1981) proposed the following two-variable model of membrane potential for a barnacle muscle fiber:

where V = membrane potential, W = fraction of open K+ channels, T = time, Cm = membrane capacitance, Iapp = externally applied current, 0 = maximum rate for closing

Table 4.7 Typical parameter values for the Morris-Lecar model.

Cm

= 20 ^F/cm2

'app =

0.06 mamp/cm2

gca

= 4.4 mS/cm2

Qk =

8 mS/cm2

Ql

= 2 mS/cm2

0 =

0.040 (ms)-1

Vi

= -1 mV

V2 =

15 mV

V3

=0

V4 =

30 mV

VCa

= 100 mV

=

-70 mV

Vl

= -50 mV

Iion(V, W) = gcM^(V)(— - VCa) +gKW(V - V0) + gL— - —L), (4.97)

Typical rate constants in these equations are shown in Table 4.7.

(a) Find a nondimensional representation of the Morris-Lecar equations in terms of the variables v = -—r, t = fKT, w = W.

VCa 2Cm

(b) Sketch the phase portrait of the nondimensional Morris-Lecar equations. Show that there is a unique steady state at v = — 0.3173,w = 0.1076 and determine its stability.

(c) Show that the Morris-Lecar equations can be reasonably well approximated by the cubic FitzHugh-Nagumo system dv

— = -k [(v - a)(v - b)(v - c) + a(w - ws)], (4.101) dw

where a = -0.317,b = -0.18,c = 0.467,ws = 0.1076,k = 10,a = 0.2,p = 0.8333, y = 0.47,0 = 0.3.

12. Does the Morris-Lecar model exhibit anode break excitation (see Exercise 7)? If not, why not?

13. The Pushchino model is a piecewise linear model of FitzHugh-Nagumo type proposed as a model for the ventricular action potential. The model has f (v,w) = F(v) - w, (4.103)

where

—30(v — 1), for v>v2, 2, for v <v1, 16.6, for v > v1,

Simulate the action potential for this model. What is the effect on the action potential of changing t(v)?

14. Perhaps the most important example of a nonphysiological excitable system is the Belousov-Zhabotinsky reaction. This reaction denotes the oxidation of malonic acid by bromate in acidic solution in the presence of a transition metal ion catalyst. Kinetic equations describing this reaction are (Tyson and Fife, 1980)

where u denotes the concentration of bromous acid and v denotes the concentration of the oxidized catalyst metal. Typical values for parameters are e & 0.01, f = 1,q & 10—4. Describe the phase portrait for this system of equations.

15. It is not particularly difficult to build an electrical analogue of the FitzHugh-Nagumo equations with inexpensive and easily obtained electronic components. The parts list for one "cell" (shown in Fig. 4.27) includes two op-amps (operational amplifiers), two power supplies, a few resistors, and two capacitors, all readily available from any consumer electronics store (Keener, 1983).

The key component is an operational amplifier (Fig. 4.25). An op-amp is denoted in a circuit diagram by a triangle with two inputs on the left and a single output from the vertex on the right. Only three circuit connections are shown on a diagram, but two more are assumed, being necessary to connect with the power supply to operate the op-amp. Corresponding to the supply voltages Vs— and Vs+, there are voltages Vr— and Vr+, called the rail voltages, which determine the operational range for the output of an op-amp. The job of an op-amp is to compare the two input voltages v+ and v—, and if v+ > v—, to set (if possible) the output voltage vo to the high rail voltage Vr+, whereas if v+ <v—, then vo is set to Vr—. With reliable electronic components it is a good first approximation to assume that the input draws no current, while the output v0 can supply whatever current is necessary to maintain the required voltage level.

dv dv

Figure 4.25 Diagram for an operational ampli fier (op-amp).

The response of an op-amp to changes in input is not instantaneous, but is described reasonably well by the differential equation es-rr = g(v+- v_) - v0. (4.109)

dv0 dt

The function g(v) is continuous, but quite close to the piecewise constant function g(v) = Vr+H(v) + Vr_H(_v), (4.110)

with H(v) the Heaviside function. The number es is small, and is the inverse of the slew-rate, which is typically on the order of 106-107 V/sec. For all of the following circuit analysis, take es ^ 0.

(a) Show that the simple circuit shown in Fig. 4.26 is a linear amplifier, with

provided that v0 is within the range of the rail voltages.

(b) Show that if R1 = 0,R2 = to, then the device in Fig. 4.26 becomes a voltage follower with v0 = v+.

(c) Find the governing equations for the circuit in Fig. 4.27, assuming that the rail voltages for op-amp 2 are well within the range of the rail voltages for op-amp 1.

di2 dt

Steps Action Potential
Figure 4.26 Linear amplifier using an op-amp.

Figure 4.27 FitzHugh-Nagumo circuit using op-amps.

where F(v) is the piecewise linear function v — Vr+, for v > aVr+,

(d) Sketch the phase portrait for these circuit equations. Show that this is a piecewise linear FitzHugh-Nagumo system.

(e) Use the singular perturbation approximation (4.55) to estimate the period of oscillation for the piecewise linear analog FitzHugh-Nagumo circuit in Fig. 4.27.

16. Simulate the Noble equations with different values of gan = 0.0, 0.075, 0.18, 0.4 mS/cm2. Explain the results in qualitative terms.

Table 4.8 Parts list for the FitzHugh-Nagumo analog circuit.

2 LM 741 op-amps (National Semiconductor)

r1 = r2 = 100kfi

r3 = 2.4q

R4 =

r5 = 10kß

C1 = 0.0VF

C2 = 0.5^F

Power supplies:

±15V for op-amp #1

±12V for op-amp #2

17. Simulate the MNT equations, and explain why the currents Isi, IK2, IX1, and Ix2 are called the slow inward, pacemaker, and plateau currents, respectively.

18. Simulate the YNI model for the SA nodal action potential. Find parameter values for the FitzHugh-Nagumo cubic model that duplicate this behavior as best possible.

19. (a) Simulate the Beeler-Reuter equations and plot each of the currents and the calcium concentration. What terms are mostly responsible for the prolongation of the action potential?

(b) Do the Beeler-Reuter equations exhibit anode break excitation?

CHAPTER 5

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