# Noble model

 Martin Fink and Denis Noble (2008), Scholarpedia, 3(2):1803. doi:10.4249/scholarpedia.1803 revision #91582 [link to/cite this article]
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Curator: Denis Noble

Figure 1: Solution for the membrane potential of the Noble Model (1962) using COR

The Noble model of 1962 was the first mathematical model of cardiac action potentials and pacemaker rhythm to be based on experimental recordings of ionic currents. It was a development from the Hodgkin-Huxley model of the action potential in the squid giant axon. Based on its formulations hundreds of other models were developed representing cardiac muscle cells in more and more detail for a number of different species – from mouse to human (see a repository of cell models).

Additionally to replicating the shape of the action potential the model was able to predict the presence of additional ionic currents which have been found later-on experimentally, but also to show how the basic rhythm generator of the mammalian heart, the sino-atrial node, can generate a rhythm without containing an explicit oscillator.

## The Origin of the Model

Figure 2: The first analysis of potassium channel currents in the heart. Redrawn from (Hall et al., 1963). The solid line shows the total membrane current recorded in a Purkinje fibre in a sodium-depleted solution. The horizontal arrow indicates the trajectory at the beginning of the action potential, while the vertical arrow indicates the time-dependent activation of $$\mathrm{i_K}$$ which initiates repolarization.

The Noble Model was developed using the mathematical model formulation for ion channels by Hodgkin-Huxley. The experimental work (Hutter & Noble, 1960; Hall et al., 1963) was done using the current clamp technique with two microelectrodes impaled into Purkinje fibres from sheep and dog hearts, one for passing current and the other for measuring changes in membrane potential. Two types of potassium currents could be shown, the inward rectifier $$\mathrm{i_{K1}}$$ and the delayed rectifier $$\mathrm{i_K}$$ ( Figure 2). The inward rectifier carries a large inward current at negative potentials, but during depolarization the current falls towards zero (horizontal arrow). The delayed rectifier current then slowly activates (upward arrow), thus inducing repolarization of the action potential .

Figure 3: Ionic channels included in the model.

The equations for $$\mathrm{i_K}$$ used slowed-down versions of the Hodgkin-Huxley K+ current equations, representing the fact that a process that takes a few ms in nerve requires hundreds of ms in heart. For $$\mathrm{i_{K1}}$$ the equations were simple empirical descriptions of the dependence of this current on voltage and on potassium concentration. It was assumed that this component was not explicitly dependent on time. The sodium current equations used the Hodgkin-Huxley equations since Weidmann (1956) had shown that the sodium inactivation process was very similar in the heart. An anion (chloride) current was included since that also had been detected in the experimental data.

## Results and Predictions – Comparison to Today’s Knowledge

### Sodium current including calcium dynamics

The first results were disappointing. Although the Hodgkin-Huxley equations predict a steady plateau sodium current, later discovered experimentally (Attwell et al., 1979), it wasn’t large enough during strong depolarizations. This calculation predicted either that the sodium current in heart differs from that in nerve or that other inward current channels were present. Both predictions are correct, but the demonstration of calcium currents had to wait until Reuter’s work (Reuter, 1967). Instead, in the 1962 model the sodium activation equations were modified to generate current over a wider range of potentials. The inward current equations in this model are therefore seriously incomplete, not only by omitting calcium channels (see models of calcium dynamics) but also because we now know that there are late components of sodium current that are not predicted by the Hodgkin-Huxley equations (Kiyosue & Arita, 1989; Maltsev et al., 1998; Sakmann et al., 2000; Zygmunt et al., 2001). Another inward current not included in the model is the "pacemaker" ("funny") If current later described in sinoatrial node and in Purkinje fibres by DiFrancesco (Brown, DiFrancesco & Noble, 1979; DiFrancesco, 1981a; DiFrancesco, 1981b). It is established today that if is responsible for generation of the diastolic depolarization in pacemaker cells, according to a mechanism by which the net inward current during pacemaker depolarization is contributed to by If activation at the termination of an action potential, rather than by a "leakage" inward current superimposed to a decaying outward current (see Noble, 1984).

Figure 4: Sodium and potassium conductance changes computed from the model. Two cycles of activity are shown. The conductances are plotted on a logarithmic scale to accommodate the large changes in sodium conductance. Note the persistent level of sodium conductance during the plateau of the action potential, which is about 2% of the peak conductance. Note also the rapid fall in potassium conductance at the beginning of the action potential. The arrows correspond to those shown in Figure 2.

### The pacemaker without an explicit oscillator

The broad outlines of the time course of potassium current was however successfully reconstructed ( Figure 4). Moreover, the slow time course of decay of $$\mathrm{i_K}$$ allowed the equations to account for pacemaker rhythm at about the right frequency. The model also illustrated an important property of the electrophysiology of repetitive activity: that pacemaker activity is an integrative characteristic of the system as a whole; there is no molecular driver. This kind of analysis is now one of the major features of the systems biology approach (Noble, 2006).

### Potassium currents and energy saving mechanisms

The 1962 model also revealed the nature of the balance evolution had struck in developing long action potentials in the heart. The inward rectifier, $$\mathrm{i_{K1}}\ ,$$ is energy-saving. By greatly reducing potassium ion flow during depolarization, it enables much smaller inward sodium and calcium currents to maintain the depolarization and so requires much less energy-expenditure by ion pumps to restore the transmembrane gradients. This is important since the energy consumption by ionic pumps is significant. These insights have stood the test of time. They are features of all subsequent cardiac cell models (Noble, 2007).

### Other major successes of the model

• reconstruction of the impedance changes during the action and pacemaker potentials,
• reconstruction of the phenomenon of all-or-nothing repolarization,
• alternation of action potential duration during repetitive firing,
• and the actions of anions on pacemaker frequency.

## The equations

The electrical potential $$V$$ across the membrane is changing due to the ionic currents

$\frac{\text{d}V}{\text{d}t} = -\frac{i_{Na}+i_{K}+i_{Leak}}{Cm}.$

The formulation for the sodium current was taken from the Hodgkin-Huxley model, but the time constants were changed as described above $\alpha_m = \frac{100(-V-48)}{\exp((-V-48)/15)-1}, \quad \beta_m = \frac{120(V+8)}{\exp((V+8)/5)-1},$ $\frac{\text{d}m}{\text{d}t} = \alpha_m (1-m)-\beta_m m,$ $\alpha_h = 170 \exp((-V-90)/20), \quad \beta_h = \frac{1000}{1+\exp((-V-42)/10)},$ $\frac{\text{d}h}{\text{d}t} = \alpha_h (1-h)-\beta_h h,$ $i_{Na} = (400000 m^3 h+140)(V-E_{Na}), \text{ where the reversal potential } E_{Na}=40.$

Potassium current was split into an instantaneous current $$i_{K1}$$ $i_{K1} = (1200 \exp((-V-90)/50)+15 \exp((V+90)/60)) (V-E_K), \text{ where the reversal potential } E_K = -100,$ and a slowly activating $$i_{K2}$$ current $\alpha_n = \frac{0.1 (-V-50)}{exp((-V-50)/10)-1}, \quad \beta_n = 2 \exp((-V-90)/80),$ $\frac{\text{d}n}{\text{d}t} = \alpha_n (1-n)-\beta_n n,$

$i_{K2} = 1200 n^4 (V-E_K).$

Futhermore, a current of anions (chloride) was introduced $i_{An} = 75 (V-E_{An}), \text{ where } E_{An} = -60.$

Parameters are given in millivolt, microsiemens and seconds.

## References

• Attwell D, Cohen I, Eisner D, Ohba M & Ojeda C. (1979). The steady state TTX sensitive ("window") sodium current in cardiac Purkinje fibres. Pflügers Archiv, European Journal of Physiology 379, 137-142.
• Brown HF, DiFrancesco D & Noble SJ. (1979) How does adrenaline accelerate the heart? Nature, 280, 235-236
• DiFrancesco D. (1981a) A new interpretation of the pacemaker current iK2 in calf Purkinje fibres. Journal of Physiology 314, 359-376
• DiFrancesco D. (1981b) A study of the ionic nature of the pacemaker current in calf Purkinje fibres. Journal of Physiology 314, 377-393
• Hall AE, Hutter OF & Noble D. (1963). Current-voltage relations of Purkinje fibres in sodium-deficient solutions. Journal of Physiology 166, 225-240.
• Hutter OF & Noble D. (1960). Rectifying properties of heart muscle. Nature 188, 495.
• Kiyosue T & Arita M. (1989). Late sodium current and its contribution to action potential configuration in guinea pig ventricular myocytes. Circulation Research 64, 389-397.
• Maltsev VA, Sabbah HN, Higgins RSD, Silverman N, Lesch M & Undrovinas AI. (1998). Novel, ultraslow inactivating sodium current in human ventricular myocytes. Circulation 98, 2545-2552.
• Noble D. (1962). A modification of the Hodgkin-Huxley equations applicable to Purkinje fibre action and pace-maker potentials. Journal of Physiology 160, 317-52 [PubMed].
• Noble D. (1984) The surprising heart: a review of recent progress in cardiac electrophysiology. Journal of Physiology 353:1-50
• Noble D. (2006). The Music of Life. OUP, Oxford.
• Noble D. (2007). From the Hodgkin-Huxley axon to the virtual heart. Journal of Physiology 580, 15-22.
• Reuter H. (1967). The dependence of slow inward current in Purkinje fibres on the extracellular calcium concentration. Journal of Physiology 192, 479-492.
• Sakmann BFAS, Spindler AJ, Bryant SM, Linz KW & Noble D. (2000). Distribution of a Persistent Sodium Current Across the Ventricular Wall in Guinea Pigs. Circulation Research 87, 910-914.
• Weidmann S. (1956). Elektrophysiologie der Herzmuskelfaser. Huber, Bern.
• Zygmunt AC, Eddlestone GT, Thomas GP, Nesterenko VV & Antzelevitch C. (2001). Larger late sodium conductance in M cells contributes to electrical heterogeneity in canine ventricle. American Journal of Physiology 281, H689-697.

Internal references

• Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.

## Recommended Reading

• Noble D. (2006). The Music of Life. OUP, Oxford.