In neurophysiology, several mathematical models of the action potential have been developed, which fall into two basic types. The first type seeks to model the experimental data quantitatively, i.e., to reproduce the measurements of current and voltage exactly. The renowned Hodgkin–Huxley model of the axon from the Loligo squid exemplifies such models. Although qualitatively correct, the H-H model does not describe every type of excitable membrane accurately, since it considers only two ions (sodium and potassium), each with only one type of voltage-sensitive channel. However, other ions such as calcium may be important and there is a great diversity of channels for all ions. As an example, the cardiac action potential illustrates how differently shaped action potentials can be generated on membranes with voltage-sensitive calcium channels and different types of sodium/potassium channels. The second type of mathematical model is a simplification of the first type; the goal is not to reproduce the experimental data, but to understand qualitatively the role of action potentials in neural circuits. For such a purpose, detailed physiological models may be unnecessarily complicated and may obscure the "forest for the trees". The FitzHugh–Nagumo model is typical of this class, which is often studied for its entrainment behavior. Entrainment is commonly observed in nature, for example in the synchronized lighting of fireflies, which is coordinated by a burst of action potentials; entrainment can also be observed in individual neurons. Both types of models may be used to understand the behavior of small biological neural networks, such as the central pattern generators responsible for some automatic reflex actions. Such networks can generate a complex temporal pattern of action potentials that is used to coordinate muscular contractions, such as those involved in breathing or fast swimming to escape a predator.
Hodgkin–Huxley model
In 1952 Alan Lloyd Hodgkin and Andrew Huxley developed a set of equations to fit their experimental voltage-clamp data on the axonal membrane. The model assumes that the membrane capacitance C is constant; thus, the transmembrane voltage V changes with the total transmembrane current Itot according to the equation
C d V d t = I t o t = I e x t + I N a + I K + I L {\displaystyle C{\frac {dV}{dt}}=I_{\mathrm {tot} }=I_{\mathrm {ext} }+I_{\mathrm {Na} }+I_{\mathrm {K} }+I_{\mathrm {L} }}
where INa, IK, and IL are currents conveyed through the local sodium channels, potassium channels, and "leakage" channels (a catch-all), respectively. The initial term Iext represents the current arriving from external sources, such as excitatory postsynaptic potentials from the dendrites or a scientist's electrode. The model further assumes that a given ion channel is either fully open or closed; if closed, its conductance is zero, whereas if open, its conductance is some constant value g. Hence, the net current through an ion channel depends on two variables: the probability popen of the channel being open, and the difference in voltage from that ion's equilibrium voltage, V − Veq. For example, the current through the potassium channel may be written as
I K = g K ( V − E K ) p o p e n , K {\displaystyle I_{\mathrm {K} }=g_{\mathrm {K} }\left(V-E_{\mathrm {K} }\right)p_{\mathrm {open,K} }}
which is equivalent to Ohm's law. By definition, no net current flows (IK = 0) when the transmembrane voltage equals the equilibrium voltage of that ion (when V = EK). To fit their data accurately, Hodgkin and Huxley assumed that each type of ion channel had multiple "gates", so that the channel was open only if all the gates were open and closed otherwise. They also assumed that the probability of a gate being open was independent of the other gates being open; this assumption was later validated for the inactivation gate. Hodgkin and Huxley modeled the voltage-sensitive potassium channel as having four gates; letting pn denote the probability of a single such gate being open, the probability of the whole channel being open is the product of four such probabilities, i.e., popen, K = n4. Similarly, the probability of the voltage-sensitive sodium channel was modeled to have three similar gates of probability m and a fourth gate, associated with inactivation, of probability h; thus, popen, Na = m3h. The probabilities for each gate are assumed to obey first-order kinetics
d m d t = − m − m e q τ m {\displaystyle {\frac {dm}{dt}}=-{\frac {m-m_{\mathrm {eq} }}{\tau _{m}}}}
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