Quantitative models of the action potential

In

where I_Na, I_K, and I_L are currents conveyed through the local sodium channels, potassium channels, and "leakage" channels (a catch-all), respectively. The initial term I_ext 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

${\displaystyle I_{\mathrm {K} }=g_{\mathrm {K} }\left(V-E_{\mathrm {K} }\right)p_{\mathrm {open,K} }}$

which is equivalent to Ohm's law. By definition, no net current flows (I_K = 0) when the transmembrane voltage equals the equilibrium voltage of that ion (when V = E_K).

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.^[9] Hodgkin and Huxley modeled the voltage-sensitive potassium channel as having four gates; letting p_n 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., p_{open, 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, p_{open, Na} = m³h. The probabilities for each gate are assumed to obey first-order kinetics

${\displaystyle {\frac {dm}{dt}}=-{\frac {m-m_{\mathrm {eq} }}{\tau _{m}}}}$

where both the equilibrium value m_eq and the relaxation time constant τ_m depend on the instantaneous voltage V across the membrane. If V changes on a time-scale more slowly than τ_m, the m probability will always roughly equal its equilibrium value m_eq; however, if V changes more quickly, then m will lag behind m_eq. By fitting their voltage-clamp data, Hodgkin and Huxley were able to model how these equilibrium values and time constants varied with temperature and transmembrane voltage.^[1] The formulae are complex and depend exponentially on the voltage and temperature. For example, the time constant for sodium-channel activation probability h varies as 3^{(θ−6.3)/10} with the Celsius temperature θ, and with voltage V as

${\displaystyle {\frac {1}{\tau _{h}}}=0.07e^{-V/20}+{\frac {1}{1+e^{3-V/10}}}.}$

In summary, the Hodgkin–Huxley equations are complex, non-linear

Because of the complexity of the Hodgkin–Huxley equations, various simplifications have been developed that exhibit qualitatively similar behavior.^[3][13] The FitzHugh–Nagumo model is a typical example of such a simplified system.^[14][15] Based on the tunnel diode, the FHN model has only two independent variables, but exhibits a similar stability behavior to the full Hodgkin–Huxley equations.^[16] The equations are

${\displaystyle C{\frac {dV}{dt}}=I-g(V),}$

${\displaystyle L{\frac {dI}{dt}}=E-V-RI}$

where g(V) is a function of the voltage V that has a region of negative slope in the middle, flanked by one maximum and one minimum (Figure FHN). A much-studied simple case of the FitzHugh–Nagumo model is the Bonhoeffer-van der Pol nerve model, which is described by the equations^[17]

${\displaystyle C{\frac {dV}{dt}}=I-\epsilon \left({\frac {V^{3}}{3}}-V\right),}$

${\displaystyle L{\frac {dI}{dt}}=-V}$

where the coefficient ε is assumed to be small. These equations can be combined into a second-order differential equation

${\displaystyle C{\frac {d^{2}V}{dt^{2}}}+\epsilon \left(V^{2}-1\right){\frac {dV}{dt}}+{\frac {V}{L}}=0.}$

This van der Pol equation has stimulated much research in the mathematics of nonlinear dynamical systems. Op-amp circuits that realize the FHN and van der Pol models of the action potential have been developed by Keener.^[18]

A hybrid of the Hodgkin–Huxley and FitzHugh–Nagumo models was developed by Morris and Lecar in 1981, and applied to the muscle fiber of barnacles.^[19] True to the barnacle's physiology, the Morris–Lecar model replaces the voltage-gated sodium current of the Hodgkin–Huxley model with a voltage-dependent calcium current. There is no inactivation (no h variable) and the calcium current equilibrates instantaneously, so that again, there are only two time-dependent variables: the transmembrane voltage V and the potassium gate probability n. The bursting, entrainment and other mathematical properties of this model have been studied in detail.^[20]

The simplest models of the action potential are the "flush and fill" models (also called "integrate-and-fire" models), in which the input signal is summed (the "fill" phase) until it reaches a threshold, firing a pulse and resetting the summation to zero (the "flush" phase).

where the integration is over the complete surface of the membrane; ${\displaystyle {\boldsymbol {\xi }}}$ is a position on the membrane, σ_inside and φ_inside are the conductivity and potential just within the membrane, and σ_outside and φ_outside the corresponding values just outside the membrane. Thus, given these σ and φ values on the membrane, the extracellular potential φ(x) can be calculated for any position x; in turn, the electric field E and current density j can be calculated from this potential field.[25]

See also

• Biological neuron models
• GHK current equation
• Models of neural computation
• Saltatory conduction
• Bioelectronics
• Cable theory

References