K+ fluxes differ in several important aspects. First, they are opposite in sign for most values of membrane potential: The Na+ flux depolarizes the neuron, whereas the K+ flux hyperpolarizes the cell. Second, the Na + flux turns on ("activates") much more quickly than the K+ flux. Third, the Na+ flux turns off ("inactivates") after a brief period of depolarization. In contrast, the K + conductance remains activated in response to a prolonged depolarizing stimulus.

A watershed event in the history of neuroscience was the development by Hodgkin and Huxley of a relatively simple mathematical model, derived from voltage-clamp studies of the giant axon of the squid, that accounts for the generation and propagation of the action potential. The Hodgkin-Huxley model describes the membrane as an electrical circuit (Fig.

4a) that includes descriptions of membrane capacitance Cm; voltage-gated, Na + - and K + -selective conductances (GNa and GK, respectively), each in series with a battery representing the appropriate equilibrium potential; and a constant "leak" conductance that passes more than one ion. Mathematically, the Hodgkin-Huxley model includes four differential equations, which describe how the derivatives of membrane potential and three "gating variables" (variables that range between 0 and 1 and determine the values of voltage-gated conductances) behave. Two time- and voltage-dependent gating variables determine the size of the Na+ conductance: The m gate captures the rapid activation of the Na+ conductance after a step depolarization, whereas the h gate describes the slower inactivation process by which the

Figure 4 The Hodgkin-Huxley model of voltage-gated Na+ and K+ conductances. (a) Electrical circuit representation of the Hodgkin-Huxley model of squid giant axon under space-clamped conditions. (b-d) Responses of the Hodgkin-Huxley gating variables to a voltage-clamp step from Vhold = —90mV to Vclamp = 0mV. The m and h gates determine the value of GNapm3h. The n gate determines the value of GKpn4. (e) Steady-state values of m, h, and n for the entire physiologically relevant range of membrane potential Vm. (f)Time constants describing how quickly the gating variables m, h, and n reach their steady-state values, plotted vs Vm. Note the log scale on the y-axis.

Figure 4 The Hodgkin-Huxley model of voltage-gated Na+ and K+ conductances. (a) Electrical circuit representation of the Hodgkin-Huxley model of squid giant axon under space-clamped conditions. (b-d) Responses of the Hodgkin-Huxley gating variables to a voltage-clamp step from Vhold = —90mV to Vclamp = 0mV. The m and h gates determine the value of GNapm3h. The n gate determines the value of GKpn4. (e) Steady-state values of m, h, and n for the entire physiologically relevant range of membrane potential Vm. (f)Time constants describing how quickly the gating variables m, h, and n reach their steady-state values, plotted vs Vm. Note the log scale on the y-axis.

Table I

Definitions and Units of Mathematical Symbols

Table I

Definitions and Units of Mathematical Symbols

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