Abstract
We hinted several times at the fact that a small excitatory synaptic input in the presence of voltage-dependent channels will lead to a local depolarization, followed by a hyperpolarization. Those of us who built our own radios will recognize such an overshooting response as being indicative of so-called RLC circuits which include resistances, capacitances as well as inductances. As a reminder, a linear inductance is defined as a circuit element whose instantaneous I—V relationship is, where L is the inductance measured in units of henry (abbreviated as H). Although neurobiology does not possess any coils or coillike elements whose voltage is proportional to the current change, membranes with certain types of voltage- and time-dependent conductances can behave as (/they contained inductances. We talk of a phenomenological inductance, a phenomenon first described by Cole (1941) and Cole and Baker (1941) in the squid axon (see Cole, 1972). Under certain circumstances, discussed further below, such damped oscillations can become quite prominent. This behavior can be obtained in an entirely linear system, as can be observed when reducing (in numerical simulations) the amplitude of the synaptic input (or step current): even though the voltage excursion around steady-state becomes smaller and smaller, the overshoot persists. It is not due to any amplification inherent in such a membrane but is caused by its time- and voltage-dependent nature. Such a linear membrane, whose constitutive elements do not depend on either voltage or time, and which behaves like a bandpass element, has been called quasi-active (Koch, 1984) to distinguish it from a truly active, that is, nonlinear membrane. In this chapter, we will explain in considerable detail how an inductance-like behavior can arise from these membranes by linearizing the Hodgkin-Huxley equations. Experimentally, this can be done by considering the small-signal response of the squid giant axon and comparing it against the theoretical predicted value, a further test of the veracity of the Hodgkin-Huxley equations, which they passed with flying colors.