The given passage discusses the relationship between voltage and current in the cell membrane of axons. It explains that within a certain range of stimulation, the axon's membrane can be represented as a passive RC circuit consisting of a resistor and capacitor. However, this simplified circuit fails to consider the spatial nature of axons and the significant resistance of the intracellular space. If there were no intracellular resistance, voltage changes across one part of the axon would instantly occur across all regions, eliminating the need for action potentials to actively propagate signals. In reality, axons have significant resistance to current flow between different regions, which complicates the relationship between intracellular resistance (Rin) and membrane resistance (rm). Understanding how current injected at one point affects the membrane potential at other points helps explain the necessity and characteristics of action potentials.

The passage further explains that when current pulses, eliciting only passive responses, are passed across the plasma membrane, the magnitude of the potential change recorded depends on the distance between the recording electrode and the point of current passage. The closer the recording electrode is to the site of current passage, the larger and steeper the potential change. However, the change in potential decreases exponentially with distance from the site of passage, reflecting passive or electrotonic conduction. These passively conducted changes in potential do not propagate far along the membrane before becoming insignificant. A graph in the passage demonstrates how an electrotonically conducted signal diminishes over a few millimeters. The length over which the potential change decreases to 1/e (37%) of its maximal value is known as the length constant or space constant. Mammalian axons, which can be over a meter long, typically have a length constant of 1 to 3 mm. This highlights the need for a mechanism to propagate information about electrical events from the soma to the far end of the axon.
The relationships between voltage and current just described show that within a certain range of stimulation, the cell membrane in one region of the axon can be modeled by a passive RC circuit. However, this model circuit, with only a single resistor and capacitor, takes no account of the fact that axons are spatially extended structures and that because of this, the resistance of the intracellular space is a significant factor in how electrical events in one region affect other regions. That is, if axons had no intracellular resistance, their intracellular space would be isoelectric, and voltage changes, like those just described, across one part of the axonal membrane would occur across all regions instantaneously. In this case, there would be no need for a
CHAPTER 5 Generation and Conduction of Action Potentials special mechanism (i.e., the action potential) to propagate signals actively down the axon. In actuality, axons (and neurons in general) are spatially extended structures with significant resistance to current flow between different regions (this is one reason the relationship of Rin and rm is complicated). Therefore, it is important to understand how current injected at one point along the axon affects the membrane potential at other points because this both helps explain why action potentials are needed and helps explain some of their characteristics.
When current pulses that elicit only passive responses are passed across the plasma membrane, the size of the change in potential recorded depends on the distance of the recording electrode from the point of passage of the current (
Fig. 5.2).
The closer the recording electrode is to the site of current passage, the larger and steeper the change in potential is. The magnitude of the change in potential decreases exponentially with distance from the 0.0 mm 0.5 mm
Change of membrane potential 1.0 mm (mV) 1.5 2.0 2.5 mm 4.5mV •Fig. 5.2 Responsesofanaxonofashorecrabtoasubthresholdrectangularcurrentpulsebyanextracellularelectrodeappliedcloselytoitssurfaceandlocatedatdifferentdistancesfromthecurrent-passingelectrode.Astherecordingelectrodeismovedfartherfromthepointofstimulation,theresponseofthemembranepotentialisslowerandsmaller.(RedrawnfromHodgkinAL,RushtonWAH:Theelectricalconstantsofacrustaceannervefibre.Proc R Soc Lond B Biol Sci. 1946;133:444-479.) site of passage of the current, and the change in potential is said to reflect passive or electrotonic conduction. Such passively conducted changes in potential do not spread very far along the membrane before they become insignificant. As shown in
Fig. 5.2 , an electrotonically conducted signal dies away over a distance of a few millimeters. The distance over which the change in potential decreases to 1/e (37%) of its maximal value is called the length constant or space constant (where e is the base of natural logarithms and is equal to 2.7182). A length constant of 1 to 3 mm is typical for mammalian axons, which can be more than a meter long, which makes obvious the need for a mechanism to propagate information about electrical events generated at the soma to the far end of the axon.