Visualizing fundamental neuronal computation for life science students.

where is the membrane time constant, is the membrane space constant, and V is the membrane potential. This equation describes the propagation of electricity along a core conductor, as some of the current escapes through a nonperfect insulator to an infinite external conductor. This cable equation was successfully adapted from the underwater telegraph to axons (4) and later to dendrites (9–12) and forms the basis for modern modeling of biophysically detailed neurons (1). This equation is simply not self-explanatory, and, in my experience, the difficulty in explaining passive cable theory to life science undergraduates is far from trivial. This difficulty is evident when browsing textbooks of neurophysiology, which offer various explanations of passive cable theory. Some provide a mathematical tracking of the subject (6), while others offer a more descriptive view (7). Over the years, I have tried these and other textbooks with equally dismal results: the students did not grasp the basics of passive cable theory. Since this subject is fundamental to the understanding of neurophysiology and appears early in the course material, the students’ failure to understand it leaves them with a knowledge gap and leaves me exasperated. In another part of the neurophysiology course, I describe the classical experiment of Hodgkin and Keynes (3), who built a physical model for ion permeation through membranes using steel balls, two connected chambers, and an electrical motor. This wonderful physical model has been able to predict that ions traverse the membrane in single file and that several of them should be found in the membrane. This led me to search for an everyday physical system that may help to better visualize the passive propagation of the electrical wave in axons and dendrites. I found my solution in drip irrigation. Unlike Hodgkin and Keynes, I did not build the actual physical system, but developed a series of thought experiments for use in the classroom while teaching passive cable theory. Drip irrigation as a model system for cable theory. Drip irrigation involves dripping water onto the soil directly on or close to the roots of a plant. The water is dripped at very low rates from plastic pipes fitted with outlets called water emitters. For readers not familiar with drip irrigation, Fig. 1 shows drip irrigation in action in my backyard as a visual introduction to the physical system. The advantage of drip irrigation over surface or sprinkler irrigation, which involves wetting entire fields homogeneously, is the application of water directly to the plant, which saves precious water. Drip irrigation has clear similarities with the passive propagation of electricity in axons and dendrites. First, it is a system with a core conductor with small resistance to the passage of the propagating wave (friction of the water with the tube walls). Second, along the wall of the tube, there are highresistance shunting points (water emitters), causing small water currents to flow through the tube wall. Clearly, drip irrigation is not a true facsimile of axons and dendrites. Principle differences are the lack of inward water current through the water emitters to the core conductor, and that water does not obey the Nernst equation. The flow of water in a system of pipes may be better described as analogous to an electrical capacitor (2). Thus the two systems are not physically and mathematically similar. However, the relative similarity can be used to generate several thought experiments, detailed below, demonstrating the basic principles of passive cable theory. Visualizing an infinite cable. Having developed Eq. 1 from basic electric circuit components (I find the steps described in Ref. 8 to be clearer for this part of the lesson), I review several solutions of the theory. The first is the case of the infinite cable under steady-state conditions. The solution of Eq. 1 in this case is: