• Swimming, flying, walking, dynamical description, energy requirements, actuators, control. Material properties of biological systems and how their structure relates to their function (e.g., wood, hair, cell membranes, cartilage).

  • Long range neuron signals; physical necessity of repeaters (e.g., nodes of Ranvier), engineering advantage of pulse coding, action potential generation, information transmission and errors.

  • Shapes of cells: force balance, hydrostatic pressure, elasticity of membrane and effects of the spatial dependence of elasticity; cytoskeletal force effects on shape.

One such effort illustrates the interactions of the engineering and science involved, and makes it clear that the subject can be examined in enough detail to teach essential ideas honestly. A “long range neural signals” section might begin with the electrical conductivity of salt water, of the lipid cell membrane, and the electrical capacitance of the cell membrane. It would next develop the simple equations for the attenuation of a voltage applied across the membrane at one end of an axon “cylinder” with distance down the axon, and the effect of membrane capacitance on signal dynamics for time-varying signals. After substituting numbers, it becomes clear that amplifiers will be essential. Real systems are always noisy and imperfect; amplifiers have limited dynamical range; and the combination of these facts makes sending of an analog voltage signal through a large number of amplifiers essentially impossible. Pulse coding information escapes that problem (all long distance communication is digital these days). How are “pulses” generated by a cell? This would lead to the power supply needed by an amplifier—ion pumps, and the Nernst potential. How are action potentials generated? A first example of the transduction of an analog quantity into pulses might be stick-slip fraction, in which a block resting on a table and pulled by a weak spring whose end is steadily moved, moves in “jumps” whose distance is always the same. This introduction to nonlinear dynamics contains the essence of how an action potential is generated. The “negative resistance” of the sodium channels in a neuron membrane provides the same kind of “breakdown” phenomenon. Stability and instabilities (static and dynamic) of nonlinear dynamical systems can be analyzed, and finally the Hodgkin Huxley equations illustrated. The material is an excellent source of imaginative laboratories involving electrical measurements, circuits, dynamical systems, batteries and the Nernst potential, and information and noise, and classical mechanics. It has great po-



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