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# Lecture 2: Review of Basics

• A review of simple circuit theory and op amps:horowitzandhill.pdf

• Resistors, pp. 1-10
• Capacitors, pp. 20-24 and see Outline of Capacitive Reactance

• Differentiators and Integrators, pp. 25-27
• Operational Amplifiers and Feedback, pp. 28-33 of (Especially "Golden Rules" for op amp circuits)
• An overview of electrophysiology:hille_chap1.pdf

• A very elementary, but useful for background, review of multi-loop circuit analysis from a Freshman Physics perspective is: multi_loop_circuits.pdf

• Note that the method of neuron equivalent circuit analysis shown in Kandel and Schwartz, page 132ff, which is common in Neuroscience approaches, can lead to confusion, as it is not the standard method of circuit analysis used in Physics and Engineering.

• Analysis of active operational amplifier circuits
• PID controllers, and practical engineering:pease.pdf

• Graph Theoretic approach to circuit analysis
• graphic approach to circuit theory:strang.pdf and also graph_circuits_flow.pdf and lecture27student.pdf. Please note the difference in the method Strang outlines and the "graph circuits" approach discussed in the latter two readings, which are more typical of the type of approach to circuit analysis that would be encountered in electrical engioneering contexts. We will use Strang's method (modified nodal analysis, MNA) in this class, which is the easiest to program (IMHO), but which requires a bit of generalization from what Strang provides in regard to reactive circuit elements (capacitors for us), which he does not adquately account for in this discussion.

# Lecture 4: Nernst Equation, Thermodynamics, Information and Entropy

• Hille Chapter 10 hillchap10.pdf from Ionic Channels of Excitable Membranes 2nd Edition, Bertil Hille

# Lecture 6: Hodgkin Huxley Theory and Simplifications

Hodgkin Huxley theory is a classic ("the" classic) paper in Computational Neuroscience. It is a combination of circuit theory: the neuron and axon are viewed as an equivalent circuit based on the Nernst batteries supplied by the ions sodium, potassium, and a "leak" channel. The key idea is that the sodium and potassium conductances are dynamically dependent on membrane potential. Hence they are "voltage controlled" conductances in circuit terms. By appropriately modelling this dependence on membrane voltage, Hodgkin and Huxley were able to get a good fit to the action potential in the giant squid axon. Later work over the following half century extended this model with additional, more exotic examples of voltage controlled channels.

One key idea in this work is the use of kinetic theory. I have made some outline notes to explain the slightly non-standard use the HH made of first order kinetics. A clear understanding of this will make the rest of the theory go by very easily. I have put a pointer to my notes on this (which are on the class wiki on the page USEFULLINK/HODGKIN-HUXLEY.

A second aspect of HH theory is the Cable Equation, which they derived to describe the propagation of an action potential down the axon. I will go over several derivations of the cable equation in class. The first is the one provided by HH in their original paper: this is a "continuum" model which provides a leaky diffusion equation to model action propagation and is usually called "the leaky cable equation". A circuit theory model of this equation is the basis of compartmental modelling, and we will discuss this discrete, lumped circuit model both in the conventional "loop-node" analysis of elementary circuit theory and also in the more elegant (and easy to program) version of modified nodal analysis. Passive compartmental models provide a finite element model of a neuron, axon, and dendrites in terms of small cylindrical compartment. Each is characterized by its radius, diameter, position, and values of specific axial, membrane resistances and capacitance. The geometry of the neuron then allows us to compute lumped values of axial and membrane resistance and membrane capacitance. At this point, it is simply a matter of circuit theory (classical or MNA) to solve for the effects of a current pulse on this neuronal structure. Finally, the model can be made "active" by inserting Hodgkin Huxley like voltage controlled conductances (and possibly more exotic and recent forms of conductances). The result is compartmental modeling, as we know it today.

voltage gated channels