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.
Advanced Optional Readings
Weyl's paper on circuit theory weyl_1923.pdf
Siggraph Tutorial on Discrete Differential Geometry SIGGRAPH_2--6_DiscreteDifferentialGeometry.pdf
For a recent application of discrete (graph theoretic) approaches to vision, see (from our lab) recent papers by Grady and Schwartz at http://eslab.bu.edu/publications/publications.php
Lecture 3: Brain, behaviour and neurons
Kandel and Schwartz (4th Edition): Kandel and Schwartz, 4th Edition Principles of Neural Science Chap 1,2,and 4 and Kandel and Schwartz, 4th Edition Principles of Neural Science Chap 17,18,19
Human Cortical sulcal pattern CorticalSulci.pdf
Lecture 4: Nernst Equation, Thermodynamics, Information and Entropy
Hille Chapter 10 hillchap10.pdf from Ionic Channels of Excitable Membranes 2nd Edition, Bertil Hille
Thermodynamics and information theory: feynman_computation.pdf
Neuronal spike energy Lennie_metabolic03.pdf
Lecture 5: Nernst-Planck and Goldman Equations-Voltage gated ion channels and synaptic transmission
Hille Chapter 13: Selective Permeability [of Ions in Solution] from Ionic Channels of Excitable, 2nd edition, Bertil Hille. This is a reference for ionic permeability and the Goldman equations from a biophysics perspective
Johnston and Wu Chapter 2: Ion Movement in Excitable Cells from Foundations of Cellular Neurophysiology by D. Johnston and S. Wu (1995) This is a reference for Nernst-Planck and Goldman Equations, including outline derivations and applications.
Kandel, Schwartz and Jessep (4th Edition) Chapters [9,10,11,12,13,14,15 This is Neuroscience background material
Notes on ion transport and resting potential W02-membranes_and_nernst_notes.pdf
Notes on ion pumps and ion channelsion_channels_ion_pumps.pdf
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.
Overview of HH Theory nelson2004electrophysiological_models.pdf
Simplified HH models: Fitzhugh-Nagumo itzi_fitzhugh_nagumo.pdf
http://www.nature.com/nature/focus/voltagesensing/index.html Recent Nature special interest page on recent work on the biophysics of
voltage gated channels
Beyond Hodgkin and Huxley and the Zoo of Ion Channels: http://neuronaldynamics.epfl.ch/online/Ch2.S3.html
Good discussion of HH system in the context of supplied MATLAB CODE from URL. Note that the URL has a link to download matlab code. I will supply a modified version, which would be better to use, but the original code can be referenced, especially if there are version skew issues (wrt to the MATLAB your are using) and the operating system (WIN,MAC) that you are using.
Lecture 7: Linear and Quadratic Integrate and Fire-Dynamical Systems
Lecture 8: Single Neuron Simulators
GPL'd Book of Genesis(Internet Edition):iBoGpdf.tar.gz
Lecture 9: Supraneuronal structures--maps and columns