Lecture1 : Overview and Introduction Spring 2016

# Lecture 2: Review of Basics

## Assigned readings

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.

### Optional Reading

- 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

## Assigned readings

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

### Optional Readings

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

## Assigned Readings

Hille Chapter 10 hillchap10.pdf from

*Ionic Channels of Excitable Membranes 2nd Edition, Bertil Hille*

## Optional Readings

Origin of membrane bioenergetics: lane2013origin.pdf and for popular account see Science Daily

Entropy and Maxwell-Boltzman law: schroedinger_maxwell.pdf and Distributing Balls into Boxes. A clearer derivation of MBE is here

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

## Assigned Readings

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 perspectiveJohnston 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

## Optional Readings

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.

## Assigned Readings

Review of Chemical Kinetics as it is used in Hodgkin Huxley Theory

Overview of HH Theory nelson2004electrophysiological_models.pdf

Original Hodgkin Huxley paper HH_perspective.pdf and Original paper

Chapter 5 of the "Book of Genesis" and also see notes and summary I've prepared on Chapter 5, especially a table of the units used in compartmental modeling

Simplified HH models: Fitzhugh-Nagumo itzi_fitzhugh_nagumo.pdf

## Optional Readings

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

## Required Readings

# Lecture 8: Single Neuron Simulators

## Required Readings

## Optional Readings

## Extra Readings

GPL'd Book of Genesis(Internet Edition):iBoGpdf.tar.gz

# Lecture 9: Supraneuronal structures--maps and columns

## Required Readings

## Optional Readings