Physics 178/278 - The biophysical basis of neurons and networks
(Thoughts without content are empty, intuitions without concepts are blind.*)
Course overview
This course explores underlying principles and design rules for the neuronal circuits that control animal behavior. The emphasis in on analytically tractable models of neuronal network dynamics and computation. We present analytical pathways that involve dimensionality reduction to go from the "exact" dynamics of single neurons and synapses to simplified but tractable aspects of network activity. One case involves networks of coupled neurons with arhythmic firing rates that form extensive recurrent connections. Here, the interactions among neurons are strong and can lead to attractor dynamics. Such circuits serve to understand motor control, sensory processing, and memory. A second case involves networks of coupled neuronal oscillators in which each neuron is rhythmically active and described by its phase in a limit cycle. Here, the interactions among neurons affect only the relative timing between neurons. These networks provide a means to understand behaviors that range from locomotion to neurovascular dynamics in fMRI. A special aspect of the class is the inclusion of ongoing efforts in connectomics to bridge prediction and experimental reality. Aspects of applied mathematics such as propability and information theory, biophysics, and statistical mechanics are presented as needed.
There is no official textbook. Detailed lecture notes, with illustrations from relevant experimental papers, are provided as handouts, along with links to background material. There are four homework assignements plus a class project. For Winter 2025, we will have circuits constructed in neuromorphic hardware, with concurrent LTspice simulation as needed, as an option for the project. We build on the ideas of Prof. Marcello Rozenberg and use fed-back thyristors as spiking elements and current-mirrors as synapses to construct small neural circuits; see:
An ultra-compact leaky-integrate and-fire model for building spiking neural networks - Rozenberg, Schneegans & Stollar, Science Reports, 2019. |
(1.6 MB PDF) |
A dynamic neuro-synaptic hardware platform for spiking neural networks - Wu, dâ€™Hollande, Du & Rozenberg, preprint. |
(16 MB PDF) |
For Winter 2025, we are also pleased to have Mr. Haidong Qin as the class teaching assistant. Mr. Qin will run a tutorial and discussion section and hold office hours; office hours with Prof. Kleinfeld are immediately after class and by appointment.
Lecture notes for Winter 2024.
(Truth is much too complicated to allow anything but approximations.**)
(Chapter 1) - Introduction: Neurons, synapses, and the tiniest circuits |
DK lecture notes |
(9.2 MB PDF) |
(Chapter 2) - Recurrent neuronal networks: Brain states and discrete attractors |
DK lecture notes |
(7.5 MB PDF) |
Statistical mechanics of neural networks - Sompolinsky reprint |
(4.0 MB PDF) |
Electrophysiology in the age of light - Scanziani & Hausser reprint |
(2.2 MB PDF) |
(Chapter 3) - Recurrent neuronal networks: Invariant tuning and continuous attractors |
DK lecture notes |
(7.2 MB PDF) |
Invariant tuning - Shapley & Sompolinsky reprint |
(1.1 MB PDF) |
Attractor dynamics in the fly brain - Kim, Rouault, Druckmann & Jayaraman reprint |
(5.3 MB PDF) |
Modeling feature selectivity in local cortical circuits - Hansel & Sompolinsky book chapter reprint |
(3.7 Mb PDF) |
(Chapter 4) - Variability is a fundamental aspect of the neuronal response |
DK lecture notes |
(7.4 MB PDF) |
van Vreeswijk & Sompolinsky reprint |
(0.8 MB PDF) |
Sanzeni, Akitake, Goldbach, Leedy, Brunel & Histed reprint |
(2.7 MB PDF) |
(Chapter 5) - Information and sensory coding |
DK lecture notes in progress |
(6.4 MB PDF) |
General Principles for Sensory Coding - Sharpee reprint |
(2.7 MB PDF) |
(Chapter 6) - Recording, information transduction, and activation of neurons |
DK lecture notes |
(14.9 MB PDF) |
(Chapter 7) - Rhythmic dynamics in neuronal networks |
DK lecture notes on Reduced Models of Spiking |
(4.2 MB PDF) |
DK lecture notes on Derivation of Weakly Coupled Phase Oscillators |
(1.0 MB PDF) |
DK lecture notes on Circuits of Weakly Coupled Phase Oscillators |
(3.7 MB PDF) |
DK lecture notes on Time Delays and Patterns and Waves with Coupled Oscillators |
(9.5 MB PDF) |
Coupled oscillators - Ermentrout & Kleinfeld reprint |
(0.6 MB PDF) |
Tutorial and discussion sections
Review of circuit equations 1 |
Review of circuit equations 2 |
Review of linear algebra |
Review of Fourier transforms |
Review of probability |
Review of spectral analysis |
Review of ordinary differential equations and stability |
Review of classic Hodgkin Huxley equations |
Project Discussions |
Homework (e-mail a type-set or scanned PDF to Ms. Ghita Guessous)
Historical lecture notes from Winter 2022.
(Lecture 1) - Recurrent neuronal networks: A tale of two cells |
DK lecture notes |
(5.3 MB PDF) |
(Lecture 2) - Recurrent neuronal networks: Associative memory 1 |
DK lecture notes |
(2.0 MB PDF) |
(Lecture 3) - Recurrent neuronal networks: Associative memory 2 |
DK lecture notes |
(3.8 MB PDF) |
(Lecture 4) - Neurotechnology to measure and manipulate |
DK lecture notes |
(14.8 MB PDF) |
(Lecture 5) - Recurrent neuronal networks: Invariant tuning and continuous attractors 1 |
DK lecture notes |
(4.6 MB PDF) |
(Lecture 6) - Recurrent neuronal networks: Invariant tuning and continuous attractors 2 |
DK lecture notes |
(0.5 MB PDF) |
(Lecture 7) - Recurrent neuronal networks: Invariant tuning and continuous attractors 3 |
DK lecture notes |
(2.6MB PDF) |
(Lecture 8) - Linear recurrent networks: Integration, line attractors and mono-stability |
DK lecture notes |
(3.9 MB PDF) |
(Lecture 9) - Biophysics of conductance-based neuronal dynamics. |
DK lecture notes |
(1.0 MB PDF) |
(Lecture 10) - Voltage scales of neuronal dynamics. |
DK lecture notes |
(3.0 MB PDF) |
(Lecture 11) - Recurrent neuronal networks: Derivation from conductance models |
DK lecture notes |
(1.0 MB PDF) |
(Lecture 12) - Variability in network dynamics. Part 1 |
DK lecture notes |
(3.9 Mb PDF) |
(Lecture 13) - Variability in network dynamics. Part 2 |
DK lecture notes |
(3.3 MB PDF) |
(Lecture 14) - Spike dynamics in brutalized conductance models |
DK lecture notes |
(4.2 MBb PDF) |
(Lecture 15) - Coupled oscillators and waves in the brain. Part 1 |
DK lecture notes |
(1.0 MB PDF) |
(Lecture 16) - Coupled oscillators and waves in the brain. Part 2 |
DK lecture notes |
(3.7 MB PDF) |
(Lecture 17) - Coupled oscillators and waves in the brain. Part 3 |
DK lecture notes |
(9.5 MB PDF) |
(Lecture 18) - Synaptic weights from space-time receptive fields |
DK lecture notes |
(3.9 MB PDF) |
(Lecture 19) - Layered networks for optimal stimulus reconstruction |
DK lecture notes |
(0.3 MB PDF) |
Simplified neuron model as a principal component analyzer - Oja reprint |
(0.3 MB PDF) |
Background material
NEUROSCIENCE: Exploring the Brain, 4th edition. Textbook by Mark Bear, Barry Connors & Michael Paradiso |
(49.6 MB PDF) |
Foundations of Mathematical Neuroscience. Textbook by Bard Ermentrout & David Therman |
(9.2 MB PDF) |
Review of diffusion. Notes and graphics of DK |
(0.3 MB PDF) |
Review of electrodiffusion across membranes. Notes and graphics of DK |
(3.8 Mb PDF) |
Review of electrotonic properties of dendrites and axons. Notes and graphics of DK. |
(0.7 MB PDF) |
Review of Hodgkin Huxley formalism: Chapter 6 from "Biophysics of Computation" by Christoff Koch. |
(28.0 MB PDF) |
Review of mammalian CNS anatomy. MBL Neuroinformatics graphics of Helen Basbas. |
(1.7 MB PDF) |
Review of linear algebra. MBL Neuroinformatics notes of DK. |
(2.1 MB PDF) |
Stability analysis of a two dimensional dynamical system. Notes of Yonitan Aljadeff. |
(0.1 MB PDF) |
Review of Fourier transforms. Notes of DK. |
(0.1 MB PDF) |
Review of Poisson distribution. Notes of John Cooper. |
(0.1 MB PDF) |
Review of neuronal variability and Poisson statistics. Notes of Yonitan Aljadef. |
(0.1 MB PDF) |
Receptive fields and predicting stimuli from spike trains. Notes of DK. |
(5.7 MB PDF) |
Review of layered networks - Chapter 6 from "Neural Networks: A Comprehensive Foundation" by Simon Haykin. |
(11.5 MB PDF) |
Reverse correlation, stimulus design, and analysis. Notes of Yonitan Aljadeff. |
(7.7 MB PDF) |
"Analysis of spike trains" - Aljadeff, Lansdell, Fairhall & Kleinfeld reprint. |
(7.8 MB PDF) |
"Spectral methods" - Kleinfeld & Mitra reprint. |
(5.3 MB PDF) |
Basic MATLAB tutorial: Notes of Douglas Rubino. |
(0.1 MB M-code) |
Advanced Matlab tutorial. |
(0.1 Mb zipped M-code) |
*Immanuel Kant, 1724 - 1804, in Critique of Pure Reason.
.
**John von Neumann, 1903-1957.