Mini Course: Algebraic geometry and topology of neural codes

Neurons encode information about the world in many ways. What is the structure of these codes? How do these codes allow animals to make inferences about structures and things outside of their skull? In this Mini Course we will develop some partial answers.

This course will incorporate some “flipped classroom” content. There is a short video you should watch before coming to each session plus an optional reading.

For an introduction about what the title of this Mini Course means, please watch the introduction video (7m28s). If you are interested, you can also watch a related introductory video from mathematician Mohamed Omar (7m58s).

Target Audience

The course is explicitly multi-disciplinary in its content, being a blend of neuroscience and mathematics. However, no prior knowledge in either discipline is expected or necessary and anyone interested is welcome to join.

Learning Outcomes

If you believe your neuroscience and mathematical knowledge is essentially zero, please don’t be afraid to come along! The course has been designed with even you in mind and we will happily answer all of your questions before/during/after.

Although having knowledge in one or both of neuroscience and mathematics is not necessary, it may lead to a deeper understanding of the content. Below are lists of learning goals associated with each session depending on your own strengths.

If your neuroscience knowledge is greater than your mathematical knowledge, from each session our learning goal for you is:

  1. Be reminded or become aware of different neural coding schemas in spiking neural networks (biological and artificial) and a fundamental problem which persists for analysing neural data no matter the encoding scheme.
  2. Learn about the existence of some mathematical topics and basic tools in abstract algebra, geometry, and topology.
  3. Understand how basic mathematical tools (introduced in session 2) can be applied to neural rate coding (discussed in session 1) to make rigorous neuroscientific deductions based on neural data.
  4. Understand how more advanced mathematical tools (introduced in session 2) can be built upon simpler ones and applied to neural rate coding (discussed in session 1) to make more complex neuroscientific deductions based on neural data.
If your mathematical knowledge is greater than your neuroscience knowledge, from each session our learning goal for you is:
  1. Learn about how neurons encode information through the activity of “spikes”/action potentials, including the variety of known coding schemas, and a fundamental problem which persists for analysing neural data no matter the encoding scheme.
  2. Be reminded or become aware of simplicial complexes and their algebraic encoding, as well as polynomial and Boolean rings, varieties, and ideals.
  3. Understand how simplicial complexes and their algebraic encodings (discussed in session 2) can be used to abstractly describe a neural coding method called rate coding (introduced in session 1) and basic geometric and topological features of the underlying space it encodes.
  4. Understand how we can extend on known algebraic structures (discussed in session 2) to generate a ‘neural ideal’ which allows us to extract many more geometric and topological details of the neural encoding.

Teacher

Tom Burns (PhD student in Neural Coding and Brain Computing Unit).

Program

This course is heavily inspired by a book chapter by mathematical neuroscientists Carina Curto, Alan Veliz-Cuba, and Nora Youngs entitled ‘Analysis of combinatorial neural codes: an algebraic approach’ (which is optional reading for session 4).

Videos should be watched before the start of each session to gain a full understanding. Optional readings will be provided as PDFs and are not necessary to gain a full understanding but may be helpful for interested students.

Date Time Room Topic Video Optional Reading
Wednesday, June 9 15:00 - 16:00 B701 Neural coding (lecture) Link (6m41s) “Is coding a relevant metaphor for the brain?” Brette (2019)
Thursday, June 10 15:00 - 17:00 A720 Algebraic geometry and topology for non-mathematicians (lecture and exercises) Link (12m24s) “Ideals, varieties, and algorithms” Cox, Little, O'Shea (2015)
Wednesday, June 16 15:00 - 17:00 B701 Simplicial complexes of neural codes (lecture and exercises) Link (7m14s) “Cell Groups Reveal Structure of Stimulus Space” Curto, Itskov (2008)
Thursday, June 17 15:00 - 17:00 A720 Neural ideals of neural codes (lecture and exercises) Link (3m59s) “Analysis of combinatorial neural codes” Curto, Veliz-Cuba, Youngs (2018)

You will find here all the material used for this course.

More information

  • What to bring: Note-taking material
  • Zoom link: if you prefer joining remotely, or if B701 exceeds 50% capacity, you can join using this link. Unfortunately, we won't be able to provide much help with the hands-on part via Zoom. 
  • Video Recording: this course might be recorded and uploaded online, only the teacher will be recorded. Contact Jeremie Gillet if you have reservations about this.
  • Drinks: There will be free coffee and tea, bring your cup!

If you are interested in the course but cannot participate to this particular event, let us know and we will contact you for any later occurrence of this course.

Thank you very much for your interest.

 
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