Lecturer: Magnus Richardson

Term(s): Term 2

Status for Mathematics students: List C

Commitment: 30 one hour lectures and 10 tutorials

Assessment: Written exam (100%)

Formal registration prerequisites: None

Assumed knowledge:

• Solutions to standard ordinary and partial differential equations as in MA133 Differential Equations and MA250 Introduction to PDEs
• 2x2-Matrix eigenvalues and eigenvectors as in MA106 Linear Algebra

Useful background:

• More experience with differential equations such as MA3G1 Theory of PDEs
• Some knowledge of probability and statistics such as MA359 Measure Theory or ST342 Mathematics of Random Events or ST202 Stochastic Processes, although basics of stochastic differential equations will be taught in the module itself.

Synergies:

• MA482 Stochastic Analysis
• MA4F7 Brownian Motion
• MA4L3 Large Deviation Theory
  
  

Content:

Experimentally verified mathematical models of synapses and neurons will first be developed with a particular emphasis on their noisy dynamics. These components, described in terms of stochastic differential equations, will be coupled to provide a probabilistic Fokker-Planck-based description of emergent phenomena at the population and network levels. Further abstractions of neurons and their synaptic weights will also be introduced to understand how patterns might be stored in attractor networks or learned in feedforward networks. These mathematical insights into natural and artificial neuronal networks will allow for a critical evaluation of whether currently proposed architectures reflect cognitive activity in the brain as we currently understand it.

Aims:

The module will cover:

• Mathematical models of the computational units of natural neuronal networks - stochastic synapses and neurons
• How detailed synaptic and neuronal models can be simplified and combined to describe activity at the level of coupled networks
• How these detailed models can be further simplified to develop abstract models of networks that can store patterns or learn from example

Objectives:

By the end of the module, students should be able to:

• Build testable mathematical models of physiological objects such as synapses and neurons
• Understand basic stochastic differential equations and develop probabilistic descriptions of their dynamics
• Understand how emergent phenomena in networks arise from the characteristics of their coupled lower-level components
• Construct abstract, artificial neuronal networks that can learn to recall or classify patterned inputs

 Books:

The course is self-contained; however, the following online books provide background reading:

Theoretical Neuroscience: Computational and Mathematical Modelling of Neural Systems, Dayan and Abbott

Neuronal Dynamics: From Single Neurons to Networks and Models of Cognition, Gerstner, Kistler, Naud and Paninski

Additional Resources