MA4M915 Mathematics of Neuronal Networks
Introductory description
This module introduces the mathematics developed to understand how natural computation might emerge in neuronal tissue and how this has inspired and is inspired by artificial neuronal networks.
Module 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 simplfied to develop abstract models of networks that can store patterns or learn from example.
Outline syllabus
This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.
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 FokkerPlanckbased 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.
Learning outcomes
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 lowerlevel components.
 construct abstract, artificial neuronal networks that can learn to recall or classify patterned inputs.
Indicative reading list
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: https://www.gatsby.ucl.ac.uk/~lmate/biblio/dayanabbott.pdf
Neuronal dynamics: From single neurons to networks and models of cognition. Gerstner, Kistler, Naud and Paninski
https://neuronaldynamics.epfl.ch/online/index.html
Interdisciplinary
This module covers the mathematics/biophysics of neuronal networks with a link to artificial neuronal networks and so spans multiple departmental disciplines.
Subject specific skills
The module will teach you:
 how to build mathematical models of stochastic synapses and neurons .
 how to construct and solve basic stochastic differential equations.
 how to model the inputoutput transformation of neurons as a function of their physiology.
 how to mathematically predict the emergent states of coupled networks of neurons.
 how abstract models of neurons are used to construct artificial attractor and feedforward neuronal networks.
Transferable skills
The transferable skills taught are:
 the ability to construct testable mathematical models of realworld systems.
 understanding of stochastic differential equations and their construction.
 how artificial neuronal networks learn input patterns.
Study time
Type  Required 

Lectures  30 sessions of 1 hour (27%) 
Tutorials  10 sessions of 1 hour (9%) 
Private study  70 hours (64%) 
Total  110 hours 
Private study description
Students are expected to work through the lecture notes, prepare for the tutorials and prepare for and sit the exam.
Costs
No further costs have been identified for this module.
You must pass all assessment components to pass the module.
Assessment group B
Weighting  Study time  

Inperson Examination  100%  40 hours 
Centrally timetabled written exam.

Assessment group R
Weighting  Study time  

Inperson Examination  Resit  100%  
Centrally timetabled written exam. 
Feedback on assessment
Students will have access to previous exams for preparation and have feedback from the lecturer.
Courses
This module is Optional for:
 Year 1 of TMAAG1PE Master of Advanced Study in Mathematical Sciences
 Year 1 of TMAAG1P9 Postgraduate Taught Interdisciplinary Mathematics
 Year 1 of TMAAG1PD Postgraduate Taught Interdisciplinary Mathematics (Diploma plus MSc)
 Year 1 of TMAAG1P0 Postgraduate Taught Mathematics
 Year 1 of TMAAG1PC Postgraduate Taught Mathematics (Diploma plus MSc)
This module is Option list A for:
 Year 2 of TMAAG1PC Postgraduate Taught Mathematics (Diploma plus MSc)

UPXAFG31 Undergraduate Mathematics and Physics (MMathPhys)
 Year 4 of FG31 Mathematics and Physics (MMathPhys)
 Year 4 of FG31 Mathematics and Physics (MMathPhys)
 Year 4 of USTAG1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)
 Year 5 of USTAG1G4 Undergraduate Mathematics and Statistics (BSc MMathStat) (with Intercalated Year)
This module is Option list B for:
 Year 2 of TMAAG1PD Postgraduate Taught Interdisciplinary Mathematics (Diploma plus MSc)
 Year 4 of UCSAG4G3 Undergraduate Discrete Mathematics
 Year 5 of UCSAG4G4 Undergraduate Discrete Mathematics (with Intercalated Year)
 Year 3 of USTAG1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)
 Year 4 of USTAG1G4 Undergraduate Mathematics and Statistics (BSc MMathStat) (with Intercalated Year)
This module is Option list C for:

UMAAG105 Undergraduate Master of Mathematics (with Intercalated Year)
 Year 4 of G105 Mathematics (MMath) with Intercalated Year
 Year 5 of G105 Mathematics (MMath) with Intercalated Year

UMAAG103 Undergraduate Mathematics (MMath)
 Year 3 of G103 Mathematics (MMath)
 Year 3 of G103 Mathematics (MMath)
 Year 4 of G103 Mathematics (MMath)
 Year 4 of G103 Mathematics (MMath)
 Year 4 of UMAAG106 Undergraduate Mathematics (MMath) with Study in Europe