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 web page

Module aims

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 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.

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 lower-level 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 input-output 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 real-world systems.
• understanding of stochastic differential equations and their construction.
• how artificial neuronal networks learn input patterns.