# MA4G4 Introduction to Theoretical Neuroscience

Lecturer: Magnus Richardson

Term(s): Term 2

Status for Mathematics students: List C for Mathematics

Commitment: 30 one-hour lectures

Assessment: 3 hour exam.

Prerequisites: Calculus and standard methods for the solution of differential equations. Basic knowledge of stochastic calculus (Langevin, Fokker-Planck and master equations) and probability theory would be an advantage

### Location and times

When and where for year 2016/2017

Mondays: 11am in B3.01
Wednesdays: 11am in D1.07
Thursdays: 11am in MS.04

Start: Monday 9th January 2017.
End: Thursday 16th March 2017.

Exam
Duration: 3 hours. No calculators allowed.
Date: To be confirmed
A Few Basic Equations

Past paper 2012 Questions
Past paper 2013 Questions
Past paper 2014 Questions
Past paper 2015 Questions
Past paper 2016 Questions

### Course details

Abstract
Mathematical Primer

Week 1 - 9th January - Basic electrophysiology

Intracellular voltage, capacitance, ionic currents, equilibrium potentials, Nernst relation, Goldman current, GHK equation, ohmic currents, resting potential, voltage equation, response to injected current waveforms, measures of capacitance and input resistance.
Lecture Notes - Questions - Answers

Week 2 - 16th January - Synaptic drive

Excitatory and inhibitory classes of synapses, AMPA, NMDA and GABA types, stochastic channel dynamics, vesicle-release statistics, PSCs and PSPs, synaptic depression.
Lecture Notes - Questions - Answers

Week 3 - 23rd January - Cable theory for passive dendrites

Derivation of the cable equation. Open and closed cables. The Rall soma-dendrite model. Calculation of total input conductance of complex dendritic trees. Decay of transients. Velocity of signals in passive structures.
Lecture Notes - Questions - Answers

Week 4 - 30th January - Subthreshold voltage-gated channels

Derivation of channel activation kinetics. Phase-plane analysis of the two-variable non-linear model. Positive feedback and bistability. Negative feedback and damped oscillations. Linearisation of the two-variable equation. Eigenvalues and phase diagram of stability.
Lecture Notes - Questions - Answers

Week 5 - 6th February - Models of spiking neurons

Hodgkin-Huxley spike-generating currents, anatomy of an action potential, two-variable reductions, excitability and spontaneous oscillations, theta/quadratic model, Fitzhugh-Nagumo model, Type I and Type II neurons.
Lecture Notes - Questions - Answers

Week 6 - 13th February - Integrate-and-fire models

Leaky, Exponential and Non-Leaky Integrate-and-Fire models, Type I and Type II integrate-and-fire models, bistability, spike-frequency adaptation.
Lecture Notes - Questions - Answers

Week 7 - 20th February - Synaptic fluctuations

Poissonian pulse arrival, Gaussian white noise models of conductance fluctuations, filtered conductance, voltage response to synaptic fluctuations, reduced response and shortened time constant in presence of synaptic input, voltage fluctuations, mean and variance.
Lecture Notes - Questions - Answers

Week 8 - 27th February - Populations of neurons

Fokker-Planck equation and current equation for a leaky IF neuron, derivation of the steady-state subthreshold voltage mean and variance, boundary conditions for the threshold case, integral form for the steady-state firing rate, the firing rate in various limits.
Lecture Notes - Questions - Answers

Week 9 - 6th March - Networks of connected neurons

Coupled single-population networks, mean-field and self-consistent solutions for the steady state, bistable excitatory networks and short-term memory, emergence of oscillations in inhibitory networks with delay, propagation of fronts of activity in neural tissue.
Lecture Notes - there are no questions this week

Week 10 - 13th March

Revision and exam questions.

Year 1 regs and modules
G100 G103 GL11 G1NC

Year 2 regs and modules
G100 G103 GL11 G1NC

Year 3 regs and modules
G100 G103

Year 4 regs and modules
G103

Past Exams
Core module averages