# MA4G4 Introduction to Theoretical Neuroscience

**Not running in 2017/18.**

**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

**Leads To: **

### Location and times

** When and where for year 2016/2017 **

Academic weeks 15-24

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.