Bayesian Models, Dynamical Sytems and the Brain
This lecture series was given in the Spring Term, 2011 at the Wellcome Trust Centre for Human Neuroimaging at UCL. The idea was to juxtapose two perspectives on human brain function:
Bayesian Inference: Lectures 1 to 5 (and 11)
Dynamical Systems: Lectures 6 to 10
All lecture notes in a single document with extra material on control theory and mathematical appendices (PDF).
1. Bayesian Inference
1. Bayesian Inference
- Bayes rule for Gaussians. Directed Acyclic Graphs
- Joint Densities and Marginalisation. Medical Decision Making.
- Perception as statistical inference. Decision Making Dynamics.
- Bayesian Sensory Integration. Explaining Away.
2. Empirical Bayes
2. Empirical Bayes
- Linear Models. Maximum Likelihood. fMRI analysis.
- Delta rule learning. Newton method.
- Weighted Least Squares. Marginal Likelihood.
- MEG source reconstruction. Empirical Bayes.
- Restricted Maximum Likelihood (ReML)
3. Sparse models
3. Sparse models
- Relevance Vector Regression. Automatic Relevance Determination.
- Recurrent Lateral Inhibition. Linear predictive coding. Hebbian Learning.
- Sparse coding of natural visual images. MAP learning. Cauchy priors.
4. Hierarchical Models
4. Hierarchical Models
- Linear Hierarchical Models. Superficial versus Deep Cortical Laminae.
- Hierarchical predictive coding. Increasing receptive field size. Simple cells.
- End Stopping. Nonlinear hierarchical models
5. Approximate Inference
5. Approximate Inference
- Information, Entropy,Source Coding Theorem, Prefix Coding,
- Kullback-Liebler Divergence, Asymmetry,
- Multimodality,Variational Free Energy, Factorised Approximations,
- Mean Field Approach, Free-Form versus Fixed Form,
- Nonlinear Regression, Adaptive Step Size, Approach to Limit.
6. The Microscopic Brain
6. The Microscopic Brain
- Linear differential equations. Matrix Exponential. Eigendecompostion.
- Nodes, Saddles, Spirals, Centres. Feeback Inhibition. Stability.
- Local linear stability analysis. Isoclines. Nonlinear Oscillations.
- Spiking Neurons. Fitzhugh-Nagumo. Hodgkin-Huxley. Rose Hindmarsh.
- Hopf and Saddle-Node Bifurcations. Type 1/2 Cells.
7. The Mesoscopic Brain
7. The Mesoscopic Brain
- Integrate and Fire Neurons.Phase Reduction.
- Phase Response Curves. Weakly Coupled Oscillators.
- Synchronization via excitation or inhibition.
- Spike Frequency Adaptation (SFA).
- Spike Time Dependent Plasticity (STDP).
- Dynamic pattern recognition via transient synchronisation.
- Spike-to-Spike versus LFP synchronization.
8. The Macroscopic Brain
8. The Macroscopic Brain
- Neural Mass Model. Alpha activity. Spike cycles. Bifurcations.
- Hierarchical cortical connectivity. Delay differential equations.
- Dynamic Causal Models (DCM) for Event Related Potentials.
- Auditory oddball. Linear Systems Analysis.
- Network kernel function. Frequency response. Cross Spectral Density.
- DCM for steady state responses. Inferring synaptic physiology.
9. Stochastic Processes
9. Stochastic Processes
- Wiener, Ornstein-Uhlenbeck (OU) and Gaussian processes.
- Stochastic Differential Equations. Stochastic chain rule.
- Mean and variance functions.
- Rodriguez-Tuckwell method for multivariate Gaussian
- Approximation of population density evolution.
- Population of Fitzhugh-Nagumo neurons.
- Fokker Planck Equation. Population of stochastic integrate and fire cells.
- 2AFC tasks. Drift Diffusion Models.
- Reaction Times and Error Rates.
- Continuum limit of discrete time Bayesian model.
10. Hierarchical Dynamic Models
10. Hierarchical Dynamic Models
- OU(p) processes. Covariance functions. MEG alpha rhythms.
- Embedding and Generalised Coordinates.
- Continuous time latent variable models.
- Filtering. Variational Energies and Actions.
- Dynamic Expectation Maximisation. Linear Convolution Model.
- Hierarchical Dynamic Models.
11. Bayesian Model Comparison
11. Bayesian Model Comparison
- Posterior model probability, Bayes factors, Odds Ratios
- Model evidence. Free Energy. Approximation.
- Accuracy and Complexity Decompositions.
- AIC and BIC. Linear Models. fMRI example simulations.
- DCM for fMRI example. Priors. Simulations.
- Inference for groups of subjects. Fixed Effects.
- Random effects generative model. Gibbs sampling.
- Exceedance probabilities.
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