**Course content and aim**

This course is intended for mathematicians interested in neuroscience and mathematically-inclined computational neuroscientists. The emphasis will be primarily on the analytical treatment of neuroscience-inspired models and algorithms. The aim of the course is to equip students with a solid technical and conceptual background to tackle research questions in mathematical neuroscience.

The course will be structured in three blocks:

*Neural dynamics. *Neural computations emerge from myriads of neuronal interactions occurring in intricate networks that have evolved over eons of time. Due to the obscuring complexity of these networks, we can only hope to uncover principles for neural computations through the lens of mathematical modeling and analysis. The main theoretical challenge is to relate quantitatively structure and activity in a tractable way, i.e. to uncover hierarchies of low-dimensional representations for the activity of high-dimensional neural systems. In this block, we will present attempts made in that direction while introducing the mathematical formalisms associated to classical models of neural dynamics.

*Information theory. *To elucidate brain structure conceptually, it is tempting to look for “design principles” that would guide the development and the evolution of neural systems. Such a putative design principle is offered by the “efficient coding hypothesis”, which states that sensory systems have evolved to optimally transmit information about the natural world given limitations on their biophysical components and constraints on energy use. In this block, we will introduce the theoretical framework suitable for investigating the efficient coding hypothesis from a mathematical standpoint.

*Machine learning. *Machine learning has allowed the realization of speech recognition, language translation, natural-object recognition, and self-driving cars. These achievements, which rival human performance, are performed by neural networks that mimic many structural features of the brain and learn how to perform tasks via biologically inspired rules, such as reinforcement learning. However, the mathematical theory underlying this computational feats is still in its infancy. This block will present the mathematical theory supporting a few machine learning methods in supervised learning, in reinforcement learning, and in unsupervised learning.

**Time & Place**

Tuesday/Thursday 09:30 a.m. 11:00 a.m. @ RLM 10.176.

**Office hours**

Monday 2:00p.m.-3:00p.m. and Wednesday 12:00p.m.-1:00p.m. @ RLM 10.148.

**Syllabus**

**Course Schedule**

Date | Topics | Resources | Homework |
---|---|---|---|

01.22.2019 | Course introduction | Introduction | |

01.24.2019 | Hodgkin-Huxley model/Reduced models | Notes_HH, Spike_HH, Atypical_Spike, Threshold_Spike | Problem_Set1 |

01.29.2019 | Introduction to bifurcation theory | NotesBifurcation | |

01.31.2019 | Center manifold reduction | NotesCMTheorem | |

02.05.2019 | No class | ||

02.07.2019 | Normal form theory | NotesNormalForm | Reading assigned in class |

02.12.2019 | Bifurcation in neuroscience | HudspethMagnasco, KopelErmentrout | |

02.14.2019 | Intensity-based neural models | NotesIntensity | PillowPaninski |

02.19.2019 | Integrate-and-fire neural models | NotesIntegrateAndFire | |

02.21.2019 | Thermodynamic mean-field limits | NotesTMF | Brunel |

02.26.2019 | Replica mean-field limits | NotesReplica | Baccelli |

02.28.2019 | Differential geometry | NotesDiffGeo | |

03.05.2019 | Information geometry 1 | NotesInfGeo1 | |

03.07.2019 | Information geometry 2 | NotesInfGeo2 | Amari, Bialek1 |

03.12.2019 | Mutual information | NotesMI | Bialek2 |

03.14.2019 | Rate-distortion theory | NotesMIOpt | Blahut |

03.19.2019 | Spring Break | ||

03.21.2019 | Spring Break | ||

03.26.2019 | Information bottleneck | NotesIB | TishbyPereiraBialek, SchwartzZivTishby, Chechik |

03.28.2019 | Variational Fisher information | NotesVar | Ganguli |

04.02.2019 | Perceptron algorithm | NotesPerceptron | ProjectRefs |

04.04.2019 | Linear separability and combinatorics | NotesLinSep | Cover |

04.09.2019 | Reproducing-kernel-Hilbert space | NotesRKHS | Aronszajn |

04.11.2019 | Support-vector machine | NotesSVM | Vapnik |

04.16.2019 | Markov decision process | NotesMDP | |

04.18.2019 | Dynamic programming | NotesDP | Dijkstra |

04.23.2019 | Q-learning algorithm | NotesQL | RobbinsMonro, Dayan |

04.25.2019 | Reduction to linear Bellman equations | NotesLBE | Todorov |

04.30.2019 | Autoencoder networks | KingmaWelling, Radford, Bengio | |

05.02.2019 | Generative adversarial networks | ||

05.07.2019 | Projects | ||

05.09.2019 | Projects |