Course Title: Introduction to Mathematical Methods and its Applications
Credit Hours: 2 (3 contact hours/week), In-person @NIAS Discussion Room (1st Floor)
Course Description:
Several disciplines of research such as neuroscience, climate research, population dynamics, economics, biomedical engineering, signal processing, communications etc. involve computational methods for analysing measurement data/time series. Methods include (but are not limited to) filtering, source localization, information processing, pattern recognition, regression, clustering, classification, causality testing, supervised and unsupervised learning methods, deep learning etc. Understanding the mathematical foundations behind these methods is vital for a sound and successful research program in those disciplines. This course attempts to address this need.
Learning Objectives:
1. To familiarize the student with the basics of mathematical methods and its applications
2. Upon completion, the student would be equipped to embark on research in the above mentioned domains by making use of the mathematical principles, computational/algorithmic methods learnt in the course.
Pre-requisites for registration:
The student is expected to be comfortable with MATLAB/Python programming. 12th grade mathematical background is expected.
Expected Student Workload:
3 contact hours/week (in-class, lecture style) and at least 6 hours/week of student preparation/study is expected. The course will have a heavy dose of MATLAB/Python assignments.
Course Duration:
Jan. - May 2024.
Modules
Module 0: Introduction to mathematical logic, deduction, induction, arguments, and proofs in mathematics, sets, functions, relations, basics of calculus, probability theory basics, (this would be more of a refresher module to help those students who had mathematics only till 12th grade) – 2 lectures
Module 1: 2x2 matrices – geometrical intuition, linear transformations, Vectors, Vector Spaces (VS), basic concepts of VS, y=Ax, covariance matrix, Singular Value Decomposition (SVD), Principal Component Analysis (PCA). MATLAB programming of PCA/SVD. Dimensionality reduction using PCA. Linear regression, Introduction to Independent Component Analysis (ICA), source localization – theory and applications on real-world data such as time series measurements (for eg., brain data). MATLAB/Python programming examples. ~ 3-4 lectures (PCA part delivered by Hema Karnam, PhD Scholar, NIAS)
Module 2: Foundations of Signal Processing - Signals and Systems basics, Analog vs. Digital, LTI systems, Convolution, Discrete Fourier Transform, Sampling theorem, Linear filters basics (low pass, high pass, band pass filter), denoising of brain signals. Filters used in time series analysis of neural signals (EEG, MRI etc.) MATLAB/Python programming examples. ~3 or 4 lectures
Module 3: Basics of Machine Learning, introduction to unsupervised (clustering) and supervised learning, dimensionality reduction methods, kNN, perceptron, logistic regression, SVM, Naïve Bayes, decision trees, random forest, artificial neural networks, deep learning, Neuroscience-inspired artificial intelligence. Python programming examples. ~3 lectures
Module 4: Basics of information theory (Shannon entropy, conditional entropy, mutual information, channel capacity) Information processing in neurons, spiking neuronal models, Hodgkin Huxley models, Integrate and Fire Neurons, Applications of Information Theory in neural coding and decoding, chaos in the brain. ~3 lectures (time permitting)
Basis for Final Grades
25% class participation and homework/programming assignments/in-class problem solving sessions
25% Mid-term examination (closed book)
50% Final examination (closed book)
Links to various materials
Module 4
Reference Material for the Modules
Strang, Gilbert. Introduction to linear algebra. Wellesley-Cambridge Press, 2022.
Haykin, Simon, and Barry Van Veen. Signals and systems. John Wiley & Sons, 2007.
Lyons, Richard G. Understanding digital signal processing, 3/E. Pearson Education India, 1997.
Rieke, F., Warland, D., Van Steveninck, R. D. R., & Bialek, W. S. Spikes: exploring the neural code. MIT press, 1999.
Jung, Alexander. Machine Learning: The Basics. Springer Nature Singapore, 2023.
Gerstner, Wulfram, et al. Neuronal dynamics: From single neurons to networks and models of cognition. Cambridge University Press, 2014.
Other research papers/reading material will be shared during the course.
