Scientific Theories of Consciousness – I: Mathematical Methods
Course Instructor: Dr. Nithin Nagaraj, NIAS Consciousness Studies Programme (Email: n i t h i n . n a g a r a j @ g m a i l . c o m )
Credit Hours: 3 hours/week (2 hours lecture + 1 hour lab session)
Course Description:
“Scientific Theories of Consciousness-I” is the first course of a two-part series. In “Part-I: Mathematical Methods”, we shall uncover the mathematical foundations that form the bedrock of several scientific theories of consciousness. Understanding ‘consciousness’ remains the final frontier of research and is increasingly becoming an interdisciplinary field of study with ideas and principles borrowed from several mathematical disciplines such as Information Theory, Signal Processing, Time Series Analysis, Chaos Theory, Complexity Measures, Brain Imaging Analysis, Network & Graph Theory. This course will equip the student with mathematical methods required to undertake basic research in scientific theories of consciousness.
Learning Objectives:
The primary objective of this course is to equip the student with the required mathematical methods, principles and techniques in order to undertake research in scientific theories of consciousness which is the subject matter of Part-II of this course to be offered in the next semester. The mathematical skills needed to build, analyze and rigorously evaluate a scientific theory of consciousness will be the key learning of this course.
Pre-requisites for registration/auditing:
Familiarity with elementary set theory and calculus with an interest in mathematics is a must. It is highly recommended that the student be comfortable with any one computer programming language of her choice (MATLAB/Python/C/any-other-equivalent-computer-language). This course will be intensive in mathematical reasoning and programming. Students will solve assignments that involve mathematical and logical thinking (including writing mathematical proofs), as well as writing computer programs as an aid to understanding the mathematical principles.
Expected Student Workload:
There will be a 2-hour lecture session and 1-hour lab session every week. The lecture session will introduce the various mathematical principles. The lab session will involve problem solving, writing mathematical proofs as well as writing computer programs. Assignments (both graded and ungraded, reading and writing) will be given extensively throughout the course.
Course Duration:
August-November 2016
Lecture Topics and Discussion
Module 1
Introduction to scientific theories of consciousness, methods of science, the role of mathematics in scientific theories with emphasis on its role in cognitive science, neuroscience, and existing scientific theories of consciousness; the unreasonable effectiveness of mathematics, and limits of mathematical reasoning; a bird’s eye-view of the various mathematical methods needed for understanding scientific theories of consciousness, the logic of quantifiers.
Module 2
Basics of linear algebra: “y=Ax”, the four fundamental spaces of linear algebra, vector spaces and linear transformations, foundations of singular value decomposition and principal component analysis; probability theory basics, introduction to random variables and stochastic processes, Markov processes, linear and nonlinear processes.
Module 3
Time series analysis basics, linear and nonlinear signal processing fundamentals, introduction to the four Fourier representations and its properties, basics of Wavelet transforms, signal processing algorithms used in neuroscience; introduction to non-linear dynamics/chaos theory and its applications in brain imaging analysis. Introduction to advanced techniques such as compressed sensing, signal processing on graphs, and dynamics on networks; basics of biostatistics, hypothesis testing, interpretation of statistical tests, and the role of statistics in scientific theories of consciousness.
Module 4
Introduction to Information Theory and complexity measures, Shannon’s coding theorems, role of information theory in the biological sciences (with emphasis to neuroscience and cognitive science), different notions of information (extrinsic, intrinsic, semantic, double-aspect information and quantum information); introduction to various measures of consciousness (such as causal density, neural complexity, differentiation-integration measures of brain complexity and dynamics, perturbational complexity index and others); introduction to Tononi’s Integrated Information Theory of Consciousness. Note: These theories will be exhaustively and rigorously dealt in Part-II of the course (to be offered in the next semester).
Course notes (scribed by Aditi Kathpaalia)
Basis for Final Grades
Class Participation and Interaction: 10%
Take-home Assignments: 30% (reading+writing, weekly)
Mid-term Exam: 30%
Final Exam: 30%
Books and References
This being a foundational mathematical methods course, there is no single text-book. Suitable handouts, papers and references will be provided for each topic. The following books (not exhaustive) are useful as references for the different modules.
Module 1:
Courant, R., Robbins, H., & Stewart, I. (1997). What is Mathematics? An elementary approach to ideas and methods (2nd ed.). USA: Oxford University Press.
Module 2:
Strang, G. (2007). Linear algebra and its applications (4th ed.). CENGAGE LEARNING (RS).
Papoulis, A., & Pillai, S. U. (2002). Probability, random variables, and stochastic processes. Boston: McGraw-Hill.
Module 3:
Haykin, S. S., & Van, V. B. (2002). Signals and systems. New York: Wiley.
Kantz, H., & Schreiber, T. (2003). Nonlinear time series analysis. Cambridge: Cambridge University Press.
Alligood, K. T., Sauer, T. D., Yorke, & J.A. (2000). Chaos: An Introduction to Dynamic Systems. Springer.
Module 4:
Cover, T. M., & Thomas, J. A. (1991). Elements of information theory. New York: Wiley.
MacKay, D. J. (2003). Information theory, inference, and learning algorithms. Cambridge, UK: Cambridge University Press.
Tononi, G. (2012). Phi: A voyage from the brain to the soul. New York: Pantheon.
Oizumi, Masafumi, Albantakis, Larissa, Tononi, & Giulio. (2014). From the Phenomenology to the Mechanisms of Consciousness: Integrated Information Theory 3.0. Public Library of Science.