Mathematical concepts in the Neurosciences (NBB 470 / BIO 470)
MWF, 10:40a–11:30a; Old Dental School, R100A
Andrei Olifer
aolifer@ emory.edu, 404-727-3981, Rollins Research Center, R2172
Content: This course is intended for NBB (Neuroscience and Behavioral Biology) and
Biology majors interested in quantitative reasoning and mathematical modeling. Several
mathematical concepts fundamentally important in multiple areas of biology will be
considered. The concepts will include differential and difference equations, information
measures, stochastic processes, and others. The concepts will be introduced in the
context of specific problems in the neurosciences to demonstrate why and how these
concepts really work. The exemplary problems will be from neuronal coding, neuronal
network dynamics, and learning in neuronal networks. Mathematical concepts will be
introduced from “scratch”, on the basis of the high school mathematics. The course will
give a foundation for quantitative reasoning and mathematical modeling in the
neurosciences and biology in general. The development of the course was funded by a
Howard Hughes Medical Institute Fellowship.
Prerequisite: NBB 301/ Bio 360 will be a useful background; knowledge of calculus is
not required but is a plus.
Textbooks: 1) F.R. Adler. Modeling the Dynamics of Life. Brooks/Cole, 2005. 2)
P.Dayan and L.F.Abbott. Theoretical Neuroscience: Computational and Mathematical
Modeling of Neural Systems. MIT Press, 2001.
Particulars: Grades will be based on the three tests (15% each), the final exam (25%),
and the homework grade (30%). There will be a discussion section every week to
ensure understanding of the course material. This course will fulfill elective credit for
NBB and Biology majors.
Homework: Homework will be assigned daily and is to be turned in at the beginning of
class on the specified day. The homework will be given a score of 0-10 and returned
promptly.
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Neuroscience Topics and Mathematical Concepts
Neuronal Coding
- Spike codes
Mathematical concepts
Functions, graphs of functions, elementary statistics.
- Entropy and information in spikes
Mathematical concepts
Probability distributions, entropy, mutual information, stochastic processes, Poisson
process.
- Population coding
Mathematical concepts
Linear algebra (operations with vectors).
Neuronal and Network Dynamics
- Neuronal models
Mathematical concepts
ODEs and their solutions. Phase space.
- Steady state neuronal dynamics
Mathematical concepts
Stable and unstable steady states of dynamical systems. Bifurcations.
- Periodic neuronal dynamics
Mathematical concepts
Stable and unstable periodic regimes of dynamical systems. Bifurcations.
- Neuronal network dynamics
Mathematical concepts
Lyapunov function.
Learning in Neuronal Networks
- Synaptic plasticity
Mathematical concepts
Time averaging. Linear algebra.
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