The practice of mathematical and computational modelling in neuroscience

1. The practice of mathematical and computational modelling in neuroscience Aapo Hyvärinen Depts of Computer Science and Psychology & Helsinki Institute for Information Technology University of Helsinki

2. Overview • Some examples of what computational neuroscience is all about • Questions that can be answered by modelling • What the word “computation” can mean in different contexts • Personal viewpoint on the very vague distinction between mathematical and computational modelling

3. The obligatory picture of the brain from Ellis, Logan & Dixon, 1991 “The brain is the most complex structure in the world ...”

4. Part I.Questions to be answered by modelling 1: What? • What is really happening in the brain? • Quantitative description ultimate goal in science? Features coded by neurons in the primary visual cortex Courtesy of Dario Ringach, UCLA

5. Questions to be answered 2: How? • How can something be computed in the brain? • Or: Is it possible to compute a given thing by a given neural system?

6. Questions to be answered 3: Why? • What are the computational goals of the brain? • Normative modelling: Given a computational goal, what should the brain be doing? • E.g. Adaptation to natural environment Vs. Two possible feature sets used in image analysis

7. Part II.Different meanings of “computational” • “Theory of computation”: Turing machines • Digital and serial • Computation as the basis of intelligence (classic AI) • “Computational science”: Floating-point operations • Groups of equations with real-valued variables • Computation (by computers) needed in modelling • Modern neuroscience: • Floating-point operations, in parallel, as basis for “intelligence” and behaviour

8. Part III.Difference between mathematical and computational modelling • System or goal is described as equations. • Sometimes you can solve them analytically, sometimes not. • Typically, you can solve part of them analytically • Reduce the solution to classic computation problem • Eigen-value decomposition • Classic differential equations • Approximations in extreme cases • Linear approximation near a stable point • Setting some parameters to, e.g., zero • Assuming that the world follows some simplified structure

9. Is there any difference? Mathematical A linear or quadratic equation Insufficient Simplest differential equations System of linear equations Maximize quadratic function Linear analysis around stable point Compromise Various approximations Special cases Most of interesting equations Incomprehensible Complete realistic models Computational

10. Conclusion • Computational neuroscience can have different goals: • What is computed ? • How can something be computed ? • Why should something be computed ? • Confusingly, “computation” in “computational neuroscience” does not always mean “computation” in the sense of neuroscience • Is computational science qualitatively new? Just an extension of pencil-and-paper mathematics?