# The IV AMMCS International Conference

## Waterloo, Ontario, Canada | August 20-25, 2017

# AMMSCS 2017 Plenary Talk

## The Mathematical Limits of Sensory Integration and Control

### André Longtin (University of Ottawa)

Theoretical studies of brain function have led to advances in a
number of areas of applied mathematics. Recent advances are
highlighted here in the context of sensory systems. These
systems provide a particularly good window onto questions of
neural dynamics, control and information processing, because
they have a well-characterized physical input. This talk will
first present recent work on information processing at the
limits of sensory detection. The probabilistic encoding of
sensory signals as modulations of intrinsically correlated
neural point processes is shown to enhance their detectability
and information content as sequences of firing or “spike”
times. Downstream neurons further aim to derive optimal
information about the position of objects in the
environment. Their sensitivity is found to peak at a
characteristic distance where the Fisher information is
maximized. This result relies on a correction to the Fisher
information for the positive correlations observed in the
stochastic firing patterns. It provides a first mathematical
description of a sensory “focus”, and this focal point
corresponds to the transition from periodic to chaotic
dynamics. Finally we discuss a stochastic optimal control
problem to precisely target the spike times of a leaky
integrate-and-fire (LIF) model of a neuron with noise. Such a
model is in fact an Ornstein-Uhlenbeck process with an absorbing
boundary, for which we are controlling the mean first passage
time. The optimal control problem is solved using dynamic
programming when the controller has access to the voltage
(closed-loop control), and using a maximum principle for the
transition density when the controller only has access to the
spike times (open-loop control).

André Longtin is the University Research Chair in Neurophysics
at the University of Ottawa, as well as Chairman of its Physics
Department. He runs the Neurophysics and Nonlinear Dynamics
Group, working at the interface of physics, applied mathematics,
biology and medicine. His main interests lie in theoretical and
computational neuroscience and the interaction of deterministic
systems with noise. He received an honours B.Sc. Physics in 1983
and M.Sc. Physics in 1985 from the Université de Montréal, and
his Ph.D. in Physics from McGill University in 1989. He joined
Los Alamos National Laboratory as an NSERC Postdoctoral Fellow
and a Los Alamos Director's Funded Fellow, where he held a joint
position in the Theoretical Division T13 (Complex Systems) and
the Center for Nonlinear Studies. He began as assistant
professor of Physics in 1992 at the University of Ottawa. He is
Professor since 2002, and cross-appointed to the Departments of
Cellular and Molecular Medicine and of Mathematics and
Statistics. He is founding co-director of the University of
Ottawa Center for Neural Dynamics and a Fellow of the American
Physical Society. He sits on the editorial board of Biological
Cybernetics, Cognitive Neurodynamics, Bulletin of Mathematical
Biology, Journal of Mathematical Neuroscience and Frontiers in
Computational Neuroscience. He was awarded a Premiers Research
Award in 1999, the inaugural award for Interdisciplinary
research at U. Ottawa in 2004 (with Len Maler), a senior
Humboldt Research Prize in 2010, and the NSERC Brockhouse Canada
Prize in Interdisciplinary Research with Len Maler in 2017 for
uncovering key features of the neural code.