Mathematical Bioengineering
September 2007– August 2008

Workshops Page

Autumn Quarter: September - December 2007

Introduction to mathematical modeling in cellular physiology and neuroscience, October 1-4, 2007:

Speaker: David Terman and Greg Smith

Topics covered include: membrane transport and diffusion, classical biophysics of the squid giant axon, Markov chain models of single channel gating, cell signal transduction, the buffered diffusion of intracellular calcium, intracellular calcium responses, and excitability, bistability, oscillations, and bursting in a physiological context. We will also consider activity patterns in networks of synaptically coupled neurons, along with specific applications including models for sleep rhythms, Parkinsonian tremor and sensory processing.

Each topic will be studied from the perspective of nonlinear dynamics (either deterministic or stochastic). Mathematical idealizations of each phenomena will be constructed and then analyzed using computer simulation (numerical integration) and graphical techniques (phase- plane analysis).

Texts:

Computational Cell Biology: An Introduction to Computer Modeling in Molecular Cell Biology. Chris Fall et al., eds. 2002.

Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems. Peter Dayan and LF Abbott. 2001.

Mathematical Physiology. James Keener and James Sneyd. 1998.

Tutorial for Workshop 2, October 18-19, 2007:

Speaker: Keith Gooch

The first session will begin with a brief history of cell, organ, and culture from the early 1900s to the present and its relationship to modern efforts in cell and tissue engineering. The focus of this hour will be a survey of current and proposed applications of cell and tissue engineering. Using these applications as a starting point, the second hour will be a survey of the recurring approaches to (paradigms) and methods evident in CTE applications. The third hour will cover major challenges in CTE and some promising approaches to dealing with them.

Winter Quarter: Metabolic Engineering and Regenerative Medicine January - March 2008
Bio-Neuro Mechanics

Joint tutorials for Workshop 3 and 4: Introductory orientation on comparative biomechanics of locomotion:

Part 1. Tutorial for Workshop 3, January 10-11
Topics include:

• Muscle: muscle physiology
• Limb: dynamics of multi-body systems, passive walking
• Brain: feedback control and state estimation

Kurt A. Thoroughman, Washington University in St. Louis
Foundations of Neural Computation and Human Motor Behavior
Presentation materials: PDF

An initial consideration of quantification of the neural basis of human motor control can be quite attractive: people are easier to talk to than animals, people can perform motor tasks per the instructions of the scientist, and scientists can analyze the performance of people. The next steps, however, contain several conundrums, enigmas, paradoxes, and dilemmas. People dislike having electrodes driven into their brains; functional imaging techniques offer limited spatial and/or temporal resolution. Emergent observable human motor behavior integrates a motley stew of predictive and reactive cortical control, subcortical and spinal circuits, and musculoskeletal biomechanics. In this tutorial I will describe the origins of my prescription for addressing these issues via a computationally-intensive theoretically-and-neurophysiologically-inspired psychophysical approach. Wide-ranging retrospective, circumspective, and prospective questions and discussions are wholeheartedly encouraged.

Background articles:

Poggio T and Bizzi E. (2004) Nature, 431:768. http://www.ncbi.nlm.nih.gov/pubmed/15483597

Sanger TD. (2003) Curr Opin Neurobiol, 13:238. http://www.ncbi.nlm.nih.gov/pubmed/12744980

Part 2. Tutorial for Workshop 4, March 27-28
Topics include:

• Mechanics: walking and running models, hybrid dynamical systems, including numerical methods
• Neurobiology: CPG models and sensory circuits state estimators
• Control and co-ordination: feedback and feedforward control

These tutorials will link with the neuroengineering workshops.

Tentative schedule:

Topics:

• Experimental methods of video tracking -- some of the tracking and filtering tools that we used that biologists are less familiar with, such as Kalman filter variants.
• Phase estimation details, with a more "hands on" orientation.
• Application of phase estimation to control hypothesis testing.
• If time allows, some details of the numerical methods I'm using with Prof. Guckenheimer -- a "methods section" for the joint talk in the workshop.

• Basic mathematical ideas: hoppers and hybrid dynamical systems.
• Piecewise holonomic constraints and partial asymptotic stability.
• Passive SLIP and LLS models.
• Muscle models.
• Bursting neurons and coupled oscillators as CPG models, phase reduction, phase response curves and averaging. Illustrated by some matlab simulation demos.

• Biological sensors and sensorimotor processing relevant for locomotion, organizational principles of CPG networks.

Spring Quarter: April - June 2008
Neuro Engineering and Mechanical Imaging

Tutorial for Workshop 5, May 8-9: Brain physiology related to movement control and epilepsy (Thursday/Friday before the workshop)

List of Participants

• Intracortical unit recording studies of normal movement
• Field potential recording studies of normal movement
• Deep brain structures and movement disorders
• Physiology and epilepsy

Paul Nunez, Ph.D., Emeritus Professor of Biomedical Engineering, Tulane University, New Orleans, LA
Fundamentals of the Relationships Between Brain Activity and EEG: Large Scale Brain Physics and Neocortical Dynamic Correlates of Conscious Experience

Spatial-temporal patterns of scalp recorded potentials (electroencephalography or EEG) are determined by the dynamic behavior of current sources in cerebral cortex and volume conduction through head tissue. Volume conduction is governed by a macroscopic version of Poisson's equation, whereas cortical source dynamics originates with delay mechanisms characterized as "local" (e.g., postsynaptic potential rise times) or "global" (finite speed of action potential propagation in cortico-cortical fibers).

All measures of brain function (fMRI, PET, etc.) are highly selective, for example, electrophysiological data recorded from inside the skull are scale-dependent, sensitive to electrode size and location. Scalp potentials are dominated by "synchronized" (phase locked) cortical sources facilitated by cortical anatomy and physiology. Cortical sources of scalp potentials are most conveniently expressed at the mesoscopic spatial scale as current dipole moment per unit volume. The integrated product of this "meso-source" with the head Green's function determines scalp potential.

Human behavior and cognition are believed to originate with cell assemblies (neural networks) embedded in the synaptic source fields that generate EEG. Based on their apparent importance to EEG dynamics, healthy brains may require the following: non-local interactions via cortico-cortical fibers, nested hierarchical structure of cerebral cortex, resonant interactions between cell assemblies at multiple scales, and a proper "balance" between functional segregation and integration controlled by (chemical) neuromodulators.

Rachael Seidler, University of Michigan, Department of Psychology, Division of Kinesiology, Neuroscience Program, & Institute of Gerontology
Fundamentals of Motor Control Theory and Underlying Neuroanatomy
PDF1, PDF2

In this tutorial session, I will cover basic motor control theory and neuroanatomy of the motor system. We will discuss methods for measurement of human movement and brain activity, with particular emphasis on techniques that are relevant for brain machine interfaces. Attendees should gain a working understanding of forward and inverse motor control models, efferent copy, state estimation, and their underlying neural correlates. If time permits, we will then delve further into motor system neuroanatomy, including the motor cortical areas (parietal cortex, premotor, supplementary, and cingulate motor areas) as well as basal ganglia thalamocortical loops.

William Stacey, Departments of Epilepsy and Bioengineering, University of Pennsylvania
Bringing Clinical EEG into the 21st Century

Clinical epileptology relies heavily on EEG for diagnosis and treatment. Current practice with EEG is based on 80 years of experience, and has derived from visual classification of the voltage patterns produced by patients with and without epilepsy. One interesting result of this method is that much of clinical EEG is based on recognition of patterns that are poorly understood physiologically. There are many EEG waveforms that have only recently been reconciled with physiology, and many more that are still unexplained. Paradoxically, epileptic seizures are one condition for which the physiology is still poorly understood. Seizure classification, therefore, is a subjective measure that relies on visual inspection and comparison with known patterns and with the patient's "typical background." A seizure is a waveform that a) deviates from the norm b) evolves in frequency and location and c) has clinical or electrical characteristics of a seizure. The subjective nature of this process, as well as the heterogeneity of seizures, makes automated seizure detection a difficult endeavor. An even more difficult problem is seizure prediction, in which early seizure biomarkers might be identified long before the actual seizure begins. Modern EEG equipment now is capable of performing complex analyses and sampling at much higher rates, opening new avenues for analysis that had never been accessible to clinicians. While clinical practice has only begun to utilize this new technology, there are tools from mathematics, engineering, and machine learning that provide intriguing new methods to tap in to this new information.