Viktoria R.T. Hsu
research statement
My primary research interest lies in mathematical biology and, more specifically, in mathematical
medicine. Beyond this area I have a general interest in dynamical systems, perturbation solutions to
ordinary and partial differential equations, scientific computing, and time series analysis. My current
research focuses on the study of electro-diffusion in application to the mechanistic modeling of electric
signal generation and modulation processes in excitable cells. I also analyse experimental data in
order to verify the qualitative and quantitative accuracy of mathematical models, and to illuminate
statistical or spectral properties of the data. In past projects, I have been exposed to the study of
reaction kinetics, metabolic networks, and signal transduction. In the following, I will briefly introduce
the specific modeling and data analysis used, including an outlook toward the future of each project:
1 Mathematical Modeling of Biological Cells in Tissue
The electro-diffusion and Poisson’s equations, also called the semiconductor device equations, model
charge-carrier transport in many fields of applications ranging from semi-conductor devices and industrial ion-exchange membranes to biological cell membranes. My interest lies, in particular, in the
accurate modeling of the electro-static potential difference as well as ion transport across biological
membranes. In this endeavor, the electro-diffusion and Poisson’s equations in 1D model charge carrier
concentrations and electro-static potential, respectively. In common notation for the concentrations,
ci , of species i and the normalized electro-static potential, ϕ,
X
∂ci
∂
∂ci
∂ϕ
∂
∂ϕ
=
Di
+ zi
ci
,
ε
=−
zi ci ,
(1)
∂t
∂x
∂x
∂x
∂x
∂x
i
m
in which the domain of interest, −L ≤ x ≤ R, contains a membrane of finite width in − m
2 ≤ x ≤ 2.
Under appropriate assumptions, the QSSA (quasi steady-state approximation) of the problem emerges
from a singular perturbation study of (1), and the system of PDEs (partial differential equations)
reduces to a system of ODEs (ordinary differential equations) and PDEs,
dciin
i
= −αin
Ji ,
dt
dciout
i
= αout
J i , and
dt
(2)
Ji
m
m
= const. for −
≤x≤ ,
(3)
i
D
2
2
i
where αin,out
are constants determined by physical properties of the system, ci (x) = ciin for −L ≤ x <
m
i
i
−m
2 , and c (x) = cout for 2 < x ≤ R. See [1] for details on the derivation and analytical results.
cix + z i ϕx ci = −
Most mathematical models for signal generation in single neurons are derived from the classic HodgkinHuxley model, and assume that the single neuron is bathed in an infinitely large buffer solution. Thus
the composition of the bath never changes. This assumption is appropriate for the comparison of
model results to in vitro studies because in these studies the cell preparation is actually bathed in a
relatively fixed environment. In their current state, such models are not able to take into account large
changes in the external environment of a cell during, e.g., ischemia (lack of metabolic supply).
The QSSA improves upon current neuron models in the sense that it models ion transport accurately,
and is able to take into account changing extracellular conditions in a single cell micro environment.
The necessity for this improvement is considering mass conservation and electro-neutrality, both of
which are incorporated in the QSSA but neither of which are incorporated in Hodgkin-Huxley type
models. Thus the classic Hodgkin-Huxley theory does not provide an accurate model of ion transport.
Despite this lack it achieves what it was designed for, that is an excellent fit to the action potential.
One possible application of the QSSA lies in the study of epilepsy, a complex neural disease in which
small groups of neurons with abnormal behavior are believed to entrain larger groups of neighboring
neurons. Once a critical number of neurons are entrained in abnormal behavior, physical consequences,
such as epileptic seizures, can be observed. Seizures have been linked to ischemia in brain tissue, during
which neurons swell, and certain ionic concentrations in the extracellular environment of the neurons
undergo large transient detours from their normal ranges.
1
Viktoria R.T. Hsu
research statement
The processes and mechanisms underlying the entrainment are still not completely understood. It is my
hope that improved modeling will clarify this problem, and open paths to more advanced treatments.
Simulation of the QSSA is achieved by the numerical solution of a system of DAEs (differential algebraic
equations) based upon (2) and (3). This has the advantage of demanding far less computation time
than the full PDE, (1), but requires the efficient numerical solution of the PNP (Poisson-NernstPlanck) system formed by (3) in connection with Poisson’s equation. The latter is accomplished by a
method developed in [2]. In contrast to previous approaches to solving the PNP system, as reviewed
in [3, 4], I use Neumann rather than Dirichlet boundary conditions on the electro-static potential.
This new approach is truly in the spirit of non-invasive techniques, since it solves for the potential
difference across the membrane instead of prescribing it. It also treats electro-neutrality in a natural,
self-regulatory way instead of explicitly enforcing it.
In the near future, I would like to consider cell swelling during ischemia by incorporating cell volume
dynamics due to osmotic forces, and apply the QSSA widely to in tissue modeling. In the far future of
this project, I see the possibility for connecting the electrical, chemical, and other metabolic signaling
processes in biological cells. This connection constitutes a key step in the development of accurate
models for perpetual, living cells and tissues.
2 Temporal Changes in Spectral Properties of Pre- and Interictal Human EEG
The main treatment options for epilepsy to date are restricted to a few medications or potential
epilepsy surgery, in which a focal center of seizure generation is removed from the brain of the patient.
In recent years, much effort has been made to develop a functional brain defibrillator, whose purpose
is the abortion or prevention of the occurrence of a seizure either directly prior to or at its onset.
The major challenge in this approach has been the automatic detection, or even prediction, of the
occurrence of epileptic seizures. To date, the best and most accurate means of locating the onset of a
seizure remains the visual inspection of an electroencephalogram (EEG) by a trained physician after
occurrence of the seizure.
In our collaboration, Mark Holmes, MD of the Harborview Medical Center, supplied me with intracranial EEG recordings of several patients and visually inspected this data for me. In recent work, I
analysed the time course of spectral properties of the data, and successfully developed some seizureindicative measures. The spectral information was obtained using the Short Time Fourier Transform
(STFT), the Stockwell Transform (ST), and, in collaboration with Dr. K. Coughlin of Northwest
Research Associates, Inc. (Bellevue, WA), the Empirical Mode Decomposition Method (EMD).
In the future, I shall continue and expand the studies on longer data sets to ensure that indicators of
seizure onset create no or very few false-positive alarms. This includes, within reason, the tailoring of
seizure detection to particular patients. I would also like to investigate the re-connection of spectral
seizure-indicators to the original data. This may lead to more efficient ways of computing seizureindicative measures, such as ways of computing these measures from the data directly instead of
having to first transform or decompose it.
[1]
Hsu V, Qian H. Asymptotic Analysis of Relaxation Kinetics to Donnan Equilibrium. SIAM Journal of Applied Mathematics (submitted 12/01/2003).
[2]
Hsu V. Almost Newton Method for Large Flux Steady-State of 1D Poisson-NernstPlanck Equations. Journal of Computational and Applied Mathematics (submitted
7/21/2003).
[3]
Kosina H, Langer E, Selberherr S. Device Modeling for the 1990s. Microelectr J
26 (2-3): 217-233 MAR 1995.
[4]
Pinnau R. A review on the quantum drift diffusion model. Transport Theor Stat
31 (4-6): 367-395 2002.
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