Design of a Simulation Toolbox of Intestinal Electrical Activity

John GoudaAdvisor: Alan Bradshaw, Ph.D.

Motivation of Project

The living state physics group have state of the art equipment to measure the magnetic field of the small intestine.

The magnetic field can be used to estimate the electric field: trans-membrane potential and slow currents.

Thus, the need arose for an educational tool to help the investigators understand the quantitative features of electrical activity in the small intestine.

The project can also serve as an educational tool for BME 251-252.

Experiment The following outlines the experimental procedures in the living state

physics group. The data was collected from rabbit small intestine using platinum

monopolar electrodes. 20 electrodes placed at 20 different location along the intestine 1 cm

apart. Data was sampled for 5 minutes per study at a rate of 20-30 Hz. Data was sampled at the three section of the small intestine

(duodenum, jejunum, ileum) and during induced ischemic conditions. The clinical goal of the study is to investigate the difference between

healthy electrical activity and pathologies: diabetic gastroparesis, mesenteric ischemia.

Squid: recording device

Design SpecificationsUser Demands

The simulation toolbox should: 1. Give the user an intuitive grasp of GI electrical activity 2. Rely on an accurate, intuitive model of GI electrical

activity 3. Relate model simulations with experimental data

Project SpecificationsUser Wishes

If possible, the simulation toolbox should: 1. Have a user-friendly interface 2. Provide the user with “on the spot” calculations and metrics

that represent the response of the model to the parameters supplied to the toolbox. Provide the user with a measure of goodness of fit with experimental data.

3. Provide the user with “on the spot” graphics that represent the response of the model to the parameters supplied to the toolbox

4. Provide the user with analysis modules that can analyze the complexity of GI activity and provide intuition into the physiologic function of GI tract.

Mathematical Background 1960 Nelson and Becker suggest that a chain of relaxation

oscillators (RO) could simulate GI electrical activity. 1971 Sarna et al. Used a modified version of the Van der Pol

oscillator to simulate GI electrical activity. We used the widely acclaimed model developed by Sarna (1971).

Relaxation Oscillators The name “relaxation oscillators” comes into place because the

“stress” accumulated during the slow buildup of charge on the membrane is “relaxed” during the sudden depolarization of the action potential.

Our model We used the widely acclaimed simple model developed by

Sarna (1971). The model consists of 16 coupled nonlinear oscillators. The

oscillators are lined along a straight line in the intestine. The user can gain an intuitive feel for the effect of model

parameters on the collective behavior of the model by utilizing the GUI interface.

The model is not intended to entirely reproduce quantitative features of GI electrical activity, but rather to show the basic characteristics qualitatively (Sarna, 1971).

Equations of the model16 coupled oscillators, each oscillator has the form

dxn/dt = alpha (e yn + f xn + g xn^2 + h xn^3 + C1 xn-1 + C2 xn+1)

dyn/dt = -1/alpha (b yn + w^2 xn + c xn^2 + d xn^3 - a)

xn= transmembrane potential

yn= slow currents

w= frequency of oscillation if the system was reduced to an undamped, unforced harmonic osc.

C1 = coupling constants with previous oscillator

C2= coupling constant with next oscillator

n = number of oscillators

Equations of Simple ModelsVan der pol

can be transformed totwo forms-->Fitzhugh

0)1( 2

xxxx

/)(

)3(*3

byaxy

zxxyx

xy

xxyx

)3(*)1(3

xyxy

yx

22 )1(

)2(

Converting a 2nd order differential equation into 2 first order differential equations

Undamped, unforced harmonic osc.

b=dampening factorm=massk=spring constantw^2=k/mwhere w=frequency

0

xm

kx

m

bx

Introduce a new variable y such that y=dx/dt

do that math and you get

xm

ky

m

by

yx

Connection with simple models The above system can be reduced to the following systems: Unforced Damped harmonic oscillator in eq. 1: set alpha=e=1,f=g=h=0, c1=c2=0 in eq. 2: set w=frequency, c=d=a=0 add a term b*y in eq.2 where b=dampening factor/mass Unforced Van der Pol oscillator in eq. 1: set e=1, f=1, g=0, h=-1/3, c1=c2=0 in eq.2: set w =frequency, c=d=a=0 Fitzhugh oscillator in eq.1: set e=1,f=1,g=0, h=-1/3, c1=c2=0, add a term z in eq. 2: add a term b*y, set w=frequency, set c=d=0

in all models, use c1 and c2 as coefficients of a forcing function

Schematic of the Model

C1=forward coupling factor C2=backward coupling factor

DemoOnly Two CommandsAt the MATLAB prompt:type ‘playmodel’ to run the simulationtype ‘plotmodel’ to plot a 3D graph of the resultstype ‘runvdp’ to simulate the Van der pol

oscillatortype ‘runharmonic’ to simulate the harmonic

oscillator

Conclusions

GI electrical activity may be modeled with coupled nonlinear oscillators

MATLAB provided a user-friendly environment that facilitated creation of a user-driven simulation and educational tool

Snapshots of the phase space of each oscillator reveal differences between the dynamics of each oscillator

Model results agree qualitatively with experimental data (Sarna, 1971)

References

Sarna SK, Daniel EE, Kingma YJ (1971): Simulation of slow-wave electrical activity of small intestine. Am J Physiol, 221(1):165-75.

Aliev RR, Richards W, Wikswo JP: A simple nonlinear model of electrical activity in the intestine. (in press)

Acknowledgements

Many thanks to:Jimmy Nguyen, B.E., VU BMERubin Aliev, Ph.D., VU PhysicsJohn Wikswo, Ph.D., VU PhysicsWilliam Richards, M.D., VU SurgeryAllison Redmond, B.E., VU Physics