EN580.223 BIOLOGICAL MODELS AND SIMULATION
The Johns Hopkins University
Due February 6, 2013
Name (print):
Members of Study Group (or write, none):
Signature:
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HW 7 due 4/10
Exam I Mon 4/22
Previously: Stability of fixed points determines local properties of trajectories
Today:
Methods to describe global properties of trajectories
Last time: Nullclines to find fixed points
Poincar-Bendixson Theorem to Prove Exis
Bifurcation and Chaos in Iterated Maps:
HW 9 due Wed May 2
A Poincare Map is an iterated map that describes successive trajectory crossings of a lower
dimensional surface: x k +1 = P (x k )
x k + 2 = P(x k +1 ) = P( P(x k )
Poincare Map of Rossler system:
Iterated maps:
HW 9 due Wed May 1
Weve mostly been considering 2D systems, as higher dim systems are less amenable to phase
plane techniques. In 3D and higher the concept of a map is very useful.
Consider the system:
x = x + y x( x 2 + y 2 )
if r<1:
xy <
HW 8 posted, due 4/17
Exam I Mon 4/22
Nullclines to find fixed points
Poincar-Bendixson Theorem to Prove Existence of Closed Orbit / Limit Cycle
- if you can create trapping region with no fixed points and all trajectories going in
- all trajectories must
Biological Models & Simulations Midterm Exam
NAME:
EN580.223 BIOLOGICAL MODELS AND SIMULATIONS
The Johns Hopkins University
March 24, 2010
Name (print; also put your name on top-right of every page):
Number of pages, including the face page:
Use pen
Close
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DEPARTMENT OF BIOMEDICAL ENGINEERING
Course 580.223, Spring 2013
Biological Models and Simulations, Part I
Dr. Aleksander S. Popel, 611 Traylor Bldg., School of Medicine, tel. 410-955-6419, e-mail apopel@jhu.edu
Teaching Assistants:
Mark Walker <mwalke49@
Biological Models & Simulations Example Exam #1
_
Name:
EN580.223 BIOLOGICAL MODELS AND SIMULATIONS
The Johns Hopkins University
February 20, 2013
Name (print; also put your name on top-right of every page):
Number of pages, including the face page: 3 pag
HW6 Due Wed 4/3
1D Autonomous:
Special cases:
a)
b)
stable
FP:
f ( x* ) = 0
Stable: f ( x * ) < 0
Unstable: f ( x * ) > 0
f ( x) f ( x* ) + f ( x* )x + f ( x* )x 2 + O(x 3 )
c)
x = x3
unstable
x = f ( x, y )
y = g ( x, y )
det( J I ) = 0
x = Ax
x = f (x)
Membrane Electrical Behavior
Dynamics of excitable cells: Much of the interesting dynamics of cells involve excitation or
spikes of intracellular cellular electric potentials, important in:
signal propagation in neurons
coordinated contraction of cardiac
Nonlinear Positive Feedback Can Produce Switch-like Response
Positive Feedback: [A*] strenghtens to forward reaction rate
signal
[S ]
kon
A
koff
*
g ([ A ])
A*
nonlinear
function
kon = k + cfw_[ S ] + g ([ A* ])
one form of feedback
which leads to
g ([ A*
HW 7 posted, due 4/10
Exam I Mon 4/22
Kermack-McKendrick model of viral infection, now with limited (logistic) population growth
x = uninfected population
y = infected population
nonlinear infection rate kxy
x grows at rate r(1-ax)
growth rate ->0 at x=1/
Review for Final (cumulative over Beers half)
Mon 5/13 9am-12pm
calculator only
nth order NLDE system of n 1st order NLDEs
existence & uniqueness of solutions to NLDEs
solutions of Linear DEs are sums of exponentials, sum of any two solns is also a soluti
Department of Biomedical Engineering
BME 580.223 Biological Models and Simulations, Part II
Instructor: Mike Beer, 573 Miller Research Bldg, mbeer@jhu.edu
TAs: Shadi Toghi Eshghi shadi@jhu.edu, Hao Dang hao.dang@jhu.edu,
Barry Leybovich bleybov1@jhu.edu
M
1
Damped Driven Oscillator
m + x + kx = Fe (t)
x
Find the general solution, since this is a inhomogeneous di eq:
m + x + kx = 0
x
t
x = ae , x = aet , x = a2 et
m2 + + k = 0
Using the quadratic equation:
2 4mk
2m
Depending on what your parameters are, th
Review for Midterm Mon 4/22:
nth order NLDE system of n 1st order NLDEs
existence & uniqueness of solutions to NLDEs
solutions of Linear DEs are sums of exponentials, sum of any two solns is also a solution
numerical solutions of IVP
x = f ( x, y )
fixe