Feed Sideward Understanding Biological Rhythms: Feed Sideward Understanding Biological Rhythms Jim Holte
1/15/2002
Sessions: Sessions Session 1 - Feed Sideward – Concepts and Examples, 1/15
Session 2 – Feed Sideward – Applications to Biological & Biomedical Systems, 1/31
Session 3 – Chronobiology, 2/12 Franz Hallberg and Germaine Cornalissen
Feed Sideward: Feed Sideward Terms Simple Example
Feed Back Reinvesting dividends
Feed Foreward Setting money aside
Feed Sideward Moving money to
another account
Introduction: Introduction Feed Sideward is a coupling that shifts resources from one subsystem to another
Feed Sideward #1 – feeds values of other variables into the specified variable
Feed Sideward #2 – feeds changes of parameters into the specified variable. (time varying parameters)
Feed Sideward #3 – feeds changes of topology by switch operations (switched systems)
Tool for global analysis
especially useful for biological systems
Phase Space: Phase Space Laws of the physical world
Ordinary differential equations
Visualization of Solutions
Understanding
Phase Space: Phase Space The Lotka-Volterra Equations for
Predator-Prey Systems
H' = b*H - a*H*P
P' = -d*P + c*H*P
H = prey abundance, P = predator
Set the parameters
b = 2 growth coefficient of prey
d = 1 growth coefficient of
predators
a = 1 rate of capture of prey per
predator per unit time
c = 1 rate of "conversion" of prey
to predators per unit time
per predator. Source: ODE Architect, Wiley, 1999
Phase Space: Phase Space The Lotka-Volterra Equations for
Predator-Prey Systems
H' = b*H - a*H*P
P' = -d*P + c*H*P
H = prey abundance, P = predator
Set the parameters
b = 2 growth coefficient of prey
d = 1 growth coefficient of
predators
a = 1 rate of capture of prey per
predator per unit time
c = 1 rate of "conversion" of prey
to predators per unit time
per predator. Source: ODE Architect, Wiley, 1999
Coupled Oscillators Model: Coupled Oscillators Model x and y represent the "phases“ of two oscillators.
Think of x and y:
angular positions of two "particles"
moving around the unit circle
a1 = 0
x has constant angular rate
a2 = 0
y has constant angular rate.
Coupling when a1 or a2 non-zero Source: ODE Architect, Wiley, 1999
Example Uncoupled Oscillators: Example Uncoupled Oscillators Source: ODE Architect, Wiley, 1999
Example Coupled Oscillators: Example Coupled Oscillators Source: ODE Architect, Wiley, 1999
Phase Resetting: Phase Resetting FUNCTION STIM(t,T1,T2,STIM_L,STIM_H)
STIM = PULSE_UP(t, T1, STIM_H) + PULSE_DOWN(t, T2, STIM_L)
RETURN STIM
END FUNCTION PULSE_UP(t, T1, STIM_H)
IF (t >= T1) THEN
PULSE_UP = STIM_H
ELSE
PULSE_UP = 0
ENDIF
RETURN PULSE_UP
END
FUNCTION PULSE_DOWN(t,T2,STIM_L)
IF (t <= T2) THEN
PULSE_DOWN = 0
ELSE
PULSE_DOWN = STIM_L
ENDIF
RETURN PULSE_DOWN
END
Example Phase Resetting: Example Phase Resetting Source: ODE Architect, Wiley, 1999
Oscillator Entrainment: Oscillator Entrainment Source: ODE Architect, Wiley, 1999 x and y represent the "phases“ of two oscillators.
Think of x and y:
angular positions of two "particles"
moving around the unit circle
a1 = 0
x has constant angular rate
a2 = 0
y has constant angular rate.
Coupling when a1 & a2 non-zero Entrainment occurs when the coupling causes
angular rate of x to
approach angular rate of y
x and y generally differ
Typical for Chronobiology
Dominant oscillator ‘entrains’ the other
Oscillator Entrainment Example: Oscillator Entrainment Example x' = w1 + a1*sin(y - x)
y' = w2 + a2*sin(x - y)
u = (x mod(2*pi)) //Wrap around the
v = (y mod(2*pi)) //unit circle
phi = (x - y)mod(2*pi)
Set the parameters
a1 =.0775*pi; a2 =.075*pi
w1 = pi/4; w2 = pi/4 - .14*pi Source: ODE Architect, Wiley, 1999
Singularities: Singularities r' = -(r-0)*(r-1/2)*(r-1) - a*STIM(t,T1,T2,STIM_L,STIM_H)
theta' = 1
x = r*cos(theta)
y = r*sin(theta)
T1 = 4
T2 = 6
a=0.0
STIM_L = -1
STIM_H = +1
Example - Singularities: Example - Singularities r' = -(r-0)*(r-1/2)*(r-1) - a*STIM(t,T1,T2,STIM_L,STIM_H)
theta' = 1
x = r*cos(theta)
y = r*sin(theta)
T1 = 4
T2 = 6
a=0.0
STIM_L = -1
STIM_H = +1 Run r a Commment
--- --- --- ---------
#1 1.25 0 approaches r=1
#2 1.0 0 stable periodic orbit
#3 0.75 0 approaches r=1
#4 0.5 0 unstable periodic orbit
#5 0.25 0 approaches r=0
#6 0 0 stable periodic orbit
#7 0.75 0.4 starts in r=1 domain,
STIM moves it to r=0 domain Source: Holte & Nolley, 2002
Feed Sideward: Feed Sideward Terms Simple Example
Feed Back Reinvesting dividends
Feed Foreward Setting money aside
Feed Sideward Moving money to
another account
Feed Sideward Example: Feed Sideward Example The Oregonator Model for Chemical
Oscillations
x' = a1*(a3*y - x*y + x*(1-x))
y' = a2*(-a3*y - x*y + f*z)
z' = x - z
smally = y/150
a1 = 25; a3 = 0.0008; a2 = 2500; f = 1 Source: ODE Architect, Wiley, 1999
Summary: Summary Feed Sideward is a coupling that shifts resources from one subsystem to another
Feed Sideward #1 – feeds values of other variables into the specified variable
Feed Sideward #2 – feeds changes of parameters into the specified variable. (time varying parameters)
Feed Sideward #3 – feeds changes of topology by switch operations (switched systems)
Tool for global analysis
especially useful for biological systems
Next Session: Next Session Session 1 - Feed Sideward – Concepts and Examples, 1/15
Session 2 – Feed Sideward – Applications to Biological & Biomedical Systems, 1/31
Session 3 – Chronobiology, 2/12 Franz Hallberg and Germaine Cornelissen
Backup: Backup
Feed Sideward - Topics (60 min): Feed Sideward - Topics (60 min) Session 1 (14 slides)
Background Concepts & Examples
Phase Space (1 slide)
Singularities (2 slides) *
Coupled Oscillators (2 slides)
Phase Resetting (2 slides) *
Oscillator Entrainment (1 slide)
Feed Sideward as modulation (3 slides) **
Summary (1 slide)
Session 2 (12 slides)
Applications to Biological Systems
Circadian & other Rhythms (2 slides)
Model & Simulation Result (2 slides)
Applications to Biomedical Systems
Blood Pressure Application (2 slides)
Model & Simulation Result (2 slides)
Summary (1 slide)
Segue to Chronobiology (1 slide)