feed sideward pres1

Feed SidewardUnderstandingBiological 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

ExampleUncoupled Oscillators: 

Example Uncoupled Oscillators Source: ODE Architect, Wiley, 1999

ExampleCoupled 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

ExamplePhase 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)