Guangyuan Liao , Casey Diekman and Amitabha Bosegl92/Files/Dynamics_Days_2020_Poster.pdf ·...
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Entrainment dynamics of forced hierarchical circadian systems. Guangyuan Liao † , Casey Diekman † and Amitabha Bose † † Department of Mathematical Science, New Jersey Institute of Technology Entrainment dynamics of forced hierarchical circadian systems. Guangyuan Liao † , Casey Diekman † and Amitabha Bose † † Department of Mathematical Science, New Jersey Institute of Technology Introduction The ability of circadian oscillators to entrain to light-dark cycles is well known. The process of circadian entrainment has been studied by various methods from dynamical systems. Recently, a tool called the entrainment map was introduced to analyze the entrainment process [1]. We developed a 2-D entrainment map to study coupled circadian oscillators. The entrainment map is quite effective to study the time and direction of entrainment. Model We first consider the situation in which a single central oscillator receives light-dark input. In turn, this central oscillator sends input to a single peripheral oscillator. To focus on the mathematical aspects of the derivation and analysis of the 2-D entrain- ment map, we will utilize the planar Novak-Tyson model [3] for both the central and peripheral oscillators. Fig. 1: Schematic figure of the model. 1 φ 1 dP 1 dt = M 1 - k f h(P 1 ) - k D P 1 - k L 1 f (t)P 1 1 φ 1 dM 1 dt = [g (P 1 ) - M 1 ] 1 φ 2 dP 2 dt = M 2 - k f h(P 2 ) - k D P 2 1 φ 2 dM 2 dt = [g (P 2 ) - M 2 + α 1 M 1 g (P 2 )]. Where g (P )= 1 1+P 4 ; h(P )= P 0.1+P +2P 2 and f (t)= Heaviside(sin( π 12 t)) Construction of the 2-D entrainment map Instead of studying the original five dimensional model directly, we reduced the model as a two dimensional discrete phase model. The method is a combination of the Poincare map and phase reduction. The reduced system is written as a two dimen- sional map: x n+1 =Π 1 (x n ,y n )= Arg [ϕ ρ(x n ,y n ) (x n ,y n )]. y n+1 =Π 2 (x n ,y n )= y n + ρ(x n ,y n ) mod 24 Fig. 2: Construct the map. Results By studying the entrainment map, determining the nature of various fixed points, together with an understanding of their stable and unstable manifolds, leads to conditions for existence and stability of periodic orbits of the circadian system. We use the map to investigate how various properties of solutions depend on parameters and initial conditions including the time to and direction of entrainment. We show that the concepts of phase advance and phase delay need to be carefully assessed when considering hierarchical systems. Fig. 3: Finding fixed points and computing entrainment time Fig. 4: Iterates and manifolds. Parameter dependance and bifurcations we calculate the 2-D map at different values of α 1 to see how the fixed points and the entrainment time depend on α 1 . In Fig. (a) (b), we show the x and y nullclines for three different α 1 values; the points with solid circle are the stable fixed points, the points with open circles are the unstable fixed points, and the starred points are saddle points. In Fig. (c), we show the heatmap of entrainment times for α 1 =1.52. In Fig. (d), we show the heatmap of entrainment times for α 1 =2.5. Note that α 1 = 2 is our canonical case, and was presented before. Increasing α 1 , in general, decreases the entrainment time as can be observed from the color scale values (yellow max value ≈ 700 for α 1 =1.52) verus 400 for α 1 =2.5. In other words, stronger coupling between the central to peripheral oscillator speeds up entrainment. Fig. 5: Parameter dependence. Conclusion We generalized the entrainment map to two dimensions by introducing the phase an- gle. Analyzed the time of entrainment and the direction of entrainment by studying the properties of the map. The direction of entrainment is not necessarily monotonic. Entrainment time calculations provide a way to locate and approximate stable and unstable manifolds. We then computed the stable and unstable manifolds by growing method and search circle method [2]. Future work Apply the entrainment map to other kinds of coupled network. Develop entrainment maps for more general models of periodic forced oscillators. Apply Floquet theory to derive the entrainment map under phase-amplitude coordinate system. Study the bifurcation of the entrainment map. References [1] Casey O Diekman and Amitabha Bose. “Entrainment maps: A new tool for understanding prop- erties of circadian oscillator models” . In: J. Biol. Rhythms 31.6 (2016), pp. 598–616. [2] James P England, Bernd Krauskopf, and Hinke M Osinga. “Computing one-dimensional stable manifolds and stable sets of planar maps without the inverse” . In: SIADS 3.2 (2004), pp. 161–190. [3] B´ ela Nov´ ak and John J Tyson. “Design principles of biochemical oscillators” . In: Nat. Rev. Mol. Cell Biol. 9.12 (2008), p. 981.
Transcript of Guangyuan Liao , Casey Diekman and Amitabha Bosegl92/Files/Dynamics_Days_2020_Poster.pdf ·...
Entrainment dynamics of forced hierarchical circadian systems.
Guangyuan Liao†, Casey Diekman† and Amitabha Bose†
†Department of Mathematical Science, New Jersey Institute of Technology
Entrainment dynamics of forced hierarchical circadian systems.
Guangyuan Liao†, Casey Diekman† and Amitabha Bose†
†Department of Mathematical Science, New Jersey Institute of Technology
Introduction
The ability of circadian oscillators to entrain to light-dark cycles is well known. Theprocess of circadian entrainment has been studied by various methods from dynamicalsystems. Recently, a tool called the entrainment map was introduced to analyze theentrainment process [1]. We developed a 2-D entrainment map to study coupledcircadian oscillators. The entrainment map is quite effective to study the time anddirection of entrainment.
Model
We first consider the situation in which a single central oscillator receives light-darkinput. In turn, this central oscillator sends input to a single peripheral oscillator. Tofocus on the mathematical aspects of the derivation and analysis of the 2-D entrain-ment map, we will utilize the planar Novak-Tyson model [3] for both the central andperipheral oscillators.
Fig. 1: Schematic figure of the model.
1
φ1
dP1
dt= M1 − kfh(P1)− kDP1 − kL1
f (t)P1
1
φ1
dM1
dt= ε[g(P1)−M1]
1
φ2
dP2
dt= M2 − kfh(P2)− kDP2
1
φ2
dM2
dt= ε[g(P2)−M2 + α1M1g(P2)].
Where g(P ) = 11+P 4; h(P ) = P
0.1+P+2P 2 and f (t) = Heaviside(sin( π12t))
Construction of the 2-D entrainment map
Instead of studying the original five dimensional model directly, we reduced the modelas a two dimensional discrete phase model. The method is a combination of thePoincare map and phase reduction. The reduced system is written as a two dimen-sional map:
xn+1 = Π1(xn, yn) = Arg[ϕρ(xn,yn)(xn, yn)].
yn+1 = Π2(xn, yn) = yn + ρ(xn, yn) mod 24
Fig. 2: Construct the map.
Results
By studying the entrainment map, determining the nature of various fixed points, together withan understanding of their stable and unstable manifolds, leads to conditions for existence andstability of periodic orbits of the circadian system. We use the map to investigate how variousproperties of solutions depend on parameters and initial conditions including the time to anddirection of entrainment. We show that the concepts of phase advance and phase delay need tobe carefully assessed when considering hierarchical systems.
Fig. 3: Finding fixed points and computing entrainment time
Fig. 4: Iterates and manifolds.
Parameter dependance and bifurcations
we calculate the 2-D map at different values of α1 to see how the fixed points andthe entrainment time depend on α1. In Fig. (a) (b), we show the x and y nullclinesfor three different α1 values; the points with solid circle are the stable fixed points,the points with open circles are the unstable fixed points, and the starred points aresaddle points. In Fig. (c), we show the heatmap of entrainment times for α1 = 1.52.In Fig. (d), we show the heatmap of entrainment times for α1 = 2.5. Note thatα1 = 2 is our canonical case, and was presented before. Increasing α1, in general,decreases the entrainment time as can be observed from the color scale values (yellowmax value ≈ 700 for α1 = 1.52) verus 400 for α1 = 2.5. In other words, strongercoupling between the central to peripheral oscillator speeds up entrainment.
Fig. 5: Parameter dependence.
Conclusion
We generalized the entrainment map to two dimensions by introducing the phase an-gle. Analyzed the time of entrainment and the direction of entrainment by studyingthe properties of the map. The direction of entrainment is not necessarily monotonic.Entrainment time calculations provide a way to locate and approximate stable andunstable manifolds. We then computed the stable and unstable manifolds by growingmethod and search circle method [2].
Future work
Apply the entrainment map to other kinds of coupled network. Develop entrainmentmaps for more general models of periodic forced oscillators. Apply Floquet theoryto derive the entrainment map under phase-amplitude coordinate system. Study thebifurcation of the entrainment map.
References
[1] Casey O Diekman and Amitabha Bose. “Entrainment maps: A new tool for understanding prop-erties of circadian oscillator models”. In: J. Biol. Rhythms 31.6 (2016), pp. 598–616.
[2] James P England, Bernd Krauskopf, and Hinke M Osinga. “Computing one-dimensional stablemanifolds and stable sets of planar maps without the inverse”. In: SIADS 3.2 (2004), pp. 161–190.
[3] Bela Novak and John J Tyson. “Design principles of biochemical oscillators”. In: Nat. Rev. Mol.Cell Biol. 9.12 (2008), p. 981.
