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Spatial trigger waves: positive feedback gets you a long way.

Gelens L, Anderson GA, Ferrell JE - Mol. Biol. Cell (2014)

Bottom Line: Trigger waves are a recurring biological phenomenon involved in transmitting information quickly and reliably over large distances.Well-characterized examples include action potentials propagating along the axon of a neuron, calcium waves in various tissues, and mitotic waves in Xenopus eggs.Here we use the FitzHugh-Nagumo model, a simple model inspired by the action potential that is widely used in physics and theoretical biology, to examine different types of trigger waves-spatial switches, pulses, and oscillations-and to show how they arise.

View Article: PubMed Central - PubMed

Affiliation: Department of Chemical and Systems Biology, Stanford University School of Medicine, Stanford, CA 94305-5174 Applied Physics Research Group, Vrije Universiteit Brussel (VUB), 1050 Brussels, Belgium.

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Different types of dynamics from the FHN model. (A, C, E) Time course; (B, D, F) phase plots. (A, B) Bistability. For b = 2, the system is bistable, with two stable steady states (B, filled circles) and one saddle point (B, open circle). For the value of v(t = 0) assumed here (v(t = 0) = –0.3), trajectories beginning above a threshold value of u (A, dashed line) go to the high-u stable steady state, whereas those beginning below the threshold go to the low-u stable steady state. In the phase plane, a separatrix (dashed curve) divides the starting points that approach the high-u stable steady state (pink area) from those that go to the low-u steady state. (C, D) Excitability. For b = 1.5, there is a single stable steady state plus a saddle point and an unstable steady state. Trajectories beginning above the threshold (C) or the separatrix (D) yield a pulse of u and circle the unstable steady state before settling down to a low steady-state value of u. Those beginning below the threshold do not yield a pulse of high u. (E, F) Oscillations. For b = 1.0, the single steady state is unstable. From all initial conditions (except starting right on the unstable steady state), the trajectories approach the same stable limit cycle, although from above the threshold, they go first to the upper limb of the u-cline, and below the threshold, they go first to the lower limb. A, C, and E are time courses; B, D, and F are phase plots. In each case, a = 0.1, ε = 0.01, v(t = 0) = −0.3, and u(t = 0) = −0.25 (red trajectories) or −0.35 (blue trajectories).
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Figure 2: Different types of dynamics from the FHN model. (A, C, E) Time course; (B, D, F) phase plots. (A, B) Bistability. For b = 2, the system is bistable, with two stable steady states (B, filled circles) and one saddle point (B, open circle). For the value of v(t = 0) assumed here (v(t = 0) = –0.3), trajectories beginning above a threshold value of u (A, dashed line) go to the high-u stable steady state, whereas those beginning below the threshold go to the low-u stable steady state. In the phase plane, a separatrix (dashed curve) divides the starting points that approach the high-u stable steady state (pink area) from those that go to the low-u steady state. (C, D) Excitability. For b = 1.5, there is a single stable steady state plus a saddle point and an unstable steady state. Trajectories beginning above the threshold (C) or the separatrix (D) yield a pulse of u and circle the unstable steady state before settling down to a low steady-state value of u. Those beginning below the threshold do not yield a pulse of high u. (E, F) Oscillations. For b = 1.0, the single steady state is unstable. From all initial conditions (except starting right on the unstable steady state), the trajectories approach the same stable limit cycle, although from above the threshold, they go first to the upper limb of the u-cline, and below the threshold, they go first to the lower limb. A, C, and E are time courses; B, D, and F are phase plots. In each case, a = 0.1, ε = 0.01, v(t = 0) = −0.3, and u(t = 0) = −0.25 (red trajectories) or −0.35 (blue trajectories).

Mentions: Here we assume a = 0.1 and e = 0.01 and change the behavior by varying the parameter b. Note that all parameters and variables are dimensionless throughout. When b is relatively large (b > 1.8), the system is bistable. Depending on the initial conditions, the system will settle down into one of two alternative stable steady states, one with a negative membrane potential (Figure 2A) and one with a positive membrane potential. For the initial value of v assumed here (v = −0.3), all trajectories that start with u > −0.3 (the threshold shown by the dashed line in Figure 2A) will approach the positive-potential steady state (Figure 2A, red curve), and all trajectories that start with u < −0.3 will end up at the negative-potential steady state (Figure 2A, blue curve). Thus a small perturbation that pushes the system across the threshold will be amplified into a large difference in the system's ultimate fate.


Spatial trigger waves: positive feedback gets you a long way.

Gelens L, Anderson GA, Ferrell JE - Mol. Biol. Cell (2014)

Different types of dynamics from the FHN model. (A, C, E) Time course; (B, D, F) phase plots. (A, B) Bistability. For b = 2, the system is bistable, with two stable steady states (B, filled circles) and one saddle point (B, open circle). For the value of v(t = 0) assumed here (v(t = 0) = –0.3), trajectories beginning above a threshold value of u (A, dashed line) go to the high-u stable steady state, whereas those beginning below the threshold go to the low-u stable steady state. In the phase plane, a separatrix (dashed curve) divides the starting points that approach the high-u stable steady state (pink area) from those that go to the low-u steady state. (C, D) Excitability. For b = 1.5, there is a single stable steady state plus a saddle point and an unstable steady state. Trajectories beginning above the threshold (C) or the separatrix (D) yield a pulse of u and circle the unstable steady state before settling down to a low steady-state value of u. Those beginning below the threshold do not yield a pulse of high u. (E, F) Oscillations. For b = 1.0, the single steady state is unstable. From all initial conditions (except starting right on the unstable steady state), the trajectories approach the same stable limit cycle, although from above the threshold, they go first to the upper limb of the u-cline, and below the threshold, they go first to the lower limb. A, C, and E are time courses; B, D, and F are phase plots. In each case, a = 0.1, ε = 0.01, v(t = 0) = −0.3, and u(t = 0) = −0.25 (red trajectories) or −0.35 (blue trajectories).
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Related In: Results  -  Collection

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Figure 2: Different types of dynamics from the FHN model. (A, C, E) Time course; (B, D, F) phase plots. (A, B) Bistability. For b = 2, the system is bistable, with two stable steady states (B, filled circles) and one saddle point (B, open circle). For the value of v(t = 0) assumed here (v(t = 0) = –0.3), trajectories beginning above a threshold value of u (A, dashed line) go to the high-u stable steady state, whereas those beginning below the threshold go to the low-u stable steady state. In the phase plane, a separatrix (dashed curve) divides the starting points that approach the high-u stable steady state (pink area) from those that go to the low-u steady state. (C, D) Excitability. For b = 1.5, there is a single stable steady state plus a saddle point and an unstable steady state. Trajectories beginning above the threshold (C) or the separatrix (D) yield a pulse of u and circle the unstable steady state before settling down to a low steady-state value of u. Those beginning below the threshold do not yield a pulse of high u. (E, F) Oscillations. For b = 1.0, the single steady state is unstable. From all initial conditions (except starting right on the unstable steady state), the trajectories approach the same stable limit cycle, although from above the threshold, they go first to the upper limb of the u-cline, and below the threshold, they go first to the lower limb. A, C, and E are time courses; B, D, and F are phase plots. In each case, a = 0.1, ε = 0.01, v(t = 0) = −0.3, and u(t = 0) = −0.25 (red trajectories) or −0.35 (blue trajectories).
Mentions: Here we assume a = 0.1 and e = 0.01 and change the behavior by varying the parameter b. Note that all parameters and variables are dimensionless throughout. When b is relatively large (b > 1.8), the system is bistable. Depending on the initial conditions, the system will settle down into one of two alternative stable steady states, one with a negative membrane potential (Figure 2A) and one with a positive membrane potential. For the initial value of v assumed here (v = −0.3), all trajectories that start with u > −0.3 (the threshold shown by the dashed line in Figure 2A) will approach the positive-potential steady state (Figure 2A, red curve), and all trajectories that start with u < −0.3 will end up at the negative-potential steady state (Figure 2A, blue curve). Thus a small perturbation that pushes the system across the threshold will be amplified into a large difference in the system's ultimate fate.

Bottom Line: Trigger waves are a recurring biological phenomenon involved in transmitting information quickly and reliably over large distances.Well-characterized examples include action potentials propagating along the axon of a neuron, calcium waves in various tissues, and mitotic waves in Xenopus eggs.Here we use the FitzHugh-Nagumo model, a simple model inspired by the action potential that is widely used in physics and theoretical biology, to examine different types of trigger waves-spatial switches, pulses, and oscillations-and to show how they arise.

View Article: PubMed Central - PubMed

Affiliation: Department of Chemical and Systems Biology, Stanford University School of Medicine, Stanford, CA 94305-5174 Applied Physics Research Group, Vrije Universiteit Brussel (VUB), 1050 Brussels, Belgium.

Show MeSH
Related in: MedlinePlus