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Mathematical Modeling in Neuroendocrinology

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ABSTRACT

Mathematical models are commonly used in neuroscience, both as tools for integrating data and as devices for designing new experiments that test model predictions. The wide range of relevant spatial and temporal scales in the neuroendocrine system makes neuroendocrinology a branch of neuroscience with great potential for modeling. This article provides an overview of concepts that are useful for understanding mathematical models of the neuroendocrine system, as well as design principles that have been illuminated through the use of mathematical models. These principles are found over and over again in cellular dynamics, and serve as building blocks for understanding some of the complex temporal dynamics that are exhibited throughout the neuroendocrine system. © 2015 American Physiological Society. Compr Physiol 5:911‐927, 2015.

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Figure 1. Figure 1. Box‐and‐arrow diagram indicating that A stimulates B and B inhibits A.
Figure 2. Figure 2. (A) With no applied current the Hodgkin‐Huxley system is at equilibrium. (B) When a brief current pulse is applied an action potential is produced. The system returns to equilibrium after the pulse. The XPPAUT software package was used to numerically solve the equations in this and other figures. Codes can be downloaded from www.math.fsu.edu/∼bertram/software/pituitary.
Figure 3. Figure 3. (A) Phase plane illustration of a bistable genetic toggle switch. The x nullcline (black), y nullcline (green), separatrix (dashed), equilibrium points (red circles and triangle), and two trajectories (orange and violet) are shown. (B) The time courses of the y variable for the two trajectories. One approaches the lower equilibrium while the other approaches the upper equilibrium. Parameter values are αx = αy = 10, β = 2, and γ = 2.
Figure 4. Figure 4. The x nullcine (black) and y nullcline (green) for the genetic toggle switch with α x = α y = 10 and β = γ. (A) β = 2, (B) β = 1.5, and (C) β = 1. The dashed curve is the line y = x and equilibria are indicated by red circles or a triangle.
Figure 5. Figure 5. Bifurcation diagrams for the genetic toggle switch model with β = γ, αy = 10, and (A) αx = 10, or (B) αx = 9. PF, pitchfork bifurcation; SN, saddle‐node bifurcation; solid curve, stable equilibria; dashed curve, unstable equilibria.
Figure 6. Figure 6. Sustained oscillations emerge in the prolactin model when the time delay is sufficiently large. (A) = 0.5 h, (B) = 1.5 h, and (C) = 3 h. PRL and DA are dimensionless variables.
Figure 7. Figure 7. (A and B) A transient response is followed by a transition to tonic spiking with two consecutive current pulses of different sizes in the Hodgkin‐Huxley model. (C and D) There is a very different response to the same final applied current if the approach is more gradual.
Figure 8. Figure 8. (A) Bifurcation diagram highlighting the subcritical Hopf bifurcation (HB1) and the saddle‐node of periodics bifurcation (SNP) in the Hodgkin‐Huxley model. (B) Same diagram, but covering a larger range of the parameter. Highlights the supercritical Hopf bifurcation (HBs). Stationary branches are in black and periodic branches are in red.
Figure 9. Figure 9. Hodgkin‐Huxley time courses with and without noise. (A) Iapp = 1 μA/cm2. (B) Iapp = 1.5 μA/cm2.
Figure 10. Figure 10. (A and C) Time course of the two variables in the planar relaxation oscillator. (B and D) Limit cycles in the phase plane, superimposed on the x nullcline (blue dashed) and the y nullcline (blue solid). Left: = 1. Right: = 0.05.
Figure 11. Figure 11. (A) Bursting electrical activity produced by the s‐model. (B) The slow variable s has a saw tooth shape and for the same value of s the voltage may be in an active phase (a) or a silent phase (s), indicating bistability in the fast subsystem.
Figure 12. Figure 12. (A) Bifurcation diagram of the Vn fast subsystem for the s‐model. Stationary (black) and periodic (red) branches are shown. SN, saddle‐node; HB, Hopf; HM, homoclinic bifurcation. (B) Fast/slow analysis of s‐model bursting. The s nullcline (blue) and burst trajectory are superimposed onto the z curve.
Figure 13. Figure 13. (A) Pseudoplateau bursting produced by a model of the pituitary lactotroph. Note the short duration and small spikes. (B) Standard fast/slow analysis of the pseudoplateau bursting. The periodic branch (red) is unstable.


Figure 1. Box‐and‐arrow diagram indicating that A stimulates B and B inhibits A.


Figure 2. (A) With no applied current the Hodgkin‐Huxley system is at equilibrium. (B) When a brief current pulse is applied an action potential is produced. The system returns to equilibrium after the pulse. The XPPAUT software package was used to numerically solve the equations in this and other figures. Codes can be downloaded from www.math.fsu.edu/∼bertram/software/pituitary.


Figure 3. (A) Phase plane illustration of a bistable genetic toggle switch. The x nullcline (black), y nullcline (green), separatrix (dashed), equilibrium points (red circles and triangle), and two trajectories (orange and violet) are shown. (B) The time courses of the y variable for the two trajectories. One approaches the lower equilibrium while the other approaches the upper equilibrium. Parameter values are αx = αy = 10, β = 2, and γ = 2.


Figure 4. The x nullcine (black) and y nullcline (green) for the genetic toggle switch with α x = α y = 10 and β = γ. (A) β = 2, (B) β = 1.5, and (C) β = 1. The dashed curve is the line y = x and equilibria are indicated by red circles or a triangle.


Figure 5. Bifurcation diagrams for the genetic toggle switch model with β = γ, αy = 10, and (A) αx = 10, or (B) αx = 9. PF, pitchfork bifurcation; SN, saddle‐node bifurcation; solid curve, stable equilibria; dashed curve, unstable equilibria.


Figure 6. Sustained oscillations emerge in the prolactin model when the time delay is sufficiently large. (A) = 0.5 h, (B) = 1.5 h, and (C) = 3 h. PRL and DA are dimensionless variables.


Figure 7. (A and B) A transient response is followed by a transition to tonic spiking with two consecutive current pulses of different sizes in the Hodgkin‐Huxley model. (C and D) There is a very different response to the same final applied current if the approach is more gradual.


Figure 8. (A) Bifurcation diagram highlighting the subcritical Hopf bifurcation (HB1) and the saddle‐node of periodics bifurcation (SNP) in the Hodgkin‐Huxley model. (B) Same diagram, but covering a larger range of the parameter. Highlights the supercritical Hopf bifurcation (HBs). Stationary branches are in black and periodic branches are in red.


Figure 9. Hodgkin‐Huxley time courses with and without noise. (A) Iapp = 1 μA/cm2. (B) Iapp = 1.5 μA/cm2.


Figure 10. (A and C) Time course of the two variables in the planar relaxation oscillator. (B and D) Limit cycles in the phase plane, superimposed on the x nullcline (blue dashed) and the y nullcline (blue solid). Left: = 1. Right: = 0.05.


Figure 11. (A) Bursting electrical activity produced by the s‐model. (B) The slow variable s has a saw tooth shape and for the same value of s the voltage may be in an active phase (a) or a silent phase (s), indicating bistability in the fast subsystem.


Figure 12. (A) Bifurcation diagram of the Vn fast subsystem for the s‐model. Stationary (black) and periodic (red) branches are shown. SN, saddle‐node; HB, Hopf; HM, homoclinic bifurcation. (B) Fast/slow analysis of s‐model bursting. The s nullcline (blue) and burst trajectory are superimposed onto the z curve.


Figure 13. (A) Pseudoplateau bursting produced by a model of the pituitary lactotroph. Note the short duration and small spikes. (B) Standard fast/slow analysis of the pseudoplateau bursting. The periodic branch (red) is unstable.
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Richard Bertram. Mathematical Modeling in Neuroendocrinology. Compr Physiol 2015, 5: 911-927. doi: 10.1002/cphy.c140034