Computational Neuroendocrinology E-Book
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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: Wiley-INF Neuroendocrinology Series
- Sprache: Englisch
Neuroendocrinology with its well defined functions, inputs, and outputs, is one of the most fertile grounds for computational modeling in neuroscience. But modeling is often seen as something of a dark art. This book aims to display the power of modeling approaches in neuroendocrinology, and to showcase its potential for understanding these complex systems.
A recurring theme in neuroendocrinology is rhythms. How are rhythms generated, and what purpose do they serve? Are these two questions inextricably intertwined?
This book is written for innocents, presuming no math beyond high school or computing beyond calculators. It seeks to lead the curious into the thinking of the modeler, providing the tools to the reader to understand models, and even develop their own, giving life to paper diagrams. The diverse chapters, from ion channels to networks, systems, and hormonal rhythms, each tell the story of a model serving to join the hard won dots of experimentation, mapping a new understanding, and revealing hidden knowledge.
- Written by a team of internationally renowned researchers
- Both print and enhanced e-book versions are available
- Illustrated in full colour throughout
This is the fourth volume in a new Series 'Masterclass in Neuroendocrinology' , a co- publication between Wiley and the INF (International Neuroendocrine Federation) that aims to illustrate highest standards and encourage the use of the latest technologies in basic and clinical research and hopes to provide inspiration for further exploration into the exciting field of neuroendocrinology.
Series Editors: John A. Russell, University of Edinburgh, UK and William E. Armstrong, The University of Tennessee, USA
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Seitenzahl: 668
Veröffentlichungsjahr: 2016
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Table of Contents
Title Page
Copyright
List of Contributors
Series Preface
Preface
About the Companion Website
Chapter 1: Bridging Between Experiments and Equations: A Tutorial on Modeling Excitability
1.1 Introducing excitability
1.2 Introducing the Morris–Lecar model
1.3 Opening XPP and triggering an action potential
1.4 Action potentials in the phase plane
1.5 Model response to sustained current injection
1.6 Reading a bifurcation diagram
1.7 Saddle node on an invariant circle (SNIC) bifurcation
1.8 Time-scale separation
1.9 Homoclinic bifurcation
1.10 Bursting
1.11 Eigenvalues and stability
1.12 Perspectives
Acknowledgement
References
Chapter 2: Ion Channels and Electrical Activity in Pituitary Cells: A Modeling Perspective
2.1 Endocrine pituitary cells are electrically active
2.2 Endocrine pituitary cell types
2.3 Voltage-gated ion channels
2.4 Nonselective cation channels
2.5 Ligand-gated ion channels
2.6 Spontaneous electrical activity and signalling
2.7 Modulation of spontaneous electrical activity by GPCRs
2.8 The dynamic clamp technique for studying the contributions of ion channels to electrical activity
2.9 Perspectives
Recommended reading
Chapter 3: Endoplasmic Reticulum- and Plasma-Membrane-Driven Calcium Oscillations
3.1 Introduction
3.2 Calcium balance equations
3.3 ER-driven calcium oscillations
3.4 Combining ER and PM oscillators
3.5 The road goes ever on
3.6 Conclusions
Acknowledgments
References
Chapter 4: A Mathematical Model of Gonadotropin-Releasing Hormone Neurons
4.1 Introduction
4.2 Previous models of GnRH neurons
4.3 A model of GnRH neurons in hypothalamic slices
4.4 Model results
4.5 Conclusions and future work
4.6 Appendix: the model equations and parameters
References
Chapter 5: Modeling Spiking and Secretion in the Magnocellular Vasopressin Neuron
5.1 Background
5.2 Modeling
5.3 Bursting in a spiking model
5.4 Modeling spike-triggered secretion
5.5 Population modeling
5.6 Conclusion
References
Further Reading
Chapter 6: Modeling Endocrine Cell Network Topology
6.1 Introduction
6.2 Networks
6.3 Step-by-step experimental and analytical protocol
6.4 Worked example
6.5 Perspectives
Further reading
References
Chapter 7: Modeling the Milk-Ejection Reflex
7.1 The milk-ejection reflex
7.2 The Model
7.3 Building the model
7.4 Model behavior
7.5 Discussion
7.6 Perspectives
Bibliography
Chapter 8: Dynamics of the HPA Axis: A Systems Modeling Approach
8.1 Introduction
8.2 Mathematically modeling the HPA axis
8.3 Unveiling the mechanism of ultradian pulsatility
8.4 Exploring model predictions experimentally
8.5 Significance of ultradian pulsatility for stress responsiveness
8.6 Discussion
8.7 Perspectives
References
Chapter 9: Modeling the Dynamics of Gonadotropin-Releasing Hormone (GnRH) Secretion in the Course of an Ovarian Cycle
9.1 Introduction
9.2 A single dynamical framework for the control of the GnRH pulse and surge generator by ovarian steroids
9.3 GnRH secretion pattern along an ovarian cycle
9.4 Reproducing known effects of ovarian steroids on the surge
9.5 Steroid challenges in the pulsatile regime
9.6 Conclusion
References
Glossary
Index
End User License Agreement
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Guide
Cover
Table of Contents
Preface
Begin Reading
List of Illustrations
Chapter 1: Bridging Between Experiments and Equations: A Tutorial on Modeling Excitability
Figure 1.1
Variety of natural excitability
. (a) Voltage responses of a mouse adrenal chromaffin cell to 10 ms current steps, recorded with whole-cell current clamp (McCobb Lab data). Action potential (AP) amplitudes were nearly invariant, and rise times varied modestly with stimulus amplitude. Voltage scale as in (b). (b) AP waveforms vary widely between cell types, ranging in duration from 180 s for a purkinje cell (orange; Bean (2007)) to ms for a cardiac muscle AP (black). Shown for comparison are spikes from a barnacle muscle cell (blue; Fatt and Katz (1953)) and a chromaffin cell (purple; McCobb Lab). (c–f) Patterns of spikes elicited with sustained current steps vary even between mouse chromaffin cells (McCobb Lab). Cell (c) would not fire more than one spike, (d) fired a train with declining frequency, amplitude, and repolarization rate, (e), an irregular volley, and (f), a very regular train at high frequency. (g, h) Pituitary corticotropes fire spontaneous bursts with features that vary between bursts and between cells, including spike amplitudes and patterns, as well as burst durations (McCobb Lab). Scale bars in (c) apply to (c–h).
Figure 1.2
Voltage clamp data
. (a, b, and c) Voltage-gated K, Ca, and Na currents, respectively, elicited with voltage steps in whole-cell voltage clamp mode applied to mouse chromaffin cells (McCobb Lab data). Outward currents are positive (upward), and inward currents are negative (downward). The K and Ca currents shown here exhibit little inactivation, though both types can inactivate in some chromaffin cells. The Na currents inactivate rapidly, and the current amplitude reverses sign when the test potential crosses the Na reversal potential. (d) K-current activation is faster at more depolarized potentials, as shown by normalizing the K currents at 0 and +100 mV from (a). (e) Current–voltage (–) plot for K currents from the cell in (a); peak current values are plotted against the corresponding test potential. (f) Conductance–voltage (–) plot; current values from (e) are divided by the driving force () and plotted against test potential. The – curve gives a summary of the voltage dependence of gating without the confounding effect of driving force.
Figure 1.3
Morris–Lecar model
. (a) Equivalent electrical circuit representation of the Morris–Lecar model cell. Membrane capacitance is in parallel with selective conductances and batteries (representing driving forces arising from ionic gradients). Arrows indicate variation (with voltage) for Ca and K conductance. (b) Normalized – curves, and , assumed for model K and Ca conductances, respectively. (c) The kinetics of voltage-dependent activation of K channels is also voltage dependent, with the normalized time constant, , assumed to peak (i.e., channel gating slowest) at the midpoint of the – curve, where channel conformational preference is weakest. (d) Simulated voltage clamped K currents at 20 mV increments up to =100 mV. Voltage clamp simulated in XPP by removing the equation for and, instead, setting as a fixed parameter. (e) Model – plots for K and Ca currents. Different reversal potentials for the two conductances ( mV) make the current traces look very different, despite similar activation functions in (b). (b)–(e) Use Hopf parameter set.
Figure 1.4
Model action potentials
. Simulated response to a stimulus pulse, using Hopf parameter set. (a) Sub- and super-threshold voltage responses to the brief current steps in (b). Current amplitudes 1–6 were 150, 460, 480, 490, 500, and 570 A, respectively. Responses 1 and 2 are below, 3 is near, and 4–6 are above firing threshold. (c) Voltages at 1 ms intervals for time course 6 in (a) (colored circles). (d) Model currents, and (e) normalized voltage-dependent conductances underlying response 6. Note the slower kinetics of K-current activation, despite similar – curves (Figure 1.3b), due to the different kinetic assumptions in the model. (f–i) Trajectories and nullclines (red and green, marking where and turn, respectively) in the phase plane. The intersection of the nullclines is a stable equilibrium; the ultimate destination of all trajectories after the brief stimulus pulse. (f) Trajectories corresponding to the voltage time courses in (a). An action potential corresponds to a counterclockwise excursion around the phase plane. (g) Trajectory 6 plotted with the direction field; vectors indicate the direction and rate of movement at any point in the phase plane. (h) Trajectory 6 plotted at 1 ms intervals, as in (c), indicating the speed of motion around the trajectory. The motion slows where V turns on the red nullcline, and where V approaches equilibrium. (i) The flow, represented by trajectories from many initial values in the phase plane. Color indicates speed of motion around the phase plane, from blue (slow) to orange (fast).
Figure 1.5
Sustained current injection with the Hopf parameter set
. Model responses to a progressive series of sustained current stimuli (amplitudes indicated), showing voltage traces in time and the corresponding phase-plane trajectories and nullclines. As stimulus amplitude begins to increase, single action potentials are elicited. At a sharply defined value close to A, repetitive trains of action potentials are elicited, corresponding to the sudden appearance of a stable limit cycle in the phase plane. At this point, the firing frequency is immediately at a characteristic non-zero value (determined by the speed around the cycle), from which it increases only modestly over a wide range of stimulus currents, as in Hodgkin's Class II neurons. At another sharply defined stimulus amplitude, just below A, the stable cycle and corresponding spike train are suddenly lost, and voltage damps to a stable equilibrium at relatively positive potential.
Figure 1.6
Bifurcation diagram for the Hopf parameter set
. The diagram summarizes the behavior illustrated in Figure 1.5. Stimulus current, , is the bifurcation parameter, and voltage, , is the behavior variable plotted. For each value of the stimulus current, there is a corresponding phase plane, as shown in Figure 1.5. The equilibria and limit cycles from all the phase planes are charted on the bifurcation diagram, to show how they vary with stimulus current, and to highlight the bifurcations. The equilibrium voltage in each phase plane is plotted on the bifurcation diagram against the corresponding stimulus current; in red if the equilibrium is stable and in black if it is unstable. A limit cycle is represented by its peak and trough voltage levels; in green if the cycle is stable and in blue if it is unstable. In the simulated experiment in Section 1.5, repetitive firing begins where the branch of unstable limit cycles, born at the subcritical Hopf bifurcation (A), appears to fold back on itself to become stable, at the saddle-node bifurcation of limit cycles (A). Firing terminates in a similar mechanism. Note the regions of bistability around the onset and offset of spiking, where a stable equilibrium and stable limit cycle coexist.
Figure 1.7
Sustained current injection with the SNIC parameter set
. (a) Series of phase-plane trajectories and corresponding voltage traces obtained with the stimulus current amplitudes indicated. For A, there are three equilibria (at the nullcline intersections). The two branches of the unstable manifold of the (middle) saddle equilibrium follow different trajectories (one sub- and one super-threshold) to the stable node, forming a “circle.” As increases, the nullclines separate and the equilibria are lost from the circle, transforming it into a stable limit cycle. Firing begins at very low frequency when the limit cycle is first created, because the motion slows almost to a stop, in memory of the lost equilibria. As the nullclines separate further, firing frequency increases. (b) Firing frequency plotted against stimulus amplitude, for the SNIC and Hopf parameter sets. The two parameter sets differ only in the voltage dependence of K-channel gating, but the associated bifurcations yield behaviors exemplifying Hodgkin's excitability Classes I and II, respectively: firing frequency starts low and grades up smoothly with increasing current in the SNIC example, but is narrowly constrained in the Hopf example. (c) Bifurcation diagram, charting the equilibria and limit cycles from each phase plane, as in Figure 1.6. The SNIC bifurcation occurs just below A, where the large stable limit cycle is created at the moment the stable equilibrium is lost in a saddle-node bifurcation. The termination of firing follows the same bifurcation structure as in the Hopf example. (d) Flow in the phase plane when A, with enlargements. (e–f) Detailed flow structure close to each equilibrium.
Figure 1.8
Changing time-scales
. Only the rate constant of K-current activation, , is varied between the graphs, so the nullclines do not change. All other parameters are from the SNIC parameter set, as in Figure 1.7, with held constant at . Arrowheads represent the direction field. (a) , so changes slowly relative to . The direction field is primarily horizontal, and the large limit cycle (resulting from a SNIC bifurcation) hugs the nullcline more tightly than in Figure 1.7, where . (b) . changes on a similar time scale to , and the cycle has a dramatically different shape, corresponding to low amplitude spikes with a high baseline. (c) . The cycle has disappeared altogether, so the spikes damp down to the attracting equilibrium.
Figure 1.9
Sustained current injection with the Homoclinic parameter set
. (a) Bifurcation diagram, showing the homoclinic bifurcation at A, and subsequent regions of bi- and tri-stabilities. (b–d) Phase-plane trajectories and nullclines and corresponding voltage traces as applied current is increased through the bifurcation. Notice the graded firing frequency response and the high spike baseline corresponding to the limit cycle position. Viewed from (d) to (b), the stable limit cycle grows till it collides with the saddle equilibrium to become a homoclinic loop and disappear in the homoclinic bifurcation. (e–g) Tri-stability when A. There is a stable limit cycle (e and f), with (e) a stable equilibrium outside and (f) a stable spiral equilibrium inside.
Figure 1.10
Bursting in the extended Morris–Lecar model
. Starting with the homoclinic parameter set, internal voltage-dependent kinetics are included for the applied current, so that slowly increases when and slowly decreases when . (a) Burst firing, with rate constant for kinetics. Behavior of the added current is shown below the voltage trace. Gray highlight indicates the recurring cycle encompassing one burst. (b) The corresponding burst trajectory is a limit cycle in three-dimensional phase space. Each burst corresponds to a complete passage around the cycle. Individual spikes within the burst are colored as in A. The spike waveforms change slightly during a burst, but the burst as a whole follows the recurring cycle. (c) Longer bursts with more spikes are elicited when the added current, , changes even more slowly (). (d and e) Details of the bifurcation diagram from Figure 1.9 onto which a two-dimensional projection of the three-dimensional burst cycles in (b) and (c), respectively, are superimposed. The projection is the view from directly “above” the cycle in (b), looking down on the – plane. During the interburst interval, so slowly increases, sweeping to the right on the bifurcation diagram. During this quiescent state, the cell tracks the low voltage stable equilibrium on the bifurcation diagram. When passes 39.6 A, the stable equilibrium disappears in a saddle-node bifurcation, and the cell, seeking an alternative stable state, jumps to the stable limit cycle and starts spiking. With the increase in voltage, changes direction to decreasing, slowly sweeping to the left. The cell continues to spike, tracking the stable cycle until the cycle collides with the saddle and disappears in a homoclinic bifurcation when A. The cell is forced to seek the quiescent state again, the voltage drops, and the entire burst cycle repeats. The slower rate constant of in (e) means there is less “momentum” (greater Time-scale separation), and the trajectory hugs the bifurcation diagram more closely.
Figure 1.11
Eigenvalues and stability
. (a and b) An equilibrium of a linear one-dimensional dynamical system has a single eigenvalue, indicating the direction and rate of flow (towards or away from the equilibrium). The eigenvalue is (a) negative when the equilibrium is stable and (b) positive when the equilibrium is unstable. (c–e) Magnified view, close to an equilibrium in a two-dimensional system. The equilibrium has two eigenvalues, organizing the direction of flow. (c) A saddle has one negative and one positive eigenvalue corresponding to flow towards and away from the saddle along eigendirections and , respectively. (d) A stable node has two negative eigenvalues, , corresponding to flow along eigendirections and , respectively. (e) A spiral equilibrium has complex eigenvalues (a complex conjugate pair). The equilibrium is unstable when the real part of the eigenvalues is positive, stable when the real part of the eigenvalues is negative. Parameter values given in the text.
Chapter 2: Ion Channels and Electrical Activity in Pituitary Cells: A Modeling Perspective
Figure 2.1
Spontaneous activity in pituitary cells
. (a) Simultaneous measurements of membrane potential () and the intracellular concentration () in pituitary somatotrophs, lactotrophs, and gonadotrophs. (b) Left: Variations in patterns of spontaneous electrical activity and associated transients in somatotrophs from the same preparation. Right: Expanded time scales, showing selected bursts labeled with asterisks on left.
Figure 2.2
Gating properties of
channels
. Steady-state activation function () and inactivation function () for the TTX-sensitive current. Shape parameters are , , , and .
Figure 2.3
Gating properties of
channels
. (a) Steady-state activation function () for the L-type current. Shape parameters are and . (b) Steady-state activation (, black) and inactivation (, red) functions for the T-type current. Shape parameters are , , , and .
Figure 2.4
Gating properties of
channels
. (a) Steady-state activation function () for the delayed rectifier current. Shape parameters are and . (b) Steady-state activation (, black) and inactivation (, red) functions for the A-type current. Shape parameters are , , , and . (c) Steady-state activation function () for the inward rectifying current. Shape parameters are and .
Figure 2.5
Gating properties of
channels
. (a) Steady-state activation function () for the SK-type or IK-type current. The shape parameter is . (b) Steady-state activation function () for the BK-type current. The shape parameters are and .
Figure 2.6
Gating properties of HCN channels
. Steady-state activation function () for the
h
-current. The shape parameters are and . Notice that the range of voltages shown here is different from that in previous figures, since the current activates at very low voltages.
Figure 2.7
Model cell simulation with three ionic currents
: , ,
and
. (a) Voltage time course with initially removed and then added at . At , a nonselective current is removed. (b) Time-dependent changes in the ionic currents (black), (red), and (green). Conductance values are , , and
Figure 2.8
Spiking versus bursting
. (a) Comparison of spontaneous spiking and bursting using the model cell with different parameter sets. The spiking parameter set is the same as that used in Figure 2.7, with the ionic currents , , and . The bursting parameter set includes two additional currents and . Parameter values for this set are , , , , , and . (b) The three currents: (red), (brown), and (blue), which are translated up by 60 pA for clarity. (c) The free cytosolic concentration is much greater during bursting than during tonic spiking.
Figure 2.9
The bursting model used in Figure
2.8
is augmented with different subthreshold currents
. (a) Addition of slows down oscillations and increases the time-averaged mean level. (b) Addition of speeds up oscillations and decreases the mean level.
Figure 2.10
The bursting model used in Figure
2.8
is augmented with currents reflecting G-protein signaling
. (a) signaling and the direct action of cAMP on HCN channels is simulated with the addition of an
h
-current. This converts bursting to spiking, and a larger value of increases the spike frequency. (b) The signaling pathway is simulated by the activation of a nonselective, depolarizing current, with increased from 0.02 to 0.3 nS. The signaling is simulated next by the activation of the current, with . This converts the spiking back to bursting. However, with too much conductance ( increased to 1 nS), the electrical activity stops completely.
Figure 2.11
Dynamic clamp for injecting a model current in an intact cell
. The cell's own BK current can be blocked pharmacologically with iberiotoxin (IBTX). In the current clamp, the recorded and digitized membrane potential (
V
) is used to calculate the BK current according to a mathematical model. This current is injected into the cell in real time. Modified with permission from Milescu
et al
. (2008).
Figure 2.12
The burst-promoting role of BK channels depends on their activation kinetics
. (a) Voltage recording from a lacto-somatotroph cell. Under control conditions, this cell exhibits a mix of spiking and bursting. (b) Pharmacological block of BK channels switches the activity pattern to spiking. (c) Addition via
dynamic clamp
of an artificial BK conductance () with fast activation () switches the electrical activity pattern back to bursting. (d) With slower kinetics (), the addition of the artificial BK conductance cannot switch the activity back to bursting.
Chapter 3: Endoplasmic Reticulum- and Plasma-Membrane-Driven Calcium Oscillations
Figure 3.1
Calcium balance
. , influx through plasma membrane; , efflux through plasma membrane; , influx through ER membrane; , efflux through ER membrane.
Figure 3.2
Illustration of the steady-state calcium theorem (Theorem
3.2.1
) using the passive model, Equations (
3.1
) and (
3.3
), with fluxes as in Equations (
3.4
)–(
3.7
)
. , cytosolic ; , ER . Parameters: M/pC; pA, pA (red), 0 (black); s; s, reduced to 0.2 s at min; s; ; .
Figure 3.3
Depleting ER
(
) increases amplitude of cytosolic
(
) response to pulses
. Black: control, Red: SERCA (sarco-endoplasmic reticulum calcium ATPase) pump inhibited 50%. Equations and parameters as in Figure 3.2 but with pulsed between and pA every 30 s.
Figure 3.4
Phase planes (a) and timecourses (b) for the closed-cell Li–Rinzel model, Equations (
3.9
)–(
3.11
)
. , fraction of IP3 receptors that are available to be opened (i.e. are not inactivated). Oscillations exist for an intermediate range of IP3 concentrations and, within that range, frequency increases with IP3. Parameters: as specified in the panels; ; M/s; 1/s; M; M; M/s; M; 1/Ms; M; .
Figure 3.6
In the open-cell Li–Rinzel model (Equations (
3.9
), (
3.13
), (
3.14
)),
influx is needed to sustain oscillations
. Parameters as in Figure 3.4 except: M; M/s; M; M; . We also add the plasma-membrane parameters: /s; M; and , which is initially 2.5 M/s and at s (
red arrow
) reduced to 0.
Figure 3.5
Bifurcation diagram for the closed-cell Li–Rinzel model
. Oscillations in cytosolic () exist for an intermediate range of IP3, demarcated by Hopf bifurcations (HB). Black: stable;
lines
: steady states;
filled circles
: min and max of oscillation; red: unstable steady states;
green
: average cytosolic during oscillations. Equations and parameters as in Figure 3.4
Figure 3.7
Bursting in combined endoplasmic reticulum–plasma membrane (ER-PM) model (Equations (
3.9
), (
3.13
)–(
3.20
))
. The PM would spike continuously on its own but is interrupted by periodic releases of from the ER, as seen in pituitary gonadotrophs (Li
et al
., 1994). The rise in cytoplasmic () thus occurs between the bursts of action potentials. Parameters are shown in Table 3.1.
Figure 3.8
ER regulates firing frequency by sourcing or sinking
. Equations as in Figure
3.7
, but
to disable ER rhythmicity and
initially set to 6.25
M/s to provide constant efflux from the ER
. is halved at s and restored at s. See Table 3.1 for other parameter changes. Bottom: traces from indicated time segments beginning at the
red arrows
in the middle panel.
Figure 3.9
Plateau or square-wave bursting
. Equations as in Figure 3.7 but with ER removed; parameters in Table 3.1. Timecourses of and cytosolic () in (a). In contrast to Figure 3.7, rises
during
the bursts of APs, not between them. Bifurcation diagram with superimposed trajectory shown in
blue
in (b). bifurcation; bifurcation; bifurcation.
Figure 3.10
(a)
–
phase plane for the model of Figure
3.9
but with
and
set to
(equivalently,
) to reduce the dimension to two and eliminate spiking
. nullclline,
red
; nullcline,
green
; trajectory,
black
. (b) Bifurcation diagram from Figure 3.9 with nullclines added. Intersection of nullcline with -curve, controlled by PMCA pump rate , determines whether behavior is silent (,
magenta
), bursting (,
green
), or continuously spiking (,
blue
).
Figure 3.11
Switching on ER fluxes in the square-wave burster of Figure
3.9
increases the period dramatically
. Note change in time base. Parameters in Table 3.1. (
lower right
) now has fast and slow components from its intrinsic kinetics and the ER, respectively. The bifurcation diagram (
upper right
) is calculated with respect to with fixed at its average value during bursting. The accumulation of spikes at the end of the periodic branch is responsible for the increased period.
Figure 3.12
(a) Pseudo-plateau bursting reminiscent of electrical activity in somatotrophs and lactotrophs
. Equations as in Figure 3.9; parameters in Table 3.1. (b) Bifurcation diagram. In contrast to the bifurcation diagram in Figure 3.9, the periodic orbits emanating from the Hopf bifurcation (HB) are unstable (max and min denoted by
filled red circles
).
Figure 3.13
(a) Bifurcation diagram for model of Figure
3.12
but with “BK” channel blocked to convert pseudo-plateau bursting to spiking, as seen in pituitary somatotrophs (Van Goor
et al
.,
2001
)
. Parameters in Table 3.1. The homoclinic bifurcation (HC) in that diagram has shifted leftward compared to diagram in Figure 3.12 and has merged with the saddle-node bifurcation (SN) at the left knee. (b) Bifurcation for electrical subsystem corresponding to Figure 3.7. The HB has shifted left so far that is has coalesced with a mirror-image HB at negative and disappeared, leaving large-amplitude spiking that continues down to .
Chapter 4: A Mathematical Model of Gonadotropin-Releasing Hormone Neurons
Figure 4.1
Recorded behavior of live GnRH neurons in the acutely prepared mouse brain slice, adapted from Lee
et al
. (2010)
. In each panel, the upper traces are the action currents, and the lower traces are the intracellular transients. (a) Spontaneous short bursts are associated with long duration (about 10 s) transients. (b) A blocker of voltage-gated channel, tetrodotoxin (TTX, 0.5 M). (c) A blocker of -dependent channels, apamin (300 nM). (d) Extracellular zero solution. (e) A blocker of inositol trisphosphate receptors, 2-aminoethoxydiphenyl borate (2-APB, 100 M). (f) An inhibitor of sarco/endoplasmic reticulum ATPase, cyclopiazonic acid (CPA, 30 M). (g) Ionotropic glutamate and receptor blockers (AP-5 20 ).
Figure 4.2
Schematic diagram of the model
.
amalgamates two different
currents, while
amalgamates three different
currents
. The currents combine with the currents to generate oscillatory AC spiking, in the manner of the Hodgkin–Huxley model. is a -sensitive and apamin-sensitive current that is activated when is raised, while is a -sensitive and time-dependent current that is activated by raised and turns off only gradually, in a -independent manner. An inward leakage current () is also included. There are five influx or efflux pathways for ; the - exchanger, the plasma membrane ATPase pump, the receptor, membrane channels and the SERCA pump. The dashed lines represent the direction of net ionic fluxes and the fluxes are denoted by solid lines.
Figure 4.3
Bursting in the original GnRH neuron model
. (a and b) Regular AC bursting and associated (; drawn as blue lines) transients (panel b shows the area boxed in panel a). Four spikes can be seen in each burst, and increases after the end of the burst.
Figure 4.4
Schematic diagram of the
channel model
. and are the two open states and is the closed state. , , and are the corresponding rate constants. This model is not based on experimental data, but is purely heuristic, being based solely on the need to have a channel that opens when increases, but then takes a long time to close.
Figure 4.5
The graph of
against
from the original GnRH neuron model
. In our simplified model, these variables are denoted by and , as discussed in the appendix. The butterfly shape of the graph indicates the fixed relationship between and . In the simplified model, we have used the simplest form, a linear equation (, the red straight line), to model this relationship.
Figure 4.6
Model simulations
. denotes the total membrane current, and denotes . (a and b) Regular AC bursting and associated transients with the indicated detail shown in panel B. Four spikes can be seen in the burst and increases after the end of the burst. (c) TTX simulation result, obtained by setting nS. (d) The solution in the presence of low extracellular , obtained by setting nS. (e and f) Apamin simulation result, obtained by setting . One burst after the addition of apamin is shown in panel f.
Figure 4.7
More model simulations
. As before, denotes the total membrane current, and denotes . (a) The addition of the SERCA pump blocker, CPA, is simulated by setting ). (b) Response to the addition of 2-APB, which blocks , SERCA pumps and membrane Ca-ATPase pumps. This is simulated by setting , and setting the rates of the SERCA pump and the membrane Ca-ATPase pumps to 0 and 0.0005 , respectively. (c) The result of blocking partially. After changing the conductance of () from 1050 to 750 nS, the bursting frequency gets much higher and the amplitudes decrease. Three bursts are shown in detail in panel d.
Chapter 5: Modeling Spiking and Secretion in the Magnocellular Vasopressin Neuron
Figure 5.1
Spiking in oxytocin and vasopressin neurons
. These traces show 10 min of spiking recorded from typical oxytocin and vasopressin neurons. The activity in the oxytocin neuron is mostly random. The vasopressin neuron shows a “phasic” pattern of long bursts and silences, with a distinctive peak at the start of each burst.
Figure 5.2
Integrate-and-fire and afterpotentials
. Model simulated excitatory and inhibitory PSPs summate and trigger a spike when the sum (the membrane potential) crosses the spike threshold. Following the spike, the HAP, DAP, and AHP are simulated by decaying exponentials with varied magnitude and half-life.
Figure 5.3
Inter-spike interval (ISI) histogram and hazard oxytocin cell example
. These show short-term patterning in the spike activity. The hazard becomes less reliable at longer intervals, but a good hazard function can usually be generated for up to 500 ms, subject to the firing rate and number of spikes in a recording.
Figure 5.4
Model fitted to oxytocin cell
. The dark shaded histogram and hazard show the basic model with an HAP fitted to the oxytocin cell of Figure 5.3. The light shade shows the effect of removing the HAP (hyper-polarizing afterpotential). Spiking is then only limited by the absolute refractory period (2 ms). The small peak early in the hazard shows a small increase in post-spike excitability due to the persistence of EPSPs (excitatory post-synaptic potentials).
Figure 5.5
Traces of the simple oxytocin cell model
. shows the summed random EPSPs and IPSPs. These are summed with the fixed to generate the membrane potential . When this crosses the model records a spike and increments the HAP, which is also summed with , generating a post-spike refractory period.
Figure 5.6
Detecting the AHP's influence using spike train analysis
. Adding an AHP to the simple model and applying spike train analysis to the spike times shows a negative correlation (a), matching the same analysis applied to a recorded cell. With no AHP the model can still match the ISI histogram and hazard, but not the negative correlation (b).
Figure 5.7
Using coloring to visualize burst detection
. If you are developing your own software then simple coloring is a very useful tool for visualizing bursts and adjusting the detection parameters. The alternating green and blue make it easier to see where detected bursts begin and end.
Figure 5.8
Oxytocin versus vasopressin hazard – post-spike influence of the DAP
. The DAP generates a peak in the hazard, observed in vasopressin, but not oxytocin neurons. The peak indicates a period of increased post-spike excitability following the decay of the HAP. At longer intervals the influence of the DAP in the vasopressin hazard competes with the AHP. In the oxytocin hazard, the AHP manifests as a very gradual positive slope following the HAP.
Figure 5.9
The phasic cell and the old and new model
. The top panel shows the target phasic pattern, as in Figure 5.1. The old model extends the oxytocin cell model by adding a fast DAP, and a simple dynorphin opposed slow DAP. It produces bursts, but without proper silent periods, or the sharp transitions observed
in vivo
(parameters in Table 5.3). The new model uses an improved form, replacing the slow DAP with a K
+
leak current based mechanism, combining the action opposing action of dynorphin and Ca
2+
(parameters in Table 5.4).
Figure 5.10
The vasopressin spike firing model
. Input is a Poisson random timed mix of excitatory and inhibitory pulses, simulating PSPs. These are summed to generate a membrane potential which is also modulated by a set of spike triggered Ca
2+
-based potentials. The HAP, fast DAP, and AHP are based on simple decaying exponentials, similar to a previous oxytocin cell model. The K
+
leak-current-based slow DAP which generates bursting is based on the mechanism of the Hodgkin–Huxley-type model of Roper
et al
. Spikes are generated when the membrane potential crosses a threshold value. Reproduced from MacGregor and Leng (2012) under Creative Commons Attribution (CC BY) license.
Figure 5.11
Model cell behavior with increasing synaptic input
. When osmotic pressure is increased
in vivo
, we see a shift to phasic firing followed by increases in intraburst firing rate and burst duration, eventually shifting to continuous firing. Here, we reproduce this in the model by increasing synaptic input. Intraburst firing rate increases fairly linearly, whereas the increase in burst duration is much more nonlinear. Silence duration shows a fairly linear decline after phasic firing is established. Reproduced from MacGregor and Leng (2012) under Creative Commons Attribution (CC BY) license.
Figure 5.12
The model fitted to four typical phasic cells recorded
in vivo
. On the left, we show pairs of matched
in vivo
and model generated spike rate data, and on the right, the fitted hazard and burst profiles. The model closely matches burst profile, mean burst length, mean silence length, intraburst firing rate, and the intraburst hazard, showing post-spike excitability and patterning. A subset of eight of the model's 21 parameters were varied to match the cells. The fit parameters vary synaptic input rate, HAP half-life, AHP magnitude, fast DAP magnitude, dynorphin magnitude, Ca
2+
magnitude, and K
+
leak conductance. The parameter values are given in MacGregor and Leng (2012), Tables 1 and 2. Adapted from MacGregor and Leng (2012) under Creative Commons Attribution (CC BY) license.
Figure 5.13
The model's burst firing mechanism
. The data here show two typical bursts from the model fitted to cell v4. The burst mechanism is driven by the spike-triggered accumulation of [Ca
2+
] and dynorphin. The [Ca
2+
] signal inhibits the hyperpolarizing K
+
leak current, increasing firing and creating a positive feedback that sustains a burst. The more slowly accumulating dynorphin signal opposes the effect of [Ca
2+
], eventually causing burst termination and driving a silent period of sustained hyperpolarization. The positive feedback combined with the two opposing effects acting on different timescales creates an emergent bistability, shown in the rapid shifts of the K
+
leak activation () and the resulting effect on membrane potential (). Reproduced from MacGregor and Leng (2012) under Creative Commons Attribution (CC BY) license.
Figure 5.14
Using simulated antidromic spikes to trigger and terminate bursts
. The data here use the model fitted to cell v4, repeated using the same random synaptic input. Antidromic stimulation is simulated by adding spikes to the model, at a specified frequency and time. In the left column, spikes are added during the silent period, attempting to trigger a burst. In the right column, spikes are added during the second burst, attempting to terminate the burst. Burst triggering occurs more likely when it is stimulated later into the silent period, or using a more intense stimulation. Generally, burst termination requires a more intense stimulation than burst triggering. Successful termination is more likely later into the burst, when there is more dynorphin accumulation, or with a more intense stimulation. The competing effects of spike-triggered increases in [Ca
2+
] and dynorphin cause a delay before termination occurs, unless the stimulation is sufficiently intense to trigger a large AHP, which immediately terminates spike firing. Reproduced from MacGregor and Leng (2012) under Creative Commons Attribution (CC BY) license.
Figure 5.15
Single-cell spike rate and secretion response to increasing synaptic stimulation
. The spiking model (parameters in Table 5.1) is coupled to the fitted secretion model (Table 5.2). Nonphasic spiking is generated by setting . (a) Examples of the four different modes of spike patterning generated by the same phasic spiking model with varied input rates. (b) The nonphasic model (left) shows a nonlinear increase in spike rate with increased input. The phasic spiking model (right), after very little response at low input rates, shows a more linear increase in spike rates. The rate of increase is initially steep as the patterning transitions from irregular firing to full phasic spiking, but this is followed by a wide range of very linear increase as bursts lengthen and intraburst firing rate increases. (c) The nonphasic model (left) shows a similarly nonlinear secretion response to increasing spike rate, showing a slow increase in secretion at low frequencies, followed by a facilitation driven rapid increase, which then slows along with the reduced spike rate response. The phasic secretion profile (right) shows an initial steep increase which slows as the intraburst spike rate reaches the optimal response frequency, and longer bursts allow less recovery from fatigue. Reproduced from MacGregor and Leng (2013) under Creative Commons Attribution (CC BY) license.
Figure 5.16
The vasopressin secretion model
. Schematic illustrating the structure of the differential-equation-based single-neuron secretion model. The model takes as input either a regular spike protocol or the output from the integrate-and-fire-based spiking model. For a single vasopressin cell, secretion occurs from about 2000 terminals and swellings; in the model, secretion is represented as coming from a single compartment; thus secretion is treated as a single continuous variable rather than many discrete stochastic variables. In the model, Ca
2+
entry is modulated by both fast () and slow () Ca
2+
variables through their modulation of axonal terminal excitability, and is also a function of spike broadening (). The secretion rate (vesicle exocytosis) is the product of the releasable pool () and the fast Ca
2+
variable (). When depleted, pool is refilled from a reserve store () at a rate dependent on the store content. Reproduced from MacGregor and Leng (2013) under Creative Commons Attribution (CC BY) license.
Figure 5.17
Facilitation and fatigue: the secretion model fitted to experimental data
. (a) Data (redrawn from Bicknell (1988)) show vasopressin secretion per spike from an isolated rat posterior pituitary stimulated with a fixed number of stimulus pulses (156) at varied frequencies, producing a frequency response profile showing an initial climb to a peak at 13 Hz followed by a slower decline. (b) The same experiment reproduced in the model with frequency ranging from 1 to 60 Hz in 1 Hz steps. The model combines frequency facilitation and competing fast negative feedback modulating Ca
2+
entry to match the
in vitro
data. (c) Experimental data (redrawn from Bicknell
et al
. (1984)) show secretion rate measured in four consecutive 18 s periods during regular stimulation at 13 Hz, showing a progressive fatigue of the secretion response. The model tested with the same protocol shows a similar decline, though fails to match the last point. The experimental data come from a series of experiments in which glands were stimulated repeatedly for different durations with different orders of presentation. (d) The same model run as C, plotted to show a detailed temporal profile. This shows the initial facilitation of secretion, followed by a slow fatigue. Reproduced from MacGregor and Leng (2013) under Creative Commons Attribution (CC BY) license.
Figure 5.18
Secretion response comparing regular and phasic stimulation with increasing frequency
. (a) Examples of regular, nonphasic, and phasic model-generated spiking used to stimulate the secretion model, all with the same 5 Hz mean rate. (b) Data (redrawn from Dutton and Dyball (1979)) show total secretion over 15-min stimulation, using regular stimulation, and stimulation triggered by recorded activity from phasic vasopressin cells. Phasic stimulation evokes much more secretion than regular stimulation at the same frequency; this is a consequence of greater facilitation of secretion at the higher intraburst firing rates, while the effects of fatigue are minimized by recovery during the silent intervals between bursts. (c) The model tested with a similar protocol matches the more optimal response to phasic patterned spiking, here also comparing randomly patterned nonphasic spiking. The nonphasic stimulus with periods of faster spiking at the same mean rate takes more advantage of facilitation than the regular patterned stimulus. Reproduced from MacGregor and Leng (2013) under Creative Commons Attribution (CC BY) license.
Figure 5.19
Homogeneous and heterogeneous 100 neuron population spike rate and secretion response to increasing synaptic stimulation
. (a) Introducing heterogeneity to the population of nonphasic cells makes little difference to the mean spike rate response. (b) In contrast, introducing equivalent heterogeneity to the phasic cell population results in an increased linearity of the mean response to stimulation. (c) Introducing heterogeneity to the population of nonphasic cells produces a modest increase in the linearity of the secretory response to stimulation. (d) In contrast, introducing heterogeneity to the population of nonphasic cells markedly enhances the linearity of secretion. Reproduced from MacGregor and Leng (2013) under Creative Commons Attribution (CC BY) license.
Figure 5.20
Heterogeneous phasic model cell population matched to
in vivo
secretion response
. The summed population secretion data of Figure 5.19d plotted on a reduced range (0–4 Hz mean spike rate) show a close match to the experimentally observed relationship between plasma osmotic pressure and vasopressin secretion in rats (Dunn
et al
., 1973). Reproduced from MacGregor and Leng (2013) under Creative Commons Attribution (CC BY) license.
Chapter 6: Modeling Endocrine Cell Network Topology
Figure 6.1
High-resolution snapshot of two intermingled pituitary networks (PRL, prolactin; GH, growth hormone) (scale bar = 20
m)
.
Figure 6.2
Common network topologies based upon degree (or edge number) distribution
. (a) A random network (above) has a Gaussian degree distribution (i.e., the proportion of nodes in a network which possess degree
k
) (below) meaning nodes, on average, possess a similar degree. (b) Degree distribution of a scale-free network (above) follows a power law (below) giving rise to a topology typified by a hub and spoke arrangement. (c) A small-world network (top) has a similar degree distribution probability to a scale-free network, but an exponent value (below) and high clustering coefficient which results in a shorter average path length. (d) A single-scale network (above) has a lattice like structure determined by its scaling factor (below).
Figure 6.3
Empirical mode decomposition (EMD) of cell signals
. In this example, the raw Ca
2+
trace has artifacts introduced by photobleaching and an axial drift. To denoise and detrend the signal, EM is performed to retrieve the intrinsic mode functions (IMF) or more simply, the static frequenices comprising the wave. The highest and lowest frequency IMFs can then be removed to result in a corrected signal which contains the identical information of interest to the source trace.
Figure 6.4
Plotting network topology to form a functional connectivity map
. The Euclidean coordinates of the nodes are retrieved and a weighted map depicting the distribution of edges between nodes constructed on the basis of association measure (in this case, correlated cell activity). Highly connected cells are displayed in green.
Figure 6.5
Determining network toplogy/function using the lactotrope cell population as an example
. (a) Pituitary slices derived from Discosoma red fluorescent protein (DsRed) transgenic animals are loaded with fura-2 and functional multicellular Ca
2+
imaging (fMCI) performed (DsRed expression is driven by the PRL promoter). Fura-2-DsRed positive cells (merge) are then delineated with a region of interest (ROI). (b) The Euclidean coordinates and intensity over time measure can be extracted from the time series. (c) The resulting traces are subjected to binarization using an arbitariliy determined threshold (dependent on various factors including cell type, experiment duration, and Ca
2+
-spiking pattern). (d) Correlation analyses are used to determine node association. Here, we have used a measure of coactivity with probability determined using a Monte Carlo statistic. However, other association measures, including Granger causality and principal component analysis, would be equally applicable.
Figure 6.6
Lactotrope network topology in virgin and lactating females
. (a) Representative functional connectivity maps demonstrate the presence of a hub and spoke structure in both virgin and lactating animals. (b) Degree distribution probability obeys a power law in both states, but the exponent value is lower during lactation indicating that more cells occupy the medium–high connectivity ranges: i.e., more hubs.
Chapter 7: Modeling the Milk-Ejection Reflex
video 1 The milk-ejection reflex in conscious rats.
Figure 7.1
Milk-ejection bursts
. Magnocellular oxytocin neurons each have one axon that projects into the neurohypophysis from where oxytocin is secreted into the general circulation. During suckling, they display intermittent high-frequency bursts of spikes every few minutes. An example of one bursts is shown – the trace is a 3-s extract from an extracellular recording.
Figure 7.2
The supraoptic nucleus (SON) of the rat hypothalamus
. (a) Oxytocin cells in the SON and paraventricular nucleus (PVN) are stained red by immunohistochemistry, in a coronal section of the rat brain. 3V=third ventricle. (b) Higher power view of the SON – the mat of fibers at the base of the nucleus are dendrites. Figure courtesy of Vicky Tobin.
Figure 7.3
Priming in oxytocin cells
. The dendrites of oxytocin cells contain many vesicles (shown as red organelles). These vesicles are normally located away from the plasma membrane, so stimuli that increase spike activity (indicated as a green stimulus) trigger release of oxytocin from axon terminals (where many vesicles are located adjacent to the plasama membrane) but not from dendrites. Some peptides can cause release from the dendrites without increasing spike activity, by triggering a mobilisation of intracellular calcium release. In addition, some peptides can prime the dendritic stores – moving vesicles close to the plasma membrane. After priming, these vesicles are available for release in response to increases in spike activity.
Figure 7.4
Spike activity in oxytocin cells
. Under background conditions, oxytocin cells discharge spikes at 1–3 spikes/s. This spiking can be characterized by measuring interspike intervals (t1, t2, etc. as shown in (a)), and constructing an interspike interval histogram. (b) Such histograms have a characteristic distribution tails of the histogram (for intervals 50 ms) that can be well fitted by a single negative exponential (red line, fitted to average of 30 cells). From this, it appears that, after a spike, oxytocin cells have a relative refractory period of about 50 ms, after which spikes arise approximately randomly. In the model, (c) spikes arise in model cells when incoming random EPSPs and IPSPs cause a fluctuation in resting potential sufficient to exceed a spike threshold. The relative refractoriness of oxytocin cells is the result of two post-spike hyperpolarizing mechanisms and (d) a short but large HAP, and a smaller but longer acting AHP (which has a major effect only after bursts). In the model, these two mechanisms are modeled as transient changes in spike threshold that occur after each spike, rather than as changes in the membrane potential – this is equivalent to changes in the membrane potential, but computationally simpler to implement.
Figure 7.5
Structure of the model network
. Oxytocin cells in the supraoptic nucleus have 1–3 large dendrites, most of which project ventrally (shown by immunocytochemistry in (a). These dendrites contain large numbers of neurosecretory vesicles (shown by electron microscopy in (b)). In the model, cells (c, in blue) have two dendrites (in red) that are coupled within bundles (indicated in yellow). The organization of the oxytocin network is shown in (d); the yellow boxes represent dendritic bundles.
Figure 7.6
The structure of a single model cell
. A single model cell receives random EPSPs and IPSPs, and its excitability is modeled as a dynamically changing spike threshold that is influenced by an HAP (parameter ), and a slower AHP (). Each cell interacts with its neighbor via two neurons by dendrites (red) that are coupled within bundles (yellow), and its excitability is increased when oxytocin is released in these bundles (). Activity-dependent production of endocannabinoids (EC) feeds back to reduce synaptic input rates.
Figure 7.7
The behavior of one model cell during a burst
. The upper two red traces show the times of occurrence of all oxytocin release events in the two dendritic bundles to which the cell is connected. Below this is the soma activity: the blue line shows the spike threshold, showing the effects of post-spike activity changes and of oxytocin; the black line
(
)
shows the impact of EPSPs and IPSPs. The bottom three traces show , and .
Figure 7.8
Synchronized bursting in model oxytocin cells
. Each row of the raster shows the spike activity of just one of the 348 cells in a network model; each bar shows the timing of a spike. Bursts are approximately synchronized, while background activity is asynchronous. The red trace below shows how changes during a burst in one of the cells; each spike causes a large transient rise in . As oxytocin is released, it causes a fall in that is offset by a slow rise caused by the AHP. Note how similar this profile is to the extracellularly recorded voltage trace of oxytocin cells in Figure 7.1.
Figure 7.9
Comparison of bursting activity in real and modeled oxytocin cells
. (a) Milk-ejection bursts triggered by i.c.v. injection of oxytocin in a urethane – anaesthetised (b) A milk-ejection burst in an oxytocin cell recorded
in vivo
(red) and a model cell (blue) plotted as instantaneous firing rate (each point is the reciprocal of the interval since the previous spike). This profile is indistinguishable to burst profiles observed
in vivo
. (c) Mean profiles of milk-ejection bursts from a living oxytocin cell (red) (Data file[g1]) and from a model cell (blue). Each profile is constructed from 17 bursts, and shows the mean S.E. instantaneous firing rate plotted for each interspike interval within the bursts.
Figure 7.10
Dependence of bursting on synaptic input
(a) In simulations of the model network, bursting behaviour is observed only within a range of values for the excitatory input. A minimum level of excitation is necessary to start the reflex. Increasing the input rate speeds up bursting until the excess of oxytocin release causes an abrupt breakdown. Bar colors correspond to varying the threshold for frequency-dependent release, defined as the maximum interspike interval allowed for dendritic release. (b) The effect of a spatially inhomogeneous input on bursting activity. Cells were subject to (balanced) inputs of rate , with drawn from a normal distribution. Plotted is the bursting frequency (based upon 50 min of dynamics; average over five trials with independently distributed rates) vs the SD of . Bars are SD; the dashed line is a linear fit.
Figure 7.11
How bursting behavior relates to network connectivity
. (a) The network can be described by a
bipartite graph
, where
N
is the set of cells,
B
is the set of bundles, and
E
is the set of connections from cells to bundles such that, for cell
a
in
N
and bundle
b
in
B
, if
a
has a dendrite in
b
. From these graphs, we can derive graphs and for connections between the cells and between the bundles. For the network to be collectively activated during a burst, it must be
connected
, that is, any two cells must be connected by some path. (b) Plots the probability that a network of 1000 cells, each with two dendrites, will be fully connected for different numbers of dendrites per bundle. Above a critical value of about 4.5, the network is almost certainly fully connected. (c) Shows how the critical value is affected by the number of cells in the network.
Figure 7.12
Impact of network topology on the propagation of bursting
. The network topology is critical for whether bursting is synchronous, or occurs as a traveling wave. (a) Schematic diagram illustrating a network with a ring structure (, top) and with random rewiring (, bottom); blue circles indicate cells, yellow boxes indicate bundles. One dendrite (in green) is randomly chosen and re-assigned; this is shown for only one cell, but the rule was applied to all cells independently. (b–e) Raster plots of spikes generated in networks with increasing probability of rewiring: (b) 0.05 (c) 0.5 (d), and 0.95 (e).
Chapter 8: Dynamics of the HPA Axis: A Systems Modeling Approach
Figure 8.1
The hypothalamic–pituitary–adrenal (HPA) axis
. The hypothalamic paraventricular nucleus (PVN) receives homeostatic/stress inputs from the brainstem and from regions of the limbic system such as the hippocampus and amygdala, as well as a circadian input from the suprachiasmatic nucleus (SCN). The PVN projects to the median eminence where it releases corticotrophin-releasing hormone (CRH) and arginine vasopressin (AVP) into the hypothalamic–pituitary portal circulation. CRH and AVP pass through this vascular route to access corticotroph cells in the anterior pituitary, which respond with the rapid release of adrenocorticotrophic hormone (ACTH) from pre-formed vesicles into the general circulation. In turn, ACTH reaches the adrenal cortex where it activates the synthesis and secretion of glucocorticoid hormones (CORT). CORT feeds back directly on the anterior pituitary to inhibit ACTH secretion, as well as acting at higher centres in the brain, including the hypothalamus and hippocampus.
Figure 8.2
Circadian and ultradian CORT rhythms in the rat
. CORT was measured in blood samples collected at 10 min intervals from a freely behaving male Sprague–Dawley rat. Shaded region indicates the dark phase. Data adapted from Walker
et al.
(2012).
Figure 8.3
Influence of constant CRH drive on the dynamics of the pituitary–adrenal system
. Computed one-parameter bifurcation diagrams showing the influence of different levels of
constant
CRH drive on the dynamics of the pituitary–adrenal system for three different values of adrenal delay (). (a) For min, the system is always at a stable steady state, independent of the level of CRH stimulation. (b) For min, sustained oscillations occur for a range of CRH levels between two supercritical Hopf bifurcation points H. (c) For min, sustained oscillations occur for an even larger range of CRH levels. Both Hopf bifurcations H have become subcritical and there are now saddle–node bifurcations of limit cycles (SL) and two very narrow regions of bistability (the regions between H and SL). (d) Period of the oscillation between the two Hopf bifurcation points in (b) as a function of CRH. (e) Period of the oscillation between the two Hopf bifurcation points in (c) as a function of CRH. In panels (a–c), black curves indicate stable solutions (either steady state, or maximum and minimum of the oscillation); gray curves indicate unstable solutions (either steady state, or maximum and minimum of the oscillation). AU, arbitrary units.
Figure 8.4
Influence of constant CRH drive and adrenal delay on the dynamics of the pituitary–adrenal system
. (a) Computed two-parameter bifurcation diagram shows that different combinations of
constant
CRH drive and adrenal delay result in qualitatively different dynamic responses from the pituitary–adrenal system. On one side of the Hopf curve (H), the pituitary–adrenal system responds to the steady-state levels of ACTH and CORT. On the other side of the bifurcation curve H, the pituitary–adrenal system responds to oscillations in ACTH and CORT. Within the region of oscillatory dynamics, the period of the oscillation is indicated by the color bar. (b–d) Numerical simulations for ACTH (gray) and CORT (black) corresponding to the points B, C, and D in panel (a). Simulations are shown for a 5-h period
after
transient dynamics have decayed. AU, arbitrary units.
Figure 8.5
Transient pituitary–adrenal dynamics in response to constant CRH infusion
. (a) Computed two-parameter bifurcation diagram showing how the qualitative dynamics of the pituitary–adrenal system depends on the level of
constant
CRH drive and adrenal delay . (b–f) Numerical simulations of the model corresponding to the points B, C, D, E, and F in panel (a). Simulations show both transient and long-term ACTH (gray) and CORT (black) dynamics in response to different CRH infusion rates: (b), (c), (d), (e), and (f). AU, arbitrary units.
Figure 8.6
Stability information for the steady state
. Real and imaginary parts of the leading eigenvalue(s) of a steady-state branch computed for and min. At the points H, the real part of the leading pair of eigenvalues crosses through zero, corresponding to Hopf bifurcations (a transition between steady-state and oscillatory dynamics) (Kuznetsov, 2004). The three top panels show the leading pair of complex conjugate eigenvalues in the complex plane for three different values of . AU, arbitrary units.
Figure 8.7
CORT oscillations induced by constant CRH infusion
. (a) Exemplar CORT response to constant 0.5 g/h CRH infusion. (b) Exemplar CORT response to constant saline infusion. Gray bars indicate the period of infusion.
Figure 8.8
Frequency comparison of CRH-induced and endogenous CORT oscillations
. (a and b) Normalized power spectra (b) of CORT oscillations (a) induced by constant CRH infusion (0.5 g/h) in an exemplar rat. (c and d) Normalized power spectra (d) of endogenous CORT oscillations (c) during the circadian peak in an exemplar rat. (e) Mean peak frequency (i.e., frequency corresponding to the maximum power in the spectrum) of CORT oscillations in response to constant CRH infusion (0.5 g/h; CRH; ), and of CORT oscillations during the circadian peak in untreated control rats (UC; ). Gray bar indicates the period of infusion. Shaded region indicates the dark phase. AU, arbitrary units. Error bars represent mean SEM.
Figure 8.9
ACTH and CORT oscillations induced by constant CRH infusion
. (a and b) Individual (a) and mean (b) ACTH and CORT oscillations in response to constant CRH infusion (0.5 g/h) (). (c and d) Individual (c) and mean (d) time course of the ACTH and CORT response to constant CRH infusion (0.5 g/h) during the initial activation phase (0–25 min) of the oscillation (). Gray bars indicate the period of infusion (starting at 0700 h); error bars represent mean SEM.
Figure 8.10
Damped oscillatory CORT responses to constant CRH infusion
. (a) Exemplar CORT response to constant 1.0 g/h CRH infusion. (b) Exemplar CORT response to constant 2.5 g/h CRH infusion. Gray bars indicate the period of infusion.
Figure 8.11
Timing of CRH stress impulse determines the magnitude of the CORT response
. (a) Profile of CRH impulse where the amplitude is scaled by the basal level of CRH. (b) Amplitude response curves of ACTH (solid blue) and CORT (solid red) computed for with varying phase of the CRH impulse. As a reference, the maximum levels of basal oscillations in ACTH (dashed blue) and CORT (dashed red) are also plotted. The shaded region indicates values of that correspond to the rising phase. Markers on the CORT amplitude response curve correspond to the time histories plotted in Figure 8.12.
Figure 8.12
Computational illustrations of the timing relationship between a CRH stress impulse and the magnitude of the CORT response
. (a) CRH impulse corresponding to for . (b–e) Time histories showing levels of ACTH (blue) and CORT (red) for fixed and values of as indicated in the panels. Vertical arrow in each panel indicates the timing of the applied CRH impulse. Levels of CORT in the absence of an impulse are shown in gray, with expected peaks indicated by vertical lines. The induced phase shift is the time separation between expected peaks (vertical lines) in the unperturbed case and the actual peak in CORT (black circles) for the perturbed case.
Figure 8.13
Parameter-dependent profiles of phase response curves
. (a) Phase response curves (PRCs) for different values of the pulse amplitude as indicated. For , the model exhibits Type-1 phase resetting (red shades) with a sharp change in phase near . For , the model exhibits Type-0 phase resetting (blue shades) with a discontinuous change in phase near . (b) Type of PRC curve plotted against .
Figure 8.14
Determining phase information from experimental stress-response data
. (a and b) Illustration of how peaks are selected in order to compute the phase information from experimental stress-response data. The time histories show levels of CORT sampled at 10-min intervals in exemplar female Sprague–Dawley (a) and female Lewis (b) rats. Shaded region indicates the period of the applied noise stress. Selected peaks () are marked red.
Figure 8.15
Comparison of theoretical PRC with experimental data confirms a Type-0 phase-resetting mechanism
. The Type-0 phase response curve for as computed with the model (black curve). The experimental data, plotted at discrete points, are shown for eight female Sprague–Dawley rats (black dots), five female Lewis rats (red diamonds), and six PVG rats (green stars). Points where two samples take the same value are circled.
Figure 8.16
Comparing the CORT response to an incoming stress between the oscillatory and steady-state regimes
. For the oscillating case, the basal level of CRH = 25, and for the steady-state case, the basal level of CRH is set such that the steady-state CORT level matches the maximum level of CORT in the oscillating case (dashed black). In the steady-state case, the response to a stress is independent of the timing of the stress (gray), whereas for the oscillatory case, we present the averaged response to an incoming stress applied at every point over the period of an oscillation (solid black). represents the magnitude of the stress. For small stressors, the response in both cases is comparable, whilst for larger stressors, the response in the oscillatory case is significantly greater.
Chapter 9: Modeling the Dynamics of Gonadotropin-Releasing Hormone (GnRH) Secretion in the Course of an Ovarian Cycle
Figure 9.1
Neuroendocrine control of ovulation.
