Mathematical Models in

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Mathematical Models in Neural and
Neuroendocrine Systems
Richard Bertram Department of Mathematics
and Programs in Neuroscience and Molecular
Biophysics Florida State University
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Whats the Point?

• Integrate biological data
• Writing down equations often identifies holes in
  biological knowledge
• Use of math models helps to determine if
• data are logically consistent
• Making testable predictions
• Models force simplification, which can help
• in the identification of essential elements

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How Much Detail Should Go Into a Model?

• Some reasons that simpler is better
• More detail means more undetermined parameters
• More equations decrease our ability to comprehend
  the behavior of the model
• Added detail can be misleading. It can give the
  false impression that all the components are
  necessary for the behavior

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How Should One Calibrate a Biological Model?

• In engineering, one does experiments to
• determine the values of each parameter

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In biological models it is rare to be able to
measure all or most of the parameters
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Calibration Method for Biological Models
Make model predictions
Test them in the lab
Modify model
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Single Cell Models
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Example Neural Excitability
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Hodgkin-Huxley Model (1952)
Conservation of charge
Gating variables
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Hodgkin-Huxley Model (1952)
Strengths
Accurate
Essential elements
Nothing extra
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Hodgkin-Huxley Model (1952)
Strengths
Difficulties
Accurate
Nonlinear
Essential elements
High dimensionality
Nothing extra
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Morris-Lecar Model (1981)
The m ODE is removed by assuming that the Ca2
channels activate instantaneously.
The h ODE is removed since the Ca2
channels dont inactivate.
The resulting system is two-dimensional or
planar. This greatly simplifies the analysis.
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How do we analyze this?
Phase plane Study the dynamics in the
Vw-plane rather than V or w versus time
Nullclines Determine the curves along which one
of the time derivatives is 0
Steady states At the intersections of the two
nullclines both derivatives are 0, so the system
is at rest
Direction arrows The nullclines divide up the
plane, and the direction of flow in each region
can be determined
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Morris-Lecar with Iap0
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Morris-Lecar with Iap0
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Morris-Lecar with Iap100 pA
Increasing the applied current translates the
V-nullcline upward, so that the nullcline
intersection is on the middle branch of the
V-nullcline. The steady state is unstable. The
stable solution is a limit cycle.
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Morris-Lecar with Iap100 pA
Motion along the limit cycle is periodic.
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Hopf Bifurcation
This transition from a stable steady state to
a stable limit cycle through variation of a
parameter is called a Hopf bifurcation. It is one
way in which periodic motion can arise from a
previously stationary system, or vice versa.
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Morris-Lecar with Iap250 pA
Increasing the applied current more puts the
intersection on the right branch of the
V-nullcline. Here the steady state is again
stable. The system has gone through a second Hopf
bifurcation.
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Bifurcation Diagrams
These different behaviors and how they change
with variation of a parameter can be summarized
with a Bifurcation Diagram.
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Stationary Branches
For each value of the bifurcation parameter Iap
plot the V value of the steady state solution.
If stable make the curve solid if unstable make
the curve dashed. Stability changes at
bifurcation points.
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Periodic Branches
Next plot the maximum and minimum V values
of periodic solutions. Stability changes at
two Saddle Node of Periodics (SNP) bifurcations.
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Bistability
For some values of Iap the system is bistable.
For a single value of the parameter, a stable
steady state and a stable limit cycle coexist.
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Bistability in the Phase Plane
The unstable limit cycle (dashed) separates
the basin of attraction of the steady state from
that of the stable limit cycle. It is the
separatrix.
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Time Courses of Bistable System
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Example 2 Calcium Dynamics
Because Ca2 plays a large role in cells,
modelers have used differential equations to
describe the changes over time of the
concentration of free Ca2 in the cytosol,
endoplasmic reticulum, and in some cases in the
mitochondria. We will derive equations for the
free concentration in the cytosol and the ER,
starting with the cytosol.
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The Ins and Outs of Calcium
Jin influx of calcium
Jout efflux of calcium
Cac free calcium concentration
fc fraction of calcium ions not bound to buffer
proteins
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Calcium Flux Terms
Influx of calcium is through ion channels, with
current ICa
? converts current to flux
Efflux of calcium is through calcium pumps in the
plasma membrane. We can approximate this pumping
as proportional to the amount of free calcium in
the cytosol
kpmca is the pump rate
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Altogether Now
This is just an expression of conservation of mass
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Calcium Equations with an ER
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ER Fluxes
Jleak is calcium leakage from the ER into the
cytosol. This is driven by the calcium
concentration difference, so
JSERCA is calcium pumping from the cytosol into
the ER through SERCA pumps. For simplicity, we
assume that the pump flux is proportional to the
level of cytosolic calcium
JIP3 is flux through IP3 channels, when activated.
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Typical Time Courses
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Mean Field Models
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What is Mean Field?
These models are often used to describe the time
dynamics of a large population of cells, where
single-cell models would be inappropriate. With
a mean field model a single variable would be
used to describe, for example, the mean firing
rate of a population of neurons. Using
single-cell models here would result in
thousands of variables and equations.
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Example Circadian Prolactin Rhythm
Daily PRL rhythm occurs for first 10 days of
pregnancy in rats.
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What is the Mechanism for This?

• Our hypothesis is that the negative feedback of
  DA from the arcuate nucleus onto lactotrophs in
  the pituitary together with the delayed positive
  feedback of PRL onto DA neurons serves as a
  rhythm generator. This feedback loop provides
  delayed negative feedback to lactotrophs.

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The Dopamine Equation
DA is mean firing rate of DA neurons PRL is the
mean PRL secretory activity
Td is tonic stimulatory drive
kpPRL?2 is the delayed positive feedback
of prolactin
-qDA is first-order recovery
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The Prolactin Equation
The first term has DA in the denominator,
reflecting the inhibitory influence of DA on PRL
secretion.
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The PRL-Oxytocin Feedback Loop
We and others have shown that Oxytocin (OT)
stimulates lactotrophs, and that PRL inhibits OT
neurons. Unlike the stimulatory action of PRL on
DA neurons, the inhibitory action of PRL on OT
neurons is rapid.
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Modified Prolactin Equation
The first term now has OT in the
numerator, reflecting its stimulatory effect on
PRL.
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The Oxytocin Equation
The first term has PRL in the denominator, since
PRL inhibits OT neurons. There is no time delay.
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Daily Pulse of VIP
VIPergic neurons from the suprachiasmatic nucleus
(SCN) innervate DA neurons of the arcuate
nucleus and provide inhibition. The VIP activity
is high in the morning, and low other times
throughout the day.
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Modified Dopamine Equation
The last term, which is negative, reflects the
inhibitory action of VIP. The VIP is non-zero
for 3 hours in the early morning.
VIP
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Model Reproduces the PRL Rhythm
First 6 days
Day 3
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Uses of the Model
We have used the PRL rhythm model to help in the
design of experiments. Several predictions have
been made and tested, and based on the results
we have modified the model. It is a work in
progress!
Our goal with this mean field model is not to
make quantitative predictions, but to make
qualitative predictions. This is all that one
can expect when thousands of cells are
represented by a single variable!
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Thanks to
Marc Freeman
Marcel Egli
The NIH and NSF for financial support