December 16, 2015
UNIVERSITY OF RHODE ISLAND
Department of Electrical, Computer and Biomedical Engineering
BME 307
Bioelectricity
Fall 2015
The Plant Model of Bursting Nerve Cells
Simulation Project
Report due on Wednesday, December 23, 2015, by 2:30 pm
Late reports: 20% will be deducted for each hour after the deadline.
The MATLAB scripts and functions you wrote as part of Homework Assignments 8, 9, and 10 are the starting point.
You will modify your scripts and/or functions, and possibly create new ones, to conduct this experiment and
analyze the results.
Report: Your report should be targeted to an audience that understands the Hodgkin-Huxley model, but not your
topic of study. The report must include a statement of the problem (or the question being studied), the methods
used to solve the problem (including equations and numerical algorithms), and the results of your investigation.
Figures or graphics may be integrated with the text or arranged sequentially immediately after the references. The
report must close with a discussion section, where the results and their implications are described. Plots must show
appropriately labeled axes, including units. Appendices will contain your scripts and any lengthy derivations. Full
citations to any reference materials used in your study must be included.
Score: The projects will be graded 15% for your analysis (the content of the report) and 5% for the style of the
report. Superior reports will include analysis beyond what is required.
Bursting nerve cells are found in the vertebrate brain, crustacean stomach, and molluscan abdomen.
These neurons can exhibit different action potentials depending on the input stimulus; the action potential
may have a single fast upstroke (like in the squid giant axon), or multiple upstrokes when “bursting.”
In certain experimental preparations, the bursting behavior has been traced to the sodium conductance;
when tetrodotoxin (TTX) is added to the cell’s environment, bursting can be abolished because TTX
blocks certain sodium channels. Plant and colleagues [1, 2, 3, 4, 5] developed a complex model of the
bursting neuron using the approach of Hodgkin and Huxley. Their formulation follows that of the squid
giant axon model: nonlinear membrane conductances are modeled using a saturation value with
activation and inactivation gates governed by voltage-dependent opening and closing rates. This model
will spontaneously generate realistic voltage spikes with no stimulus current. A TTX-sensitive sodium
current regulates the firing rate. The purpose of this study is to implement the Plant bursting neuron
model and characterize its response to TTX.
The Plant model uses six state variables:
1. Vm , membrane potential
4. n, K+ activation gate
2. m, Na+ activation gate
5. x, Ca2+ activation gate
3. h, Na+ inactivation gate
6. [Ca2+ ]i , intracellular Ca2+ concentration
These state variables are handled much the same way as those in the Hodgkin-Huxley model, except for
the transport of calcium. The simulation will generate a membrane action potential (a non-propagating
action potential at a point).
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BME 307
December 16, 2015
Figure 1: The action potential Vm (left) and intracellular calcium concentration [Ca2+ ]i (right) from the
Plant bursting nerve cell model. No stimulus current was applied, and g Na = 3.0 mS/cm2 .
Modify the Hodgkin-Huxley scripts to implement the Plant model. The first simulation should cover
30 seconds using a time step ∆t = 0.1 milliseconds. Start with a zero stimulus current and a sodium
conductance of g Na = 3.0 mS/cm2 ; this should generate the action potential and calcium concentration
shown in Figure 1.
With the stimulus conditions above, create plots of the membrane potential, currents, gates, and the
intracellular calcium concentration. Compute the action potential amplitude, the duration at 90%
repolarization (APD90 ) and the maximum upstroke velocity, dV /dtmax (measure this on the first spike in
action potential).
Determine how the repetitive firing rate changes with the value of the sodium conductance. Decrease the
sodium conductance from 4.0 to 0.0 mS/cm2 by intervals of 0.4. Plot the spike firing frequency versus the
sodium conductance. Below g Na = 1.37 mS/cm2 you should observe an underlying “slow wave” that the
spiking action potentials appear to “ride.” Are the frequency and amplitude of this slow wave sensitive to
the sodium conductance? In what way(s)? You may want to consult the original research paper [4] for
more information.
[1] Richard E Plant and M Kim. On the mechanism underlying bursting in the Aplysia abdominal
ganglion R15 cell. Math Biosci, 26(3–4):357–375, 1975.
[2] RE Plant and M Kim. Mathematical description of a bursting pacemaker neuron by a modification of
the Hodgkin-Huxley equations. Biophys J, 16(3):227–244, Mar 1976.
[3] RE Plant. The effects of calcium++ on bursting neurons: A modeling study. Biophys J,
21(3):217–237, Mar 1978.
[4] Richard E Plant. Bifurcation and resonance in a model for bursting nerve cells. J Math Biol,
11(1):15–32, Jan 1981.
[5] Gerda de Vries and Richard E Plant. Plant model. Scholarpedia, 2(10):1413, 2007. Accessed at
http://www.scholarpedia.org/article/Plant_model on November 6, 2014.
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BME 307
December 16, 2015
The Plant Model of Bursting Nerve Cells
Currents are given in µA/cm2 , conductances in mS/cm2 , and potentials in mV. Note that l’Hôpital’s rule
must be applied to αm and αn (with respect to Vs ).
Jion = JNa + JNaS + JK + JK,Ca + Jleak
ion current
JNa = g Na · m3∞ · h · (Vm − ENa )
TTX-sensitive, fast Na+ current
slow Na+ current
JNaS = g T · x · (Vm − ENa )
JK = g K · n4 · (Vm − EK )
JK,Ca = g K Ca ·
fast K+ current
[Ca2+ ]i
· (Vm − EK )
0.5 + [Ca2+ ]i
slow, [Ca2+ ]i dependent K+ current
Jleak = gL · (Vm − EL )
leak current
Nernst potentials, conductances, and membrane capacitance:
ENa
EK
EL
ECa
Cm
g Na = 4.00
g K = 0.30
gL = 0.003
g K Ca = 0.030
g T = 0.01
= 30
= −75
= −40
= 140
= 1.0 µF
The initial values of the state variables are:
[Ca2+ ]i = 0.75841 mM
n = 0.05042
x = 0.22710
Vm = −60 mV
m = 0.00483
h = 0.95627
The intracellular calcium concentration [Ca2+ ]i is governed by the state equation
d
[Ca2+ ]i = ρ Kc · x · (ECa − Vm ) − [Ca2+ ]i
dt
where:
Kc = 0.0085 mM/mV
ρ = 0.0003 msec−1
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BME 307
December 16, 2015
The gates are governed by the opening and closing rates (in msec−1 ):
αm = 0.1
αh =
50 − Vs
exp [(50 − Vs )/10] − 1
βm = 4.0 exp [(25 − Vs ) /18]
0.07
exp [(25 − Vs ) /20]
12.5
αn = 0.01
55 − Vs
12.5 exp [(55 − Vs ) /10] − 1
βh =
1
12.5 exp [(55 − Vs ) /10] + 1
βn =
0.125
exp [(45 − Vs ) /80]
12.5
Note that the m gate does not directly activate JNa ; instead, it is replaced with its “steady state” value
computed by:
αm
m∞ =
αm + βm
and Vs = (127Vm + 8265) /105, not Vs = (127Vm − 8265) /105 as stated in the original paper [5]. The
calcium x gate is governed by the state equation
dx
x∞ − x
=
dt
τx
where x∞ =
1
and τx = 235 msec
exp [−0.15 (Vm + 50)] + 1
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