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MODELING VOLTAGE-DEPENDENT CONDUCTANCES
TO INVESTIGATE ION CHANNEL BEHAVIOR
DURING AN ACTION POTENTIAL
Angie Pronger
December 5, 2007
Abstract
Introduction
Cellular behavior is greatly influenced by the conductances of its membrane. In
neural signaling, the conductances partially determine when particular ion channels are
activated or inactivated and when maximum activation occurs. By analyzing the shape of
a particular conductance curve, it is possible to determine which phase of neural signaling
is most affected. For example, the steepest part of the voltage-dependent activation curve
for calcium conductance pertaining to the B-current occurs at voltages that correspond to
the peak voltages of an action potential (Bower & Beeman). Therefore, the calcium
channels pertaining to the B-current achieve maximum activation during and immediately
after the peak of the action potential. The curve also demonstrates that it is not possible to
activate these calcium channels in the absence of an action potential since activation does
not occur at the corresponding voltages.
However, identification of channel activity at particular voltages through
examination of conductance curves result in misleading conclusions about neural
behavior. Although voltage-dependent conductances help to define the activity of ion
channels, neural activity is shaped by various factors. The resting membrane potential,
inactivation and activation properties, and the concentration of extracellular and
intracellular ions all interrelate and jointly affect the ionic current flow through the
membrane. Moreover, these factors show great variation between different species as
well as between different kinds of cells within an organism. In glomus cells of the rabbit
carotid body, the resting membrane potential is around -48 mv (Overholt & Ficker,
2000), in epicardial cells of the left ventricle, the potential is around -81 mV (Fedida &
Giles, 1991). Additionally, neural signaling is highly dependent on chemical and
electrical gradients. Firing can also be influenced by ion channel density and the presence
and conformation of gates. In the Shaker potassium channel, a change in tilt of the S4
domain is suggested to control the gate of a separate pore domain or remove an arginine
residue from the voltage-sensitive domain to allow the influx of ions through the channel
(Tambola et al, 2007). The Hodgkin-Huxley model of the action potential incorporates
these factors in the four characteristics of action potentials: threshold, all-or-none
response, undershoot and the refractory period. The Hodgkin-Huxley equations are as
follows (Nelson & Rinzel, 1994):
(i)(a) I= CM dV/dt
+
___________
capacitive current
gkn4 (Vm-Vk) + gNam3h(Vm-VNa) + gl(Vm-Vl)
_____________
K+
current
_________________
Na+
current
___________
leak current
(primarily Cl-)
Or stated otherwise:
(b) dV/dt = 1/C [Injection current - ∑gi(V) (Vi-V)]
Time and voltage dependence:
(ii) dn/dt = an (1-n) – bnn
 gK+ activation
(iii) dm/dt = am (1-m) – bmm  gNa+ activation
(iv) dh/dt = ah (1-h) – bhh
 gNa+ inactivation
(n, m, and h are gating parameters)
Thus, the activity of each channel can be interpreted by its voltage dependent
conductance of the form gi(v) = gonk, where n (the probability of channels opening), k
(the number of gates per channel or cooperative index) and go are changeable parameters.
The aim of this project is to assess how the dynamics of each type of ion channel on
neuronal firing by creating an electrical circuit that models neural activity.
Design
The electrical equivalent of the Hodgkin-Huxley model is based on the principle
that when current flows across the membrane, some of it is used to charge the membrane
capacitance and some of it is used to carry ions across. the membrane. Thus, the ionic
current is divided into three components: a sodium, potassium, and small leak current that
consists primarily of chloride ions. The circuit proposed by Maeda and Makino was
modified in the present study and a spike frequency adaptation component was added.
(Figure 1).
To best explain what is happening at the electrical level and how this models
physiological phenomenon, it is helpful to divide the diagram into three main parts. The
capacitor labeled Cm models the membrane capacitance. The greater this value, the
longer the circuit takes to charge to reach the voltage provided by the battery the
threshold of firing determined by the transistors in the electrical circuit modeling fast
Na+ channel. Rleak models the inward leak current. The greater the resistance, the
greater smaller the magnitude of the outward flow of negative chloride ions is. The fast
inward current of sodium ions is modeled in the first portion of the circuit by transistors
BJT1, Digi-Key part no 2N3904, and BJT2, Digi-Key part no 2N3906, and resistors
RNa1, RNa2, and RNa3. Modification of these components will determine the shape of
the of the depolarization voltage phase of the action potential. The delayed outward
potassium current is modeled in the second portion of the circuit by transistor BJT3,
Digi-Key part no 2N3904, resistors RK1, RK2, and capacitor CK1. Modification of these
components will determine the shape of the action potential during repolarization. The
spike frequency adaptation circuit was added to change the firing pattern of the action
potential over time.
Properties
A.
Figure 1: Electrical circuit
modeling the spike-adapting
neuron. A. Circuit schematic. The
three ionic channels are
highlighted by the red rectangles.
From left to right: fast sodium
current, delayed potassium
current, spike-frequencyadaptation circuit. B. Photograph
of the circuit implementation.
Corresponding parts of the circuit
were also highlighted: blue membrane capacitor and leakage
resistor, red - fast Na+ current,
dark cyan - delayed K+ current,
light cyan - spike frequency
adaptation circuit, lack - voltage
divider that allows to linearly
change the supply voltage
between 0 and 12 V. .
B.
Figure 2 placeholder: a single action potential zoomed-in, side-by-side with a biological
action potential - you can steal a pic of that off the web, or from Jimmy-and-Jeff's paper,
attached.
Resistors
A resistor is an electrical component that resists current by creating a voltage drop
between its terminals. In a voltage divider, two resistors are placed in series with a
movable pin in between. As the pin moves up or down, one resistor becomes longer and
the other becomes shorter. The longer the wire is, the greater the resistance. Once the pin
is adjusted, the voltage divider creates a stable output voltage which is proportional to the
input voltage. Furthermore, voltage dividers can create precise reference voltages which
are commonly used as inputs to devices of high impedance, so as not to put a high load
on the divider.
Capacitors
A capacitor is an electrical component that consists of two parallel plates, each capable of
accruing charge. When a voltage source is provided, electrical charges of equal
magnitude and opposite polarity build up on each plate and the capacitor begins to store
energy. The capacitor will then continue to charge until it has reached the source voltage
or until there is somewhere else for the current to go. The voltage across the capacitor is
1 e t
et
expressed by: V C t
during charging and Vc t
,
where
is the time constant, equal to the circuits resistance times the
capacitance R C and
is the electromotive force, the maximum voltage
value of the capacitor.
Transistors
Current within a transistor can be controlled by changing the energy barrier
between the p- and n-enriched regions. The energy barrier for the current between
collector and emitter is determined by the control voltage on the base. In a neuron, the
state of a given channel depends on its previous state of activation. The rate at which it
changes between states is dependent upon the energy difference between the states.
Therefore, transistors can mimic neural behavior by establishing a threshold under which
current will not pass (Mead, 1989).
asdfasdfas
Figure 3. Input-output characteristic of a bipolar junction transistor 2N3904. The output
voltage Vce (collector-emitter), measured through a load resistor ??, rises exponentially
once the input voltage Vg (gate voltage)
Neuron Threshold
exceeds the threshold (in this case 0.5
V).
450
400
350
300
250
200
150
100
50
0
0
0.5
1
sadfasdf
1.5
Operational Amplifier - LOOK UP IN
WIKIPEDIA
-low impedance
Compares the reference voltage to input
of neuron, -> fires
Principles of Operation
Current flows from a point of higher voltage to a point of lower voltage. When the
6V power source is turned on, the positive voltage will flow toward the top of the circuit,
the inside membrane, through RNa3. Because the membrane voltage at this point is less
than the threshold of the diode, current will flow only through Cmem and Rleak. Rleak
provides resistance to model leak current of neurons, particularly of chloride ions. Cmem
models the membrane capacitance of a neuron. As the capacitor charges, the voltage
across the membrane and ground, which models extracellular space, begins to rise.
Eventually the membrane voltage becomes great enough to overcome the threshold
before the diode. Current will then begin to flow through the diode to BJT1. Once the
threshold of BJT1 is reached, current can flow to BJT2. When the threshold for BJT2 is
overcome, the opening of its gate will permit current to flow from the battery, through
RNa2 and through BJT2 toward the membrane. The opening of this gate creates a
positive feedback loop, similar to the one in opening of Na+ voltage-dependent ion
channels in the upshoot phase of the action potential (Figure 2). The higher is the voltage
differential between the Cmem and the ground, the more the BJT2 gate opens, and the
higher is the current that flows through RNa2, thus further increasing V(Cmem).
Eventually this voltage becomes high enough to open the gate on BJT3 and activate the
delayed K+ current that terminates the action potential (downshoot, Figure 2).
In-Circuit Experiments
First the neural model implemented by Maeda and Makino (Figure 1 minus the spike
frequency adaptation circuit) was constructed. An input voltage of 6.0 volts (AC) was
used. In order to investigate the behavior of the circuit various parameters were changed:
resistors, capacitors, and number of diodes used. For each adjusted parameter, the firing
frequency (Hz) was measured, as indicated by the oscilloscope, and the action potential
width (sec), the total amplitude of the peak (mV) and total period (s) were assessed.
Additional diodes whose resistances were recorded were then inserted in series before the
diode of the fast inward current. In order to obtain a more drastic increase in resistance at
the point of that diode, resistors were then used instead of diodes. The effect on the
threshold (mV) was observed. Finally, the components of the delayed outward current
were modified and the firing frequency (Hz) and action potential amplitude (mV) were
recorded.
Results
Effects of R2 in Na Channel
2500
180
2000
160
140
Frequency (Hz)
Amplitude (mV)
Effects of R2 in Na Channel
1500
1000
500
120
100
80
Frequency, Hz
60
40
20
0
0
200
400
600
800
1000
0
1200
0
Resistance (kOhm)
100
A
.
200
300
400
500
R2 (kohm)
B
.
Figure 4: Effects of the circuit parameter values on the action potential. A. As the
resistance of RNa1 increases, the amplitude of the spike increases and reaches a
maximum around 2250mV. B. In response to an increasing resistance, the firing
frequency is greatly reduced then begins to taper. Correspondingly, the period of each
action potential increases. This ranges from about 8.5 seconds at 56kOhm to 20 seconds
at 438 kOhm (data not shown).
Angie's neuron parameter space, time dimension
1400
1200
1000
total spike amplitude,
mV
charge-up phase height,
mV
800
600
400
200
0
10.5
11
11.5
12
input voltage, V
12.5
measured parameters, msec
measured parameetrs, mV
Angie's neuron parameter space, voltage dimension
30
25
20
AP width, msec
15
total period, msec
10
5
0
10.5
11
11.5
12
12.5
input voltage, V
Figure 5:
While the charge-up phase of the action potential was unaffected by an increasing input
voltage, the spike amplitude increased greatly.
Blue squares:
Pink squares:
Correspondingly, the increasing input voltage results in a decreasing action potential
width and period.
Capacitance and Firing Frequency
45
40
Firing freq
35
30
25
20
15
10
5
0
0
1
2
3
4
5
K capacitor, m icroF
Figure 6.
By increasing the capacitance of KC, the firing frequency decreases due to its ability to
hold a greater amount of charge.
Voltage Across Capacitor
0.7
voltage (mV)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
12
time (ms)
Spike Frequency Adaptation
7
amplitude (mV)
6
5
4
3
2
1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
time (ms)
Figure 7. Spike frequency adaptation. Panel A: Voltage across Csfa (spike frequency
adaptation capacitor) rises with every spikes as the neuron fires. Once the opening
voltage of the transistor BJT4 is reached (marked by arrow), it begins to drain current
from the membrane capacitor, thus slowing down the firing (panel B).
Discussion
Spike frequency adaptation is commonly exhibited in neurons of humans as well
as other species. ?? >check references, elaborate here<
Discuss firing patterns, i.e. “Action-Potential Broadening and Endogenously Sustained
Bursting Are Substrates of Command Ability in a Feeding Neuron of Pleurobranchaea”
*how transferable to robots
*further application (combing project)
*spike-frequency adaptation
*postinhibitory rebound ?
Sources: Jones2001, Angstadt2005.pdf, data and model Sekirnjak2002.pdf , some
overview of pacemaking, not sure how useful Saint1998.pdf are modeled using ion
channels. model them in hhsim.
Acknowledgments
Appendices
Works Cited
HHSim: http://www.cs.cmu.edu/~dst/HHsim/
SNNAP (simulator for neural networks and action potentials): http://snnap.uth.tmc.edu/
HHSim online w/ equations: http://afodor.net/HHModel.htm
Ion Channels in Bursting Neurons:
Bower, J. & Beeman, D.
Ch. 7 Ion Channels in Bursting Neurons
Digital Neurons (Analog and Digital Neural Models)
http://www.nbb.cornell.edu/neurobio/land/PROJECTS/NeuralModels/index.html
VLSI (pdf):
http://www.biology.ucsd.edu/~gert/courses/bggn260/2006/BGGN260_2006_aVLSI.pdf
Vertebrate values:
http://jn.physiology.org/cgi/content/full/87/2/995
Ion Channels in Human Axon:
http://jn.physiology.org/cgi/content/abstract/70/3/1274
Overholt, J., Ficker, E., Yang, T., Shams, H., Bright, G., Prabhakar N. “HERG-Like
potassium current regulates the resting membrane potential in glomus cells of the
rabbit carotid body.” 2000. Journal of Neurophysiology, 1150-7.
Fedida, D. & Giles, W. “Regional variations in action potentials and transient outward
current in myocytes isolated from rabbit left ventricle.” 1991. Journal of
Physiology, 191-209.
Tambola, F., Pathak, M., Gorostiza, P., Isacoff, E. “The twisted ion-permeation pathway
of a resting voltage-sensing domain.” 2007. Nature, 546-549.
Nelson, M. & Rinzel, J. “The Hodgkin-Huxley Model.” “The Book of GENESIS:
Exploring Realistic Neural Models with the GEneral NEural SImulation System.”
Bower, J. & Beeman, D. Sringer-Verlag, New York, 29-49.
Mead, C. “Analog VLSI and Neural Systems.” Addison-Wesley. 1989.
Neuroscience book
Brain and Neuron book
NOTES
Put in graphs, captions
Discussion- SFA
Tomorrow: principles of operation, discussion: robots, further application