Potassium Channels in Neuron Axons

1
Potassium Channels in Neuron Axons
Sean Ewen Faculty Advisor Dr. Schvartsman Center
for Applied Mathematics University of St.
Thomas August 30, 2005
2
Introduction

• The nervous system comprises the electrical
  network of the human body
• It is responsible for our thoughts, senses, and
  movements

• Our nervous system transmits information
  incredibly fast
• What processes account for this transmission?
• What mechanisms drive these processes?
• What does Math have to do with this?

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The Basic Anatomy of a Neuron
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Anatomy of an Axon

• Think of the axon as a cable or electrical wire
• They both have a membrane or insulation
• However, axons have many channels that allow
  ions to pass through the membrane
• Axons have intracellular space where cables
  would have bundled wires

Vertical Cross-section of Axon
Membrane
Intracellular Space
Ion Channels
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The Action Potential

• Axons act as electrical wires, carrying current
  and charge
• Most human axons have a resting potential at -70
  millivolts (mV)
• When at resting potential, the axon acts as a
  capacitor
• Electrical charge moves through the axon like a
  wave
• This is called the action potential
• The level of charge dictates which and how many
  ions come through channels

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The Action Potential

• This graph shows an action potential at any
  given point

Threshold Potential
Resting Potential
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The Action Potential

• The action potential is driven by the
  differences in potential that occur along the
  axon
• The movement of ions in and out of the axon
  drives these differences in potential
• The following animation is a good description
  http//www.blackwellpublishing.com/matthews/channe
  l.html

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Speed of Action Potentials

• Potentials propagate anywhere from 0.1 to 100
  meters per second
• Speed depends on
• Higher temperature faster speed
• Bigger axon diameter faster speed
• Myelin sheath (insulation) faster speed

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Nervous System Reaction Times

• Varies person to person
• On a self-test my average was 390 milli-seconds
• This test involved a motor function after a
  visual stimulus

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A First-Order Equation

• We are studying potassium in axons because
• Potassium is much less dependent on potential
• Its actions are subtle and sensitive to changes

• A first-order equation reflects properties of
  kinetics
• It can model population growth and other common
  practical applications
• In our case, it applies specifically to
  potassium channels

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A First-Order Equation

• Based on concentration (between 0 and 1) of open
  channels and time

Rate channels go from open to closed
Rate channels go from closed to open
dn a(V) (1-n)?1 - ß(V) (n) ?2
? concentration
dT
? time
Concentration of closed channels
Concentration of open channels

• Concentration of open channels (n) is higher at
  higher voltages

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A First-Order Equation

• The equation can be represented as an
  equilibrium
• Closed Channels Open Channels

a
ß

• In a first-order equation, ?1 ?21, according
  to the rules of kinetics
• Our model may or may not be first-order

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Finding Change in Time

• At high voltages ( gt 40mV) ß(V) is close to or
  is zero, so we can drop the term - ß(V) (n)
• Next we isolate dT to get dT dn

a(V)(1-n)?

• Take the integral and set the limit between 0
  and 1

1
T dn
a(V)(1-n)?
0

• Since its possible the equation is not first
  order, gamma may not be equal to one

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Finding Change in Time

• This is a graph of the alpha and beta rates
  along different voltages

Alpha
Beta
Rate
Voltage (mV)
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Finding Change in Time

• Graph of T(?) at 40mV

Time (ms)
?

• Best fit seems to be ?1/8 (or 0.125) as curve
  begins to level out

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Finding Change in Time

• T(40mV) 3.62 milliseconds
• This is time it takes for channels to become
  completely open at 40mV
• Change in time seems to have an inverse
  relationship with voltage
• As voltage rises, change in time decreases
• As voltage decreases, change in time increases

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Conclusion

• We now know what carries the signal - the action
  potential
• But can this account for the incredible speed of
  the nervous system?
• A quantum hypothesis of brain function and
  consciousness suggests
• There may be interactions that occur at quantum
  levels
• There may be other processes that scientists are
  not aware of yet

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Future Research

• How do other ions play a role in the action
  potential?
• Further modeling with the Hodgkin-Huxley
  Equation
• How can synaptic transmission be modeled?
• What other associative processes influence the
  action potential?