Amirkabir University of Technology MATHEMATICAL MODEL FOR ACTION
Amirkabir University of Technology MATHEMATICAL MODEL FOR ACTION POTENTIAL Supervisor: Dr Gharibzadeh Designed by Yashar Sarbaz 1
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Action Potential 4
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In the real world, neurons have a variety of additional channels that shape their action potentials 7
Attention 8
A. L. HODGKIN and A. F. HUXLEY The Nobel Prize in Physiology or Medicine 1963 (with Eccles): "for their discoveries concerning the ionic mechanisms involved in excitation and inhibition in the 9 peripheral and central portions of the nerve cell membrane"
Separation of Current into its Na and K Components 10
HH Experiments in two Case n Normal Seawater n Low Na Seawater: Replace 90% sodium chloride by choline chloride while K and remaining chloride ions are unchanged 11
Three Assumption of HH 1. T: Time of peak inward current 2. Same voltage clamp but different 3. 12
Calculating Current of Na and K 13
HH Equations 14
Conductance Changes with Time 15
The Hodgkin-Huxley Model n Central concept of model: Define three state variables that represent (or “control”) the opening and closing of ion channels n n n m controls Na channel opening h controls Na channel closing n controls K channel opening 16
The Potassium Channel n The potassium has 4 similar sub units n Each subunit can be either “open” or “closed” (Protein 3 D Configurations) n The channel is open if and only if all 4 subunits are open 17
The Potassium Channel n The probability of a subunit being open: n The probability of the channel being open: n The conductance of a patch of membrane to K+ when all channels are open: (Constant obtained by experiments) n The conductance of a patch of membrane to K+ when the probability of a subunit being open is n: 18
The Kinetics of Potassium Channel Subunits 19
Dependence of the Potassium Channel Parameters to the Membrane Potential The potassium channel is closed in the resting membrane potential 20
Mathematical Model for K : Fraction of Open Channels : Conductance When all Channels are Open 21
Calculating n n Assuming n to Obey First Order Kinetics: 22
Solving n Equation 23
Curve Fitting for Rate Constants 24
Na+ Channels Have Two Gates F 8 -1525
The Sodium Channel n n The potassium has 3 similar fast subunits and a single slow subunit Each subunit can be either “open” or “closed” (Protein 3 D Configurations) n The channel is open if and only if all 4 subunits are open 26
The Sodium Channel n The probability of a fast subunit being open: n The probability of a slow subunit being open: n The probability of the channel being open: n n The conductance of a patch of membrane to Na+ when all channels are open: (Constant obtained by experiments) The conductance of a patch of membrane to Na+ : 27
The Kinetics of Sodium Channel Subunits 28
Dependence of the Sodium Channel Parameters to the Membrane Potential The slow subunit is open in the resting potential The fast subunit is closed in the resting potential The Sodium Channel is closed in the resting potential 29
Comparison of Voltage Dependence of channel kinetics 30
Mathematical Model for Na 31
Mathematical Model for Na 32
Solving m, h Equations 33
Solving m, h Equations 34
Border Condition For Na Channels in n n In the Steady State Conductance of Na is Near the Zero and Since m is increasing Function, then: At the Rest Conductance of Na is relatively Slow, So: 35
Main Relation for 36
Curve Fitting for Rate Constants 37
Curve Fitting for Rate Constants 38
Obtaining H for all V 39
H As Membrane Potential 40
Total Current of Membrane 41
Simulation of Action Potential 42
Calculation Changes in Membrane Potential 43
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- Slides: 44