MATHEMATICAL MODEL FOR ACTION POTENTIAL Amirkabir University of Technology MATHEMATICAL MODEL FOR ACTION POTENTIAL Supervisor: Dr Gharibzadeh Designed by Yashar Sarbaz

Action Potential

In the real world, neurons have a variety of additional channels that shape their action potentials

Attention

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 peripheral and central portions of the nerve cell membrane"

Separation of Current into its Na and K Components

HH Experiments in two Case Normal Seawater Low Na Seawater: Replace 90% sodium chloride by choline chloride while K and remaining chloride ions are unchanged

Three Assumption of HH 1. T: Time of peak inward current 2. Same voltage clamp but different 3.

Calculating Current of Na and K

HH Equations

Conductance Changes with Time

The Hodgkin-Huxley Model Central concept of model: Define three state variables that represent (or “control”) the opening and closing of ion channels m controls Na channel opening h controls Na channel closing n controls K channel opening

The Potassium Channel The potassium has 4 similar sub units Each subunit can be either “open” or “closed” (Protein 3D Configurations) The channel is open if and only if all 4 subunits are open

The Potassium Channel The probability of a subunit being open: The probability of the channel being open: The conductance of a patch of membrane to K+ when all channels are open: (Constant obtained by experiments) The conductance of a patch of membrane to K+ when the probability of a subunit being open is n:

The Kinetics of Potassium Channel Subunits

The potassium channel is closed in the resting membrane potential Dependence of the Potassium Channel Parameters to the Membrane Potential The potassium channel is closed in the resting membrane potential

Mathematical Model for K :Fraction of Open Channels :Conductance When all Channels are Open

Calculating n Assuming n to Obey First Order Kinetics:

Solving n Equation

Curve Fitting for Rate Constants

Na+ Channels Have Two Gates F8-15

The Sodium Channel The potassium has 3 similar fast subunits and a single slow subunit Each subunit can be either “open” or “closed” (Protein 3D Configurations) The channel is open if and only if all 4 subunits are open

The Sodium Channel The probability of a slow subunit being open: The probability of a fast subunit being open: The probability of a slow subunit being open: The probability of the channel being open: 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+ :

The Kinetics of Sodium Channel Subunits

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

Comparison of Voltage Dependence of channel kinetics

Mathematical Model for Na

Mathematical Model for Na

Solving m, h Equations

Solving m, h Equations

Border Condition For Na Channels in 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:

Main Relation for

Curve Fitting for Rate Constants

Curve Fitting for Rate Constants

Obtaining H for all V

H As Membrane Potential

Total Current of Membrane

Simulation of Action Potential

Calculation Changes in Membrane Potential

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