Model neurons ! !The membrane equation!
Suggested reading:!
Chapter 5.1-5.3 in Dayan, P. & Abbott, L., Theoretical Neuroscience, MIT Press, 2001.!
Model neurons: The membrane equation!
Contents:
•! Ion channels •! Nenst equation •! Goldman-Hodgkin-Katz equation •! Ion concentrations •! Membrane capacity •! RC-circuit Model neurons: The membrane equation! 3! The neurons in the brain!
Examples of neurons in the brain.!
Neurons are already extremely complicated devices.!
They transfer signals by means of ion exchanges through channels made of proteins.!
Model neurons: The membrane equation! 4! Terms and definitions! Model neurons: The membrane equation! 5! Cell membrane with ion channels!
The cell membrane keeps a! voltage difference between! the cell and the surrounding by selectively allowing different ions in and out of the cell.!
Typical ions:! Calcium [Ca2+]! Potassium [K+]! Sodium [Na+]! Cloride [Cl-]! Membrane Potential:! Magnesium [Mg2+]!
Vm = Vi (t) !Ve (t)
Model neurons: The membrane equation! 6! Ion transporters and ion channels!
Ion transporters and ion channels are responsible for ionic movements across membranes. Transporters create ion concentration differences by actively transporting ions against their chemical gradients.! Channels take advantage of these concentration gradients, allowing selected ions to move, via diffusion, down their chemical gradients.! Model neurons: The membrane equation! 7! Electrochemical equilibrium! (A) A membrane permeable only to K+ separates compartments 1 and 2, which contain the indicated concentrations of KCl. !
(B) Increasing the KCl concentration in compartment 1 to 10 mM initially causes a small movement of K+ into compartment 2 until the electromotive force acting on K+ balances the concentration gradient, and the net movement of K+ Potassium [K+] ! becomes zero.! Cloride [Cl-]!
Model neurons: The membrane equation! 8! Nernst equation! Each ion has an equilibrium potential associated with whereby the diffusive forces and the electrical forces balance. This can be ! expressed by equal probabilities of an ion to cross the membrane:!
Pconcentration gradient = Pthermal energy
RT [C]out kBT [C]out Veq = Vi !Ve = ln = ln zF [C]in qz [C]in
T: the absolute temperature (273.16 + °C)! R: the universal gas constant (8.31451 joules/mol K°)! F: Faraday constant (96485.3 C/mol)! z: valence of the ion!
kB: Boltzman constant! Model neurons: The membrane equation! 9! Nernst equation! Each ion has an equilibrium potential associated with whereby the diffusive forces and the electrical forces balance. This can be ! expressed by equal probabilities of an ion to cross the membrane:!
Pconcentration gradient = Pthermal energy
RT [C]out kBT [C]out Veq = Vi !Ve = ln = ln zF [C]in qz [C]in
The Nernst equation only applies when the channels that generate a particular conductance allow only one type of ion pass through them.!
Model neurons: The membrane equation! 10! Explanation of the Nernst equation!
From the theory of thermodynamics, it is known that the probability that a molecule takes a state of energy E is proportional to the Boltzmann factor. ! !E p(E) " e kBT Electrical energy:! E(x) = zqu(x) Interpret probability as! ion density:!
zqu(x1 )!zqu(x2 ) +! ! [C] u(x) e kBT = in [C]out
E = "u = u(x1) ! u(x2 ) Model neurons: The membrane equation! 11! Goldman-Hodgkin-Katz equation!
The Nernst equation only applies when the channels that generate! a particular conductance allow only one type of ion pass through! them. In the presence of several different ions, the equilibrium! of the cell depends on the relative permeability P of the ions.! Potassium! Sodium! Chloride! + + " RT PK [K ]out + PNa [Na ]out + PCl[Cl ]in Veq = ln + + " F PK [K ]in + PNa[Na ]in + PCl [Cl ]out
The permeability of an ion depends on a number of factors such as! the size of the ion, its mobility, etc. ! E.g. Squid giant axon: P : P : P = 1 : 0.03 : 0.1! ! k Na Cl 1(10) + 0.03(460) + 0.1(40) V = 58log = !70mV eq 1(400) + 0.03(50) + 0.1(540)
Model neurons: The membrane equation! 12! The Goldman-Hodgkin-Katz equation can be linearized using! conductances and individual ion potentials.!
gK EK + g Na ENa + gCl ECl Veq = gK + g Na + gCl
Often this equilibrium potential is not computed explicitly
but defined as an independent leakage potential EL and determined by the resting potential given the experimental data (free parameter).!
I L = gL (V ! EL ) Model neurons: The membrane equation! 13! Extracellular and intracellular ion concentrations!
ENa pos.!
Model neurons: The membrane equation! 14! Simplified membrane capacity !
Q = CVm V m dV (t) I = C m C dt ! Model neurons: The membrane equation! 15!
The capacitance and membrane resistance of a neuron considered as a single compartment with area A!
The insulation of the membrane is modeled as a capacitance and the pores are described by a conductance. !
Model neurons: The membrane equation! 16! RC-Circuit if we inject a current I! Model neurons: The membrane equation! 17! RC-Circuit!
V !V I = m rest R R
Kirchhoffs law:! dV (t) V (t) !V C m + m rest = I (t) dt R e
Membrane Equation:! ! = RC
dVm (t) with units! " = !Vm (t) +Vrest + RI e (t) dt !F = sec
Model neurons: The membrane equation! 18!
Example: Inject current I0 at dV (t) " m = !V (t) +V + RI (t) t=0:! dt m rest e !t " Vm (t) = v0e + v1 Solution of the ODE! Equilibrium!(t " #) v1 = Vrest + RI 0 0 = "Vm + Vrest + RI0(t)
V (t = 0) = v + v = V V (t = 0) = V m 0 1 rest ! m rest ! Def.:! V! = RI0 !t # Vm (t) = V" (1! e ) +Vrest Model neurons: The membrane equation! 19!
Example: Inject current I0:!
Synaptic input into RC-Circuit:!
!t !(t!toff ) # # Vm (t) = V" (1! e ) +Vrest Vm (t) = V"e +Vrest
Appendix: Model neurons ! !!!!Numerical integration! Appendix: Model Neurons, Numerical integration! Analytic solution of the ODE!
Solution of an ordinary differential equation!
dr I " r i = i dt # Separate the variables!
t dri dt t " = ! ( 1) ln I r ln k # # # " # " i = " I " ri = ke I " ri $ $
t " # The constant k is determined ri = I " ke ! ! from the! boundary conditions!
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Appendix: Model Neurons, Numerical integration! Numeric solution of the ODE!
r(t1) = m(t1 " t0 ) + r(t0) Or more general using the stepsize h:! r(t + h) = mh + r(t) ! dr r(t + h) " r(t) = m = dt h
The Euler-integration method! approximates the unknown function r(t) piecewise by lines, by computing the slope m of the function in each interval. It then estimates the next value from the previous and the slope. ! The Euler method is simple,! but allows only for small stepsizes to be sufficiently exact.! ! min Rule of the thump for h:! h = 10 Appendix: Model Neurons, Numerical integration! Example! d " r = #r + I dt i i r (t + h) # r (t) " i i = #r + I h i ! h ri (t + h) = (#ri (t) + I) + ri (t) " !
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