Axonal Excitability Workshop
Antalya, December 2012
Modelling myelinated axons: from Hodgkin-Huxley to the Barrett & Barrett model
Passive electrical components of axon membrane: i ENa
EK
ECl C
V GNa
GK
GCl
Total current = ionic current + capacity current i = iNa + iK + iCl + C.dV/dt i = GNa(V-ENa) + GK(V-EK) + GCl(V-ECl) + C.dV/dt
Active electrical components: Ionic conductances are time and voltagedependent variables: i ENa
EK
ECl C
V GNa
GK
GCl
Total current = ionic current + capacity current i = iNa + iK + iCl + C.dV/dt i = GNa(V-ENa) + GK(V-EK) + GCl(V-ECl) + C.dV/dt
Hodgkin-Huxley model of squid axon membrane I = Iionic + Icapacity
I = INa + IK + ILk + C.dV/dt INa = GNa(E-ENa) =m3h GNa(E-ENa)
I ENa
EK
IK = GK(E-EK) = n4 GK(E-EK)
ELk C
GNa
GK
ILk = GLk(E-ELk)
GLk dm/dt = αm(1-m) - βmm dh/dt = αh(1-h) - βhh dn/dt = αn(1-n) – βnn
Where ‘α’s and ‘β’s are empirical functions of membrane potential
Model
Squid axon
Model used to simulate saltatory conduction in demyelinated axons cf. Fitzhugh, 1962; Goldman & Albus, 1968
Node
ENa
EK
ENa
ELk CM
CN
V GNa
GK
Outside
Myelin
GLk
RM
RAx
EK
ELk
CM RM
CN GNa
GK
GLk
RAx Inside
1. Paranodal demyelination was modelled by increasing nodal GK and CN. 2. Continuous conduction across a demyelinated internode was simulated by replacing each segment of myelin with a segment of internodal axolemma, with a low density of Na and K channels
Membrane current contours (solid lines indicate inward current, dotted lines outward current)
Normal rat ventral root fibre
Bostock, 1995
Bostock, 1995
Normal rat ventral root fibre
Membrane current contours and action potentials computed for model axon (contours computed for 200Âľm electrode separation)
A
A. Recording from rat ventral root fibre, 6 days after diphtherial toxin injection, showing continuous conduction over a single internode
Bostock, 1995
A
B
C
A. Recording from rat ventral root fibre, 6 days after diphtherial toxin injection, showing continuous conduction over a single internode B,C. Currents and potentials in model nerve with one segment demyelinated and sufficient internodal sodium channels to support continuous conduction (PNa is 4x that in previous figure)
Bostock, 1995
Model used to simulate saltatory conduction in demyelinated axons cf. Fitzhugh, 1962; Goldman & Albus, 1968
Node
ENa
EK
ENa
ELk CM
CN
V GNa
GK
Outside
Myelin
GLk
RM
RAx
EK
ELk
CM RM
CN GNa
GK
GLk
RAx Inside
1. Paranodal demyelination was modelled by increasing nodal GK and CN. 2. Continuous conduction across a demyelinated internode was simulated by replacing each segment of myelin with a segment of internodal axolemma, with a low density of Na and K channels
Fig. 1. Action potentials and depolarizing afterpotentials recorded intracellularly from lizard myelinated peripheral axon.
If membrane potential is kept constant, DAP amplitude does not depend on extracellular K+, as would be expected if it were due to extracellular K+ accumulation.
A
B
Recordings made during penetration of single rat ventral root axon by microelectrode. A: dc potential with respect to bath, B: afterpotential of conducted impulse. After penetration, fibre repolarised as it recovered from damage (a-d) and depolarising afterpotential increased. After a short 200 Hz train, microelectrode slipped out of axon into periaxonal space (e), where action potential amplitude was unchanged, but there was no resting potential and afterpotential was inverted. (Grafe & Bostock, unpublished)
Interaction between GKs and GKf
1 = Control (TTX) 2 = Block of GKs 3 = Block of GKs + GKf
1 = Control (TTX) 2 = Block of GKf 3 = Block of GKf + GKs
“In each case, the drug applied second has the greater effect on the slow electrotonus. This is only to be expected when two parallel conductance pathways are blocked in turn.” Baker et al., 1987
Revised electrical model, based on Barrett-Barrett and electrotonus
Outside Myelin GBB
CM
Node ENa
ENap
EKf
EKs
EKf
EKs
EH
ELk
CN GNa
GNap
GKf
GKs
CI GKf
GKs
GH
GLk
Internode Inside
A
B (i)
(ii)
Modelling electrotonus in rat spinal roots. A: Responses of rat vental root in TTX to different 50 ms current pulses, showing different components of electrotonus. B: Responses of model axon to currents indicated, (i) potential across nodal membrane, (ii) potential across internodal membrane. (Bostock, 1995)
1 = Control (TTX) 2 = Block of GKs 3 = Block of GKs + GKf
1 = Control (TTX) 2 = Block of GKf 3 = Block of GKf + GKs
Modelling electrotonus in rat spinal roots. A,B: Responses of rat dorsal roots in TTX to 50 ms current pulses, showing effects of blocking fast and slow potassium currents. C,D: Matching responses of model axon (Bostock, 1995)
Modelling action potential at human node of Ranvier, and role of GKs in limiting repetitive firing during sustained depolarization
Model (with GKs)
Recorded
Model Model (no GKf)
15
ENa
EKf
EKs
GKS (nS)
Model (without GKs)
ELk CN
GNa
GKf
GKs
GLk
Schwarz et al., 1995
John Rothwell
Latent addition
Latent addition
Principal of the method
Method of recording
Latent addition in motor and sensory axons of normal subject C, D: Fit of 1 and 2 exponentials to 90% hyperpolarizing responses Bostock & Rothwell, 1997
Modelling latent addition
Model 1
Model 2
Model 3
Model 4
Passive RC membrane
Rat node Schwarz Eikhof, 1987
Human node Schwarz et al., 1995
Human node + persistent Na channels
Bostock & Rothwell, 1997
Electrical model of node and internode based on Barrett-Barrett, electrotonus and latent addition
Outside Myelin GBB
CM
Node ENa
ENap
EKf
EKs
EKf
EKs
EH
ELk
CN GNa
GNap
GKf
GKs
CI GKf
GKs
GH
GLk
Internode Inside