CompNeuro_HodgkinHuxley
Dr. Cian O’Donnell
cian.odonnell@bristol.ac.uk
The Hodgkin-Huxley model
COMS30127: Computational Neuroscience
mailto:cian.odonnell@bristol.ac.uk?subject=
Questions you may have that we will answer today
• “What is the Hodgkin-Huxley model and why do I
need to know about it?”
• “Who were Hodgkin and Huxley?”
• “What does the model consist of?”
• “What does it do?”
• “What does it not do?”
What is the Hodgkin-Huxley model?
• The original Hodgkin-Huxley model is a mathematical model of the
electrical dynamics of the ‘giant’ axon of the squid Loligo forbessi.
• Its key success was to demonstrate that two voltage-gated
membrane conductances were sufficient to explain the action
potential.
• These days people often use the term “Hodgkin-Huxley style
model” more loosely to mean any mathematical model of any
neuron that is built using conductance-based dynamics.
• The Hodgkin-Huxley model stands as one of the outstanding
successes of computational neuroscience.
Hodgkin & Huxley, J Physiol (1952e)
Model
Data
Loligo forbessi
(Squid) giant axon Giant squid (axon)
Who were Hodgkin and Huxley?
Alan Hodgkin &
Andrew Huxley
• Physiologists based at Cambridge and Plymouth.
• Published a series of five landmark papers on the squid axon model of the action
potential in 1952.
• Began working together in 1938/9 but were interrupted for seven years by WW2.
• Awarded the 1963 Nobel Prize in Physiology or Medicine (along with John Eccles)
“for their discoveries concerning the ionic mechanisms involved in excitation and
inhibition in the peripheral and central portions of the nerve cell membrane”
What does the model consist of?
The HH model
CM
dV
dt
= INa + IK + Il
I
x
= g
x
(E
x
� V ) …where x is Na, K or l
g
x
=? How do we model the conductances?
gK = ḡKn
4(V, t)
gNa = ḡNam
3(V, t)h(V, t)
How do we model the conductances?
Using time and voltage-dependent gating variables.
dm
dt
=
m1(V )�m
⌧m(V )
How do the gating variables evolve in time?
How do the steady-state values and time constants depend on voltage?
m1(V ) =
↵m(V )
↵m(V ) + �m(V )
⌧m(V ) =
1
↵m(V ) + �m(V )
How do the forward and backward rate constants depend on voltage?
Hodgkin and Huxley fit them to match their voltage-clamp data.
Hodgkin & Huxley, J Physiol (1952e)
↵m(V ) =
0.1(V + 40)
1� e�(V+40)/10
�m(V ) = 4e
�(V+65)/18
↵h(V ) = 0.07e
�(V+65)/20 �h(V ) =
1
1 + e�(V+35)/10
↵n(V ) =
0.01(V + 55)
1� e�(V+55)/10
�n(V ) = 0.125e
�(V+65)/80
Gating variables steady-state values and time constants as a function of voltage
Dayan and Abbott (2001)
What does the HH model do?
Hodgkin & Huxley, J Physiol (1952e)
Data
Model
Koch (1999)
What does the HH model do?
What else does the HH model do?
• It is a “Type 2” model neuron
• Discontinuous fi-curve (unlike the integrate-and-
fire model).
What does the HH model not do?
• It is unlike the action potentials in mammalian neurons:
– different ion channels
– different waveform
– energy inefficient
– extremely leaky resting conductance
• Not a good model for myelinated axons
• It is deterministic.
We now know that ion channels are discrete (Neher and Sakmann) and noisy.
• Description of multiple independent gates per channel type is biophysically
unrealistic.
The ion channel zoo
Yu et al., Pharm Rev (2005)
Energy efficiency of mammalian axons
Alle et al., Science (2009)
Neher & Sakmann, Nature (1976)Hille (1992)
O’Donnell & Nolan (2014)
Volume 71 December 1996
TABLE I Parameters used in the theory and simulations
C Membrane capacitance 1 AF/cm2
VL Leakage reversal potential -54.4 mV
gL Leakage conductance 0.3 mS/cm2
VK Potassium reversal potential -77 mV
gK Maximal potassium conductance 36 mS/cm2
PK Potassium ion channel density 18 channels/jxm2
NK Number of potassium channels
Nnk Number of potassium channels in state nk
VNa Sodium reversal potential 50 mV
gNa Maximal sodium conductance 120 mS/cm2
PNa Sodium ion channel density 60 channels/,um2
NNa Number of sodium channels
Nmihj Number of sodium channels in state mihj
where m3h1 corresponds to the open state where the three
activating m gates and the inactivating h gate are open.
For the classic parameters of Table 1 the voltage-depen-
dent rate constants have the form
0.0l(V + 55)
an= e(V+55)/1O 13n = 0.125e-
O.1I(V±+ 40) V6)1
am = 1 -(V+40)/1O’ 13m = 4e-
ah = 0.07e-(V+65)/2 Ph = 1 + e-(V+35)/10
(5)
(6)
(7)
where QK and QNa are the number of “open” potassium and
sodium channels; NK and NNa are the total number of
potassium channels, given by the equations NK = PK x Area,
and NNa = PNa X Area, where PK and PNa are the K+ and Na+
channel densities, respectively; and gK and gNa give the max-
imum conductance densities when all the channels are open.
Hodgkin and Huxley (1952) found that the electrical
properties of the squid giant axon were well modeled by
considering conductance to be governed by the states of a
finite number of independent binary “gates.” In the modern
view, one could consider these gates to be relevant at the
single channel level. A given channel would only conduct
when all of its composite gates were in the open position. In
this scheme, the potassium channel is composed of four
identical gates, while the sodium channel is composed of
three identical “activating” gates and one “inactivating”
gate. The opening and closing of these gates are inherently
probabilistic. In classical analyses of voltage-clamp data,
only the mean values of the fraction of open channels are
considered. This is a valid approximation for large numbers
of channels in a large patch of membrane. However, for a
small patch, statistical fluctuations will play a role.
Assuming a simple Markov process for four identical
gates with an opening rate an and a closing rate of o3n, the
kinetic scheme for the potassium channel is given by (Skau-
gen and Wall0e, 1979; Hille, 1992; Strassberg and De
Felice, 1993)
where V has units of mV and the rates have units of ms-1.
In the continuous limit for the classic Hodgkin-Huxley
equations, the conductances satisfy
gK = gKn gNa = gNamh, (8)
where
dn
-= an(l -n)- gnn,
dm
dt = am(I -m) – Pmm,dt
dh
-=-ah(l -h)-P3hh.dt
(9)
(10)
(11)
Here m, h, and n are mean gate fractions.
For a fixed membrane potential these equations approach
steady-state rest values of
n(V) = n,(V) = n
am
m(V) =m (V) am + Pm
ahh(V)= h.m(V-ah+Ph’
(12)
(13)
(14)
4an 3an 2an an
no > ‘ n I ~ n2—-fn3 > n4, (3)
13n 29n 31n 4fn
where n4 corresponds to the open state where all four gates
are open. The Markov kinetic scheme for the Na+ channel
is given by
3am 2am am
m0h1 m1h1 > m2h1 m3hl
g3m 2pm 33m
ah TIPh ah UjPh ah TPPh ah T |Ph (4)
3am 2am am
moho > mlho ‘ m2ho ‘MO=
gm 29m 39m
STOCHASTIC THEORY
We want to analyze Eq. 1 with channel kinetics given by the
Markov schemes Eqs. 3 and 4 for subthreshold current
injection. In particular we wish to examine the generation of
spontaneous action potentials due to the fluctuations in the
ion channels. Our strategy is to coarse-grain the problem
into a continuous time stochastic problem. We will trans-
form the membrane equation into a Langevin equation. The
probabilistic nature of the channels will appear as a noise
source in the stochastic equation. This will then be analyzed
using standard techniques of nonequilibrium statistical me-
chanics. We will then compare with numerical simulations.
Our analysis depends on the separation of time scales
present in the dynamics. We will use approximations that
3014 Biophysical Journal
2-state:
8-state:
Ion channels are discrete and stochastic
Spontaneous spikes from ion channel noise
Strassberg & DeFelice, Neural Comput (1993)
Faisal et al., Curr Biol (2005)
A lower limit on axon diameter
End