an introduction to the mathematical modeling of...
TRANSCRIPT
An Introduction to the Mathematical
Modeling of Neurons: The Hodgkin-
Huxley Equations
Louis Tao 陶乐天 taolt @ mail.cbi.pku.edu.cn, letaotao @ pku.edu.cn
北京大学 生命科学学院 生物信息中心 定量生物中心
Center for Bioinformatics, College of Life Sciences
Center for Quantitative Biology
Peking University
交大致远学院 自然科学研究院 计算神经科学 短期班
17 December 2012
Neuronal Networks Are Complex
~1011 neurons & 1015 connections
104 cells & 1 km wiring in 1 mm3 of cortex
Neurons 神经元
• Information processing units 信息处理加工单元
• 1010-1013 neurons in mammalian brains 哺乳动物大脑
• 104 cell bodies and roughly 1 km of ‘wiring’ per mm3
• Different shapes, sizes, functions, … 不同形状、大小、功能
• Spiking vs. Analog neurons 锋电位(动作电位) (“analog neurons,” e.g., bipolar and amacrine cells in retina,
sensory-motor neurons of invertebrates, …
视网膜里的双极细胞与无长突细胞,无脊椎动物的感觉-运动 神经元,等)
• Many other cells (e.g., glia cells 胶质细胞) in cortex: to supply
energy, to provide structural stability, …, and not directly
involved in information processing
Computational Neuroscience
• What “computations” are performed by neurons & neuronal
networks?
• How are these computations done?
• What?
– Feature detection (visual systems, olfactory system, …)
– Coincidence / timing (auditory system)
– Memory (hippocampus)
– Sensory-motor (eye saccades, …)
– Neural Code: firing rate, spike timing
• How?
– Cell level: molecular and biophysical
– Network & systems level
• What & How to Study?
– Cellular: membrane potential, ion channels, synaptic mechanisms
– Extracellular: firing rates, spike times, statistics of spike trains, …
– Systems: fMRI, optical imaging, …
A cell at ‘rest’:
• ionic currents in dynamical equilibrium: net zero current
across membrane potential
• maintenance requires huge amounts of energy, it has
been estimated that half the energy consumed by brain is
used to maintain ionic charge gradients
• Vrest ~ -30 mV to -90 mV (depends on concentration
ratios of ions, e.g. Cl- )
Action Potentials 动作电位
• Action potential (spikes): voltage pulses responding to ‘strong’
input
• All-or-None electrical events, initiated near the cell body,
propagates along axons (at roughly constant velocity and
amplitude)
• Hodgkin & Huxley (1950s) studied various ionic currents
individually (using pharmacological blocks); used squid giant
axon (1/2 mm diameter)
• Action Potential generation involves 2 major, voltage-
dependent currents: sodium and potassium; individual ionic
current obeys Ohm’s law
,
,
1 ln
ion inside
ion ion ion syn ion
ion ion outside
cRTI G V V G V
R zF c
2
2
m mm Na K Leak syn inj
V VdC I I I I I
t R x
Hodgkin Huxley Neuron Model
1963 Nobel Prize
Squid
Giant Axon
2
2
m mm Na K Leak syn inj
V VdC I I I I I
t R x
3 4 Na Na Na K K KI g m h V V I g n V V
Hodgkin Huxley Neuron Model
2
2
2 2
55 mV, 115 mS/cm
90 mV, 36 mS/cm
65 mV, 0.1 mS/cm , 1 F/cm
/ 10 msec
Na Na
K K
Leak Leak m
m m Leak
V g
V g
V g C
C g
syn syn m synI G t V V
leak leak m restI G V V
Synaptic current: induced by
action potentials of other neurons
Leak current:
2
2
m mm Na K Leak syn inj
V VdC I I I I I
t R x
Hodgkin Huxley Neuron Model
Voltage Clamp
Delcomyn (1997)
Device to measure the current necessary to keep the desired V
2
2
m mm Na K Leak syn inj
V VdC I I I I I
t R x
Hodgkin Huxley Neuron Model
+25mV Vrest
voltage step command
KK
K
Ig
V E
nndt
dn
ngg
n
KK
4
Estimate n and n
from time course
Hodgkin Huxley Neuron Model
Vrest
Estimate n(V) and n(V)
for different voltage steps
n
0
1
V
Modeling the K-current
2
2
m mm Na K Leak syn inj
V VdC I I I I I
t R x
Hodgkin Huxley Neuron Model
3
Na Na
m
h
g g m h
dmm m
dt
dhh h
dt
Estimate m(V), m(V), h(V), h(V)
for different voltage steps V
Modeling the Na-current
2
2
m mm Na K Leak syn inj
V VdC I I I I I
t R x
Hodgkin Huxley Neuron Model
3
Na Na
m
h
g g m h
dmm m
dt
dhh h
dt
Estimate m(V), m(V), h(V), h(V)
for different voltage steps V
m
h
V
V 0
1
0
1 Modeling the Na-current
0.1 25
exp 0.1 25 1
4exp18
m
m
VV
V
VV
0.07exp20
1.0
exp 0.1 30 1
h
h
VV
VV
0.01 10
exp 0.1 10 1
0.125exp80
n
n
VV
V
VV
Gating Variable Kinetics:
• Action Potential generation involves 2 major, voltage-dependent currents
• Functional forms guessed by Hodgkin & Huxley to fit experimental data!!!
1 1 1m m h h n n
dm dh dnV m V m V h V h V n V n
dt dt dt
m 1 - m m
m
3 4 Na Na Na K K KI g m h V V I g n V V
K-channel, activation Na-channel, activation & inactivation
mm Na K Leak inj
VC I I I I
t
3 4 Na Na Na K K KI g m h V V I g n V V
Hodgkin Huxley Neuron Model
2
2
2 2
55 mV, 115 mS/cm
90 mV, 36 mS/cm
65 mV, 0.1 mS/cm , 1 F/cm
/ 10 msec
Na Na
K K
Leak Leak m
m m Leak
V g
V g
V g C
C g
syn syn m synI G t V V
leak leak m restI G V V
Synaptic current: induced by
action potentials of other neurons
Leak current:
Action Potentials 动作电位
• Weak input, no action potential, voltage deflection linear in
applied current
• Strong input, action potential generated
• Each ‘action potential’ is characterized by a stereotypical V-
trajectory
• Absolute and relative refractory periods
• Time for HH Numerical Simulations
Reference: F.C. Hoppensteadt & C.S. Peskin, Modeling and
Simulation in Medicine and the Life Sciences, Springer-Verlag,
2002, Chap 3 Matlab files at http://www.math.nyu.edu/faculty/peskin/ModSimPrograms/ch3/
HH Model Phenomenology
Weak applied current: membrane potential deflection linear in current
Current step applied between t=10 and 11
HH Model Phenomenology
Existence of Threshold in Action Potential (AP) Generation:
Applied current has to be sufficiently strong
Current step applied between t=10 and 11
8.28
ThresI
A
HH Model Phenomenology
Absolute Refractory Period after Action Potentials:
Time during which no amount of current can lead to another AP
Current step applied between t=5 and 6
HH Model Phenomenology
Relative Refractory Period after Action Potentials:
Time during which a larger than threshold current is needed for AP
Current step applied between t=10 and 11
8.28
ThresI
A
Recall
Some References:
• A. L. Hodgkin and A. Huxley, J. Physiol. (1952) series of
papers
• Jane Cronin, “Mathematical Theory of the Hodgkin-Huxley
Equations”
• Christof Koch, “Biophysics of Computation”
• Peter Dayan and Lawrence Abbott, “Theoretical
Neuroscience: Computational and Mathematical Modeling of
Neural Systems”