LECTURE 3
Single Neuron Models (1)
I. Overview
II. Single-Compartment Models
− Integrate-and-Fire Models
− Firing rate models
− The Hodgkin-Huxley Model
− Synaptic conductance description
− The Runge-Kutta method
III. Multi-Compartment Models
− Two-Compartment Models
Detailed descriptions involving thousands of
coupled differential equations are useful
for channel-level investigation
Greatly simplified caricatures are useful for
analysis and studying large
interconnected networks
From compartmental models to point
neurons
Axon hillock
I. Overview
II. Single-Compartment Models
− Integrate-and-Fire Models
− Firing rate models
− The Hodgkin-Huxley Model
− Synaptic conductance description
− The Runge-Kutta method
III. Multi-Compartment Models
− Two-Compartment Models
The equivalent circuit for a generic
one-compartment model
H-H model
cm
I
dV
 im  e
dt
A
cmV  Q
Ie
dV
cm
 im 
dt
A
(…/cm2)
Passive or leaky integrate-and-fire model
I. Overview
II. Single-Compartment Models
− Integrate-and-Fire Models
− Firing rate models
− The Hodgkin-Huxley Model
− Synaptic conductance description
− The Runge-Kutta method
III. Multi-Compartment Models
− Two-Compartment Models
• Maybe the most popular neural model
• One of the oldest models (Lapicque 1907)
(Action potentials are generated when the integrated sensory or synaptic
inputs to a neuron reach a threshold value)
• Although very simple, captures almost all of the
important properties of the cortical neuron
• Divides the dynamics of the neuron into two
regimes
– Sub- Threshold
– Supra- Threshold
Ie
dV
cm
  g L (V  EL ) 
dt
A
dV
m
 EL  V  Rm I e
dt
(τm = RmCm = rmcm)
• Sub Threshold:
－ Linear ODE
－ Without input (
at ( V  E )
L
Ie  0
), the stable fixed point
• Supra- Threshold:
– The shape of the action potentials are more or less
the same
– At the synapse, the action potential events translate
into transmitter release
– As far as neuronal communication is concerned, the
exact shape of the action potentials is not important,
rather its time of occurrence is important
• Supra- Threshold:
– If the voltage hits the threshold at time t0:
• a spike at time t0 will be registered
• The membrane potential will be reset to a reset
value (Vreset)
• The system will remain there for a refractory period
(t ref)
t0
V
Vth
Vreset
t
Formula summary
dV
m
 EL  V  Rm I e
dt
dV
if V(t)  th :  m
 EL  V  Rm I e
dt
t  registered spikes     (t  t k )

k
if V(t)  Vth 
V ([t , t  t ref ])  Vreset
I. Overview
II. Single-Compartment Models
− Integrate-and-Fire Models
− Firing rate models
− The Hodgkin-Huxley Model
− Synaptic conductance description
− The Runge-Kutta method
III. Multi-Compartment Models
− Two-Compartment Models
Under the assumption:
The information is coded by the firing rate of the
neurons and individual spikes are not important
We have:
dV
m
 EL  V  Rm I e
dt
dV
if V(t)  th :  m
 EL  V  Rm I e
dt
t  registered spikes     (t  t k )

k
if V(t)  Vth 
V ([t , t  t ref ])  Vreset
• The firing rate is a function of the membrane
voltage
f
g
Sigmoid function
• g is usually a monotonically increasing function.
These models mostly differ in the choice of g.
• Linear-Threshold model:
dV
m
 EL  V  Rm I e ,
dt
if V  Vth
0
g (V )  
aV , a  0 if V  Vth
f  g (V )
f
V
• Based on the observation of the gain function
in cortical neurons:
f
100 Hz
Physiological
Range
I
I. Overview
II. Single-Compartment Models
− Integrate-and-Fire Models
− Firing rate models
− The Hodgkin-Huxley Model
− Synaptic conductance description
− The Runge-Kutta method
III. Multi-Compartment Models
− Two-Compartment Models
Nobel Prize in Physiology or
Medicine in 1963
• Combination of experiments,
theoretical hypotheses, data fitting
and model prediction
• Empirical model to describe
generation of action potentials
• Published in the Journal of
Physiology in 1952 in a series of 5
articles (with Bernard Katz)
Stochastic channel
A single ion channel (synaptic receptor channel) sensitive to the neurotransmitter
acetylcholine at a holding potential of -140 mV
.
(From Hille, 1992)
Single-channel probabilistic formulations
Macroscopic deterministic descriptions
i  gi (V  Ei )
gi  gi Pi
(μS/mm2  mS/mm2)
the conductance of an open channel
× the density of channels in the membrane
× the fraction of channels that are open at that time
Persistent or noninactivating
conductances
PK = nk
(k = 4)
a gating or an activation variable
Activation of the conductance: Opening of the gate
Deactivation: gate closing
Channel kinetics
closing rate
dn
  n (V )(1  n)   n (V )n
dt
opening rate
For a fixed voltage V, n
approaches the limiting value
n∞(V) exponentially with time
constant τn(V)
dn
 n (V )  n (V )  n
dt
1
 n (V ) 
 n (V )   n (V )
 n (V )
n (V ) 
 n (V )   n (V )
For the delayed-rectifier K+ conductance
 n (V )
open
n
 n (V )
closed
(1-n)
Transient conductances
PNa = mkh
activation variable
(k = 3)
inactivation variable
dz
  z (V )(1  z )   z (V ) z
dt
m or h
The Hodgkin-Huxley Model
Ie
dV
cm
 im 
dt
A
dz
 z (V )  z (V )  z
dt
Gating equation
The voltage-dependent functions of the
Hodgkin-Huxley model
deinactivation
activation
inactivation
deactivation
Improving Hodgkin-Huxley Model
Connor-Stevens Model (HH + transient
A-current K+) (EA~ EK)
－type I behavior (continuous firing rate)
transient Ca2+ conductance
(L, T, N, and P types.
ECaT = 120mV)
－ Ca2+ spike, burst spiking, thalamic relay neurons
Ca2+-dependent K+ conductance
－spike-rate adaptation
I. Overview
II. Single-Compartment Models
− Integrate-and-Fire Models
− Firing rate models
− The Hodgkin-Huxley Model
− Synaptic conductance description
− The Runge-Kutta method
III. Multi-Compartment Models
− Two-Compartment Models
Synaptic conductances
is  g s (V  Es )
Synaptic open
probability
Transmitter release
probability
Two broad classes of synaptic
conductances
Metabotropic:
Many neuromodulators including serotonin, dopamine,
norepinephrine, and acetylcholine. GABAB receptors.
γ-aminobutyric acid
Ionotropic:
AMPA, NMDA, and GABAA receptors
Glutamate, Es = 0mV
Inhibitory and excitatory synapses
Inhibitory synapses: reversal potentials being
less than the threshold for action potential
generation (GABAA , Es = -80mV)
Excitatory synapses: those with more
depolarizing reversal potentials (AMPA,
NMDA, Es = 0mV)
The postsynaptic conductance
T = 1ms
A fit of the model to the average EPSC recorded
from mossy fiber input to a CA3 pyramidal cell
in a hippocampal slice preparation
(Dayan and Abbott 2001)
NMDA receptor conductance
1. When the postsynaptic neuron is near its resting potential, NMDA
receptors are blocked by Mg2+ ions. To activate the conductance,
the postsynaptic neuron must be depolarized to knock out the
blocking ions
2. The opening of NMDA receptor channels requires both pre- and postsynaptic
depolarization (synaptic modification)
(Dayan and Abbott 2001)
Synapses On Integrate-and-Fire Neurons
dV
m
 EL  V  Rm I e
dt
I. Overview
II. Single-Compartment Models
− Integrate-and-Fire Models
− Firing rate models
− The Hodgkin-Huxley Model
− Synaptic conductance description
− The Runge-Kutta method
III. Multi-Compartment Models
− Two-Compartment Models
The Runge-Kutta method (simple and robust)
An initial value problem:
Then, the RK4 method is given as follows:
where yn + 1 is the RK4 approximation of y(tn + 1), and
Program in Matlab or C
作业及思考题
1. 已知参数 EL = Vreset =−65 mV, Vth =−50 mV, τm =
10 ms, and Rm = 10 MΩ，在step 电流及其他不同电流
注射下，计算模拟整合－发放神经元模型。
2. 写出 Hodgkin-Huxley Model方程，说明各参数生物学
意义。
3. NMDA 受体电导有哪些特性?