single neuron models (1) lecture 3. i.overview ii.single-compartment models − integrate-and-fire...
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Single Neuron Models (1)
LECTURE 3
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 genericone-compartment model
A
Ii
dt
dVc e
mm
A
Ii
dt
dVc
QVc
emm
m
H-H model
Passive or leaky integrate-and-fire model(…/cm2)
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
• Sub Threshold:
- Linear ODE - Without input ( ), the stable fixed point
at ( )LEV
0eI
emLm IRVEdt
dV
A
IEVg
dt
dVc e
LLm )(
(τm = RmCm = rmcm)
• 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
Vth
Vreset
V
t
resetref
kk
th
emLm
VtttV
tttVV(t)
IRVEdt
dVth : V(t)
]) ,([
)(spikes registered if
if
emLm IRVEdt
dV
Formula summary
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:
resetref
kk
th
emLm
VtttV
tttVV(t)
IRVEdt
dVth : V(t)
]) ,([
)(spikes registered if
if
emLm IRVEdt
dV
• The firing rate is a function of the membrane voltage
• g is usually a monotonically increasing function. These models mostly differ in the choice of g.
f g
Sigmoid function
if 0,
if 0)(
th
th
VVaaV
VVVg
V
f
I
f
100 HzPhysiological
Range
• Linear-Threshold model:
)( , VgfIRVEdt
dVemLm
• Based on the observation of the gain function in cortical neurons:
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
(μS/mm2 mS/mm2)
)( ii EVgi
iii Pgg
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
a gating or an activation variable
Activation of the conductance: Opening of the gate
Deactivation: gate closing
(k = 4)
Channel kinetics
nVnVdt
dnnn )()1)((
)()(
1)(
VVV
nnn
nVndt
dnVn )()(
)()(
)()(
VV
VVn
nn
n
opening rate
closing rate
For a fixed voltage V, n approaches the limiting value n∞(V) exponentially with time constant τn(V)
open closed n (1-n))(Vn
)(Vn
For the delayed-rectifier K+ conductance
Transient conductances
PNa = mkh
activation variable
(k = 3)
inactivation variable
zVzVdt
dzzz )()1)((
m or h
The Hodgkin-Huxley Model
A
Ii
dt
dVc e
mm
zVzdt
dzVz )()( Gating equation
The voltage-dependent functions of the Hodgkin-Huxley model
deinactivation
inactivation
activation
deactivation
Improving Hodgkin-Huxley ModelImproving Hodgkin-Huxley Model
Connor-Stevens Model (HH + transient
A-current K+) (EA~ EK)
transient Ca2+ conductance
(L, T, N, and P types.ECaT = 120mV)
Ca2+-dependent K+ conductance
- spike-rate adaptation
- type I behavior (continuous firing rate)
- Ca2+ spike, burst spiking, thalamic relay neurons
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
Synaptic open probability
Transmitter release probability
)( sss EVgi
Two broad classes of synaptic conductances
Metabotropic: Many neuromodulators including serotonin, dopamine, norepinephrine, and acetylcholine. GABAB receptors.
Ionotropic: AMPA, NMDA, and GABAA receptors
γ-aminobutyric acid
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
emLm IRVEdt
dV
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)
Then, the RK4 method is given as follows:
An initial value problem:
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 ?