1. Single Neuron Models (1)
LECTURE 3
Single Neuron Models (1)

2. Single-Compartment Models − Integrate-and-Fire Models
Overview
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

3. 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

4. From compartmental models to point neurons
Axon hillock

5. Single-Compartment Models − Integrate-and-Fire Models
Overview
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

6. The equivalent circuit for a generic one-compartment model
H-H model
Passive or leaky integrate-and-fire model
(…/cm2)

7. Single-Compartment Models − Integrate-and-Fire Models
Overview
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

8. 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

9. Sub Threshold: － Linear ODE (τm = RmCm = rmcm)
－ Without input ( ), the stable fixed point
at ( )

10. 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

11. Supra- Threshold: If the voltage hits the threshold at time t0: V t
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) V t0
Vth
Vreset t

12. Formula summary

13. Single-Compartment Models − Integrate-and-Fire Models
Overview
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

14. Under the assumption:
The information is coded by the firing rate of the neurons and individual spikes are not important
We have:

16. The firing rate is a function of the membrane voltage
Sigmoid function
g is usually a monotonically increasing function. These models mostly differ in the choice of g.

17. Linear-Threshold model:f V
Based on the observation of the gain function in cortical neurons: f
100 Hz
Physiological
Range I

18. Single-Compartment Models − Integrate-and-Fire Models
Overview
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

19. 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)

20. Stochastic channel
A single ion channel (synaptic receptor channel) sensitive to the neurotransmitter acetylcholine at a holding potential of -140 mV .
(From Hille, 1992)

21. Single-channel probabilistic formulations
Macroscopic deterministic descriptions

22. (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

23. 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

24. Channel kinetics closing rate opening rate
For a fixed voltage V, n approaches the limiting value n∞(V) exponentially with time constant τn(V)

25. For the delayed-rectifier K+ conductance
open closed
n (1-n)

26. Transient conductances
PNa = mkh
(k = 3)
activation variable
inactivation variable

27. m or h

28. The Hodgkin-Huxley Model
Gating equation

29. The voltage-dependent functions of the Hodgkin-Huxley model
deinactivation
activation
inactivation
deactivation

30. 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

31. Single-Compartment Models − Integrate-and-Fire Models
Overview
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

32. Synaptic conductances
Synaptic open probability
Transmitter release probability

33. 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

34. 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)

35. The postsynaptic conductance
T = 1ms

36. 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)

37. NMDA receptor conductance
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)

38. (Dayan and Abbott 2001)

39. Synapses On Integrate-and-Fire Neurons

40. Single-Compartment Models − Integrate-and-Fire Models
Overview
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

41. 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

42. Program in Matlab or C

43. 作业及思考题
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。
写出 Hodgkin-Huxley Model方程，说明各参数生物学意义。
NMDA 受体电导有哪些特性?