Single Neuron Models (1)

1. LECTURE 3 Single Neuron Models (1)

2. Overview • Single-Compartment Models • −Integrate-and-Fire Models • − Firing rate models • −The Hodgkin-Huxley Model • −Synaptic conductance description • −The Runge-Kutta method • III. Multi-CompartmentModels • − Two-CompartmentModels

3. Detailed descriptions involving thousands of coupled differential equationsare 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. Overview • Single-Compartment Models • −Integrate-and-Fire Models • − Firing rate models • −The Hodgkin-Huxley Model • −Synaptic conductance description • −The Runge-Kutta method • III. Multi-CompartmentModels • − Two-CompartmentModels

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

7. Overview • Single-Compartment Models • − Integrate-and-Fire Models • − Firing rate models • −The Hodgkin-Huxley Model • −Synaptic conductance description • −The Runge-Kutta method • III. Multi-CompartmentModels • − Two-CompartmentModels

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 reacha 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. (τm = RmCm = rmcm) • Sub Threshold: － Linear ODE － 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, ratherits time of occurrence is important

11. t0 • 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) V Vth Vreset t

12. Formula summary

13. Overview • Single-Compartment Models • −Integrate-and-Fire Models • − Firing rate models • −The Hodgkin-Huxley Model • −Synaptic conductance description • −The Runge-Kutta method • III. Multi-CompartmentModels • − Two-CompartmentModels

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 f g 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. Overview • Single-Compartment Models • −Integrate-and-Fire Models • − Firing rate models • −The Hodgkin-Huxley Model • −Synaptic conductance description • −The Runge-Kutta method • III. Multi-CompartmentModels • − Two-CompartmentModels

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

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. Overview • Single-Compartment Models • −Integrate-and-Fire Models • − Firing rate models • −The Hodgkin-Huxley Model • −Synaptic conductance description • −The Runge-Kutta method • III. Multi-CompartmentModels • − Two-CompartmentModels

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 Inhibitorysynapses: reversal potentials being less than the threshold for action potential generation (GABAA ,Es = -80mV) Excitatorysynapses: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. Overview • Single-Compartment Models • −Integrate-and-Fire Models • − Firing rate models • −The Hodgkin-Huxley Model • −Synaptic conductance description • −The Runge-Kutta method • III. Multi-CompartmentModels • − Two-CompartmentModels

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. Programin 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?