计算神经科学讲义

[K+ ]o mV [K+ ]i [Cl− ]o −61.54 log10 [Cl− ]i mV
ENa = 61.54 log10 ECa =
[Na+ ]o [Na+ ]i [Ca2+ ]o 30.77 log10 [Ca2+ ] i
mV mV
Goldman Equation and Reversal Potential
Example Calculate the longitudinal resistance of a segment 100 µm long with a radius of 2 µm): RL = rL L 1kΩmm × 100µm = ≈ 8MΩ. 2 πa π × 4µm2
Example Calculate a single channel’s conductance (6 nm long with a cross-sectional area 0.15 nm2 ): g= πa2 0.15nm2 = ≈ 25 × 10−12 S = 25pS. rL L 1kΩmm × 6nm
dV = −V + EL + Rm Ie dt where Rm = rm /A is the total membrane resistance and τm is the time constant. τm
Firing Rate w.r.t Constant Input
The subthreshold potential V (t) is obtained by solving the basic model as V (t) = EL + Rm Ie + (V (0) − EL − Rm Ie ) exp(−t/τm ) Suppose V (0) = Vreset and the neuron will fire an action potential at time t = tisi again, then V (tisi ) = Vth = EL + Rm Ie + (Vreset − EL − Rm Ie ) exp(−tisi /τm ) 1 = ⇒risi = tisi
Chapter 5 – Model Neurons I: Neuroelectronics
Lecturer Xiaolin Hu
Updated on Mar. 10, 2011
Outline
1
5.1 Electrical Properties of Neurons
2
5.2 Single-Compartment Models
3
5.3 Integrate-and-Fire Models
Ions and Molecules in a Neuron
A cubic micron of cytoplasm might contain, e.g., 1010 water molecules, 108 ions, 107 small molecules such as amino acids and nucleotides, and 105 proteins. Important ions: K+ , Na+ , Ca2+ , Cl−
Lipid bilayer
Ion Channels and Pumps
Movement of Ions
Diffusion
Electricity
Equilibrium and Reversal Potentials
[KA]i =20[KA]o
Ca2+ ? Cl− ? Net charges are distributed on the membrane surfaces ⇒ Membrane = capacitor. Ei : Equilibrium potential Ei can be calculated by Nernst equation, given the concentration difference
Definition Depolarization: make the membrane potential less negative. Hyperpolarization: make the membrane potential more negative. Na+ and Ca2+ tend to depolarize a neuron, K+ tends to hyperpolarize a neuron.
In this case, equilibrium potentials for individual ions are called “reversal potentials”.
Definition Depolarization: make the membrane potential less negative. Hyperpolarization: make the membrane potential more negative. Na+ and Ca2+ tend to depolarize a neuron,
Membrane Current
Membrane current per unit area (Positive-outward) im =
i
gi (V − Ei )
Ei Reversal potential for the ion labeled as i V : membrane potential, equal to about -65 mV at resting gi : conductance per unit area V − Ei : driving forces Question As both V and Ei are constants at resting potential, will the ion moves always in the same direction? PS: Leakage current gL (V − EL ), where gL and EL are free ¯ parameters
2
5.2 Single-Compartment Models
3
5.3 Integrate-and-Fire Models
Simplify Modeling
1
2
3
4
Model every detail of the membrane potential dynamics ⇓ Model only subthreshold membrane potential dynamics and ignoring modeling realistic action potentials. ⇓ All active membrane conductances are ignored ⇓ Leaky integrate-and-fire model cm or dV Ie = −¯L (V − EL ) + g dt A
−Vreset mI τm ln RRmeI+ELL −Vth e +E 0, otherwise. −1
Outline
1
5.1 Electrical Properties of Neurons
2
5.2 Single-Compartment Models
3
5.3 Integrate-and-Fire Models
The Model and Its Equivalent Circuit
cm
dV Ie = −im + dt A
Intracellular Resistance
IL : longitudinal current RL : longitudinal resistance rL : intracellular resistivity (1-3 kΩ mm) gL 1/RL : longitudinal conductance (unit: Siemens)
Membrane Capacitance and Resistance
For electrotonically compact neurons
Cm
dV dQ = dt dt
A: neuron surface area (0.01-0.1 mm2 ) Cm : membrane capacitance (0.1-1 nF) cm : Cm /A, specific membrane capacitance (10nF/mm2 ) Rm : membrane resistance (10-100 MΩ) rm : Rm × A, specific membrane resistance (1MΩ mm2 ) Rm Cm = rm cm : time constant (10-100 ms)
where im = i gi (V − Ei ) + gL (V − EL ): membrane current ¯ (positive-outward) Ie : input current (positive-inward).
Outline
1
5.1 Electrical Properties of Neurons
Membrane Current
Membrane current per unit area (Positive-outward) im =
i
gi (V − Ei )
Ei Reversal potential for the ion labeled as i V : membrane potential, equal to about -65 mV at resting gi : conductance per unit area V − Ei : driving forces