Electrical properties of neurons Rub

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Electrical properties of neuronsRubén
Moreno-Bote
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Galvani frogs legs experiment
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• Overview
• Passive properties of neurons (resting potential)
• Action potential (generation and propagation).
• Synaptic currents (AMPA, GABA, NMDA).
• 4. Reduced models of neurons (LIF, QIF, LNP).
• 5. Neuronal networks (balance and chaos).

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1. Passive properties of the neuron membrane 1.1
Membrane potential.
Current, I
Vout
Vin
Vin-Vext -RI Ohms law Igt0, inward
current, which means that Vin is negative, Vin
-70 mV (we define Vout0) I, current-gt Amperes,
AC/s (order of magnitude 10 nA, 10 µA) V,
potential-gt Volts (100mV, action potential
0.1-1mV, postsynaptic current) R, resistance-gt
Ohms, ?, V/A (1 M?) g, conductance, g1/R-gt
Siemens, S (1/ ?) (order µS). Ohms law with
conductances I-g(Vin-Vout)
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1.1 Membrane potential. Currents, resistances and
capacitors.
Current, I
Vout
capacitor
Vin
Membrane is impermeable to ions and creates the
voltage difference (Capacitor). QCV C,
Capacitance-gt Faradays (order 1 nF) Extra and
intracellular fluid is electrically neutral.
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1.2 Ions and Ion Channels
Ca2
Cl-
Na
K
K
Na
Ca2
Cl-
Cations Anions - Channels are selective to
particular ions. Passive vs. Active
channels. Permeability is very high to K and Na,
medium to Cl and very low to big
anions. Kin20 Kout Naout10 Nain
Main question How is the membrane potential is
related to charges in and out?
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1.3 Equilibrium potential for one ion

• Two competing forces
• Diffusion by concentration gradient.
• Motion by voltage gradient.

K
K
Diffusion
Voltage difference
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1.3 Equilibrium potential for one ion
Diffusion Flux Jdif(x) -D dK(x) / dx D,
diffusion coefficient-gt DµkT/q µ, mobility k,
Boltzmann constant q, ion charge
K
x
K
Voltage difference Flux Jelec(x) -µz K(x)
dV(x) / dx µ, mobility z, ion valence, /-1,
/-2, etc.
Equilibrium happens when Jdif(x) Jelec(x) 0,
which leads to the Nernst equation E K kT /
zq ln Kout / Kin -75,-90 mV E K is the
potential necessary to maintain the concentration
gradient Kout / Kin
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1.3 Equilibrium potential for one ion
K
E K kT / zq ln Kout / Kin -75,-90
mV E Na 55 mV E Ca2 150 mV E Cl-
-60,-65mV
x
K
Na
K
K
Na
Vm -70 mV E K lt Vm lt E Na
Compensated by Na-K pump.
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1.4 Equilibrium potential with K and Na channels
Equilibrium I K I Na 0
g K(Vm-E K) g Na(Vm-E Na) 0
g Na
g K
EL Vm (g KE K g NaE Na) / (g K g
Na) EL -69 mV
I Na
I K
-

E K
E Na
-

Leak Current IL I K I Na
Vm
IL g L (Vm - EL) g L g K g Na
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1.5 RC circuit for the passive membrane
Leak Current IL g L (V m - EL)
Capacitor Current IC C dV m /dt ( Q
C V )
g L



I C
I L
C

-
-
-
External Stimulation IC IL Iext(t)
E L
-
RC passive membrane equation C dVm / dt -gL (V
m - EL) Iext(t) ?m C / gL 20nF / 1µS
20ms
Vm
Vm
Iext(t)
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2 Action Potential 2.1. Active ion channels.
Active membrane
P, prob of being active can depend on several
factors. Active Channels -Voltage-gated (Na, K,
etc) -Extracellular ligand gated (e.g. synaptic
receptors) -Intracellular ligand gated (e.g.
Ca-depenent channel)
5 pA
500 ms
Patch-clamp technique (E. Neher and B. Sakmann,
1976)
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2.1 Active ion channels
K
Vm -70 mV E K lt Vm lt E Na
Voltage-clamp
Voltage-dependent K channel (Persistent)
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2.1 Active ion channels
Na
Vm -70 mV E K lt Vm lt E Na
Voltage-dependent Na channel (Transient)
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2.2 Dynamics of ion channels
n4 is the probability that the potassium
channel is open m3h is the probability that the
sodium channel is open
activation gates
inactivation gates
a is the probability a closed gate will open ß
is the probability an open gate will close
?(V)
Open
Close
?(V)
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2.3 Hodgkin and Huxley equations
Na Channels GNa (1/RNa) and ENa55mV K
Channels GK (1/RK) and EK-80mV Ca2 Channels
GCa (1/RCa) and ECa Leak Channels GL (1/RL) and
EL-70mV
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2.3 Hodgkin and Huxley equations
Steady State
Time const.
Spike Generation Iapp ? ? V ? ? m ?
(quickly) while n ? and h ? (slowly) Thus V
goes up quickly toward ENa until h shuts off Na
channels and K inhibition dominates
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2.3 Hodgkin and Huxley equations
dn/dtan(V)(1-n)-bn(V)n an(V) opening
rate bn(V) closing rate dm/dtam(V)(1-m)-bm(
V)m am(V) opening rate bm(V) closing
rate dh/dtah(V)(1-h)-bh(V)h ah(V)
opening rate bh(V) closing rate
an(0.01(V55))/(1-exp(-0.1(V55)))
bn0.125exp(-0.0125(V65)) am(0.1(V40))/(1-exp(-
0.1(V40))) bm4.00exp(-0.0556(V65)) a
h0.07exp(-0.05(V65))
bh1.0/(1exp(-0.1(V35)))
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2.4 Spatially distributed neuron models
point neuron model
The spatial distribution of ion channels is
almost completely unknown, so any
multi-compartment model is highly speculative
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2.4 Propagation of the AP in a passive and active
axon
Attenuation of 70 in 1mm, and very slow (0.2m/s)
propagation
Axon
Electrodes
V
time
Time dt
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2.4 Propagation of the AP in a passive and active
axon
propagation
Na
Axon
Electrodes
V
time
Time dt
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3 Synaptic conductances
-Excitatory -Inhibitory
Synaptic Current Is g(t) (Vm - Es)
EPSC AMPA (fast), NMDA (slow) IPSC GABAA (fast)
EPSC g(t) is an exponential
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4 Reduced models of neurons. Leaky Integrate and
Fire.
Models Stereotyped After Hyperpolarization
Potential
Models Stereotyped effects of incoming spikes
Models synaptic channels g(t)
A new spike occurs at time tnew if the threshold
is reached V is reset and integration begins
again
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4 Reduced models of neurons. Leaky Integrate and
Fire.
Two spiking regimes sub- and supra-threshold
regimes
Supra-threshold regime
Sub-threshold regime
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4 Conductance-based IF neuron
Models Stereotyped After Hyperpolarization
Potential
Models stereotyped excitatory channels
Models stereotyped inhibitory channels
Few solutions were known for this model. But see
recent developments by M. Richardson et al,
Destexhe et al, and R. Moreno-Bote et al.
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4 Spike response neuron

• Good approximation of IF neuron model, but only
  with noisy inputs.
• Spikes are generated randomly (Poisson) given the
  input u(t).

Models Stereotyped After Hyperpolarization
Potential
Models stereotyped post-synaptic potentials
f
u
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5 Neuronal networks
r(t)
rate
input
t dr/dt -r f( W r(t) W0 r0(t))
r0(t )
Exc Inh pops.
E
I
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5 Neuronal networks. Balanced regime
Balanced regime experimentally found that firing
is low and irregular. Excitation in cortex is
large. Then, excitation must be cancelled out by
strong inhibition.
Gerstein and Mandelbrot (1964), Van Vreeswijk and
Sompolinsky (1996), Shadlen and Newsome (1998)
only exc
rE,out rE,in
rE,out
balanced exc/inh
rE,in
Low variability regime
High variability regime
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5 Neuronal networks. Balanced regime
Itotal (NEJErE - NIJIrI)?m Threshold N
10000 JE 0.2 mV r 2-5Hz If
(80000.22-2000JI5)0.02020mV -gt JI 0.22
mV If JI 0.25 mV, then Itotal 14 mV (No
firing!) If JI 0.19 mV, then Itotal 26 mV
(Saturation!)
only exc
rE,out rE,in
rE,out
balanced exc/inh
rE,in
Problem it requires fine-tuning of the network
parameters (e.g., N, J)
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5 Neuronal networks. Balanced regime
rate
input
N, neurons K, connections
Take the large N limit, with 1ltltKltltN, and in
particular
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5 Neuronal networks. Balanced regime
N, neurons K, connections
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• Overview
• Passive properties of neurons (resting potential)
• Action potential (generation and propagation).
• Synaptic currents (AMPA, GABA, NMDA).
• 4. Reduced models of neurons (LIF, QIF, LNP).
• 5. Neuronal networks (balance and chaos).

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