Electrical properties of neurons
Rubén Moreno-Bote
Galvani frog’s legs experiment
Overview:
1. Passive properties of neurons (resting potential)
2. Action potential (generation and propagation).
3. Synaptic currents (AMPA, GABA, NMDA).
4. Reduced models of neurons (LIF, QIF, LNP).
5. Neuronal networks (balance and chaos).
1. Passive properties of the neuron membrane1.1 Membrane potential.
Current, I
Vin
Vout
Vin-Vext = -RI Ohm’s law
I>0, inward current, which means that Vin is negative, Vin = -70 mV (we define Vout=0)
I, current-> Amperes, A=C/s (order of magnitude: 10 nA, 10 µA)
V, potential-> Volts (100mV, action potential; 0.1-1mV, postsynaptic current)
R, resistance-> Ohms, , V/A (1 M)
g, conductance, g=1/R-> Siemens, S (1/ ) (order: µS).
Ohm’s law with conductances: I=-g(Vin-Vout)
Outside
Inside
1.1 Membrane potential. Currents, resistances and capacitors.
Current, I
capacitor
Vin
Vout
Membrane is impermeable to ions and creates the voltage difference (=Capacitor).
Q=CV
C, Capacitance-> Faradays (order 1 nF)
Extra and intracellular fluid is electrically neutral.
Outside
Inside
1.2 Ions and Ion Channels
[K+]
[K+] [Na+] [Ca2+] [Cl-]
[Cl-][Ca2+][Na+]
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.
[K+]in=20 [K+]out
[Na+]out=10 [Na+]in
Main question: How is the membrane potential is related to [charges] in and out?
Outside
Inside
1.3 Equilibrium potential for one ion
[K+]
[K+]
Two competing forces:1. Diffusion by concentration gradient.2. Motion by voltage gradient.
Diffusion Voltage difference
Outside
Inside
1.3 Equilibrium potential for one ion
[K+]
[K+]
Diffusion Flux: Jdif(x)= -D d[K+](x) / dx
D, diffusion coefficient-> D=µkT/q
µ, mobilityk, Boltzmann constantq, ion charge
x
Voltage difference Flux: Jelec(x)= -µz [K+](x) dV(x) / dx
µ, mobilityz, ion valence, +/-1, +/-2, etc.
Equilibrium happens when Jdif(x) + Jelec(x) = 0, which leads to the Nernst equation:
E K+ = kT / zq ln [K+]out / [K+]in = -75,-90 mV
E K+ is the potential necessary to maintain the concentration gradient [K+]out / [K+]in
1.3 Equilibrium potential for one ion
[K+]
[K+]
x
E K+ = kT / zq ln [K+]out / [K+]in = -75,-90 mV
E Na+ = +55 mV
E Ca2+ = +150 mV
E Cl- = -60,-65mV
K+ Na+
Vm = -70 mV
E K+ < Vm < E Na+
Compensated by Na-K pump.
K+ Na+
1.4 Equilibrium potential with K and Na channels
E K+
Vm
E Na+
g Na+ g K+ I Na+ I K+
Equilibrium:
I K+ + I Na+ = 0
g K+(Vm-E K+) + g Na+(Vm-E Na+) = 0
EL = Vm = (g K+E K+ + g Na+E Na+) / (g K++ g Na+)
EL = -69 mV
IL = g L (Vm - EL)
g L = g K++ g Na+
Leak Current:
IL = I K+ + I Na+
--
+
+
1.5 RC circuit for the passive membrane
E L
Vm
C
g L I C I L
Leak Current:
IL = g L (V m - EL)
--+ - -
+ + +
Capacitor Current:
IC = C dV m /dt ( Q = C V )
External Stimulation:
IC + IL = Iext(t)
VmIext(t) RC passive membrane equation:
C dVm / dt = -gL (V m - EL) + Iext(t)
m = C / gL = 20nF / 1µS = 20ms
I
V
2 Action Potential 2.1. Active ion channels. Active membrane
Patch-clamp technique(E. Neher and B. Sakmann, 1976)
5 pA
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)
500 ms
2.1 Active ion channels
Voltage-clamp Voltage-dependent K+ channel (Persistent)
K+
Vm = -70 mV
E K+ < Vm < E Na+
Outside
Inside
2.1 Active ion channels
Voltage-dependent Na+ channel (Transient)
Na+
Vm = -70 mV
E K+ < Vm < E Na+
Outside
Inside
4
3
K K
Na Na
g g n
g g m h
n4 is the probability that the potassium channel is open
m3h is the probability that the sodium channel is open
( ) 1 ( )n n
dnV n V n
dt
α is the probability a closed gate will open
β is the probability an open gate will close
2.2 Dynamics of ion channels
activation gates inactivation gates
OpenClose(V)
(V)
Na+ Channels: GNa (1/RNa) and ENa=55mV
K+ Channels: GK (1/RK) and EK=-80mV
Ca2+ Channels: GCa (1/RCa) and ECa
Leak Channels: GL (1/RL) and EL=-70mV
2.3 Hodgkin and Huxley equations
4 3( ) ( ) ( )L L K K Na Na app
dVC g E V g n E V g m h E V I
dt
, ,
( ) ( ) ,
x n m h
dxV x V x
dt
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
4 3( ) ( ) ( )L L K K Na Na app
dVC g E V g n E V g m h E V I
dt
2.3 Hodgkin and Huxley equations
dn/dt=an(V)(1-n)-bn(V)n an(V) = opening rate bn(V) = closing rate
dm/dt=am(V)(1-m)-bm(V)m am(V) = opening rate bm(V) = closing rate
dh/dt=ah(V)(1-h)-bh(V)h ah(V) = opening rate bh(V) = closing rate
an=(0.01(V+55))/(1-exp(-0.1(V+55))) bn=0.125exp(-0.0125(V+65))
am=(0.1(V+40))/(1-exp(-0.1(V+40))) bm=4.00exp(-0.0556(V+65))
ah=0.07exp(-0.05(V+65)) bh=1.0/(1+exp(-0.1(V+35)))
2.3 Hodgkin and Huxley equations
4 3
( , ) ( ) ( , ) ( , )
( , ) ( , ) ( , ) ( , ) ( , )
( ) ( , ) L L app
K K Na Na
dC V x t g x E V x t I x t
dt
g n x t E V
d dD x V x t
dx dx
x t g m x t h x t E V x t
The spatial distribution of ion channels is almost completely unknown, so any multi-compartment model is highly speculative
2.4 Spatially distributed neuron models
point neuron model
V
time
Electrodes
Time + dt
2.4 Propagation of the AP in a passive and active axon
propagation
Axon
Attenuation of 70% in 1mm, and very slow (0.2m/s)
V
time
Electrodes
Time + dt
2.4 Propagation of the AP in a passive and active axon
propagation
AxonNa+
3 Synaptic conductances
-Excitatory-Inhibitory
Synaptic Current:
Is = g(t) (Vm - Es)
EPSC: g(t) is ~ an exponentialEPSC: AMPA (fast), NMDA (slow)
IPSC: GABAA (fast)
4 Reduced models of neurons. Leaky Integrate and Fire.
( ) ( )L L ahp last j jj
dVC g E V I t t w R t t
dt
( )newV t Models Stereotyped After Hyperpolarization Potential
Models Stereotyped effects of incoming spikes
A new spike occurs at time tnew if the threshold is reached:V is reset and integration begins again
Models synaptic channels g(t)
4 Reduced models of neurons. Leaky Integrate and Fire.
( ) ( )L L ahp last j jj
dVC g E V I t t w R t t
dt
Two spiking regimes: sub- and supra-threshold regimes
Supra-threshold regime Sub-threshold regime
, ,
( )
( ) ( )
L L ahp last ahp
j E E j E j I I j Ij j
dVC g E V g t t E V
dt
w g t t E V w g t t E V
Models Stereotyped After Hyperpolarization Potential
Models stereotyped excitatory channels
Models stereotyped inhibitory channels
4 Conductance-based I&F neuron
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.
• Good approximation of I&F neuron model, but only with noisy inputs.
• Spikes are generated randomly (Poisson) given the input u(t).
( ) ( ) ( , )
Pr( | ( )) ( )
last j last jj
new
u t t t w R t t t t
t t u t f u t
Models Stereotyped After Hyperpolarization Potential
Models stereotyped post-synaptic
potentials
f
u
4 Spike response neuron
τ dr/dt = -r + f( W r(t)+ W0 r0(t))
r(t)
r0(t )
5 Neuronal networks
input
rate
E I
0 ,0 0
0 ,0 0
( )
( )E E E EE E I EI I E
I I E IE E I IE I I
r f K J r K J r K J r
r f K J r K J r K J r
Exc + Inh pops.:
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)
rE,in
rE,out
rE,out = rE,in
only exc
balanced exc/inh
Low variabilityregime High variability
regime
5 Neuronal networks. Balanced regime
Itotal = (NEJErE - NIJIrI)m ~ Threshold
N = 10000
JE = 0.2 mV
r = 2-5Hz
If (8000×0.2×2-2000×JI×5)×0.020=20mV
-> 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!)
rE,in
rE,out
rE,out = rE,in
only exc
balanced exc/inh
Problem: it requires fine-tuning of the network parameters (e.g., N, J…)
5 Neuronal networks. Balanced regime
N, neuronsK, connections
Take the large N limit, with 1<<K<<N,and in particular 1/J K
0 ,0 0
0 ,0 0
0
0
0
0
0
0
( )
( )
( ( ))
( ( ))
(1/ )
(1/ )
E E E EE E I EI I E
I I E IE E I IE I I
E E E E I
I I E I I
E E I
E I I
I EE
E I
IE I
r f K J r K J r K J r
r f K J r K J r K J r
r f K r J r Er
r f K r J r Ir
r J r Er O K
r J r Ir O K
J E J Ir r
J J
E Ir r
J J
input
rate
5 Neuronal networks. Balanced regime
N, neuronsK, connections
Overview:
1. Passive properties of neurons (resting potential)
2. Action potential (generation and propagation).
3. Synaptic currents (AMPA, GABA, NMDA).
4. Reduced models of neurons (LIF, QIF, LNP).
5. Neuronal networks (balance and chaos).