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).