•Excitability

•Information processing in the retina

•Artificial neural networks

Introduction to Neurobiology - 2004

Regular firing

A burster

Firing mode of thalamic neurons

Delayed Burst:

Rebound from hyperpolarization

QuickTime™ and aMicrosoft Video 1 decompressorare needed to see this picture.

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QuickTime™ and aMicrosoft Video 1 decompressorare needed to see this picture.

R C

I =Cdvdt

+VR

τm =RC

Isopotential model for passive neuron

V =IR(1−e−t/τm)

Isopotential model for excitable neuron

Vth =Visi =IR(1−e−tisi

τm)

IR−Vth =IR* e−tisi

τm

IR−Vth

IR=e

−tisiτm

ln(IR −Vth

IR) =−

tisi

τm

τm ln(IR

IR −Vth

) =tisi

f =1tisi

=[τm ln(IR

IR −Vth

)]−1

Integrate - and - fire (I&F) model (Lapicque - 1907)

Vth

I

tisi

Integrate - and - fire (I&F) model with fluctuating input

I(nA)

f(H

z)

Cortical neuron

I&F model neuron

Spike-rate adaptation

cdvdt

+VR

+gSRA(V −EK ) =I

τSRA

dgSRA

dt=−gSRA

Each spike: gsra = gsra +gsra

Integrate - and - fire (I&F) model with adaptation

I&F

I&F + adaptation

H&H model + “A” current

The squid - H&H model

I(nA)

f(H

z)f(

Hz)

The Hodgkin & Huxley Model

J. Physiol. London (1952, a,b,c,d)

Space-clamped (“membrane”) action potential (H&H 1952)

Gating of membrane channels

sensor

Persistent conductance

Transient conductance

sensor

Persistent conductance K-conductance (delayed

rectifier)

PK =n4;0≤n≥1

n - activation (or gating) variable

n - probability of subunit gate to be open

1- n probability of subunit gate to be close

nαn( V)

← ⏐ ⏐ ⏐ ⏐βn (V) ⏐ → ⏐ ⏐ ⏐ 1−nOpen

Close

dep

ola

riza

tion

dndt

=αn(V)(1−n) −βn(V)n

Dividing by

αn(V) +βn(V)

τn

dndt

=n∞(V) −n

n∞(V) =αn(V)

αn(V) +βn(V)

τn(V) =1

αn(V)+βn(V)

nαn( V)

← ⏐ ⏐ ⏐ ⏐βn (V) ⏐ → ⏐ ⏐ ⏐ 1−n;αn(V);βn(V)...1/sec

For a fixed voltage V

n(t) =n0 −(n0 −n∞)(1−e−t /τn )

τn(V)dndt

=n∞(V)−n

n approaches exponentially with time-constant

n∞

τ∞

n∞(V) =αn(V)

αn(V) +βn(V)

τn(V) =1

αn(V)+βn(V)

αn(V) =n∞(V) /τn(V)

βn(V)=(1−n∞(V))/τn(V)

Calculating n and n

Time-course of potassium conductance (H&H 1952)

nαn( V)

← ⏐ ⏐ ⏐ ⏐βn (V) ⏐ → ⏐ ⏐ ⏐ 1−n

Transient conductance Na-conductance

m - activation (or gating) variable

h - inactivation (or gating) variable

mαm( V)

← ⏐ ⏐ ⏐ ⏐βm(V) ⏐ → ⏐ ⏐ ⏐ 1−m

hαh( V)

← ⏐ ⏐ ⏐ ⏐βh (V) ⏐ → ⏐ ⏐ ⏐ 1−h

dep

ola

riza

tion

time

PK =m3h;0≥m,h≤1

Time-course of sodium conductance (H&H 1952)

Time-course of n,m,h following voltage step

ImgL(V EL ) gKn4 (V EK ) gKm

4h(V ENa )

dmdt

=αm(V)(1−m)−βm(V)m

dhdt

=αh(V)(1−h) −βh(V)h

dndt

=αn(V)(1−n) −βn(V)n

The Hodgkin & Huxley Equations

gK =36ms/cm2

gNa =120ms/cm2

Time-course of n,m,h during “membrane” action potential

Time-course of underlying conductances during

“membrane” action potential (H&H 1952)

Note the small % of ion conductance (channels) used during the action potential

Simulated (top) versus experimental “membrane” action potential (H&H 1952)

Temperature effect on action potential

Simulated (b) versus experiments (top)

(H&H 1952)

* Amplitude decreases

* Speed increases

* no propagation for T > 330

C

Good fit with:

multiply by

e

Stochastic opening of voltage-gated ion-channels

(underlying excitability)

Holding potential

Sakmann and Neher, 1991

The “soup” of diverse excitable ion channels

(beyond H&H and the squid giant axon)

Kinetics of the “A” (K+) current

Transient K+ current; blocked by 4-AP (not by TEA)

-100 mV 50 mV

1nA40 msec

Acti

vati

on

msec

inactivation 20-30 msec

Im =gL(V −VL)+gKn4(V−VK ) +gNam

3h(V −VNa)+gAmh(V −VK )

Function of the “A” (K+) current

1. Delays onset of AP2. Enables very-low firing rate for weak depolarizing input (due to fast activation and slow inactivation)3. Enables high-frequency for large inputs (strong inactivation)

1 2

3

“A” (K+) current enables low-firing rates

Fast activation - delays 1st spike

Prevents Vm from reaching threshold

Inactivaes and enables Vm to reach threshold

“IT” (Ca+2) current produces burst of Na + spikes

Release from prolong hyperpolarization:

IT de-inactivates (h=1)

Na spikes riding on “Ca spike”

Kinetics of the variety of excitable ion channels

Function of variety of excitable ion channels