excitability
DESCRIPTION
Introduction to Neurobiology - 2004. Excitability. Information processing in the retina. Artificial neural networks. Firing mode of thalamic neurons. Regular firing. A burster. Delayed Burst: Rebound from hyperpolarization. Isopotential model for passive neuron. R. C. - PowerPoint PPT PresentationTRANSCRIPT
•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
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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