Neuroni: Effetti di coerenza indotti dal rumore (Noise ...atorcini/ARTICOLI/lezione5-kreuz.pdf ·...
Transcript of Neuroni: Effetti di coerenza indotti dal rumore (Noise ...atorcini/ARTICOLI/lezione5-kreuz.pdf ·...
Firenze, 05.05.2006
Corso di dottorato: Modelli semplici di interesse biologico -
Dalle proteine ai neuroni
Neuroni:Effetti di coerenza indotti dal rumore(Noise induced coherence resonance)
Thomas Kreuz (Email: [email protected])
Content
• Repetition: Information transfer between neurons
• Simple models of neuronal spiking - Integrate and fire (IF) - Leaky integrate and fire (LIF)
• Characterization of spike trains
• Coherence Resonance (CR)
• CR with correlated synaptic input - FitzHugh-Nagumo (FHN)
Morphology of neurons
• Cell Body: A globular compartment with a variety of organelles including the nucleus
• Axon: A cellular extension that projects to the dendrites of other neurons (OUT)
• Dendrites: Extend from the cell body and receive input from other neurons (IN)
Communication between neurons
Neurons communicate through action potentials.These are waves of electrical discharge that travelalong the membrane of a cell.
This signal is transferred from one neuron to the other via a synapse, where the axon terminal of one cell impinges upon a dendrite (typically).
Two types of synapses:
1. Electrical synapses (direct conductive junctions between cells)
2. Chemical synapses (communicate via neurotransmitters)
Chemical synapses
Action potential travel down the axon to the presynaptic ending where an electrical impulse (Ca++) triggers the migration of vesicles toward the presynaptic membrane. The vesicle fuses with the presynaptic membrane releasing neurotransmitters into the synaptic cleft. On the postsynaptic ending the neurotransmitter molecules bind with receptor sites to influence the membrane potential of the postsynaptic neuron.
Three basic parts:
1. Presynaptic ending Contains neurotransmitters, mitochondria and other cell organelles
2. Synaptic cleftSpace between the presynaptic and postsynaptic endings, ~20 nm
3. Postsynaptic endingContains receptor sites for neurotransmitters.
The membrane potential at rest
Semi-permeable membrane with selective ion channels
Important ions:
Sodium Na +, potassium K+, chloride Cl -, proteins A-
At rest (equilibrium) the inside is negative relative to the outside:The neuron is polarized.
More sodium ions outside and more potassium ionsinside the neuron
The resting membrane potential of a neuron is about -70 mV.
The postsynaptic potential
Beyond the synaptic gap receptors respond by opening ion channels causing a change of the local membrane potential.
This is called a postsynaptic potential (PSP).
PSPs change the postsynaptic cell's excitability:It makes the postsynaptic cell either more or less likely to fire.
The result is excitatory (EPSP), in the case of depolarizing currents, or inhibitory (IPSP) in the case of hyperpolarizing currents.
If the number of EPSPs is sufficient, an action potential is fired.
Neuronal integration:
Synaptic input(PSPs)
Neuronal output(Action potential)
The action potential
A sufficient depolarization (Threshold-voltage Vthr~ -55 mV) caused by EPSPs leads to an action potential (spike).
“All or None”-principle: The size of the action potential is always the same. Either the neuron does not reach the threshold or a full action potential is fired.
1. Depolarization (Na+ in)2. Repolarization (K+ out)3. Hyperpolarization (stillK+out)
Hyperpolarization, Exhaustion: Refractoriness
(All sodium channels inactivated)
Single neuron models: Basic ingredients
• Variable of interest: Membrane potential Vm of postsynaptic neuron (State)
• Neuron receives excitatory and inhibitory postsynaptic potentials (Input)
• Neuron emits action potentials (Output)
- “All or none” behavior (Action potentials stereotyped)
- Threshold behavior (VThr )
- Resting potential (VR)
- Refractoriness
• Neuron without spatial extension (Point neuron) (Multi-compartment models)
Real neurons: - Temporal delays due to propagation of PSPs from dendrites to cell body.
• PSPs have constant amplitude
Real neurons: - Amplitude of PSP depends on voltage and on position of synapse
• Neurons without memory (Reset mechanism: All spikes are independent)
Real neurons: - Bursting - Adaptation - …
Single neuron models: Simplifications
Simple neuronal model: Integrate & Fire
Basic assumptions:
Input: Excitatory and inhibitory post-synaptic potentials (Kicks)
Output: Action potentials (Spikes)
Neuron acts as integrator (Electrical equivalent: Membrane Capacity Cm)
PSPs as instantaneous jumps in the voltage
tdt
t)
d VCI m
m
)(( = mti
∫ti + 1
thrVCd ttI =)(Spike times:
Here: Threshold VThr=8mV, Reset potential VR=0 mV (absorbing)
Integrate & Fire
0 10 20 30 40 50 60 70 80 90 100
Inh
Exc
U_R = 0
2
4
6
U_Thr = 8
Output
U [
mV
]
Time [s]
• Distribution of Inter-Spike-Intervals (ISI)
• Average ISI:
• Firing rate:
Characterization of spike trains I
∑ −=k
ktttx )()( δ
=r>< I S I
1
>< I S I
Spike train: Series of Delta-Functions
Aim: Characterization of output in dependence on input
Further simplifications:
No inhibition, no refractoriness, periodic excitatory kicks (~ constant positive current)
P [I
SI]
ISI [s]
0 5 10 15 20 25 30 35 40
Inh
Exc
U_R = 0
2
4
6
U_Thr = 8
Output
U [
mV
]
Time [s]
Integrate & Fire: Periodic excitation
0 5 10 15 20 25 30 35 40
Inh
Exc
U_R = 0
2
4
6
U_Thr = 8
Output
U [
mV
]
Time [s]
Integrate & Fire: Increasing rate (current)
0 5 10 15 20 25 30 35 40
Inh
Exc
U_R = 0
2
4
6
U_Thr = 8
Output
U [
mV
]
Time [s]
Integrate & Fire: And so on
Gain function
0 0.2 0.4 0.6 0.8 10
50
100
150
200
Input current [nA]
Fir
ing
rat
e [H
z]IF without refractoriness
thrmVC
IIr =)(
Integrate & Fire: Refractory time
0 5 10 15 20 25 30 35 40
Inh
Exc
U_R = 0
2
4
6
U_Thr = 8
Output
U [
mV
]
Time [s]
0 0.2 0.4 0.6 0.8 10
50
100
150
200
Input current [nA]
Fir
ing
rat
e [H
z] IF without refractoriness IF with refractoriness
Gain function II
ItVC
IIr
refthrm +=)(
reft
1
Maximumfrequency:
Leaky Integrate & Fire (LIF)
Real neurons: Existence of leakage channels which remain always open (no gating)
K+ out, N+ and Cl- in (Leak current); Net efflux of positive charge (Hyperpolarization)
Electrical equivalent: Resistance R
tut( C
RI
dt
td u )()() +=
CR III +=)(t +τ )( tR Iu
dt
duRC −=→=τ
New variable: Time constant of membrane
Leaky Integrate & Fire
0 5 10 15 20 25 30 35 40
Inh
Exc
U_R = 0
2
4
6
U_Thr = 8
Output
U [
mV
]
Time [s]
Leaky Integrate & Fire
0 5 10 15 20 25 30 35 40
Inh
Exc
U_R = 0
2
4
6
U_Thr = 8
Output
U [
mV
]
Time [s]
Gain function III
0 0.2 0.4 0.6 0.8 10
50
100
150
200
Input current [nA]
Fir
ing
rat
e [H
z]
IF without refractoriness IF with refractorinessLIF with refractoriness
>−−
≤= −
Thrm
thrmref
Thr
IIIR
Vt
IIIr , )1log(
, 0 )( 1τ
Two types of neuronal models
0 0.2 0.4 0.6 0.8 10
50
100
150
200
Input current [nA]
Fir
ing
rat
e [H
z]
Type I neuron
0 0.2 0.4 0.6 0.8 10
50
100
150
200
Input current [nA]
Fir
ing
rat
e [H
z]
Type II neuron
Random input (Noise)
So far:
• Periodic input (Constant current)
• Characterization in terms of firing rate
Periodic output (completely regular)
Now:
• Random input (Noise)
• Characterization in terms of regularity
Question: What makes a neuron spike regular/irregular???
ISI [s]
P [
ISI ]
P [I
SI]
ISI [s]
Firing rate:
• Coefficient of variation:
• Autocorrelation function:
- Correlation time:
Characterization of spike trains II
V=C >< I S I
I S I s t d )(
= )( dtt∞
∫0
2Ccτ
(((( 2))) ><−>+=< txtxtx τ)C τ
=r>< I S I
1
P [
ISI]
ISI [s]
-1
0
1
C (
τ)
0 5 10 15 20 25 30 35 400
0.5
1
C2 (
τ)
ττ
Input: Poisson distributionThree conditions:
Every kick is generated
• randomly
• independently of other kicks
• with a uniform probability of occurrence in time
Properties:
• Exponential distribution
• Autocorrelation function flat:
• Coefficient of variation:
ISI
P [
ISI ]
0 50
0
1
τC
( τ)
)0()( δτ =C
><−=
ISI
tC ref
V 1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Inh
Exc
U_R = 0
2
4
6
U_Thr = 8
Output
U [
mV
]
Time [s]
Leaky Integrate & Fire (LIF) Low time constant: Coincidence detection (rather irregular)
0 5 10 15 20 25 30 35 40
Inh
Exc
U_R = 0
2
4
6
U_Thr = 8
Output
U [
mV
]
Time [s]
Integrate & FireHigh amplitudes: 1:1 synchronization (Output follows input)
Integrate & Fire
0 2 4 6 8 10 12 14 16 18
Inh
Exc
U_R = 0
2
4
6
U_Thr = 8
Output
U [
mV
]
Time [s]
Refractoriness: Higher regularity
0 5 10 15 20 25 30 35 40
Inh
Exc
U_R = 0
2
4
6
U_Thr = 8
Output
U [
mV
]
Time [s]
Integrate & FireAveraging over many kicks: Higher regularity
0 10 20 30 40 50 60 70 80 90
15
15-2
0
T=0.4
2
Time [ms]
Inp
ut
Ou
tpu
t
Exc
Ini
V(t)
What causes regularity under more general conditions?
Coherence resonance
Maximum regularity of neuronal response for an intermediate noise strength
Low noise: Activation process (Poissonian ISI-Distribution)
High noise: Diffusion with threshold (Inverse Gaussian ISI-Distribution)
In between: Spiking most regular
Indications:
• Minimum of coefficient of variation CV
• Maximum of correlation time τc
σ
Hodgkin-Huxley:
Correlated input
wNwxx
wxx
x
xNw
xx CCwNw
Np −−
−= )1(
)!(!!
Correlated kicks:
- Low frequency
- High amplitude
Increase of variance :
Correlated kicks:
Increase in amplitude
σ
Uncorrelated kicks:
Increase in frequency
Shared input: Different neurons fire together
Correlation coefficient Cxx: Average fraction of shared neurons
Intervals between kicks: Poissonian distribution
Kick amplitudes: Binomial distribution:
)10( ≤≤ xxC
0 10
0.02
0.04
0.06
0.08
Correlated input (Different correlations)
0 0.2 0.4 0.6 0.8 1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time [s]
Co
rrel
atio
n
FitzHugh-Nagumo (FHN)
)( tIaV( )
d t
t d W++=
) d t
t(d V
3)(
3
WV
V −−Φ=
• Balanced neuron: Total amount of excitation = Total amount of inhibition
• Three different cases:
- Only correlation in the excitation
- Only correlation in the inhibition
- [ No correlation ]
• For each correlation: Dependence of CV on the noise strength
Two-dimensional single neuron model:
σ
Inh
Exc
Inh
Exc
Inh
Exc
Full excitatory correlation
Correlated excitatory kicks:- Low frequency- High amplitude
Increase of variance :- Correlated excitatory kicks: Increase in amplitude- Uncorrelated inhibitory kicks: Increase in frequency
Inh
Exc
10-4
10-3
10-2
10-0.7
10-0.5
10-0.3
10-0.1
iic1.0eic0.0eec1.0
σ
VC
σ
0 20 40 60 80 100 120 140 160 1800
9-2
0
T=0.4
2
Time [s]
Inp
ut
Ou
tpu
t
Exc
Ini
Full excitatory correlation: Low noiseActivation
0 20 40 60 80 100 120 140 160 1800
13-2
0
T=0.4
2
Time [s]
Inp
ut
Ou
tpu
t
Exc
Ini
Full excitatory correlation: Medium noiseHighest regularity (1:1!!!)
0 20 40 60 80 100 120 140 160 1800
315-2
0
T=0.4
2.1
Time [s]
Inp
ut
Ou
tpu
t
Exc
Ini
Full excitatory correlation: High noiseStill 1:1, but very short refractory time
Full excitatory correlations: ISI-Distributions
0 50 100 150 200 250 300 350 400 450 500
0
0.01
0.02
P [
ISI]
a)
0 5 10 15 20 25
0
0.01
0.02
0.03
P [I
SI]
b)
0 5 10 15 20 25
0
0.01
0.02
0.03
P [
ISI]
c)
ISI
High noise: Still 1:1, but shorter refractory time(very large kicks )
Intermediate noise: Exc. kicks big enough to trigger spikes (1:1 synchronization, shorter tail)
Low noise: Activation (long tail)
Always Poissonian
Full excitatory correlation
Inh
Exc
10-4
10-3
10-2
10-0.7
10-0.5
10-0.3
10-0.1
iic1.0eic0.0eec1.0
σ
VC
Correlated inhibitory kicks:- Low frequency- High amplitude
Increase of variance :- Correlated inhibitory kicks: Increase in amplitude-Uncorrelated excitatory kicks: Increase in frequency
Full inhibitory correlation
Inh
Exc
10-4
10-3
10-2
10-0.7
10-0.5
10-0.3
10-0.1
iic1.0eic0.0eec1.0
σ
VC
σ
Full inhibitory correlation: Low noise
0 20 40 60 80 100 120 140 160 180
550
-2
0
T=0.4
2
Time [s]
Inp
ut
Ou
tpu
t
Exc
Ini
Activation
Full inhibitory correlation: Medium noise
0 20 40 60 80 100 120 140 160 180
1770
-2.1
0
T=0.4
2
Time [s]
Inp
ut
Ou
tpu
t
Exc
Ini
Repetitive firing
Full inhibitory correlation: High noise
0 20 40 60 80 100 120 140 160 180
3150
-2.2
0
T=0.4
2
Time [s]
Inp
ut
Ou
tpu
t
Exc
Ini
Disturbed repetitive firing
Full inhibitory correlations: ISI-Distributions
0 50 100 150 200 250 300 350 400 450 500
0
0.01
0.02
0.03
P [
ISI]
a)
0 5 10 15 20 25
0
0.02
0.04
P [
ISI]
b)
0 5 10 15 20 25
0
0.02
0.04
0.06
P [
ISI]
c)
ISI
Low noise: Activation (Poissonian with long tail)
Intermediate noise: Excitatory background high enoughto reach repetitive firing
High noise: Very high kicks lead to multiple peaks in the histogram
Full inhibitory correlation
Inh
Exc
10-4
10-3
10-2
10-0.7
10-0.5
10-0.3
10-0.1
iic1.0eic0.0eec1.0
σ
VC
All correlations: Smooth transition
10-4
10-3
10-2
10-0.8
10-0.7
10-0.6
10-0.5
10-0.4
10-0.3
10-0.2
10-0.1
100
σ
CV
iic1.0iic0.8iic0.6iic0.4iic0.2eic0.0eec0.2eec0.4eec0.6eec0.8eec1.0
Double coherence resonance
0.0001 0.001 0.01
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
σ
IIC
EEC
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ReferencesBiophysical background:
C. Koch: Biophysics of computation, Oxford Univ. Press, New York, 1999
Simple neuronal models (e.g., Integrate and Fire):
W. Gerstner and W.M. Kistler: Spiking neuron models, Cambridge Univ. Press, 2002
Correlated noise:
E. Salinas and T.J. Sejnowski: Correlated neuronal activity and the flow of neural information, Nature Rev Neurosci 2 (2001) 539.
T. Kreuz, S. Luccioli, A. Torcini: Coherence resonance due to correlated noise in neuronal models (submitted),contact: [email protected]