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: thomas.kreuz@fi.isc.cnr.it)

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: thomas.kreuz@fi.isc.cnr.it