Convergence and stability in networks with spiking neurons

Convergence and stability in networks with spiking neurons

Stan GielenDept. of Biophysics

Magteld ZeitlerDaniele Marinazzo

Overview• What’s the fun about synchronization ?• Neuron models• Phase resetting by external input• Synchronization of two neural oscillators• What happens when multiple oscillators are coupled ?• Feedback between clusters of neurons• Stable propagation of synchronized spiking in neural

networks• Current problems

The neural code

Firing rate

Recruitment

Synchronous firingNeuronalassembly

Neuronal assemblies are flexible

Why flexible synchronization ?

Stimulus driven; bottom-up processFrom Fries et al. Nat Rev Neurosci.

Synchronization of firing related to attention

Riehle et al. Science, 1999

Evidence for Top-Down processes on coherent firing

Coherence between sensori-motor cortex MEG and muscle EMG

Schoffelen, Oosterveld & Fries, Science in press

Before and after a visual warning signal for the “go” signal to start a movement

Functional role of synchronization

Schoffelen, Oosterveld & Fries, Science in press

Questions regarding initiation/disappearence of temporal

coding

• Bottom-up and/or top-down mechanisms for initiation of neuronal synchronization ?

• Stability of oscillations of neuronal activity• functional role of synchronized neuronal

oscillations



Leaky-Integrate and Fire neuron)()()( tRItV

dttdV

m

For constant input I mm tt eRIVeRItV /0/ )1()(

Small input Large input

Conductance-based Leaky-Integrate and Fire neuron

)()()( tRItVdttdV

m

Membrane conductance is a function of total input, and so is the time-constant.

With increasing synaptic input, the neuron changes from an integrator to a co-incidence detector.

Synaptic processes


)()()( tRItVdttdV

m

Membrane conductance is a function of total input, and so is the time-constant.


mmmmm GCCR /



τ = 40 ms τ = 2 ms

)()()( tRItVdttdV

m

Hodgkin-Huxley neuron

V mV

0 mV

V mV

0 mV

ICINa

Membrane voltage equation

-Cm dV/dt = gmax, Nam3h(V-Vna) + gmax, K n4 (V-VK ) + g leak(V-Vleak)

K

V (mV)

mmOpen Closedm

mmProbability:

State:

(1-m)

Channel Open Probability:

dt

dm m)1( m m m

hhdtdh

hh )1(

Gating kinetics

m.m.m.h=m3h

mm

mm

mm

1

Actionpotential

Simplification of Hodgkin-HuxleyFast variables• membrane potential V• activation rate for Na+

m

Slow variables• activation rate for K+ n• inactivation rate for

Na+ h-C dV/dt = gNam3h(V-Ena)+gKn4(V-EK)+gL(V-EL) + I

dm/dt = αm(1-m)-βmm

dh/dt = αh(1-h)-βhh

dn/dt = αn(1-n)-βnn

Morris-Lecar model

Phase diagram for the Morris-Lecar model

Phase diagram for the Morris-Lecar model

Linearisation around singular point :

WV

bV

WV

dtd

1)1( 2

*

*

WWW

VVV

Phase diagram

Phase diagram of the Morris-Lecar model



Neuronal synchronization due to external inputT

ΔTΔ(θ)= ΔT/T

Synaptic input

Neuronal synchronizationT

ΔTΔ(θ)= ΔT/T

Phase shift as a function of the relative phase of the external input.

Phase advance

Hyperpolarizing stimulus

Depolarizing stimulus


ΔTΔ(θ)= ΔT/T

Suppose:

• T = 95 ms

• external trigger: every 76 ms

• Synchronization when ΔT/T=(95-76)/95=0.2

• external trigger at time 0.7x95 ms = 66.5 ms

ExampleT=95 ms

P=76 ms = T(95 ms) - Δ(θ)

For strong excitatory coupling, 1:1 synchronization is not unusual. For weaker coupling we may find other rhythms, like 1:2, 2:3, etc.


ΔTΔ(θ)= ΔT/T

Suppose:

• T = 95 ms

• external trigger: every 76 ms

• Synchronization when ΔT/T=(95-76)/95=0.2

• external trigger at time 0.7x95 ms = 66.5 ms

StableUnstable

Convergence to a fixed-point Θ* requires

Substitution of and expansion near gives

Convergence requires

and constraint gives

TPnnn /)(1

TPnnn /)(1

TP /)( * |||| **

1 nn

n= n* *

nn 1 nn TP

)(/)()( *

1)(1

n

n n

n

1 <1

-1< 1)(<1 and so –2 < )(<0

T

P



Excitatory/inhibitory interactions

excitation-excitation

inhibition-inhibition

excitation-inhibition

Behavior depends on synaptic strength ε and size of delay Δt

Excitatory interactions


Mirollo and Strogatz (1990) proved in a rigorous way that excitatory coupling without delays always leads to in-phase synchronization.

Stability for two excitatory neurons with delayed coupling

Return map)(1 kk R

For two neurons with excitatory coupling strength = 0.1 and time delay = 0.2

Ernst et al. PRL 74, 1995

)()()(: kBkAkBk ttt

if tk is time when oscillator A fires

Summary for excitatory coupling between two neurons

• In-phase behavior for excitatory coupling without time delays

• tight coupling with a phase-delay for time delays with excitatory coupling.

Inhibitory interactions


Inhibitory couplingfor two identical leaky-integrate-and-fire neurons

Out-of-phase stable In-phase stable

Lewis&Rinzel, J. Comp. Neurosci, 2003

Phase-shift function for neuronal synchronizationT

ΔTΔ(θ)= ΔT/T

Phase shift as a function of the relative phase of the external input.

Phase advance

Hyperpolarizing stimulus

Depolarizing stimulus

Phase-shift functionfor inhibitory coupling

0)( *

d

dG

for stable attractor

Increasing constant input to the LIF-neurons

I=1.2

I=1.4

I=1.6

Bifurcation diagram for two identical LIF-neurons with inhibitory coupling

Bifurcation diagram for two identical LIF-neurons with inhibitory coupling

Time constant for inhibitory synaps

Summary for inhibitory coupling

Stable pattern corresponds to• out-of-phase synchrony when the time

constant of the inhibitory post synaptic potential is short relative to spike interval

• in-phase when the time constant of the inhibitory post synaptic potential is long relative to spike interval

Inhibitory coupling with time delays

Stability for two inhibitory neurons with delayed coupling

Return map)(1 kk R



Stability for two excitatory neurons with delayed coupling

Return map)(1 kk R



)()()(: kBkAkBk ttt

if tk is time when oscillator A fires

Summary about two-neuron coupling with delays

• Excitation leads to out-of-phase behavior• Inhibition leads to in-phase behavior



A network of oscillators with excitatory coupling

Winfree model of coupled oscillators

0

P(Θj) is effect of j-th oscillator on oscillator I (e.g. P(Θj) =1+cos(Θj)

R(Θi) is sensitivity function corresponding to contribution of oscillator to mean field.

N-oscillators with natural frequency ωi

Ariaratnam & Strogatz, PRL 86, 2001

Averaged frequency ρi as a function of ωi

65.0locking partial locking

incoherence partial deathslowest oscillators stop

Frequency range of oscillators Ariaratnam & Strogatz, PRL 86, 2001

Phase diagram assuming uniform distribution of natural frequencies

SummaryA network of spontaneous oscillators with different natural

frequencies can give – locking– partial locking– incoherence– partial death

depending on strength of excitatory coupling and on distribution of natural frequencies.

Excitatory coupling can cause synchrony and chaos !

Role of excitation and inhibition in neuronal synchronizationin networks with excitatory and inhibitory neurons

Borgers & Kopell

Neural Computation 15, 2003

Role of excitation and inhibition in neuronal synchronizationin networks with excitatory and inhibitory neurons

gEE

gIE

gII

gEI

Mutual synchronizationAll-to-all connectivity Sparse connectivity

IE=0.1; II=0; gEI=gIE=0.25; gEE=gII=0; τE=2 ms; τI=10 ms

gEE

gIE

gII

gEI

Main messageSynchronous rhythmic firing results from• E-cells driving the I-cells• I-cells synchronizing the E-cellsSynchronization is obtained for• Continuous drive to E-neurons• Relatively strong EI connections• Short time decay of inhibitory post synaptic

potentials

Simple (Theta) neuron modelNeuronal state represented as phase on the unit-circle

with input I (in radians) and membrane time-constant τ.

When I<0: two fixed points :

unstable

stable

I<0 I=0 I>0

Saddle-node bifurcation

: spiking neuron

unstable

stable

I<0 I=0 I>0

Spiking neuron

Saddle-node bifurcation

Time interval between spikes

-π

π

-π

π

=

Sufficient conditions for synchronized firing

E=>I synapses too weak External input to I-cells

I=>E synapses too weakRhythm restored by adding I=>I synapses

Conditions for synchrony

•E-cells receive external input above threshold

• I-cells spike only in response to E-cells; Relatively strong EI connections

• I=>E synapses are short and strong such that I-cells synchronize E-cells



Possible role of feedback

Data from electric fish

Correlated input

Uncorrelated input

Doiron et al. Science, 2004

Possible role of feedback

feedbacknoise

Doiron et al. Science, 2003Doiron et al. PRL, 93, 2004

Feedback to retrieve correlated input

data model

Experimental data

cAxxxx jjjj 22*,0,0

* )()()()(

Linear response analysis gives

Power in range

2-22 Hz

40-60 Hz

What happens to our analytical formula when FB comes from a LIF neuron?

2

22222*

,0,0*

|1|||)(2||)()()()(

gKAgKAgKAAcAxxxx jjjj

0 50 100 150

15

20

25

30

freq (Hz)

S (s

pike

s2 /s)

FB from a linear unit

FB from LIF neuron G

D = 18 msec

g = - 1.2G = 6 msec, but we obtain the same line with G = 18 msec

τDelay= 18 ms

Feedback gain = -1.2τLIF = 6 ms (longer time constant, same result)

)()( * jj xx

Paradox with between results by Kopell (2004) and Doiron (2003, 2004) ?

• Börgers and Kopell (2003): spontaneous synchronized periodic firing in networks with excitatory and inhibitory neurons

• Doiron et al: Feedback serves to detect common input: no common input no synchronized firing.

0 20 40 60 80 100 120

0

100

200

300

400

500

600

700

800Spectrum with FB from LIF neuron and C1=0

Short time constant for inhibitory neuron: τLIF = 2 ms

LIFSynchronized firing even without correlated input

C=0

0 20 40 60 80 100 1200

50

100

150

200

250Spectrum with FB from LIF neuron and C1 =1

0 20 40 60 80 100 1200

100

200

300

400

500

600

700

800Spectrum with FB from LIF neuron and C1=0

LIF

Uncorrelated input Fully correlated input

For small time constant of LIF-neuron, network starts spontaneous oscillations of synchronized firing

Sufficient conditions for synchronized firing in the Kopell model

E=>I synapses too weak External input to I-cells

I=>E synapses too weakRhythm restored by adding I=>I synapses

Conditions for synchrony

•E-cells receive external input above threshold

• I-cells spike only in response to E-cells; Relatively strong EI connections

• I=>E synapses are short and strong such that I-cells synchronize E-cells

10 20 30 40 50 60 70 80 90 100 110 1200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1FB to all

CI 60 Hz to P1CI 45 Hz to P1CI 90 Hz to P1

10 20 30 40 50 60 70 80 90 100 110 1200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9no FB

CI 60 Hz to P1CI 45 Hz to P1CI 90 Hz to P1

No feedback; input at 50, 60 or 90 Hz With feedback; input at 50, 60 or 90 Hz

LIF LIFLIF with small τ

Time constant of inhibitory neuron is crucial !

Short time constant: neuron is co-incidence detectorBörgers and Kopell (2003): spontaneous synchronized

periodic firing in networks with excitatory and inhibitory neurons

Long time constant:Doiron et al: Feedback serves to detect common input:

no common input no synchronized firing.



Propagation of synchronous activity

Diesman et al., Nature, 1999

•Leaky integrate-and-fire neuron:

•20,000 synpases; 88% excitatory and 12 % inhibitory•Poisson-like output statistics

‘Synfire chain’

i

iIVdtdV

Propagation of synchronous activity

Activity in 10 groups of 100 neurons each.

Under what conditions

•preservation

•extinction

of synchronous firing ?

Critical parameters• Number of pulses in volley (‘activity’)• temporal dispersion σ• background activity• integration time constant for neuron ( τ = 10 ms)

•activity a

•dispersion σ

Spike probability versus input spike number as function of σ

Temporal accuracy versus σin for various input spike numbers

Temporal accuracy versus σin for various input spike numbers

Output less precise

Output more precise

State-space analysis

Model parameters:

# 100 neurons

Stable attractor

Transmission function for pulse-packet for group of 100 neurons.

Evolution of synchronous spike volley

State-space analysis

Model parameters:

# 100 neurons

Attractor

Saddle point

Dependence on size of neuron groups

• a minimum of 90 neurons are necessary to preserve synchrony

•fixed point depends on a, σ and w.

N=80 N=90 N=100 N=110

a-isocline

σ-isocline

Summary

• Stable modes of coincidence firing.• Attractor states depend on number of

neurons involved, firing rate, dispersion and time constant of neurons.

Further questions

• What happens for correlated input from multiple groups of neurons ?

• What is the effect of (un)correlated excitation and inhibition ?

• What is the effect of lateral interactions ?• What is the effect of feedback ?

Summary• Excitatory coupling leads to chaos; inhibition lead to

synchronized firing.• Synchronization can easily be obtained by networks of

coupled excitatory and inhibitory neurons. In that case the frequency of oscillations depends on the neuronal dynamics and delays, not on input characteristics

• There is no good model yet, which explains the role of input driven (bottom-up) versus top-down processes in the initiation of synchronized oscillatory activity.

• The role and relative contribution of feedforward (stochastic resonance) and feedback in neuronal synchronization is yet unknown.

Documents

Convergence and stability in networks with spiking neurons