Transcript - Lorentz Center

Convergence and stability in
networks with spiking neurons
Stan Gielen
Dept. of Biophysics
Magteld Zeitler
Daniele 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
Neuronal
assembly
Synchronous firing
Neuronal assemblies are flexible
Why flexible synchronization ?
Stimulus driven; bottom-up process
From Fries et al. Nat Rev Neurosci.
Synchronization of firing related to attention
Evidence for Top-Down processes on coherent firing
Riehle et al. Science, 1999
Coherence between sensori-motor cortex
MEG and muscle EMG
Before and
after a visual
warning signal
for the “go”
signal to start a
movement
Schoffelen, Oosterveld & Fries, Science in press
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
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
Leaky-Integrate and Fire neuron
m
dV (t )
 V (t )  RI (t )
dt
For constant input I
Small input
V (t )  RI (1  e t / m ) 
V0 t / m
e
RI
Large input
Conductance-based LeakyIntegrate and Fire neuron
m
dV (t )
 V (t )  RI (t )
dt
Membrane conductance is a function
of total input, and so is the timeconstant.
With increasing synaptic input, the
neuron changes from an integrator to a
co-incidence detector.
Synaptic processes
Conductance-based LeakyIntegrate and Fire neuron
m
dV (t )
 V (t )  RI (t )
dt
Membrane conductance is a function
of total input, and so is the timeconstant.
  R C C
m
m
m
m
/ Gm
With increasing synaptic input, the
neuron changes from an integrator to a
co-incidence detector.
Conductance-based LeakyIntegrate and Fire neuron
m
dV (t )
 V (t )  RI (t )
dt
τ = 40 ms
With increasing synaptic input, the
neuron changes from an integrator to a
co-incidence detector.
τ = 2 ms
Hodgkin-Huxley neuron
Membrane voltage equation
0 mV
0 mV
IC
INa
K
V mV
V mV
-Cm dV/dt = gmax, Nam3h(V-Vna) + gmax, K n4 (V-VK ) + g leak(V-Vleak)
Gating kinetics
m
State:
Open
Probability:
m
m
dm  (1 m)   m
 m
m
dt
dh   (1 h)   h
h
h
dt
m
Closed
m
(1-m)
m 
m
 m  m
1

 m  m
Channel Open Probability: m.m.m.h=m3h
V (mV)
Actionpotential
Simplification of Hodgkin-Huxley
Fast 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
V  V  V *
Linearisation around singular point :
W  W  W *
d  V   (1  V 2 )  1  V 


  


dt  W   
 b  W 
Phase diagram
Phase diagram
of the MorrisLecar model
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
Neuronal synchronization due to
external input
T
ΔT
Synaptic input
Δ(θ)= ΔT/T
Neuronal synchronization
T
Δ(θ)= ΔT/T
ΔT
Depolarizing
stimulus
Phase
advance
Hyperpolarizing
stimulus
Phase shift as a function of the
relative phase of the external
input.
Neuronal synchronization
T
ΔT
Δ(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
T=95 ms
Example
For strong excitatory
coupling, 1:1
synchronization is not
unusual. For weaker
coupling we may find
other rhythms, like 1:2,
2:3, etc.
P=76 ms = T(95 ms) - Δ(θ)
Neuronal synchronization
T
Δ(θ)= ΔT/T
ΔT
Suppose:
Unstable
• T = 95 ms
Stable
• external trigger: every 76 ms
• Synchronization when
ΔT/T=(95-76)/95=0.2
• external trigger at time 0.7x95
ms = 66.5 ms
 n 1   n  ( n )  P / T
P
 n 1   n  ( n )  P / T
Convergence to a fixed-point Θ* requires
T
(* )   P / T
|  n 1  * ||  n  * |
Substitution of n = *  n and expansion near  * gives
*
n1 n   ( ) 
 ( )
 ( )
n  P / T 
n


Convergence requires
 n 1 ( )

 1 and
n

-1<
constraint
( )
( )
 1 <1 and so –2 <
<0


n1
<1
n
gives
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
Excitatory/inhibitory interactions
excitation-excitation
inhibition-inhibition
excitation-inhibition
Behavior depends on synaptic strength ε and size of delay Δt
Excitatory interactions
excitation-excitation
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
 k 1  R( k )
 k :  B (t k )   A (t k )   B (t k )
if tk is time when oscillator A fires
For two neurons with
excitatory coupling
strength = 0.1 and time
delay
= 0.2
Ernst et al. PRL 74, 1995
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
excitation-excitation
Inhibitory coupling
for 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
synchronization
T
Δ(θ)= ΔT/T
ΔT
Depolarizing
stimulus
Phase
advance
Hyperpolarizing
stimulus
Phase shift as a function of the
relative phase of the external
input.
Phase-shift function
for inhibitory coupling
dG( * )
0
d
for stable attractor
Increasing constant
input to the LIFneurons
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
 k 1  R( k )
For two neurons with
excitatory coupling
strength = 0.1 and time
delay
= 0.2
Ernst et al. PRL 74, 1995
Stability for two excitatory neurons
with delayed coupling
Return map
 k 1  R( k )
 k :  B (t k )   A (t k )   B (t k )
if tk is time when oscillator A fires
For two neurons with
excitatory coupling
strength = 0.1 and time
delay
= 0.2
Ernst et al. PRL 74, 1995
Summary about two-neuron coupling
with delays
• Excitation leads to out-of-phase behavior
• Inhibition leads to in-phase behavior
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
A network of oscillators with
excitatory coupling
Winfree model of coupled oscillators
N-oscillators with natural frequency ωi
 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.
Ariaratnam & Strogatz, PRL 86, 2001
Averaged frequency ρi as a function of ωi
locking
partial locking
incoherence
partial death
  0.65
slowest oscillators
stop
Frequency range of oscillators
Ariaratnam & Strogatz, PRL 86, 2001
Phase diagram
assuming uniform distribution of natural frequencies
Summary
A 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 synchronization
in networks with excitatory and inhibitory neurons
Borgers & Kopell
Neural Computation 15, 2003
Role of excitation and inhibition
in neuronal synchronization
in networks with excitatory and inhibitory neurons
gIE
gEE
gII
gEI
Mutual synchronization
All-to-all connectivity
Sparse connectivity
gIE
gEE
IE=0.1; II=0; gEI=gIE=0.25; gEE=gII=0; τE=2 ms; τI=10 ms
gII
gEI
Main message
Synchronous rhythmic firing results from
• E-cells driving the I-cells
• I-cells synchronizing the E-cells
Synchronization is obtained for
• Continuous drive to E-neurons
• Relatively strong EI connections
• Short time decay of inhibitory post synaptic
potentials
Simple (Theta) neuron model
Neuronal state represented as phase on the unit-circle
with input I (in radians) and membrane time-constant τ.
When I<0: two fixed points :
Saddlenode
bifurcation
unstable
stable
I<0
I=0
I>0 : spiking neuron
I=0
I<0
unstable
I>0
Saddlenode
bifurcation
stable
Spiking neuron
Time interval between spikes
π
π
=
-π
-π
Sufficient conditions for synchronized
firing
E=>I synapses too weak
External input to I-cells
Conditions for synchrony
•E-cells receive external
input above threshold
Rhythm restored by adding
I=>I synapses
I=>E synapses too weak
• I-cells spike only in
response to E-cells;
Relatively strong EI
connections
• I=>E synapses are short
and strong such that I-cells
synchronize E-cells
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
Possible role of feedback
Data from electric fish
Correlated input
Uncorrelated input
Doiron et al. Science, 2004
Possible role of feedback
noise
feedback Doiron et al. Science, 2003
Doiron et al. PRL, 93, 2004
Feedback to retrieve correlated
input
data
model
Experimental data
Linear response analysis gives
 x j () x*j ()  x0, j () x0*, j ()   2 A2  c
Power in range
2-22 Hz
40-60 Hz
What happens to our analytical formula when
FB comes from a LIF neuron?
2( gKA) | gKA |2
 x j ( ) x ( )  x0, j ( ) x ( )   A  c | A |
| 1  gKA |2
*
j
*
0, j
2
2
2
2
FB from a linear unit
FB from LIF neuron G
30
D ==
1818
msec
τDelay
ms
g = - 1.2
 x j () x () 
S (spikes2/s)
*
j
25
G = 6 msec,
but we
Feedback
gain
=obtain
-1.2 the same line with G = 18 msec
τLIF = 6 ms (longer time constant, same result)
20
15
0
50
100
freq (Hz)
150
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.
Short time constant for inhibitory neuron:
τLIF = 2 ms
C=0
Spectrum with FB from LIF neuron and C1=0
800
700
LIF
Synchronized
firing even
without correlated
input
600
500
400
300
200
100
0
0
20
40
60
80
100
120
LIF
For small time constant of LIF-neuron,
network starts spontaneous oscillations of
synchronized firing
Uncorrelated input
Fully correlated input
Spectrum with FB from LIF neuron and C1 =1
Spectrum with FB from LIF neuron and C1=0
250
800
700
200
600
500
150
400
100
300
200
50
100
0
0
20
40
60
80
100
120
0
0
20
40
60
80
100
120
Sufficient conditions for synchronized
firing in the Kopell model
E=>I synapses too weak
External input to I-cells
Conditions for synchrony
•E-cells receive external
input above threshold
Rhythm restored by adding
I=>I synapses
I=>E synapses too weak
• I-cells spike only in
response to E-cells;
Relatively strong EI
connections
• I=>E synapses are short
and strong such that I-cells
synchronize E-cells
LIF with
small τ
LIF
No feedback; input at 50, 60 or 90 Hz
LIF
With feedback; input at 50, 60 or 90 Hz
no FB
FB to all
0.9
1
CI 60 Hz to P1
CI 45 Hz to P1
CI 90 Hz to P1
0.8
CI 60 Hz to P1
CI 45 Hz to P1
CI 90 Hz to P1
0.9
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0
0.1
10
20
30
40
50
60
70
80
90
100
110
120
0
10
20
30
40
50
60
70
80
90
100
110
120
Time constant of inhibitory neuron is
crucial !
Short time constant: neuron is co-incidence detector
Börgers 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.
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
Propagation of synchronous activity
•Leaky integrate-and-fire neuron:

dV
 V   I i
dt
i
•20,000 synpases; 88% excitatory and 12 % inhibitory
•Poisson-like output statistics
‘Synfire chain’
Diesman et al., Nature, 1999
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
Output less precise
Temporal
accuracy
versus σin for
various input
spike
numbers
Output more precise
State-space analysis
Model parameters:
# 100 neurons
Transmission
function for pulsepacket for group
of 100 neurons.
Evolution of
synchronous spike
volley
Stable attractor
State-space analysis
Model parameters:
# 100 neurons
Attractor
Saddle point
Dependence on size of neuron groups
N=80
N=90
N=100
N=110
• a minimum of 90 neurons are necessary to preserve synchrony
•fixed point depends on a, σ and w.
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.