What weakly coupled oscillators can tell us about networks and cells

1
What weakly coupled oscillators can tell us
about networks and cells

• Boris Gutkin
• Theoretical Neuroscience Group, DEC, ENS
• College de France

2
Game Plan

• Overview of mathematical framework weakly
  coupled oscillators, phase models, coupling
  functions, phase response curves
• Shunting inhibition and synchrony in pairs of
  neurons
• Adaptation and synchrony effects of cholinergic
  modulation
• Dendrites and oscillations neuron as a network

3
Game Plan

• Overview of mathematical framework weakly
  coupled oscillators, phase models, coupling
  functions, phase response curves
• Shunting inhibition and synchrony in pairs of
  neurons
• Adaptation and synchrony effects of cholinergic
  modulation
• Dendrites and oscillations neuron as a network

4
How to compute H? By averaging, or equivalently
(Kopell Ermentrout 91) by formal method due to
Kuramoto (82)
Consider 2 weakly coupled oscillators
2.1
2.6
Let uncoupled oscillators have a stable limit
cycle with period T
Where X is the solution to the linearised adjoint
Then solution can be described by
2.7
2.2
For two coupled neurons, the coupling is
synaptic G(Vpre,V)spre(t)(Es-V)
Letting
We get
2.3
2.8
Phase locked solutions when
Where
If
2.4
2.9
So the phase locked solution f is stable when
(Hansel et al. 1995)
2.5
5
How to compute H? By averaging, or equivalently
(Kopell Ermentrout 91) by formal method due to
Kuramoto (82)
Consider 2 weakly coupled oscillators
2.1
2.6
Let uncoupled oscillators have a stable limit
cycle with period T
Where X is the solution to the linearised adjoint
Then solution can be described by
2.7
2.2
For two coupled neurons, the coupling is
synaptic G(Vpre,V)spre(t)(Es-V)
Letting
We get
2.3
2.8
Phase locked solutions when
Where
If
2.4
2.9
So the phase locked solution f is stable when
(Hansel et al. 1995)
2.5
6
How to compute H? By averaging, or equivalently
(Kopell Ermentrout 91) by formal method due to
Kuramoto (82)
Consider 2 weakly coupled oscillators
2.1
2.6
Let uncoupled oscillators have a stable limit
cycle with period T
Where X is the solution to the linearised adjoint
Then solution can be described by
2.7
2.2
For two coupled neurons, the coupling is
synaptic G(Vpre,V)spre(t)(Es-V)
Letting
We get
2.3
2.8
Phase locked solutions when
Where
If
2.4
2.9
So the phase locked solution f is stable when
(Hansel et al. 1995)
2.5
7
How to compute H? By averaging, or equivalently
(Kopell Ermentrout 91) by formal method due to
Kuramoto (82)
Consider 2 weakly coupled oscillators
2.1
2.6
Let uncoupled oscillators have a stable limit
cycle with period T
Where X is the solution to the linearised adjoint
Then solution can be described by
2.7
2.2
For two coupled neurons, the coupling is
synaptic G(Vpre,V)spre(t)(Es-V)
Letting
We get
2.3
2.8
Phase locked solutions when
Where
If
2.4
2.9
So the phase locked solution f is stable when
(Hansel et al. 1995)
2.5
8
How to compute H? By averaging, or equivalently
(Kopell Ermentrout 91) by formal method due to
Kuramoto (82)
Consider 2 weakly coupled oscillators
2.1
2.6
Let uncoupled oscillators have a stable limit
cycle with period T
Where X is the solution to the linearised adjoint
Then solution can be described by
2.7
2.2
For two coupled neurons, the coupling is
synaptic G(Vpre,V)spre(t)(Es-V)
Letting
We get
2.3
2.8
Phase locked solutions when
Where
If
2.4
2.9
So the phase locked solution f is stable when
(Hansel et al. 1995)
2.5
9
How to compute H? By averaging, (Kopell
Ermentrout 91)or by formal method due to
Kuramoto (82)
Consider 2 weakly coupled oscillators
2.1
2.6
Let uncoupled oscillators have a stable limit
cycle with period T
Where X is the solution to the linearised adjoint
Then solution can be described by
2.7
2.2
For two coupled neurons, the coupling is
synaptic G(Vpre,V)spre(t)(Es-V)
Letting
We get
2.3
2.8
Phase locked solutions when
Where
If
2.4
2.9
So the phase locked solution f is stable when
(Hansel et al. 1995)
2.5
10
How to compute H? By averaging, (Kopell
Ermentrout 91)or by formal method due to
Kuramoto (82)
Consider 2 weakly coupled oscillators
2.1
2.6
Let uncoupled oscillators have a stable limit
cycle with period T
Where X is the solution to the linearised adjoint
Then solution can be described by
2.7
2.2
For two coupled neurons, the coupling is
synaptic G(Vpre,V)spre(t)(Es-V)
Letting
We get
2.3
2.8
Phase locked solutions when
Where
If
2.4
2.9
So the phase locked solution f is stable when
(Hansel et al. 1995)
2.5
11
Phase Response Curve
12
Game Plan

• Overview of mathematical framework weakly
  coupled oscillators, phase models, coupling
  functions, phase response curves
• Shunting inhibition and synchrony in pairs of
  neurons
• Adaptation and synchrony effects of cholinergic
  modulation
• Dendrites and oscillations neuron as a network

Jeong and Gutkin, Neural Comp 2007
13
Synchrony with hyperpolarizing inhibition
depolarizing
hyperpolarizing
stable
Phase difference
unstable
Synaptic speed
Synaptic speed
Van Vreeswijk, Abbott, Ermentrout 1994
14
Phase Locking with Shunting Inhibition
Type II
Type I
15
Phase Locking with Shunting Inhibition
Type II
Type I
16
Direct simulations confirm analysis
Esyn-60
Esyn-96
17
Type II
Type I
Asynch
Synchrony
Bistable
Asynch
Asynch
Synchrony
18
Shunting Inhibition/Excitability

• Type I
• Low firing rate and fast depolarizing GABA leads
  to in-phase synchronization for Type I
  oscillators.
• Only the anti-phase locked solution is stable in
  the shunting region.
• Hyperpolarizing GABAergic synapses cause the
  phase dynamics to
• have two stable solutions.
• Type II
• Asynchrony with hyperpolarizing GABA
• Synchrony with depolarising GABA
• Bistable regime for shunting GABA -- possible
  appearance of clusters
• Key how is the reversal potential related to the
  voltage trajectory of the neuron

19
Game Plan

• Overview of mathematical framework weakly
  coupled oscillators, phase models, coupling
  functions, phase response curves
• Shunting inhibition and synchrony in pairs of
  neurons
• Adaptation and synchrony effects of cholinergic
  modulation
• Dendrites and oscillations neuron as a network

Ermentrout, Pascal, Gutkin, Neural Comp
2002 Stiefel, Gutkin, Sejnowski (in prep)
20
Spike Frequency Adaptation changes type I to type
II dynamics when the I-K slow is voltage
dependent (low threshold).
Adaptation increasing
Ermentrout, Pascal, Gutkin 2001
21
Type 1 neurons with adaptation synchronize with
excitation
22
Spike Frequency adaptation changes the shape of
the PRC
Adaptation increasing
23
Blocking M-current with Cholinergic agonist
changes PRC shape!
24
Complex interactions between adaptation currents
Blocking I-M converts type II to type I
Blocking I-AHP uncovers type II
25
I-AHP and I-M have different sensitivity to
Acetylcholine
I-AHP I-M
I-AHP I-M
26
Extended cell structure and PRC
27
Cholinergic effects are local
28
Game Plan

• Overview of mathematical framework weakly
  coupled oscillators, phase models, coupling
  functions, phase response curves
• Shunting inhibition and synchrony in pairs of
  neurons
• Adaptation and synchrony effects of cholinergic
  modulation
• Dendrites and oscillations neuron as a network

Remme, Lengyel, Gutkin (in prep)
29
Intrinsic Oscillations in Dendritic Trees Michiel
Remme, GNT Mate Lengyel, Cambridge
30
(No Transcript)
31
(No Transcript)
32
Example Morris-Lecar oscillators
Dynamics of dendritic oscillators

• Morris-Lecar (Type II) oscillators coupled via
  passive cable

33
(No Transcript)
34
(No Transcript)
35
(No Transcript)
36
(No Transcript)
37
(No Transcript)
38
(No Transcript)
39
(No Transcript)
40
(No Transcript)
41
Hodd and Bifurcation diagram for 40 Hz
Influence of GABA reversal Ho Young Jeong,
Gatsby, UCL
Traub neuron DC injection to get 40 Hz w/o
coupling synapses with time scales compatible
with fast GABA
42
Influence of Adaptation Firing rate

• Low firing rate can lead to in-phase
  synchronization but adaptation has much
• stronger effect.
• - The system could be bistable in the region of
  hyperpolarization but the shunting effect
• tends to have the stable anti-phase locked
  solution.

43
Direct simulations confirm analysis splay state
5 coupled neurons (Esyn -20mV, b0.1)
44
Type II regime
Bifurcation Diagram
PRC
45
Hodd for type II neuron
46
Stability diagram for type II neurons as a
function of GABA reversal
47
Extension to Large Network
Populations of globally coupled oscillators
Order parameters

• Order parameters Z characterize the collective
  behavior of the N-neurons
• - The instantaneous degree of collective
  behaviors can be described by the square modulus
• of Z
• Synchronous state R1 R2 -gt 1 Asynchronous
  state R1 R2 -gt 0
• Two equal size cluster R1 -gt0 R2-gt1

• H is the interaction function defined from two
  oscillator phase dynamics
• - H can be approximated by using the Fourier
  expansion because it is the T-periodic function
• This approximated function ( ) used for
• simulations

48
Extension to Large Network
H function Order parameters
Rastergram
Depolarizing GABA with Adaptation
sync
Hyperpolarizing GABA with Adaptation
async
Hyperpolarizing GABA without Adaptation
2-cluster
100 globally coupled phase models
49
PRCs from the canonical model with adaptation
With adaptation
subHopf
Without adaptation
SNIC
50
The Phase Model Approach

• Suppose that we couple two oscillators Where
  G is a coupling function.
• They can be changed into a phase model using
  the averaging method. Where H is a T-periodic
  interaction function. - The phase difference
  is introduced to see the stability of the
  phase-locked solutions.- When ,
  the phase-locked solution is stable.
• For membrane models,

51
(No Transcript)
52
(No Transcript)