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. Consider 2 weakly coupled oscillators
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. Consider 2 weakly coupled oscillators
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. Consider 2 weakly coupled oscillators
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. Consider 2 weakly coupled oscillators
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. Consider 2 weakly coupled oscillators
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. Consider 2 weakly coupled oscillators
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. Consider 2 weakly coupled oscillators
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
Type I
Type II
Hodd

12. Jeong and Gutkin, Neural Comp 2007
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 I
Type II

15. Phase Locking with Shunting Inhibition
Type I
Type II

16. Direct simulations confirm analysis
Key - where is Esyn wrt
minimal voltage
Esyn=-60
Esyn=-96

17. GABA is hyperpolarizing
GABA is depolarising
GABA is hyperpolarizing
GABA is shunting
Type II
Type I
Asynch
Synchrony
Asynch
Bistable
Synchrony
Asynch

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. Ermentrout, Pascal, Gutkin 2001
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. Remme, Lengyel, Gutkin (in prep)
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
Hot spot --- Unexciting cable -- Hot spot
If the electrotonic distance is large -- weak interactions
Can view the dendritic tree as a network of weakly coupled oscillators

32. Example: Morris-Lecar oscillators
Dynamics of dendritic oscillators
Example: Morris-Lecar oscillators
Morris-Lecar (Type II) oscillators coupled via passive cable
- effect van L in algemeen
- passive cable
- spiking oscillator

41. Hodd and Bifurcation diagram for 40 Hz
Influence of GABA reversal
Ho Young Jeong, Gatsby, UCL
Hodd and Bifurcation diagram for 40 Hz
Hodd
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
No adaptation (40Hz)
No adaptation (10Hz)
Adaptation (10Hz)
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, b=0.1)

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 -> 1 Asynchronous state : R1 & R2 -> 0
Two equal size cluster: R1 ->0 & R2->1
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
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,