What weakly coupled oscillators can tell us about networks and cells Boris Gutkin Theoretical Neuroscience Group, DEC, ENS; College de France

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

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

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

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

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

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

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

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

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

Phase Response Curve Type I Type II Hodd

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

Synchrony with hyperpolarizing inhibition depolarizing hyperpolarizing stable Phase difference unstable Synaptic speed Synaptic speed Van Vreeswijk, Abbott, Ermentrout 1994

Phase Locking with Shunting Inhibition Type I Type II

Phase Locking with Shunting Inhibition Type I Type II

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

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

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

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)

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

Type 1 neurons with adaptation synchronize with excitation

Spike Frequency adaptation changes the shape of the PRC Adaptation increasing

Blocking M-current with Cholinergic agonist changes PRC shape!

Complex interactions between adaptation currents Blocking I-M converts type II to type I Blocking I-AHP uncovers type II

I-AHP and I-M have different sensitivity to Acetylcholine I-AHP; I-M I-AHP; I-M

Extended cell structure and PRC

Cholinergic effects are local

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)

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

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

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

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.

Direct simulations confirm analysis: splay state 5 coupled neurons (Esyn= -20mV, b=0.1)

Type II regime Bifurcation Diagram PRC

Hodd for type II neuron

Stability diagram for type II neurons as a function of GABA reversal

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

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

PRCs from the canonical model with adaptation subHopf Without adaptation SNIC

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,