Quasiperiodic partial synchronization and …...Quasiperiodic partial synchronization and...
Transcript of Quasiperiodic partial synchronization and …...Quasiperiodic partial synchronization and...
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Quasiperiodic partial synchronization and macroscopic chaos in
populations of inhibitory neurons with delay
Ernest Montbrió
Center for Brain and Cognition-Univestitat Pompeu Fabra
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Introduction
● Inhibitory networks are responsible for the generation of fast neuronal oscillations (> 50 Hz)
● Oscillations due to inhibition and synaptic delays ● Individual neurons do not fire at the freq of the fast, mean
field oscillations● Challenge for canonical, analytically tractable, phase
oscillator models (Kuramoto model)● Increasing interest for more complex forms of
synchronization (quasiperiodic partial synchronization, chimera states...)
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Kuramoto model with time delay
Coupling Strength
Time delayNatural frequencies
IncoherenceAsynchronous state
Full synchronization
Yeung, Strogatz, Phys Rev Lett (1999)
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Kuramoto model with time delayphase diagram
J
Full SynchronyIncoherence
Sync/Incoh
Same dynamics for Excitation and for Inhibition
Yeung, Strogatz, Phys Rev Lett (1999)
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Neuronal Mean field modelsHeuristic Firing Rate Equations
Single excitatory population:
Wilson and Cowan, 1972; Amari 1974
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Fast oscillations in HFREInhibitory population with delay
Roxin, Brunel, Hansel, PRL (2005)Roxin, Montbrió, Phys D (2011)
If D = 5 ms, freqs. between 50 and 100 Hz
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Quadratic Integrate-and-fire neuron
Excitable dynamics:
Oscillatory dynamics:
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Ensemble of recurrently coupled QIF neuronswith synaptic time delay
● Coupling: J>0: Excitation; J<0: Inhibition
● Mean synaptic activity ( sD=s(t-D) ):
● Fast synapses (s->0):
Population-Averaged Firing Rate
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Correspondence of QIF, Winfree and Theta models
When: Vpeak=-Vreset→infty :
● Inter-spike Interval self-oscillatory neurons (>0,J=0):
● Winfree Model (identical, self-oscillatory neurons):
● Theta-Neurons (=1):
Ermentrout and Kopell,SIAM J Appl Math 1986
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Numerical simulations, QIF neuronsJ=-1.65, D=2.5 (=)
Neurons display quasiperiodic dynamics for inhibbitory (J<0) coupling only
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Numerical simulations (II)J=-1.85, D=2.5 (=)
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Numerical simulations (III)Macroscopic chaos?
J=-3.8, D=3 (=)
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Derivation of exact Firing Rate Equations
● Spiking neurons: QIF● All-to-all coupling● Quenched heterogeneity (no noise)● Exact in the thermodynamic limit
Montbrió, Pazó and Roxin, PRX 2015; Pazó and Montbrió, Submitted
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Thermodynamic limitContinuous formulation
The Continuity Equation is
Fraction of neurons with V between V and V+dVand parameter at time t
PDF of the currents
For each value of Then the total density at time t is given by:
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Lorentzian Ansatz
Center Width
Equivalence btw the LA and the Ott-Antonsen ansatz
● The LA is the Poisson Kernel in the positive semi-plane (x>0)
● The OA ansatz is Poisson Kernel in the unit disk (R<=1).
Ott and Antonsen, Chaos, 2008
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Lorentzian Ansatz Firing Rate & Mean Membrane potential
Firing Rate: Prob flux at threshold:
Firing Rate
Mean Membrane potential
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2D Firing Rate equations (FRE)
Lorentzian distribution of currents
Cauchy Residue's theorem to solve
Substituting the LA in the continuity equation:
without loss of generality
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Numerical simulations using FREJD
2D
→ Periodic activity of collective variables
→ Quasiperiodic dynamics at microscopic level
Quasiperiodic partial synchronization in inhibitory networks
Van Vresswijk, PRE (1996); Mohanti, Politi, J. Phys A (2006);Rosenblum, Pikovsky, PRL 2007; Pikovsky, Rosenblum Physica D (2009);Politi, Rosenblum, PRE (2015)
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Increasing inhibition...JD
2D
Period of oscillations remains constant
Limit cycle is symmetric v → -v
Using FRE for =0, this symmetry implies that:
The symmetry is broken at period doubling bif...
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Macroscopic chaos through quasiperiodic partial sync
JD
The QPS undergoes a succession of period doubling bifs leading to macroscopic chaos(using the FRE: Largest Lyapunov exp. 0.055)
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Analysis of FRE
For identical neurons, the only fixed point is:
Linearizing around the f.p. and imposing the cond. of marginal stab: = i
Hopf boundaries:
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Analysis of the fully synchronized stateWinfree model
And by the evently spaced lines:
Stability of the fully sync state in the the Winfree model:
We find the boundaries:
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Phase diagram for identical neurons
Shaded regions: Asynch/Incoherent state STABLE
Dashed regions: Full sync UNSTABLE
Hopf(sub-cr)
Hopf(super-cr)Sync UNST
Incoh UNST
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Onset of QPS and heterogeneity
TC bifs. are not robust
bistability btw two partially sync states remains, though
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Macroscopic chaos in heterogeneous networks
No chaosat microscopic
level!
Sync plateaus
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Thanks!
ITN project: Complex Oscillatory Systems: Modeling and Analysis
Diego PazóInstituto de Física de
Cantabria (CSIC-Universidad Cantabria)
Fedrico Devalle
Poster: Solvable model for a network of spiking neurons with delays