SLIDE 1 Irmantas Ratas, Kęstutis Pyragas
irmantas.ratas@gmail.com Loughborough 2018 09 04
Chimera-like states in two interacting populations of heterogeneous quadratic integrate-and-fire neurons
Center for Physical Sciences and Technology, Lithuania
http://ratas.ndlab.lt
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- Single neuron
- Single population of neurons
– Macroscopic field equations – Bifurcations
- Two interacting identical populations
– Symmetric solutions – Non-symmetric solutions
Overview
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Quadratic integrate-and-fire neurons
Excitable Spiking Equations:
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Quadratic integrate-and-fire neurons
Excitable Spiking Theta representation Equations:
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Interaction
Neurons interact synaptically
Modeled by Heaviside function Neuron effects other neurons only, when its potential exceed threshold value.
SLIDE 6 Macroscopic variables
Microscopic model Infinite size network limit enables analytical approach. Macroscopic variables:
- Mean membrane potential
- Firing rate
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Continuity density function: number of neurons between and .
Continuity equation
Continuity equation Trivial stationary solution:
SLIDE 8 Continuity equation
Lorentzian ansatz
- E. Montbrio, D. Pazo, A. Roxin , Phys. Rev. X 5, 021028 (2015)
- population firing rate
- average potential
distribution of parameter
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Macroscopic equations
If external currents are distributed according to Lorentz function with width and center . (Network consists both excitable and spiking neurons) Relation with Kuramoto order parameter:
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Macroscopic equations
If external currents are distributed according to Lorentz function with width and center Spiking rate Average potential Coupling strength
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Network
internal external interactions
1st Network 2nd Network
Identical populations
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Network
internal external interactions Macroscopic equations:
1st Network 2nd Network
Identical populations
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Symmetric solutions
Transverse and longitudinal coordinates: Symmetric solutions: Equations for Q and M are identical to eq. of a single population with a modified coupling strength J=Jin+Jex (equilibrium points and limit cycles)
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Symmetric solutions
Transverse and longitudinal coordinates: Symmetric solutions: Equations for Q and M are identical to eq. of a single population with a modified coupling strength J=Jin+Jex (equilibrium points and limit cycles)
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Symmetric solutions
Transverse and longitudinal coordinates: Equations for Q and M are identical to eq. of a single population with a modified coupling strength J=Jin+Jex (equilibrium points and limit cycles) stationary points limit cycles Symmetric solutions:
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Non-symmetric solutions
Macroscopic equations:
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Non-symmetric solutions
Splay state Chimera-like Chaotic
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Non-symmetric solutions
Chimera-like state for external excitatory coupling
SLIDE 19 Conclusions
Competition of neural interactions within and between the populations may lead to a rich variety of nonsymetrical patterns, including splay state, antiphase periodic oscillations, chimera like states and chaotic oscillations as well as bistabilities between them.
- I. Ratas and K. Pyrgas, Symmetry breaking in two interacting populations of quadratic integrate-and-fire
neurons, Phys. Rev. E 96, 042212 (2017).
- I. Ratas and K. Pyragas, Macroscopic self-oscillations and aging transition in a network of synaptically
coupled quadratic integrate-and-fire neurons, Phys. Rev. E 94, 032215 (2016).
SLIDE 20 ACKNOWLEDGMENT
This work was supported by Grant No. S-MIP-17-55
- f the Research Council of Lithuania
