Stability, bifurcation and phase-locking of time-delayed excitatory-inhibitory neural networks

Hwayeon Ryu; Sue Ann Campbell; Hwayeon Ryu; Sue Ann Campbell

doi:10.3934/mbe.2020403

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 6: 7931-7957. doi: 10.3934/mbe.2020403

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Stability, bifurcation and phase-locking of time-delayed excitatory-inhibitory neural networks

Hwayeon Ryu ^{1
,
,},
Sue Ann Campbell ²

1.
Department of Mathematics and Statistics, Elon University, 50 Campus Drive, Elon, NC 27244, USA
2.
Department of Applied Mathematics and Centre for Theoretical Neuroscience, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

Received: 31 August 2020 Accepted: 03 November 2020 Published: 10 November 2020

We study a model for a network of synaptically coupled, excitable neurons to identify the role of coupling delays in generating different network behaviors. The network consists of two distinct populations, each of which contains one excitatory-inhibitory neuron pair. The two pairs are coupled via delayed synaptic coupling between the excitatory neurons, while each inhibitory neuron is connected only to the corresponding excitatory neuron in the same population. We show that multiple equilibria can exist depending on the strength of the excitatory coupling between the populations. We conduct linear stability analysis of the equilibria and derive necessary conditions for delay-induced Hopf bifurcation. We show that these can induce two qualitatively different phase-locked behaviors, with the type of behavior determined by the sizes of the coupling delays. Numerical bifurcation analysis and simulations supplement and confirm our analytical results. Our work shows that the resting equilibrium point is unaffected by the coupling, thus the network exhibits bistability between a rest state and an oscillatory state. This may help understand how rhythms spontaneously arise in neuronal networks.
- neural networks,
- phase-locking,
- coupling delays,
- Hopf bifurcation
Citation: Hwayeon Ryu, Sue Ann Campbell. Stability, bifurcation and phase-locking of time-delayed excitatory-inhibitory neural networks[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 7931-7957. doi: 10.3934/mbe.2020403

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Abstract

We study a model for a network of synaptically coupled, excitable neurons to identify the role of coupling delays in generating different network behaviors. The network consists of two distinct populations, each of which contains one excitatory-inhibitory neuron pair. The two pairs are coupled via delayed synaptic coupling between the excitatory neurons, while each inhibitory neuron is connected only to the corresponding excitatory neuron in the same population. We show that multiple equilibria can exist depending on the strength of the excitatory coupling between the populations. We conduct linear stability analysis of the equilibria and derive necessary conditions for delay-induced Hopf bifurcation. We show that these can induce two qualitatively different phase-locked behaviors, with the type of behavior determined by the sizes of the coupling delays. Numerical bifurcation analysis and simulations supplement and confirm our analytical results. Our work shows that the resting equilibrium point is unaffected by the coupling, thus the network exhibits bistability between a rest state and an oscillatory state. This may help understand how rhythms spontaneously arise in neuronal networks.

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