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This paper derives the first phase model for weakly coupled Izhikevich neurons with chemical synapses, overcoming limitations of applying continuous oscillator theory to this discontinuous model. The phase model is then used to analyze the existence and stability of phase-locked states across parameter variations. A key finding reveals a direct correlation between the stability of anti-phase states in the phase model and the bifurcation behavior of the uncoupled Izhikevich neuron.
Weakly coupled Izhikevich neurons can exhibit stable phase-locked states, a phenomenon previously unquantifiable due to the model's discontinuous nature.
The Izhikevich model is a computationally efficient neuron model that can exhibit a wide range of firing patterns observed in the brain. However, it is a discontinuous dynamical system, which means the methods for applying weakly coupled oscillator theory developed for continuous dynamical systems cannot be applied. Therefore, the collective behaviour of coupled Izhikevich model has not been fully studied. To our knowledge, we carry out the first computation of the phase model for an Izhikevich neuronal model with chemical synapses. Using the phase model we study the existence and stability of phase-locking states and how these vary with parameters. Moreover, we find an interesting connection between the stability of anti-phase states of the phase model and the bifurcation behaviour of the uncoupled Izhikevich model itself. The accuracy of the prediction of the phase model is supported by simulations of two weakly coupled Izhikevich neurons.