Macroscopic Multistability and Bifurcations in Theta-Neuron Networks with Distributed Delays

We study an all-to-all coupled network of identical theta neurons with synaptic interaction mediated by a distributed time delay.

Using the Watanabe-Strogatz reduction and passing to the thermodynamic limit under the assumption of uniformly distributed constants of motion, we derive a single delay differential equation for the complex order parameter.

The delay is modeled by a family of delay kernels with prescribed mean delay, allowing discrete and distributed delays to be treated in a unified framework.

The equilibria of the reduced system can be classified into two geometrically distinct families: type 1 equilibria on the unit circle and type 2 equilibria on the real axis.

For both families, the local stability problem reduces to scalar characteristic equations involving the Laplace-Stieltjes transform of the delay kernel.

We obtain stability criteria for admissible kernels and explicit Hopf bifurcation conditions for the Dirac kernel, with additional comparison to weak and strong Gamma kernels.

The results show that the delay may either preserve stability, destabilize equilibria, or produce stability switching, depending on the equilibrium branch, parameter regime, and choice of kernel.

Numerical simulations for the discrete-delay case support the analytical results and illustrate the corresponding phase portraits, basins of attraction, coexistence of attractors, and delay-induced periodic dynamics.