Andrey Shilnikov

This paper studies quasi-periodicity phenomena appearing at the transition from spiking to bursting activities in the Pernarowski model of pancreatic beta cells. Continuing the parameter, we show that the torus bifurcation is responsible for the transition between spiking and bursting. Our investigation involves different torus bifurcations, such as supercritical torus bifurcation, saddle torus canard, resonant torus, self-similar torus fractals, and torus destruction. These bifurcations give rise to complex or multistable dynamics. Despite being a dissipative system, the model still exhibits KAM tori, as we have illustrated. We provide two scenarios for the onset of resonant tori using the Poincaré return map, where global bifurcations happen because of the saddle-node or inverse period-doubling bifurcations. The blue-sky catastrophe takes place at the transition route from bursting to spiking.

This is the first part of a review of the scientific works of L.P. Shilnikov. We group his papers according to 7 major research topics: bifurcations of homoclinic loops; the loop of a saddle-focus and spiral chaos; Poincare homoclinics to periodic orbits and invariant tori, homoclinic in noautonous and infinite-dimensional systems; Homoclinic tangency; Saddlenode bifurcation — quasiperiodicity-to-chaos transition, blue-sky catastrophe; Lorenz attractor; Hamiltonian dynamics. The first two topics are covered in this part. The review will be continued in the further issues of the journal.

We study the emergence of in-phase and anti-phase synchronized rhythms in bursting networks of Hodgkin–Huxley–type neurons connected by inhibitory synapses.We show that when the state of the individual neuron composing the network is close to the transition from bursting into tonic spiking, the appearance of the network’s synchronous rhythms becomes sensitive to small changes in parameters and synaptic coupling strengths. This bursting-spiking transition is associated with codimension-one bifurcations of a saddle-node limit cycle with homoclinic orbits, first described and studied by Leonid Pavlovich Shilnikov. By this paper, we pay tribute to his pioneering results and emphasize their importance for understanding the cooperative behavior of bursting neurons. We describe the burst-duration mechanism of inphase synchronized bursting in a network with strong repulsive connections, induced by weak inhibition. Through the stability analysis, we also reveal the dual property of fast reciprocal inhibition to establish in- and anti-phase synchronized bursting.

The methods of qualitative theory of slow-fast systems applied to biophysically realistic neuron models can describe basic scenarios of how these regimes of activity can be generated and transitions between them can be made. We demonstrate how two different codimension-one bifurcations of a saddle-node periodic orbit with homoclinic orbits can explain transitions between tonic spiking and bursting activities in neuron models following Hodgkin–Huxley formalism. In the first case, we argue that the Lukyanov–Shilnikov bifurcation of a saddle-node periodic orbit with non-central homoclinics is behind the phenomena of bi-stability observed in a model of a leech heart interneuron under defined pharmacological conditions. This model can exhibit two coexisting types of oscillations: tonic spiking and bursting. Moreover, the neuron model can also generate weakly chaotic trains of bursting when a control parameter is close to the bifurcation value. In the second case, the transition is continuous and reversible due to the blue sky catastrophe bifurcation. This bifurcation provides a plausible mechanism for the regulation of the burst duration which may increases with no bound as $1/\sqrt{\alpha-\alpha_0}$, where $\alpha_0$ is the transitional value, while the inter-burst interval remains nearly constant.

Asymptotic normal forms for equilibria with a triplet of zero characteristic exponents in systems with $\mathbb{Z}_q$-symmetry are listed.