We consider a heterogeneous, globally coupled population of excitatory quadratic integrate-and-fire neurons with excitability adaptation due to a metabolic feedback associated with ketogenic diet, a form of therapy for epilepsy. Bifurcation analysis of a three-dimensional mean-field system derived in the framework of next-generation neural mass models allows us to explain the scenarios and suggest control strategies for the transitions between the neurophysiologically desired asynchronous states and the synchronous, seizure-like states featuring collective oscillations. We reveal two qualitatively different scenarios for the onset of synchrony. For weaker couplings, a bistability region between the lower- and the higher-activity asynchronous states unfolds from the cusp point, and the collective oscillations emerge via a supercritical Hopf bifurcation. For stronger couplings, one finds seven co-dimension two bifurcation points, including pairs of Bogdanov–Takens and generalized Hopf points, such that both lower- and higher-activity asynchronous states undergo transitions to collective oscillations, with hysteresis and jump-like behavior observed in vicinity of subcritical Hopf bifurcations. We demonstrate three control mechanisms for switching between asynchronous and synchronous states, involving parametric perturbation of the adenosine triphosphate (ATP) production rate, external stimulation currents, or pulse-like ATP shocks, and indicate a potential therapeutic advantage of hysteretic scenarios.

Citation: Eydam, S., Franović, I., & Kang, L. (2024). Control of seizure-like dynamics in neuronal populations with excitability adaptation related to ketogenic diet. Chaos: An Interdisciplinary Journal of Nonlinear Science, 34(5), 053128. https://doi.org/10.1063/5.0180954

We study the collective dynamics in a population of excitable units (neurons) adaptively interacting with a pool of resources. The resource pool is influenced by the average activity of the population, whereas the feedback from the resources to the population is comprised of components acting homogeneously or inhomogeneously on individual units of the population. Moreover, the resource pool dynamics is assumed to be slow and has an oscillatory degree of freedom. We show that the feedback loop between the population and the resources can give rise to collective activity bursting in the population. To explain the mechanisms behind this emergent phenomenon, we combine the Ott-Antonsen reduction for the collective dynamics of the population and singular perturbation theory to obtain a reduced system describing the interaction between the population mean field and the resources.

Citation: Franović, I., Eydam, S., Yanchuk, S., & Berner, R. (2022). Collective Activity Bursting in a Population of Excitable Units Adaptively Coupled to a Pool of Resources. Frontiers in Network Physiology, Volume 2 - 2022. https://doi.org/10.3389/fnetp.2022.841829

We study an excitable active rotator with slowly adapting nonlinear feedback and noise. Depending on the adaptation and the noise level, this system may display noise-induced spiking, noise-perturbed oscillations, or stochastic bursting. We show how the system exhibits transitions between these dynamical regimes, as well as how one can enhance or suppress the coherence resonance or effectively control the features of the stochastic bursting. The setup can be considered a paradigmatic model for a neuron with a slow recovery variable or, more generally, as an excitable system under the influence of a nonlinear control mechanism. We employ a multiple timescale approach that combines the classical adiabatic elimination with averaging of rapid oscillations and stochastic averaging of noise-induced fluctuations by a corresponding stationary Fokker–Planck equation. This allows us to perform a numerical bifurcation analysis of a reduced slow system and to determine the parameter regions associated with different types of dynamics. In particular, we demonstrate the existence of a region of bistability, where the noise-induced switching between a stationary and an oscillatory regime gives rise to stochastic bursting.

Citation: Franović, I., Yanchuk, S., Eydam, S., Bačić, I., & Wolfrum, M. (2020). Dynamics of a stochastic excitable system with slowly adapting feedback. Chaos: An Interdisciplinary Journal of Nonlinear Science, 30(8), 083109. https://doi.org/10.1063/1.5145176