Recently, in the past decade, high-frequency oscillations (HFOs), very high-frequency oscillations (VHFOs), and ultra-fast oscillations (UFOs) were reported in epileptic patients with drug-resistant epilepsy. However, to this day, the physiological origin of these events has yet to be understood. Our study establishes a mathematical framework based on bifurcation theory for investigating the occurrence of VHFOs and UFOs in depth EEG signals of patients with focal epilepsy, focusing on the potential role of reduced connection strength between neurons in an epileptic focus. We demonstrate that synchronization of a weakly coupled network can generate very and ultra high-frequency signals detectable by nearby microelectrodes. In particular, we show that a bistability region enables the persistence of phase-shift synchronized clusters of neurons. This phenomenon is observed for different hippocampal neuron models, including Morris-Lecar, Destexhe-Paré, and an interneuron model. The mechanism seems to be robust for small coupling, and it also persists with random noise affecting the external current. Our findings suggest that weakened neuronal connections could contribute to the production of oscillations with frequencies above 1000 Hz, which could advance our understanding of epilepsy pathology and potentially improve treatment strategies. However, further exploration of various coupling types and complex network models is needed.

We have built a mathematical framework to examine how a reduced neuronal coupling within an epileptic focus could lead to very high-frequency (VHFOs) and ultra-fast oscillations (UFOs) in depth EEG signals. By analyzing weakly coupled neurons, we found a bistability synchronization region where in-phase and anti-phase synchrony persist. These dynamics can be detected as very high-frequency EEG signals. The principle of weak coupling aligns with the disturbances in neuronal connections often observed in epilepsy; moreover, VHFOs are important markers of epileptogenicity. Our findings point to the potential significance of weakened neuronal connections in producing VHFOs and UFOs related to focal epilepsy. This could enhance our understanding of brain disorders. We emphasize the need for further investigations of weakly coupled neurons.

Behavioral and Social Neuroscience Research Group Central European Institute of Technology Masaryk University Brno Czech Republic

Brno Epilepsy Center Dept of Neurology St Anne's Univ Hospital and Faculty of Medicine Masaryk University Brno Czech Republic member of the ERN EpiCARE

Department of Applied Mathematics Techmed Centre University of Twente Enschede The Netherlands

Department of Mathematics and Statistics Faculty of Science Masaryk University Brno Czech Republic

Institute of Scientific Instruments The Czech Academy of Sciences Brno Czech Republic

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