Mathematicians Reveal the Mechanism behind Neuron Synchronisation: Hyperchaos

Scientists of the International Laboratory of Dynamic Systems and Applications at HSE Campus in Nizhny Novgorod have described a rare case of synchronisation in a system of chemically coupled neuron models. The study findings enable a mathematical description of atypical brain functioning modes, including those associated with neurodegenerative diseases. The study has been published in Regular and Chaotic Dynamics.

The research was financed by a megagrant from the Russian Government as part of the 'Science and Universities' National Project and a grant from the Russian Science Foundation.

Neurons are non-linear entities, and this characteristic plays a crucial role in their functioning. Their non-linearity is manifested in how they process and transmit information. Under specific conditions, neurons can synchronise, giving rise to collective rhythms or waves of activity in the brain. This form of synchronisation can be regarded as ensembles of oscillators.

Ensembles of oscillators are interconnected and interacting oscillating entities. Particularly interesting are their nonlinear interactions, where a shift in the 'behaviour' of one oscillator can elicit a complex response in others. 

One of the most noteworthy nonlinear effects is the phenomenon of synchronisation. In earlier literature, systems have been described as individually displaying bursting oscillations—alternating periods of activity and rest—and collectively synchronising in antiphase, where the activity of one element corresponds to the state of rest of the other.

This is a relatively typical mechanism of activity within the neuronal systems. However, there are specific parameters at which oscillators can be in-phase, meaning that both subsystems are active or 'silent' simultaneously. This atypical synchronisation mode attracted the interest of HSE researchers. By employing numerical modelling and solving a system of equations that describes the chemical interaction between two neurons, the scientists were able to describe the scenario and mechanism underlying the occurrence of such in-phase oscillations.

A fundamentally new aspect of our work is demonstrating that the in-phase behaviour corresponds to hyperchaos. We successfully described a mechanism behind the occurrence of in-phase synchronisation by revealing the manifestation of saddle cycles with a two-dimensional unstable manifold in the subsystem dynamics.

Natalya Stankevich
Co-author of the paper, Associate Professor, HSE Campus in Nizhny Novgorod

Hyperchaos in the oscillator system represents an auto-oscillatory mode with an infinite period, where the system behaves unpredictably and the instability is multidimensional. According to the scientists, this is a more complex form of chaos, since there are multiple directions along which closely positioned phase trajectories can diverge.

In the hyperchaos mode, oscillations are chaotic within each subsystem. They are absolutely not identical to each other. However, their cycles of rest and activity occur in-phase. Using numerical modelling, we identified specific regions within the parameter space of the model where in-phase synchronisation can be observed.

Natalya Stankevich
Co-author of the paper, Associate Professor, HSE Campus in Nizhny Novgorod

The findings from the study have potential application in neuroscience research. Neuronal synchronisation modes play a central role in shaping both normal and atypical patterns of activity in different brain regions. Thus, epilepsy is frequently linked to unconventional patterns of neuronal activity and to their synchronisation. Epileptic seizures result from the collective activity of neural cells and are often referred to as pathologically hypersynchronised states. Studies focusing on neurosynchronisation can be instrumental in identifying the specific changes in brain activity that lead to epileptic seizures.
IQ

March 19