Period addition cascade in a model of neuron-glial interaction

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We analyze a system of four differential equations that describe the dynamics of a neuron-glial network using the mean field approximation [1, 2]:

(τ=–E+α ln(1+e^{1/α(JuxE+I0)}),

=(1–x)/τ_D–uxE,(1)

=U(y)–u/τF+U(y)(1–u)E,

=–y/τ_y+βσ(x),

where E(t) is the average activity, x(t) is the fraction of available neurotransmitter released into the synaptic gap with a probability u(t); y(t) is the fraction of the gliatransmitter released by the astrocyte. Sigmoidal functions U(y) and σ(x) correspond to changes in the base probability level u(t) during the release of the gliatransmitter and activation of astrocytes during neurotransmitter release, respectively. The input inhibitory current corresponds to a bifurcation parameter with a negative value of I₀ <0, while the remaining parameters have positive and fixed values. The rest of the parameters are positive and fixed. For a detailed description of the model, including information on the type of functions and parameter values, refer to works [1, 2].

For a constant value of U(x)=const, the first three equations in system (1) represent the Tsodyks–Markram model, which explains the short-term synaptic plasticity phenomenon [1]. The model was enhanced with a fourth equation for y in [2], incorporating the influence of astrocytes via the concept of a tripartite synapse [3].

Model (1) illustrates a wide range of dynamic behavior including quiescence, regular tonic activity, and chaotic bursting activity. These behaviors correspond to various sets in the phase space, such as stable equilibrium states, limit cycles of period 1, limit cycles of any period n∈N, and chaotic attractors. Changing the I₀ l parameter causes sets to bifurcate, resulting in the loss of stability of certain attractors and the emergence of others, leading to a shift in the dynamic regime. Therefore, in terms of dynamics, the conditions for bifurcation and the characteristics of the newly formed attractors are crucial.

In this presentation, we have obtained a series of numerical bifurcations in system (1) that correspond to the shift from tonic activity to burst activity, resulting in subsequent modifications to the bursts. Specifically, our findings demonstrate that an increase in the number of spikes per burst is determined by a period adding cascade where the aforementioned limit cycle of period n becomes unstable, allowing for a previously established stable cycle of period n+1 to occupy the position of the “main” attractor. This process culminates in the vanishing of the orbit with an endless period due to the saddle-node bifurcation of cycles, followed by the creation of a dependable cycle with a period of 1.

The main properties of the cascade were reproduced in our model one-dimensional piecewise-smooth map

$\overline{z}=\left\{\begin{array}{l}1-z^6,forz\text{ < }\mathrm{0,}\\ \mu -1-\mu {(z-1)}^6,forz\text{ > }\mathrm{0,}\end{array}\right.$

where z∈R¹, μ is a bifurcational parameter. The map’s results suggest that an increase in current I₀ i in model (1) may lead to the emergence and disappearance of quasi-strange attractors (quasi-attractors), implying chaotic behavior in connection with burst variation.

ADDITIONAL INFORMATION

Funding sources. This study was supported by the Russian Scientific Foundation (grant No. 22-12-00348).

Competing interests. The author declares that he has no competing interests.

About the authors

N. V. Barabash

Author for correspondence.
Email: barabash@itmm.unn.ru
Russian Federation, Nizhny Novgorod; Nizhny Novgorod

References

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