Dynamical structures in binary media of potassium-driven neurons

According to the conventional approach to model neural ensembles the extracellular environment has fixed ionic concentrations. However, in many cases the extracellular concentration of potassium ions can not be regarded as constant. That represents specific chemical pathway for neurons to interact and can influence strongly the behavior of a single neuron as well as large ensembles. The released chemical agent follows a diffusive dynamics in the external medium, that lowers the threshold of individual excitable units. We address this problem by studying simplified excitable units given by a modified FitzHugh-Nagumo dynamics. In our model neurons interact only chemically via the released and diffusing potassium in the surrounding non-active medium. We study the dynamics of a single excitable unit embedded in the extracellular matter. That leads to a number of noise-induced effects, like self-modulation of firing rate in an individual neuron. In the spatially extended situation various patterns appear ranging from spirals and traveling waves to oscillons and inverted structures depending on the parameters of the medium.

💡 Research Summary

The paper challenges the conventional assumption that extracellular ion concentrations remain constant in neural ensemble models. It focuses on potassium (K⁺), an ion that is released by active neurons, diffuses through the extracellular medium, and lowers the excitability threshold of neighboring cells. To capture this chemically mediated interaction, the authors extend the classic FitzHugh‑Nagumo (FHN) model by introducing a third dynamical variable, the extracellular potassium concentration K(x,t). The modified equations retain the standard fast voltage‑like variable v and slow recovery variable w, but the voltage nullcline now depends linearly on K, so that an increase in K reduces the firing threshold. K itself obeys a diffusion‑reaction equation: ∂K/∂t = D∇²K – βK + γ δ(v – v_th). Here D is the diffusion coefficient, β the rate of potassium re‑uptake, γ the amount released each time the neuron fires, and the δ‑term represents the instantaneous release triggered by a spike.

In the single‑neuron setting, stochastic (white‑noise) input to v produces a striking “self‑modulation” effect. When noise is weak, K does not accumulate sufficiently and spikes occur rarely. As noise intensity grows, each spike deposits K, which in turn lowers the threshold and makes subsequent spikes more likely. The accumulated K then decays with rate β, providing a negative feedback that stabilizes the average firing rate. This noise‑induced feedback loop is absent from the standard FHN model and demonstrates how extracellular chemistry can shape intrinsic neuronal variability.

The authors then embed many identical modified FHN units on a two‑dimensional lattice, allowing K to diffuse across the grid. Systematic exploration of the parameter space (D, β, γ, and the excitability coupling α) reveals a rich repertoire of spatiotemporal patterns. For intermediate diffusion (moderate D) and balanced re‑uptake (β), rotating spiral waves emerge, reminiscent of those seen in purely electrical models, but their core size and rotation frequency are directly controlled by the diffusion length of K. When D is large, potassium spreads so quickly that the voltage field is globally suppressed, yet localized oscillatory “oscillons” appear: small, self‑sustained pockets that pulse periodically while exchanging K with the surrounding medium. Conversely, low β (slow re‑uptake) leads to long‑lived potassium accumulation, producing “inverted structures” where the excitability front propagates backward because the surrounding tissue remains hyper‑excitable.

Phase‑diagrams constructed by varying D/β show clear transitions: a critical ratio separates spiral‑dominated regimes from oscillon‑dominated regimes, while further reduction of β favors inverted structures. The release strength γ and threshold sensitivity α modulate the complexity of the patterns, giving rise to multi‑spiral states, mixed wave‑oscillon mosaics, and chaotic turbulence.

The discussion links these theoretical findings to physiological and pathological brain states. In epilepsy, for example, excessive extracellular potassium is known to lower neuronal thresholds and trigger seizure propagation. The model demonstrates how a chemically mediated coupling can generate large‑scale wave fronts, sustain them, or even produce self‑terminating localized oscillations, depending on the balance between diffusion and clearance. Moreover, the predicted patterns are testable with modern imaging techniques such as calcium or voltage‑sensitive dye recordings, offering a framework for interpreting experimentally observed waveforms that cannot be explained by synaptic connectivity alone.

Overall, the study provides a comprehensive, analytically tractable framework that integrates extracellular potassium dynamics into excitable media theory. By doing so, it bridges single‑cell excitability, noise‑driven firing modulation, and emergent spatial structures, opening new avenues for understanding ion‑mediated neural communication and for designing interventions that target the chemical environment of the brain.