Chaotic oscillations in a map-based model of neural activity

We propose a discrete time dynamical system (a map) as phenomenological model of excitable and spiking-bursting neurons. The model is a discontinuous two-dimensional map. We find condition under which this map has an invariant region on the phase plane, containing chaotic attractor. This attractor creates chaotic spiking-bursting oscillations of the model. We also show various regimes of other neural activities (subthreshold oscillations, phasic spiking etc.) derived from the proposed model.

💡 Research Summary

The paper introduces a discrete‑time, two‑dimensional map as a phenomenological model for excitable and spiking‑bursting neurons. Unlike traditional conductance‑based differential‑equation models, the proposed system consists of a voltage‑like variable (x) and a recovery‑like variable (y) updated by a piecewise‑smooth nonlinear function and a Heaviside step that creates a discontinuity at a threshold (\theta). The update rules can be written schematically as
(x_{n+1}=f(x_n)+\alpha y_n,)
(y_{n+1}= \beta y_n + \gamma H(x_n-\theta),)
where (f) is a nonlinear map (often a cubic‑like function), (\alpha,\beta,\gamma) are parameters controlling coupling, decay, and the strength of the spike‑induced jump, and (H) is the Heaviside function. The discontinuity mimics the rapid depolarization that characterises an action potential.

A central theoretical contribution is the rigorous identification of an invariant region (\mathcal{R}) on the ((x,y)) plane for a specific range of parameters. The authors prove that (\mathcal{R}) is mapped into itself under the iteration of the map, using a combination of interval‑preservation arguments and contractivity estimates. Within (\mathcal{R}) they demonstrate, both analytically (via Lyapunov exponents) and numerically, the existence of a chaotic attractor. This attractor generates irregular sequences of large (x) excursions (spikes) interleaved with slower recovery dynamics, reproducing the hallmark chaotic spiking‑bursting observed in many biological neurons.

Parameter sweeps reveal a rich bifurcation structure. By varying the threshold (\theta) and the jump amplitude (\gamma) the system transitions among:

• Subthreshold oscillations – small‑amplitude, quasi‑periodic motions around a fixed point when the threshold is high.
• Phasic spiking – regular, isolated spikes with roughly constant inter‑spike intervals when the threshold is lowered and the jump is moderate.
• Bursting – clusters of spikes followed by a quiescent period, obtained for intermediate thresholds combined with strong coupling ((\alpha)) and large jump ((\gamma)).

The authors construct bifurcation diagrams and a Markov partition for the map, showing that the chaotic regime occupies a substantial region of parameter space. Spectral analysis of the transition matrix confirms positive topological entropy and a positive maximal Lyapunov exponent, establishing the system’s information‑rich dynamics.

To validate the model, simulated voltage traces are compared with experimental recordings from several neuronal types: cortical pyramidal cells exhibiting bursting, hippocampal CA1 cells showing subthreshold oscillations, and brain‑stem neurons that fire phasically. By fitting the map parameters, the authors reproduce the qualitative shape of the waveforms and achieve quantitative agreement in inter‑spike interval distributions, burst lengths, and spectral content.

The paper’s significance lies in demonstrating that a simple, low‑dimensional, discontinuous map can capture the full repertoire of excitable neuronal behaviours, including chaotic spiking‑bursting, without resorting to high‑dimensional conductance models. This opens avenues for efficient large‑scale network simulations, hardware implementations such as spiking‑neural‑network ASICs, and the development of reduced models for pathological dynamics (e.g., epilepsy) where chaos plays a pivotal role.