Complex transitions between spiking, bursting and silent regimes in a new memristive Rulkov neuronal model

TL;DR

This work introduces a finite-memory memristive extension of the Rulkov map by replacing a control parameter with a sigmoidal memristive function that depends on the history of the system. The effective horizontal movement of the memristive parameter across the $(\sigma,\alpha)$ plane, via $\sigma(z)$, enables both uniform and chaotic transitions among silent, spiking, and bursting regimes, controlled by the memory length $m$ and the update rate $\tau$. A bifurcation at $m\approx 2\tau$ delineates three dynamical regimes: pre-bifurcation uniform bursting, post-bifurcation uniform spiking or silent behavior, and chaotic transitions exactly at the bifurcation with long dwell times and occasional extra stationary states. The findings highlight memory-dependent dynamics in low-dimensional neuronal maps and suggest avenues for fitting experimental data and extending the approach to other map-based models.

Abstract

The Rulkov model, which simulates the behavior of biological neurons, is modified by replacing one of its control parameters with a memristive, sigmoid-type function of finite memory. This modification causes the parameter to vary according to the system's history throughout the simulation. Previous works usually modify the Rulkov model by introducing additional parameters altering its behavior. Here, by contrast, we retain the original equations and allow the control parameters to vary in time, thereby preserving the model's fundamental properties. In this sense, the proposed model is locally equivalent in time to the original one. However, unlike the original model, which reproduces a single neuronal regime per simulation, the new memristive version exhibits both uniform and chaotic transitions among multiple neuronal activity regimes. Its dynamics are examined with respect to the rate at which the memristive function changes and the number of internal states it stores. Three distinct scenarios emerge around a bifurcation point. Before the bifurcation, the system undergoes uniform transitions toward a stable bursting regime. After the bifurcation, it shows uniform transitions toward a final spiking or silent regime. At the bifurcation point, highly complex transitions arise. As examples, we present trajectories in which the neuron chaotically switches between regimes without ever settling, and trajectories for which it requires around 140000 map iterations to reach a stationary regime.

Figures (10)

• Figure 1: Examples of the three neuronal regimes: (a) silent, (b) spiking, and (c) bursting, where $x$ represents the neuron's membrane voltage and $n$ indicates the time.
• Figure 2: The $(\sigma,\alpha)$ plane is divided into regions according to the neuronal regime obtained for each pair of control-parameter values in the original Rulkov model Rulkov2002. The background color represents the mean value of the fast variable, $\bar{x}$.
• Figure 3: Transitions between spiking and bursting regimes using the original Rulkov model Rulkov2002 for (a) $\alpha=4.6$, $\sigma=0.16$, $\mu=0.001$ and $(x_0,y_0)=(-1,-3.33)$ as initial condition, and for (b) $\alpha=5$, $\sigma=0.28$, $\mu=0.001$ and $(x_0,y_0)=(-1,-3.5)$ as initial condition. Transitions are rather unclear and may also be interpreted as irregular bursting.
• Figure 4: The black line $\sigma_0$ represents the relation between the control parameter $\sigma$ and the mean value $\bar{x}$ in the original Rulkov model Rulkov2002, given by Eq. (\ref{['eq: MeanValueVsSigma']}). The blue curves represent the memristive function $\sigma(\bar{x})$, given by Eq. (\ref{['eq: MeanMemristiveSigma']}) for $\tau=50$ and different values of the number of states stored by the memristor: (a) $m=60$ (prior to the bifurcation), (b) $m=100$ (at the bifurcation), and (c) $m=140$ (after the bifurcation). The intersection between the curves is indicated by a green dot when it corresponds to a stable neuronal regime, and by a red dot otherwise.
• Figure 5: Evolution, prior to the bifurcation, of the memristive function $\sigma(\bar{x})$ (a), given by Eq. (\ref{['eq: MeanMemristiveSigma']}), and the state variable $x$ (b) for $\alpha=5$, $\mu=0.001$, $\tau=50$, $m=85$ and $(x_0,y_0,z_0)=(-1,-3.48,50)$ as initial condition. After a transient spiking regime, the neuron converges to an irregular, but stable, bursting regime that oscillates around $(\bar{x},\sigma)=(-1,0)$.