Coupled Chialvo Maps

• Coupled Chialvo Maps are discrete-time dynamical systems featuring interconnected neuron models with fast activation and slow recovery dynamics.
• They reveal diverse phenomena such as noise-optimal synchronization, chimera states, and complex spatiotemporal patterns influenced by network topology and coupling strength.
• Analytical and numerical studies highlight the impact of heterogeneity, delayed coupling, and memristive effects on multistability and fractal basin boundaries.

Coupled Chialvo maps are discrete-time dynamical systems where multiple Chialvo neuron maps are interconnected via electrical or more complex coupling schemes. These networks, parameterized by local nonlinearity and coupling topology, serve as canonical models for investigating synchronization, multistability, pattern formation, and noise-induced phenomena in synthetic neuronal ensembles. The prototypical Chialvo map is a two-dimensional system modeling excitable or bursting behavior via a fast activation variable and a slow recovery variable, updated by a nonlinear map with tunable parameters. When such maps are coupled, the resultant networks display a diversity of collective phenomena including noise-optimal synchronization, chimeras, multistable attractors with fractal basin boundaries, and rich spatiotemporal patterns—each sensitive to coupling topology, strength, heterogeneity, and stochastic drive.

1. The Single Chialvo Map and Its Elementary Coupling

The canonical single-neuron Chialvo map updates membrane potential $x_t$ and recovery variable $y_t$ in discrete time: $$\begin{aligned} x_{t+1} &= x_t^2 \exp(y_t - x_t) + I + D\xi_t \ y_{t+1} &= a y_t - b x_t + c \end{aligned}$$ where $a < 1$ is the recovery constant, $b<1$ is the strength of activation–recovery coupling, $I$ is a bias current, $c$ is an offset, $D$ is white noise intensity, and $\xi_t$ is unit-variance Gaussian noise. This system, for varying $b$, admits fixed points, invariant circles (Neimark–Sacker bifurcation), chaos (positive Lyapunov exponent), and periodic windows.

The basic coupling in a network of $N$ Chialvo neurons, indexed by $i$, is formulated as a diffusive term in the $x$-component: $$x_{i,t+1} = x_{i,t}^2 \exp(y_{i,t} - x_{i,t}) + I + (k / N_i) \sum_{j=1}^N \delta_{ij} \left[x_{j,t-1} - x_{i,t-1}\right] + D\xi_{i,t}$$ where $k$ is the global coupling strength, $N_i$ is the in-degree of node $i$, and the adjacency matrix $\delta_{ij} \in \{+1, -1, 0\}$ encodes excitatory, inhibitory, or null couplings, respectively (Used et al., 2024).

These equations generalize readily to more complex schemes, including asymmetric coupling and delayed or time-varying interconnections.

2. Topologies: Small-World, Ring-Star, and 2D Lattices

The emergent dynamics in coupled Chialvo systems are critically determined by network topology:

• Small-world network: Constructed by random rewiring of a regular ring lattice (Watts–Strogatz model), this configuration exhibits rapid synchronization transitions as the rewiring probability $p$ increases—reducing the critical coupling $k_c$ necessary for global synchrony and expanding the synchrony domain in the noise–coupling parameter plane (Used et al., 2024).
• Ring and ring-star topologies: In ring–star networks, peripheral neurons are arranged on a ring, each connected to a central hub; each edge can have time-varying coupling strength and can be modulated by electromagnetic flux (via memristive variables), with both spatial and temporal heterogeneity (Ghosh et al., 2022). Such topologies give rise to chimera states, solitary clusters, and traveling wave patterns, contingent on both deterministic and stochastic link modulation.
• Two-dimensional lattices: In 2D square lattices, coupling is typically nearest-neighbor, implemented as nonlinear (diffusive or quadratic) mixing of neighbor dynamics. These settings enable the spontaneous formation of expanding rings, spiral waves, turbulence, and spatially complex chimera states (Joshi et al., 29 Jan 2026, P et al., 8 May 2025).
• Time-variation and noise in topology: Networks wherein links switch probabilistically at each timestep or coupling strengths fluctuate (additive noise) exhibit temporal heterogeneity, providing an explicit control of coherence and complexity.

3. Synchronization, Heterogeneity, and Noise-Optimality

Coupled stochastic Chialvo maps display rich synchronization phenomena, shaped by parameter heterogeneity, coupling, and noise:

• Synchronization order parameter: For $N$ neurons, the synchronization is quantified by

$$R(x) = \frac{\langle \overline{x}^2 \rangle_t - \langle \overline{x} \rangle_t^2}{\langle\, \langle x^2 \rangle_i - \langle x \rangle_i^2\,\rangle_t},\quad \overline{x}(t) = \frac{1}{N}\sum_{i=1}^N x_{i,t}$$

with $R \simeq 1$ signifying global synchrony.

• Noise-induced and noise-optimal synchronization: There exist critical values of the noise intensity $D$—notably $D^\ast \approx 8 \times 10^{-4}$ and $2 \times 10^{-3}$—where noisy fluctuations enhance synchrony, independent of coupling strength $k$, degree of parameter mismatch, or proportion of inhibitory links. This delineates coherent "tongues" in the $(k,D)$ phase diagram (Used et al., 2024).
• Impact of heterogeneity and inhibition: Parameter mismatch (heterogeneity in $b_i$ up to $\pm 1\%$) degrades the maximum attainable $R$ and raises critical coupling thresholds. Inhibitory links, especially beyond 5% of all connections, rapidly destroy synchrony and suppress the characteristic noise-optimal bands. Purely excitatory or low-inhibition networks, by contrast, retain global coherence over larger regions of parameter space.
• Firing statistics: The mean inter-spike interval (ISI) decreases monotonically with noise, and its network-wide standard deviation stays low except in large $p$ or high-inhibition regimes.

4. Spatiotemporal Patterns, Chimera States, and Complexity Measures

Chialvo networks, especially with nontrivial topology and electromagnetic flux, exhibit diverse spatiotemporal behaviors:

• Chimera states: In ring and ring–star Chialvo networks, portions of the network synchronize while others remain incoherent. Parameter ranges promoting chimeras align with intermediate diffusive coupling, heterogeneity, and moderate flux or noise (Ghosh et al., 2022, Muni et al., 2022).
• Traveling waves and solitary states: Time-varying topology, electromagnetic flux, and spatially distributed noise can nucleate traveling wavefronts and cause single (or small clusters of) nodes to desynchronize (“solitary states”), observable in bifurcation continuations as alternating synchrony windows.
• Pattern formation in 2D: Under nonlinear neighbor coupling, expanding rings or persistent spiral waves form, with transitions to spatiotemporal turbulence at large coupling (Joshi et al., 29 Jan 2026). Persistence (fraction of sites never changing Okubo–Weiss discriminant sign) decays via stretched exponentials or power-laws, quantifying the robustness of coherent domains.
• Complexity quantifiers: Synchrony can be measured via global cross-correlation coefficients and error norms; complexity is captured by sample entropy computed on spatial means or local time series, peaking in disordered regimes and minimizing in regular oscillatory or synchronized phases.

5. Multistability, Fractal Basins, and Final State Sensitivity

Finite coupled Chialvo systems can be multistable with coexistence of periodic and chaotic attractors:

• Coexisting attractors: Two asymmetrically coupled (but individually nonchaotic) Chialvo maps can display both periodic (near-synchronous alternating burst) and chaotic (out-of-sync bursts) attractors, controlled by the structure of the electrical coupling (Lamb et al., 5 Nov 2025).
• Fractal basin boundaries: The separatrix between these attractors is fractal with uncertainty exponent $\mathfrak{u}\approx 0.18$, yielding a basin boundary of dimension $d\approx 3.82$ in the four-dimensional state space. Final state sensitivity grows super-exponentially: to reduce initial-condition uncertainty by one digit, six extra digits of precision are required.
• Biological relevance: Such extreme sensitivity and unpredictability in final network state—synchrony versus desynchrony—has implications for understanding the dynamical fragility of neuronal circuits, especially under small intrinsic heterogeneity or noise.

6. Bifurcation Structure and Analytical Techniques

The global dynamical landscape of coupled Chialvo maps is shaped by a complex bifurcation architecture:

• Codimension-1 bifurcations: Saddle-node, period-doubling, and Neimark–Sacker bifurcations occur as coupling or local parameters are varied, marking transitions among fixed points, limit cycles, invariant circles, and chaos (Ghosh et al., 2024, Muni et al., 2022).
• Codimension-2 points and resonance: Two-parameter continuation identifies special points where bifurcation curves collide (e.g., flip–Neimark–Sacker, fold–flip, and 1:1, 1:2 resonance) (Ghosh et al., 2024). These organize the transition zones supporting bistability and partial synchronization.
• Linear stability analysis: At synchronous fixed points, the block-diagonalization of the network Jacobian allows for explicit calculation of transverse eigenvalues, providing critical values for the onset of synchrony or instability as coupling terms are tuned.
• Numerical Lyapunov exponents: Calculation of maximal Lyapunov exponents across trajectories phasespaces discriminates chaotic from periodic regions and quantifies internal complexity.

7. Extensions: Noise Types, Delay, and Nonstandard Coupling

Variants on the standard coupled Chialvo map setup expand the observed dynamical repertoire:

• Lévy (α-stable) noise: Heavy-tailed fluctuations induce standing and traveling waves, chimera states, and abrupt desynchronization regimes, shifting the boundaries of regular–bursting–wave transitions, demonstrating that noise statistics can be an effective macroscopic control parameter in Chialvo networks (P et al., 8 May 2025).
• Delayed coupling: Introducing time-lagged coupling in lattice topologies generates transitions among labyrinthine patterns, rotating spirals, and coherent waves, with delay acting as a bifurcation parameter (P et al., 8 May 2025).
• Electromagnetic flux and memristive coupling: Networks with local memristors alter the global synchronization properties, supporting chimera and multi-cluster states responsive to both static and time-modulated flux strength (Muni et al., 2022, Ghosh et al., 2022).

These extensions underscore the universality and adaptability of the coupled Chialvo map framework for probing neuronal synchrony, pattern formation, and complexity transitions under biologically relevant constraints and perturbations.

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Key References:

(Used et al., 2024, Ghosh et al., 2022, Used et al., 2024, Ghosh et al., 2024, Lamb et al., 5 Nov 2025, P et al., 8 May 2025, Muni et al., 2022, Joshi et al., 29 Jan 2026)