Drive-Response Coupling Framework

• Drive-response coupling framework is a system where a master subsystem drives a slave, establishing unidirectional forcing as seen in FitzHugh–Nagumo oscillators.
• The framework employs mathematical models and numerical techniques (ODEs, PDEs, Lyapunov analysis) to reveal synchronization phenomena, resonance tongues, and chaos.
• It has practical applications in excitable media and neural circuits, providing insights into pattern formation, control, and robust information encoding.

The drive-response coupling framework encompasses a class of dynamical systems architectures where one subsystem (the "drive", or master) exerts unilateral or asymmetric influence over another ("response", or slave), with the response system's temporal evolution being directly forced by the state or output of the drive. This paradigm is foundational in the study of coupled excitable systems, including neural, chemical, and electronic models, and serves as a canonical template for phenomena such as synchronization, entrainment, multistability, and chaos, particularly in the context of FitzHugh–Nagumo (FHN) oscillators and their extensions (Cebrián-Lacasa et al., 2024, Hoff et al., 2015, Jalnine, 2013). The mathematical and computational infrastructure of this framework allows systematic investigation of how unidirectional, time-varying, or functionally structured couplings mediate complex spatiotemporal dynamics.

1. Mathematical Formulation and Canonical Models

In its archetypal instantiation for excitable media, the drive-response framework is formalized as a system of ODEs or PDEs with explicit coupling terms:

$$\begin{aligned} \dot x_\mathrm{drive} &= F(x_\mathrm{drive}) \ \dot x_\mathrm{resp} &= G(x_\mathrm{resp}) + H(x_\mathrm{drive}, x_\mathrm{resp}; \gamma) \end{aligned}$$

where $F$, $G$ define the intrinsic nonlinear dynamics of the drive and response, and $H$ specifies the coupling, typically parameterized by strength $\gamma$ and selected to reflect physical connectivity, e.g., $H=\gamma(x_\mathrm{drive} - x_\mathrm{resp})$ (diffusive), $H=\gamma f(x_\mathrm{drive})$ (nonlinear, threshold-like), or more general operator structure.

In the context of FHN oscillators, the minimal drive-response system for two units takes the form (Hoff et al., 2015):

$$\begin{aligned} \dot v_1 &= v_1 - \frac{v_1^3}{3} - w_1 + \gamma(v_2 - v_1)\ \dot w_1 &= \varepsilon(v_1 + a - b w_1) \ \dot v_2 &= v_2 - \frac{v_2^3}{3} - w_2 \ \dot w_2 &= \varepsilon(v_2 + a - b w_2) \end{aligned}$$

with $v_2$ (drive) evolving autonomously, while $v_1$ (response) receives forced input via $\gamma(v_2 - v_1)$. For finite lattices and spatial continua, this approach extends to partial differential equations with spatially nonlocal or delayed drive-response couplings (Cebrián-Lacasa et al., 2024).

2. Bifurcation Structure and Dynamical Phenomena

Drive-response coupled systems display a spectrum of global bifurcation patterns and nonlinear behaviors that are unattainable in symmetric or mutually coupled systems. The principal phenomena include:

• Arnold Tongues and Resonance Locking: In unidirectionally coupled FHN oscillators, the master acts as a periodic or quasi-periodic forcing on the slave. The response system exhibits a hierarchy of synchronization tongues ("Arnold tongues")—domains in the $(\gamma, \text{param})$ plane where $p$:1 locking between the drive and response occurs. These tongues are organized in a Stern–Brocot (period-adding) tree structure, and their emergence is tightly linked to Neimark–Sacker and saddle-node bifurcation loci (Hoff et al., 2015).
• Quasi-Periodic and Chaotic Regimes: Between tongues, the response oscillator can exhibit quasi-periodic tori and chaotic attractors, sometimes of Smale–Williams solenoid type if the drive is itself modulated in antiphase. Rigorous diagnostics via Lyapunov exponents and angle-distribution tests distinguish hyperbolic chaos (structurally stable, robust to parameter changes) from soft chaos or quasiperiodicity (Jalnine, 2013).
• Multistability and Basin Fractalization: Coexistence of multiple attractors (different periodic orbits, tori, chaos) is generic in drive-response networks. The basins of attraction are often strongly intermingled, with fractal boundaries, so small perturbations in initial conditions or noise realize abrupt transitions between qualitatively distinct spiking patterns (Hoff et al., 2015).
• Phase-Locking Cascades in Feedforward Motifs: In chains or trees of drive–response-connected FHN oscillators, higher-order nesting of phase-locking and period-adding behavior is observed, with progressively more complex locking and mixed-mode oscillations as network depth increases (Davison et al., 2018).

3. Spatiotemporal Pattern Formation and Wave Propagation

The drive-response concept extends naturally to spatially distributed systems. In FHN reaction–diffusion frameworks, localized or extended regions can serve as organizing "drives" for secondary response subpopulations or spatial domains. Examples include:

• Pacemaker-Driven Target Patterns: Embedding an autonomous oscillatory patch (the drive) within a surrounding excitable medium induces spherical or spiral waves propagating outward. The wave speed and envelope are functions of spatial coupling and drive-response matching (Cebrián-Lacasa et al., 2024).
• Delayed and Nonlocal Response: Incorporation of transmission delay or spatially nonlocal coupling in the drive-response architecture yields rich pattern formation scenarios, including delayed-induced oscillations, wave trains, and reentrant (spiral) phenomena (Cebrián-Lacasa et al., 2024, Krupa et al., 2014).
• Chimera States: Nonlocally coupled rings under partial heterogeneity and phase-lag realize drive-response motifs at the level of coherently and incoherently oscillating domains, providing a mesoscopic mechanism for partial synchronization (Cebrián-Lacasa et al., 2024).

4. Computational and Numerical Methodologies

The analysis of drive-response frameworks leverages advanced computational tools:

• Lyapunov and Isoperiodic Diagrams: Systematic scanning of parameter planes via Lyapunov exponents ($\lambda_1$) visualizes regions of synchrony (locked, $\lambda_1\approx0$), periodicity, chaos ($\lambda_1>0$), and equilibrium. Isoperiodic plots quantify periodicity and help resolve primary versus secondary Arnold tongues (Hoff et al., 2015).
• Numerical Continuation: Bifurcation curves (saddle-node, Hopf, Neimark–Sacker, period-doubling) are traced in parameter spaces, revealing the intricate substructuring of drive-induced resonances and their collision loci. Tools such as MATCONT automate detection and classification of local and global bifurcations (Hoff et al., 2015).
• Stochastic Path-Integral and Action Methods: For noise-driven or stochastic drive–response systems, large-deviation theory and Hamiltonian dynamics of most-probable activation paths provide quantitative predictions of escape times and fluctuation-induced transitions (Franović et al., 2015).

5. Physiological and Computational Significance

Within neuroscience and related fields, drive-response coupling encapsulates the essential organizing principle of neural circuits wherein directional or hierarchical information flow underlies computation and pattern generation:

• Unidirectional Connections in Neural Microcircuits: Real neurons often display asymmetric synaptic connectivity, with forward-driving pyramidal neurons controlling inhibitory or secondary units (as in feedforward inhibition or relay pathways).
• Information Encoding and Coding Robustness: Hyperbolic chaos and robust locking under drive-response mechanisms afford stable yet flexible coding motifs resistant to parameter variability and noise, with direct implications for brain-inspired (neuromorphic) computing architectures (Jalnine, 2013).
• Control and Modulation: External (drive) signals—ranging from periodic electrical fields, pacemaker regions, to computational input layers—can dynamically set network oscillatory modes, induce desired synchronization or chaos, and implement switching between distinct computation regimes (Nguessap et al., 12 Feb 2025).

6. Extensions, Limitations, and Open Problems

Despite their generality, drive-response frameworks have limitations and intrinsic complexity:

• Suppression of Multistability Under Mutual Coupling: Bidirectional/diffusive coupling inverts many of the fine features of unidirectional drive-response systems: Arnold tongue organization collapses, quasi-periodicity is suppressed, and global network synchronization dominates (Hoff et al., 2015).
• Model Reduction and Approximation: Many canonical models employ dimension reduction (e.g., timescale separation, reduction to phase maps), but the hierarchy of emerging dynamical phenomena—including canard explosions, mixed-mode oscillations, and multistability—demands careful local-global analysis (Davison et al., 2018, Krupa et al., 2014).
• High-Dimensional and Realistic Connectomics: Extending the drive-response formalism to realistic, heterogeneous, and high-dimensional neural circuits with feedback, adaptation, and noise—while preserving analytic tractability—is a central ongoing research challenge.

---

References

• "Six decades of the FitzHugh-Nagumo model: A guide through its spatio-temporal dynamics and influence across disciplines" (Cebrián-Lacasa et al., 2024)
• "Numerical bifurcation analysis of two coupled FitzHugh-Nagumo oscillators" (Hoff et al., 2015)
• "Hyperbolic and non-hyperbolic chaos in a pair of coupled alternately excited FitzHugh-Nagumo systems" (Jalnine, 2013)
• "Modulation of Neuronal Firing Modes by Electric Fields in a Thermosensitive FitzHugh-Nagumo Model" (Nguessap et al., 12 Feb 2025)
• "Activation process in excitable systems with multiple noise sources: One and two interacting units" (Franović et al., 2015)
• "Mixed mode oscillations and phase locking in coupled FitzHugh-Nagumo model neurons" (Davison et al., 2018)
• "Complex oscillations in the delayed Fitzhugh-Nagumo equation" (Krupa et al., 2014)