The notion of morph (from greek μορφη for form) extends beyond static geometry to encompass system dynamics. In complex systems, form and state are inseparably linked: the visible appearance (morph) emerges from the underlying conditions and interactions that define a system’s state. Changes in state manifest as transformations of form, a process sometimes described as “morphing”, in case of smooth dynamic transitions.

Here, morphostate is defined the state of form a system occupies at a given time, expressing a temporal equilibrium between internal emergent processes and external constraints. Tangent to both self‑organization and morphogenesis, it represents form shaped by endogenous dynamics yet conditioned by environmental flows, boundary conditions, and historical trajectories. In this way, the morphostate serves as a conceptual bridge, linking a system’s internal emergent processes with the evolution of its form while acknowledging the role of external reality in shaping both state and structure.

Morphostate reflects the intention to understand, model, and shape the form and function of complex systems by numerically simulating their dynamics and applying AI-based tools to guide their evolution toward desired structures and behaviors—an essential challenge in contemporary research and industrial applications that demand adaptive, optimized structures and functions.

An example of a model that is used across many disciplines to describe excitable dynamics is the FitzHugh–Nagumo (FHN) model whose mathematical structure appears across many fields. The FHN model appears in many real systems: In neuroscience, the FHN model is used to capture excitable dynamics, such as threshold-dependent spike generation, oscillations, and propagating waves, that emerge from its solutions for appropriate choices of parameters and coupling terms in the model equations, reproducing phenomena observed experimentally; in chemistry to capture the generation of autocatalytic pulses, oscillatory reactions, and propagating reaction–diffusion waves that emerge from the system’s reaction kinetics and diffusion properties; in physics, the model is applied to simulate and analyze nonlinear waves, including plasma instabilities, optical pulses, fluid convection patterns, and chemical–physical reaction fronts, providing insight needed to predict and control instabilities in optical devices, in plasma environments involving heating and transport, the onset of turbulence in fluids, circulation fronts in climate and ocean models, and combustion instabilities in engines and turbines. In biology, the model is used to investigate self‑organization phenomena, pattern formation, and cell signaling dynamics. In social and economic systems, it supports the analysis of rumor propagation, collective behavior cascades, and threshold‑driven adoption of new trends and ideas, which can be interpreted as social analogues of excitable spikes.
The broad applicability of the model ensures that improvements in analytical techniques or numerical solution methods in one field can be leveraged in other fields, facilitating cross‑disciplinary knowledge transfer. When a new numerical solver, stability analysis, bifurcation technique, reaction–diffusion discretization, or simulation scheme is developed, the resulting innovation is immediately applicable across all fields that employ the same model. This cross‑disciplinary transferability is a hallmark of universality in nonlinear dynamics.

Artificial intelligence increasingly supports the simulation of excitable‑media models such as the FitzHugh–Nagumo system. Beyond advanced architectures like neural operators, a wide range of simpler machine‑learning approaches, including feedforward networks, recurrent neural networks, autoencoder‑based reduced models, and physics‑informed neural networks, can approximate the system’s dynamics or accelerate numerical solvers. Because these methods are largely model‑agnostic, methodological improvements developed in one application domain often transfer directly to others, enhancing stability, accuracy, and computational efficiency across disciplines. Within this spectrum, Echo State Networks provide a particularly lightweight and effective approach: their reservoir‑based dynamics can learn the temporal evolution of the FitzHugh–Nagumo equations from data, enabling fast prediction, real‑time simulation, and the augmentation or replacement of traditional numerical schemes.

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