Amplitude Equations in Pattern Dynamics

• Amplitude Equations (AEs) are reduced-order PDEs/ODEs that capture slow spatio-temporal evolution of amplitude variables near bifurcation in complex, non-local systems.
• They are derived via multiple-scale expansions that project dynamics onto stored spatial patterns, resulting in coupled parabolic equations with cubic nonlinearities and diffusion.
• AEs model pattern selection, completion, and front propagation in neural networks and distributed memory systems with finite-range nonlocal interactions.

Amplitude equations (AEs) are reduced-order PDEs or ODEs that describe the slow spatio-temporal evolution of amplitude-like variables near criticality or bifurcation in complex, often non-locally coupled, multistable systems. In the context of distributed systems with distance-constrained non-local interactions and multiple stored spatially heterogeneous states—such as neural networks or autoassociative memories—the AE framework yields a coupled system of parabolic equations for the amplitudes of memory patterns embedded in the connectivity structure.

1. Formulation of the Underlying Distributed System

Consider a scalar field $u(r,t)$ on a domain $\Omega$, subject to both local dynamics $f(u)$ and non-local (distance-restricted) coupling,

$$\partial_t u(r, t) = f(u) + \gamma \int_{\Omega} J(r, r')\, g[u(r', t)] \, dr',$$

where the connectivity kernel is constructed as

$$J(r, r') = \Phi(|r - r'|) \sum_{k=1}^{M} \frac{1}{M} \mu_k(r) \mu_k(r').$$

Here, $\Phi(|r-r'|)$ encodes the spatial range of non-local interactions, and each $\mu_k(r)$ is a spatially heterogeneous “pattern” (memory) stored in the system’s connectivity.

The nonlinearity $f(u)$ is assumed odd (i.e., $f(u) = -f(-u)$) for technical convenience; $g(u)$ is expanded analogously.

2. Perturbation Hierarchy and Slow Modulation Ansätze

The base state $U_0 = u_0$ is assumed to destabilize at a critical parameter value. A multiple scale expansion is performed,

$$u(r, t) = U_0 + \varepsilon U_1 + \varepsilon^2 U_2 + \cdots,$$

where the first-order correction is projected onto the stored patterns: $$U_1 = \sum_{k=1}^M A_k(\tilde{r}, \tilde{t}) \mu_k(r).$$ Amplitudes $A_k$ are functions of slow spatial and temporal variables, $\tilde{r} = \varepsilon r$ and $\tilde{t} = \varepsilon^2 t$, capturing pattern envelopes on long spatiotemporal scales.

Expansions of the nonlinear terms $f(u)$ and $g(u)$ in Taylor series (centered at $u_0$) provide explicit cubic nonlinearities, while expansions in the spatial variable yield diffusive contributions from the distance-constrained kernel.

3. Derivation of Coupled Amplitude Equations

At $O(\varepsilon)$, the linearized equation in the slow variables leads to a solvability requirement that sets the critical coupling constant $\gamma_k$ for each stored pattern. At $O(\varepsilon^3)$, the solvability condition yields for each amplitude,

$$\partial_t A_k = \varepsilon^2 \sum_m \alpha_{km} A_m + \sum_{ijm} \beta_{kijm} A_i A_j A_m + \sum_m D_{km} \nabla^2 A_m,$$

where:

• $\alpha_{km}$ quantify linear growth and inter-pattern coupling,
• $\beta_{kijm}$ encode cubic saturation and mode interactions,
• $D_{km}$ are effective pattern-dependent diffusion coefficients, with

$$\alpha_{km} = \gamma_k b_1 \left\langle \int_{\Omega} J(r, r') \mu_m(r') dr',\, \mu_k(r) \right\rangle,$$

$$\beta_{kijm} = \left\langle \int_{\Omega} [a_3 + \gamma_k b_3 J(r, r')] \mu_i(r') \mu_j(r') \mu_m(r') dr',\, \mu_k(r) \right\rangle,$$

$$D_{km} = \gamma_k b_1 \left\langle \int_{\Omega} J(r, r') (r'-r)^2 \mu_m(r') dr',\, \mu_k(r) \right\rangle.$$

Here, $a_1 = f'(u_0)$, $a_3 = f'''(u_0)/6$, $b_1 = g'(u_0)$, $b_3 = g'''(u_0)/6$.

These equations form a coupled, spatially extended system for $A_k$. Off-diagonal couplings (e.g., $\alpha_{km}$ for $k\neq m$) mediate pattern competition, enforcing selection dynamics.

4. Emergence of Pattern Completion, Selection, and Patterning Fronts

The gradient structure of the amplitude dynamics,

$\partial_t A = -\nabla V[A],$

with an associated Lyapunov functional $V[A]$, implies convergence to stable fixed points corresponding to retrieval of a single memory pattern (pattern selection). The cubic terms typically enforce winner-take-all dynamics: only one $A_k$ tends to stabilize at finite amplitude, while others are suppressed.

Due to finite-range interactions ($\Phi(\cdot)$), diffusion terms are present in each amplitude equation, yielding spatial spread of activation. This facilitates the emergence of patterning fronts: interfaces across which $A_k$ transitions from zero to its asymptotic value,

$c_k^* = \varepsilon \sqrt{2\alpha_{kk} D_{kk}},$

where $c_k^*$ is the minimal front speed for invasion of the $k$-th memory pattern. These fronts realize the physical process of pattern completion: partial or localized cues of a pattern can trigger global retrieval via propagating activation fronts.

5. Effect of Non-locality and Diffusion on Spatio-temporal Pattern Dynamics

Distance-constrained coupling encoded by $\Phi(|r - r'|)$ ensures that evolving amplitude dynamics are not globally coupled but manifest finite, pattern-dependent diffusion. This constraint is critical for modeling physical or neural systems where true global interactions are impossible.

Spatial propagation in the envelope variables,

$\sum_m D_{km} \nabla^2 A_m,$

permits memory retrieval from incomplete cues and determines the speed and structure of invading fronts—features central to both pattern completion and dynamic selection.

The inclusion of nonlocality also enables the description of complex spatio-temporal phenomena (e.g., persistent spatially heterogeneous memories, invasion of one pattern into another, competition at fronts) not captured by classical globally-coupled AE models.

6. Theoretical and Application Significance

• The derivation extends the envelope equation approach to networks storing arbitrary spatially heterogeneous patterns (not just periodic or uniform states).
• These amplitude equations unify the treatment of memory storage, recall, and spatial propagation dynamics within a rigorous multiscale perturbation framework.
• The resulting model provides a mathematically controlled approach to paper pattern selection, completion, and interference, relevant to neural tissue, distributed computation devices, and designed materials with embedded memory.
• This formalism accommodates analysis of stability, speed, and interaction of patterning fronts in high-dimensional heterogeneous spaces with constrained coupling—a key aspect in modern distributed information systems.

7. Summary Table: Key Elements in the AE Framework for Spatially Heterogeneous Patterns

Feature	Mathematical Formulation	Physical/Functional Role
Pattern embedding	$J(r,r')$ via $\mu_k(r)$	Memory storage in connectivity
Amplitude variable	$A_k(\tilde{r},\tilde{t})$	Slow evolution of stored pattern $k$
Dynamical equation	$\partial_t A_k = \cdots$	Envelope modulation (growth, competition, front)
Diffusion term	$D_{km} \nabla^2 A_m$	Finite-range spatial spread (fronts)
Cubic nonlinearity	$\beta_{kijm}$	Saturation, selection, and pattern stability
Front speed	$c_k^* = \varepsilon \sqrt{2\alpha_{kk} D_{kk}}$	Pattern completion rate

This amplitude-equation-based theory provides a rigorous and tractable description of how distributed systems with distance-constrained nonlocal connectivity can store, select, and dynamically retrieve spatially structured memories through the evolution and interactions of their pattern amplitudes (Houben, 16 Jun 2025).