Allee-Based Nonlinear Plasticity Model

• Allee-based nonlinear plasticity is a framework that integrates threshold-dependent regulation in both neural and ecological systems.
• It employs nonlinear terms and bifurcation analysis to model dynamics such as bistability, monostability, and extinction based on critical thresholds.
• The model provides insights for robust synaptic memory formation and controlled population spread, with applications in computational neuroscience and ecology.

The Allee-based nonlinear plasticity model is a framework that extends concepts from population dynamics—specifically, the Allee effect—into neural and ecological systems characterized by threshold-dependent growth, dispersal, and adaptation. In both biological and artificial contexts, it formalizes how populations (or synaptic strengths) require a minimal threshold to persist, with dynamics strongly influenced by nonlinearities arising from density- or activity-dependent feedback.

1. Mathematical Formulation of Allee-Based Nonlinear Plasticity

The canonical formulation introduces a density- or norm-dependent regulation term reflecting the Allee effect. In neural synaptic plasticity, the weight update rule is modeled as:

$$\frac{d\mathbf{W}}{dt} = \mathbf{v}^T\left[\mathbf{u} - \frac{1}{K}\mathbf{W}\mathbf{v} \right] \left[1 - \frac{A}{\mathbf{W}^T\mathbf{W}}\right]$$

Here, $\mathbf{W}$ is the weight vector, $\mathbf{u}$ presynaptic input, $\mathbf{v}$ postsynaptic output, $K$ a saturation parameter, and $A$ the Allee threshold (Kwessi, 2022, Kwessi, 11 Aug 2025). If $||\mathbf{W}||^2 < A$, the update becomes negative or drives rapid decay, expressing absence of plasticity; above the threshold, strengthening and competition can persist. The regulation term $1 - A/\mathbf{W}^T\mathbf{W}$ is the central nonlinear feature responsible for threshold-guided synaptic adaptation.

In spatial ecological models, analogous nonlinear terms modulate population growth and motility as a function of local density, leading to reaction–diffusion equations such as:

$$\partial_t \rho = \partial_x \big[D(\rho) \partial_x \rho \big] + F(\rho)$$

where

$$F(\rho) = R_b \cdot r_s(\rho) \cdot \rho(1 - \rho) - R_d \rho,\qquad r_s(\rho) = \frac{1}{2}[1 + \tanh(\kappa(\rho - \theta))]$$

(Böttger et al., 2014). Here, $\kappa$ tunes the plasticity regime, and an appropriate choice (e.g., $\kappa > 0$) generates an Allee effect with extinction below a critical density.

2. Threshold Dynamics, Bifurcations, and Stability

The principal consequence of the Allee-based nonlinearities is the existence of multiple dynamic regimes—monostable persistence, bistable criticality, and extinction. Mathematical analysis reveals:

• Bistability: For attractive plasticity ($\kappa > 0$ in ecological models, $A > 0$ in neural models), at low density or weight, the per capita growth rate is negative, enabling extinction if the system starts below threshold.
• Monostability: For repulsive plasticity ($\kappa < 0$), motility or plasticity is favored at low density or small weights, ensuring inevitable persistence regardless of initial conditions.
• Saddle-node and transcritical bifurcation: Varying the Allee parameter ($A$, $\kappa$, or corresponding ecological $m$) induces bifurcations where stable and unstable equilibria merge or disappear; this governs transition between extant and extinct states (Xia et al., 2023, Kwessi, 11 Aug 2025).

This bifurcation structure underpins robust memory formation and critical transitions in both biological and computational systems. In neural models, by tuning $A$ or $K$, multiple attractors corresponding to stable memories can be sustained, while subthreshold signals decay ("forgetting").

3. Reaction–Diffusion Interactions and Spatial Propagation

In ecological domains, the Allee-based model manifests in nonlinear dispersal mechanisms. Typical equations for density $u(x,t)$ include nonlinear diffusion terms ($u^m$) and a reaction component exhibiting a weak or strong Allee effect ($f(u) = r u^\beta(1-u)$, $\beta > 1$):

$\partial_t u = \partial_{xx}(u^m) + r u^\beta(1-u)$

(Alfaro et al., 2017).

The interplay between the diffusion exponent ($m$), tail heaviness of the initial condition ($\alpha$), and Allee strength ($\beta$) determines propagation speed:

• Heavy initial tails and fast diffusion (small $m$) enable accelerating invasions, sometimes with "infinite" speed.
• Strong Allee effects (large $\beta$) counteract acceleration, producing constant-speed spreading.
• If $\beta < 1 + 1/\alpha$ (porous medium) or $\beta < \min\{1 + 1/\alpha, m + 2/\alpha\}$ (fast diffusion), acceleration occurs; otherwise, propagation is linear in time.

This separation establishes conditions for rapid spread versus containment/extinction—a dynamics mirrored in patch dispersal models with density-dependent plastic migration (Xia et al., 2023).

4. Biological and Computational Implications

Biologically, the Allee-based nonlinear plasticity models represent mechanistic links between cell/environment state and population or synaptic survival. In tumor modeling (Böttger et al., 2014), a density-dependent phenotypic switch (go-or-grow) can induce an Allee effect; enhancing contact inhibition of migration (CIM) corresponds to attractive plasticity, potentially driving small tumors to extinction, whereas repulsive regimes yield uncontrolled growth.

In computational neuroscience, introducing Allee-inspired rules in synaptic plasticity embeds a lower threshold. Synapses with $||\mathbf{W}||^2 < A$ lose efficacy, which is interpreted as absence of plasticity (Kwessi, 2022, Kwessi, 11 Aug 2025). This yields:

• Synaptic normalization: Bounded plasticity avoids runaway growth.
• Competition and decorrelation: High postsynaptic activity depresses competing weights. Recurrent networks with appropriate Allee regulation suppress redundant correlations.
• Memory capacity: Multiple attractors (stable states) arise, increasing storage reliability compared to classic Hebbian/Oja models.

5. Extension: Temporal Dynamics and Memory Traces

Beyond static plasticity, time-dependent effects (eligibility traces, oscillatory inputs) have been incorporated to model biological memory more closely (Kwessi, 11 Aug 2025). The extended equations include:

$$\tau_{\mathbf{v}} \frac{d\mathbf{v}}{dt} = -\mathbf{v} + G(W^T u + W^T v) + \kappa e^{-\Delta t/\tau_1}$$

$$\tau_W \frac{dW}{dt} = v^T [u - K^{-1}WV] [1 - A/(W^T W)] + \lambda e^{-\Delta t/\tau_2}$$

Eligibility traces ($e^{-\Delta t/\tau}$) introduce a memory of recent activity, enhancing retrieval accuracy and resilience under noise. Analysis reveals emergent rhythmic (Hopf) bifurcations, further enriching the dynamical repertoire (Kwessi, 11 Aug 2025).

6. Ecological Applications and Persistence/Extinction Criteria

In ecological predator-prey systems, emergent Allee effects may arise via nonlinear maturation rates (e.g., $\varphi(x) = \kappa/(1 + x)$), where juvenile populations below critical thresholds cannot sustain adult populations, resulting in extinction (Granados et al., 21 Mar 2025). Stochastic perturbations in mortality rates further modulate persistence/extinction dynamics. Sufficient conditions for extinction or persistence are established rigorously:

• Prey extinction: $\beta < \min\{\mu_x, \mu_y\}$
• Stability of extinction equilibrium: $\mu_y \geq \beta$, additional noise conditions
• Prey persistence: Existence of stationary distributions under bounded predation, with recovery possible if maturation is high and noise intensities are moderate

Numerical simulations confirm these regimes and illuminate how the Allee effect (via nonlinear plasticity and environmental stochasticity) controls critical transitions in population survival.

7. Comparative Analysis and Model Impact

The Allee-based nonlinear plasticity framework synthesizes diverse phenomena across biological and artificial systems:

• Enforces extinction thresholds and bounded adaptation
• Generates bifurcation structures (saddle-node, transcritical, Hopf)
• Enhances robustness to noise and external perturbations
• Facilitates competitive, decorrelated, and stable memory formation in neural networks
• Models spatial rescue effects, population spread, and persistence in fragmented habitats

By systematically connecting threshold-dependent dynamics to nonlinear plasticity in both populations and synapses, these models offer a unified methodology for understanding and regulating adaptation, survival, and memory in complex systems.