Developmental approaches to neural architecture search grow functional networks from compact genomes through self-organisation, but the resulting networks operate with fixed post-growth weights. We characterise Hebbian and anti-Hebbian plasticity across 50,000 morphogenetically grown recurrent controllers (5M+ configurations on CartPole and Acrobot), then test whether co-evolutionary experiments -- where plasticity parameters are encoded in the genome and evolved alongside the developmental architecture -- recover these patterns independently. Our characterisation reveals that (1) anti-Hebbian plasticity significantly outperforms Hebbian for competent networks (Cohen's d = 0.53-0.64), (2) regret (fraction of oracle improvement lost under the best fixed setting) reaches 52-100%, and (3) plasticity's role shifts from fine-tuning to genuine adaptation under non-stationarity. Co-evolution independently discovers these patterns: on CartPole, 70% of runs evolve anti-Hebbian plasticity (p = 0.043); on Acrobot, evolution finds near-zero eta with mixed signs -- exactly matching the characterisation. A random-RNN control shows that anti-Hebbian dominance is generic to small recurrent networks, but the degree of topology-dependence is developmental-specific: regret is 2-6x higher for morphogenetically grown networks than for random graphs with matched topology statistics.
Figures & tables
Figure 1. Developmental dynamics of a MorphoNAS network on a 10×10 grid. Background colour encodes morphogen concentrations (RGB = 3 morphogens); white squares = neurons; white lines = connections. From a single progenitor cell (Step 2), reaction-diffusion dynamics drive cell division, differentiation, and chemotactic axon growth, producing a 7-neuron, 13-connection recurrent controller by Step 200. Three grid panels labelled Step 2, Step 10, and Step 200 (final), showing a developing MorphoNAS network. Each grid uses a coloured background to encode three diffusing morphogen concentrations as RGB channels: pale at Step 2 (one neuron, no connections), mixed greens and pinks at Step 10 (6 neurons, 7 connections), saturated purple-blue gradient at Step 200 (7 neurons, 13 connections). White squares mark neurons; white lines mark connections.
Stratum
N
Oracle Δr
% Impr.
Best η
Regret
Weak
47,638
+2.2
15%
+0.05
59%
Low-mid
1,235
+35.7
50%
−0.05
52%
High-mid
256
+60.6
93%
−0.05
86%
Near-perf.
102
+22.5
84%
−0.005
89%
Perfect
769
+7.7
20%
—
100%
Table 1. CartPole plasticity impact by performance stratum under per-network oracle parameters; regret as defined in the text.
Figure 2. CartPole High-mid stratum: mean Δr across the η×λ grid (248-point extended grid). Anti-Hebbian ( η<0 ) with moderate decay ( λ=0.01 ) yields the best results. Hebbian ( η>0 ) is consistently harmful. Heatmap of mean reward improvement on a 248-point grid: decay lambda on the x-axis (8 values from 0 to 0.1) and learning rate eta on the y-axis (31 values from -0.5 at the top to +0.5 at the bottom). Each cell shows a numerical delta-r value with a diverging colour scale: warm tones (red) for negative values where plasticity hurts, cool tones for positive values where plasticity helps. Highest positive values cluster around eta from -0.1 to -0.005 with decay 0.01-0.02; the most negative values appear at the |eta| > 0.2 extremes in either direction.
Figure 3. CartPole: Cohen’s d (anti-Hebbian − Hebbian) by stratum. Anti-Hebbian significantly outperforms Hebbian for competent strata ( d=0.53 – 0.64 , all p<0.001 ). The effect reverses for Weak and Perfect networks with small magnitudes. Horizontal bar chart of Cohen's d (positive = anti-Hebbian advantage) across five performance strata listed top to bottom as Perfect, Near-perf., High-mid, Low-mid, Weak. Three blue bars extend rightward: Near-perf. 0.56, High-mid 0.64, Low-mid 0.53. Two short pink bars extend leftward: Perfect -0.07, Weak -0.14. The horizontal axis spans from -0.75 to +1.00.
Figure 4. CartPole adaptation premium (non-stationary plasticity benefit minus static plasticity benefit) for two perturbation types. Gravity-2 × yields significant positive premiums for competent strata (*** p<0.001 ), indicating genuine adaptation beyond static fine-tuning. Heavy-pole shows near-zero premiums (n.s.). Vertical grouped bar chart of CartPole adaptation premium (reward, y-axis from -30 to +20) across four performance strata on the x-axis: Low-mid, High-mid, Near-perf., Perfect. Each stratum has two bars: blue for Gravity 2x, pink for Heavy pole 10x. Low-mid is strongly negative for both variants (-29.0 *** blue and -20.1 *** pink). High-mid through Perfect show positive significant Gravity 2x premiums (+8.8, +18.7, +8.6, all ***) and small or non-significant Heavy pole 10x values (-5.2 ns, ns, -2.8 ns).
Stratum
N
% Impr.
Cohen’s d
NS % Helped
Low-mid
71
95.8%
+0.48 ***
88.7%
High-mid
74
93.2%
+0.31 ***
95.9%
Near-perf.
281
93.6%
+0.15 ***
86.8%
Perfect
87
96.6%
+0.09 *
95.4%
*** p<0.001 , * p<0.05
Table 2. Acrobot-v1 plasticity characterisation. % Impr.: per-network oracle improvement rate on the combined grid (coarse + micro- η ). Cohen’s d : anti-Hebbian vs Hebbian (coarse grid). NS: non-stationary oracle rate (combined grid). Regret is 100% under any fixed (η,λ) .
Figure 5. Acrobot fraction of episodes unsolved over time under non-stationarity (2 × link mass at step 50). Solid: per-network oracle plasticity; dashed: no plasticity. Lower is better. Oracle plasticity accelerates solving across all strata—most visibly for High-mid and Near-perfect, where the unsolved fraction at step 300 drops from 66%/31% (no plasticity) to 40%/18% (oracle plasticity). Single-panel plot of fraction-of-episodes-unsolved (y-axis 0 to 1) over timestep (x-axis 0 to 500) for Acrobot under non-stationarity. A vertical dashed black line at timestep 50 marks the perturbation switch (2x lower-link mass). Four colour-coded stratum curves: Low-mid (blue), High-mid (pink), Near-perf. (green), Perfect (yellow). Solid lines (Plasticity) drop faster than dashed lines (No plasticity) across all strata, with the largest separation in the High-mid and Near-perfect groups.
Figure 6. Co-evolution (Condition C) evolved η over 200 generations on CartPole (top) and Acrobot (bottom). Left: best individual’s η ; right: population mean η (median ± IQR across 30 runs). CartPole: strong anti-Hebbian drift ( η<0 ). Acrobot: η hovers near zero with mixed signs, matching the characterisation finding. Four-panel plot (2 rows by 2 columns) of evolved learning rate eta over 200 GA generations. Top row: CartPole; bottom row: Acrobot. Left column: best individual eta per generation; right column: population mean eta with median (solid green line) and interquartile range (shaded band) across 30 runs. A red dashed reference line at eta = 0 separates Hebbian (positive) from anti-Hebbian (negative). CartPole panels show clear drift to eta around -0.1 to -0.2 (anti-Hebbian); Acrobot panels show eta fluctuating very close to zero with small mixed-sign deviations.
Biological neural networks continuously adapt and modify themselves in response to experiences throughout their lifetime - a capability largely absent in artificial neural networks. Hebbian plasticity offers a promising path toward rapid adaptation in changing environments. Here, we introduce Hebbian Attractor Networks (HAN), a class of plastic neural networks in which local weight update normalization induces emergent attractor dynamics. Unlike prior approaches, HANs employ dual-timescale plasticity and temporal averaging of pre- and postsynaptic activations to induce either co-dynamic limit cycles or fixed-point weight attractors. Using simulated locomotion benchmarks, we gain insight into how Hebbian update frequency and activation averaging influence weight dynamics and control performance. Our results show that slower updates, combined with averaged pre- and postsynaptic activations, promote convergence to stable weight configurations, while faster updates yield oscillatory co-dynamic systems. We further demonstrate that these findings generalize to high-dimensional quadrupedal locomotion with a simulated Unitree Go1 robot. These results highlight how the timing of plasticity shapes neural dynamics in embodied systems, providing a principled characterization of the attractor regimes that emerge in self-modifying networks.
Alexander Dittrich, Fuda van Diggelen, Dario Floreano
Laboratory of Intelligent Systems, EPFL, Switzerland
We introduce the Tan-HWG framework (Hebbian-Wasserstein-Geometry), a geometric theory of Hebbian plasticity in which memory states are modeled as probability measures evolving through Wasserstein minimizing movements. Hebbian learning rules are formalized as Hebbian energies satisfying a sequential stability condition, ensuring well-posed fiberwise JKO updates, optimal-transport realizations, and an energy descent inequality. This variational structure induces a fundamental separation between internal and observable dynamics. Internal memory states evolve along Wasserstein geodesics in a latent curved space, while observable quantities, such as effective synaptic weights, arise through geometric projection maps into external spaces. Simplicial projections recover classical affine schemes (including exponential moving averages and mirror descent), while revealing synaptic competition and pruning as geometric consequences of mass redistribution. Hilbertian projections provide a geometric account of phase alignment and multi-scale coherence. Classical neural networks appear as flat projections of this curved dynamics, while the framework naturally accommodates richer distributional representations, including structural weights and embedding memories, and their spectral extensions in complex internal spaces. Under mild Lipschitz regularity assumptions, including a quasi-stationary "sleep-mode" regime, we establish the existence of continuous-time limit curves. This yields a variational formulation of memory consolidation as a perturbed Wasserstein gradient flow. The framework thus provides a unified geometric foundation for synaptic plasticity, representation dynamics, and context-dependent computation.
Can lifetime learning expand behavioral diversity over evolutionary time, rather than collapsing it? Prior theory predicts that plasticity reduces variance by buffering organisms against environmental noise. We test this in a competitive domain: chess agents with eight NEAT-evolved neural modules, Hebbian within-game plasticity, and a desirability-domain signal chain with imagination. Across 10seeds per Hebbian condition, a variance crossover emerges: Hebbian ON starts with lower cross-seed variance than OFF, then surpasses it at generation34. The crossover trend is monotonic (\r{ho} = 0.91, p < 10^{-6): plasticity's effect on behavioral variance reverses over evolutionary time, initially compressing diversity (consistent with prior predictions) then expanding it as evolved Perception differences are amplified through imagination -- a feedback loop that mutation alone cannot sustain. The result is structured behavioral divergence: evolved agents select different moves on the same positions (62% disagreement), develop distinct opening repertoires, piece preferences, and game lengths. These are not different sampling policies -- they are reproducible behavioral signatures (ICC > 0.8) with interpretable signal chain configurations. Three regimes appear depending on opponent type: exploration (Hebbian ON, heterogeneous opponent), lottery (Hebbian OFF, elitism lock-in), and transparent (same-model opponent, brain self-erasure). The transparent regime generates a falsifiable prediction: self-play systems may systematically suppress behavioral diversity by eliminating the heterogeneity that personality requires. \textbf{Keywords: Baldwin Effect, neuroevolution, NEAT, Hebbian learning, chess, cognitive architecture, personality emergence, imagination