DYNAMIC PLASTIC CONTINUOUS NEURAL NETWORKS

Learning lives in the connections—and the medium

Explore 200 finite neuro-elements: one neuro cell is connected to each cell of a 20 × 10 finite-element mesh. The base medium is a single continuous layer of 3D elements. The raised neural markers and their individual connections show local displacement sensing and force feedback.

Dynamic plastic continuous neural network trainingA single layer of 200 connected solid-looking finite element cells is viewed obliquely. Every finite element has exactly one raised neuro cell and a local feedback connection. All 200 neuro cells are active. Select a cell to inspect its four corner displacement inputs and force feedback. Material scales at elements 100 and 120 change in the final stage. All fields are illustrative.01 / INPUT → WAVE PROPAGATION → NEURAL FEEDBACK20 × 10 × 1 element layer · 200 FE cells ↔ 200 neuro cells · illustrative extensionDistributed harmonic inputONE layer · shared faces · no gapsO(t)Output reactionat node 126MATERIAL REFINEMENTσελ₁₀₀ = 1.00λ₁₂₀ = 1.00σnew(ε) = λ σbase(ε)Bilinear backbone · schematic02 / OUTPUT APPROACHES A TARGETIllustrative normalized signals · not digitized paper data– – Target— Network outputTraining progression is illustrative.
● Teal: wave / displacement● Violet: 200 local neuro cells● Gold: input and returned forcesGold / violet cells: material variables 100 / 120

INSIDE THE SELECTED NEURON

z = Σ wᵢuᵢ
f(z) = 5 × 10⁷ [2 / (1 + e⁻ᵝᶻ) − 1]
Returned nodal force: Fⱼ = wout,j f(z)

The sigmoid form follows the paper (β = 500). This 200-cell extension connects each neuro cell to the four corner nodes of its corresponding FE cell, using normalized signals and illustrative weights. It is an explanatory mapping, not the paper’s original three-neuron layout.

ORIGINAL PAPER · THREE-NEURON STUDY

1.19 × 10¹³Error after one-neuron training
4.57 × 10¹¹Error after three-neuron training
2.20 × 10¹¹Error after material refinement
≈52% lowerAdditional error reduction from material refinement

Error E = (0.5 / n) Σ[T(tᵢ) − O(tᵢ)]², following the paper. These error values belong to the paper’s original one-to-three-neuron numerical example. They are not results of this 200-cell extension.

HOW THE NETWORK LEARNS

1. Perturb a connection weight and repeat the dynamic analysis.
2. Estimate error sensitivity sᵢ = ΔE / Δxᵢ.
3. Apply a bounded sensitivity-based update.
4. In this extension, every cell has its own local neuron from the start.
5. Illustrate material refinement of λ₁₀₀ and λ₁₂₀ near the output.

xᵢ ← xᵢ − Δmax [2/(1 + e⁻ξsᵢ) − 1]

WHAT MAKES THE MEDIUM USEFUL?

Wave propagation provides spatial coupling and a time delay. Plastic deformation supplies history dependence. Adjusting the base material adds learning variables alongside the synaptic weights.

The paper uses an undamped nonlinear dynamic finite-element model with von Mises plasticity and isotropic hardening. It studies a constructed input–target problem to demonstrate the concept.

Conceptual 200-neuro-cell extension requested for this demonstration. The paper grows a one-to-three-neuron network; this view instead assigns one neuro cell to every FE cell. The medium contains exactly one layer of connected 3D-looking elements; raised dots are neural markers, not a second FE layer. This is not a reproduction of the finite-element solver or the training algorithm. Wave fields, output curves, intermediate learning states, and the smooth λ progression are illustrative. The reported final scales (λ₁₀₀ = 1.5, λ₁₂₀ = 0.5) and error milestones come from the paper. No biological-brain model or real-time learning performance is claimed.

Joghataie & Shafiei Dizaji (2016), Neuroplasticity in dynamic neural networks comprised of neurons attached to adaptive base plate, Neural Networks 75, 77–83.