Columnar-Embedder: A Biologically Inspired Cortical Architecture for Binary Sparse Distributed Graph Representations

Mohamed Abidalrekab; Dan Hammerstrom

cs.NEAug 12, 2026

Columnar-Embedder: A Biologically Inspired Cortical Architecture for Binary Sparse Distributed Graph Representations

Authors: Mohamed Abidalrekab, Dan Hammerstrom

Organizations: Portland State University, Electrical and Computer Engineering, Portland, 97223, USA

Abstract

Finding a representative description of graph entities that captures their structural roles and homophily is a challenging goal for graph embedding techniques due to the non-Euclidean nature of graphs. Traditionally, Graph embeddings achieve top performance via random-walk methods and graph neural networks. However, these methods are transductive and utilize an expensive global optimization via softmax or a dense representation trained in an end-to-end pipeline with gradient descent. Nonetheless, other variants of GNNs can map to unseen nodes; they still rely on iterative message passing and backpropagation, incurring high computational and memory costs. Conversely, the mammalian cortex solves structurally similar problems by learning to map its input stream of patterns into a compact representation for downstream regions. We present the biologically inspired Columnar-Embedder architecture for learning binary Sparse Distributed Representations (SDRs) of graph nodes. The learning is driven by a local Bienenstock-Cooper-Munro (BCM) Hebbian rule modulated by positive pointwise mutual information (PPMI) computed from online streams of random walks. Continuous learning from streaming random-walk pairs without labels, backpropagation, or supervision enables the architecture to exhibit natural resistance to catastrophic forgetting. Across five graph benchmarks, the performance of SDRs is competitive with that of real-valued dense embeddings on node classification and link prediction, while the architecture exhibits portability, resilience to noise, and robustness to data corruption.

Figures & tables

Dataset	N	Edges	Classes	Mean Degree	Source
Cora	2,485	5,069	7	4.08	Citation network [ 82 ]
Citeseer	2,120	3,717	6	3.51	Citation network [ 82 ]
Amazon-Photo	7,487	119,081	8	31.8	Co-produce network [ 83 ]
PubMed	19,717	88,648	3	4.5	Citation network [ 82 ]
Co-Author Physics	34,493	495,924	5	14.38	Coauthor Physics networks [ 83 ]

Table 1: Graph Datasets Benchmark

Figure 1: Node Classification accuracy on three datasets: Cora, Citeseer, and Amazon-Photo. †GCN on Amazon-Photo is high-variance (0.902 ± 0.037). ‡GraphSAGE on Amazon-Photo is bistable; a converged run (0.924) is the best run.

Figure 2: Link Prediction across the datasets ( Leakage-free, 15% held out)

Figure 3: Operator Sweep of link Prediction task on datasets (leakage-free and 15% held out)

Figure 4: 28-bits SDR fidelity metrics over Cora, Citeseer, and Amazon-Photo. (a) SDR correlation with n2v, walks statistics, and modulator signal. (b) shows class discriminability of SDR on benchmark datasets.

Figure 5: 28-bits SDR overlap vs graph distance (hops), indicating strong geometric correlations between graph nodes’ proximity and shared bits in the SDRs.

Figure 6: Robustness to corruption ( bit flipping fraction $\delta$ ) vs n2v with d =64

Figure 7: The class separability under corruption. Same-class AUC in each representation’s native metric (SDR overlap, n2v cosine similarity)

Figure 8: SDR overlap vs n2v geometry. The Similarity matrix panel shows that it sampled roughly $600$ nodes reordered by class. The hexbin R values are computed on the full 3-million-pair set.

Figure 9: The Columnar-Embedder architecture scaling to larger graphs with no architectural or parameter changes. The two graphs of PubMed, and CoAuthor-Physics are 7.9×, and 13.9× the size of Cora.

Figure 10: The proposed feedforward Columnar-Embedder Architecture of three main layers: an input encoding layer, cortical columnars layer, and Readout/Olfactory layer.

Rule	Update	$\eta$	operates on	Role
BCM (PPMI-modulated)	$\Delta W\propto\eta_{BCM}\cdot L_{4}^{\intercal}\cdot r_{post}\cdot(r_{post}-\theta_{M})\cdot m_{PPMI}(v,u)$	0.010	$W(800\times 1500)$	Class-coherent column selection (Accuracy)
Anti-CE	$\Delta W=\eta_{ACE}\cdot L_{4}^{\intercal}\cdot(S_{5a}^{soft}-L_{2/3})\cdot col_{mask}$	0.002	Active columns of W(:,i)	Within-columns neuron diversification (uniqueness at $L_{2/3}$ )
Anti-Hebbian (lateral inhibition)	$\Delta L_{col}(j,i)\propto-\eta\cdot col_{active}(i)\cdot col_{active}(j)$	0.005	$L_{col}\in\mathbb{R}^{(300\times 300)}$	Sparse column selection ( $M_{active}=8$ winners)
Intrinsic Plasticity (IP)	$\Delta V_{th}(c)\propto\eta_{IP}\cdot(\bar{r}_{rate}(c)-r_{target})$	0.05	Per $L_{2/3}$ neuron	Homeostatic threshold stability
Outstar (Grossberg rule)	$\Delta W_{r}(j,i)=\eta_{OF}\cdot L_{5a}(j)\cdot(L_{2/3}(i)-W_{r}(j,i))$	0.01	$W_{r}\in\mathbb{R}^{(300\times 1500)}$	$L_{2/3}$ -> $L_{5a}$ Olfactory readout.

Table 2: Columnar-Embedder learning Rules

Figure 11: SDRs disambiguation process of two identical clusters representation of nodes at L2/3.

Graph Embedding

Graph embedding maps nodes, edges, or whole graphs into vectors that capture structure for prediction, clustering, retrieval, and analysis.

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Graph Representation Learning Graph Embedding Topological Representation Learning

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Graph Representation Learning Graph Embedding Spectral Graph Embedding Kernel Methods Random Feature Methods

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Graph Embedding

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