Deep learning works. But it does not work the way the brain works — there is no backpropagation in biological neural circuits, no global loss function, no gradient-based optimizer. My interest comes from being tired of pseudoscientific fluff about how cognition, language, and the brain work. Neuromorphic and dynamical systems drew me in because they offered a mechanistic way to approach analog reality. I was also introduced to complex networks by now retired Clarkson physics professor Daniel ben-Avraham (Google Scholar). Daniel even published with network science pioneer Albert Barabasi, whom I met at an MIT talk, and who encouraged me to pursue research in dynamical network theory. The theory models neural as a substrate for higher cognition — a base layer, not the whole story. The brain computes with assemblies: stable ensembles of neurons that form through Hebbian plasticity (neurons that fire together wire together), compete through local inhibition, and compose through a small set of operations. This project implements the Assembly Calculus (Papadimitriou et al., PNAS 2020) — a formal model of this computation — and extends it with Mitropolsky & Papadimitriou's NEMO model (2025) for biologically plausible language acquisition.
The question
How do assemblies of neurons represent word meanings? What synaptic dynamics enable syntactic composition — “cat” + “runs” becoming the structured thought the cat runs — without a parser, a grammar file, or gradient descent? Can Hebbian plasticity alone discover grammatical categories from exposure to sentences, the way a child does? These questions were previously intractable because nobody had the computational infrastructure to test them at brain scale. The Assembly Calculus provides the theory; this project provides the simulation platform.
Assembly Calculus in brief
An assembly is a stable set of k neurons in a brain area that emerges from repeated stimulation through Hebbian plasticity and winner-take-all competition. The calculus defines a small set of operations — projection, association, merge — that compose assemblies into representations of arbitrary complexity. Papadimitriou et al. (2020) formalized these operations and proved convergence guarantees — projection stabilizes in O(log N) rounds, assemblies remain distinct under competition. Dabagia et al. (2023) added sequences and long-range inhibition, achieving Turing completeness — meaning assembly circuits can simulate any computation a digital computer can, using only biologically plausible mechanisms.
The key constraint is biological plausibility: everything runs on Hebbian plasticity, sparse coding, local competition, and top-k winner-take-all. No backpropagation, no attention heads, no global optimizer. Assemblies are the proposed computational primitive bridging raw neural activity and high-level cognition — and the operations that compose them are the vocabulary of brain computation. For the full treatment of the operations, convergence properties, and Turing completeness, see the Assembly Calculus foundations page.
What we have found
The research is organized questions-first: each experiment tests a specific hypothesis about how assemblies compute. The findings that have emerged so far:
- N400 = global pre-k-WTA energy. In our simulation experiments (not yet published), the N400 event-related potential — the canonical 400ms electrophysiological signature of semantic processing — maps to the total synaptic input summed across all neurons before winner-take-all selection. When a prime is semantically related to the target, their assemblies share neurons; the overlapping inputs saturate rather than summing, producing lower total energy. Unrelated primes activate disjoint neurons, so inputs add without saturation — higher total energy, larger N400. The effect is massive in simulation (Cohen's d = −25, reflecting the deterministic nature of the model rather than noisy human EEG) and robust across parameter sweeps, vocabulary sizes, and both CPU and GPU engines. See the N400 finding for the full analysis.
- Competition is essential for distinctiveness. When two stimuli are trained in separate brains (independent random initialization, no shared competition), the resulting assemblies overlap by 94% — they are effectively indistinguishable. When trained in the same brain with winner-take-all , overlap drops to 3%. Competition is not a parameter choice; it is the mechanism that forces assemblies to mean different things.
- Assemblies converge in O(log N) rounds. Projection stabilizes with 100% convergence, mean 2.4 steps. Convergence time decreases as the network grows (R² = 0.989) — larger brains form assemblies faster, not slower.
- Hebbian plasticity does not cause forgetting. Unlike standard neural networks, where learning new tasks can overwrite previous knowledge, assembly-based learning achieves retrieval accuracy of 1.0 across sequential learning episodes. The sparse, competitive dynamics naturally partition the representational space.
- Grammar can be learned, not hardcoded. The NEMO model (2025) shows that the same assembly architecture acquires SVO word order and SOV order from grounded sentences alone — the same code, with only the training data changed, learns constituent order for typologically different languages. No language-specific rules, no grammar file, no optimizer. See language & NEMO.
Scale
The simulation platform can model up to 100 billion neurons at 4.2 trillion neurons per second, with 57,000:1 memory compression — brain-scale experiments on consumer hardware. This is possible because of hash-based implicit connectivity (25 MB instead of 40 GB for explicit weight matrices) and custom CUDA kernels for projection, Hebbian updates, and batched operations. The GPU-accelerated engine achieves 40–54× speedup over the CPU baseline. For the full architecture and performance benchmarks, see architecture & scaling.
Scale matters because the theoretical predictions of Assembly Calculus depend on network size: convergence time, assembly stability, and the capacity for distinct representations all change with the number of neurons. Testing these predictions requires simulation at biologically relevant scales — tens of thousands to millions of neurons per area, with realistic connectivity densities. The summer 2025 scaling push validated linear scaling from 1 million to 100 billion neurons with biologically realistic mechanisms running at 1ms biological timescale.
Why this matters
Assembly Calculus is not trying to beat transformers at text generation. It is asking a different question: how does the brain actually compute? The answer matters for neuroscience (understanding the computational role of neural ensembles), for AI (building systems that learn from less data and compute more efficiently), and for the long-term trajectory of machine intelligence (whether there are computational principles we are missing by building exclusively on gradient descent).
There is a structural parallel with the AI civilizations research: both study how populations of simple units produce structured collective behavior. In multi-agent systems, the units are LLM agents coordinating through explicit communication; in assembly networks, the units are neurons coordinating through synaptic competition. The mechanisms are different, but the questions overlap — how does specialization emerge, how are conflicts resolved, what makes representations stable? The parallel is suggestive, not formal. Whether a deeper mathematical connection exists is an open question.
Where things stand
Core operations — projection, association, merge — are implemented and validated. Sequence memorization, ordered recall, and long-range inhibition from Dabagia et al. (2023) are implemented; FSM learning is the remaining piece from that paper. NEMO can learn word semantics, noun/verb classification, and constituent order from grounded sentences. GPU scaling and CUDA acceleration are production-ready. The N400 finding (from our simulation experiments, not yet published) is validated across multiple experimental configurations.
What remains open: cross-linguistic validation beyond SVO and SOV orders (the 2025 paper notes that OSV and OVS orders require significantly more training data, matching their rarity in natural languages), scaling the lexicon beyond the current experimental sizes, the morphological agreement framework, and the long-term goal of multimodal grounding — vision, audio, proprioception bound through the same assembly operations that handle language.
Foundational papers
- Papadimitriou et al. (2020) — “Brain Computation by Assemblies of Neurons.” PNAS 117(25):14464–14472. The Assembly Calculus: projection, association, merge; proof of universal computation.
- Dabagia et al. (2023) — “Computation with Sequences of Assemblies in a Model of the Brain.” ALT 2024 / Neural Computation. Sequences, FSM simulation, Turing completeness through long-range inhibition.
- Mitropolsky & Papadimitriou (2023) — “The Architecture of a Biologically Plausible Language Organ.” NEMO architecture: the language organ design.
- Mitropolsky & Papadimitriou (2025) — “Simulated Language Acquisition in a Biologically Realistic Model of the Brain.” First biologically plausible system for grammar acquisition from grounded sentences.
Deep dives
- Assembly Calculus foundations — The operations, convergence properties, Turing completeness, and why competition is the mechanism that makes assemblies mean different things.
- Language & NEMO — How the brain might learn grammar: the NEMO model, N400/P600 findings, ERP triple dissociation, cross-linguistic experiments, and developmental curriculum.
- Architecture & scaling — Compute engines, GPU acceleration, CUDA kernels, and the path from 10,000 neurons to 100 billion.