Neuron Co-Activation Graph

• Neuron co-activation graphs are mathematical models that depict synchronous neuronal activations as weighted edges between nodes, facilitating analysis of circuit modularity and functional dependencies.
• They are constructed by thresholding activation data using methods like Pearson correlation and graphical lasso to produce interpretable adjacency matrices.
• Their applications span deep learning interpretability, resource optimization, and biological network analysis, revealing network bottlenecks and dynamic structure.

A neuron co-activation graph is a mathematical and algorithmic framework for representing the functional relationships between neurons—artificial or biological—via the analysis of their activation patterns. In this object, each node corresponds to a neuron, and (typically weighted) edges quantify the strength or frequency with which pairs of neurons activate simultaneously or within specific relational contexts. The graph abstraction enables quantitative analysis using graph theory and network science, yielding critical insights into circuit modularity, bottlenecks, information flow, and functional dependencies in neural systems ranging from deep networks to neuronal populations in vivo and in vitro.

1. Formal Definitions and Mathematical Foundations

A neuron co-activation graph is formally defined as $G = (V, E, W)$, where $V$ is a set of neuronal nodes, $E \subseteq V \times V$ is a set of (possibly directed) edges, and $W : E \to \mathbb{R}_+$ assigns a nonnegative weight to each edge, reflecting the strength of co-activation between the corresponding neurons (Gross et al., 6 Jan 2025). For artificial neural networks, $G$ is most commonly undirected and weighted; for biological networks, directionality may reflect synaptic coupling (Brochini et al., 2016).

For a feed-forward network of $N$ neurons and a dataset $D = \{x_1, \dots, x_M\}$, one first chooses an activation threshold $\tau$ and evaluates, for each input $x \in D$ and each ordered neuron-pair $(i, j)$, the indicator: $$I_{ij}(x) = \begin{cases} 1 & \text{if } a_i(x) > \tau \text{ and } a_j(x) > \tau \ 0 & \text{otherwise} \end{cases}$$ where $a_i(x)$ is the post-nonlinearity activation of neuron $i$ on input $x$. The co-activation frequency $w_{ij}$ is then

$$w_{ij} = \frac{1}{|D|} \sum_{x \in D} I_{ij}(x) \in [0,1]$$

and the adjacency matrix $A$ is symmetric, $A_{ij} = w_{ij}$, with zeros along the diagonal (Gross et al., 6 Jan 2025).

Variants in neuroscience include defining $W_{ij}$ by Pearson correlation, dot-product, or conditional dependence via sparse inverse covariance estimation (graphical lasso) on multivariate time series (Beede et al., 8 Feb 2024, Nelson et al., 2020), or by more complex metrics derived from statistical models of spike trains or population activity (Santis et al., 2021, Chen et al., 2019, Chang et al., 2021). Directionality and sign can encode excitatory/inhibitory functional links.

2. Methodologies for Constructing Co-Activation Graphs

Construction methodologies depend on both the nature of the data (artificial vs. biological networks; time-series vs. activation matrix) and the scientific question:

• Artificial Networks (feed-forward/relu): Forward-pass all $x \in D$ through the network, binarize activations above threshold $\tau$, and count all unordered active neuron pairs to accumulate $c_{ij}$, then normalize by $|D|$ to obtain $w_{ij}$ (Gross et al., 6 Jan 2025). Optionally, prune low-weight edges by threshold $\epsilon$.
• Neuroscience (calcium imaging/spike trains):
  • Pearson correlation: Compute $r_{ij}$ for all neuron pairs and threshold or rank to produce a weighted/binary graph (Nelson et al., 2020).
  • Precision matrix (graphical lasso): Estimate the sparse inverse covariance matrix $\Theta$ of the time-series data $X \in \mathbb{R}^{n \times d}$ via:

  $$\hat \Theta = \arg\min_{\Theta \succ 0} -\log\det(\Theta) + \operatorname{tr}(S\Theta) + \lambda \|\Theta\|_{1, \text{off}}$$

  and set edges where $\hat\Theta_{ij} \ne 0$ (Beede et al., 8 Feb 2024). - Models for extremes: Fit penalized pseudo-likelihood for the Subbotin graphical model, associating edges with recovered nonzero $\theta_{ij}$ (Chang et al., 2021). - Spike train models: Assess influence via conditional probability changes, sensitivity of firing rates to past patterns, or pairwise algorithms that label directed edges as excitatory, inhibitory, or absent by thresholding empirical contrasts (Brochini et al., 2016, Santis et al., 2021).

A selection of common formulations is summarized below:

Domain	Edge Definition	Notable References
Deep RL/ANN	$\mathbb{P}(a_i(x)>\tau, a_j(x)>\tau)$	(Gross et al., 6 Jan 2025, Wang et al., 25 Oct 2024)
Calcium imaging	Pearson $r_{ij}$, Graphical Lasso $\Theta_{ij}$	(Beede et al., 8 Feb 2024, Nelson et al., 2020)
Spike trains	Directed: effect of $j$ on $i$ firing	(Brochini et al., 2016, Santis et al., 2021)

3. Applications for Interpretability, Safety, and Biological Networks

Neuron co-activation graphs yield interpretable summaries of network function across domains:

• Deep RL and Policy Analysis: Clusters of tightly co-activating neurons indicate modular computation; high-centrality ("hub") neurons suggest key information bottlenecks. Comparing co-activation graphs for "safe" vs. "unsafe" RL states isolates functional subcircuits responsible for unsafe behavior (Gross et al., 6 Jan 2025).
• System Optimization: For resource-constrained inference (e.g., LLMs on smartphones), co-activation graphs guide storage organization; neurons frequently co-activated are placed contiguously to maximize I/O efficiency (Wang et al., 25 Oct 2024).
• Neuronal Population Analysis: In calcium imaging or spike train data, co-activation graphs recover direct functional associations (conditional dependencies, directed causal links), reveal cell-type clusters, and expose structural motifs such as small-worldness or highly modular communities (Beede et al., 8 Feb 2024, Nelson et al., 2020, Santis et al., 2021).

Interpretability is further enhanced by leveraging graph-theoretic tools: community detection, centrality ranking, motif analysis, and cluster visualization. Spectral regularization enforces interpretable structure during ANN training by penalizing activation variability along the co-activation graph (Tong et al., 2018).

4. Algorithmic Workflows and Concrete Examples

A generic workflow to compute a co-activation graph from activations or time-series comprises the following steps (Gross et al., 6 Jan 2025, Beede et al., 8 Feb 2024, Nelson et al., 2020):

1. Data Acquisition: Collect activation matrices ($a_i(n)$ for ANNs or $X\in \mathbb{R}^{n\times d}$ for calcium signals) or spike trains.
2. Preprocessing: Filter, normalize, or detrend as appropriate.
3. Similarity Metric: Select and compute co-activation metric (e.g., joint thresholding, Pearson $r_{ij}$, graphical lasso).
4. Edge Construction: Form $w_{ij}$ (normalized if required), produce adjacency matrix $A$.
5. Thresholding/Sparsification: Remove weak/uninformative edges to yield a manageable, interpretable graph.
6. Visualization and Analysis: Apply graph algorithms for clustering, centrality, motif counting, or spectral embedding.

An explicit toy example: for three artificial neurons across three samples, with activations and $\tau=0.5$,

Input $x$	$a_1$	$a_2$	$a_3$	Active Neurons	Active Pairs
$x^1$	0.6	0.8	0.2	1, 2	(1,2)
$x^2$	0.7	0.1	0.9	1, 3	(1,3)
$x^3$	0.5	0.6	0.7	2, 3	(2,3)

Normalized weights: $w_{12}=w_{13}=w_{23}=1/3$, with adjacency matrix $A$ as in Section 1 (Gross et al., 6 Jan 2025).

5. Extensions: Directed, Signed, and High-Order Graphs

Several research streams address extensions beyond symmetric, undirected graphs:

• Directed and Signed Graphs: Algorithmic frameworks such as those in (Brochini et al., 2016, Santis et al., 2021) recover directed graphs with edge sign, quantifying excitatory (positive weight) or inhibitory (negative weight) relationships from spike-resolved data.
• Population Graphs with Learned Weights: Neural message passing models learn edge weights $K_{ij}$ reflecting context-dependent coupling, facilitating spatiotemporally-resolved reasoning in population spikes (Chen et al., 2019).
• High-Order Dependencies: Some frameworks embed not only pairwise but triplet or motif-level co-activations to better capture structure in highly recurrent or modular networks, often via tailored regularization or model-based approaches (Tong et al., 2018).
• Sparse, Decentralized Engram Formation: Biologically-inspired storage and memory models treat the weakly connected components of co-activation subgraphs as the carriers of distinct engrams, harnessing local competition and self-organization for robust, scalable storage in large networks (Wei et al., 2023).

6. Empirical Performance, Practical Considerations, and Limitations

Empirical analysis demonstrates the specificity and explanatory power of co-activation graphs:

• Edge Recovery and Biological Validity: In both ANN and biological models, strong within-class or within-type edges correspond to known functional modules or cell types; performance metrics (MAE, variance explained, motif frequencies) surpass baseline and correlation-based approaches (Chen et al., 2019, Tong et al., 2018, Beede et al., 8 Feb 2024).
• Scalability: With precomputed activation matrices, the graph construction is highly parallelizable; graphical lasso and spectral regularization scale to thousands of nodes (Beede et al., 8 Feb 2024).
• Selection of Thresholds and Regularization: Pruning parameters ($\epsilon, \lambda$) and regularization strengths ($\alpha$) directly affect graph sparsity, interpretability, and biological fidelity (Beede et al., 8 Feb 2024, Tong et al., 2018). Empirical or cross-validation-based tuning is generally needed.
• Limitations: Standard co-activation graphs capture correlation, not causality; static graphs may miss temporal and directional aspects unless extended via point-process or time-resolved models (Santis et al., 2021). Estimation under partial observability requires careful design to avoid false positives due to unrecorded neurons (Brochini et al., 2016).
• Bioviability, Fault Tolerance, and Decentralization: Models of autonomous node behavior and decentralized co-activation subgraphs confer high robustness and capacity under node/edge failure, in contrast to globally optimized but brittle structures (Wei et al., 2023).

7. Synergies Across Domains and Future Directions

Neuron co-activation graphs unify methodologies and insights across deep learning, computational neuroscience, and neuromorphic systems:

• Unification of Graph-Based Analysis: Whether in artificial policy networks, cellular imaging, or memory engram theory, the abstraction of neurons as nodes and functional co-activation as edges enables transfer of algorithms and metrics across fields (Gross et al., 6 Jan 2025, Nelson et al., 2020).
• Optimization Beyond Interpretation: As in the Ripple approach, co-activation graphs also serve as a tool for resource allocation, persistent storage, and efficient inference, demonstrating ongoing fusion of learning, inference, and systems considerations (Wang et al., 25 Oct 2024).
• Dynamic, Context-Specific Graphs: Future approaches incorporate time dependency, context shifts, and high-order interactions via advances in dynamic graph learning, causality inference, and message-passing frameworks (Chen et al., 2019).
• Integrated Tools and Standardization: The neurofuncon package exemplifies turnkey pipelines for inferring and visualizing large-scale co-activation graphs, highlighting the trajectory toward wider adoption in both experimental and computational contexts (Beede et al., 8 Feb 2024).

Neuron co-activation graphs are therefore a foundational construct bridging statistical analysis, interpretability, algorithmic optimization, and biological plausibility in the paper of neural information processing.