PAULA: A Computational Substrate for Self-Organizing Biologically-Plausible AI

1. Introduction: Why a New Approach?

In traditional artificial neural networks, a neuron is a static mathematical function: it takes inputs, applies weights, sums them up, and passes the result through an activation function. A biological neuron, however, is something far more complex: a dynamic, adaptive agent that learns from purely local information and continuously adjusts to its environment (Moore et al., 2024; Liu et al., 2025; Patel et al., 2025). This biological premise forms the foundation of the ALERM framework (Architecture, Learning, Energy, Recall, and Memory), for which PAULA serves as the primary empirical validation.

This divergence suggests that current AI is solving the wrong problem (Chen et al., 2025). Rather than merely approximating static functions (input/output mappings), biological intelligence is fundamentally about constructing and maintaining dynamical states that mirror environmental structure.

"Intelligence is not programmed; it emerges through continuous interaction with an environment. Any persistent system must model its environment to survive."

- Derived from Karl Friston's Free Energy Principle

Biological neural networks operate under fundamentally different principles:

• Local Learning: Neurons learn through local synaptic plasticity without global error signals
• Homeostatic Regulation: They maintain stability through self-regulation (Niemeyer et al., 2022; Shen et al., 2024)
• Metaplasticity: They adapt their learning rates based on activity history (Kuśmierz et al., 2017)

A fundamental challenge in temporal pattern learning is the temporal ambiguity problem: different temporal sequences can appear identical when compressed into static representations. Consider handwritten digit recognition: digit 8 (top loop → connector → bottom loop) and digit 0 (single loop) can look similar in static images, yet their temporal production differs substantially. Standard neural networks struggle with such distinctions because they lack explicit temporal representation.

We frame this as a dimensionality expansion problem: to disentangle temporally ambiguous patterns, a system must project low-dimensional temporal signals into high-dimensional representations where patterns become linearly separable. This is analogous to the kernel trick in classical machine learning, where input data is implicitly mapped to higher dimensions to enable linear classification. However, in the temporal domain, this expansion must be self-organizing: adapting to signal statistics without explicit design.

This paper makes three contributions:

1. A self-organizing temporal dimensionality expansion mechanism that amplifies temporal variance through multiplicative weight dynamics and projects signals into multi-scale representations through adaptive metaplasticity.
2. Empirical validation of mechanism necessity showing that complex temporal patterns collapse by 25% when multi-scale integration is prevented, while simple patterns remain unaffected, proving that expansion adapts to signal complexity.
3. Integration with recent advances in homeostatic plasticity and three-factor learning, positioning the work within the emerging convergence on energy-efficient predictive coding in spiking neural networks.

We developed PAULA to realize these contributions through a novel formal model for a network of dynamic, self-regulating computational units that learn continuously based on principles of local prediction (N'dri et al., 2025). This model synthesizes a wide range of neuroscientific discoveries into a single, coherent mathematical framework.

2. The PAULA Model

At its core, PAULA defines a "smart neuron": a biologically-inspired computational unit that goes beyond traditional artificial neural network primitives. In this section, we present a comprehensive specification of the model's architecture.

2.1 Graph Architecture

We formally define each neuron as a weighted, directed graph \(G = (V, E, d)\), where:

• Node Sets: The node set \(V\) is the union of three distinct processing sets:
  • \(V_{P'_{in}}\): The set of all postsynaptic processing nodes
  • \(V_{P'_{out}}\): The set of all presynaptic processing nodes
  • \(\{H\}\): The axon hillock node
• Complete Node Set: \(V = V_{P'_{in}} \cup \{H\} \cup V_{P'_{out}}\)
• Edges: \(E \subseteq V \times V\) represents the directed connections between nodes
• Weights: A distance function \(d: E \to \mathbb{N}\) that assigns a travel time (in ticks) to each edge

2.2 Fundamental Units

• Input Unit (\(i\)): A postsynaptic receptor. The \(I_{adapt}\) set is a family of abstract functional subtypes:
  • \(I_{adapt, Excitability}\)
  • \(I_{adapt, Metaplasticity}\)
• Output Unit (\(o\)): A presynaptic neurotransmitter release. The \(O_{mod}\) set includes:
  • \(O_{mod, Excitability}\)
  • \(O_{mod, Metaplasticity}\)

2.3 Integrated Interaction Points (\(P′\))

Postsynaptic Point (\(P'_{in}\)): Its state is defined by a vector of receptor efficacies. Each synapse maintains multiple distinct strengths: one for information flow and others for adapting to excitability and metaplasticity signals:

\[\vec{u_i} = (u_{info}, u_{adapt, Excitability}, u_{adapt, Metaplasticity}, ...)\]

Presynaptic Point (\(P'_{out}\)): Its output vector includes corresponding neuromodulator types. When a neuron fires, it releases signals that carry information and also modulate the behavior of the receiving neurons:

\[\vec{u_o} = (u_{info}, u_{mod, Excitability}, u_{mod, Metaplasticity}, ...)\]

2.4 System State Variables and Parameters

Tick-Dependent Variables:

• \(S(t)\): Internal potential at the axon hillock \(H\)
• \(O(t)\): Output signal function from the hillock \(H\)
• \(t_{last\_fire}\): Timestamp of the last somatic firing event
• \(\bar{F}(t)\): The long-term average firing rate
• \(\vec{M}(t)\): The neuromodulatory state vector (\(\vec{M}(t) = (M_{Excitability}(t), M_{Metaplasticity}(t), ...)\))
• \(r(t), b(t)\): The dynamic primary and post-cooldown thresholds
• \(t_{ref}(t)\): The plastic causal time window for learning

Constant Parameters:

• \(\delta_{decay}\): Per-tick potential decay factor
• \(\beta_{avg}\): Decay factor for the average firing rate EMA
• \(\eta_{post}\): The postsynaptic learning rate
• \(\eta_{retro}\): The presynaptic learning rate for retrograde adaptation
• \(c\): The activation cooldown period in ticks
• \(\lambda\): The membrane time constant for the axon hillock
• \(p\): The constant magnitude of every output signal
• \(r_{base}, b_{base}\): Baseline values for the dynamic thresholds
• \(\vec{\gamma}\): Vector of decay factors for each component of \(\vec{M}(t)\)
• \(\vec{w}_r, \vec{w}_b, \vec{w}_{tref}\): Sensitivity vectors for dynamic parameters

2.5 State Transitions (The "Tick" and Graph-Based Dynamics)

The system evolves over discrete time steps \(dt\).

A. Neuromodulation and Dynamic Parameter Update:

1. Aggregate Neuromodulatory Input: For each modulator type \(k\), calculate the total weighted input signal. This sums up all the external "control signals" (like excitability and metaplasticity) arriving at each synapse, weighted by how sensitive that synapse is to each type:

   \[\text{total\_adapt\_signal}_{k}(t) = \sum_{v \in V_{P'_{in}}} \left( O_{ext, mod, k}^{(v)}(t) \cdot u_{i, adapt, k}^{(v)}(t) \right)\]

2. Update Neuromodulatory State Vector: The internal "state of mind" of the neuron updates as an exponential moving average; it slowly forgets past modulation while incorporating new signals:

   \[M_k(t+dt) = \gamma_k \cdot M_k(t) + (1-\gamma_k) \cdot \text{total\_adapt\_signal}_k(t)\]

3. Update Average Firing Rate: The neuron tracks how often it fires over time. This running average is key for homeostasis: it lets the neuron notice if it is too active or too quiet:

   \[\bar{F}(t+dt) = \beta_{avg} \cdot \bar{F}(t) + (1-\beta_{avg}) \cdot O(t)\]

4. Calculate Final Dynamic Parameters: These equations set the firing thresholds of the neuron based on its neuromodulatory state. The thresholds \(r\) and \(b\) shift up or down depending on the accumulated modulation:

   \[\begin{align*} r(t+dt) &= r_{base} + \vec{w}_r \cdot \vec{M}(t+dt) \\ b(t+dt) &= b_{base} + \vec{w}_b \cdot \vec{M}(t+dt) \end{align*}\]

   Adaptive Refractory Period (Multi-Scale Projection): The plasticity window \(t_{ref}\) creates heterogeneous temporal integration windows that enable multi-scale projection:

   \[\begin{align*} t_{ref\_homeostatic}(t) &= \text{upper\_bound} - (\text{upper\_bound} - \text{lower\_bound}) \cdot (\bar{F}(t) \cdot c) \\ t_{ref}(t+dt) &= t_{ref\_homeostatic}(t) + \vec{w}_{tref} \cdot \vec{M}(t+dt) \end{align*}\]

   Mechanism: High-firing neurons (large weights, high variance) → short \(t_{ref}\) (~90 ticks). Low-firing neurons (small weights, low variance) → long \(t_{ref}\) (~150 ticks).

   Function: Different neurons become selective to different temporal scales. Short windows capture fast transitions (e.g., connector strokes in digit 8), while long windows maintain global context (e.g., full loop structure). Together, they project input into high-dimensional multi-scale representations. This provides the projection mechanism itself.

B. Input Processing & Propagation:

When a signal arrives at a synapse, it gets multiplied by the strength of that synapse: this is how learning shapes perception:

1. When an external signal vector \(\vec{O}_{ext}\) arrives at a node \(v \in V_{P'_{in}}\), the local voltage is computed as the signal times the synaptic weight:

   \[V_{local}(t) = O_{ext, info} \cdot u_{info}^{(v)}(t)\]

2. This \(V_{local}\) potential is scheduled for arrival at the axon hillock \(H\) at \(t_{arrival} = t + d(v, H)\). The delay models realistic signal propagation time along the dendrite.

C. Signal Integration at Axon Hillock:

All incoming signals are summed at the cell body, but each is weakened by how far it had to travel (longer paths = more decay):

\[I(t) = \sum V_{local}(t) \cdot (\delta_{decay})^{d(v, H)}\]

D. Somatic Firing (Model H):

The core spiking dynamics. The membrane potential rises toward input and decays toward rest, like a leaky bucket:

1. State Evolution: The membrane potential \(S\) evolves as a leaky integrator; it accumulates input while constantly draining toward zero:

   \[S(t+dt) = S(t) + \frac{dt}{\lambda} (-S(t) + I(t))\]

2. Firing Condition: A spike occurs if \(S(t+dt) \ge T_{active}(t)\) and \(t+dt \ge t_{last\_fire} + c\). The neuron must exceed threshold AND wait out its refractory cooldown.
3. Instantaneous Firing Dynamics: If a spike occurs at \(t_{fire}\), apply Model H reset rules (membrane resets, threshold jumps up to prevent immediate re-firing).
4. Threshold Reset: If \(S(t)\) decays to near-zero, \(T_{active}(t)\) resets to \(r(t)\). This prepares the neuron for the next cycle of input.

E. Output and Learning:

1. Presynaptic Release: A somatic spike propagates to all \(v \in V_{P'_{out}}\), causing release of \(\vec{u_o}^{(v)}\)
2. Dendritic Computation & Plasticity: At a synapse \(v \in V_{P'_{in}}\) receiving \(\vec{O}_{ext}\):
   1. Step 1: Directional Error Calculation: The mismatch between what the synapse expected (its weight) and what actually arrived. This local error drives learning:

      \[\vec{E}_{dir}(t) = \vec{O}_{ext}(t) - \vec{u_i}^{(v)}(t)\]

   2. Step 2: Temporal Correlation (Plasticity Gating) : The key STDP-like mechanism. If the input arrived within the plasticity window after the neuron fired, strengthen the connection ("neurons that fire together, wire together"). If too late, weaken it:
      • Calculate \(\Delta t = t - t_{last\_fire}\) (time since last spike)
      • If \(\Delta t \le t_{ref}(t)\), set direction = +1 (LTP: long-term potentiation)
      • If \(\Delta t > t_{ref}(t)\), set direction = -1 (LTD: long-term depression)
   3. Step 3: Postsynaptic Update: The actual learning: adjust the synaptic weight proportionally to the error magnitude, the current weight (multiplicative update), and the temporal direction (LTP vs LTD):

      \[\Delta u_{info}^{(v)}(t+dt) = \eta_{post} \cdot \text{direction} \cdot ||{E}_{dir}(t)|| \cdot u_{info}^{(v)}(t)\]

      Variance Amplification: This multiplicative form creates intentional variance amplification. Large weights receive larger updates, causing neurons to diverge in their firing rates. High-firing neurons (processing high-variance temporal patterns) develop distinct properties from low-firing neurons, creating the heterogeneity necessary for multi-scale processing.

   4. Step 4: Retrograde Signal and Presynaptic Update : The postsynaptic neuron sends a directed error signal back to the presynaptic neuron, allowing bilateral learning without global supervision. The upstream neuron adjusts its output accordingly:

      \[\begin{align*} \vec{O}_{retro}(t) &= \text{direction} \cdot \vec{E}_{dir}(t) \\ \Delta \vec{u_o}(t+dt) &= \eta_{retro} \cdot \vec{O}_{retro}(t) \end{align*}\]

3. Experimental Methodology

Testing a new neuron model like PAULA required a carefully designed experimental setup. We could not simply train it like a regular neural network. We needed to create an entire research pipeline that treats the PAULA network as a living system we can observe, measure, and understand.

Our approach is analogous to studying a new organism in the wild: rather than placing it in a cage to see if it survives, we set up careful observations, controlled experiments, and multiple ways to measure its behavior.

3.1 Network Construction

We built deep perceptron-style networks made entirely of PAULA neurons. These are fully-connected layers where each neuron connects to every neuron in the next layer, but we systematically varied the connection density to see how sparsity affects learning and performance.

Each network was designed as a directed graph: neurons as nodes, synapses as edges with propagation delays. The interactive builder let us experiment with different layer sizes, depths, and connection patterns to find the sweet spot for biological plausibility and computational power.

Code: Network Builder

Interactive tool for designing PAULA networks with real-time visualization

code View Source

3.2 Activity Recording

We fed MNIST images to the networks one by one, recording everything that happened inside the network during processing, not just the final answer.

For each image, we captured: membrane potentials (how excited each neuron was over time), plasticity windows (the dynamic learning sensitivity), neuromodulatory signals (internal control mechanisms), and spike timing (exact moments when neurons fired). This gave us a complete temporal fingerprint of how the network "thought" about each digit.

Code: Activity Recorder

Simulates network response to image sequences and logs all internal dynamics

code View Source

3.3 Preparing Data for Analysis

Raw activity logs are messy, containing gigabytes of temporal data per network. We needed to extract meaningful features that a classifier could learn from. We focused on the key dynamical signatures: average membrane potentials over time, firing rates, and temporal patterns in the plasticity parameters.

This transformed the network's continuous internal dynamics into structured datasets that could be fed into standard machine learning pipelines for analysis and evaluation.

Code: Data Processor

Converts raw activity logs into PyTorch datasets with temporal feature extraction

code View Source

3.4 Training the Pattern Decoder

To measure what the PAULA network actually learned, we trained an external spiking neural network classifier to decode the activity patterns. This serves as an "EEG" device that reads brain activity and translates it into recognizable categories.

The classifier used a simple but effective architecture: input layer for the extracted features, a couple of hidden layers with leaky integrate-and-fire neurons, and an output layer that predicts digit classes. We trained it using standard techniques: cross-entropy loss, Adam optimizer, and a learning rate of 0.0005. The extracted features were genuine signatures of neural computation that happened inside PAULA units.

Code: Pattern Decoder

Trains SNN classifier to decode PAULA network activity patterns into digit predictions

code View Source

3.5 Real-Time Evaluation

Final evaluation went beyond simple accuracy metrics. We ran the networks in real-time, watching how their "thinking" evolved over time. Some networks made quick decisions, others took longer to deliberate, revealing cognitive dynamics that static classifiers would miss.

The "thinking mode" experiments were particularly revealing: by giving networks extra processing time, we could see which digits required fast recognition versus deep deliberation, mirroring how humans process familiar versus complex stimuli.

Code: Live Evaluator

Real-time performance testing with temporal analysis and extended processing capabilities

code View Source

3.6 Evaluation Protocol

To ensure comprehensive and reproducible results, we established a systematic evaluation protocol across all experiments. Every network architecture was evaluated under five distinct "thinking mode" configurations, where the network is given varying amounts of processing time (measured in simulation ticks) to integrate evidence:

• No thinking mode: Standard processing with base time allocation
• Thinking ×2: Extended processing time at 2× base duration
• Thinking ×3: Extended processing time at 3× base duration
• Thinking ×5: Extended processing time at 5× base duration
• Thinking ×7: Extended processing time at 7× base duration

This evaluation matrix was applied consistently across three distinct network architectures (single neuron, sparse networks, and dense networks), yielding 15 unique experimental conditions. All results reported in this paper are drawn from this comprehensive evaluation set, ensuring that observed phenomena are robust across different temporal integration scales and architectural configurations.

3.7 Structural Analysis

A significant insight came from analyzing what emerged inside the networks during learning. We used clustering algorithms to find functional groups of neurons that specialized in different aspects of the task, dynamical systems analysis to map the network's behavioral landscape, and performance analysis to understand scaling relationships.

This multi-modal approach revealed that PAULA networks self-organize into sophisticated computational architectures that mirror biological neural systems.

Analysis Tools

Scripts for analyzing emergent network properties and visualization

code Cell Assembly Analysis code Attractor Landscapes code Network Dynamics

4. Key Results

4.1 Single Neuron Computational Power

Our first experiments aimed to establish the computational capacity of a single PAULA neuron. The relevant control is not random chance (10%); it is what a single neuron with frozen weights (our ablation condition) achieves. A static single-unit produces near-chance discrimination, because it cannot adapt its temporal window to the input. In contrast, a single adaptive neuron with homeostatic metaplasticity achieves remarkable performance on MNIST, demonstrating that the adaptive dynamics (not the parameter count) are doing the work. This proves that single neurons represent powerful computational units capable of solving complex tasks (Jones & Kording, 2020).

Analysis of the neuron's temporal dynamics reveals that it encodes digit identity through characteristic membrane potential trajectories rather than static response magnitudes. Different digits produce different temporal patterns in the integrated response, with the external decoder learning to read these trajectories. This demonstrates that computational richness emerges from temporal dynamics and adaptive mechanisms, not from parameter counts, validating the hypothesis that neurons are dynamical systems rather than static functions.

Single Neuron Performance with Extended Processing Time

Condition	Top-1 Accuracy	Top-3 Accuracy
No thinking	24.1%	53.6%
Thinking ×2	31.3%	51.8%
Thinking ×3	35.6%	55.5%
Thinking ×5	37.5%	53.5%
Thinking ×7	38.1%	53.5%

Beyond classification accuracy, the single neuron demonstrates remarkable dimensionality reduction: compressing 784-dimensional input into a 2-dimensional dynamical state (⟨S⟩, ⟨t_ref⟩) while preserving ~40% of class-discriminative information.

4.2 Systemic Properties & Architectural Principles

Next, we tested the behavior of multi-neuron networks, uncovering key design principles that emerge from PAULA's local learning dynamics.

Detailed Performance Metrics

Sparsity	Best Top-1	Best Top-3	Best Time	Avg Ticks to Correct	Prediction Stability
25%	86.80%	94.20%	79	31.02	96.88%
50%	81.80%	94.30%	95	29.22	95.01%
100%	78.60%	92.40%	140	36.18	92.47%

Network sparsity critically influences homeostatic dynamics and performance. The 25% sparse network achieves 86.8% accuracy, outperforming both 50% sparse (80.9%) and 100% dense (78.6%) configurations. Analysis of homeostatic parameters reveals the mechanism: dense networks show Layer 1 saturation, with neurons clustered at the \(t_{ref}\) lower bound (88-90 ticks), while sparse networks maintain \(t_{ref}\) diversity (88-96 ticks). This pattern indicates that full connectivity creates excessive input drive, pushing all neurons toward maximal selectivity and preventing the heterogeneity necessary for specialization. Sparse connectivity, by contrast, allows neurons to operate across the full homeostatic range, enabling differentiated responses and emergent cell assemblies.

4.3 "Thinking Mode" and Temporal Integration

To further characterize the network's cognitive dynamics, we applied extended processing time, what we call "thinking mode", to the best-performing sparse network:

The model is not a static classifier. It is a dynamic decision-making engine that demonstrably integrates evidence over time, mirroring cognitive models like the Drift-Diffusion Model.

4.4 Emergent Cell Assemblies

Our clustering analysis, analogous to fMRI in neuroscience, revealed that PAULA networks spontaneously self-organize into 4-6 distinct functional "cell assemblies". These are groups of neurons that encode similar features:

This emergent organization mirrors biological cortical columns, where neurons spanning multiple layers form functional units based on shared response properties. Critically, PAULA produces this organization without explicit architectural constraints or supervised grouping. It arises purely from homeostatic regulation interacting with task statistics.

The performance of the "EEG", referring to accuracy metrics, was shown to be a direct, predictable consequence of the "fMRI", which refers to the underlying cell assembly structure. For example, the model's difficulty with digit '7' was directly explained by the fact that no specialized cell assembly for it had emerged.

4.5 Attractor Landscapes

We analyzed network state dynamics in (⟨S⟩, ⟨t_ref⟩) space, revealing digit-specific topographies. The table below shows the full comparison across sparsity levels:

Figure 5: Click any image to enlarge. Attractor landscapes in (⟨S⟩, ⟨t_ref⟩) state space for different digit classes across sparsity levels.

Each panel shows the energy landscape (KDE-smoothed density) for a different digit class. Dark blue regions indicate the network's preferred states (low energy basins); yellow regions represent rarely-visited configurations. Digit 1 exhibits a compact, centered attractor, while Digit 0 shows extended horizontal structure with multiple striations.

These distinct landscape geometries support the processual representation hypothesis: digits are encoded as characteristic dynamical regimes rather than static weight patterns. The network does not store 'digit 0' as a weight pattern but rather as the capacity to enter a specific oscillatory state.

5. Ablation Study: Validating Temporal Dimensionality Expansion

5.1 Rationale

To test whether the coupled multiplicative-homeostatic dynamics are necessary for learning complex temporal patterns, we systematically ablated individual mechanisms. If temporal dimensionality expansion is indeed the core innovation, removing multi-scale integration should cause collapse on complex patterns while sparing simple patterns.

5.2 Methods

We trained seven network configurations on MNIST (1000 samples/digit):

1. Baseline: Full PAULA (all mechanisms active)
2. Frozen t_ref: Adaptive refractory disabled (fixed at initial value)
3. Frozen weights: Synaptic plasticity disabled (random initialization frozen)
4. Frozen thresholds: Excitability adaptation disabled
5. No retrograde: Presynaptic learning disabled
6. No STDP: Directional learning disabled (LTP-only)
7. All ablations: All above ablated together

We measured:

• Performance: Top-1 and Top-3 accuracy
• Variance: Per-digit performance variance (stability metric)
• Decision time: Ticks to first correct answer

5.3 Results

Overall Performance

Configuration	Top-1 Acc	Top-3 Acc	Variance	Decision Time
Baseline	84.3%	95.6%	53.82	32.1 ticks
No retrograde	84.1%	95.1%	71.13	42.3 ticks
Frozen thresholds	83.7%	94.1%	72.53	47.1 ticks
Frozen weights	83.3%	95.0%	42.77	19.6 ticks
No STDP	82.4%	95.5%	76.70	28.6 ticks
All ablations	82.2%	95.1%	110.05	47.1 ticks
Frozen t_ref	80.4%	94.8%	250.22	46.0 ticks

Removing adaptive t_ref caused the largest performance loss (-3.9%) and variance explosion (+365%).

Variance Stability Analysis

Variance measures attractor stability: low variance indicates consistent performance across patterns; high variance indicates chaotic, unreliable dynamics.

The 4.6× variance increase when homeostatic regulation is removed demonstrates that multi-scale projection is essential for attractor stability. Without it, multiplicative learning creates chaotic dynamics that prevent any stable states from forming.

Pattern Complexity Analysis

We analyzed performance loss by digit complexity:

Digit	Complexity	Baseline	Frozen t_ref	Loss
1	Simple (vertical line)	97.6%	99.2%	-1.6%
0	Simple (single loop)	94.1%	94.1%	0.0%
8	Complex (loop-connector-loop)	74.2%	48.3%	+25.8%
9	Complex (loop-stem)	77.7%	52.1%	+25.5%

Thinking Time Requirements

The percentage of samples requiring extended deliberation (>50 ticks):

Digit	Baseline	Frozen t_ref	Interpretation
1	3.2%	0.8%	Simple = fast decisions
8	62.9%	97.8%	Complex = needs multi-scale

Without adaptive t_ref, 98% of complex patterns require extended processing (vs. 63% with multi-scale), a 35-point increase demonstrating that unstable attractors significantly slow recall.

5.4 Discussion

The ablation results validate three key hypotheses:

1. Multiplicative weights create exploratory variance
Frozen weights have lowest variance (42.77) because random initialization is more balanced than learned multiplicative weights. Learning intentionally creates variance to make temporal structure explicit. This amplification requires gain control (adaptive t_ref) to prevent runaway.

2. Adaptive t_ref enables multi-scale projection
Not merely "compensating for instability" but providing the projection mechanism itself. Creates heterogeneous temporal windows: short (~90 ticks) for fast features, long (~150 ticks) for global context. Enables linear separability of temporally ambiguous patterns.

3. Expansion degree self-organizes based on signal statistics
Simple patterns (low variance): Single scale sufficient, -1.6% loss without expansion
Complex patterns (high variance): Multi-scale essential, -25% collapse without expansion
System automatically adjusts: high-variance signals → high firing → short t_ref → selective windows

6. Qualitative Observations on Homeostatic Stability

While standard classification operates effectively within short timeframes (e.g., 240 ticks per pattern), we conducted extended observational runs to understand the long-term thermodynamic behavior of the topologies, comparing sparse and dense architectures.

Limit Cycle Attractors vs Chaotic Fluctuations

Rather than tracking quantitative classification metrics over these extended timescales, we observed the internal phase-space trajectory of the plasticity gate (\(t_{ref}\)). In the sparse configurations, the \(t_{ref}\) variable naturally settled into a stable sinusoidal oscillation—a limit cycle attractor.

This directly parallels biological cortical development. Early postnatal cortex exhibits excessive synaptic density, which undergoes activity-dependent pruning during critical periods. Our results suggest this developmental trajectory is a necessary path to stable neural dynamics.

7. Discussion

7.1 Temporal Dimensionality Expansion as Core Mechanism

The ablation results reveal that PAULA's primary innovation is the coupled dynamics of its mechanisms implementing temporal dimensionality expansion. This reframes our understanding of how biological-inspired SNNs learn temporal patterns.

Traditional interpretations view multiplicative weight updates as problematic, creating instability that requires homeostatic compensation. From this perspective, adaptive mechanisms seem like "nice to have" improvements that add modest performance gains of only 2-3%.

Our ablation results fundamentally challenge this view. Multiplicative weights are not a bug requiring compensation; they are an intentional variance amplifier. By scaling updates proportionally to current weight (Δw ∝ w), they create divergent dynamics where different neurons develop distinct firing rates. This divergence acts as exploration pressure that forces neurons to specialize.

Similarly, adaptive refractory periods convert exploration into productive specialization. High-firing neurons, having developed large weights through high-variance patterns, develop short refractory periods that make them selective to fast temporal features. Low-firing neurons develop long refractory periods that make them integrative of slow global structure. Together, this heterogeneity creates the multi-scale representation essential for complex pattern learning.

Together, these mechanisms appear to implement a temporal kernel trick. The system begins with compressed temporal patterns that are temporally ambiguous. Multiplicative weights amplify variance across neurons, creating divergent firing rates where different neurons develop different response properties. Adaptive refractory periods then project these signals into multi-scale temporal windows, creating heterogeneous temporal integration that captures both fast transitions and slow global structure. The result is a high-dimensional multi-scale representation where temporally ambiguous patterns become linearly separable.

The self-organizing nature is crucial. For simple patterns like digit 1, low input variance produces low amplification and relatively homogeneous temporal windows, making single-scale processing sufficient with only minor performance loss (-1.6%) when multi-scale integration is removed. For complex patterns like digit 8, high input variance drives high amplification and heterogeneous temporal windows, making multi-scale integration essential; without it, performance collapses by 25%.

These results carry significant biological implications. Cortical heterogeneity (neurons with different temporal properties, from fast-spiking interneurons to regular-spiking pyramidal cells) is fundamentally functional. This diversity enables multi-scale temporal processing necessary for complex pattern recognition. The brain creates multiple parallel temporal views of the same input, each emphasizing different timescales, allowing complex temporal patterns to be disentangled through integration across scales.

7.2 The Processual Representation Hypothesis

The finding that different digits drive the network into different dynamical regimes supports a processual view of representation. The network stores 'digit 0' not as a pattern in weight space, but as the capacity to enter an oscillatory state characterized by 15-tick periodicity and growing variance amplitude. Classification is re-enactment: the network recreates the dynamics it previously associated with that input. This aligns with enactivist cognitive science and predictive coding frameworks, while providing computational instantiation with measurable state variables.

This processual view suggests rethinking how we design learning systems. Rather than asking 'what weights represent X?', we should ask 'what state corresponds to X?' and 'how does the system transition between states?'

• Memory becomes the landscape of attractors, not a database of patterns
• Learning becomes niche construction in state space, not gradient descent in weight space
• Forgetting becomes inability to access attractors, not weight overwriting, suggesting solutions to catastrophic forgetting through state-space partitioning

Extending this approach toward artificial general intelligence requires augmenting perceptual classification with goal-directed behavior and self-monitoring. Our broader research program proposes supervisory networks that observe the system's own state dynamics, creating meta-level representations. These supervisors could detect anomalies, specifically states that do not match expectations, regulate homeostatic setpoints based on task demands, and potentially give rise to self-awareness through recursive observation of observation. The current work establishes that state-based representations are computationally viable; future work must demonstrate they support flexible, goal-directed cognition.

7.3 Architectural Priors and the Variance Constraint

Our experiments reveal a critical relationship between network topology and input statistics: while the PAULA neuron is entirely modality-agnostic and will dynamically model any continuous input stream, the network's macroscopic ability to separate temporal trajectories relies on spatial variance. With flattened RGB inputs in a dense Multilayer Perceptron (MLP) topology (e.g., CIFAR-10), performance is limited due to lower pixel-level variance compared to the high-contrast edges characteristic of MNIST.

This is a strict requirement for topological alignment. This architectural variance constraint yields a falsifiable theoretical prediction: a flat PAULA topology will succeed on inputs with high intrinsic variance, but will struggle to resolve low-variance, dense sensory streams without hierarchical feature extraction. When arranged in a spatial hierarchy featuring the appropriate inductive priors, the model successfully projects low-variance density into highly separable temporal domains, as demonstrated in our 96.55% convolutional result.

7.4 Interpretability Challenges

An interesting paradox emerged from our analysis: functional specialization is evident through population analysis but resists traditional single-unit ablation studies. This actually validates the biological realism of the substrate. Like natural neural systems, our model exhibits graceful degradation (Result 3) and distributed representation, requiring neuroscience-appropriate analysis methods rather than standard ML interpretability techniques.

Unlike brittle engineered systems, PAULA networks exhibit biological-like damage tolerance, which precludes traditional lesion-based interpretability. Instead, we employ population-level analysis methods from neuroscience, demonstrating functional specialization through activity correlation rather than causal necessity. This property, often viewed as a limitation in artificial systems, is a defining characteristic of robust biological computation.

7.5 Global Context for Local Learning

While this work validates the core predictive learning and homeostatic metaplasticity mechanisms (Kuśmierz et al., 2017), the model's neuromodulation framework (\(\vec{M}(t)\) dynamics) enables future investigation of context-dependent learning, attention-like modulation, and task-switching, capabilities essential for embodied agents in dynamic environments.

Specifically, the model explicitly supports arbitrary neuromodulator types which can be seamlessly integrated in any internal process of a single neuron, from regulating the postsynaptic integration to affecting the decay factors of traveling signals and more.

7.6 Synaptic Weight Dynamics

Our validation establishes that the local learning mechanisms produce functional networks, but characterizing the emergent weight structures, learning trajectories, and the specific contribution of retrograde error correction requires dedicated analysis. Such investigation would reveal how the network's internal representations develop over training and could inform optimal initialization strategies.

Crucially, our model establishes that contrary to traditional AI systems, synaptic weights are not the only storage for information in neural networks. When the complex structure of a biological neuron is respected, it allows for a different type of information storage which does not rely solely on synaptic weight distribution or spike trains in SNNs. The dimensionality reduction power of a single computational unit (784 synapses reduced down to membrane potential and plasticity window with ~40% of preserved information) strongly supports that internal neuron dynamics and structure play a major role in bio-plausible computational systems (Ding et al., 2025; Intrator & Cooper, 1992).

The single neuron's capacity for intelligent dimensionality reduction suggests that biological neurons may function as nonlinear embedding units, projecting high-dimensional sensory input onto low-dimensional manifolds that preserve task-relevant structure. This spatiotemporal dimensionality reduction parallels recent work on nonlinear transforms for compression, which demonstrated that neural representations can achieve massive compression by transforming data into domains where redundancy is more efficiently encoded.

7.7 Sparsity Scaling Law and Network Architecture

The sparsity scaling law suggests a broader architectural principle: representation quality requires constraints on information flow. In dense networks, these constraints come from depth (vertical segregation); in shallow networks, they must come from sparsity (horizontal segregation). This explains both the success of deep learning architectures and the efficiency of biological sparse cortical networks, which employ both strategies simultaneously.

7.8 Architecture Search via Homeostatic Stability

Our long-term dynamics findings suggest a novel principle for neural architecture design: architectures should be optimized for homeostatic stability rather than immediate task performance. This criterion, selecting networks that achieve dynamical equilibrium within feasible timescales, provides a biologically-grounded alternative to gradient-based neural architecture search (NAS) approaches.

Traditional NAS methods optimize architectures through task-specific performance metrics, requiring extensive labeled data and computational resources for evaluating each candidate. In contrast, stability-based selection requires only observation of intrinsic dynamics under unlabeled input, as homeostatic convergence is a property of the network's response to input statistics, not task labels. An architecture that fails to stabilize, as we observed with dense networks, cannot form reliable internal representations regardless of task, filtering it out before expensive task evaluation.

The biological plausibility of this approach is evident in cortical development. Early cortex exhibits random, excessive connectivity, which undergoes activity-dependent refinement through synaptic pruning, homeostatic scaling, and inhibitory tuning. These mechanisms do not optimize for specific tasks (the infant does not yet know what tasks it will encounter) but rather establish stable, adaptive computational substrates that can subsequently learn diverse skills.

The observed scaling of stabilization effort with input dimensionality provides qualitative intuition for architecture design:

• Small inputs (e.g., MNIST): phase-space stabilization is readily observable in reasonable timeframes.
• Moderate inputs (e.g., CIFAR-10): extended observation reveals longer trajectories before limit cycles emerge.
• Large inputs (e.g., ImageNet): monolithic architectures likely face extreme stabilization demands, suggesting the need for hierarchical processing where lower layers stabilize on local receptive fields before deeper layers integrate globally.

7.9 Biological Parallels: Development, Pruning, and Stability

Our finding that sparse networks achieve stability while dense networks remain perpetually unstable provides computational support for a long-standing puzzle in neurodevelopment: why does cortex initially overproduce synapses, only to eliminate 40-60% during postnatal critical periods? Traditional explanations emphasize metabolic efficiency or Darwinian selection of "useful" connections. Our results suggest a complementary mechanism: pruning is necessary for dynamical stability.

In early postnatal cortex, synaptic density peaks at approximately twice the adult level, then undergoes activity-dependent pruning during critical periods. This developmental trajectory parallels our findings: dense initial connectivity (analogous to 100% networks) creates unstable dynamics where homeostatic mechanisms cannot converge. Pruning to sparse adult levels (analogous to 25% networks) enables stabilization. The brain may not "know" which specific connections to eliminate; instead, activity-dependent mechanisms remove connections until the network enters a stable dynamical regime where homeostatic regulation can converge.

This interpretation is consistent with experimental observations that synaptic pruning is triggered by neural activity patterns and regulated by homeostatic mechanisms. Networks with excessive connectivity show heightened excitability and instability. Pruning reduces this instability, enabling refinement of functional circuits. Disruptions to this process, as occur in neurodevelopmental disorders where pruning is impaired, result in networks with persistent instability, potentially underlying the sensory hypersensitivity and information processing deficits observed in autism spectrum disorders and schizophrenia.

The cross-layer functional organization that emerges in our stabilized sparse networks further parallels cortical development. Orientation columns in V1, for instance, develop through spontaneous activity-driven self-organization before visual experience, with neurons across layers coordinating based on response properties rather than anatomical proximity. Our model reproduces this emergent organization through purely local homeostatic mechanisms, supporting theories that cortical columns arise from stabilization of initially random connectivity under homeostatic constraints.

Cortical Multi-Scale Processing: The ablation results provide a functional explanation for cortical neuronal diversity. Fast-spiking interneurons (short integration windows) and regular-spiking pyramidal neurons (long integration windows) may constitute a biological implementation of temporal dimensionality expansion. Our finding that complex patterns collapse 25% without multi-scale integration proves this diversity is a computational necessity.

The 90-150 tick range in PAULA aligns with cortical temporal dynamics. The slowdown in recall without heterogeneous t_ref (98% of complex patterns requiring extended processing vs. 63% with multi-scale) mirrors the speed-accuracy tradeoffs observed in human perceptual decisions. This positions PAULA as a computational framework for investigating principles of neural development and multi-scale temporal processing.

7.10 Resistance to Catastrophic Forgetting

A persistent challenge in modern Artificial Neural Networks (ANNs) is catastrophic forgetting, where learning new information rapidly overwrites previously learned representations because backpropagation fundamentally relies on global scalar updates to synaptic weights.

In contrast, PAULA provides a natural, physical resistance to catastrophic forgetting. Because the architecture relies on physical temporal delays and localized attractors (limit cycles), memories are structurally bound to specific topological basins in the network's state space. This structural binding ensures that novel sensory data cannot instantly overwrite established temporal representations, offering a biologically-plausible mechanism for continual learning.

8. Limitations and Future Work

While PAULA demonstrates significant temporal dimensionality expansion, the experimental validation presented in this work utilizes static initializations for specific temporal parameters to isolate the effects of adaptive \(t_{ref}\). Consequently, the current proof-of-concept pipeline requires careful initial tuning of the \(\lambda\) integration constants to match expected stimulus lengths, and relies on fixed, randomly initialized spatial delays (\(d\)) to construct its temporal topology.

We emphasize that these static variables are artifacts of the current experimental initialization, rather than theoretical limitations of the ALERM framework. Because the architecture supports an arbitrary number of concurrent neuromodulators, future work may implement dynamic meta-plasticity rules where \(\lambda\) is continuously regulated by dedicated neuromodulatory state vectors, allowing individual neurons to autonomously tune their integration windows to shifting input frequencies. Similarly, future iterations will explore dynamic routing and adaptive delays (\(d(t)\)), systematically quantifying the cross-seed topological variance currently observed in static initializations.

9. Conclusion

This work makes three contributions:

1. A formal model of homeostatic metaplasticity integrating dynamic learning windows with vector synapses and local predictive learning
2. Empirical validation suggesting this mechanism enables competitive performance (86.8% MNIST) with purely local rules
3. Discovery that different input classes drive networks into distinct dynamical regimes with characteristic temporal signatures

A network of these adaptive neurons is a self-organizing system whose primary emergent function is the construction of a robust internal model. The model's Hebbian learning rule naturally gives rise to cell assemblies: interconnected groups of neurons that represent stable concepts, objects, and causal relationships in the environment. Information is therefore not stored in any single neuron but is encoded in the distributed patterns of activity across these populations.

A critical challenge in any plastic network is maintaining stability. This model addresses the problem from the bottom up through homeostatic plasticity. Each neuron's metaplasticity rule (Kuśmierz et al., 2017), which adjusts its learning window based on its average firing rate, provides a powerful local stabilizing force. If a circuit becomes over-excited, its constituent neurons automatically reduce their learning sensitivity, preventing runaway potentiation and ensuring the network's internal representations remain stable.

This model's architecture provides a framework for addressing several key open problems across multiple disciplines:

In AI and machine learning, it offers a plausible, local alternative to the credit assignment problem solved by backpropagation, using a retrograde signaling loop to provide targeted feedback. It addresses continual learning and catastrophic forgetting through its inherent homeostatic plasticity, and its sparse, event-driven nature as a Spiking Neural Network (SNN) makes it a candidate for highly energy-efficient computation.

In neuroscience, the model provides a unified mathematical framework for exploring how different forms of plasticity, Hebbian, homeostatic, and neuromodulatory, can coexist and interact. Its graph-based structure provides a testable platform for investigating the role of dendritic computation, and its use of sensitivity vectors offers a quantitative hypothesis for the function of neuromodulatory diversity.

In cognitive science and philosophy, the model provides a plausible mechanistic substrate for several theories. The temporal synchronization of its neurons can address the "binding problem" of perception. Its self-organizing nature aligns with theories of internal representation where the brain builds a predictive model of its environment. And its capacity for information integration, global coordination, and self-reflection aligns with key principles of scientific theories of consciousness, such as Integrated Information Theory (IIT) and Global Workspace Theory (GWT).

These results demonstrate that local learning with homeostatic regulation is viable for complex tasks and open paths toward neuromorphic implementations and continual learning systems, bridging the gap towards nonbiological life forms.

Future architectural extensions appear promising. Preliminary convolutional implementations (in preparation) achieve 96.55% MNIST and 87.1% FashionMNIST accuracy (unoptimized first runs) that exceed the classic Diehl & Cook (2015) purely-local benchmark. Top-3 accuracies of 99.3% and 98.3% indicate that PAULA's representations are highly discriminative; remaining gaps reflect decision confidence rather than perceptual failure. Grayscale CIFAR-10 reaches ~50% accuracy on a 600-neuron network (in preparation). Together with systematic hyperparameter optimization, these results suggest PAULA's principles extend beyond the MLP architecture demonstrated here.

10. Related Work

This work builds upon and relates to several important research directions in neuroscience-inspired AI. While the full list of references is below, three specific comparisons highlight PAULA's unique contributions:

• BCM Theory: While classical BCM theory achieves homeostasis by modulating synaptic efficacy via a sliding spatial activity threshold, PAULA achieves it by dynamically expanding or contracting the temporal integration window ($t_{ref}$). This directly links plasticity to temporal dimensionality rather than just spike rate.
• Biological Capacity: Jones & Kording (2020) demonstrated that biological dendritic neurons can solve MNIST through compartmentalized spatial computation. Our single PAULA neuron achieves comparable capacity through temporal dimensionality expansion, suggesting spatial dendritic branching and temporal graph delays may be functionally equivalent substrates for single-unit computation.
• Local Learning: While models like DECOLLE utilize local surrogate gradients with fixed learning dynamics, PAULA introduces an adaptive temporal gate ($t_{ref}$) that entirely bypasses the need for surrogate gradient estimations during local weight updates.

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11. Code Availability

All code, simulation parameters, and evaluation scripts required to reproduce the temporal dimensionality expansion and network benchmarks presented in this paper are available in the open-source repository at https://github.com/arterialist/neuron-model.