Biological and Hebbian Plasticity Rules

• Biological and Hebbian Plasticity Rules are mechanisms that adjust synaptic efficacy through timing-based and activity-dependent updates, such as STDP.
• They integrate complex dynamics including bidirectional updates, homeostatic stabilization, and modulatory signals to prevent instability.
• These principles inform computational models and neuromorphic systems, enabling rapid adaptation, robust memory, and efficient learning.

Biological and Hebbian Plasticity Rules refer to the neurobiologically grounded mechanisms that regulate synaptic efficacy in neural circuits, thereby enabling learning, memory storage, and adaptation. These rules govern how individual synapses are strengthened or weakened based on local and global signaling, including correlations in spike timing, activity rates, and modulatory factors. While the canonical Hebbian principle states that the coincidence of pre- and postsynaptic firing potentiates synaptic strength (“cells that fire together, wire together”), decades of experimental and theoretical work have shown that biological plasticity encompasses a family of rules with rich dynamics, including bidirectional updates, homeostatic stabilization, nonlinearities, and complex forms of credit assignment.

1. Fundamental Forms of Hebbian Plasticity in Biological Networks

Hebbian plasticity remains foundational in both biological and computational neuroscience. The basic rate-based rule is

$\Delta w_{ij} = \eta \, x_j y_i$

where $\eta$ is a positive learning rate, $x_j$ is presynaptic firing, and $y_i$ is postsynaptic firing. This mathematical framework captures the essence of synaptic potentiation by correlating local activities (Tyulmankov, 7 Dec 2024). However, biological synapses can undergo both potentiation and depression (LTP and LTD). Bidirectional plasticity appears in models such as

$$\Delta w_{ij} = \begin{cases} A_p \exp\left(\frac{t_j - t_i}{\tau_p}\right) & \text{if } t_i > t_j \ -A_d \exp\left(\frac{t_j - t_i}{\tau_d}\right) & \text{if } t_i < t_j \end{cases}$$

linking STDP to experimentally observed spike order effects (Borges et al., 2016, Tyulmankov, 7 Dec 2024).

The diversity within Hebbian rules encompasses higher-order versions (triplet, quadruplet, or nonlinear Hebbian rules) and anti-Hebbian plasticity (negative learning rate), as well as contextual gating by external signals. This heterogeneity is necessary to account for the wide range of plasticity profiles observed across brain areas and developmental stages.

2. Spike-Timing-Dependent Plasticity and Higher-Order Synaptic Dynamics

STDP specifies that synaptic updates depend on the precise timing between pre- and postsynaptic spikes. In the excitatory case, the canonical form is

$$\Delta\epsilon_{ij} = \begin{cases} A_1 \exp(-\Delta t_{ij}/\tau_1) & \Delta t_{ij} \geq 0\ -A_2 \exp(\Delta t_{ij}/\tau_2) & \Delta t_{ij} < 0 \end{cases}$$

with $\Delta t_{ij} = t_{pos} - t_{pre}$, regulating the temporal window and polarity of plasticity (Borges et al., 2016).

Generalized nonlinear Hebbian rules, as described in (Ocker et al., 2021), include power-law nonlinearities: $$J_i(t+dt) = \frac{J_i(t) + \frac{dt}{\tau} (n^a x_i^b J_i^c)}{\|J(t)\|_p}$$ where $n$ is the postsynaptic activity, $x_i$ is the presynaptic activity, and exponents $a, b, c$ shape sensitivity to higher-order correlations. For example, increasing $a$ or $b$ enables extraction of higher-order input statistics (tensor decomposition), explaining nonlinear dependencies observed in real neural recordings (Ocker et al., 2021).

STDP also generalizes to models combining mean firing rate and spike-timing information, such as calcium-based rules that integrate both event-based and rate-based signals. These models reproduce empirical observations of potentiation and depression frequency- and timing-dependence in both feedforward and recurrent spiking networks (Girão et al., 9 Apr 2025).

3. Biological Constraints: Stabilization, Homeostasis, and Modular Structure

Hebbian rules alone are unstable due to runaway positive feedback. Biological systems incorporate homeostatic and normalization mechanisms, including:

• Synaptic scaling: multiplicative adjustment to keep overall synaptic drive within bounds.
• Competition: “winner-take-all” effects, as captured in optimization approaches (e.g., noisy gradient/mirror descent on the probability simplex, (Dexheimer et al., 15 May 2025)), ensure that only the strongest inputs maintain potentiated connections, formalizing the principle that one postsynaptic neuron specializes to the most informative presynaptic afferents.
• Structural plasticity: homeostatic control over the number of synaptic elements—axonal or dendritic—adjusts connectivity. In networks where only firing rate homeostasis is enforced locally, associative (Hebbian-like) clusters emerge as a network-level effect. Neurons deviating from the target rate grow (or prune) synaptic elements, preferentially connecting to others with similar activity history (Gallinaro et al., 2017).

An additional layer is provided by rules inspired by the Allee effect, enforcing lower bounds for synaptic weights and suppressing plasticity when weights or activity fall below a critical threshold: $$\tau_W \frac{dW}{dt} = v (u - \frac{v}{K}W) \left(1 - \frac{A}{W^2}\right)$$ promoting robust stabilization and dynamic decorrelation (Kwessi, 2022).

4. Three-Factor and NeoHebbian Plasticity: Gating and Behavioral Timescales

Biological synaptic plasticity often requires conjunctive conditions beyond pre- and postsynaptic activity, introducing the concept of “three-factor” rules:

• Eligibility traces: temporary flags that store a “memory” of pre–post co-activation. Represented as

$$\frac{de_{ij}}{dt} = \eta \, x_j \, g(y_i) - \frac{1}{\tau_e} e_{ij}$$

• Gating by third factors: Neuromodulatory or behavioral signals (e.g., reward, surprise, novelty) are delivered with delay, converting the eligibility trace into a lasting synaptic change:

$\frac{dw_{ij}}{dt} = e_{ij} \cdot M_{3rd}(t)$

where $M_{3rd}$ denotes neuromodulator level at time $t$ (Gerstner et al., 2018).

This mechanism extends plasticity from millisecond to behavioral timescales (seconds to minutes) and aligns with reinforcement learning paradigms such as temporal difference (TD) learning where delayed rewards “gate” the eligibility of previous activations.

5. Modulatory and Inhibitory Plasticity: Credit Assignment and Global Constraints

Complex behaviors require plasticity rules beyond pairwise Hebbian learning, both to solve credit assignment and to maintain balanced network operation. Several key principles emerge:

• Neuromodulation: Acetylcholine (ACh) and noradrenaline (NE) act as global gates, targeting specific or broad plasticity changes respectively. Formally, target-specific updates (ACh) and broad potentiation (NE) are described by

$$(\Delta w_{ei})_A = \eta_A \alpha_A (y - \bar{f}) [X^{e}_{\mu, i} - \beta_A \frac{f}{1-f} (1 - X^{e}_{\mu, i})]$$

$$(\Delta w_{ei})_N = \eta_N \alpha_N (y - \bar{f}) X^{e}_{\mu, i}$$

, with gating variables and parameters controlling the balance of potentiation and depression (Aljadeff et al., 2019).

• Inhibitory plasticity: Ensures detailed E/I balance. Homeostatic plasticity at inhibitory synapses ensures that excitatory and inhibitory currents are tightly matched (tilted balance line: $c^i_\mu = a c^e_\mu + b$). Proper E/I balance reduces the dimensionality of the associative learning problem and allows imperfect neuromodulatory signals to efficiently guide learning close to theoretical memory capacity (Aljadeff et al., 2019).

Biologically, factors such as retrograde signaling (e.g., via nitric oxide, neurotrophic factors) provide local “credit” signals that propagate error information, allowing multilayer networks to implement backpropagation-like gradient descent while only relying on local computation and multiple timescales of neurosignaling (Fan et al., 23 May 2024). This approach replicates chain-rule based backpropagation for layered architectures, connecting global objectives to local plasticity with testable cellular mechanisms.

6. Computational and Functional Consequences: Network Topology, Learning Behavior, and Meta-Plasticity

Plasticity rules directly shape emergent network topology and functional capabilities:

• STDP and Hebbian plasticity can transform all-to-all random graphs into structured topologies—preferentially directed, modular, or clustered based on neuronal firing rate differences or external perturbations. These dynamics support the co-existence of synchronous and asynchronous cell assemblies, a motif repeatedly observed in biological networks (Borges et al., 2016).
• Meta-plasticity, or the plasticity of plasticity parameters themselves, enables agents to “learn how to learn.” Optimizing both baseline weights and plasticity coefficients (e.g., via gradient backpropagation through local Hebbian traces or by evolutionary strategies over rule parameters) endows artificial neural networks with rapid adaptation, one-shot learning, reversal learning, and continual updating capabilities—functions fundamental to biological cognition and robust memory (Miconi, 2016, Najarro et al., 2020, Pedersen et al., 2021, Wang et al., 2021).
• Homeostatic, structural, and genomic bottlenecks: Limiting the number of meta-parameters describing plasticity rules (e.g., by decomposing into neuron-specific components or by clustering rules via K-means to encourage parameter sharing) promotes generalization and OOD robustness, mirroring evolutionary constraints on biological genomes (Pedersen et al., 2021, Wang et al., 2021).
• Practical implementations in neuromorphic and artificial hardware leverage these principles for online learning, energy efficiency, and computational adaptability across domains, from low-level pattern recognition (e.g., MNIST digits) to sequence memory, cross-modal association, and language tasks (Limbacher et al., 2022, Girão et al., 9 Apr 2025).

7. Mathematical Formalisms and Theoretical Frameworks

The mathematical structure of biological and Hebbian plasticity rules is broad, ranging from explicit update prescriptions to optimization-theoretic interpretations:

• Gradient and mirror descent: STDP rules can be cast as noisy gradient descent flows or entropic mirror descent on the probability simplex—with explicit Lyapunov functions ensuring convergence to specialized, winner-take-all synaptic patterns (Dexheimer et al., 15 May 2025).
• Tensor eigenvector decomposition: Nonlinear Hebbian rules are rigorously shown to perform tensor decomposition, extracting robust components of higher-order input correlation tensors; dominant components have maximal basin size, explaining the stability and selectivity of emergent receptive fields (Ocker et al., 2021).
• Surrogate loss gradients: Modern machine learning frameworks can implement classical Hebbian, Oja’s, and instar rules via the gradients of specifically designed losses, allowing Hebbian updates to be seamlessly integrated into convolutional architectures with high performance (Miconi, 2021).

Complications such as runaway potentiation, catastrophic forgetting, and the need for stabilization are addressed through competition, weight normalization, decay terms, or by evolutionary/meta-learning constraints.

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In summary, biological and Hebbian plasticity rules encompass a diverse suite of synaptic modification mechanisms, from fundamental correlation-based updates through STDP and homeostatic plasticity, to neoHebbian three-factor learning, E/I balance-constrained adaptation, and meta-plasticity frameworks. These mechanisms underpin the flexibility, stability, and computational power of biological learning systems and inspire robust, adaptable, and efficient learning architectures in artificial neural networks.