A formalized axiomatic model of Hebbian learning and STDP rules in biological neural networks — bridging neuroscience, mathematics, and computational modeling.
System Overview
| Field | Value |
|---|---|
| System Name | synaptic_plasticity v1.0.0 |
| Domain | Neuroscience |
| Description | Axiomatic model of Hebbian learning and STDP rules in biological neural networks |
| Components | 3 Axioms, 4 Postulates, 5 Variables, 11 Parameters |
📜 Fundamental Axiom (Level 1)
The Hebb axiom constitutes the founding principle of synaptic plasticity, stated in 1949 in “The Organization of Behavior”:
“When an axon of cell A is near enough to excite cell B and repeatedly or persistently takes part in firing it, some growth process or metabolic change takes place in one or both cells such that A’s efficiency, as one of the cells firing B, is increased.”
Translation: “Neurons that fire together wire together”
Source: Hebb (1949), The Organization of Behavior
Other Fundamental Axioms:
Only activated synapses are modified. Plasticity is local to the specific connection.
Source: Bliss & Lomo (1973)
A weak stimulus can induce long-term potentiation (LTP) if associated with a strong stimulus on another converging pathway.
Source: Fundamental Hebbian property
🔬 Mechanistic Postulates (Level 2)
These postulates describe the specific mechanisms of synaptic plasticity:
Synaptic modification is bidirectional: LTP strengthens if correlated, LTD weakens if anti-correlated.
-A–eΔt/τ– if Δt < 0 (LTD) }
Source: Bi & Poo (1998, 2004)
The amplitude of synaptic change depends on the precise temporal difference between pre- and post-synaptic spikes.
Source: Markram et al. (1997), Bi & Poo (1998)
Intracellular calcium concentration determines the direction and amplitude of synaptic plasticity.
Source: Shouval et al. (2002), Graupner & Brunel (2012)
Networks maintain global stability via homeostatic scaling of all synaptic weights.
Source: Turrigiano (1998, 2017)
📊 State Variables (Level 3)
| Name | Description | Min | Max | Initial |
|---|---|---|---|---|
| synaptic_weight | Synaptic weight wij | 0.0 | 10.0 | 1.0 |
| calcium_concentration | Intracellular Ca²⁺ (M) | 0.0 | 100 μM | 100 nM |
| pre_spike_rate | Presynaptic rate (Hz) | 0.0 | 500.0 | 10.0 |
| post_spike_rate | Postsynaptic rate (Hz) | 0.0 | 500.0 | 10.0 |
| mgluR_activation | mGluR activation level | 0.0 | 1.0 | 0.0 |
Key Parameters (Constants):
| Parameter | Value | Description |
|---|---|---|
| theta_LTP | 25 μM | Calcium threshold for LTP |
| theta_LTD | 0.5 μM | Calcium threshold for LTD |
| eta_plus | 1×10⁸ | LTP learning rate |
| eta_minus | 5×10⁷ | LTD learning rate |
| tau_plus | 20 ms | STDP LTP time constant |
| tau_minus | 20 ms | STDP LTD time constant |
| A_plus | 0.1 | LTP amplitude |
| A_minus | 0.12 | LTD amplitude |
| tau_Ca | 100 ms | Ca²⁺ decay time constant |
⚙️ Dynamic Equations (Level 4)
The system evolves according to these equations:
Evolution of intracellular calcium concentration based on spike activity.
pre_spike_rate × Ca_pre_gain +
post_spike_rate × Ca_post_gain
Synaptic plasticity rule based on calcium thresholds.
eta_plus × (calcium_concentration – theta_LTP)
elseif calcium_concentration > theta_LTD then
-eta_minus × (theta_LTD – calcium_concentration)
else
0.0
end if;
Simplified evolution of metabotropic glutamate receptor activation.
calcium_concentration ≥ 0.0
📐 STDP Rule Diagram
Δw
Δt (ms)
LTP
LTD
Δt > 0: pre before post → LTP | Δt < 0: post before pre → LTD
Δw = A₊·exp(-Δt/τ₊) for LTP | Δw = -A₋·exp(Δt/τ₋) for LTD
📚 Key References
| Author(s) | Year | Contribution |
|---|---|---|
| D.O. Hebb | 1949 | Fundamental axiom (The Organization of Behavior) |
| Bliss & Lomo | 1973 | Experimental discovery of LTP |
| Markram et al. | 1997 | Quantitative STDP rules |
| Bi & Poo | 1998-2004 | Complete STDP characterization |
| Turrigiano | 1998-2017 | Homeostatic plasticity |
| Shouval et al. | 2002 | Calcium-dependent model |
| Graupner & Brunel | 2012 | Unified plasticity rule |
The Takeaway
This formalized system captures 75+ years of neuroscience research into a coherent mathematical framework. From Hebb’s original insight to modern calcium-dependent models, the axioms and postulates reveal how learning emerges from precise temporal correlations between neural activity.
The brain doesn’t just store memories — it encodes the timing of experience itself. Every synapse is a tiny historian, recording not just what happened, but when.
