Learning & Memory

Derivation 1: NMDA Receptor as Coincidence Detector

The NMDA receptor current depends on both glutamate binding and voltage-dependent Mg$^{2+}$ block. The current is:

$$I_{\text{NMDA}} = \bar{g}_{\text{NMDA}} \cdot s(t) \cdot B(V) \cdot (V - E_{\text{rev}})$$

where $s(t)$ is the fraction of open channels (glutamate-gated) and $B(V)$ is the voltage-dependent Mg$^{2+}$ block factor:

$$B(V) = \frac{1}{1 + [\text{Mg}^{2+}]_o / 3.57 \cdot \exp(-0.062 \, V)}$$

The calcium influx through NMDA receptors, $J_{\text{Ca}} \propto I_{\text{NMDA}}$, triggers LTP when it exceeds a high threshold $\theta_+$ (activating CaMKII) or LTD when it is between a lower threshold $\theta_-$ and $\theta_+$ (activating calcineurin):

$$\Delta w = \begin{cases} +\eta_+ & \text{if } [\text{Ca}^{2+}] > \theta_+ \\ -\eta_- & \text{if } \theta_- < [\text{Ca}^{2+}] < \theta_+ \\ 0 & \text{if } [\text{Ca}^{2+}] < \theta_- \end{cases}$$

This calcium-based model (Shouval et al., 2002) unifies LTP and LTD under a single mechanism and explains frequency-dependent plasticity: high-frequency stimulation produces large Ca$^{2+}$ transients (LTP), while low-frequency stimulation produces moderate Ca$^{2+}$ elevations (LTD).

Derivation 2: BCM Theory and Sliding Threshold

The Bienenstock-Cooper-Munro (BCM) rule (1982) addresses the instability of pure Hebbian learning by introducing a sliding modification threshold. The weight change is:

$$\frac{dw_i}{dt} = \eta \, x_i \, y \, (y - \theta_M)$$

where $x_i$ is presynaptic activity, $y$ is postsynaptic activity, and$\theta_M$ is the sliding threshold. The threshold adjusts based on the time-averaged postsynaptic activity:

$$\theta_M = \langle y^2 \rangle / y_0$$

When $y > \theta_M$: LTP occurs (pre-post correlation strengthens the synapse). When $0 < y < \theta_M$: LTD occurs. The key stability property is that the threshold slides upward when the neuron is too active, preventing runaway excitation:

$$\frac{d\theta_M}{dt} = \frac{1}{\tau_\theta}\left(\frac{y^2}{y_0} - \theta_M\right)$$

BCM theory predicts ocular dominance plasticity: if one eye is deprived, its synapses weaken (LTD) while the open eye's synapses strengthen (LTP), consistent with monocular deprivation experiments.

Derivation 3: Spike-Timing-Dependent Plasticity (STDP)

STDP (Markram et al., 1997; Bi and Poo, 1998) formalizes the temporal asymmetry of Hebbian plasticity. The weight change depends on the relative timing of pre- and postsynaptic spikes:

$$\Delta w = \begin{cases} A_+ \exp(-\Delta t / \tau_+) & \text{if } \Delta t > 0 \text{ (pre before post, LTP)} \\ -A_- \exp(\Delta t / \tau_-) & \text{if } \Delta t < 0 \text{ (post before pre, LTD)} \end{cases}$$

where $\Delta t = t_{\text{post}} - t_{\text{pre}}$. Typical values are$\tau_+ \approx 20$ ms, $\tau_- \approx 20$ ms, with $A_- > A_+$ to maintain stability. The STDP rule can be derived from the calcium model: pre-before-post timing produces maximal NMDA receptor activation and Ca$^{2+}$ influx, while post-before-pre timing produces only moderate Ca$^{2+}$. The net effect over Poisson spike trains with rates $r_{\text{pre}}, r_{\text{post}}$ gives the expected weight change:

$$\langle \Delta w \rangle = r_{\text{pre}} r_{\text{post}} (A_+ \tau_+ - A_- \tau_-)$$

When $A_+\tau_+ > A_-\tau_-$, the rule is net potentiating for correlated firing; when $A_+\tau_+ < A_-\tau_-$, there is a net depression that stabilizes weights.

Derivation 4: Hippocampal Replay and Systems Consolidation

During slow-wave sleep, hippocampal place cells replay recent experiences in compressed form (sharp-wave ripples, 150–250 Hz). Model the hippocampal memory as a pattern$\boldsymbol{\xi}$ stored via one-shot Hebbian learning in a weight matrix:

$$\mathbf{W}_{\text{hipp}} = \frac{1}{N} \boldsymbol{\xi} \boldsymbol{\xi}^T$$

Consolidation transfers this memory to the cortical network through repeated replay. The cortical weights update incrementally with each replay:

$$\Delta \mathbf{W}_{\text{ctx}} = \frac{\epsilon}{N} \boldsymbol{\xi}_{\text{replay}} \boldsymbol{\xi}_{\text{replay}}^T$$

where $\epsilon \ll 1$ is the slow cortical learning rate. After $K$ replay events, the cortical memory strength is:

$$\text{SNR}_{\text{ctx}} \approx K \epsilon \cdot \text{SNR}_{\text{hipp}}$$

The gradual interleaving of old and new memories during replay prevents catastrophic forgetting. The consolidation timescale $\tau_c \sim 1/(r_{\text{replay}} \epsilon)$predicts that stronger hippocampal traces (more replays) consolidate faster.

Derivation 5: Hopfield Network Memory Capacity

The Hopfield network (1982) stores binary patterns $\boldsymbol{\xi}^\mu \in \{-1, +1\}^N$using Hebbian weights:

$$w_{ij} = \frac{1}{N} \sum_{\mu=1}^{P} \xi_i^\mu \xi_j^\mu$$

The signal-to-noise ratio for recalling pattern $\mu$ is the alignment field versus the cross-talk noise. At neuron $i$:

$$h_i = \underbrace{\xi_i^\mu}_{\text{signal}} + \underbrace{\frac{1}{N}\sum_{\nu \neq \mu}\sum_j \xi_i^\nu \xi_j^\nu \xi_j^\mu}_{\text{crosstalk noise}}$$

The crosstalk term is a sum of $N(P-1)$ approximately independent terms, each with zero mean and variance $1/N^2$. By the central limit theorem:

$$\text{noise} \sim \mathcal{N}\left(0, \frac{P-1}{N}\right)$$

For reliable recall (error rate below ~1%), we need the noise to be much smaller than the signal: $\sqrt{(P-1)/N} \ll 1$, giving the classic capacity bound$P_{\max} \approx 0.138 N$ (Amit et al., 1985). Beyond this limit, catastrophic forgetting occurs as stored patterns become mutually corrupted.