Synaptic Transmission

Derivation 1: The Binomial Model of Vesicle Release

At a synapse with $N$ readily releasable vesicles, each with release probability $p$ per action potential, the number of released quanta $k$ follows a binomial distribution:

$$P(k) = \binom{N}{k} p^k (1-p)^{N-k}$$

The mean quantal content is $m = Np$ and the variance is$\sigma^2 = Np(1-p)$. The coefficient of variation is:

$$\text{CV} = \frac{\sigma}{m} = \sqrt{\frac{1-p}{Np}}$$

When $N$ is large and $p$ is small (as in central synapses), the distribution approaches Poisson:

$$P(k) \approx \frac{m^k e^{-m}}{k!}, \quad \text{where } m = Np$$

The relationship between mean $m$ and variance allows estimation of $N$ and$p$ from the mean-variance relationship across conditions that change $p$(e.g., varying extracellular calcium):

$$\sigma^2 = m \cdot q - \frac{m^2}{N} \cdot q$$

where $q$ is the quantal amplitude. Plotting $\sigma^2$ vs $m$ yields a parabola from which $N$ and $q$ can be extracted. At the NMJ,$N \sim 200\text{--}300$ and $p \sim 0.1\text{--}0.3$; at central synapses,$N \sim 1\text{--}20$ and $p \sim 0.1\text{--}0.9$.

Derivation 2: Synaptic Conductance and the Reversal Potential

The postsynaptic current through a ligand-gated channel is:

$$I_{\text{syn}} = g_{\text{syn}}(t) \cdot (V - E_{\text{syn}})$$

where $g_{\text{syn}}(t)$ is the time-dependent synaptic conductance and$E_{\text{syn}}$ is the synaptic reversal potential. For glutamatergic AMPA receptors,$E_{\text{syn}} \approx 0$ mV (excitatory); for GABA$_A$ receptors,$E_{\text{syn}} \approx -70$ mV (inhibitory). The conductance time course is commonly modeled as a difference of exponentials:

$$g_{\text{syn}}(t) = \bar{g} \cdot \frac{\tau_d \tau_r}{\tau_d - \tau_r}\left(e^{-t/\tau_d} - e^{-t/\tau_r}\right)$$

where $\tau_r$ is the rise time and $\tau_d$ is the decay time. For AMPA:$\tau_r \approx 0.2$ ms, $\tau_d \approx 2$ ms. For NMDA:$\tau_r \approx 5$ ms, $\tau_d \approx 50\text{--}100$ ms.

The postsynaptic potential is found by integrating the cable equation. For a point neuron with membrane time constant $\tau_m$ and input resistance $R_{\text{in}}$:

$$V(t) = E_L + R_{\text{in}} \int_0^t g_{\text{syn}}(t') \cdot (V(t') - E_{\text{syn}}) \cdot e^{-(t-t')/\tau_m} \, dt'$$

In the subthreshold regime where $g_{\text{syn}} \ll g_L$, the PSP amplitude is approximately $\Delta V \approx \bar{g} \cdot R_{\text{in}} \cdot (E_L - E_{\text{syn}}) \cdot \tau_{\text{syn}}$. Typical EPSP amplitudes at central synapses are 0.1–2 mV, requiring summation of many inputs to reach threshold.

Derivation 3: The Tsodyks-Markram Model of Short-Term Plasticity

Tsodyks and Markram (1997) developed a phenomenological model with three state variables representing the fraction of synaptic resources in recovered ($x$), active ($y$), and inactive ($z$) states, with $x + y + z = 1$:

$$\frac{dx}{dt} = \frac{z}{\tau_{\text{rec}}} - u \cdot x \cdot \delta(t - t_{\text{sp}})$$

$$\frac{dy}{dt} = -\frac{y}{\tau_{\text{in}}} + u \cdot x \cdot \delta(t - t_{\text{sp}})$$

$$\frac{dz}{dt} = \frac{y}{\tau_{\text{in}}} - \frac{z}{\tau_{\text{rec}}}$$

where $u$ is the utilization parameter (analogous to release probability),$\tau_{\text{rec}}$ is the recovery time from depression, and $\tau_{\text{in}}$is the inactivation time. The effective synaptic strength is proportional to $y$.

For facilitation, $u$ itself becomes dynamic:

$$\frac{du}{dt} = -\frac{u - U}{\tau_{\text{fac}}} + U(1-u) \cdot \delta(t - t_{\text{sp}})$$

where $U$ is the baseline utilization and $\tau_{\text{fac}}$ is the facilitation decay time. At steady state for regular firing at frequency $f$, the average response amplitude is $\bar{y} = U \cdot x_{\text{ss}} / (1 - (1-U) e^{-1/(f\tau_{\text{rec}})})$. Depression dominates at high frequencies when $\tau_{\text{rec}}$ is long; facilitation dominates when $\tau_{\text{fac}} > \tau_{\text{rec}}$.

Derivation 4: Calcium-Based Plasticity Model (BCM Theory)

The Bienenstock-Cooper-Munro (BCM) theory proposes that the sign and magnitude of synaptic change depend on postsynaptic activity relative to a sliding threshold. The synaptic weight$w$ evolves according to:

$$\frac{dw}{dt} = \eta \cdot \phi(c) \cdot r_{\text{pre}}$$

where $\eta$ is the learning rate, $r_{\text{pre}}$ is the presynaptic rate, and$\phi(c)$ is the plasticity function that depends on postsynaptic calcium concentration $c$:

$$\phi(c) = \begin{cases} -A_{\text{LTD}} & \text{if } \theta_{\text{LTD}} < c < \theta_{\text{LTP}} \\ A_{\text{LTP}}(c - \theta_{\text{LTP}}) & \text{if } c > \theta_{\text{LTP}} \\ 0 & \text{otherwise} \end{cases}$$

The LTP threshold $\theta_{\text{LTP}}$ slides with average postsynaptic activity:

$$\theta_{\text{LTP}} = \theta_0 + \alpha \langle c \rangle^2$$

This sliding threshold is essential for stability: it prevents runaway potentiation by raising the bar for LTP when a neuron is very active, and lowering it when inactive. The BCM rule has been confirmed by experiments varying stimulation frequency at hippocampal synapses, showing LTD at low frequencies (1–5 Hz) and LTP at high frequencies (>10 Hz), with the crossover frequency shifting with prior activity history.

Derivation 5: The STDP Learning Rule and Its Stability

The classic STDP window is described by:

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

where $\Delta t = t_{\text{post}} - t_{\text{pre}}$, and typical parameters are$\tau_+ \approx 20$ ms, $\tau_- \approx 20$ ms,$A_+ \approx 0.01$, $A_- \approx 0.012$. The slight asymmetry ($A_- > A_+$) ensures net depression for uncorrelated firing, providing stability.

For Poisson pre- and postsynaptic neurons firing at rates $r_{\text{pre}}$ and$r_{\text{post}}$, the expected weight change per unit time is:

$$\frac{d\bar{w}}{dt} = r_{\text{pre}} r_{\text{post}} (A_+ \tau_+ - A_- \tau_-)$$

If pre- and postsynaptic spikes have a correlation function $C(\Delta t)$, then:

$$\frac{d\bar{w}}{dt} = \int_{-\infty}^{\infty} W(\Delta t) \cdot C(\Delta t) \, d(\Delta t)$$

where $W(\Delta t)$ is the STDP window function. This integral formulation shows that STDP selectively strengthens synapses whose presynaptic activity consistently predicts postsynaptic firing (positive correlations at short lags), implementing a causal Hebbian learning rule at the millisecond timescale.