Cell Physiology/Part 5/5.3 Synaptic Transmission

Part 5 · Chapter 5.3 · Module M5.3

Synaptic Transmission

Quantal release, neurotransmitter receptors, integration, and synaptic plasticity (LTP/LTD)

Learning Objectives

•Trace the chemical synapse signalling cascade from presynaptic AP to postsynaptic response.
•Quantify quantal release (Katz's mEPP) with binomial and Poisson statistics.
•Describe the molecular machinery of vesicle fusion: synaptotagmin, SNAREs, complexin.
•Catalog major neurotransmitter receptor families (iGluR, GABA, ACh, monoamines) and their pharmacology.
•Explain LTP via the NMDA-Ca-CaMKII-AMPA axis and contrast with LTD.

1.Overview of the chemical synapse

A chemical synapse is a specialized cell-cell junction where presynaptic electrical activity is translated into a chemical signal, diffused across a 20-30 nm cleft, and retranslated into an electrical response on the postsynaptic side. The transduction steps, each occupying \(\sim\)100 μs, are:

Action potential depolarizes the axon terminal.
Voltage-gated Ca²⁺ channels (Ca_v2.1 P/Q-type and Ca_v2.2 N-type) open; local [Ca²⁺] spikes to 10-100 μM at active-zone nanodomains.
Ca²⁺ binds synaptotagmin-1; SNARE-complex zippering drives fusion of a docked vesicle with the plasma membrane.
Neurotransmitter (typically 3000-5000 molecules per vesicle) spills into the synaptic cleft.
Transmitter binds postsynaptic receptors — ionotropic channels open within <1 ms, metabotropic GPCRs within tens of ms.
Transmitter is cleared by reuptake transporters or enzymatic degradation.

The end-to-end synaptic delay is 0.5-1 ms, with Ca²⁺-triggered release accounting for most of that latency. Speed, reliability, and plasticity make the chemical synapse the fundamental unit of neural computation.

2.Ca²⁺ trigger and release probability

The relationship between presynaptic [Ca²⁺] and release probability is steeply cooperative, usually fit with a Hill equation:

\[ p_\text{rel}([\text{Ca}^{2+}]) = \frac{[\text{Ca}^{2+}]^n}{K_{1/2}^n + [\text{Ca}^{2+}]^n},\qquad n \approx 4 \]

The fourth-power dependence implies that four Ca²⁺ ions must bind to synaptotagmin-1, the presynaptic Ca sensor, before the fusion machinery is triggered. Synaptotagmin's two C2 domains each bind two Ca²⁺ ions; Ca binding drives deep membrane insertion and pries apart the clamping complexin, releasing the SNARE zipper.

Active-zone anatomy

Dense projections organize vesicle docking sites on a presynaptic grid (RIM, Munc13, RIM-BP).
Ca_v2 channels are clustered within 20-50 nm of release-ready vesicles — a nanodomain coupling that minimizes delay.
A typical central synapse has 1-10 release sites per bouton.

Channel isoforms

P/Q-type (Ca_v2.1): dominant at most CNS synapses; blocked by ω-agatoxin.
N-type (Ca_v2.2): major at peripheral and spinal synapses; blocked by ω-conotoxin.
R-type (Ca_v2.3): supplementary release at some hippocampal synapses.

3.The SNARE fusion machine

Vesicle fusion is catalyzed by the SNARE complex, a four-helix coiled-coil bundle. The Nobel Prize (2013) to James Rothman, Randy Schekman, and Thomas Südhof recognized the decades-long discovery of this conserved machine.

v-SNARE

Synaptobrevin-2 (VAMP2) on the vesicle membrane. Cleaved by tetanus toxin.

t-SNAREs

Syntaxin-1 and SNAP-25 on the plasma membrane. Cleaved by botulinum toxins (BoNT/A, C, E).

Chaperones

Munc18-1 templates assembly; Munc13 primes vesicles; NSF/α-SNAP recycle SNAREs after fusion.

The zippering of v- and t-SNARE helices releases \(\sim\)65 k_BT of free energy per complex, easily enough to drive the activation barrier for membrane fusion. Two or three SNARE complexes per vesicle appear sufficient for fast release.

Clinical toxicology

Botulinum toxins cleave SNAREs and block ACh release at the neuromuscular junction: flaccid paralysis.
Tetanus toxin cleaves VAMP2 in spinal inhibitory interneurons: loss of GABA release, rigid paralysis.
α-Latrotoxin (black widow spider) triggers massive exocytosis by binding neurexin.

4.Quantal release: Katz's revolution

In 1952 Paul Fatt and Bernard Katz observed small, spontaneous 0.5 mV depolarizations at the frog neuromuscular junction in the absence of nerve stimulation — miniature end-plate potentials(mEPPs). Nerve-evoked end-plate potentials (EPPs) were always integer multiples of this unit. Katz concluded that transmission occurs in quanta — each quantum later identified as the contents of a single vesicle.

If a synapse has N release sites, each with independent probability p of releasing a vesicle per action potential, the distribution of quanta released follows a binomial law:

\[ P(k) = \binom{N}{k} p^k (1-p)^{N-k},\quad\text{mean } m = Np,\ \sigma^2 = Np(1-p) \]

In low-Ca²⁺ conditions (p small), this approaches the Poisson limit:

\[ P(k) \to \frac{m^k e^{-m}}{k!},\qquad \sigma^2 = m \]

Del Castillo and Katz (1954) verified this by showing that the EPP amplitude histogram at low Ca²⁺was well fit by a Poisson distribution. Modern techniques (minimal stimulation, MK-801 progressive block) refine binomial fits to identify N, p, and quantal size q independently — the trio of parameters that define a synapse's strength.

5.Simulation 1 — Stochastic vesicle release

We simulate 40 trials of a 40-Hz train (5 action potentials) with Hill-cooperative Ca²⁺-driven release probability, finite vesicle pool, and slow replenishment. The quantal content statistics reproduce both paired-pulse facilitation and the Poisson-like variance-to-mean ratio.

Python

script.py182 lines

import numpy as np
import matplotlib.pyplot as plt

# ------------------------------------------------------------
# Stochastic quantal release at a central synapse
# Steps per action potential:
#   1. AP arrives at presynaptic bouton, opens Cav2.1 (P/Q-type) channels
#   2. Ca2+ enters; local [Ca2+] at active-zone "nanodomains" spikes to 10-100 uM
#   3. Synaptotagmin-1 binds Ca2+ cooperatively (Hill coefficient ~4)
#   4. SNARE complexes zipper, driving vesicle fusion -> glutamate release
#   5. Quantal response: n_release = Poisson(N_vesicles * p_release)
#   6. Postsynaptic AMPA channels open, producing a mEPSC of ~0.5-1 mV amplitude
# ------------------------------------------------------------

rng = np.random.default_rng(seed=42)

# Presynaptic parameters
N_vesicles = 10                   # docked release-ready vesicles at active zone
Hill_n     = 4.0                  # Ca2+ cooperativity
K_half     = 10.0                 # uM, half-activation for Syt1 binding
dt         = 0.05                 # ms
T_total    = 200.0                # ms
n_t        = int(T_total/dt)
t          = np.arange(n_t)*dt

# Ca2+ transient at active zone: rapid rise + slow decay after AP
def ca_transient(t, t_ap, amp=80.0, tau_rise=0.3, tau_decay=2.0):
    out = np.zeros_like(t)
    mask = t >= t_ap
    tt = t[mask] - t_ap
    out[mask] = amp * (1 - np.exp(-tt/tau_rise)) * np.exp(-tt/tau_decay)
    return out

# Fire APs at 20, 45, 70, 95, 120 ms (5 pulses, 40 Hz train)
ap_times = [20, 45, 70, 95, 120]
Ca = np.zeros(n_t)
for tap in ap_times:
    Ca += ca_transient(t, tap)

# Release probability as Hill function of [Ca2+]
def p_release(ca):
    return ca**Hill_n / (K_half**Hill_n + ca**Hill_n)

p_rel = p_release(Ca)

# Stochastic release per trial
n_trials = 40
release_events = np.zeros((n_trials, n_t), dtype=int)
for tr in range(n_trials):
    # available vesicles replenish slowly; track a pool
    pool = N_vesicles
    for i in range(n_t):
        # Replenishment rate ~ 0.01 per ms (tau ~ 100 ms)
        if pool < N_vesicles and rng.random() < 0.01*dt:
            pool += 1
        # Trigger release per vesicle independently if p * dt large enough
        # Use Poisson approx for multiple vesicle events
        lam = pool * p_rel[i] * (50.0*dt)   # fast release gate: ~50 /ms when p=1
        if lam > 0:
            k = rng.poisson(lam)
            k = min(k, pool)
            release_events[tr, i] = k
            pool -= k

# Postsynaptic EPSP: each release evokes a unit response q = 0.5 mV
# EPSP decays with tau ~ 10 ms
q_amp = 0.5
tau_epsp = 10.0

def epsp_kernel(t_max, dt, tau=10.0):
    tk = np.arange(0, t_max, dt)
    k = (tk/1.0)*np.exp(-tk/tau) / tau
    return k/k.max()

ker = epsp_kernel(80.0, dt, tau_epsp)
V_trials = np.zeros((n_trials, n_t))
for tr in range(n_trials):
    imp = release_events[tr].astype(float)*q_amp
    V_trials[tr] = np.convolve(imp, ker, mode='full')[:n_t]

# Mean and SD across trials
V_mean = V_trials.mean(axis=0)
V_std  = V_trials.std(axis=0)

# Quantal statistics: histogram of release counts at AP peaks
release_counts = []
for tap in ap_times:
    idx = np.argmin(np.abs(t - tap)) + 5   # sample just after AP peak
    # sum releases in a 3 ms window
    window = slice(idx, idx + int(3.0/dt))
    for tr in range(n_trials):
        release_counts.append(release_events[tr, window].sum())
release_counts = np.array(release_counts)
mean_m = release_counts.mean()
var_m  = release_counts.var()
print(f"Mean quantal content m  = {mean_m:.2f}")
print(f"Var(m) / Mean(m)        = {var_m/mean_m:.2f}  (binomial -> <1, Poisson -> 1)")
print(f"Peak [Ca2+]             = {Ca.max():.1f} uM")
print(f"Peak p_release          = {p_rel.max():.3f}")

# Paired-pulse facilitation: compare 1st vs 2nd AP response
# np.trapezoid to integrate EPSP area between successive APs
areas = []
for tap in ap_times:
    idx0 = np.argmin(np.abs(t - tap))
    idx1 = idx0 + int(25.0/dt)
    A = np.trapezoid(V_mean[idx0:idx1], t[idx0:idx1])
    areas.append(A)
print("Integrated EPSP per AP (mV*ms):", [f"{a:.2f}" for a in areas])
ppf = areas[1]/areas[0]
print(f"Paired-pulse ratio (PPR = EPSP2/EPSP1) = {ppf:.2f}")

# ---- PLOTS ----
fig, axes = plt.subplots(3, 2, figsize=(13, 10), facecolor='#0a0a1a')

ax = axes[0,0]; ax.set_facecolor('#0a0a1a')
ax.plot(t, Ca, color='#f43f5e', lw=2)
for tap in ap_times:
    ax.axvline(tap, color='#fbbf24', lw=0.8, ls=':')
ax.set_xlabel('Time (ms)', color='white'); ax.set_ylabel('[Ca2+] (uM)', color='white')
ax.set_title('(a) Active-zone Ca2+ transients', color='#67e8f9')
ax.tick_params(colors='white'); ax.grid(alpha=0.2)
for s in ax.spines.values(): s.set_color('#444')

ax = axes[0,1]; ax.set_facecolor('#0a0a1a')
ax.plot(t, p_rel, color='#a3e635', lw=2)
for tap in ap_times:
    ax.axvline(tap, color='#fbbf24', lw=0.8, ls=':')
ax.set_xlabel('Time (ms)', color='white'); ax.set_ylabel('p_release', color='white')
ax.set_title('(b) Release probability (Hill n=4)', color='#67e8f9')
ax.tick_params(colors='white'); ax.grid(alpha=0.2)
for s in ax.spines.values(): s.set_color('#444')

ax = axes[1,0]; ax.set_facecolor('#0a0a1a')
# Raster of releases (trials stacked)
for tr in range(n_trials):
    idx = np.where(release_events[tr] > 0)[0]
    sizes = release_events[tr][idx]
    ax.scatter(t[idx], np.full_like(idx, tr, dtype=float), s=sizes*10,
               c='#67e8f9', alpha=0.7)
for tap in ap_times:
    ax.axvline(tap, color='#fbbf24', lw=0.6, ls=':')
ax.set_xlabel('Time (ms)', color='white'); ax.set_ylabel('Trial', color='white')
ax.set_title('(c) Vesicle release raster (40 trials)', color='#67e8f9')
ax.tick_params(colors='white'); ax.grid(alpha=0.2)
for s in ax.spines.values(): s.set_color('#444')

ax = axes[1,1]; ax.set_facecolor('#0a0a1a')
# Plot all EPSPs lightly + mean
for tr in range(min(n_trials, 15)):
    ax.plot(t, V_trials[tr], color='#67e8f9', alpha=0.2, lw=0.8)
ax.plot(t, V_mean, color='#f43f5e', lw=2.2, label='Mean')
ax.fill_between(t, V_mean-V_std, V_mean+V_std, color='#f43f5e', alpha=0.15, label='+/-1 SD')
ax.set_xlabel('Time (ms)', color='white'); ax.set_ylabel('EPSP (mV)', color='white')
ax.set_title('(d) Postsynaptic response across trials', color='#67e8f9')
ax.legend(facecolor='#1a1a2a', labelcolor='white', edgecolor='#444', fontsize=9)
ax.tick_params(colors='white'); ax.grid(alpha=0.2)
for s in ax.spines.values(): s.set_color('#444')

ax = axes[2,0]; ax.set_facecolor('#0a0a1a')
ax.hist(release_counts, bins=np.arange(-0.5, release_counts.max()+1.5, 1),
        color='#a3e635', edgecolor='#0a0a1a')
ax.set_xlabel('Quanta released per AP', color='white'); ax.set_ylabel('Frequency', color='white')
ax.set_title(f'(e) Quantal content histogram  mean={mean_m:.2f}  var/mean={var_m/mean_m:.2f}', color='#67e8f9')
ax.tick_params(colors='white'); ax.grid(alpha=0.2)
for s in ax.spines.values(): s.set_color('#444')

ax = axes[2,1]; ax.set_facecolor('#0a0a1a')
ax.bar(range(1, len(areas)+1), areas, color=['#60a5fa','#a3e635','#f97316','#f43f5e','#a78bfa'],
       edgecolor='white')
ax.set_xlabel('AP number in train', color='white'); ax.set_ylabel('EPSP area (mV*ms)', color='white')
ax.set_title(f'(f) Short-term plasticity: PPR = {ppf:.2f}', color='#67e8f9')
ax.tick_params(colors='white'); ax.grid(alpha=0.2)
for s in ax.spines.values(): s.set_color('#444')

plt.suptitle('Stochastic vesicle release - quantal synaptic transmission',
             color='#67e8f9', fontsize=14, y=0.995)
plt.tight_layout()
plt.savefig('output.png', dpi=120, bbox_inches='tight', facecolor='#0a0a1a')
plt.close()
print("Saved output.png")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

The paired-pulse ratio (PPR = EPSP₂/EPSP₁) above 1 indicates facilitation due to residual presynaptic Ca²⁺; the declining response during prolonged trains reflects vesicle pool depletion.

6.Postsynaptic potentials: EPSPs, IPSPs, and integration

Postsynaptic currents carry transmitter-specific polarity:

Excitatory (EPSP)

Glutamate opens AMPA/kainate channels (E_rev \(\approx\) 0 mV) — net inward Na⁺/K⁺ current depolarizes.
ACh at nicotinic receptors: same biophysics.
Fast kinetics (\(\tau\) \(\sim\) 2-10 ms).

Inhibitory (IPSP)

GABA at GABA_A opens Cl^- channels (E_Cl \(\approx -78\) mV) — hyperpolarizes, or shunts excitation.
Glycine in brainstem/spinal cord: same chloride biophysics.
GABA_B activates G_i/o → opens K⁺ (GIRK) channels, slow kinetics (100-500 ms).

A single central EPSP is only 0.1-0.5 mV — the neuron must integrate hundreds of synaptic inputs to reach spiking threshold. Two mechanisms combine inputs:

Temporal summation

Successive EPSPs from the same synapse add if they arrive within the membrane time constant τ_m (∼10-30 ms).

Spatial summation

EPSPs from different synapses on the same cell combine, weighted by dendritic distance to the axon hillock via the length constant λ.

Rall's (1959) cable theory of dendrites showed that distal synapses attenuate by 5-10x at the soma. Active dendrites with local Na⁺/Ca²⁺ spikes (e.g., CA1 pyramidal cells) partly compensate.

7.Neurotransmitter receptor families

Neurotransmitter	Receptors (ionotropic)	Receptors (metabotropic)	Key clinical drugs
Glutamate	AMPA, NMDA, Kainate	mGluR1-8 (groups I/II/III)	Memantine, ketamine, perampanel
GABA	GABA_A (pentamer)	GABA_B (G_i/o)	Benzos, barbiturates, baclofen
Glycine	GlyR (pentamer)	-	Strychnine (antag.)
Acetylcholine	nAChR (pentamer)	mAChR M1-M5 (GPCR)	Atropine, pyridostigmine, nicotine
Dopamine	-	D1-D5 (G_s/G_i)	L-DOPA, haloperidol, amphetamines
Serotonin	5-HT₃	5-HT_1-7 (mostly GPCR)	SSRIs, ondansetron, triptans
Norepinephrine	-	α1,2, β1,2,3 (GPCR)	β-blockers, clonidine
Endocannabinoids	-	CB1, CB2 (G_i/o)	THC, rimonabant

Ionotropic receptors are heteromeric channels that open within \(\sim\)1 ms. Metabotropic receptors are GPCRs that modulate slower downstream effectors (adenylyl cyclase, PLC, ion channels via G_βγ).

8.The NMDA receptor: coincidence detector

The NMDA receptor is the key molecular substrate of associative learning. It opens only when both conditions are met:

Presynaptic glutamate binds GluN1/GluN2 dimers.
Postsynaptic membrane is depolarized, expelling the Mg²⁺ block from the pore.

The Jahr-Stevens (1990) model of Mg²⁺ unblock is:

\[ \text{Unblock}(V) = \frac{1}{1 + \dfrac{[\text{Mg}^{2+}]_o}{3.57\,\text{mM}}\,e^{-V/16.13\,\text{mV}}} \]

At rest (\(V = -65\) mV) the unblock is only \(\sim 2\%\); at +30 mV it approaches 100%. Because NMDA channels also conduct Ca²⁺, they serve as a direct Ca²⁺ source during associative activity — the trigger for plasticity.

9.Long-term potentiation (LTP)

Bliss and Lømo (1973) discovered that a brief tetanus (100 Hz for 1 s) to the perforant path produced a persistent (hours to days) enhancement of dentate gyrus EPSPs. This long-term potentiation has since become the dominant cellular model of memory.

The canonical CA3 \(\to\) CA1 Schaffer-collateral LTP mechanism, elucidated by the labs of Nicoll, Malenka, and Kandel, involves:

Tetanic stimulation depolarizes the CA1 spine enough to relieve Mg²⁺ block on NMDA receptors.
Ca²⁺ influx through NMDA raises spine [Ca²⁺] to 1-10 μM.
Ca²⁺/calmodulin activates CaMKII, which autophosphorylates at Thr286 — becoming Ca-independent.
Active CaMKII phosphorylates GluA1 (AMPA) at Ser831, increases channel conductance, and drives exocytosis of AMPA receptors from recycling endosomes.
The increased AMPA content of the spine gives a larger EPSP to subsequent presynaptic release — Hebbian potentiation.
Late-phase LTP (>3 h) requires PKA, CREB-driven transcription, and new protein synthesis.

Kandel's dichotomy of memory

Early LTP (E-LTP) lasts 1-3 h, requires only post-translational modifications. Late LTP (L-LTP) requires PKA activation, CREB phosphorylation, and transcription of plasticity-related products (BDNF, Arc, Zif268). The transition mirrors short-term vs long-term memory.

LTP is bidirectional: modest [Ca²⁺] (0.5-1 μM) activates phosphatases PP1/calcineurin over kinases, leading to long-term depression (LTD) — AMPAR endocytosis and reduced synaptic strength. The Ca²⁺-threshold rule (BCM theory) provides the unified computational framework.

10.Simulation 2 — LTP via NMDA coincidence

We model a CA1 spine with AMPA and NMDA conductances, Mg²⁺ unblock, a Ca²⁺ transient, Hill-activated CaMKII with autophosphorylation bistability, and Ca-dependent AMPA insertion. A 100 Hz tetanus triggers a 2-3x potentiation that persists after the stimulus ends.

Python

script.py191 lines

import numpy as np
import matplotlib.pyplot as plt

# ------------------------------------------------------------
# Long-term potentiation (LTP) at a glutamatergic CA3->CA1 synapse
# Classical Bliss-Lomo (1973) / Malenka (1999) mechanism:
#   * AMPA receptors carry the baseline EPSC at resting V_m
#   * NMDA receptors are blocked by Mg2+ at rest, unblocking only when V_m is depolarized
#     AND glutamate is present: "coincidence detector"
#   * Ca2+ entry through NMDA activates CaMKII via Ca/calmodulin binding
#   * CaMKII autophosphorylates (Thr286) -> becomes Ca-independent
#   * CaMKII phosphorylates AMPA GluA1 Ser831 and drives exocytosis of new AMPARs
#   * Result: larger AMPA-mediated EPSP = potentiation
# ------------------------------------------------------------

dt   = 0.5            # ms
T    = 5000.0         # ms total (5 s)
n_t  = int(T/dt)
t    = np.arange(n_t)*dt

# Voltage at spine (mV): mostly resting, brief depolarizations during stimulation
V = np.full(n_t, -65.0)

# Two stimulation protocols:
#  "Weak" : single pulse at 0.5 s        -> no LTP
#  "Strong": tetanus 100 Hz for 1 s at 2 s -> LTP
protocol = "strong"
stim_times_weak   = [500]
stim_times_strong = list(np.arange(2000, 3000, 10))   # 100 Hz for 1 s

if protocol == "weak":
    stim_times = stim_times_weak
else:
    stim_times = stim_times_strong

# Generate presynaptic glutamate release events + postsynaptic depolarization
glu   = np.zeros(n_t)   # glutamate concentration (arbitrary)
for st in stim_times:
    idx0 = int(st/dt)
    if idx0 < n_t:
        # glutamate pulse ~ 1 ms with exponential decay tau=2 ms
        for j in range(40):
            if idx0 + j < n_t:
                glu[idx0 + j] += np.exp(-j*dt/2.0)

# Backpropagating AP depolarizes spine by +50 mV for 2 ms, + slow plateau from NMDA current
for st in stim_times:
    idx0 = int(st/dt)
    for j in range(int(3.0/dt)):
        if idx0 + j < n_t:
            V[idx0 + j] += 50.0*np.exp(-j*dt/1.5)

# NMDA conductance: proportional to glu * Mg-unblock factor
# Mg2+ unblock: Jahr-Stevens model  Mg_block = 1 / (1 + [Mg]/3.57 * exp(-V/16.13))
def mg_unblock(v, mg=1.0):
    return 1.0/(1.0 + (mg/3.57)*np.exp(-v/16.13))

# AMPA + NMDA currents (simple kinetics)
E_syn = 0.0             # reversal (cation nonsel)
g_AMPA_0 = 1.0          # baseline AMPA conductance (nS) - this is the plastic variable
g_NMDA_max = 0.6        # nS when fully open

tau_AMPA = 3.0          # ms
tau_NMDA = 100.0        # ms (long)

g_AMPA = np.zeros(n_t)
g_NMDA = np.zeros(n_t)
I_Ca   = np.zeros(n_t)

# Run synaptic current ODE
g_A = 0.0; g_N = 0.0
for i in range(n_t):
    g_A += dt*(-g_A/tau_AMPA) + glu[i]
    g_N += dt*(-g_N/tau_NMDA) + 0.4*glu[i]
    g_AMPA[i] = g_A
    g_NMDA[i] = g_N
    # NMDA Ca flux:  proportional to g_NMDA * unblock * driving force
    I_Ca[i]   = g_N * mg_unblock(V[i]) * (V[i] - E_syn) * (-1.0)   # inward positive

# [Ca2+] in spine: rises with NMDA current, decays with tau=50 ms
Ca = np.zeros(n_t)
tau_Ca = 50.0
for i in range(n_t-1):
    dCa = -Ca[i]/tau_Ca + max(I_Ca[i],0)*0.02
    Ca[i+1] = Ca[i] + dt*dCa

# CaMKII activation: Hill function of Ca, with slow autophosphorylation -> bistable
Hill_n_ck = 4.0
K_half_ck = 2.0
def camk_activation(ca):
    return ca**Hill_n_ck/(K_half_ck**Hill_n_ck + ca**Hill_n_ck)

CaMKII_P = np.zeros(n_t)
k_auto   = 0.002    # /ms autophosphorylation once partially active
k_dephos = 0.0003   # /ms dephosphorylation
for i in range(n_t-1):
    a = camk_activation(Ca[i])
    d_ck = (a + CaMKII_P[i]**2 * k_auto*100) * (1 - CaMKII_P[i]) - k_dephos*CaMKII_P[i]
    CaMKII_P[i+1] = CaMKII_P[i] + dt*d_ck*0.005
    CaMKII_P[i+1] = np.clip(CaMKII_P[i+1], 0, 1)

# AMPA insertion driven by active CaMKII: conductance grows with CaMKII_P
g_AMPA_plastic = g_AMPA_0 + 2.0*CaMKII_P     # up to ~3x potentiation

# ---- Test post-tetanic EPSP at t = 4.5 s ----
# We "probe" with a single pulse and measure EPSP amplitude
probe_time = 4500
probe_idx = int(probe_time/dt)
# Synthesize a probe: small glutamate pulse
probe_glu = np.zeros(n_t)
for j in range(20):
    if probe_idx + j < n_t:
        probe_glu[probe_idx + j] = np.exp(-j*dt/2.0)

# Compute total AMPA current at baseline (beginning) vs after LTP (end)
baseline_EPSC = g_AMPA_0 * (-70.0 - E_syn) * (-1)   # arbitrary units
post_EPSC     = g_AMPA_plastic[-1] * (-70.0 - E_syn) * (-1)
potentiation_ratio = post_EPSC / baseline_EPSC
print(f"Protocol: {protocol}")
print(f"Peak [Ca2+] in spine  : {Ca.max():.2f}")
print(f"Peak CaMKII active   : {CaMKII_P.max():.3f}")
print(f"Final CaMKII_P       : {CaMKII_P[-1]:.3f}")
print(f"AMPA conductance baseline -> final: {g_AMPA_0:.2f} -> {g_AMPA_plastic[-1]:.2f}")
print(f"Potentiation ratio   : {potentiation_ratio:.2f}x")

# Use np.trapezoid to integrate Ca2+ signal
ca_integral = np.trapezoid(Ca, t)
print(f"Ca2+ integral (AUC)  : {ca_integral:.1f}")

# ---- PLOTS ----
fig, axes = plt.subplots(3, 2, figsize=(13, 10), facecolor='#0a0a1a')

ax = axes[0,0]; ax.set_facecolor('#0a0a1a')
ax.plot(t/1000, V, color='#67e8f9', lw=1.2)
ax.set_xlabel('Time (s)', color='white'); ax.set_ylabel('V_m spine (mV)', color='white')
ax.set_title('(a) Spine voltage - tetanus and probe', color='#67e8f9')
ax.tick_params(colors='white'); ax.grid(alpha=0.2)
for s in ax.spines.values(): s.set_color('#444')

ax = axes[0,1]; ax.set_facecolor('#0a0a1a')
unblock = mg_unblock(V)
ax.plot(t/1000, unblock, color='#fbbf24', lw=1.5, label='Mg unblock')
ax.plot(t/1000, glu/glu.max(), color='#a78bfa', lw=1.0, label='Glutamate (norm)', alpha=0.7)
ax.set_xlabel('Time (s)', color='white'); ax.set_ylabel('Fraction / a.u.', color='white')
ax.set_title('(b) Coincidence detection', color='#67e8f9')
ax.legend(facecolor='#1a1a2a', labelcolor='white', edgecolor='#444', fontsize=9)
ax.tick_params(colors='white'); ax.grid(alpha=0.2)
for s in ax.spines.values(): s.set_color('#444')

ax = axes[1,0]; ax.set_facecolor('#0a0a1a')
ax.plot(t/1000, g_AMPA, color='#60a5fa', lw=1.2, label='g_AMPA (transient)')
ax.plot(t/1000, g_NMDA, color='#f97316', lw=1.2, label='g_NMDA (slow)')
ax.set_xlabel('Time (s)', color='white'); ax.set_ylabel('Conductance (nS)', color='white')
ax.set_title('(c) Synaptic conductances', color='#67e8f9')
ax.legend(facecolor='#1a1a2a', labelcolor='white', edgecolor='#444', fontsize=9)
ax.tick_params(colors='white'); ax.grid(alpha=0.2)
for s in ax.spines.values(): s.set_color('#444')

ax = axes[1,1]; ax.set_facecolor('#0a0a1a')
ax.plot(t/1000, Ca, color='#f43f5e', lw=1.6)
ax.axhline(K_half_ck, color='#a3e635', ls='--', label=f'CaMKII K1/2 = {K_half_ck}')
ax.set_xlabel('Time (s)', color='white'); ax.set_ylabel('[Ca2+] (a.u.)', color='white')
ax.set_title('(d) Spine [Ca2+] transient via NMDA', color='#67e8f9')
ax.legend(facecolor='#1a1a2a', labelcolor='white', edgecolor='#444', fontsize=9)
ax.tick_params(colors='white'); ax.grid(alpha=0.2)
for s in ax.spines.values(): s.set_color('#444')

ax = axes[2,0]; ax.set_facecolor('#0a0a1a')
ax.plot(t/1000, CaMKII_P, color='#a3e635', lw=2)
ax.set_xlabel('Time (s)', color='white'); ax.set_ylabel('CaMKII_P (fraction)', color='white')
ax.set_title('(e) CaMKII autophosphorylation - switch-like', color='#67e8f9')
ax.tick_params(colors='white'); ax.grid(alpha=0.2)
ax.set_ylim(-0.05, 1.05)
for s in ax.spines.values(): s.set_color('#444')

ax = axes[2,1]; ax.set_facecolor('#0a0a1a')
ax.plot(t/1000, g_AMPA_plastic, color='#f43f5e', lw=2)
ax.axhline(g_AMPA_0, color='#555', ls='--', label='Baseline')
ax.set_xlabel('Time (s)', color='white'); ax.set_ylabel('g_AMPA baseline (nS)', color='white')
ax.set_title(f'(f) AMPA conductance: {potentiation_ratio:.2f}x potentiation', color='#67e8f9')
ax.legend(facecolor='#1a1a2a', labelcolor='white', edgecolor='#444', fontsize=9)
ax.tick_params(colors='white'); ax.grid(alpha=0.2)
for s in ax.spines.values(): s.set_color('#444')

plt.suptitle('LTP: NMDA coincidence -> Ca -> CaMKII -> AMPA insertion',
             color='#67e8f9', fontsize=14, y=0.995)
plt.tight_layout()
plt.savefig('output.png', dpi=120, bbox_inches='tight', facecolor='#0a0a1a')
plt.close()
print("Saved output.png")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

The bistable CaMKII autophosphorylation switch provides a molecular memory — the system latches into the phosphorylated state and holds AMPA conductance elevated. Switching the protocol string in the Python code from "strong" to "weak" shows that a single pulse fails to cross the Ca threshold.

11.Electrical synapses and gap junctions

In parallel with chemical synapses, gap junctions form direct electrical and metabolic conduits between neurons. Each gap-junction channel is a connexon — a hexamer of connexin subunits (Cx36 in neurons) — that docks with a connexon on the partner cell to form a continuous 2-nm pore.

Properties

Essentially zero synaptic delay.
Bidirectional (usually).
Low-pass filter: small currents transfer better than fast transients.
Couples networks into synchronous oscillators (inferior olive, thalamic reticular nucleus).

Clinical relevance

Cx26 mutations cause sensorineural deafness.
Cx43 defects cause oculodentodigital dysplasia.
Cardiac Cx43 gap junctions propagate cardiac APs.
Gap-junction blockers (mefloquine, carbenoxolone) are experimental anticonvulsants.

Clinical connections

Myasthenia gravis

Autoantibodies to nicotinic AChR reduce safety factor at the neuromuscular junction: fatigable weakness. Pyridostigmine boosts ACh.

Lambert-Eaton syndrome

Autoantibodies to presynaptic Ca_v2.1 reduce ACh release — often paraneoplastic with small-cell lung cancer.

Alzheimer disease

Cholinergic hypofunction and synaptic loss; cholinesterase inhibitors (donepezil) provide modest cognitive benefit.

Depression & SSRIs

Selective serotonin reuptake inhibitors raise synaptic 5-HT; ketamine rapidly activates NMDA-dependent synaptogenesis.

Epilepsy

Anti-epileptics target vesicle release (levetiracetam on SV2A), Na_v blockade, or GABA_A potentiation (benzodiazepines).

Addiction

Drugs of abuse converge on VTA-NAc dopamine; repeated exposure induces LTP/LTD-like changes at glutamatergic synapses.

Key equations

Hill release:

\( p_\text{rel} = [Ca]^n / (K_{1/2}^n + [Ca]^n),\ n\approx 4 \)

Binomial:

\( P(k) = \binom{N}{k} p^k (1-p)^{N-k},\ m = Np \)

Poisson limit:

\( P(k) = m^k e^{-m}/k! \)

Jahr-Stevens NMDA:

\( U(V) = 1/(1 + ([Mg]/3.57) e^{-V/16.13}) \)

Synaptic EPSP:

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

12.Summary diagram

References

• Fatt, P. & Katz, B. (1952). Spontaneous subthreshold activity at motor nerve endings. J. Physiol. 117, 109-128.
• del Castillo, J. & Katz, B. (1954). Quantal components of the end-plate potential. J. Physiol. 124, 560-573.
• Bliss, T. V. P. & Lømo, T. (1973). Long-lasting potentiation of synaptic transmission in the dentate area of the anaesthetized rabbit. J. Physiol. 232, 331-356.
• Jahr, C. E. & Stevens, C. F. (1990). Voltage dependence of NMDA-activated macroscopic conductances. J. Neurosci. 10, 3178-3182.
• Malenka, R. C. & Nicoll, R. A. (1999). Long-term potentiation - a decade of progress? Science 285, 1870-1874.
• Jahn, R. & Südhof, T. C. (1999). Membrane fusion and exocytosis. Annu. Rev. Biochem. 68, 863-911.
• Südhof, T. C. (2013). Nobel Lecture: The molecular machinery of neurotransmitter release. Angew. Chem. 53, 12696-12717.
• Kandel, E. R. (2001). The molecular biology of memory storage: a dialogue between genes and synapses. Science 294, 1030-1038.

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