— AMPA, NMDA & GABA Kinetic Models

§ 01

Interactive Simulation

Var 1

—

Var 2

—

Input

—

Output

—

Count

—

Mode

—

Time

Playback

Preset

Parameters

Parameter 15.0

Parameter 25.0

Input I5.0

T sim (ms)500

Overlays

Grid Legend

§ 02

The Idea, Step by Step

▸ Three Kinds of Doors, Three Kinds of Timing

START — Three Different Doors (middle school)

When one brain cell talks to the next, it does not send electricity straight across the gap. It squirts a tiny puff of chemical messenger that pops open little doors — ion channels — in the receiving cell. But the doors are not all alike. AMPA doors snap open and slam shut in a blink, like a fast text message. NMDA doors are slow and picky: they only open if two things happen at the same time. And GABA doors carry the "calm down" message that quiets the cell instead of exciting it. Same idea — a chemical opens a door — but the timing and the meaning are different for each.

BUILD — Counting the Open Doors (high school)

We track the fraction of doors that are open, called $r(t)$, running from 0 (all shut) to 1 (all open). Transmitter $[T]$ pushes doors open at rate $\alpha[T]$ and they fall shut at rate $\beta$, so $\frac{dr}{dt}=\alpha[T](1-r)-\beta r$. The current that flows is $I=g_{\max}\,r\,(V-E)$. One worked number for AMPA with $\alpha=1.1$ and $\beta=0.19$ ms⁻¹ during a brief 1 mM pulse: the peak open fraction is $\frac{\alpha[T]}{\alpha[T]+\beta}=\frac{1.1}{1.29}\approx0.85$, and once the pulse ends the doors close with time constant $\tau=1/\beta\approx5$ ms.

DEEPEN — Voltage Gates and Sticky States (AP / college)

NMDA adds a twist: a magnesium ion plugs the open pore at negative voltages, so its conductance is itself voltage-dependent, $g\propto 1/\!\left(1+[\mathrm{Mg}^{2+}]/K\;e^{-V/V_0}\right)$. The cell must already be depolarised and glutamate must be bound, which makes NMDA a coincidence detector — the molecular basis of learning. GABA$_A$ adds desensitisation: channels slip into a sticky third state $D$ during prolonged exposure, $\frac{dd}{dt}=\alpha_2 r-\beta_2 d$, which weakens repeated signals. On the control panel, Parameter 1 and Parameter 2 act as the opening and closing rates, Input I scales the transmitter pulse, and T sim sets the time window you watch.

TRY THIS — In the Sim Above

(1) Raise the closing rate (Parameter 2) and watch the response sharpen toward a fast AMPA-like blink; lower it and see the long, slow NMDA-like tail emerge. (2) Shrink Input I and notice the peak open fraction never reaches 1 — some doors always stay shut when the puff is weak. (3) Stretch T sim and send repeated inputs to see a desensitising GABA$_A$-style response shrink with each successive pulse, the postsynaptic side of short-term depression.

§ 03

Equation Derivation

▸ Kinetic Models of Ionotropic Receptors

Synaptic receptors are ligand-gated ion channels modelled as Markov chains. Each state (Closed, Open, Desensitised) transitions with voltage- or ligand-dependent rate constants.

$$\text{AMPA (simple 2-state): } C \underset{\beta}{\overset{\alpha[T]}{\rightleftharpoons}} O, \quad I_{\text{AMPA}} = g_{\text{AMPA}}\cdot r(t)\cdot(V-E_{\text{AMPA}})$$ $$\text{where } r(t) \text{ obeys: } \frac{dr}{dt} = \alpha [T](1-r) - \beta r$$ $$\text{NMDA (Mg²⁺ block): } I_{\text{NMDA}} = g_{\text{NMDA}}\cdot s(t)\cdot\frac{V-E_{\text{NMDA}}}{1+[Mg^{2+}]/K_{Mg}\cdot e^{-V/V_{Mg}}}$$ $$\text{GABA}_A\text{: } \frac{ds}{dt} = \alpha_1[GABA](1-s-d) - \beta_1 s, \quad \frac{dd}{dt} = \alpha_2 s - \beta_2 d \quad \text{(with desensitisation)}$$

▸ Receptor Parameter Table

Parameter	AMPA	NMDA	GABA_A	GABA_B
E_rev (mV)	0	0	−70	−90
τ_rise (ms)	0.2–0.5	5–10	0.5–1	50–100
τ_decay (ms)	2–5	50–200	5–15	200–500
Peak conductance	0.5–1 nS	0.3–0.8 nS	0.5–1.5 nS	0.1–0.5 nS
Key blocker	CNQX, DNQX	AP5, MK-801, Ketamine	Bicuculline	CGP 55845
Key modulator	Cyclothiazide (slows deactivation)	Glycine (co-agonist required)	Benzodiazepines, Barbiturates	Baclofen (agonist)

▸ Derivation Steps

STEP 1 — 2-State AMPA Kinetics

The simplest AMPA model: C (closed) ↔ O (open). Opening rate α[T] depends on transmitter concentration [T] (brief pulse on presynaptic spike, duration ~0.3 ms). Closing rate β is voltage-independent. The fraction of open channels r(t) follows: dr/dt = α[T](1−r) − βr. Solution for brief pulse: r(t) = r_peak × e^{−βt} for t > pulse duration, with r_peak = α[T]/(α[T]+β). Time constant τ = 1/β ≈ 3–5 ms for AMPA. This is the minimal model used in most large-scale simulations.

STEP 2 — NMDA: 4-State Model with Mg²⁺

NMDA requires: glutamate binding + glycine co-agonist + voltage relief from Mg²⁺. A minimal 4-state model: C (unbound) → C_bound (glutamate bound) → O (open) ↔ O_blocked (Mg²⁺-blocked). The Mg²⁺ blocking/unblocking rates depend on voltage: k_off = k_off_0 × e^{+δ×F×V/RT}, k_on = k_on_0 × e^{−(1−δ)×F×V/RT}. The simplified Boltzmann voltage-dependence: g_NMDA(V) × 1/(1 + [Mg]/K × e^{−V/V0}) captures the essential voltage-dependence.

STEP 3 — GABA_A Desensitisation

GABA_A receptors desensitise (current decreases during prolonged GABA exposure): C → O → D (desensitised). The 3-state model: dr/dt = α₁[GABA](1−r−d) − β₁r; dd/dt = α₂r − β₂d. d accumulates when [GABA] remains high. Desensitisation explains: (1) smaller IPSCs during high-frequency stimulation (synaptic depression); (2) faster apparent decay of the IPSC (desensitised channels stop contributing); (3) recovery time after a burst (~100 ms). Pharmacologically, benzodiazepines increase α₁ (more opening), barbiturates prolong the open state directly.

STEP 4 — Benzodiazepine Mechanism

Benzodiazepines (diazepam, lorazepam) bind to the GABA_A receptor at the α-γ interface — a site distinct from the GABA binding site. They act as positive allosteric modulators: they increase the FREQUENCY of channel opening (increase α₁ in the kinetic model) without affecting the single-channel conductance or maximal open probability. This is why benzodiazepines require GABA to work — they are modulators, not direct agonists. Clinically, this explains their anxiolytic effect: enhancing inhibitory GABA transmission in limbic circuits reduces anxiety without causing the respiratory depression of barbiturates (which increase open duration, not frequency).

▸ Worked Example — AMPA EPSC from Kinetic Model

AMPA: α = 1.1 ms⁻¹mM⁻¹, β = 0.19 ms⁻¹, [T] = 1 mM for 0.3 ms, then [T]=0. At t=0.3 ms:

$$r_{\text{peak}} \approx \frac{\alpha[T]}{\alpha[T]+\beta} = \frac{1.1}{1.1+0.19} = 0.85$$ $$\text{Decay: } r(t) = 0.85\,e^{-\beta(t-0.3)} = 0.85\,e^{-0.19(t-0.3)} \;\Rightarrow\; \tau_{\text{decay}} = 1/\beta = 5.3 \;\text{ms}$$ $$I_{\text{AMPA}} = g_{\text{max}}\times r(t)\times(V-E_{\text{AMPA}}) = 1 \;\text{nS}\times 0.85\times(-65) = -55 \;\text{pA at peak}$$

▸ References

[DA01]	Dayan & Abbott — Theoretical Neuroscience, MIT Press, 2001. Ch. 5.6–5.8
[Des94]	Destexhe et al. — An efficient method for computing synaptic conductances based on kinetic model, Neural Comput. 6:14, 1994
[Hil01]	Hille — Ion Channels of Excitable Membranes (3rd ed.), Sinauer, 2001. Ch. 6–9

§ 04

Frequently Asked Questions

🔬 SimulationWhat does the Kinetic Scheme tab show for AMPA vs NMDA vs GABA?▼

The Kinetic Scheme tab visualises the state-transition diagram for each receptor type as an animated Markov chain. Circles represent states (C=closed, O=open, D=desensitised, B=blocked); arrows show transitions with rate constants. At each timestep, the thickness of each arrow changes proportionally to the flux between states. For AMPA: C⇌O (2 states, simple). For NMDA: C→C_glu→O⇌O_Mg (4 states including Mg²⁺ block). For GABA_A: C⇌O→D (3 states with desensitisation). Inject a virtual presynaptic spike and watch probability flow through the states in real time.

Key takeaway: Kinetic scheme = Markov chain visualisation. Each state's occupancy changes in real time as ligand binds/unbinds and channels open/close. AMPA: 2 states. NMDA: 4. GABA_A: 3 (with desensitisation).

🧠 ConceptualWhy does NMDA require glycine as a co-agonist — what is the 'glycine site'?▼

The NMDA receptor has TWO ligand-binding sites that must both be occupied for the channel to open: (1) the glutamate site (on the GluN2 subunit) — binds glutamate released from presynaptic terminals; (2) the glycine/serine site (on the GluN1 subunit) — binds glycine or d-serine. Both must be occupied simultaneously for channel gating. In vivo, the glycine site is thought to be tonically saturated by ambient glycine/d-serine in the extracellular space. However, during intense activity, d-serine released by astrocytes can modulate NMDA transmission through the glycine site. Pharmacologically, drugs targeting the glycine site (D-cycloserine, full agonist) are used as cognitive enhancers in schizophrenia treatment; strychnine blocks glycine RECEPTORS (not the NMDA co-agonist site).

Key takeaway: NMDA co-agonist (glycine/d-serine) site on GluN1 must be occupied in addition to the glutamate site on GluN2. Both sites must be occupied for gating. Ambient d-serine from astrocytes regulates this site.

🌍 AppliedHow do anaesthetics and recreational drugs act on AMPA, NMDA, and GABA receptors?▼

Pharmacology of synaptic receptors: Ketamine: open-channel blocker of NMDA (enters through the open pore and blocks it). Sub-anaesthetic doses produce dissociative effects and are used as antidepressants (rapid-acting, via AMPA receptor upregulation). Propofol: potentiates GABA_A (increases τ_decay of the IPSC, similar to barbiturates). At low doses: sedation; at high doses: anaesthesia via enhanced inhibition. Alcohol (ethanol): potentiates GABA_A AND inhibits NMDA — dual mechanism for CNS depression. Benzodiazepines: enhance GABA_A frequency of opening. MDMA: no direct ionotropic receptor action, but elevates extracellular serotonin/dopamine, indirectly modulating AMPA expression. Phencyclidine (PCP): open-channel NMDA blocker, produces schizophrenia-like symptoms by eliminating NMDA-dependent coincidence detection.

Key takeaway: Most CNS drugs target GABA_A (potentiation → sedation/anaesthesia) or NMDA (block → dissociation, schizophrenia model, antidepressant). Understanding kinetic mechanisms explains clinical effects and side effects.

💡 Non-ObviousWhy does synaptic desensitisation cause short-term synaptic depression?▼

Short-term synaptic depression (STD) has TWO components: (1) presynaptic vesicle depletion: the readily releasable pool (RRP) of vesicles is exhausted by high-frequency stimulation; (2) postsynaptic desensitisation: AMPA and GABA_A receptors enter desensitised states during ained transmitter exposure. During a high-frequency burst (50 Hz, 20 ms inter-spike), the second EPSC is smaller because: (a) fewer vesicles released (RRP depleted); (b) AMPA receptors still partially desensitised from the first EPSC. Recovery requires ~100–500 ms (receptor recovery time) and ~1 s (vesicle replenishment). Paradoxically, cyclothiazide (blocks AMPA desensitisation) dramatically increases EPSC amplitude during repetitive stimulation — showing that postsynaptic desensitisation is a major factor.

Key takeaway: STD = presynaptic vesicle depletion + postsynaptic receptor desensitisation. Both components recover on different timescales (ms for receptors, seconds for vesicles). Cyclothiazide eliminates the postsynaptic component.

📐 ComputationalWhat is the 'exponential' approximation for kinetic models and when does it fail?▼

The full kinetic model (Markov chain) can be approximated by the double-exponential g(t) = g_max × (e^{−t/τ_1} − e^{−t/τ_2}) when: (1) the transmitter pulse is brief compared to the channel kinetics; (2) there is no desensitisation; (3) there is no voltage-dependence (OK for AMPA, not NMDA). This approximation fails for: NMDA (Mg²⁺ block introduces voltage-dependence, making g(V,t) not factorisable); GABA_A with desensitisation during repetitive stimulation; scenarios where [T] remains elevated (spillover from nearby synapses). For large-scale simulations (>10,000 neurons), the double-exp approximation is preferred for speed; for single-synapse studies, use the full kinetic model.

Key takeaway: Double-exp ≈ kinetic model for brief [T] pulses, no desensitisation, no voltage-dependence. Fails for NMDA (Mg²⁺ block), GABA_A with desensitisation, or prolonged [T] exposure (spillover).

🎓 DeepWhat are GluN2 subunit variants of NMDA, and why do they matter for development?▼

NMDA receptors are tetramers: 2× GluN1 (required for glycine binding) + 2× GluN2 (A,B,C,D variants — glutamate binding). The GluN2 subunit determines kinetics and Mg²⁺ sensitivity. GluN2B (neonatal, fetal cortex): τ_decay ≈ 300 ms, high Mg²⁺ sensitivity. GluN2A (mature cortex): τ_decay ≈ 100 ms, lower Mg²⁺ sensitivity. The GluN2B→2A switch occurs during postnatal development (2–4 weeks in mice), driven by sensory experience. The switch shortens τ_NMDA, reducing the temporal window for LTP induction — this is thought to close the critical period for visual cortex plasticity. NMDA antagonists (AP5) can re-open the critical period by preventing the 2A upregulation.

Key takeaway: GluN2B (neonatal): τ_decay ≈ 300 ms, long LTP window. GluN2A (mature): τ_decay ≈ 100 ms, shorter window. The developmental 2B→2A switch closes critical periods. This is the molecular basis of experience-dependent plasticity.

§ 04 Best Resources

Gerstner et al. — Neuronal Dynamics, Cambridge, 2014. neuronaldynamics.epfl.ch
Dayan & Abbott — Theoretical Neuroscience, MIT Press, 2001
Izhikevich — Dynamical Systems in Neuroscience, MIT Press, 2007. dynamicalsystems.org
Hille — Ion Channels of Excitable Membranes (3rd ed.), Sinauer, 2001

§ 05

Misconceptions & Common Errors

Sub-block A — Conceptual Misconceptions

❌"All inhibition is hyperpolarising — GABA always pushes V more negative."

✅GABA_A inhibition is only hyperpolarising when V > E_Cl. When V ≤ E_Cl (e.g., E_Cl = −60 mV in immature neurons, or when V has been hyperpolarised by another current), GABA_A is depolarising. In early development (before KCC2 co-transporter is expressed), E_Cl ≈ −40 mV — GABA is strongly depolarising and even excitatory. This is why neonatal seizures respond poorly to benzodiazepines (GABA is excitatory). Additionally, shunting inhibition (E_Cl ≈ V_rest) is inhibitory via conductance (not voltage) — it prevents charge accumulation even without hyperpolarisation.

📖 Hille — Ion Channels (3rd ed.), Ch. 9; Ben-Ari (2002) — GABA in developing neurons, Nat. Rev. Neurosci.

Sub-block B — Numerical Errors

❌Ignoring the Mg²⁺ blocking factor in NMDA current: I_NMDA = g_NMDA × r(t) × (V − E_NMDA) — missing the voltage-dependent denominator.

✅NMDA current must include the Mg²⁺ block: I_NMDA = g_NMDA × r(t) × (V − E_NMDA) / (1 + [Mg]/3.57 × exp(−0.062V)). Without the blocking factor, the simulated NMDA current is ~10× too large at V_rest (−65 mV), and the model shows unrealistic excitability. The Mg²⁺ block reduces NMDA conductance by ~85% at V = −65 mV. Forgetting this produces models where NMDA alone can drive the postsynaptic neuron to fire — which is biologically incorrect under normal conditions.

🔍 Why: Missing Mg²⁺ block makes NMDA current ~10× too large at rest. Always include: g_eff = g_NMDA / (1 + [Mg²⁺]/K × exp(−V/V₀)).

§ 05 References

Gerstner et al. — Neuronal Dynamics, Cambridge, 2014. neuronaldynamics.epfl.ch
Dayan & Abbott — Theoretical Neuroscience, MIT Press, 2001
Hille — Ion Channels of Excitable Membranes (3rd ed.), Sinauer, 2001
Izhikevich — Dynamical Systems in Neuroscience, MIT Press, 2007