— Synaptic Conductance

§ 01

Interactive Simulation

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Parameter 15.0

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Input I5.0

T sim (ms)500

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§ 02

The Idea, Step by Step

▸ From a Squirt of Chemical to a Current

START — Cracking a Window (middle school)

When one neuron talks to the next, it does not send electricity straight across the gap. It squirts tiny chemical messengers that briefly pop open little doors — ion channels — in the receiving cell. Each open door lets charged particles trickle through, then the door slowly swings shut. Picture cracking a window: open it and a little air rushes in, then it eases closed. How far those doors are open, moment by moment, is the synaptic conductance. More doors open at once means a bigger whoosh of charge; as they close, the whoosh fades away.

BUILD — Naming the Opening (high school)

Call the opening $g_{\text{syn}}(t)$, measured in nanosiemens (nS). After a spike it shoots up fast, peaks, then fades. The simplest description is the alpha function $g_{\text{syn}}(t)=g_{\max}\,\frac{t}{\tau}\,e^{\,1-t/\tau}$. One worked number: with $g_{\max}=1$ nS and $\tau=2$ ms, the conductance peaks exactly at $t=\tau=2$ ms and reaches $g_{\max}=1$ nS. The current that actually flows is $I_{\text{syn}}=g_{\text{syn}}\,(V-E_{\text{syn}})$ — large when the doors are wide open and when the membrane voltage $V$ sits far from the channel's reversal voltage $E_{\text{syn}}$.

DEEPEN — Two Clocks and a Sign (AP / college)

Real synapses open and close on two different clocks, so we use a double exponential $g_{\text{syn}}(t)\propto e^{-t/\tau_1}-e^{-t/\tau_2}$ with $\tau_1>\tau_2$: the fast $\tau_2$ sets the rise, the slow $\tau_1$ sets the decay. The sign of $(V-E_{\text{syn}})$ decides the synapse's job. With $E_{\text{syn}}=0$ mV (AMPA) and $V=-65$ mV the current is inward and excitatory; with $E_{\text{syn}}\approx-70$ mV (a GABA-A synapse) near rest there is little voltage push, so it mostly shunts. On the panel, Parameter 1 and Parameter 2 act as the rise and decay constants, Input I scales the peak $g_{\max}$, and T sim sets how long a window you watch.

TRY THIS — In the Sim Above

(1) Make the rise and decay constants nearly equal and watch the waveform turn symmetric — that is the alpha-function limit. (2) Push the decay constant much larger and see the long, slow tail of an NMDA-like synapse, the kind that sustains working memory. (3) Move the reversal toward the resting voltage and notice the current nearly vanishes even while the doors are still open — that is shunting inhibition in action.

§ 03

Equation Derivation

▸ Synaptic Conductance Models

Synaptic transmission is modelled as a transient conductance change. Two standard waveform models are the alpha function and the double-exponential (biexponential) model.

$$\text{Alpha function:}\quad g_{\text{syn}}(t) = g_{\max}\,\frac{t}{\tau}\,e^{1-t/\tau}\,\Theta(t)$$ $$\text{Double exponential:}\quad g_{\text{syn}}(t) = g_{\max}\,\frac{\tau_1\tau_2}{\tau_1-\tau_2}\left(e^{-t/\tau_1}-e^{-t/\tau_2}\right)\Theta(t), \quad \tau_1 > \tau_2$$

Synaptic current into postsynaptic neuron:

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

Synapse type	E_syn (mV)	τ_rise (ms)	τ_decay (ms)	Function
AMPA	0	0.2–0.5	2–5	Fast excitation
NMDA	0	5–10	50–200	Slow excitation, coincidence detection, LTP
GABA_A	−70	0.5–1	5–10	Fast inhibition (shunting or hyperpolarising)
GABA_B	−90	50–100	200–500	Slow inhibition (K⁺ channel)

▸ Derivation Steps

STEP 1 — Alpha Function Origin

The alpha function g(t) = g_max (t/τ) e^{1−t/τ} Θ(t) is the impulse response of a first-order kinetic scheme: C → O with rate α, O → C with rate β. At t=0 (presynaptic spike), channels begin opening with time constant 1/α. They reach peak conductance at t=τ, then decay as channels close with time constant 1/β ≈ τ. Peak value = g_max at t = τ (verified: d/dt[(t/τ)e^{1-t/τ}] = 0 at t=τ). The factor e ensures g_max is achieved exactly at t=τ.

STEP 2 — Double Exponential: Separate Rise and Decay

The alpha function assumes τ_rise = τ_decay. Real synapses have separate rise and decay timescales. The double-exponential: g(t) ∝ e^{-t/τ_1} − e^{-t/τ_2} (τ_1 > τ_2) has: peak time t_peak = τ_1τ_2 ln(τ_1/τ_2)/(τ_1−τ_2), rise driven by τ_2, decay driven by τ_1. For AMPA: τ_rise ≈ 0.3 ms, τ_decay ≈ 3 ms. For NMDA: τ_rise ≈ 8 ms, τ_decay ≈ 100 ms. The normalisation factor τ_1τ_2/(τ_1−τ_2) ensures g(t_peak) = g_max.

STEP 3 — NMDA Voltage Dependence (Mg²⁺ Block)

NMDA receptors have a unique property: they are blocked by extracellular Mg²⁺ at hyperpolarised potentials. The conductance is: g_NMDA(V) = g_NMDA_max × g_syn(t) × 1/(1 + [Mg²⁺]/3.57 × exp(−0.062 V)). At V = −70 mV: Mg²⁺ block ≈ 85% reduction. At V = 0 mV: Mg²⁺ block ≈ 0%. This makes NMDA a coincidence detector: it requires BOTH presynaptic glutamate release (to bind glutamate site) AND postsynaptic depolarisation (to relieve Mg²⁺ block). This is the cellular basis of Hebbian LTP.

STEP 4 — Synaptic Integration

The postsynaptic membrane integrates multiple synaptic inputs. For the LIF model: C_m dV/dt = −g_L(V−E_L) + Σ_i g_i(t)(V−E_i). This is now a non-autonomous linear ODE (coefficients change with time). The effective time constant τ_eff = C_m/(g_L + Σ g_i(t)) becomes shorter when many synapses are active (conductance shunting). Large synaptic inputs can "clamp" V near E_syn — a phenomenon called conductance-based inhibition that is stronger than the simple current model.

▸ Worked Example — EPSC Peak and Charge

AMPA synapse: g_max = 1 nS, E_syn = 0 mV, V_rest = −65 mV, τ_1 = 3 ms, τ_2 = 0.3 ms. Peak time:

$$t_{\text{peak}} = \frac{\tau_1\tau_2}{\tau_1-\tau_2}\ln\frac{\tau_1}{\tau_2} = \frac{3\times0.3}{3-0.3}\ln\frac{3}{0.3} = \frac{0.9}{2.7}\times 2.303 = 0.767 \;\text{ms}$$ $$g_{\text{peak}} = 1 \;\text{nS}, \quad I_{\text{peak}} = g_{\text{peak}}\times(V-E_{\text{syn}}) = 1\times(-65-0) = -65 \;\text{pA (inward)}$$ $$Q = \int_0^\infty g(t)(V-E_{\text{syn}})dt \approx g_{\max}(V-E_{\text{syn}})(\tau_1-\tau_2) \approx -65\times 2.7 \approx -175 \;\text{fC}$$

▸ References

[Rall67]	Rall — Distinguishing theoretical synaptic potentials, J. Neurophysiol. 30:1138, 1967 (alpha function origin)
[DA01]	Dayan & Abbott — Theoretical Neuroscience, MIT Press, 2001. Ch. 5.5–5.7
[Ger14]	Gerstner et al. — Neuronal Dynamics, Cambridge, 2014. Ch. 3. neuronaldynamics.epfl.ch

§ 04

Frequently Asked Questions

🔬 SimulationWhat does the Synapse Waveform tab show, and how do I compare alpha vs double-exponential?▼

The Synapse Waveform tab shows g_syn(t) as a function of time after a presynaptic spike at t=0. The alpha function (single-τ) produces a symmetric rise-and-decay around the peak. The double-exponential (τ_rise ≠ τ_decay) produces an asymmetric waveform — fast rise, slow decay, which more accurately matches recorded EPSCs. Toggle between the two models and observe: for τ_1/τ_2 = 10 (e.g., AMPA: 3/0.3 ms), the double-exp shows a sharp onset followed by gradual decay. The NMDA tab shows the Mg²⁺ block voltage dependence — the conductance rises as V depolarises, implementing coincidence detection.

Key takeaway: Alpha function: rise τ = decay τ (symmetric). Double-exp: separate τ_rise and τ_decay (asymmetric, more realistic). NMDA: additionally voltage-gated by Mg²⁺ block.

🧠 ConceptualWhy is NMDA called a 'coincidence detector' — what exactly does it detect?▼

NMDA channels require TWO simultaneous conditions to conduct: (1) glutamate bound to the receptor (from presynaptic release — 'the presynaptic neuron fired'); (2) the postsynaptic membrane depolarised (to >−50 mV) to relieve Mg²⁺ block — 'the postsynaptic neuron is also depolarised.' Condition 2 is met when the postsynaptic neuron is also firing (or receiving many simultaneous EPSPs). Thus NMDA responds to the logical AND of (pre fires) AND (post fires). This AND gate is exactly Hebb's synaptic learning rule — it detects the co-activity of pre and post that drives LTP. NMDA is the molecular implementation of Hebbian plasticity.

Key takeaway: NMDA = molecular AND gate: conducts only when pre fires (glutamate) AND post is depolarised (Mg²⁺ block relieved). This coincidence detection is the cellular basis of Hebbian LTP.

🌍 AppliedHow do synaptic time constants affect network oscillations and computation?▼

Synaptic time constants set the timescale of neural integration. Fast AMPA (τ_decay ≈ 3 ms) drives fast gamma oscillations (30–80 Hz) because the E-I loop completes in ~20 ms. Slow NMDA (τ_decay ≈ 100 ms) enables working memory — the slow decay creates persistent activity ('reverberant excitation') that can maintain a memory trace for hundreds of ms without continuous input. GABA_A (τ_decay ≈ 8 ms) provides fast feedback inhibition that sharply times spike windows. GABA_B (τ_decay ≈ 300 ms) provides slow hyperpolarisation that gates cortical down-states. In BCI applications, selective NMDA antagonists (ketamine) produce a pharmacological model of schizophrenia by disrupting working memory, demonstrating the critical role of synaptic time constants in cognition.

Key takeaway: AMPA fast → gamma; NMDA slow → working memory; GABA_A fast → sharp timing; GABA_B slow → down-states. Synaptic τ sets the timescale of information integration and storage.

💡 Non-ObviousWhat is 'conductance-based inhibition' and why is it stronger than current-based inhibition?▼

In the current-based (LIF) model, inhibition subtracts a fixed current: I_inh = −g_inh × (V−E_inh) evaluated at a fixed V. But in the conductance-based model, g_inh(t) changes the effective input resistance: C_m dV/dt = −(g_L + g_inh)(V − E_eff) where E_eff = (g_L E_L + g_inh E_inh)/(g_L + g_inh). Large g_inh clamps V close to E_inh ≈ −70 mV AND reduces the membrane time constant τ_eff = C_m/(g_L + g_inh). This shortens the integration window — excitatory inputs arriving during strong inhibition produce smaller EPSPs (shunting inhibition). Shunting inhibition is thus more powerful than subtraction: it not only shifts V negative but also 'shorts' the membrane, preventing charge accumulation.

Key takeaway: Conductance-based inhibition shunts the membrane resistance, reducing both V and the integration time constant. It is more powerful than current subtraction: it prevents charge accumulation even when E_inh ≈ V_rest.

📐 ComputationalWhat is the most efficient way to implement thousands of synapses in simulation?▼

For large networks, compute ALL synaptic currents as a single matrix operation. Represent synaptic weights as a matrix W (N×N), presynaptic spike trains as a vector S(t) (N×1), and synaptic conductances as G(t) = W × S(t) convolved with the alpha/double-exp kernel. Using sparse matrix representations (most W[i,j] = 0 for unconnected pairs), the cost scales with number of synapses, not N². For the alpha function update: at each timestep, G += −G/τ × dt; when a pre-spike occurs: G += w_ij. This is the event-driven update used in NEST and Brian2 simulators — only active synapses require computation.

Key takeaway: Large networks: represent synapses as sparse weight matrix, use event-driven updates (only update G when a spike occurs). Cost = O(N_spikes × N_connections/neuron), not O(N²) per timestep.

🎓 DeepHow does synaptic conductance relate to the cable equation — do distal synapses contribute less?▼

Yes — cable theory predicts that a distal synapse delivers less voltage to the soma (e^{−L} attenuation). But conductance-based synapses have an additional effect: a distal conductance change also reduces the local membrane resistance. This means: a distal EPSC produces both less somatic voltage (attenuated) AND a local shunt that reduces the space constant λ, further reducing signals from even-more-distal synapses. Rall showed that the effective charge transfer from a distal synapse to the soma scales as Q_soma ≈ Q_syn × e^{−L} × (R_soma/R_soma + R_path). This 'impedance mismatch' means distal synapses have much lower current injection efficiency at the soma than proximal ones — motivating the dendritic amplification mechanisms (NMDA spikes, voltage-gated channels) that compensate.

Key takeaway: Distal synapses attenuate both voltage (e^{−L}) and effective charge transfer (impedance mismatch). Active dendritic conductances (NMDA, Ca²⁺) compensate this by amplifying distal EPSPs locally.

§ 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

❌"EPSP and EPSC are the same thing — one is just the current, the other the voltage."

✅EPSP and EPSC are related but distinct: EPSC (postsynaptic current) = g_syn(t) × (V−E_syn) — depends only on conductance waveform and driving force, measured in voltage clamp at fixed V. EPSP (postsynaptic potential) = the voltage change driven by the EPSC through the membrane impedance — measured in current clamp, depends on g_syn(t), V(t), R_m, C_m, and the dendritic location of the synapse. EPSP shape ≠ EPSC shape: the EPSP is slower and smaller than the EPSC because C_m filters the fast EPSC current. The EPSC is a rectangle integrated by the membrane RC circuit to produce the smoother EPSP.

📖 Dayan & Abbott — Theoretical Neuroscience, Ch. 5.5; Johnston & Wu — Foundations of Cellular Neurophysiology, Ch. 8

Sub-block B — Numerical Errors

❌Using E_syn = −70 mV for GABA_A inhibition always — at V_rest = −65 mV, GABA_A depolarises rather than hyperpolarising.

✅GABA_A acts through Cl⁻ channels with E_Cl ≈ −70 mV. When V_rest > E_Cl (e.g., V_rest = −65 mV): GABA_A produces a small depolarising current (Cl⁻ flows out, positive charge out → depolarises slightly). When V_rest < E_Cl (e.g., V_rest = −75 mV): GABA_A hyperpolarises. In many neurons and in early development, E_Cl ≈ V_rest or even > V_rest → GABA_A is depolarising. The inhibitory effect of GABA_A at V_rest ≈ E_Cl comes from shunting (conductance increase reduces R_m), not hyperpolarisation. Always check whether V_rest < or > E_Cl before assuming GABA_A hyperpolarises.

🔍 Why: GABA_A E_syn depends on intracellular Cl⁻ concentration via Nernst — not fixed at −70 mV. In immature neurons and some interneurons, E_Cl > V_rest → GABA is depolarising.

§ 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