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Short-Term Synaptic Plasticity

Tsodyks-Markram Model · Facilitation · Depression · Frequency Filtering

🧠 Tier: Standard Undergraduate · Synaptic Dynamics · Temporal Filtering
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§ 01
Interactive Simulation
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§ 02
The Idea, Step by Step
▸ From a snack bowl at a party to the Tsodyks-Markram equations

Think of a synapse as a snack bowl that the sending neuron refills slowly. Every time a signal (a spike) arrives, the neuron scoops some snacks across the gap to the next cell. If spikes come in a fast burst, the bowl runs low and each later scoop is smaller — the synapse gets weaker the more it is used. That is short-term depression. Surprisingly, some synapses do the opposite: a quick run of spikes "warms up" the release so each scoop gets bigger — that is facilitation. Either way, the synapse briefly remembers how recently and how often it fired, for a few milliseconds up to about a second.

Two numbers capture this. Let $x$ be how full the bowl is (the fraction of ready-to-release vesicles, between 0 and 1) and let $u$ be how big a scoop each spike takes (the release probability). The signal delivered to the next neuron is just their product, $R = u\,x$. Each spike removes $u\,x$ of the resources, so $x$ drops; between spikes the bowl refills toward full with time constant $\tau_D$. Try a number: start full ($x=1$) with a medium scoop $u=0.5$. The first spike delivers $R = 0.5\times1 = 0.5$ and leaves $x = 0.5$. If the next spike lands before the bowl recovers (say $x\approx0.53$), it delivers only $R\approx0.27$ — about half the first. That shrinking train is depression you can see.

Precisely, between spikes $\frac{dx}{dt}=\frac{1-x}{\tau_D}$ and $\frac{du}{dt}=-\frac{u}{\tau_F}$, while every spike updates $u \leftarrow u + U(1-u)$ and then $x \leftarrow x - u\,x$. Which behaviour wins is a race between two clocks: a long $\tau_D$ with a tiny $\tau_F$ gives depression (a low-pass filter that favours slow inputs), whereas a tiny $\tau_D$ with a long $\tau_F$ lets $u$ pile up spike after spike — facilitation (a high-pass filter, or "burst detector"). The steady-state strength at firing rate $f$ falls off as $1/(1+f\,U\,\tau_D)$, with a cutoff near $f_c = 1/(U\,\tau_D)$.

Try this in the sim above: (1) Switch the preset to Default (a depressing synapse) and push Input I up — higher drive stands in for a faster spike train, where depression bites hardest. (2) Compare a Variant preset and open the Time Series tab to watch whether successive responses shrink (depression) or grow (facilitation). (3) Lengthen T sim and notice how, given enough rest between bursts, the bowl refills and the very next response bounces back to full — the "novelty" the synapse reacts to.

§ 03
Equation Derivation
▸ Tsodyks-Markram (TM) Model (1997)

The Tsodyks-Markram model describes short-term synaptic plasticity using three variables: synaptic resources (x), utilisation (u), and effective synaptic strength (R_eff = u×x).

$$\text{Between spikes:}\quad \frac{dx}{dt} = \frac{1-x}{\tau_D}, \quad \frac{du}{dt} = -\frac{u}{\tau_F}$$ $$\text{At each presynaptic spike: } \quad u \leftarrow u + U(1-u), \quad x \leftarrow x - u\,x, \quad R_{\text{eff}} = u\,x$$

The postsynaptic conductance pulse amplitude is proportional to R_eff. Three synapse types:

TypeUτ_D (ms)τ_F (ms)Behaviour
Depressing (D)0.58000First EPSC largest; subsequent smaller
Facilitating (F)0.0401000EPSCs grow with each spike in a burst
Intermediate (I)0.2570020Initial depression, then recovery
▸ Derivation Steps
STEP 1 — Resource Variable x
x ∈ [0,1] represents the fraction of releasable vesicles (readily releasable pool, RRP). At each spike, a fraction u×x is released: x decreases by u×x → x_new = x(1−u). Between spikes, x recovers toward 1 with time constant τ_D (recovery from depletion): dx/dt = (1−x)/τ_D. This gives exponential recovery: x(t) = 1 − (1−x_min) × e^{−t/τ_D}. Large τ_D (slow recovery) → strong depression at high frequencies.
STEP 2 — Utilisation Variable u
u is the utilisation of synaptic efficacy — the fraction of the available pool released per spike (release probability). At each spike u jumps up toward 1: u_new = u + U(1−u), so the first spike of a rested synapse uses exactly u = U (this is why the worked example below starts at u = U). Between spikes u decays back toward its resting level (≈0) with time constant τ_F: du/dt = −u/τ_F → u(t) = u_spike × e^{−t/τ_F}. When τ_F is large (slow decay), the u left over from one spike has not yet decayed when the next arrives, so u accumulates spike after spike → facilitation. When τ_F ≈ 0 (fast decay), u collapses between spikes and every spike merely re-sets it to U → no facilitation, and transmission is governed purely by depletion of x. (Note the convention: u relaxes toward 0, matching du/dt = −u/τ_F in §03 above; the equivalent Mongillo–Tsodyks form relaxes u toward U instead.)
STEP 3 — Frequency Filtering
The TM model implements a frequency-dependent filter on synaptic transmission. Depressing synapses (large τ_D, small τ_F): low-frequency inputs pass through strongly; high-frequency bursts are attenuated (x depletes). Depressing synapses act as low-pass filters. Facilitating synapses (small τ_D, large τ_F): low-frequency inputs are weak (small u); high-frequency bursts facilitate (u accumulates). Facilitating synapses act as high-pass filters. This differential frequency filtering enables synapses to compute temporal features of their presynaptic spike trains.
STEP 4 — Steady-State Response
For a regular spike train at frequency f: at steady state, x_ss × u_ss = U / (1 + f × U × τ_D). As f → ∞: R_eff → 0 (complete depression). As f → 0: R_eff → U (resting release probability). The steady-state effective strength normalised by its single-pulse value: R_ss = 1/(1 + f × U × τ_D). This gives a cutoff frequency f_c = 1/(U × τ_D) below which synaptic transmission is near maximum.

▸ Worked Example — Depressing Synapse Train

Depressing synapse: U=0.5, τ_D=800 ms, τ_F=0, I.S.I.=50 ms (20 Hz). Initial conditions: x=1, u=U=0.5.

$$\text{Spike 1: }R_1 = u_1\cdot x_1 = 0.5\times 1.0 = 0.5, \quad x_1^+ = 1-0.5 = 0.5$$ $$\text{Recovery (50 ms): }x_2^- = 1-(1-0.5)e^{-50/800} = 1-0.5\times 0.94 = 0.530$$ $$\text{Spike 2: }R_2 = 0.5\times 0.530 = 0.265, \quad \text{(53\% of R}_1\text{)}$$ $$\text{Steady state: }R_{ss} = \frac{U}{1+20\times 0.5\times 0.8} = \frac{0.5}{1+8} = 0.056 \;\text{(11\% of R}_1\text{)}$$
▸ References
[TM97]Tsodyks & Markram — The neural code between neocortical pyramidal neurons depends on neurotransmitter release probability, PNAS 94:719, 1997
[Mar98]Markram et al. — Differential signaling via the same axon of neocortical pyramidal neurons, PNAS 95:5323, 1998
[Abb04]Abbott & Regehr — Synaptic computation, Nature 431:796, 2004
§ 04
Frequently Asked Questions
🔬 SimulationWhat does the Frequency Filter tab show for depressing vs facilitating synapses?▼
The Frequency Filter tab plots steady-state effective synaptic strength R_ss as a function of presynaptic firing rate f. Depressing synapse (D-type): R_ss decreases monotonically with f — high-frequency inputs are attenuated (low-pass filter). Cutoff f_c ≈ 1/(U×τ_D) ≈ 1/(0.5×0.8s) = 2.5 Hz. Above 2.5 Hz, each subsequent EPSC is smaller than the last. Facilitating synapse (F-type): R_ss increases with f, saturating at high frequency — low-frequency inputs barely activate the synapse (u stays near U=0.04), but high-frequency bursts accumulate u → strong transmission (high-pass filter). The Burst Detection tab shows how facilitating synapses specifically respond to bursts: single spikes → near-zero; burst of 5 → large EPSC.
Key takeaway: Depressing = low-pass filter (attenuates high frequency, passes low). Facilitating = high-pass / burst detector (weak single spikes, strong bursts). Both implement temporal filtering of presynaptic spike trains.
🧠 ConceptualWhat is the computational function of short-term synaptic depression?▼
Short-term depression (STD) implements gain control and novelty detection. Gain control: depressing synapses automatically reduce their strength during ained activity, preventing saturation of the postsynaptic neuron. The effective gain scales as 1/(1 + f/f_c) — a divisive normalisation. This matches the observed response compression in sensory neurons. Novelty detection: the first stimulus in a sequence evokes the largest response; subsequent identical stimuli are attenuated. Novel stimuli restore the response (vesicle recovery during silence). This is the synaptic basis of 'mismatch negativity' and oddball detection in auditory cortex — unexpected sounds produce a larger response than expected ones because expected stimuli depress their synapses.
Key takeaway: STD implements divisive gain control (prevents saturation) and novelty detection (first event is largest; repeated events are suppressed). Novelty restores the response when stimulus changes.
🌍 AppliedHow is short-term plasticity relevant to working memory, motor control, and psychiatric disorders?▼
Facilitating synapses (L5 pyramidal → L5 pyramidal) are abundant in prefrontal cortex and appear critical for working memory: the facilitation builds up a 'memory trace' in the synapse during a burst, allowing the synapse to 'remember' recent activity. Loss of this facilitation (via stress-induced CRF receptor activation) has been proposed as a synaptic mechanism of working memory impairment in PTSD and schizophrenia. In motor control, depressing synapses at corticospinal→spinal motor neuron synapses implement gain control during repetitive movements — high-frequency cortical commands are automatically scaled down to prevent motor overflow. In epilepsy, loss of depressing plasticity at excitatory synapses removes a natural brake on high-frequency bursting.
Key takeaway: Facilitating synapses in PFC support working memory traces; depressing synapses implement gain control in sensory/motor systems. Pathological loss of STD removes natural brakes on network excitability.
💡 Non-ObviousWhy does a facilitating synapse respond to the RATE of firing rather than just the total number of spikes?▼
u (utilisation) decays with time constant τ_F between spikes. If spikes are separated by intervals >> τ_F, u resets completely to U — each spike is 'forgotten.' Only spikes within τ_F of each other accumulate u. This makes the synapse sensitive to the temporal pattern, not just total spike count. Two input patterns with 10 spikes each: (a) evenly spaced at 1 Hz → each spike has same small R_eff; (b) 10 spikes in 100 ms burst → u builds up, R_eff increases dramatically. The synapse effectively computes a temporal integral of recent spiking weighted by τ_F — a temporal matched filter for bursts. This is why facilitating synapses are often called 'burst detectors.'
Key takeaway: Facilitating synapses detect temporal clustering, not spike count. Two trains with same spike count but different temporal pattern produce different postsynaptic responses because u accumulates only within a τ_F window.
📐 ComputationalHow do you implement the TM model efficiently in a large network simulation?▼
The TM model requires tracking (u, x) per synapse. For N neurons with K synapses each: N×K state variables. Update: (1) between spikes, integrate dx/dt = (1−x)/τ_D and du/dt = (U−u)/τ_F exactly: x(t+Δt) = 1 − (1−x) × e^{−Δt/τ_D}; u(t+Δt) = U + (u−U) × e^{−Δt/τ_F}. These exponential updates are exact regardless of Δt — no numerical error. (2) At each spike: u += U(1−u); R_eff = u × x; x −= u × x; postsynaptic conductance += w × R_eff. For sparse networks (K ≪ N), track only the (u,x) of active synapses. NEST and Brian2 implement TM models natively.
Key takeaway: TM model: exact exponential update for (u,x) between spikes; event-driven update at each spike. O(1) per spike per synapse. Exact integration eliminates numerical error in the exponential recovery.
🎓 DeepWhat is the connection between short-term plasticity and the Bayesian brain hypothesis?▼
Depressing synapses implement a form of predictive coding: when the same stimulus repeats, its synaptic transmission weakens, as if the brain is 'expecting' it and allocating less processing. Novel stimuli (unexpected) have full synaptic strength. This can be formalised: a depressing synapse computes the empirical mean of recent input: the postsynaptic current ∝ input − (predicted input from history). This is the basis of the 'repetition suppression' or 'mismatch negativity' signals used as neural signatures of prediction error in Bayesian brain models. The TM model has been shown to implement the optimal Bayesian filter for tracking a Poisson input process with unknown rate — the steady-state R_eff is proportional to 1/(1 + f_estimated × U × τ_D), which is the inverse of the estimated input rate.
Key takeaway: Depressing synapses implement predictive coding: repeated inputs are suppressed; novel inputs are amplified. Mathematically equivalent to a Bayesian filter estimating input rate. The mismatch negativity (EEG) reflects synaptic depression-based prediction error.
§ 04 Best Resources
§ 05
Misconceptions & Common Errors
Sub-block A — Conceptual Misconceptions
❌"Short-term depression and long-term depression (LTD) are the same phenomenon — both weaken synapses."
✅STP (short-term plasticity) and LTP/LTD (long-term plasticity) are mechanistically and functionally distinct. STP: lasts milliseconds to seconds; purely presynaptic (vesicle depletion + receptor desensitisation); reversible on the same timescale; does not require protein synthesis or gene expression. LTD: lasts hours to days/lifetime; requires NMDA activation + specific Ca²⁺ signalling cascades + AMPA receptor internalisation + sometimes gene expression; not reversible without LTP induction. STP is a dynamic filter; LTD is a persistent memory trace. A strongly depressing synapse (large STP depression) does NOT have reduced 'baseline' weight — it is still capable of full transmission after sufficient rest.
📖 Tsodyks & Markram (1997) PNAS; Abbott & Regehr (2004) Nature 431:796
Sub-block B — Numerical Errors
❌Updating x before u at each spike: x_new = x − U×x; u_new = u + U(1−u) — wrong order.
✅In the TM model, u must be updated FIRST at each spike (u reflects the release probability that WILL BE used), then x is updated: (1) u += U×(1−u); (2) R_eff = u × x; (3) x −= u × x. If you update x first, the depletion uses the OLD u value, but then u is updated — the effective release calculated for the postsynaptic response uses the wrong u. The correct order: update u (new utilisation) → compute R_eff = u × x (how much is released) → deplete x by R_eff. This order is explicitly specified in Tsodyks & Markram (1997) and Markram et al. (1998).
🔍 Why: In TM model: update u first, then compute R = u×x, then deplete x by R. Wrong order: x first → R calculated with old u → incorrect EPSC amplitude.
§ 05 References