Neurotransmitter Diffusion Calculator

Inputs

Choose geometry and release mode, then set diffusion, distance, and time. Continuous release uses numerical integration for a realistic pulse.

Example data table

Illustrative outputs using the instantaneous model, k = 0, and typical microscale parameters.

Formula used

This calculator uses the diffusion equation with optional first‑order decay:

∂C/∂t = D ∇²C − k C

For an instantaneous point release in n dimensions:

C(r,t) = Q · (4πDt)^−n/2 · exp(−r²/(4Dt)) · exp(−kt)

For continuous release at constant rate R over duration τ:

C(r,t) = ∫₀^min(t,τ) R · (4πD(t−u))^−n/2 · exp(−r²/(4D(t−u))) · exp(−k(t−u)) du

The spread scale is summarized by the RMS displacement:

r_RMS = √(2 n D t)

How to use this calculator

1. Select geometry that matches your diffusion environment.
2. Choose instantaneous or continuous release mode.
3. Enter D, distance r, and time t with proper units.
4. Add decay rate k if enzymatic breakdown is relevant.
5. Provide Q for instantaneous, or R and τ for continuous.
6. Press Calculate to view results above the form.
7. Use CSV or PDF buttons to export the latest result.

Professional article

1) Why diffusion modeling matters

Neurotransmitter signaling is fast, spatially localized, and strongly shaped by transport physics. A diffusion model translates release size, distance, and time into an expected concentration profile. That profile connects to receptor binding probability, sensor response, and timing jitter in electrophysiology, imaging, and microfluidic assays.

2) Synaptic length scales

Typical synaptic cleft widths are on the order of tens of nanometers, while receptor clusters and perisynaptic regions extend into the sub‑micrometer range. Because the Gaussian term uses r²/(4Dt), changing distance from 0.1 µm to 0.5 µm can shift peak concentration by orders of magnitude at early times.

3) Diffusion coefficients and units

Small transmitters in aqueous environments often fall around D ≈ 0.2–0.8 µm²/ms, while larger peptides can be lower depending on viscosity and crowding. This calculator accepts µm²/ms, µm²/s, cm²/s, or m²/s and converts internally to SI units to keep the mathematics consistent.

4) Geometry choices: 1D, 2D, and 3D

Geometry changes the normalization (4πDt)^−n/2. In 3D, concentration dilutes most rapidly because volume expands with radius. In 2D, dilution is slower, and in 1D the spread is along a line. Use 3D for bulk extracellular space, 2D for membrane‑adjacent diffusion, and 1D for channel‑like constraints.

5) First‑order decay and clearance

Enzymatic breakdown, uptake, and binding can be approximated by a first‑order decay term exp(−kt). A decay rate of k = 10 s⁻¹ corresponds to a 1/e lifetime of 0.1 s, while k = 1000 s⁻¹ represents millisecond‑scale clearance. This factor reduces concentrations uniformly in time at every distance.

6) Instantaneous versus continuous release

An instantaneous release models a vesicle‑like packet with total amount Q. Continuous release models a constant rate R over a duration τ, which is useful for sustained exocytosis, uncaging, or microinjection pulses. The continuous solution is computed with numerical integration over the release window, capturing how later‑released molecules have less time to spread.

7) Threshold interpretation and time‑to‑hit

The threshold tool estimates when a target level is reached at distance r for the instantaneous model. In 3D, you can specify thresholds in mol/m³ or nM; for reference, 1 nM equals 10⁻⁶ mol/m³. If the curve does not cross the threshold within the search window, adjust Q, r, D, or the threshold value.

8) Practical workflow and sensitivity

Start with your best D estimate, then explore sensitivity by varying D ±25% and distance across the receptor field. Compare the RMS displacement √(2nDt) to your geometry scale; when r is much larger than RMS, early concentrations will be extremely small. Export CSV/PDF to document assumptions alongside experimental conditions.

FAQs

1) What does the calculator output represent?

It returns the predicted concentration at distance r and time t for your chosen geometry and release mode. Outputs include SI units and a microscale unit, plus RMS displacement as a spread summary.

2) Which geometry should I pick for synapses?

Use 3D for free diffusion in extracellular space. Use 2D when diffusion is effectively confined near a surface. Use 1D for narrow channels or strong constraints along one axis.

3) How do I choose D?

Pick D from literature or calibration experiments for your transmitter and medium. If unsure, sweep a plausible range (for example 0.2–0.8 µm²/ms) and report sensitivity rather than a single number.

4) What does the decay rate k model?

k approximates clearance processes such as uptake, enzymatic degradation, or irreversible binding. Increasing k reduces concentration over time at every distance. If you do not expect strong clearance, set k to zero.

5) Why is continuous mode slower to compute?

Continuous mode integrates contributions from many release times between 0 and min(t,τ). More integration steps improve accuracy, especially for short times, but require more computation in the browser request.

6) Can I interpret 3D concentration as molarity?

Yes. In 3D, molecules/m³ are converted to mol/m³ and mol/L (M). The calculator also reports nM for convenience. For 1D and 2D, “concentration” is per length or area.

7) Why might the time‑to‑threshold show “Not bracketed”?

The solver needs the concentration curve to cross your threshold within its search range. If it never crosses, reduce the threshold, increase Q, decrease r, increase D, or include less decay.