The Accumulator Model Simulator

This page summarises the models implemented in the accumulator model simulator, why you might use each one, how the parameters map to psychologically meaningful concepts, and what the parameter sliders do. Have models you want included? Email paul.garrett@unimelb.edu.au.

Open The Accumulator Model Simulator

1 Descriptive RT/Accuracy Models

Descriptive models provide the shape of a response time (RT) or error distribution without relying on psychologically interpretable parameters.

1.1 Bernoulli–Weibull (shifted Weibull RT + Bernoulli accuracy)

Why/when to use.
Generate RT distributions following a flexible hazard (Weibull) function with independent accuracy. Useful for baseline fits, explaining RT or accuracy concepts, and when you want a clean RT shape without committing to a decision process.

What’s unique/useful.
Treats speed and accuracy separately. The reaction time is described by a flexible curve (a Weibull), which can show people getting faster or slower to respond over time depending on the task.

Parameters (app)

k (Weibull shape) — Distribution of RTs (controls the hazard function). If k < 1, people are more likely to respond early; if k ~ 1, responses are steady over time; if k > 1, responses become more likely the longer you wait.
λ (Weibull scale) — Stretches RTs
t0 (non-decision) — Pure shift (encoding/motor)
p (accuracy) — Bernoulli success probability

1.2 Poisson Counter (Gamma race)

Why/when to use.
A discrete-event race: each response channel waits for (N) countable events at rate (λ). Great for showing race logic and how changing the rate or the threshold affects both RT and accuracy.

What’s unique/useful.
You can reason from simple math: faster rates or smaller (N) produce faster responses. Finishing times follow a Gamma distribution, so it’s easy to visualise and teach.

Parameters (app)

N (threshold counts) — Number of events needed to finish (Gamma shape)
λ_c, λ_e (rates) — Event rates for the correct vs error channels
t0 (non-decision) — Adds a fixed time for perception/motor stages

2 Noisy Sequential Sampling Models

Noisy sequential sampling models, like diffusion decision models, accumulate noisy evidence over time. The evidence is one-part deterministic (represented by the drift-rate: quality of information being sampled) and one-part random (represented by Brownian noise added to the deterministic evidence sampled at every step). This evidence noisily accumulates towards a decision boundary, representing the correct or alternative (e.g., error) outcome. The time a boundary is crossed provides the decision finishing time, and which boundary is crossed tells us what decision you made.

2.1 Random Walk (discrete drift to fixed bounds)

Why/when to use.
The ancestor of diffusion models. Shows evidence growing by small steps until hitting an upper/lower bound.

What’s unique/useful.
Transparent control of the speed–accuracy trade-off via step bias and bound height.

Parameters (app)

p (step-up probability) — Bias toward the upper (correct) bound
h (step size) — Size of each evidence step
a (bound) — Decision threshold from 0 to a
z/a (start proportion) — Starting bias as a fraction of a
dt (step time) — Time per step (temporal grain)

2.2 Diffusion Decision Model (DDM; fixed bounds)

Why/when to use.
The most popular 2-choice RT/accuracy model. Jointly explains RT and accuracy distributions.

What’s unique/useful.
One-dimensional noisy accumulation with clean psychological meanings: drift = evidence quality, boundary = caution.

Parameters (app)

v (drift) — Mean evidence rate (quality)
s (diffusion SD) — Within-trial noise scale
a (bound) — Response caution/threshold
z/a (start bias) — Starting point as a fraction of a

2.3 Wiener DDM with across-trial variability

Why/when to use.
Captures skew/heavy tails by letting starting point, drift, and non-decision time vary across trials.

What’s unique/useful.
Keeps the within-trial process the same while adding realistic between-trial variation.

Parameters (app)

a, v, s, z/a, t0 — As in the basic DDM
sz (start range) — Uniform variability in z across trials
sv (drift SD) — Gaussian variability in v across trials
st0 (non-decision range) — Uniform variability in t0

2.4 DDM with Hyperbolically Collapsing Bound(s)

Why/when to use.
Models urgency or deadlines: the bound shrinks over time so late decisions need less evidence to trigger a response.

What’s unique/useful.
Hyperbolic collapse gives a simple, interpretable urgency function with face validity – it matches how people anecdotally describe the feeling of urgency.

Parameters (app)

v, s — As DDM
a0 (initial bound) and a_min (floor) — Starting height and minimum height of the bound
k (collapse rate) — How quickly urgency grows
z/a0 (start bias) — Starting point relative to a0
dt (integration step) — Simulation time step
±a(t) (checkbox) — Collapse both bounds (±) or only the upper correct bound

3 Linear Ballistic Accumulator (LBA) Family

Ballistic models, like the LBA, discard the within-trial noisy steps seen in Noisy Sampling Models and replace this with a linear trajectory that is extremely fast to simulate and analytic to compute. Ballistic models capture the fast and slow error patterns most decision makers care to explain using between-trial noise in the drift rate.

3.1 LBA (fixed bound)

Why/when to use.
A deterministic-within-trial race with only across-trial variability — very fast, closed-form, and easy to explain.

What’s unique/useful.
Linear (ballistic) growth from random starts and random drifts; no within-trial noise. Considered the ‘simplest complete decision model’. Ideal for close-to real-time model fits.

Parameters (app)

v (mean drift of correct channel) — Evidence strength; in this app the error channel uses 1−v (didactic 2-accumulator setup)
s (drift SD) — Across-trial variability (truncated normal)
b (threshold) — Common decision bound
a (start max) — Uniform start range ([0,a])

3.2 LBA with Linear Collapsing Bounds

Why/when to use.
Adds urgency to LBA by lowering the bound over time. In contrast to the hyperbollic DDM, this has been operationalized to collapse linearly (at a fixed rate).

What’s unique/useful.
Provides the simplest mechanism for understanding the effect of urgency on decision outcomes.

Parameters (app)

v, s — As LBA
b0 (initial bound), b_min (floor) — Start and minimum bound heights
k (collapse rate) — How quickly the bound drops
a (start max) — Uniform start range

4 Leaky, Competing, Timed & Mean-Reverting Models

Leaky models describe how evidence stored in memory may decay over time. Competing models describe how multiple options can interact to inhibit or facilitate our choices. Timed models describe how a sense of urgency may build and, in competition with our need to gather information, lead us to a poorly informed decision. Mean-reverting models describe how bias towards one choice or state, can facilitate or inhibit our ability to reach a decision.

4.1 Leaky Competing Accumulator (LCA; 2 accumulators)

Why/when to use.
When options inhibit each other (e.g., competition and distraction) and activation decays (e.g., if evidence stored in memory gets weaker over time).

What’s unique/useful.
Two coupled accumulators with leak (κ) and lateral inhibition (β). Rectification at 0 forces negative numbers to zero, ensuring activation is always positive. Allows decision to time-out when leak or inhibition stop decisional evidence from reaching a boundary.

Parameters (app)

v (target input) and (1−v) (competitor input) — Relative drive to each accumulator
s (noise SD) — Within-trial noise
b (bound), a (max start) — Decision threshold and start range
kappa (leak κ) — Pulls activation back toward 0
beta (inhibition β) — How strongly accumulators suppress each other
dt (step); Max time — Plotting time and max decision time

4.2 MDFT: Multialternative Decision Field Theory

Why/when to use.
Models how people choose from among several options with different attributes, especially when context and similarity matter (e.g., why adding a “decoy” option can change preferences).

What’s unique/useful.
Extends the LCA idea to many channels and includes:

Leak back toward neutral (phi)
Lateral inhibition between options (beta)
A similarity matrix (rho, sigma_s) so similar items compete more
Option-specific drifts and noise (v1, vC, sv, s)

Parameters (app)

M (number of options)
b (bound) — Decision threshold
phi (leak) — Pull back toward 0
beta (inhibition) — Competition strength
rho, sigma_s (similarity) — How much similar options suppress each other
v1, vC, sv (drifts) — Mean evidence for focal and competitor options, and across-trial variability
s (noise SD) — Random fluctuation each step
dt, t0, max_t, dpos — Step size, non-decision time, max-time and choice spacing.

4.3 Ornstein–Uhlenbeck (OU) with fixed bounds

Why/when to use.
Mean-reverting accumulation toward a set-point. Useful for contrasting mean reversion with constant-drift diffusion.

What’s unique/useful.
When reversion is set to zero it behaves like a standard DDM; for larger reversion values, the state is attracted toward a set-point. Reversion is useful to model forgetting, decay, or a drawing-back to baseline, sometime seen in attention and memory tasks.

Parameters (app)

θ (mean reversion rate) — Strength of pull toward μ
μ/a (set-point) — Target level as a proportion of a
s (noise SD) — Within-trial noise
a, z/a, t0, dt — As above

4.4 LDLIV Dual-Race (state-dependent noise)

Why/when to use.
Smith’s Linear Drift, Linear Infinitesimal Variance model explores what happens when the amount of noise changes with the amount of evidence. Each choice channel builds up evidence at a steady rate but also “leaks” back toward baseline. The random variation gets bigger as the evidence grows, so early on it’s fairly stable but later it can wobble more. The process stays at zero or above, so evidence never goes negative - just like in the brain.

What’s unique/useful.
It shows multiplicative noise (noise that scales with the signal), making it ideal for situations where uncertainty increases with stronger signals or counts.

Parameters (app)

b (bound) — Decision threshold
q_c, q_e (inputs) — Drive for correct vs error channels
k (leak) — Pull back toward baseline
σ (noise scale) — Scales sqrt noise
z/b (start proportion), t0, dt — Start bias, non-decision, step size

4.5 TRDM (Timed Racing Diffusion Model)

Why/when to use.
When there is both an evidence process and an internal timer/urgency process racing to decide. TRDM captures decisions that can end either because evidence accumulated to a threshold or because a timer “go” process hit its bound (leading to a guess).

What’s unique/useful.
Three one-boundary diffusions race on each trial: A correct evidence channel, an incorrect (error) evidence channel, and a timer channel. The first channel to cross a decision boundary wins (first passage wins). Because one-boundary Brownian motion with drift has Inverse-Gaussian hitting times, TRDM draws evidence and timer finishing times directly, picking the earliest. This makes it fast and very transparent. It is also distinct from other urgency models that assume your decision boundary changes (see collapsing LBA and DDM for comparison).

Parameters (app).

mE (mean evidence drift) — Baseline drive common to both evidence channels.
dE (evidence sensitivity) — Separates correct vs error drifts.
sE (evidence noise σ_E) — Within-trial diffusion for both evidence channels.
bE (evidence threshold) — Bound for the evidence race.
t0E (evidence non-decision) — Adds a pure time shift to evidence finishing times.
vT (timer drift μ_T) — Average timer speed.
sT (timer noise σ_T) — Within-trial diffusion for the timer.
bT (timer threshold) — Bound for the timer race.
t0T (timer onset) — Timer starts after this latency (e.g., 50 ms).
(Internal) gT (timer guess probability) — If the timer wins, the response is marked correct with probability gT (fixed at 0.5 in the app for 2AFC).

5 Circular/Angular Models

5.1 Circular Diffusion Model (vector 2D diffusion to a circle)

Why/when to use.
For continuous-report/angle tasks. Evidence drifts in 2D toward a direction and stops when it hits a circle; the hit angle is the response.

What’s unique/useful.
Naturally couples a response angle with its RT from the same process.

Parameters (app)

v (drift speed) — Strength of drift toward θ
s (noise SD) — 2D noise
R (radius) — Circular decision boundary
θ (degrees) (target angle) — Direction of evidnece drift

5.2 SCDM: Sinusoidal Field to Bound (didactic variant)

Why/when to use.
A model of population tuning: a sinusoidal/von-Mises template over angle gets drift + noise; decide when the crest reaches b.

What’s unique/useful.
Shows how tuning sharpness (κ) and noise change the growth of a directional “wave” toward a threshold. Allows for multiple racing processes and is not restricted to a circle: it can be applied to any continuous response format.

Parameters (app)

v (drift amplitude) — Strength of the template drive
s (noise SD) — Additive field noise
b (threshold) — Circular bound in activation space
κ (tuning) — κ=0 gives a cosine; larger κ sharpens the peak
M (angular bins) — Discretisation of 0..2π
θ (deg) (target angle) — Template centre
dt (integration step) — Simulation time step

--- title: "The Accumulator Model Simulator" format: html toc: true number-sections: true --- > This page summarises the models implemented in the accumulator model simulator, why you might use each one, how the parameters map to psychologically meaningful concepts, and what the parameter sliders do. Have models you want included? Email paul.garrett@unimelb.edu.au. <div class="launch-row"> <a class="btn btn-primary" href="https://drpaulmgarrett.shinyapps.io/accumlatormodels/" target="_blank" rel="noopener"> Open The Accumulator Model Simulator </a> </div> # Descriptive RT/Accuracy Models Descriptive models provide the shape of a response time (RT) or error distribution without relying on psychologically interpretable parameters. ## Bernoulli–Weibull (shifted Weibull RT + Bernoulli accuracy) **Why/when to use.** Generate RT distributions following a flexible hazard (Weibull) function with independent accuracy. Useful for baseline fits, explaining RT or accuracy concepts, and when you want a clean RT shape without committing to a decision process. **What’s unique/useful.** Treats speed and accuracy separately. The reaction time is described by a flexible curve (a Weibull), which can show people getting faster or slower to respond over time depending on the task. **Parameters (app)** - `k` *(Weibull shape)* — Distribution of RTs (controls the hazard function). If k < 1, people are more likely to respond early; if k ~ 1, responses are steady over time; if k > 1, responses become more likely the longer you wait. - `λ` *(Weibull scale)* — Stretches RTs - `t0` *(non-decision)* — Pure shift (encoding/motor) - `p` *(accuracy)* — Bernoulli success probability  --- ## Poisson Counter (Gamma race) **Why/when to use.** A discrete-event race: each response channel waits for \(N\) countable events at rate \(λ\). Great for showing race logic and how changing the **rate** or the **threshold** affects both RT and accuracy. **What’s unique/useful.** You can reason from simple math: faster rates or smaller \(N\) produce faster responses. Finishing times follow a Gamma distribution, so it’s easy to visualise and teach. **Parameters (app)** - `N` *(threshold counts)* — Number of events needed to finish (Gamma *shape*) - `λ_c`, `λ_e` *(rates)* — Event rates for the correct vs error channels - `t0` *(non-decision)* — Adds a fixed time for perception/motor stages  --- # Noisy Sequential Sampling Models Noisy sequential sampling models, like diffusion decision models, accumulate noisy evidence over time. The evidence is one-part deterministic (represented by the drift-rate: quality of information being sampled) and one-part random (represented by Brownian noise added to the deterministic evidence sampled at every step). This evidence noisily accumulates towards a decision boundary, representing the correct or alternative (e.g., error) outcome. The time a boundary is crossed provides the decision finishing time, and which boundary is crossed tells us what decision you made. ## Random Walk (discrete drift to fixed bounds) **Why/when to use.** The ancestor of diffusion models. Shows evidence growing by small steps until hitting an upper/lower bound. **What’s unique/useful.** Transparent control of the speed–accuracy trade-off via step bias and bound height. **Parameters (app)** - `p` *(step-up probability)* — Bias toward the upper (correct) bound - `h` *(step size)* — Size of each evidence step - `a` *(bound)* — Decision threshold from 0 to `a` - `z/a` *(start proportion)* — Starting bias as a fraction of `a` - `dt` *(step time)* — Time per step (temporal grain)  --- ## Diffusion Decision Model (DDM; fixed bounds) **Why/when to use.** The most popular 2-choice RT/accuracy model. Jointly explains RT and accuracy distributions. **What’s unique/useful.** One-dimensional noisy accumulation with clean psychological meanings: drift = evidence quality, boundary = caution. **Parameters (app)** - `v` *(drift)* — Mean evidence rate (quality) - `s` *(diffusion SD)* — Within-trial noise scale - `a` *(bound)* — Response caution/threshold - `z/a` *(start bias)* — Starting point as a fraction of `a`  --- ## Wiener DDM with across-trial variability **Why/when to use.** Captures skew/heavy tails by letting starting point, drift, and non-decision time vary across trials. **What’s unique/useful.** Keeps the within-trial process the same while adding realistic between-trial variation. **Parameters (app)** - `a, v, s, z/a, t0` — As in the basic DDM - `sz` *(start range)* — Uniform variability in `z` across trials - `sv` *(drift SD)* — Gaussian variability in `v` across trials - `st0` *(non-decision range)* — Uniform variability in `t0`  --- ## DDM with Hyperbolically Collapsing Bound(s) **Why/when to use.** Models urgency or deadlines: the bound shrinks over time so late decisions need less evidence to trigger a response. **What’s unique/useful.** Hyperbolic collapse gives a simple, interpretable urgency function with face validity -- it matches how people anecdotally describe the feeling of urgency. **Parameters (app)** - `v, s` — As DDM - `a0` *(initial bound)* and `a_min` *(floor)* — Starting height and minimum height of the bound - `k` *(collapse rate)* — How quickly urgency grows - `z/a0` *(start bias)* — Starting point relative to `a0` - `dt` *(integration step)* — Simulation time step - `±a(t)` *(checkbox)* — Collapse both bounds (±) or only the upper correct bound  --- # Linear Ballistic Accumulator (LBA) Family Ballistic models, like the LBA, discard the within-trial noisy steps seen in Noisy Sampling Models and replace this with a linear trajectory that is extremely fast to simulate and analytic to compute. Ballistic models capture the fast and slow error patterns most decision makers care to explain using between-trial noise in the drift rate. ## LBA (fixed bound) **Why/when to use.** A deterministic-within-trial race with only across-trial variability — very fast, closed-form, and easy to explain. **What’s unique/useful.** Linear (ballistic) growth from random starts and random drifts; no within-trial noise. Considered the 'simplest complete decision model'. Ideal for close-to real-time model fits. **Parameters (app)** - `v` *(mean drift of correct channel)* — Evidence strength; in this app the error channel uses `1−v` (didactic 2-accumulator setup) - `s` *(drift SD)* — Across-trial variability (truncated normal) - `b` *(threshold)* — Common decision bound - `a` *(start max)* — Uniform start range \([0,a]\)  --- ## LBA with Linear Collapsing Bounds **Why/when to use.** Adds urgency to LBA by lowering the bound over time. In contrast to the hyperbollic DDM, this has been operationalized to collapse linearly (at a fixed rate). **What’s unique/useful.** Provides the simplest mechanism for understanding the effect of urgency on decision outcomes. **Parameters (app)** - `v, s` — As LBA - `b0` *(initial bound)*, `b_min` *(floor)* — Start and minimum bound heights - `k` *(collapse rate)* — How quickly the bound drops - `a` *(start max)* — Uniform start range  --- # Leaky, Competing, Timed & Mean-Reverting Models Leaky models describe how evidence stored in memory may decay over time. Competing models describe how multiple options can interact to inhibit or facilitate our choices. Timed models describe how a sense of urgency may build and, in competition with our need to gather information, lead us to a poorly informed decision. Mean-reverting models describe how bias towards one choice or state, can facilitate or inhibit our ability to reach a decision. ## Leaky Competing Accumulator (LCA; 2 accumulators) **Why/when to use.** When options inhibit each other (e.g., competition and distraction) and activation decays (e.g., if evidence stored in memory gets weaker over time). **What’s unique/useful.** Two coupled accumulators with **leak** \(κ\) and **lateral inhibition** \(β\). Rectification at 0 forces negative numbers to zero, ensuring activation is always positive. Allows decision to time-out when leak or inhibition stop decisional evidence from reaching a boundary. **Parameters (app)** - `v` *(target input)* and `(1−v)` *(competitor input)* — Relative drive to each accumulator - `s` *(noise SD)* — Within-trial noise - `b` *(bound)*, `a` *(max start)* — Decision threshold and start range - `kappa` *(leak κ)* — Pulls activation back toward 0 - `beta` *(inhibition β)* — How strongly accumulators suppress each other - `dt` *(step)*; `Max time` — Plotting time and max decision time  --- ## MDFT: Multialternative Decision Field Theory **Why/when to use.** Models how people choose from among several options with different attributes, especially when context and similarity matter (e.g., why adding a “decoy” option can change preferences). **What’s unique/useful.** Extends the LCA idea to *many* channels and includes: - Leak back toward neutral (`phi`) - Lateral inhibition between options (`beta`) - A similarity matrix (`rho`, `sigma_s`) so similar items compete more - Option-specific drifts and noise (`v1`, `vC`, `sv`, `s`) **Parameters (app)** - `M` *(number of options)* - `b` *(bound)* — Decision threshold - `phi` *(leak)* — Pull back toward 0 - `beta` *(inhibition)* — Competition strength - `rho`, `sigma_s` *(similarity)* — How much similar options suppress each other - `v1`, `vC`, `sv` *(drifts)* — Mean evidence for focal and competitor options, and across-trial variability - `s` *(noise SD)* — Random fluctuation each step - `dt`, `t0`, `max_t`, `dpos` — Step size, non-decision time, max-time and choice spacing.  --- ## Ornstein–Uhlenbeck (OU) with fixed bounds **Why/when to use.** Mean-reverting accumulation toward a set-point. Useful for contrasting mean reversion with constant-drift diffusion. **What’s unique/useful.** When reversion is set to zero it behaves like a standard DDM; for larger reversion values, the state is attracted toward a set-point. Reversion is useful to model forgetting, decay, or a drawing-back to baseline, sometime seen in attention and memory tasks. **Parameters (app)** - `θ` *(mean reversion rate)* — Strength of pull toward `μ` - `μ/a` *(set-point)* — Target level as a proportion of `a` - `s` *(noise SD)* — Within-trial noise - `a`, `z/a`, `t0`, `dt` — As above  --- ## LDLIV Dual-Race (state-dependent noise) **Why/when to use.** Smith’s **Linear Drift, Linear Infinitesimal Variance** model explores what happens when the amount of noise changes with the amount of evidence. Each choice channel builds up evidence at a steady rate but also “leaks” back toward baseline. The random variation gets bigger as the evidence grows, so early on it’s fairly stable but later it can wobble more. The process stays at zero or above, so evidence never goes negative - just like in the brain. **What’s unique/useful.** It shows *multiplicative noise* (noise that scales with the signal), making it ideal for situations where uncertainty increases with stronger signals or counts. **Parameters (app)** - `b` *(bound)* — Decision threshold - `q_c`, `q_e` *(inputs)* — Drive for correct vs error channels - `k` *(leak)* — Pull back toward baseline - `σ` *(noise scale)* — Scales sqrt noise - `z/b` *(start proportion)*, `t0`, `dt` — Start bias, non-decision, step size  --- ## TRDM (Timed Racing Diffusion Model) **Why/when to use.** When there is **both** an evidence process **and** an internal **timer/urgency** process racing to decide. TRDM captures decisions that can end either because evidence accumulated to a threshold *or* because a timer “go” process hit its bound (leading to a guess). **What’s unique/useful.** Three **one-boundary** diffusions race on each trial: A correct evidence channel, an incorrect (error) evidence channel, and a timer channel. The first channel to cross a decision boundary wins (first passage wins). Because one-boundary Brownian motion with drift has **Inverse-Gaussian** hitting times, TRDM draws evidence and timer finishing times directly, picking the earliest. This makes it fast and very transparent. It is also distinct from other urgency models that assume your **decision boundary** changes (see collapsing LBA and DDM for comparison). **Parameters (app).** - `mE` *(mean evidence drift)* — Baseline drive common to both evidence channels. - `dE` *(evidence sensitivity)* — Separates correct vs error drifts. - `sE` *(evidence noise σ_E)* — Within-trial diffusion for both evidence channels. - `bE` *(evidence threshold)* — Bound for the evidence race. - `t0E` *(evidence non-decision)* — Adds a pure time shift to evidence finishing times. - `vT` *(timer drift μ_T)* — Average timer speed. - `sT` *(timer noise σ_T)* — Within-trial diffusion for the timer. - `bT` *(timer threshold)* — Bound for the timer race. - `t0T` *(timer onset)* — Timer starts after this latency (e.g., 50 ms). - *(Internal)* `gT` *(timer guess probability)* — If the **timer** wins, the response is marked correct with probability `gT` (fixed at 0.5 in the app for 2AFC). --- # Circular/Angular Models ## Circular Diffusion Model (vector 2D diffusion to a circle) **Why/when to use.** For continuous-report/angle tasks. Evidence drifts in 2D toward a direction and stops when it hits a circle; the hit angle is the response. **What’s unique/useful.** Naturally couples a **response angle** with its **RT** from the same process. **Parameters (app)** - `v` *(drift speed)* — Strength of drift toward `θ` - `s` *(noise SD)* — 2D noise - `R` *(radius)* — Circular decision boundary - `θ (degrees)` *(target angle)* — Direction of evidnece drift  --- ## SCDM: Sinusoidal Field to Bound (didactic variant) **Why/when to use.** A model of population tuning: a sinusoidal/von-Mises template over angle gets drift + noise; decide when the crest reaches `b`. **What’s unique/useful.** Shows how **tuning sharpness** (κ) and **noise** change the growth of a directional “wave” toward a threshold. Allows for multiple racing processes and is not restricted to a circle: it can be applied to any continuous response format. **Parameters (app)** - `v` *(drift amplitude)* — Strength of the template drive - `s` *(noise SD)* — Additive field noise - `b` *(threshold)* — Circular bound in activation space - `κ` *(tuning)* — κ=0 gives a cosine; larger κ sharpens the peak - `M` *(angular bins)* — Discretisation of 0..2π - `θ (deg)` *(target angle)* — Template centre - `dt` *(integration step)* — Simulation time step