Perception#

Bistable perception: how an ambiguous stimulus flips between two interpretations.

A Necker cube, a rotating silhouette or an ambiguous object seen by a robot’s camera is perceived as one of two interpretations, which alternate. The time spent in one interpretation (the dwell time) is the standard measurement.

Models.

QuantumZenoBistableModel (Atmanspacher, Filk and Römer, Biological Cybernetics 90, 33-40, 2004): the percept is a qubit whose two basis states are the two interpretations. Between two “observations” (internal sampling of the percept, every \(\Delta t\)) the state rotates freely, \(U = e^{-i g \Delta t\, \sigma_x}\); each observation collapses it. The probability of a switch at one observation is \(q = \sin^2(g\,\Delta t)\), so dwell times are geometric with mean \(\Delta t / \sin^2(g \Delta t) \approx 1 / (g^2 \Delta t)\): more frequent observation slows the switching (the quantum Zeno effect).
MarkovSwitchingModel: classical baseline; switches occur at a constant rate \(r\), so dwell times are exponential and do not depend on the observation interval.
GammaRenewalModel: the standard empirical description of dwell times (Levelt), a gamma distribution with shape \(k\) and scale \(\theta\).

What distinguishes them. With one observation interval the Zeno and Markov models predict the same (geometric or exponential) shape; the gamma model fits the typical unimodal histograms better. The distinctive prediction of the Zeno model is the dependence on \(\Delta t\): halving the observation interval should double the mean dwell time. Design conditions with different intervals (for example different presentation or attention-sampling rates) to test it.

Design. {condition: (dt, bin_edges)} with the observation interval and the histogram bins (in seconds); built with dwell_time_design(). Outcomes: the probability of a dwell time falling in each bin (multinomial, with a final bin for longer dwell times). Data: dwell-time counts per bin, built from raw dwell times with dwell_counts().

Classes

`QuantumZenoBistableModel`	Quantum Zeno model of bistable perception.
`MarkovSwitchingModel`	Classical baseline: switches at a constant rate (exponential dwell times, no Zeno effect).
`GammaRenewalModel`	Classical baseline: gamma-distributed dwell times (the usual empirical description).

Functions

`dwell_time_design`	Design with equal-width dwell-time bins for several observation intervals.
`dwell_counts`	Histogram of dwell times, with a final bin for times beyond the last edge.
`mean_dwell_time`	Mean dwell time predicted by a bistable-perception model.

class QuantumZenoBistableModel(**params)[source]#

Bases: Model

Quantum Zeno model of bistable perception.

Parameters:: g (float) – Free switching frequency (rad/s) of the unobserved percept.

switch_probability(dt)[source]#

Probability of a switch at one observation.

Parameters:: dt (float) – Observation interval.
Returns:: float – \(\sin^2(g\,\Delta t)\).

mean_dwell(dt)[source]#

Mean dwell time \(\Delta t / \sin^2(g \Delta t)\).

Parameters:: dt (float)
Returns:: float

predict(design=None)[source]#: Dwell-time bin probabilities for each condition (geometric in units of dt).

class MarkovSwitchingModel(**params)[source]#

Bases: Model

Classical baseline: switches at a constant rate (exponential dwell times, no Zeno effect).

Parameters:: rate (float) – Switching rate (1/s); the mean dwell time is 1 / rate.

mean_dwell(dt=None)[source]#

Mean dwell time 1 / rate (independent of the observation interval).

Parameters:: dt (float, optional) – Ignored.
Returns:: float

predict(design=None)[source]#: Dwell-time bin probabilities (exponential distribution) for each condition.

class GammaRenewalModel(**params)[source]#

Bases: Model

Classical baseline: gamma-distributed dwell times (the usual empirical description).

Parameters:

shape (float) – Gamma shape \(k\) (k = 1 is the exponential).
scale (float) – Gamma scale \(\theta\) (seconds); the mean is \(k\theta\).

mean_dwell(dt=None)[source]#

Mean dwell time \(k\theta\) (independent of the observation interval).

Parameters:: dt (float, optional) – Ignored.
Returns:: float

predict(design=None)[source]#: Dwell-time bin probabilities (gamma distribution) for each condition.

dwell_time_design(dts, max_time=20.0, n_bins=20)[source]#

Design with equal-width dwell-time bins for several observation intervals.

Parameters:

dts (dict or sequence) – {condition: dt} or a sequence of intervals (conditions named 'dt=<value>').
max_time (float, optional) – Upper edge of the last finite bin, in seconds.
n_bins (int, optional) – Number of finite bins (one more bin collects longer dwell times).

Returns:

dict – {condition: (dt, bin_edges)}.

dwell_counts(dwell_times, edges)[source]#

Histogram of dwell times, with a final bin for times beyond the last edge.

Parameters:

dwell_times (array_like) – Observed dwell times in seconds.
edges (array_like) – Bin edges.

Returns:

numpy.ndarray – Counts, length len(edges).

mean_dwell_time(model, dt)[source]#

Mean dwell time predicted by a bistable-perception model.

Parameters:

model (QuantumZenoBistableModel, MarkovSwitchingModel or GammaRenewalModel)
dt (float) – Observation interval (ignored by the Markov and gamma models).

Returns:

float – Seconds.