Our sensory systems transform external signals into neural activity, thereby
producing percepts. We are endowed with an intuitive notion of similarity
between percepts, that need not reflect the proximity of the physical
properties of the corresponding external stimuli. The quantitative
characterization of the geometry of percepts is therefore an endeavour that
must be accomplished behaviorally. Here we characterized the geometry of color
space using discrimination and matching experiments. We proposed an
individually tailored metric defined in terms of the minimal chromatic
difference required for each observer to differentiate a stimulus from its
surround. Next, we showed that this perceptual metric was particularly adequate
to describe two additional experiments, since it revealed the natural symmetry
of perceptual computations. In one of the experiments, observers were required
to discriminate two stimuli surrounded by a chromaticity that differed from
that of the tested stimuli. In the perceptual coordinates, the change in
discrimination thresholds induced by the surround followed a simple law that
only depended on the perceptual distance between the surround and each of the
two compared stimuli. In the other experiment, subjects were asked to match the
color of two stimuli surrounded by two different chromaticities. Again, in the
perceptual coordinates the induction effect produced by surrounds followed a
simple, symmetric law. We conclude that the […]
In the mean field integrate-and-fire model, the dynamics of a typical neuron
within a large network is modeled as a diffusion-jump stochastic process whose
jump takes place once the voltage reaches a threshold. In this work, the main
goal is to establish the convergence relationship between the regularized
process and the original one where in the regularized process, the jump
mechanism is replaced by a Poisson dynamic, and jump intensity within the
classically forbidden domain goes to infinity as the regularization parameter
vanishes. On the macroscopic level, the Fokker-Planck equation for the process
with random discharges (i.e. Poisson jumps) are defined on the whole space,
while the equation for the limit process is on the half space. However, with
the iteration scheme, the difficulty due to the domain differences has been
greatly mitigated and the convergence for the stochastic process and the firing
rates can be established. Moreover, we find a polynomial-order convergence for
the distribution by a re-normalization argument in probability theory. Finally,
by numerical experiments, we quantitatively explore the rate and the asymptotic
behavior of the convergence for both linear and nonlinear models.