Mathematical Neuroscience and Applications |

Our sensory systems transform external signals into neural activity, therebyproducing percepts. We are endowed with an intuitive notion of similaritybetween percepts, that need not reflect the proximity of the physicalproperties of the corresponding external stimuli. The quantitativecharacterization of the geometry of percepts is therefore an endeavour thatmust be accomplished behaviorally. Here we characterized the geometry of colorspace using discrimination and matching experiments. We proposed anindividually tailored metric defined in terms of the minimal chromaticdifference required for each observer to differentiate a stimulus from itssurround. Next, we showed that this perceptual metric was particularly adequateto describe two additional experiments, since it revealed the natural symmetryof perceptual computations. In one of the experiments, observers were requiredto discriminate two stimuli surrounded by a chromaticity that differed fromthat of the tested stimuli. In the perceptual coordinates, the change indiscrimination thresholds induced by the surround followed a simple law thatonly depended on the perceptual distance between the surround and each of thetwo compared stimuli. In the other experiment, subjects were asked to match thecolor of two stimuli surrounded by two different chromaticities. Again, in theperceptual coordinates the induction effect produced by surrounds followed asimple, symmetric law. We conclude that the individually-tailored notion […]

In the mean field integrate-and-fire model, the dynamics of a typical neuronwithin a large network is modeled as a diffusion-jump stochastic process whosejump takes place once the voltage reaches a threshold. In this work, the maingoal is to establish the convergence relationship between the regularizedprocess and the original one where in the regularized process, the jumpmechanism is replaced by a Poisson dynamic, and jump intensity within theclassically forbidden domain goes to infinity as the regularization parametervanishes. On the macroscopic level, the Fokker-Planck equation for the processwith 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, withthe iteration scheme, the difficulty due to the domain differences has beengreatly mitigated and the convergence for the stochastic process and the firingrates can be established. Moreover, we find a polynomial-order convergence forthe distribution by a re-normalization argument in probability theory. Finally,by numerical experiments, we quantitatively explore the rate and the asymptoticbehavior of the convergence for both linear and nonlinear models.