Response of the difference-of-Gaussians model to circular drifting-grating patches.

Forty years ago R.W. Rodieck introduced the Difference-of-Gaussians (DOG) model, and this model has been widely used by the visual neuroscience community to quantitatively account for spatial response properties of cells in the retina and lateral geniculate nucleus following visual stimulation. Circular patches of drifting gratings are now regularly used as visual stimuli when probing the early visual system, but for this stimulus type the mathematical evaluation of the DOG-model response is significantly more complicated than for moving bars, full-field drifting gratings, or circular flashing spots. Here we derive mathematical formulas for the DOG-model response to centered circular patch gratings. The response is found to be given as the difference between two summed series, where each term in the series involves the confluent hypergeometric function. This function is available in commonly used mathematical software, and the results should thus be readily applicable. Example results illustrate how a strong surround suppression in area-summation curves for iso-luminant circular spots may be reversed into a surround enhancement for circular patch gratings. They also show that the spatial-frequency response changes from band-pass to low-pass when going from the full-field grating situation to the situation where the patch covers only the receptive-field center.

Linear mechanistic models for the dorsal lateral geniculate nucleus of cat probed using drifting-grating stimuli.

Experiments with sinusoidal visual stimuli in the early visual pathway have traditionally been interpreted in terms of descriptive filter models. We present an alternative mechanistic approach for interpretation of this type of data recorded from X cells in the dorsal lateral geniculate nucleus (dLGN) of cat. A general, linear, rate-based mathematical expression for the geniculate transfer ratio, i.e. the ratio between the first-harmonic components of the output of a geniculate relay cell and its retinal input, is derived. In linear theory this ratio is independent of the signal processing occurring at the retinal level. Further, the ratio is straightforwardly accessible in experiments due to the presence of S-potentials, representing the retinal input, in extracellular recordings from dLGN. The expression accounts for feedforward inputs from retina and intrageniculate interneurons as well as feedback inputs from cortex and the thalamic reticular nucleus and can be used to experimentally test different mechanistic models for the geniculate circuitry. Two examples of this are considered: a purely feedforward model incorporating relay cell inputs from retinal ganglion cells and interneurons, and a model including cortical feedback inhibition of relay cells via intrageniculate interneurons.

Stochastic resonance in neuron models: endogenous stimulation revisited.

The paradigm of stochastic resonance (SR)-the idea that signal detection and transmission may benefit from noise-has met with great interest in both physics and the neurosciences. We investigate here the consequences of reducing the dynamics of a periodically driven neuron to a renewal process (stimulation with reset or endogenous stimulation). This greatly simplifies the mathematical analysis, but we show that stochastic resonance as reported earlier occurs in this model only as a consequence of the reduced dynamics.

Noise in integrate-and-fire neurons: from stochastic input to escape rates.

We analyze the effect of noise in integrate-and-fire neurons driven by time-dependent input and compare the diffusion approximation for the membrane potential to escape noise. It is shown that for time-dependent subthreshold input, diffusive noise can be replaced by escape noise with a hazard function that has a gaussian dependence on the distance between the (noise-free) membrane voltage and threshold. The approximation is improved if we add to the hazard function a probability current proportional to the derivative of the voltage. Stochastic resonance in response to periodic input occurs in both noise models and exhibits similar characteristics.

Markov analysis of stochastic resonance in a periodically driven integrate-and-fire neuron.

We model the dynamics of the leaky integrate-and-fire neuron under periodic stimulation as a Markov process with respect to the stimulus phase. This avoids the unrealistic assumption of a stimulus reset after each spike made in earlier papers and thus solves the long-standing reset problem. The neuron exhibits stochastic resonance, both with respect to input noise intensity and stimulus frequency. The latter resonance arises by matching the stimulus frequency to the refractory time of the neuron. The Markov approach can be generalized to other periodically driven stochastic processes containing a reset mechanism.