A multiscale continuum model of the vertebrate outer retina: The temporal dynamics of background-induced flicker enhancement.

The retina is a part of the central nervous system that is accessible, well documented, and studied by researchers spanning the clinical, experimental, and theoretical sciences. Here, we mathematically model the subcircuits of the outer plexiform layer of the retina on two spatial scales: that of an individual synapse and that of the scale of the receptive field (hundreds to thousands of synapses). To this end we formulate a continuum spine model (a partial differential equation system) that incorporates the horizontal cell syncytium and its numerous processes (spines) within cone pedicles. With this multiscale modeling approach, detailed biophysical mechanisms at the synaptic level are retained while scaling up to the receptive field level. As an example of its utility, the model is applied to study background-induced flicker enhancement in which the onset of a dim background enhances the center flicker response of horizontal cells. Simulation results, in comparison with flicker enhancement data for square, slit, and disk test regions, suggest that feedback mechanisms that are voltage-axis modulators of cone calcium channels (for example, ephaptic and/or pH feedback) are robust in capturing the temporal dynamics of background-induced flicker enhancement. The value and potential of this continuum spine approach is that it provides a framework for mathematically modeling the input-output properties of the entire receptive field of the outer retina while implementing the latest models for transmission mechanisms at the synaptic level.

Drift-diffusion simulation of the ephaptic effect in the triad synapse of the retina.

Experimental evidence suggests the existence of a negative feedback pathway between horizontal cells and cone photoreceptors in the outer plexiform layer of the retina that modulates the flow of calcium ions into the synaptic terminals of cones. However, the underlying mechanism for this feedback is controversial and there are currently three competing hypotheses: the ephaptic hypothesis, the pH hypothesis, and the GABA hypothesis. The goal of this investigation is to demonstrate the ephaptic hypothesis by means of detailed numerical simulations. The drift-diffusion (Poisson-Nernst-Planck) model with membrane boundary current equations is applied to a realistic two-dimensional cross-section of the triad synapse in the goldfish retina to verify the existence of strictly electrical feedback, as predicted by the ephaptic hypothesis. The effect on electrical feedback from the behavior of the bipolar cell membrane potential is also explored. The computed steady-state cone calcium transmembrane current-voltage curves for several cases are presented and compared with experimental data on goldfish. The results provide convincing evidence that an ephaptic mechanism can produce the feedback effect seen in experiments. The model and numerical methods presented here can be applied to any neuronal circuit where dendritic spines are invaginated in presynaptic terminals or boutons.

A simple spatiotemporal rabies model for skunk and bat interaction in northeast Texas.

We formulate a simple partial differential equation model in an effort to qualitatively reproduce the spread dynamics and spatial pattern of rabies in northeast Texas with overlapping reservoir species (skunks and bats). Most existing models ignore reservoir species or model them with patchy models by ordinary differential equations. In our model, we incorporate interspecies rabies infection in addition to rabid population random movement. We apply this model to the confirmed case data from northeast Texas with most parameter values obtained or computed from the literature. Results of simulations using both our skunk-only model and our skunk and bat model demonstrate that the model with overlapping reservoir species more accurately reproduces the progression of rabies spread in northeast Texas.

Electrodiffusion model simulation of the potassium channel.

The drift-diffusion (Poisson-Nernst-Planck) model is applied to the potassium channel in a biological membrane plus surrounding solution baths. Two-dimensional cylindrically symmetric simulations of the K channel in KCl solutions are presented which show significant boundary layers at the ends of the channel and display the spreading of charge into the bath regions. The computed current-voltage curve shows excellent agreement with experimental measurements. In addition, the response of the K channel to time-dependent applied voltages is investigated.

Electrodiffusion model simulation of rectangular current pulses in a voltage-biased biological channel.

Numerical methods are presented for simulating stochastic-in-time current pulses for an electrodiffusion model of the biological channel, with a fixed applied voltage across the channel. The electrodiffusion model consists of the parabolic advection-diffusion equation coupled either to Gauss' law or Poisson's equation, depending on the choice of boundary conditions. The TRBDF2 method is employed for the advection-diffusion equation. The rectangular wave shape of previously simulated traveling wave current pulses is preserved by the full set of partial differential equations for electrodiffusion.