Ionic current model of rabbit retinal horizontal cell

A multi-scale simulation of retinal physiology

We present a detailed physiological model of the (human) retina that includes the biochemistry and electrophysiology of phototransduction, neuronal electrical coupling, and the spherical geometry of the eye. The model is a parabolic-elliptic system of partial differential equations based on the mathematical framework of the bi-domain equations, which we have generalized to account for multiple cell-types. We discretize in space with non-uniform finite differences and step through time with a custom adaptive time-stepper that employs a backward differentiation formula and an inexact Newton method. A refinement study confirms the accuracy and efficiency of our numerical method. Numerical simulations using the model compare favorably with experimental findings, such as desensitization to light stimuli and calcium buffering in photoreceptors. Other numerical simulations suggest an interplay between photoreceptor gap junctions and inner segment, but not outer segment, calcium concentration. Applications of this model and simulation include analysis of retinal calcium imaging experiments, the design of electroretinograms, the design of visual prosthetics, and studies of ephaptic coupling within the retina.

A general pattern of non-spiking neuron dynamics under the effect of potassium and calcium channel modifications

Electrical activity of excitable cells results from ion exchanges through cell membranes, so that genetic or epigenetic changes in genes encoding ion channels are likely to affect neuronal electrical signaling throughout the brain. There is a large literature on the effect of variations in ion channels on the dynamics of spiking neurons that represent the main type of neurons found in the vertebrate nervous systems. Nevertheless, non-spiking neurons are also ubiquitous in many nervous tissues and play a critical role in the processing of some sensory systems. To our knowledge, however, how conductance variations affect the dynamics of non-spiking neurons has never been assessed. Based on experimental observations reported in the biological literature and on mathematical considerations, we first propose a phenotypic classification of non-spiking neurons. Then, we determine a general pattern of the phenotypic evolution of non-spiking neurons as a function of changes in calcium and potassium conductances. Furthermore, we study the homeostatic compensatory mechanisms of ion channels in a well-posed non-spiking retinal cone model. We show that there is a restricted range of ion conductance values for which the behavior and phenotype of the neuron are maintained. Finally, we discuss the implications of the phenotypic changes of individual cells at the level of neuronal network functioning of the C. elegans worm and the retina, which are two non-spiking nervous tissues composed of neurons with various phenotypes.

Survey of electrically evoked responses in the retina - stimulus preferences and oscillation among neurons

Electrical stimulation is an important tool in neuroscience research and clinically. In the retina, extensive work has revealed how the retinal ganglion cells respond to extracellular electrical stimulation. But little is known about the responses of other neuronal types, and more generally, how the network responds to stimulation. We conducted a survey of electrically evoked responses, over a range of pulse amplitudes and pulse widths, for 21 cell types spanning the inner two layers of the rabbit retina. It revealed: (i) the evoked responses of some neurons were charge insensitive; (ii) pulse-width sensitivity varied between cell types, allowing preferential recruitment of cell types; and (iii) 10-20 Hz damped oscillations across retinal layers. These oscillations were generated by reciprocal excitatory / inhibitory synapses, at locations as early as the cone-horizontal-cell synapses. These results illustrate at cellular resolution how a network responds to extracellular stimulation, and could inform the development of bioelectronic implants for treating blindness.

Simulation of rabbit A-type retinal horizontal cell that generates repetitive action potentials

Rabbit A-type retinal horizontal cells are classified into at least two types: cells that generate repetitive action potentials and those that do not. A mathematical model of these two types of cells based on the ionic current mechanisms has been proposed. Although the response of the former cell to 1-s positive current injection has so far been investigated by both electrophysiological and computational approaches, how this cell responds to much longer current injection has not been investigated. In the present study, I use this model to investigate the response of the former cell to a much longer current injection. Computer simulation indicates that when the stimulation period is relatively short, the membrane potential returns to the resting potential after current injection. In this case, the cell can repeatedly generate repetitive action potentials in response to stimulation. In contrast, when the stimulation period is relatively long, the membrane potential does not return to the resting potential but maintains the depolarized level even after current injection. In this case, the cell cannot repeatedly generate repetitive action potentials. When the present results were compared with those of the previous study, the difference in the response pattern after stimulation between the two types of horizontal cells was revealed.

Modulation of A-type potassium currents in retinal horizontal cells by extracellular calcium and zinc

Extracellular Ca2+ and Zn2+ influence many aspects of retinal function. Here, we examined the effect of external Ca2+ and Zn2+ on potassium channels of retinal horizontal cells. When extracellular Ca2+ was lowered from 3 mM to 0.3 mM, horizontal cell transient outward currents elicited by voltage steps from resting membrane potential (−70 mV) were decreased by approximately 50%, whereas the sustained currents remained unchanged. This effect was due to a hyperpolarizing shift in the steady-state inactivation curve of A-type K+ currents when extracellular Ca2+ concentration was lowered. The mean half inactivation potential of the steady-state inactivation curves was hyperpolarized from −56.3 ± 4.7 mV in 3 mM Ca2+ to −76.4 ± 3.9 mV in 0.3 mM Ca2+. Neither the state-steady activation curve nor the kinetics of inactivation was significantly changed in low extracellular Ca2+. The addition of 30 μM Zn2+ restored peak outward currents in 0.3 mM Ca2+. The half inactivation voltages were depolarized from −70 ± 2.8 mV in 0.3 mM Ca2+ to −56 ± 2.6 mV in 0.3 mM Ca2+ plus 30 μM Zn2+. Taken together, the results indicate that external Ca2+ and Zn2+ maintain the activity of A-type potassium channels in retinal horizontal cells by influencing the voltage dependence of steady-state inactivation.

Simulation analysis of receptive-field size of retinal horizontal cells by ionic current model

The size of the receptive field of retinal horizontal cells changes with the state of dark/light adaptation. We have used a mathematical model to determine how changes in the membrane conductance affect the receptive-field properties of horizontal cells. We first modeled the nonlinear membrane properties of horizontal cells based on ionic current mechanisms. The dissociated horizontal cell model reproduced the voltage–current (V–I) relationships for various extracellular glutamate concentrations measured in electrophysiological studies. Second, a network horizontal cell model was also described, and it reproduced theV–Irelationship observedin vivo. The network model showed a bell-shaped relationship between the receptive-field size and constant glutamate concentration. The simulated results suggest that the calcium current is a candidate for the bell-shaped length constant relationship.