Analysis of “a computational Model of Electrical Stimulation of the Retinal Ganglion Cell” Introduction



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Lorne Nelson, Daniel Wilhelm, Mike Scott, Adam Lugowski

Math 490N

Prof. Cowen

Analysis of “A Computational Model of Electrical Stimulation of the Retinal Ganglion Cell”

Introduction


Approximately 1.2 million people suffer from retinitis pigmentosa, a type of blindness characterized by photoreceptor degeneration (Greenberg et al. 1999). The clinical treatment of blindness generally relies on the use of alternate, functioning sensory modalities including audition and touch to compensate for the loss of vision. There has been some experimentation in grafting functioning photoreceptors into the retina (Humayun et al. 1996). However, due to the large number and complex connectivity of photoreceptors, as well as technical challenges in the actual procedure, this treatment currently appears unfeasible. Further, an alternative experimental approach has been to stimulate portions of the visual cortex (Humayun et al. 1996). This procedure also presents limitations, as the processing of visual information takes place in various locations of the cortex, and surgical access of the brain imposes its own risks. In cases of blindness resulting from the loss of photoreceptor function, the retinal ganglion cells remain intact. It has been shown that even without consistent stimulation the retinal ganglion cells experience little or no transsynaptic degeneration (Humayun et al. 1996). This allows for the possibility of electrical stimulation of existing ganglion cells by inserting a fine stimulating electrode into the eye positioned proximally to ganglion cells to elicit vision (Humayun et al. 1996).

It has been shown that a resolution of approximately 25x25 pixels is capable of eliciting a restricted visual field that is sufficient to permit mobility (Humayun et al. 1996). However, the anatomical organization of the ganglion cells on the retina complicates the ability to determine which ganglion cells are preferentially stimulated. The large area covered by ganglion cells with respect to the retina and the axonal convergence focused to the optic disc inhibits purposeful selection of a specific ganglion cell. Alternatively, determination of the threshold value at various locations on a retinal ganglion cell provides a method to determine which portion of the cell is most easily stimulated. The ability to discern which segments of the ganglion cells are most easily stimulated is crucial to achieve experimental repeatability and essential resolution. In the paper A Computational Model of Electrical Stimulation of the Retinal Ganglion Cell by R.J. Grrenberg, T.J. Velte, M.S. Humayun, G.N. Scarlatis, and E. de Juan Jr., a number of computational retinal ganglion cell models are employed to more precisely resolve which segments of retinal ganglion cells are most easily stimulated.

Essential to the development of these computational models of a retinal ganglion cells are the cellular properties that influence threshold values. The threshold value is the value of the membrane potential at which the influx of sodium exceeds the efflux of potassium resulting in an action potential. In general, this value is represented by a depolarization of ten to twenty millivolts. This value can be influenced by various neuronal properties. The density of voltage gated sodium ion channels for a given area of membrane dictates that area’s contribution to achieve the threshold depolarization. The greater the density of such ion channels, the greater the contribution to reaching the threshold depolarization. Generally, the axon hillock is considered to possess the greatest density of voltage gated sodium ion channels. Additionally important is the sensitivity of voltage gated sodium channels to a change in membrane voltage. The greater the sensitivity, the more rapidly the channel will be modulated and drive the membrane toward the threshold potential. The presence of myelin at or near the point of external stimulation can inhibit a threshold depolarization in response to an external electrical stimulation. In this particular model, myelin is not important, as ganglion cells within the retina are not myelinated. Utilizing these biological factors in conjunction with computational constructs, three models were developed and investigated. These three distinct models, closely based on actual cell morphology, include a linear passive model, a Hodgkin-Huxley active model with passive dendrites, and an all-active model utilizing five nonlinear ion channels with varying distributions and densities.

Specifics of the Model


A retinal ganglion cell with a long axon and a large dentritic field was chosen. For the majority of the calculations and simulations, a soma with 24μm diameter was used. However, some testing with a 10μm diameter soma was done in order to see if the size of the soma changed the ease with which an action potential could be elicited when externally stimulated. Empirical data was recorded regarding some of the neurons inherent physical attributes. These attributes are considered constant with regard to spatial and temporal changes within the cell. These values are as follows: membrane capacitance (Cm = 1μF/cm2) and membrane resistance (Rm = 50KΩcm2) (Greenberg et al. 1999).

In order to determine the part of the neuron that is easiest to stimulate, a method of dividing the neuron into different segments was necessary. For this reason the soma and cell membrane were segmented into compartments. An illustration of this compartmentalizing is nicely shown in Figure 1. The retinal ganglion cell was segmented into more than 9000 such compartments (Greenberg et al. 1999). The actual sizes of the compartments, as well as the time increments used in the math model for each compartment, were altered with respect to threshold. Compartment size and time increments were separately reduced until cross-compartment changes in threshold did not exceed 1%. Average compartment size was 1μm and average time increments of 25μs were used.





Figure 2

Figure 1



Each compartment was modeled as shown in Figure 3. The compartments consist of a membrane mechanism in parallel with a membrane capacitance. A resistance found between compartments is labeled as Ra. As mentioned earlier in this paper three models were used to describe the membrane mechanisms: a linear passive model, a Hodgkin-Huxley model with passive dendrites (HH), and an all active model (FCM) with five nonlinear ion channels distributed at varying densities. The linear passive mechanism simply reduces each compartment to a simple RmCm circuit with a leak modeled as a battery at -70mV (Greenberg et al. 1999). The second type of mechanism, HH, is the typical description given by Hodgkin and Huxley included in the simulation package NEURON. For this particular mechanism the dendrites are considered passive so Hodgkin-Huxley channel equations are only applied to the axon and soma.


Figure 3


The third mechanism model, FCM model, consists of five ion channels dealing mainly with sodium, calcium and potassium. Each of the five channels has a conductance value associated with it based on what kind of channel it is. The five conductances are: a sodium conductance gNa, a calcium conductance gCa, a delayed rectifier potassium conductance gK, an inactivating potassium conductance gA, and an activating calcium activated potassium conductance gK,Ca. All of the channels except for the channel with conductance gK,Ca are voltage gated (Greenberg et al. 1999). For this mechanism, the membrane potential was modeled according to equation (1) where the parameter E represents the membrane potential. The rate constants m, h, c, n, a, and hA, vary with time and are described by equation (2). The internal calcium concentration, which controls the gK,Ca channel also varies with time and is described in equation (3) (Greenberg et al. 1999).





In order to apply a voltage to the neuron, two types of sources were used across all models: a monopolar point source and a disk electrode source. In both cases the pulse was administered to the neuron at a height of 30μm for 100μs. Each of the three membrane mechanisms were simulated with both sources.



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