Conceived and designed the experiments: RP RFO ACR. Performed the experiments: RP RFO. Analyzed the data: RP ACR. Wrote the paper: RP RFO ACR.
The authors have declared that no competing interests exist.
Recent works suggest that one of the roles of gap junctions in sensory systems is to enhance their dynamic range by avoiding early saturation in the first processing stages. In this work, we use a minimal conductance-based model of the ON rod pathways in the vertebrate retina to study the effects of electrical synaptic coupling via gap junctions among rods and among AII amacrine cells on the dynamic range of the retina. The model is also used to study the effects of the maximum conductance of rod hyperpolarization activated current I
One of the mechanisms used by the retina to operate over a wide range of brightness conditions is signal segregation into distinct pathways, all of which converge to the output layer of ganglion cells. In the mammalian retina, two different rod pathways are responsible for carrying rod signals to on-center (ON) ganglion cells
Experimental and modeling works on photoreceptors of vertebrate species have shown that electrical coupling among rods improves the signal-to-noise ratio and extends the dynamic range of the rod output to rod bipolar cells
Theoretical works on a sensory layer of electrically coupled excitable elements like e.g. the photoreceptor layer in the retina have shown that coupling enhances the dynamic range of the layer and the response of this system is well-fitted by a power law, where the output is proportional to a power (
In this paper, we used a minimal conductance-based model of the two ON rod processing pathways of the vertebrate retina to investigate the effects of cell coupling via gap junctions with different connectivity degrees on the dynamic range of the retina. In particular, we studied the effects of variable connectivity degrees among two different cell populations of the retina, namely rods and AII amacrine cells. In our model, each cell population is represented by a two-dimensional square array and the connectivity degree of each layer is defined as the number of connections, on the average, that each cell makes with its first neighbors.
We used conductance-based neuronal models because they allow the investigation of the interacting effects of network connectivity degree and cellular properties on the dynamic range of the retina. The rod hyperpolarization activated current I
We studied the effects of combinations of different connectivity degrees of rods (
The retina model is made of conductance-based models of the following cells: rods, cones, rod and cone bipolar cells, AII amacrine cells and a ganglion cell. These models were adapted from previously published models. The values of passive properties and ionic current parameters for these cell models, which were modified by us for this work, are given in
Diameter | Lenght | C |
V |
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Rod | 8 (µm) | 8 (µm) | 20 (pF) | −38 (mV) |
Cone | 8 (µm) | 8 (µm) | 20 (pF) | −42 (mV) |
Bipolar cell | 8 (µm) | 8 (µm) | 10 (pF) | −38 (mV) |
AII amacrine cell | 7 (µm) | 7 (µm) | 20 (pF) | −69(mV) |
Ganglion cell | 25 (µm) | 25 (µm) | 20 (pF) | −65 (mV) |
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The rod model is a modified version of a single compartment model described by us elsewhere
(A) Rod responses to the five simulated photocurrents. From top to bottom, traces correspond to photocurrents from 10 pA to 50 pA. (B) Photocurrent time course for five different photocurrent amplitudes, from bottom to top: 10 pA, 20 pA, 30 pA, 40 pA and 50 pA.
The photocurrent is injected in the rod compartment to simulate changes in the dark current caused by the light transduction process. The rod model responses to the five photocurrents waveforms are given in
The cone model is a modified version of the single compartment model of
The rod bipolar cell model is a modified version of the single compartment model of
The cone bipolar cell model is entirely similar to the rod bipolar cell model, but without the I
The third neuron in the rod pathway is the AII amacrine cell. Evidences show that most of these cells have only sodium and potassium voltage-gated channels, and can produce spikes under specific
For the ganglion cell we used the single compartment model of Fohlmeister and Miller
All electrical synapses in the model were modeled as a single resistance connecting two neighboring cells
The chemical synapses between rods and rod bipolar cells, rod bipolar cells and AII amacrine cells, and cone bipolar cells and ganglion cells are glutamatergic ribbon synapses. They are graded synapses specialized to continuously release glutamate as the stimulus intensity changes
Our adapted chemical synaptic model simulates both the AMPA and the mGluR6 glutamate receptors present in the primary and secondary retina rod pathways. AMPA receptors are found in AII amacrine and ganglion cells
2.56 (nS) | 0 (mV) | 10 (ms) | 10 (mV) |
The retina network model consists of a set of two dimensional rectangular grids representing a small area of the retina (
Rod bipolar cells in the RB layer are connected by chemical synapses with AII amacrine cells in the AII layer. AII amacrine cells in the AII layer are connected among themselves by electrical synapses and with cone bipolar cells in the CB layer. Cone bipolar cells in the CB layer receive chemical synapses from cones in the C layer and send chemical synapses to the ganglion cell.
The number of cells on each layer was chosen to preserve, approximately, the convergence factors of each cell type to a ganglion cell for a region of the cat retina located at about 0.4–0.5 mm from the area centralis (which corresponds to the fovea in primates)
Each rod in the R layer makes an electrical synapse with each one of its first neighbors with probability
For the electrical synapses between cones and rods, we considered that each cone in the C layer is coupled by gap junctions with
Electrical coupling among cells in the AII layer was determined in the same fashion as for the R layer. The average number of connections between each AII amacrine cell and its first neighbors was given by a connectivity index
For the electrical synapses between AII amacrine cells and cone bipolar cells, we considered that each AII amacrine cell is electrically coupled with two randomly chosen cells from the CB layer. The conductances of these electrical synapses also had their values fixed at 0.2 nS
According to the topology of each grid, cells belonging to the borders of the grids make a number of electrical synapses equal to
The pattern of connections by chemical synapses was determined from the divergence factors between cell layers given by Sterling
For each simulation we assumed that a light flash of a given strength was presented to the entire R layer. However, to account for the fact that photon absorption is probabilistic
To obtain the response of the ganglion cell for each flash intensity, we simulated the application of the corresponding photocurrent to the R layer for a period of 5 seconds as described in the previous paragraph. The firing frequency was calculated by counting the number of spikes generated by the ganglion cell during this 5-s period and dividing it by this period. We consider 5 seconds as a period suficiently long for a reliable estimation of ganglion cell firing frequency with a single realization of each experiment.
The dynamic range
(A) Linear-log plot. The dots correspond to the experimental measures and the straight lines were included only to guide the eyes. (B) Log-log plot. The gray line shows the best-fit power law curve with exponent
The simulations were performed in NEURON 6.0
The retina model was used to study the effects of
In our first experiment, we fixed
The dynamic ranges calculated for all combinations of
A gray scale was used to indicate the dynamic range
The dynamic range maxima in the diagram of
The second experiment was performed to assess the effect of the connectivity index of AII amacrine cells on the dynamic range of the retina. In this case, we fixed
In this case,
To quantify the dynamic range variability as a function of the connectivity index for a given layer, we define the line-averaged dynamic range percent variation in relation to the maximum line-averaged value (
The use of
In this work, we have used a minimal model of the primary and secondary ON rod processing pathways of the vertebrate retina to investigate the effects of some possible dynamic range-enhancing parameters, namely electrical coupling by gap junctions among rods and AII amacrine cells and maximal conductance
The main motivation for our study were some recent theoretical works
Our results show that electrical coupling among rods and AII amacrine cells produces, as a general effect, an increase in the dynamic range of the vertebrate retina. This confirms the predictions of the theoretical works that motivated this work.
An important result is that the increase in dynamic range is much larger for the case of rod coupling than for the case of coupling among AII amacrine cells. This result is consistent with the idea that the greater part of the dynamic range enhancement along a sensory processing pathway should be implemented as peripherically as possible to avoid early saturation effects
The results also show that the dynamic range-enhancing effects of either
This latter result may be due to the fact that for the
The response curve of the ganglion cell for the range of inputs considered and for the parameter configuration which gave the largest dynamic range observed by us (
Moreover, our result may be related to the prediction from these models that sensory systems operate at or near a critical point of a phase transition. Recent evidence for power laws in the developing retina
The
Our biologically detailed model of the rod pathways in the vertebrate retina gave results which are compatible with predictions from more abstract models. These predictions are consistent with experimental data
Further investigations are required in order to advance our understanding of the roles of cell connectivity and membrane properties on the dynamic range of the retina. We conclude that both simplified and realistic models, which respect the anatomy and physiology of the retina, will play a distinctive role in this endeavor.
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Equations for the rod photoreceptor model.
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Equations for the cone photoreceptor model.
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Equations for the bipolar cell model.
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Equations for the AII amacrine cell model.
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Equations for the ganglion cell model.
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