The authors have declared that no competing interests exist.
Conceived and designed the experiments: RP ACR. Performed the experiments: RP CCC. Analyzed the data: RP CCC. Contributed reagents/materials/analysis tools: RP ACR CCC. Wrote the paper: RP ACR.
The vertebrate retina has a very high dynamic range. This is due to the concerted action of its diverse cell types. Ganglion cells, which are the output cells of the retina, have to preserve this high dynamic range to convey it to higher brain areas. Experimental evidence shows that the firing response of ganglion cells is strongly correlated with their total dendritic area and only weakly correlated with their dendritic branching complexity. On the other hand, theoretical studies with simple neuron models claim that active and large dendritic trees enhance the dynamic range of single neurons. Theoretical models also claim that electrical coupling between ganglion cells via gap junctions enhances their collective dynamic range. In this work we use morphologically reconstructed multicompartmental ganglion cell models to perform two studies. In the first study we investigate the relationship between single ganglion cell dynamic range and number of dendritic branches/total dendritic area for both active and passive dendrites. Our results support the claim that large and active dendrites enhance the dynamic range of a single ganglion cell and show that total dendritic area has stronger correlation with dynamic range than with number of dendritic branches. In the second study we investigate the dynamic range of a square array of ganglion cells with passive or active dendritic trees coupled with each other via dendrodendritic gap junctions. Our results suggest that electrical coupling between active dendritic trees enhances the dynamic range of the ganglion cell array in comparison with both the uncoupled case and the coupled case with cells with passive dendrites. The results from our detailed computational modeling studies suggest that the key properties of the ganglion cells that endow them with a large dynamic range are large and active dendritic trees and electrical coupling via gap junctions.
One of the many important features of the vertebrate retina is the capacity to respond to signals over a wide range of intensities with a dynamic range of several orders of magnitude
A single neuron characteristic, which is claimed to be fundamental for enhancing the neuronal dynamic range in general is the size and complexity of the neuronal dendritic tree with active conductances
There is evidence in favor of a relationship between properties of ganglion cell dendritic trees and their firing behavior, however not exactly as predicted by the above theory. Computational studies with morphologically reconstructed models of ganglion cells of the salamander retina
From the point of view of network properties, a mechanism that may contribute to enhance the collective dynamic range of the network is electrical coupling between cells via gap junctions
Ganglion cells of the vertebrate retina are coupled by electrical synapses via dendrodendritic gap junctions
In this work we use morphologically reconstructed, multicompartmental models of ganglion cells of the vertebrate retina with realistic distributions of ion channels to perform two computational studies on the dynamic range of ganglion cells. The first is concerned with the dynamic range of isolated ganglion cells and its objective is to compare the effects of active and passive dendrites on the cells’ dynamic range and to assess what measure of size of dendritic arbor correlates better with the ganglion cell dynamic range: total dendritic surface area or number of dendritic branches. We measure the correlation between dynamic range and these two measures of dendritic tree size for a population of cell models with either passive or active dendrites. Our results show that active dendrites enhance the dynamic range in comparison with passive dendrites and when dendrites are active the dynamic range of isolated ganglion cells is positively correlated with either measure of dendritic tree size, though more strongly with total dendritic surface area than with number of dendritic branches.
The second study is aimed at assessing the role of ganglion cell coupling by gap junctions on the dynamic range of the ganglion cell population. We construct a network of ganglion cells by coupling them via dendrodendritic gap junctions with realistic conductance values. The network simulates a small area of the ganglion cell layer. We consider different configurations of the network, with passive/active dendrites and different values of the electrical synapse conductance. The dynamic range of the network is measured either directly, by the average firing rate of all neurons in the network, or indirectly, by the firing rate of a lateral geniculate nucleus pyramidal neuron model coupled by chemical synapses to all ganglion cells in the network. As far as we know, this is the first computational investigation of the dynamic range of a neural cell layer using reconstructed neurons with full morphologies and realistic ion channel distributions. Our results show unequivocally that electrical coupling, especially when dendrites are active, increases the dynamic range in comparison with the uncoupled case.
Our two results put together imply that to maximize the dynamic range of a population of vertebrate ganglion cells the best configuration would be cells with large and active dendritic trees coupled by gap junctions.
We worked with a sample of 20 morphologically and biophysically detailed models of ganglion cells from the tiger salamander (for details see Methods). Cells belonged to four different morphological groups (5 per group), based on the size and complexity of their dendritic trees
The dynamic range of each cell in the sample was determined from its FI curve (see Methods). To obtain the FI curve of a cell model, we submitted it to somatic step current injections ranging from 10^{1} to 2.10^{3} pA in steps of 10 pA. In
Current clamp response for a sample cell from the SS group. The sampled cell has 47 branches and total dendritic area of 1284.67 μm2. (A) Voltage response to a current clamp of 10 pA amplitude. (B) FxI curve for the same cell model with inputs varying from 10 pA to 100 pA. The dashed lines indicate the minimum and maximum current amplitudes used to obtain the dynamic range of the cell.
The transition from tonic firing to rest observed in
(A) Amplitude of the action potential as a function of increasing current step. The current clamp is applied for 1000 ms and its amplitude increases linearly with time (I = 0.01t). (B) Twodimensional phase diagram showing membrane potential in the horizontal axis and the activation variable (
Plots in the first row are for cells with active dendrites and plots in the second row are for cells with passive dendrites. The plots also show the Pearson correlation coefficients and pvalues. (A) Scatter plot of dynamic range
A comparison between plots in the first row of
Mean (dB)  Median (dB)  SD (dB)  

18.25  17.98  1.93 

11.29  12.49  4.41 
The positive correlation between dynamic range and size of the dendritic tree is lost when dendrites become passive.
The strong correlation between dynamic range and total dendritic surface area motivated us to do another experiment to investigate the effect of dendritic surface area alone on the dynamic range. Since dendritic surface area and number of dendritic branches are related, we designed an experiment in which only dendritic surface area varied while the number of dendritic branches remained constant. We chose a ganglion cell model from the SS group with passive parameters described in
Length (μm)  Diameter (μm)  Axial resistance (Ωm)  Leakage conductance density (mS/cm^{2})  Leakage reversal potential (mV)  
Soma      110  8.10^{−3}  −62.5 
Axon  5340  1  110  8.10^{−3}  −62.5 
Initial segment  40  1  110  8.10^{−3}  −62.5 
Narrow segment  90  0.4  110  8.10^{−3}  −62.5 
Current type  Soma (mS/cm^{2})  Dendrites (mS/cm^{2})  Axon (mS/cm^{2}) 
Sodium  80  25  70 
Calcium  1.5  2  0 
Potassium  18  12  18 
Atype inactivating potassium  54  36  0 
Calcium dependent potassium  0.065  0.008  0.065 
The results of this experiment are given in
(A) Dynamic range of the ganglion cell model as function of the area factor (see main text for a definition) of the extra attached compartment. Black dots indicate active extra compartment, and black squares indicate passive extra compartment. (B) Difference Γ between the dynamic ranges for active and passive cases as a function of the area factor.
The second study was designed to assess the effect of electrical coupling between ganglion cells on the collective dynamic range of the cells. We simulated a 3×3 square array of ganglion cells coupled via dendrodendritic gap junctions as shown in
Ganglion cells are placed in the vertices of a 3×3 square grid and are coupled with their first neighbors via dendrodendritic gap junctions. Each ganglion cell makes an excitatory chemical synapse with a pyramidal cell from the LGN. Only the central cell of the array (indicated by an arrow) receives external input in the form of current clamps of varying amplitudes.
In our experiments, step current inputs were applied to the central neuron of the array with amplitudes varying from 10^{1} to 10^{5} pA. This central neuron excites the other neurons via gap junctions so that we can relate the steady state firing frequency of each one of the nine neurons in the array to the amplitude of the applied current input.
In all cases, to calculate steady state firing frequencies, the input current was applied for 0.3 seconds. (A) Ganglion cells with passive dendritic trees. (B) Ganglion cells with active dendritic trees. The dashed line represents the maximum current value for the cell with response curve indicated in blue in Figure A. The factor
Each coupled cell exhibits a FI pattern similar to the one of the single cell in
Although the enhancement was similar for the simulations with passive and active trees, the maximum dynamic range was obtained for the coupled array of ganglion cells with active dendrites (
A quantitative estimate of the contribution of active dendrites to the further enhancement of the dynamic range of a cell in the coupled array is provided by factor
An indirect way to assess the dynamic range of the coupled array of ganglion cells is by measuring the dynamic range of the pyramidal neuron that receives input from them.
Firing frequency of the pyramidal neuron of the LGN as a function of input current applied to the central cell of the ganglion cell array. The vertical dashed line gives the input current for which the central ganglion cell in the array stops firing (580 pA). The dynamic range of the FxI curve is 34.8 dB.
The main result of our simulation studies with morphologically reconstructed ganglion cell models is that active dendrites enhance the dynamic range of the cells. The other two important results are that (1) for isolated ganglion cells, the dynamic range has positive and significant correlation with the size of the active dendritic tree (the correlation is stronger with total dendritic surface area than with number of dendritic branches), and (2) by coupling ganglion cells via gap junctions the dynamic range of the coupled cells is further enhanced, being greater than the average dynamic range of the individual cells.
Why a cell with active dendrites has a larger dynamic range than a morphologically similar cell with passive dendrites? Passive dendrites act as current sinks. Current injected at the soma escapes to dendrites reducing the efficiency of the input current in making the cell fire. Active dendrites, on the other hand, allow somagenerated spikes to propagate into the dendritic tree. These in turn generate dendritic spikes, which interact nonlinearly across the dendritic arbor leading to creation and annihilation of spikes and the consequent enhancement of the cell’s dynamic range
This explains the lower values of dynamic range for ganglion cells with passive dendrites in comparison with ganglion cells with active dendrites. It also explains the positive correlation between dynamic range and size of the dendritic tree for cells with active dendrites and the negative correlation between dynamic range and size of the dendritic tree for cells with passive dendrites. Since passive dendrites act as current sinks, the larger the dendritic tree, the more space for current to sink. On the other hand, larger dendritic trees have more active ion channels to support spike creation and summation.
And why dendritic surface area correlates better with dynamic range than with number of dendritic branches? The stronger correlation of dynamic range with dendritic surface area than with number of dendritic branches means that dendritic surface area is a better predictor of dynamic range than number of dendritic branches. As commented above, the critical factor for the enhancement of the dynamic range of a cell is to have active ion channels distributed over its dendrites. So, surface area correlates better with dynamic range simply because it is a better estimator of the number of ion channels in a dendritic tree than number of dendrites. This is because the number of ion channels in a dendritic tree is determined by channel densities
A demonstration that the key factor to enhance the dynamic range of a ganglion cell is the number of active ion channels in its dendrites is given by the results shown in
Our results show that when ganglion cells are coupled by gap junctions their dynamic range is much higher than the average dynamic range of isolated ganglion cells. Even ganglion cells with passive dendrites, when coupled by gap junctions, have a larger dynamic range than isolated ganglion cells with active dendrites. The average dynamic range of these latter is 11.30 dB and the dynamic range of the former is 31.9 dB. When the coupled cells have active dendrites, the dynamic range of the array is a little higher: 34.7 dB.
Why coupled cells with passive dendrites have a larger dynamic range than isolated ganglion cells with active dendrites? The reason for this is that dendrodendritic coupling by gap junctions interlinks the somata of the cells, which have active ion channels, transforming the system into a spatially extended excitable medium. The mechanism responsible for the enhancement of the dynamic range of the coupled ganglion cells with passive dendrites is, therefore, the same one responsible for the enhancement of the dynamic range of an isolated ganglion cell with active dendrites, namely nonlinear summation of spikes
Based on our results on the dynamic range of coupled ganglion cells we can make two predictions: (1) blockade of gap junctional coupling of ganglion cells in the vertebrate retina should strongly reduce (approximately by 40%) the output dynamic range of the retina; and (2) selective suppression of dendritic (but not somatic) spiking of coupled ganglion cells in the vertebrate retina should reduce the output dynamic range of the retina by a much smaller factor (approximately 9%). These reductions could be verified by simultaneous recording from lateral geniculate nucleus pyramidal cells.
In a previous work, we used a detailed model of the scotopic pathways that convey information from rods to a single ganglion cell of the vertebrate retina to study the effect of coupling by gap junction at the first stages of these pathways on the dynamic range of the ganglion cell
We further predict, based on our results, that it is unlikely that any type of ganglion cell would have a passive tree or a very low channel density at the dendrites. We also predict that, if there are uncoupled ganglion cells in the retina, these are distributed over the retina so that cells with large dendritic trees (which imply large dynamic ranges) are able to integrate signals from circuits mediated by rods and cones responding to dim and bright light conditions. On the other hand, ganglion cells with small dendritic trees should be specialized to photopic or scotopic conditions. These predictions could be experimentally confirmed in the future with a detailed study on the distribution of morphologically distinct ganglion cells over the vertebrate retina.
Future investigations can provide a better understanding on the roles of cell connectivity and membrane properties on the dynamic range of the retina. The model can be further improved with a more realistic synaptic input distribution over the dendrites of ganglion cells and also extended to include the main circuits involved with dim and bright light processing in the retina.
We worked with a sample of 20 morphologically reconstructed, threedimensional ganglion cell models from the tiger salamander (
The same set of active ion channels were placed in all ganglion cell models. Each model has four voltagedependent channels (Na, Ca, K, and K_{A}), one calciumdependent channel (K_{Ca}). The dynamics and parameters of the calcium current were able to fit the highvoltage activated component of the calcium current (Ltype) described in a previous experimental work
To obtain the FI curve of an isolated ganglion cell we submitted it to steps of somatic current clamp of fixed amplitudes. The duration of each step was 300 ms and the current amplitudes varied from 10 pA to 1000 pA. The duration of 300 ms was chosen because, based on our studies, it is sufficient for a reliable estimate of the steady state firing frequency of a ganglion cell to a step current.
For all simulations, the dynamic range (Δ) was calculated as:
The single compartment model of a lateral geniculate nucleus pyramidal neuron used to obtain
We used experimental evidence of dendrodendritic bidirectional gap junctions connecting ganglion cells
The excitatory chemical synapse between the axon of a ganglion cell and the LGN pyramidal cell was modeled by a closed/open gating scheme (
The network model consisted of 9 ganglion cells and a single LGN pyramidal cell. The ganglion cells were arranged in a 3×3 square grid and connected by gap junctions as shown in
To obtain the average response of the ganglion cells and the pyramidal neuron for a given amplitude of current clamp, we stimulated the network with the current for 0.3 seconds and counted the number of spikes during this period. We used current amplitudes in the range from 10^{1} pA to 10^{5} pA separated by steps of 10 pA. We consider 0.3 seconds a period sufficiently long for a reliable estimate of the ganglion cells’ firing frequency and to obtain the dynamic range of the ganglion cells and the pyramidal neuron. The simulations were performed in NEURON 7.1