Figure 1.
A, Cartoon of simulated retinocollicular projection. Each retina extends axons to the contralateral colliculus. The model simulated the development of individual RGC axons from when they first grew through the colliculus until a refined arbor was produced. Each axon was assumed to have a retinotopically correct target in the colliculus that was indicated by guidance molecules. B, Conceptual organization of the model. Constitutive dendritic release of growth and repulsive factors (e.g., neurotrophins and pro-neurotrophins) influenced the growth and retraction of axons and the local growth of synapses on these axons. Regulated postsynaptic release of trophic factors stabilized synapses and enhanced axon and synapse growth on the presynaptic axon near point of receipt. Vesicle release destabilized synapses, resulting in their eventual retraction if they did not receive sufficient trophic feedback. Trophic factor was provided when a postsynaptic spike followed within tens of milliseconds of vesicle release. The chemoaffinity of an axon segment to its surroundings, regulated by ephrin and Eph receptors and ligands in vivo [23], modulated the efficacy of growth and repulsive factors on each axon based on the co-localization of these molecules [55]–[59]. This meant that growth and trophic factors had higher efficacy on an axon in the retinotopically correct area of the colliculus compared to one further removed, and vice versa for repulsive factors. C, Simulated RGC axons in the colliculus were composed of segments 11 long that could each extend, branch and retract. Collicular neurons were densely packed (167/mm, 27,900/mm2) and each had a dendritic field 50
in diameter. Development was represented in two dimensions and each dendritic field was treated as a disk. Axon segments could generate synapses with any dendritic field that it overlapped with. D, Cartoon of axon, showing axon segments, extension and branching. Axon extension occurred at axon tips (i.e., segments that had no children, in blue) and branching occurred in segments that had already extended but that did not have any branches (red). Axon retraction occurred only at axon tips. Extending axons grew in-line with the existing axonal trajectory and branching occurred in a orthogonal direction (D adapted from [36]).
Figure 2.
A, Time-lapse development of a group of axons from a specific retinal location. The retina is represented by the circle at lower-right and the colored dot(s) indicate the location of RGCs whose axons are displayed in the colliculus. Development was regulated only by molecular guidance for the first 48 hours whereafter activity-dependent feedback (i.e., regulated trophic release) contributed to simulated development over the next 60 hours. Axon arbors from five retinal locations are shown in the last frame. B, Arborizations from 21 points on the retina show retinotopic order in axonal projections. Developmental paradigm is identical to ().
, Axon development over 144 hours driven only by molecular guidance. Activity-dependent feedback was required for refinement.
, Developmental sequence as in (
), but with molecular guidance blocked along nasal-temporal retinal axis; anterior-posterior collicular axis. Organization is significantly disrupted.
, Axon arbor development when activity-dependent feedback contributed to development from the time when axons first began interstitial branching (i.e., T = 0 hr in A). Development was qualitatively normal and not dominated by ectopic projections as previously predicted [36], indicating that molecular guidance and activity-dependent mechanisms are able to simultaneously guide and refine development. Data from a single simulation of retinocollicular development are shown in each of (A)-(E). In all results presented here and elsewhere, three or more simulations were performed (typical runtime 3 days each) and no qualitative differences were observed.
Figure 3.
Eye-specific and ON/OFF segregation.
, Spatial segregation of retinal afferents is observed when two retinas with independent patterns of activity project to the same colliculus (tectum). As noted in the main text, the model is based on mouse retinocollicular development but is used generically to explore segregation phenomena observed in other retinal afferent pathways. Colors indicate synaptic drive from either eye (red from left, green from right, yellow equal from both eyes).
, RGCs from a single retina segregate onto different target neurons when RGC activity is spatially correlated but is temporally offset, such as occurs between ON and OFF RGCs during development.
, In the present implementation of the model, synapse survival is based both on how much trophic factor is received by the synapse and how much is received by nearby synapses on the axon. This results in synapse survival being a co-operative effort among nearby synapses, but with some degree of independence (arbitrary 75%/25% split). Eliminating this independence so that survival is purely cooperative results in strong spatial segregation.
, When synapse survival is a purely independent process, such that each synapse's survival depends on how much trophic feedback it receives, segregation occurs but it is not spatially organized.
, Distribution of segregation under different paradigms. Horizontal axis shows ratio of innervating synapses onto each collicular neuron as a function of retina of origin and vertical axis shows relative number of cells (all normalized to have same area under the curve). The red line shows eye-specific segregation (same data as
) with target neurons becoming highly selective and driven by one or the other retina. When segregation occurs in the binocular paradigm (e.g., A) but is followed by a period of development when activity in both retinas becomes synchronized, segregation is reversed (blue lines), as is observed experimentally [34].
Figure 4.
Interpretation of ephrin-A and EphA gradient manipulations.
Cartoon depicts chemoaffinity of RGC axons in simulated wild-type and mutant development. Horizontal axes show retinal location, nasal (N) to temporal (T), and the relative magnitude of ephrinA and EphA gradients. Vertical axes show collicular location, anterior (A) to posterior (P), and the relative magnitude of collicular EphA and ephrinA gradients. Circle locations denote the retinotopically correct termination zone and sizes indicate the relative chemospecificity, with larger circles indicating axons having reduced chemospecifity, or chemoaffinity with a broader collicular area. , In the simulated wild-type case, all axons are assumed to have similar levels of chemospecificity in retinotopically appropriate areas of the colliculus.
, Eliminating ephrinA2 was interpreted as reducing repulsion of RGCs from posterior colliculus, in turn broadening chemoaffinity (reducing chemospecificity) of axons normally repelled from there.
, The guidance molecule ephrinA5 is expressed in both posterior colliculus and nasal retina [23]. Its elimination was assumed to reduce chemospecificity of all RGCs, through loss of repulsion to posterior colliculus like in ephrinA2 knock-out, and loss of repulsion from anterior colliculus due reduced repulsion to collicular EphA.
, Mutations which upregulate EphA3 in a spatially distributed subset of RGCs [6], [9] results in a single retinal location having RGCs with maximal chemoaffinity for two different collicular locations. Unaltered RGCs have the same preferred targeting as wild-type (gray, as in
) while RGCs with upregulated EphA3 have stronger repulsion from posterior colliculus and are pushed anteriorly (black), forming a second map in anterior colliculus. Heightened repulsion and the compressed map were assumed to increase chemospecificity, as a converse to how ephrinA knockouts reduced it. The model presented here describes the hypothesized functional effect of altering molecular guidance expression. There are many ways that nature might achieve this.
Figure 5.
, Arbor development from RGCs at four retinal locations in simulated EphA knock-in experiments (Isl-EphA
), where 50% of RGCs had higher simulated EphA representation and thus stronger repulsion from posterior colliculus. Dual maps are formed.
, Arbors from the same four retinal locations as in (A), but in a control (WT) retina.
, Arborization location in colliculus as function of retinal location. The WT projection (gray) is linear, indicating a single contiguous projection of retina onto colliculus. The EphA projection is split into two contiguous maps, with blue indicating the TZ location of RGCs having upregulated EphA and black showing the locations of unaltered RGCs. Arborizations of unaltered RGCs normally targeting anterior colliculus are shifted posteriorly in simulated mutants (also visible comparing
and B).
, TZ location from eight points in nasal and temporal retina in simulated ephrinA2 knockouts. Arborizations are disrupted in anterior colliculus but are normal in posterior colliculus.
, Simulated ephrinA5 knockouts. Organization is disrupted throughout the colliculus.
, Chart showing relative number of disrupted projections in simulated WT, A2 and A5 retinas, as a function of retinal location. Axons were marked in four locations in seven dorsal-ventral bands of the retina (n = 3 simulations of each type) and the number of arbors having ectopic projections from each band was counted (maximum = 12). While ectopic projections were observed in simulated WT near collicular boundaries, the frequency and degree of these disruptions was minor in comparison to those observed in simulated ephrinA2 and A5 mutants in (D) and (E).
,
Simulation results from ephrinA5 knock-out (
) and blocking all ephrinA/EphA mediated guidance (H). Manipulating molecular guidance on only the anterior-posterior axis can cause perturbations along the orthogonal axis, a phenomenon observed experimentally [7].
Figure 6.
Different mathematical models of the same general description and structure give rise to different behaviors.
In early visual processing, visually responsive neurons in the midbrain and cortex typically become selective to input from a single retina (e.g., [34], [45], [46]). In the center of the image is a cartoon showing axons from each retina targeting the same midbrain structure, in this case a simulated superior colliculus. Collicular neurons are color-coded to show which retina they receive their primary input from (green is left retina, red is right, and yellow is equal from both). , When the previous model (GES-2009, [36]) was modified to run in this binocular paradigm, it exhibited segregation when two simulated retinas projected to the same target structure.
, When eliminating a model assumption affecting synapse stability, segregation was perturbed very little. This assumption was that the stability of a synapse was influenced by the stability of other nearby synapses on an axon – i.e., synapse survival was co-operative or synapses were independent.
, The model in the present study similarly exhibited segregation.
, When the same assumption was dropped in the model in this study, the characteristics of segregation changed significantly. This observation in the present study leads to the prediction that altering synapse stability mechanisms can influence patterns of spatial segregation, although this prediction is not fully supported by GES-2009.
Figure 7.
Arborization size relative to collicular area.
Relative size of axon arbors to collicular area changes with colliculus size when collicular cell density is held constant (scale bars = 100). All colliculi scaled to same display size to show relative arbor coverage. Colliculi have 6772 cells (A, 2515 RGCs), 25K cells (B, 10K RGCs) and 50K cells (C, 20K RGCs). All results shown are based on the same simulation parameters, other than number of RGCs and collicular cells.
,
, Same colliculi in (A) and (B), respectively, shown to same scale of (C). Simulation results can thus be influenced by the number of neurons represented, indicating that the scale of a computational model can influence its quantitative results.
Figure 8.
Description of model structure.
The behaviors described in Fig. 1B are implemented through different model equations. Axon retraction was governed by the relative concentrations of growth and repulsive factors (Eq. 11).
Axon growth was influenced both by the relative concentrations of growth and repulsive factors and by the total arbor size (Eq. 10). Synapse formation was influenced by the axon's local exposure to trophic factor and the number of existing axonal synapses (Eq. 13) and the target cell's firing rate and total number of dendritic synapses (Eq. 14).
,
Synapse survival was governed by the total amount of trophic factor received by the synapse, the number of existing axonal synapses, the firing rate of the postsynaptic cell, and optionally the trophic factor received by neighboring axonal and dendritic neighbors (Eqs. 16,17)
Growth and trophic factors were continuously released into the extracellular space and diffused locally (Eqs. 6,7), based on each cell's firing rate and it's relative maturity (Eq. 3).
The chemoaffinity and chemospecificity modulating axon and synapse growth was based on how far an axon segment was to its retinotopically correct termination zone (Eq. 2).
Trophic feedback was provided to the presynaptic terminal when a spike in the postsynaptic cell followed a spike in the presynaptic cell within tens of milliseconds (Eq. 12). The activity of simulated RGCs was governed by a phenomenological model of retinal wave activity [64] and postsynaptic neural activity was based on an integrate and fire neuron (Eqs. 4,5).