The mass-action contribution (Sperry)
Our model is designed to predict the locations of terminations of retinal axons. More exactly, our model traces the behavior of synapses formed by axons. The synapses are defined by the weight matrix Wij
, where the index j
describes the number of retinal axon, while the index i
is the number of the dendrite with which given synapse is formed. The weight matrix therefore describes the strength of connection between the axon number j
and the dendrite number i
. For simplicity we assume that each axon can form a single synapse with a dendrite and each dendrite can form a single synapse with an axon. We then define the affinity potential that is a function of the weight matrix. The affinity potential is a sum of the chemoaffinity (Sperry) and correlated activity-dependent (Hebb) contributions, as postulated by equation (1). The affinity potential is similar to the one used by us before [45
] with the expression levels of chemical labels modified to address experiments in Isl2/EphA3 mutants [46
]. The Sperry contribution is
Here indexes α and β describe the chemical labels (α = EphA or B, β = ephrin-A or -B). The sum
defines the total number of receptors bound by ligands for a given pair. It depends on receptor and ligand expression levels
of axon number i
and dendrite number j
respectively. It also depends on the dissociation constant for the pair of molecules Kαβ
. The total number of bound receptor-ligand pairs B
) is determined by the receptor occupancy that can be derived from the mass-action law
This function describes receptor saturation by the ligand and vice versa. For example, the number of bound receptors cannot exceed the total number of ligand molecules present. In agreement with this, B → L when the level of receptor is very high, i.e. R → ∞. Conversely, B → R when L → ∞. For small levels of receptor and ligand, much smaller than the saturation concentration K, the number of bound receptor-ligand pairs is determined by a simpler expression
This expression has been used by us in the previous studies. In this study, we adopt this simpler expression for all receptor-ligand pairs with the single exception of retinal EphA receptor bound by collicular ephrin-A for which we use equation (3). This is to account for the saturation of EphA signaling that could explain the relative signaling in EphA3 knockin mice as suggested by [48
]. For this case we use the value of saturation concentration K
= 7. For the interactions between EphB and ephrin-B we assumed no saturation, i.e. K
Matrix Mαβ defines the effects of receptor/ligand binding on retinal axons. Thus if Mαβ is positive the interaction of receptors and ligands of given types is repulsive. For negative Mαβ , the interaction is attractive. The absolute value of Mαβ determines the strength of the effect of receptor α activation by ligand β on axons.
In our model we assumed that the dissociation constants between different EphA receptors and different ephrin-A ligands are the same. This was an approximation that was used to simplify our model. The same approximation was adopted for EphB/ephrin-B receptors/ligands. Although some in vitro
data indicates that dissociation constants may differ within a family [11
], conclusive data on the strength of binding in vivo
is missing. Similarly, we assumed that the effects of receptor binding Mαβ
are the same within each receptor/ligand family. Because of these approximations, the receptor/ligand concentration was combined into a single number for every family (A and B).
Derivation of the mass-action expression (3)
For a receptor-ligand pair the following equations describe the chemical equilibrium between bound and unbound receptors:
Here [R], [L] are the levels of free (unbound) receptor and ligand, and [LR] is the level of bound receptor. R and L are the total levels of receptor and ligand present in an axon or on the substrate. By solving the system of equations (5) for B = [RL] we obtain equation (3).
The Hebbian contribution
We adopted the Hebbian contribution from the previous studies:
Here Cij is the correlation in electric activity between axons number i and j. This function describes the strength of similarity between axons as a function of their location is retina. The arbor function Uml on the other hand describes the strength of Hebbian interaction as a function of dendrite position in the target. The affinity potential defined by equations (1) through (6) is minimized using the stochastic procedure defined below.
RGCs are arranged in retina on a square array restricted within a circle of radius equal to 48. These cells establish connections with a matching in the number of recipient cells square array of collicular dendrites. The square array is restricted to the oval area shown in the figures. Each axon is constrained to make connections with one and only one collicular dendrite for simplicity. We therefore assume that Wmi = 1 for the pair of cells m and l that are connected and 0 for unconnected cells. This assumption implements the competition constraint described in the text.
We begin from a random set of connections that reflects the broad initial distribution of axons and their synapses in the target [68
]. To minimize the affinity potential (1) we use an iterative stochastic optimization procedure. On each step of the algorithm two cells are chosen randomly. The cells are not necessarily neighboring in the target or in the retina. We then calculate the potential change in the affinity potential for the modification of retinocollicular connectivity in which these two cells exchange their positions in the target. This change in potential is defined by ΔE
. The modification of connectivity is then implemented with probability
Thus, if the potential is decreased as the result of this modification (ΔE < 0) the probability to accept this attempt is more than 1/2, leading therefore to the bias towards minimizing the overall value of potential. This step is repeated 107 times. The number of iterations is chosen to ensure the algorithm's convergence for the wild type distribution of the molecular labels.
Receptor and ligand distributions
The parameters of the model are as follows. The distributions of molecular labels are
Here the horizontal and vertical coordinates in retina are x and y, while the collicular coordinates are denoted by x' and y'. All coordinates vary between 1 and N. The additional level of expression of EphA3 receptor ΔR is equal to 0, 0.45, and 0.9 for wild-type, heterozygous, and homozygous cases respectively. 50% of axons were chosen randomly to express EphA3 in each case. The level of EphA4 receptor R4 was equal to 2, 1, and 0 in EphA4+/+, +/-, and -/- mice correspondingly.
To model reverse signaling (Figure ) we used the distribution of labels indicated in Figure . The reverse signaling strength was 1/10th of that for the forward signaling. This estimate for the reverse signaling strength was derived from recent experiments on p75 receptor (a co-receptor mediating ephrin-A-based reverse signal) mutant mice that displayed ~10% rostral shifts of the termination zones [59
]. Because direct interaction between retinal p75 and collicular BDNF cannot be ruled out, 10% provides an upper bound for the reverse signaling effect.
The matrix of affinities Mαβ in equation (2) is
Negative/positive values of the matrix of affinities describe chemoattraction/repulsion. The zero values imply that there is no direct interaction between the "A" and "B" families of receptors and ligands [11
is the dissociation concentration [equation (2)]. K
= 7 and K
= ∞ was used for EphA/ephrin-A and EphB/ephrin-B interactions respectively. We therefore excluded saturation from the latter interaction for simplicity.
The parameters in equation (6) are as follows
is the range of correlations in the retina [37
] while b
and γ = 0.25 are the range and the strength of Hebbian attraction in SC [45
Position of bifurcation
Here we calculate the location of the bifurcation point from the balance between Hebbian and Sperry contribution to the affinity potential. We will assume a simple form of mass-action law without saturation that is given by equation (4). The bifurcation in the map is associated with the interface between the single-valued and doubled maps. In the doubled map the Sperry contribution to the affinity functional is minimized, while the Hebbian contribution is increased by
Here n is the density of neurons (n = 1 in our model), x is the location of the interface, UH ~ γnb2 is the Hebbian potential per neuron. The Hebbian potential is estimated here up to the numerical factor which depends on the exact geometry of the problem. In the single-valued part of the map the Hebbian contribution has the minimum possible value while the Sperry contribution is increased. The total increase in the Sperry contribution is
Here Usp is the Sperry contribution per neuron. The total affinity is minimum if
Due to equations (16) and (17) this implies that
To evaluate the increase in the Sperry contribution in the area occupied by the single-valued contribution we notice that it is equal to
Here Λ is the shift of the axons in the single-valued map from the location minimizing Sperry contribution. This shift is therefore equal to the separation between two branches of the doubled map. The gradients of the wild-type levels of receptor and ligand are denoted by R
. Because this correction to potential per neuron describes deviation from the minimum it is quadratic in Λ. The assumption under which (20) is true is that Λ <<N
i.e. maps separation in the doubled map is smaller than the size of the map.
To find Λ we notice that for doubled map R(x + Λ) = R(x) + ΔR. This implies that the wild-type EphA3- axons with the level of EphA expression R(x + Λ) terminate at the same location as the knockin EphA3+ axons with the receptor levels of R(x) + ΔR. Thus the separation between two maps is
Combining (19), (20), and (21) we obtain the equation for the location of the point of doubling x
The single-valued are of the map is defined by the condition
which implies that the Hebb contribution is large. The doubled map region is defined by the opposite to equation (23) condition. This confirms the qualitative understanding that we derived from Figure that collapse should occur more readily at location where the gradient of ligand is small and gradient of wild-type level of receptor is large i.e. in temporal retina, as observed experimentally (Figure ).