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In the presence of several user categories or transportation modes, or when transportation costs on each arc of a network depend on the flows on adjacent arcs, the traffic equilibrium problem may be expressed as a variational problem. Methods for determining traffic equilibria are then adaptations of techniques for solving variational inequalities. In this paper, we present a new convex optimization formulation for the general traffic equilibrium problem, and propose a simple iterative method for calculating traffic equilibria, which essentially involves postoptimizing a linear sub-problem at each iteration. Preliminary computational results are reported.

Netrin has been implicated in retinal ganglion cell (RGC) axon pathfinding in a number of species. In Xenopus laevis, RGC axons reaching their target in the optic tectum can be repelled by a netrin-1 gradient in vitro, suggesting that netrin may also function in wiring events that follow successful axon pathfinding. Here, we examined the contribution of netrin to RGC axon arborization and synapse formation at the target. Time-lapse confocal microscopy imaging of individual RGC axons coexpressing GFP-synaptobrevin and DsRed in the intact Xenopus brain demonstrated a role for deleted in colorectal cancer (DCC)-mediated netrin signaling. Microinjection of netrin-1 into the tectum induced a rapid and transient increase in presynaptic site addition that resulted in higher presynaptic site density over a 24 h observation period. Moreover, netrin induced dynamic axon branching, increasing branch addition and retraction; a behavior that ultimately increased total branch number. In contrast, microinjection of DCC function-blocking antibodies prevented the increase in presynaptic site number normally observed in control axons as well as the associated increase in branch number and axon arbor growth. Dynamic analysis of axon arbors demonstrated that the effects of anti-DCC on axon morphology and presynaptic connectivity were attributable to a specific decrease in new synapse and branch additions, without affecting the stability of existing synapses and branches. Together, these results indicate that, in the absence of DCC signaling, RGC axons fail to branch and differentiate, and support a novel role for netrin in later phases of retinotectal development.

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ca, , }This work pleads for the use of the concept of strategies, and their network-theoretic representation as hyperpaths, for modeling network assignment problems. While this concept describes adequately the behavior of users in transit systems, we show that it can apply as well to networks where arc capacities are rigid. This opens up a whole new field of research and raises several questions, from both the theoretical and computational points of view. These are investigated in the paper.

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