The Neural Marketplace: I. General Formalism and Linear Theory

We propose a mathematical theory for how networks of neurons in the brain self-organize into functional networks, similarly to the self-organization of supply networks in a free-market economy. The theory is inspired by recent experimental results showing how information about changes to output synapses can travel backward along axons to affect a neuron's inputs. In neuronal development, competition for such ``retroaxonal'' signals determines which neurons live and which die. We suggest that in adults, an analogous form of competition occurs between neurons, to supply their targets with appropriate information in exchange for a ``payment'' returned to them backward along the axon. We review experimental evidence suggesting that neurotrophins may constitute such a signaling pathway in the adult brain. We construct a mathematical model, in which a small number of ``consumer'' neurons receive explicit fast error signals while a larger number of ``producer'' neurons compete to supply them with information, guided by retroaxonal signals from the consumers and from each other. We define a loss function to measure network performance and define the ``worth'' of a producer to be the increase in loss that would result if that neuron were to fall silent. We show how slow retroaxonal signals can allow producers to estimate their worth, and how these estimates allow the network to perform a form of parallel search over multiple producer cells. We validate our approximations and demonstrate the proposed learning rule using simulations.