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In 1972, 1973 Wilson and Cowan introduced a mathematical model for the population dynamics of nets of interacting neurons, based on a mean field description of the activity. In 1978 Cowan (unpublished) showed that such a mean field can be derived from a master equation, and in 1991 that such a master equation can be conveniently represented in terms of the annihilation and creation operators of a quantum field theory. In effect, neural nets can be represented in terms of quantum spins. The resulting theory (Ohira & Cowan 1993) proved difficult to calculate. Recently Buice & Cowan (2006, submitted for publication) have found a more tractable way to represent such quantum spins, and to calculate (perturbatively) the effects of fluctuations and correlations in such nets beyond the range of mean field theory. Major results of this formulation include a role for critical branching in such nets and the demonstration that there exist non-equilibrium phase transitions in neural nets which are in the same universality class as directed percolation. |