Reduction of Conductance-Based Models with Slow Synapses to Neural Nets

The method of averaging and a detailed bifurcation calculation are used to reduce a system of synaptically coupled neurons to a Hopfield type continuous time neural network. Due to some special properties of the bifurcation, explicit averaging is not required and the reduction becomes a simple algebraic problem. The resultant calculations show one how to derive a new type of “squashing function” whose properties are directly related to the detailed ionic mechanisms of the membrane. Frequency encoding as opposed to amplitude encoding emerges in a natural fashion from the theory. The full system and the reduced system are numerically compared.