Statistical mechanics for networks of graded-response neurons

A general statistical mechanical analysis is presented for networks of graded-response neurons whose dynamics is described by a system of differential RC-charging equations. The analysis requires that the dynamics is governed by a Lyapunov function, a condition that is met for networks whose synaptic matrix is symmetric, and whose neurons have monotonically increasing input-output relations may be arbitrary. In particular, they may vary from neuron to neuron. As examples, we study networks with synaptic couplings as in the Hopfield model: two homogeneous networks consisting of neurons with a sigmoidal or a piecewise linear input-output characteristics. Apart from this, the input-output relation, and a network containing a random mixture of these two neuron types.