Amplification, Attenuation, and Integration

9 have long been used to model networks of interacting neurons (Er10 mentrout, 1998; PHASE PLANE ANALYSIS OF NEURAL NETS). The 11 activity of neuron i is represented by a single dynamical variable 12 xi, and its input-output characteristics by a single transfer function 13 f. There are more biophysically realistic descriptions of neural net14 works that include many dynamical variables per neuron, in order 15 to explicitly model dendritic integration, action potential genera16 tion, and synaptic transmission. Nevertheless, simplified models 17 like that in Equation 1 have been useful for understanding how the 18 computational properties of neural networks are related to their 19 synaptic organization. 20 The parameter Wij in Equation 1 represents the strength of the 21 synapse from neuron j to i. These synapses are termed recurrent, 22 as they connect to other neurons in the same network. Feedforward 23 synaptic input from outside the network is implicit in the bias bi. 24 The feedforward synapses could be made explicit by writing 25 , where za are input neuron activities, Via the 0 b b V z i i a ia a 26 strengths of the feedforward synapses, and any intrinsic ten0 bi 27 dency of neuron i to be active. But the feedforward connections 28 will be left implicit in the following, so as to focus on the com29 putational role of the recurrent connections. 30 Accordingly, the biases bi in Equation 1 will be regarded as the 31 inputs to the network, while the activities xi are the outputs. If there 32 were no recurrent synapses (Wij 0 for all i and j), then each 33 neuron i would respond by low-pass filtering the signal f (bi) with 34 time constant s. When there are recurrent synapses, a general char35 acterization of the response properties of a network is difficult, but 36 the situation is greatly simplified when nonlinearity is neglected. 37 Putting the transfer function f (u) u in Equation 1 yields the linear 38 network