- An odd sigmoid function is defined by

where denotes a hyperbolic tangent. The limiting values of this second sigmoid function are and . Show that the derivate of with respect to is given by

What is the value of this derivate at the origin? Suppose that the slope parameter is made infinitely large. What is the resulting form of ? - Show that the McCulloch-Pitts formal model of a neuron may be approximated by a sigmoidal neuron (i.e., neuron using a sigmoid activation function with large synaptic weights).
- Show that a linear neuron may be approximated by a sigmoidal neuron with small synaptic weights.

- Construct a fully recurrent network with 5 neurons, but with no
self-feedback.
- Consider a multilayer feedforward network, all the neurons of
which operate in their linear regions. Justify the statement that such
a network is equivalent to a single-layer feedforward network.
- Figure 1(a) shows the signal-flow graph of a recurrent network made up of two neurons. Write the nonlinear difference equation that defines the evolution of or that of . These two variables define the outputs of the top and bottom neurons, respectively. What is the order of this equation?
- Figure 1(b) shows the signal-flow graph of a recurrent network consisting of two neurons with self-feedback. Write the coupled system of two first-order nonlinear difference equations that describe the operation of the system.

Jarkko Venna 2005-04-13