2_19_09_LinearRecurrentNetworks_1

2_19_09_LinearRecurrentNetworks_1 - A full feedforward...

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A full feedforward network has vector inputs and outputs connected by a weight matrix.
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A recurrent network is a feedforward network with a recurrent synaptic weight matrix.
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For a feedforward network: For a recurrent network: τ r dv a dt = v a + F W ab b = 1 N a u b r d v dt = v + F W u ( ) r d v dt = v + F W u + M v ( )
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Assuming that the activation function F is linear, that is, F(x)=x, and denoting the input as h = W . u τ r d v dt = v + h + M v r d v dt = I v + h + M v r d v dt = M I ( ) v + h r d v dt = v + F W u + M v ( )
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τ r d v dt = M I ( ) v + h General solution of homogeneous equation Particular solution of full equation, including h Solution is linear combination of
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The homogeneous equation describes a vector field, with curved and straight trajectories.
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The straight-line solutions are the simplest, as for them: d Y dt = A Y Y t ( ) = f t ( ) Y 0 df t ( ) dt Y 0 = A f t ( ) Y 0 ( ) = f t ( ) A Y 0 df t ( ) dt f t ( ) = λ A Y 0 =
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2_19_09_LinearRecurrentNetworks_1 - A full feedforward...

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