Nếu ma trận trọng lượng thường xuyên có một giá trị riêng chính xác bằng một, λ1 = 1, và tất cả các giá trị riêng khác đáp ứng λν | 20 Network Models network integration Input Integration If the recurrent weight matrix has an eigenvalue exactly equal to one À1 1 and all the other eigenvalues satisfy kV 1 a linear recurrent network can act as an integrator of its input. In this case c1 satisfies the equation e1 h dt obtained by setting À1 1 in equation . For arbitrary time-dependent inputs the solution of this equation is C1 t C1 0 1 f dt e1 h t . Tr Jo If h t is constant C1 t grows linearly with t. This explains why equation diverges as À1 1. Suppose instead that h t is nonzero for a while and then is set to zero for an extended period of time. When h 0 equation shows that CV 0 for all V 1 because for these eigenvectors ÀV 1. Assuming that C1 0 0 this means that after such a period the firing-rate vector is given from equation and by v t e dte1 h fi . Tr Jo This shows that the network activity provides a measure of the running integral of the projection of the input vector onto e1. One consequence of this is that the activity of the network does not cease if h 0 provided that the integral up to that point in time is nonzero. The network thus exhibits sustained activity in the absence of input which provides a memory of the integral of prior input. Networks in the brain stem of vertebrates responsible for maintaining eye position appear to act as integrators and networks similar to the one we have been discussing have been suggested as models of this system. As outlined in figure eye position changes in response to bursts of activity in ocular motor neurons located in the brain stem. Neurons in the medial vestibular nucleus and prepositus hypoglossi appear to integrate these motor signals to provide a persistent memory of eye position. The sustained firing rates of these neurons are approximately proportional to the angular orientation of the eyes in the horizontal direction and activity persists at an approximately constant rate when the eyes are held
