In order to obtain some idea of the properties of the solution of equations (27) the first approximation to them will be that of linearisation. Absorbing the derivatives of the output functions into the strengths of the connection matrices, and dropping the input J and the lateral connection matrix , the equations may be reduced to the single equation
For the case that I is purely an initiator of activity, the growth of is exponential in time, with behaviour
and so has time constant , where a and b are the diagonal elements of and (when they are diagonal). In the case the behaviour (29) will be of exponential increase, until the non-linearity in the neuronal responses, especially that of saturation, becomes important. For , the activity would initially be expected to die away. In that case, it is necessary to take more careful account of the basal ganglia contribution. Then, again in the diagonal case, and reducing to the largest eigenvalue of , for the behaviour of as , there results the solution
In the case that all of a, b, c and d are positive, which seems supported by the neurobiology, then it is still possible to have in (30) even though . This is so for any value of provided that is large enough.
Given that it is always possible to have an increasing activity on the frontal cortex, for suitably large basal ganglia to thalamus connection weights, it is then possible to see that this activity will saturate when the non-linear response function in equation (27) is used. To see this, it is only necessary to include the non-linearity of the thalamic output. Equation (27) then becomes
The asymptotic value will therefore satisfy the equation (dropping the external input )
This has a bounded solution, for any connection matrix. The general form of the temporal development of the activities of the modules will thus, for large enough strength of connectivity from basal ganglia to thalamus (as noted above), be of an initial exponential rise of activity, followed by saturation for a period depending on the external input. If that latter acts in an inhibitory manner at a certain time value, then it is possible to see that this will cause the activity of equation (32) to die away. This would agree with the general form of the experimental activities shown, for example, on the left hand column of Figure 5 of Zipser .
The presence of more than one attractor is clear from the relation of (31) to a BAM [24,30], to which it is identical when , . More generally a number of attractors will be expected, with noise causing transitions between them [25,53].
The training necessary to achieve such active memory behaviour is obtainable through simple Hebbian learning on the basal ganglia connection weights and , since as these increase they will enable the lifetime of the neuronal activity of the cortical nodes to increase, according to equation (30). This has already been investigated in , to which the reader is referred.
An important feature of active memory is whether or not the ACTION net acts as a flip-flop for non-zero input. Thus the simplest ACTION net
in the limit that is the Heaviside step function , is turned on permanently by an input , but only transiently for . Replacing by a smooth sigmoid function in (33) will then lead to a smoothly increasing and saturating response to a large enough input, but a transient decaying response to a smaller one. The first of these two possibilities are observed in the cells of Figure 1 of Zipser et al. , and in the growing population vector in Figure 11 of Georgopoulos et al. ; the latter is presented in Fuster .
It is possible to model the growth of neuronal activity in the Georgopoulos et al. results as follows. For N neurons, each with optimal angular direction for the i'th neuron, let that cell have input arising from some directional input with information that it involves the vector (angular) direction. For the linear case, with t=0 in (32) (and diagonal connection matrices), the asymptotic activity of each ACTION cortical neuron becomes
(where it is assumed that ). Then the population vector output points along the desired direction, since this vector has value
where only has been used to have a positive output from the neurons in (34), and the sum over random directions i in that interval for the cross terms may be shown to be zero for large enough N. The time constant to rise to saturation of about 200 msec  would be expected to be obtainable for a suitable range of connection weights and thresholds.