# Effective Behavioural Dynamic Coupling through Echo State Networks
## 5 Properties of the Echo-State Network and Human-Machine Interface
### Geometrical Properties of the Input  
Figure 4 shows a detail of three patterns from the condition _Dataset Testing_ correctly recognised by the ESN:\(CW\), which stands for a clockwise circle pattern performed by the user, \(ACW\) an anticlockwise circle pattern, and \(Up-Down\), an up and down movement of the hand of the user, as recorded by the Leap Motion device. By observing the first segment of the graph and delimited by the first vertical line in the figure around \(Time100\), we can see that the network correctly recognises a \(CW\) gesture in input (the \(Value\) of \(CW\) in the figure reaches \(1.0\)). Interestingly, the \(ACW\) gesture at the same time shows a negative value. This observation can be explained by the fact that the \(ACW\) pattern is 'opposite' to the \(CW\) patter. Therefore, it suggests that geometrical properties of the input are retained and the spatial relationship between the two signals is captured and embedded from the network in its output signals.  
Similarly, in the next section the input behaviour changes from \(CW\) to a mixture of both \(CW\) and \(Up-Down\), and ultimately to just \(Up-Down\) around \(Time\)\(130\). This transition is also reflected to the output of the network, but, besides the two behaviours being mixed, \(ACW\) remains always negative and opposing the values of \(CW\). This observation indicates that, although it is possible to mix behaviours, it remains impossible to do so with geometrically opposite ones. This is an interesting feature that, to our knowledge, cannot be found in other models. Similar dynamics can also be observed in the following section of the pattern, where the \(Up-Down\) recognition settles around \(0\), the user is performing a \(ACW\) gesture and, as expected, the opposite \(CW\) pattern is negative.  
Figure 4: Usage of a trained ESN. The plot highlights how the geometrical properties of the input sequences are retained on the output of the network. The antagonistic behaviour between _Clockwise_ (CW) and _Anticlockwise_ (ACW) behaviours is shown, while the _Up-Down_ motion recognition remains unaffected. In the graph the output of the network is depicted, with each colour and line style representing a pattern recognised in the input. All values are plotted against time.  
Such observation indicates that geometrical properties of the input are propagated to the output. In our example, indeed, the clockwise and an anti-clockwise motion inhibit each other. By assuming that the network has been trained to control a moving robots, it is possible to grasp the importance of this feature. For example, lets assume the \(CW\) motion is mapped to the robot moving forward and the \(ACW\) backwards. Having opposite behaviours being interpreted as 'opposite' by the system, it provides the network with an 'insight': The user cannot perform two opposite behaviours at the same time, but it can perform the \(Up-Down\) gesture in combination with any of the above. At the same time, the fact that the two behaviours are opposite is also maintained in the output. Assuming that the robot behaviours are combined in a linear fashion based on the network outputs, the recognition of'move forwards' implicitly means for the system that'move backwards' will hold opposite values (and negative in the specific implementation presented here).