Sets of neurons that have similar characteristics and are connected to other neurons in similar ways are called “layers” or “slabs”.
Concerning connection topologies that define the direction of data flows between the input, hidden and output neurons, these can be classified into two different types of network architectures, so-called feedforward network and recurrent network. A feedforward network has a layered structure. Each layer consists of neurons which receive their input from neurons in a layer directly below and send their output to neurons in a layer directly above.
This network does not have internal feedback; in other words, there exist only forward connections which produce forward activities of neurons. Therefore, feedforward networks are static; that is, their output depends only on the current inputs and the networks represent just simple nonlinear input–output mappings (Fig. 3.5, left).
On the contrary, if feedback exists within the connection structure which allows cyclic propagation of activity (or backward activities), the network is called a recurrent network (Fig. 3.5, right). The output of the network depends on the past inputs; thus, the network can represent various dynamical properties (e.g., hysteresis, oscillation and even deterministic chaos). Some dynamical behaviors of the network are useful for signal processing and robot control being the approach of the book. Therefore, this book is concentrated on applying recurrent neural networks together with their dynamical behavior to create so-called versatile artificial perception–action systems described in Chap. 5. However, there are two exceptions of applying the network to create the neural controller of the system, where input neurons can receive only input from the outside world (sensor data) and the number of the input and output neurons is determined by the number of sensors and motors used, respectively.
3.2 Discrete Dynamics of the Single Neuron
The single neuron with a self-connection, namely a recurrent neuro-module, has several interesting (discrete) dynamical properties which have been inves-tigated by F. Pasemann [151, 152] and others [13, 88, 89, 170]. From these investigations, the single neuron with an excitatory self-connection has a hys-teresis effect while the stable oscillation with period-doubling orbit can be observed for an inhibitory self-connection. However, both phenomena occur for specific parameter domains of an input and a self-connection weight.
In this book, the hysteresis effect is utilized for preprocessing sensor signals as well as robot control (described in Sect. 5.1). By now, the recapitulation of the used dynamical property of a recurrent neuro-module is discussed by em-ploying the single neuron model presented in the previous section. The corre-sponding dynamics is parameterized by the inputIand the self-connectionw.
The discrete dynamics of the single neuron with a self-connection is given by:
Output 1 Output 2 Output 1 Output 2
Input 1 Input 2 Input 3 Input 1 Input 2 Input 3
Output layer
Hidden layer
Input layer
Forward activity Backward activity
Motors Motors
Sensors Sensors
Fig. 3.5.Examples of a feedforward network (left) and a recurrent network (right).
Generally for robot control, the input for the networks comes from the sensors while their output is sent to control the motors
a(t+ 1) =wf(a(t)) +θ, (3.6)
with the hyperbolic transfer function
f(a) = tanh(a) = 2
1 +e−2a −1, (3.7)
where the parameter θ stands for the sum of the fixed bias term b and the variable total inputI of the neuron. The model neuron with a self-connection for the investigation is presented in Fig. 3.6.
I O
b
w
+ + +
Fig. 3.6. The model neuron with a self-connection
3.2 Discrete Dynamics of the Single Neuron 39 As mention above, the hysteresis effect is observed only for an excitatory self-connection with the specific parameter domains; thus, the dynamics of an excitatory self-connection is shown here while the dynamics of an inhibitory self-connection as well as the detail of mathematical proof are referred to [151, 152].
By simulating the dynamical behavior of varying the excitatory self-connection w together with the input θ, two different domains in the (θ, w)-space are observed (Fig. 3.7).
a b
Fig. 3.7.The dynamics of a neuron with an excitatory self-connection.Left: Param-eter domains for one stable fixed point (I), two stable and one unstable fixed points (II).Right: The cusp catastrophe with respect to the dynamics of the neuron. One can compare between the left andright diagrams to obviously see two stable fixed points and one unstable fixed point which exist in region II. There are also tran-sition states shown on the right diagram where the system changes from one (low) stable fixed point to another (high) stable fixed point and vice versa (F. Pasemann 2005, personal communication)
In region I, there exists a unique stable equilibrium (one fixed point at-tractor) for the system while three stationary states (one unstable fixed point and two coexisting fixed point attractors which are the low and high points) are found in region II. In fact, the hysteresis effect of the output appears when the input θ crosses the region I and II; e.g.,θ sweeps over the input interval between−2 and +2 for a fixedw= 2 (see also anarrow linebetween c and din Fig. 3.7). Ifw is increased to 4 while θ still varies over the input interval (between −2 and +2); i.e., θ does not pass back and forth through the region I and II (see also an arrow line between a and b in Fig. 3.7).
Instead, it varies inside the regionII; consequently, the outputOwill stay at one fixed point attractor (either high or low fixed point attractor) depending
on where the input starts. Furthermore, the width of the hysteresis loop is defined by the strength of the self-connection w >+1; i.e., the stronger the self-connection, the wider the loop is [96]. The comparison of the width of the hysteresis loop is presented in Fig. 3.8.
6.0 4.0 2.0
Fig. 3.8.Comparison of the “hysteresis effects” between the inputθand outputO forw= 2.0, 4.0 and 6.0, respectively (F. Pasemann 2005, personal communication)
One can utilize such different sizes of a hysteresis loop for robot control, e.g., the turning angle of a mobile robot for avoiding obstacles can be deter-mined by the width of a hysteresis loop. In other words, the wider the loop, the larger the turning angle is (see also Sect. 5.1.3). Additionally, there is an example situation shown in Fig. 3.9 where the hysteresis effect depends on the frequency of a dynamic input, e.g., slowly and quickly varying inputs.
2.0
Fig. 3.9. The example of the dynamics of the recurrent neuron with the dynamic input.Left: The dynamic inputθvaries between≈ −1 which is in regionIand≈0.5 which is near to the border where the system can jump from one fixed point (low) to another fixed point (high). Right: The hysteresis effect of the dynamic input for a fixedw= 2, see text for details (F. Pasemann 2005, personal communication)
3.3 Evolutionary Algorithm 41