While
in signal representation, we represent the input signal itself, in
signal transformation, we represent some function of the input signal.
Here, the system was utilized to neuromorphically perform squaring of an
input sinusoidal signal (Figure 5, A ). The transformation
results with one positively encoded neuron and eight bounded neurons are
shown in Figure 5, B . While one positively encoded neuron
cannot account for the input signal’s negative phase, it provides a
temporal output only at the positive sinus phase. With eight neurons,
however, the results show accurate transformation (Figure 5,
C) . We measured transformation error with bounded, uniform, and pure
activations with one to eight neurons. The results show superior
performance for a uniform distribution of neuron tuning (Figure
5, D ). The neuromorphic system presented herein continually modulates
neuronal weights. Weight tuning for each of the eight spiking neurons
during transformation is shown in Figure 5, E .
Dynamics
Neuromorphic representation and transformation are the first two main
pillars of NEF. The third is the realization of a dynamical system.
Here, we used our circuit design to implement an induced leaky
oscillator (Equation 9) . The system schematic is shown inFigure 5, F . This system utilizes our 2D representation scheme,
described above in Figure 4 . We traced the system’s two
dimensions (\(x_{1}\) and \(x_{2}\)) throughout time, following
induction with a single three mSec pulse (driven to \(x_{1}\)). The
resulting oscillatory dynamic is shown in Figure 5, G .
Oscillation slowly converges back to zero at a rate determined by the
hardware’s leaky characteristic. When induced again, oscillation can be
maintained, as demonstrated in Figure 5, H , where two five mSec
pulses are spaced by two seconds.
Worst-case Analysis