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