# Neuromorphic Neural Engineering Framework-Inspired Online Continuous Learning with Analog Circuitry
## 3 Results
### Circuit Simulation
#### 3.2.1 Representation  
In NEF-driven representation, input signals are distributively _encoded_ with neurons as spikes (following each neuron's tuning) and _decoded_ by either calculating a set of decoders (Equation (2)) or learning a set of weights (Equation (5)) via PES learning (Equation (6)). In both cases, neuromorphic representation entails a reference signal (supervised learning).  
We utilized our proposed learning block (Figure 2B,C) to realize neuromorphic representation with PES learning, using the input signal itself as a reference signal (Figure 3A). We used the system to encode and decode exponential and sinusoidal signals with two, four, and eight OZ neurons (Figure 3B-D). As expected, following Equation (4), as the number of neurons increases, the learning system's performance improves. Our hardware simulation-derived results (Figure 3D, red traces) closely follow Nengo's NEF-based software simulation (Figure 3D, purple traces), with a cross-correlation similarity (sliding dot product) of 0.872 \(\pm\) 0.032. We show that an analog learning system comprising only 8 OZ neurons can accurately represent the input with a swift convergence toward the represented value.  
As described above, representation is highly dependent on neuron tuning. The results shown in Figure 3B-D were derived using neurons with a bounded activation distribution. We further represented the sinusoidal input with neurons characterized by uniform and pure activations, following Figure 1C. The results are shown in Figure 3E. We evaluated this representation using the three activation schemes with one to eight neurons by calculating the error's root mean square (RMS). Our results demonstrate the superior performance for  
Figure 3: **Neuromorphic representation.** (**A**) System schematic for a neuromorphic representation of input values; (**B–D**) Representation of exponential (**top**) and sinusoidal (**bottom**) input signals with two, four, and eight neurons. Representation was based on neurons with a bounded distribution of activation (Figure 1C, middle); (**E**) Representation of a sinusoidal input with eight neurons featuring uniform (Figure 1C, top) and pure (Figure 1C, bottom) distributions of activation; (**F**) Representational error RMS with bounded, uniform, and pure activations with one to eight neurons; (**G**) Continual weight tuning of each of the eight neurons during representation.  
a bounded distribution of neuron tuning (Figure 3F). The continually changing weights of each neuron are shown in Figure 3G, demonstrating continual online learning.  
An essential characteristic of NEF is the ability to represent high-dimensional mathematical constructs with high-dimensional neurons. OZ neurons can be driven with high-dimensional signals (using few weighted inputs), featuring high-dimensional tuning [15]. The analog learning system schematic is shown in Figure 4A. By setting a similar weight for the two incoming signals, we derive the neuron's tuning shown in Figure 4B. We used the circuit described in Figure 4A, with eight 2D spiking neurons to represent a 2D signal, wherein one dimension follows an exponentially rising signal and the other, a sinusoidal wave. Representation results are shown in Figure 4C, and the error traces are shown in Figure 4D.