# Neuromorphic Neural Engineering Framework-Inspired Online Continuous Learning with Analog Circuitry
## 4 Discussion  
NEF is one of the most utilized frameworks in neuromorphic computing. It has been utilized to implement a wide range of neuromorphic applications, ranging from robotics [14] to the most comprehensive functional cognitive model [30]. NEF is based on three main pillars: representation of high-dimensional mathematical constructs, the transformation of one representation to another, and the two's utilization to build dynamical systems. In this work, we propose a fully analog circuit design, which realizes these three main pillars. Our design comprises the realization of PES, a neuromorphic online learning rule, and the utilization of OZ, a spiking neuron model tailored to the NEF configuration. Previous designs utilized NEF to implement neuromorphic functional circuits. For example, Intel's Loihi can be programmed to work with the NEF-based Nengo compiler [18], realizing, among other features, PES learning [31]. Although Loihi, TrueNorth [32], and SpiNNaker [33] can be programmed with Nengo, they are digital circuits. NEF was also compiled to work on hybrid neuromorphic designs, such as the NeuroGrid [19] and its later successor, the BrainDrop [11]. While digital and hybrid neuromorphic were used to support a wide array of neuromorphic applications, analog designs have been tremendously successful in signal processing due [19]. Analog components, however, are particularly vulnerable to fabrication variability. There are several techniques to reduce the process variation; for example, adding dummy transistors to pad the operational transistors in the layout stage. Fortunately, heterogeneous neuronal tuning is desirable with NEF, as it provides the variability in activation needed for spanning a representation space. Circuit variability was shown to be essential for adequately spanning a representation space [19], NEF's computational approach, therefore, embraces variability. However, we show that relying on process variation alone may require a large number of neurons. We demonstrated that by programming neuron tuning, we could better span a representation space per a given number of neurons. More importantly, even though a postsilicon calibration sequence can be used to compensate for process variation during neuron programming, we show  
Figure 8: **Application in adaptive neurorobotics. ** (**A**) A simplified schematic of the REACH model for adaptive robotic control; (**B**) A screenshot from the simulated arm and a designated target point in space; (**C**) Reaching eight target points while an external force is applied without adaptation; (**D**) Reaching eight target points while an external force is applied with adaptation using Nengo-based simulation (**left**) and OZ emulator (**right**). that our learning circuit can inherently compensate for it. Our circuit design can therefore learn to adapt to changes within itself. To conclude, in this work, we propose a design of a novel analog circuitry for online learning, implementing the PES neuromorphic learning rule using NEF-specified spiking neurons (OZ). We demonstrated for the first time the representation and transformation of high-dimensional mathematical constructs using NEF-specified spiking neurons using our online learning analog circuitry. We illustrate the design of dynamical oscillatory dynamics by combining NEF-driven neuromorphic representation and transformation with analog circuitry. We further propose a new circuit emulator (written with Python), allowing us to efficiently simulate our analog circuitry on a large scale (thousands of spiking neurons and learning circuits). Finally, we demonstrate adaptive control of a robotic arm using a physics-aware robotic framework (MuJoCo) and using analog spiking neurons, featuring representation, transformation, and our circuit emulator. The next design improvement to be further considered is our naive control circuit. The control circuit presented herein is limited, as the resistance values governing the neurons' tuning curves are constant.