The effect of the non-linear function on system dynamics within delay-feedback reservoirs

Delay-feedback reservoirs are a subset of reservoir computers characterized by a hardware-efficient architecture that trades spatial complexity for temporal processing. It employs a single non-linear node, a delay line, and a time-multiplexed input signal to generate a network of "virtual nodes," effectively emulating a larger spatial neural network. One of the most powerful aspects of delay-feedback reservoirs is their versatility. Our previous work found that the non-linear node performs two mathematical functions, a non-linear transform and integration. The non-linear transform can be represented by any number of non-linear functions, making it difficult to optimize a delay-feedback reservoir to solve a specific computational task. This work explores different non-linear functions in order to determine their effect on the dynamics of the reservoir, in order to provide insight into this optimization problem. Five different non-linear functions are compared in terms of performance, metrics, and utilization: Mackey-Glass, sine squared, double sinusoids, Tan, and Tanh. Our results find that the Mackey-Glass non-linear function shows limited system dynamics, performing well on non-linear tasks but performing poorly on memory intensive tasks. We then demonstrate the distinct system dynamics within the other four non-linear functions. We found that sine squared shows limited overall performance, double sinusoid performs well in non-linear tasks, Tan resembles an odd valued exponent Mackey-Glass reservoir but with greater parameter sensitivity, and tanh offers balanced performance across both task types. We find that modifying the system dynamics of a reservoir is an important step toward optimizing a delay-feedback reservoir for specific computational tasks.