## 4 Result  
### Time-Constant Effects  
This section presents the results of using the G-CTRNN for human activity classification. We built the G-CTRNN and compared it to the standard RNN. We also examine the importance of training the time constant parameter and its significant impact on the accuracy of the G-CTRN model. Toward this objective, we conducted different experiments. The initial investigation involved training the model by setting a time constant value equal to a constant value of one in all sixteen cells of the first layer, which is the standard practice in the literature. The time constant values were next initialized in the subsequent experiment using a uniform random distribution with a range of [0, 2]. Finally, for the first time, the time constant of each cell in G-CTRNN is trained using backpropagation through time. The models were trained for 50 epochs in all scenarios, and the G-CTRNN results were compared to the standard RNN. The RNN model shows poor performance with inconsistent accuracy as the number of epochs increases despite all the models using the same optimizer and starting with the same learning rate. The G-CTRNN model with the time constant set to one shows an accuracy of 68%, and the model successfully converged smoothly throughout 50 epochs. The G-CTRNN model had a better learning opportunity when the time constant was randomly initialized (resulting in varying time constant values for each cell). This is because different time constant values imply different response speeds for each cell, essential for extracting knowledge from the sequence data. The accuracy in this scenario improved to 76%, as depicted in Figure 5. Finally, an accuracy of 81.9% was achieved when the G-CTRNN time constants were included in the training. In summary, Figure 5 shows that G-CTRNN exhibits smoother convergence and improved stability across epochs compared to traditional RNNs, which suffer from inconsistent learning and catastrophic forgetting. The ability to adjust time constants dynamically contributes to G-CTRNN's robust and efficient convergence behavior. Finally, Table 1 summarizes our findings concerning the different time-constant scenarios. All models started with the same learning rate to ensure a fair comparison: 0.001, the default value for ADAM algorithms in Keras 2.8. In addition to the time-constant variations, we conducted internal ablation studies to validate key design choices. Removing the MINMAX constraint on the time constant resulted in numerical instability and occasional divergence during training, confirming the  
Figure 5: Validation dataset accuracy for different G-CTRNN configurations and RNN model: the training time constant is the best, random initialization is the second best, and static time constant is the worst, compared to the RNN, which suffers from catastrophic forgetting. necessity of enforcing physical bounds. Similarly, using standard discrete-time backpropagation instead of continuous-time BPTT led to significant performance degradation and failure to converge when training the time constant, underscoring the importance of using continuous-time gradient propagation for temporal consistency and parameter sensitivity. Finally, it is worth noting that while our experimental comparison centers on traditional RNNs due to their structural similarity to the G-CTRNN, we note that other lightweight models such as Gated Recurrent Units (GRUs), Echo State Networks (ESNs), and Spiking Neural Networks (SNNs) have also been proposed for efficient temporal modeling. GRUs simplify gating mechanisms to reduce parameter count, ESNs offload learning to the readout layer but require careful reservoir tuning, and SNNs operate with biologically inspired spike timing but often demand event-based hardware. In contrast, G-CTRNN offers a compact continuous-time formulation amenable to analog or MEMS-based hardware, providing a balance between trainability and low-power implementation. Future work will expand experimental comparisons to include these models.