## 2 Materials and Methods
### Image Classification Methods
#### 2.3.2 Recurrent Neural Networks  
The RNN is a class of artificial neural networks designed to process sequential data, such as time series, text, speech, and video [30]. Unlike feedforward neural networks, which process inputs independently, RNNs maintain an internal memory, allowing them to capture temporal dependencies and patterns within the data. At each time step \(t\), an RNN receives an input \(x_{t}\) and the hidden state \(h_{t-1}\) from the previous time step. It then computes a new hidden state \(h_{t}\) using a set of learnable parameters (weights and biases) and an activation function. This hidden state \(h_{t}\) serves as a representation of the input sequence up to time step t. The process can be mathematically represented as follows:  
\[h_{t}=f(W_{hx}x_{t}+W_{hh}h_{t-1}+b_{h}), \tag{1}\]  
where  
* \(x_{t}\) is the input at time step \(t\);
* \(h_{t-1}\) is the hidden state from the previous time step;
* \(W_{hx}\) and \(W_{hh}\) are weight matrices;
* \(b_{h}\) is the bias vector;
* \(f\) is the activation function (e.g., sigmoid, tanh, ReLU).  
With the hidden state calculated from the previous time step, the output of an RNN unit can be calculated as follows:  
\[y_{t}=f(Wh_{t}+b_{o}), \tag{2}\]  
where  
* \(y_{t}\) is the output at time step \(t\);
* \(h_{t}\) is the hidden state from the previous time step;
* \(W\) is the weight matrix connecting the hidden to the output layer;
* \(b_{0}\) is the bias for the output layer;
* \(f\) is the activation function applied element-wise to the weighted sum \((Wh_{t}+b_{o})\).  
RNNs are optimized using the backpropagation through time (BPTT) [31] algorithm, which extends the standard backpropagation algorithm to account for the temporal dependencies in sequential data. This makes RNNs particularly effective for tasks involving sequential or time-varying data, such as natural language processing, speech recognition, and time series prediction.  
However, training RNNs can be challenging due to the issue of vanishing or exploding gradients, where gradients become too small or too large as they propagate backward through time. This problem can hinder the learning process, especially in long sequences. To mitigate this issue, techniques like gradient clipping and specialized architectures such as LSTM networks and Gated Recurrent Units (GRUs) have been developed.  
LSTM networks are a type of RNN architecture designed to address the vanishing gradient problem of traditional RNNs and effectively capture long-term dependencies in sequential data [32]. They achieve this by introducing specialized memory cells and gating mechanisms that regulate the flow of information through the network over time.  
Figure 4: Structure of a typical convolutional neural network.  
This research will involve an extensive training and comparison of the designed model with a hybrid method that uses LSTM networks. The goal is to validate the effectiveness of Liquid Time-Constant (LTC) networks in comparison to LSTM networks. The same LSTM-based approach was utilized in studies referenced in [15] and [17]. This comparison aims to assess the benefits of using the LTC in terms of accuracy, computational efficiency, and robustness, as opposed to previous studies that focused on the LSTM.