## 2 Materials and Methods
### Image Classification Methods
#### 2.3.5 Closed-Form Continuous-Time Neural Network  
The Closed-form Continuous-time Neural Network (CfC) model extends the capabilities of LTC networks by providing a closed-form solution to the differential equations that define their dynamics [36]. This advancement addresses the computational challenges posed by traditional numerical solvers used in the LTC, enabling more efficient real-time processing and scalability in applications such as time-series prediction and sequential decision-making.  
The CfC model builds on the foundational architecture of LTC networks, where neurons and synapses interact over continuous time. A significant improvement is the ability to translate these continuous interactions into a closed-form expression, which eliminates the need for iterative numerical solvers and, thus, significantly enhances computational efficiency.  
The mathematical formulation of LTCs is based on the differential equation:  
\[\frac{dx(t)}{dt}=-\bigg{[}\frac{1}{\tau}+f(x(t),I(t),t;\theta)\bigg{]}x(t)+f(x (t),I(t),t;\theta)A, \tag{6}\]  
where  
* \(x(t)\) is the hidden state at time t;
* \(I(t)\) is the input;
* \(f(x(t),I(t),t;\theta)\) is a neural network function with parameter \(\theta\) that determines the rate of change of \(x(t)\) with respect to time;
* \(\tau\) represents a time constant;
* \(A\) is an adjacency matrix representing connections.  
For piecewise constant inputs, the differential equation can be solved in a closed form. Given a sequence of time points \(\{t_{i}\}\) and assuming \(I(t)\) is constant between \(t_{i}\) and \(t_{i+1}\), the hidden state \(x(t)\) can be approximated as follows:  
\[x(t_{i+1})=e^{-\frac{1}{t}(t_{i+1}-t_{i})}x(t_{i})+\bigg{(}\int_{t_{i}}^{t_{i+ 1}}e^{-\frac{1}{t}(t_{i+1}-s)}ds\bigg{)}f(x(t_{i}),I(t_{i}),t_{i};\theta)A, \tag{7}\]  
This expression can be further simplified to  
\[x(t_{i+1})=e^{-\frac{1}{t}\Delta t}x(t_{i})+\frac{1}{\tau}\Big{(}1-e^{-\frac{ 1}{t}\Delta t}\Big{)}f(x(t_{i}),I(t_{i}),t_{i};\theta)A, \tag{8}\]  
where \(\Delta t=t_{i+1}-t_{i}\).  
For continuous inputs, the integral can be approximated by  
\[x(t)\approx e^{-\frac{1}{t}t}x(0)+\frac{1}{\tau}\Big{(}1-e^{-\frac{1}{t}t} \Big{)}f(x(0),I(0),0;\theta)A \tag{9}\]This closed-form approximation ensures a tight bound on the approximation error, providing high accuracy in modeling temporal dynamics.  
By transforming the integral solution of LTCs into a closed form, the CfC model significantly reduces the computational complexity associated with numerical solvers, resulting in up to a 100-times speed increase in both training and inference times. The CfC model scales efficiently compared to traditional ODE-based networks, allowing for larger network architectures without a proportional increase in computational resources. Despite the simplification, the CfC model retains the expressive power of the LTC, performing exceptionally well in time-series prediction tasks. Empirical results demonstrate that the CfC model closely approximates the dynamics of ODE-based LTC networks with minimal error.  
This model is particularly useful in real-time applications, such as autonomous driving, where rapid and accurate processing of sequential data is critical. The closed-form solution enables real-time decision-making and control without sacrificing accuracy. By leveraging the strengths of LTC and eliminating their computational drawbacks, the CfC model represents a significant advancement in continuous-time neural network design, offering a practical solution for real-time applications that require both speed and accuracy.