The ability to perceive and learn timing is crucial for animals to interact with the world. Experiments have shown that during motor planning, certain neurons in multiple brain regions exhibit slow ramping activity that gradually increases until an action begins. This ramping activity is believed to be the neural basis of timing. However, the mechanisms by which these ramping activities emerge from network dynamics and how they are learned through reinforcement learning remain poorly understood. We propose a spiking neural network model and a biologically realistic learning rule to learn this ramping behavior that can last on the order of seconds. Our model consists of a randomly connected recurrent neural network (RNN) that integrates input from a decision-making network. Synaptic connections are trained using an eligibility trace-based learning rule. Using mean-field theory, we analyze the RNN to determine the conditions necessary for it to function effectively as an integrator. This model qualitatively reproduces the ramping patterns observed in mouse brains during decision-making tasks at the single-neuron and population levels. Furthermore, an extended version of our model accounts for additional experimental findings in mouse decision-making tasks.