This article studies the expressive power of spiking neural networks with firing-time-based information encoding, highlighting their potential for future energy-efficient AI applications when deployed on neuromorphic hardware. The computational power of a network of spiking neurons has already been studied via their capability of approximating any continuous function. By using the Spike Response Model as a mathematical model of a spiking neuron and assuming a linear response function, we delve deeper into this analysis and prove that spiking neural networks generate continuous piecewise linear mappings. We also show that they can emulate any multi-layer (ReLU) neural network with similar complexity. Furthermore, we prove that the maximum number of linear regions generated by a spiking neuron scales exponentially with respect to the input dimension, a characteristic that distinguishes it significantly from an artificial (ReLU) neuron. Our results further extend the understanding of the approximation properties of spiking neural networks and open up new avenues where spiking neural networks can be deployed instead of artificial neural networks without any performance loss.