The Hodgkin and Huxley neuron model describes the complex behavior of biological neurons. However, due to the complexity of these computations, the Hodgkin and Huxley models are impractical for use in large-scale networks. In contrast, Izhikevich introduced a simpler model capable of producing various firing patterns typical of cortical neurons. This study proposes a novel model of memcapacitive-based neurons that offers a potential implementation of spiking neurons with energy efficiency due to the inherent storage nature of memcapacitive devices. The findings demonstrate that memcapacitive neurons can produce 23 firing patterns similar to Izhikevich neurons but at significantly higher firing rates. Memcapacitive neurons exhibit firing patterns associated with excitatory, inhibitory, and thalamocortical neurons. Similar to Izhikevich neurons, pulse-coupled neural networks of memcapacitive neurons display collective behaviors, such as synchronous and asynchronous responses, which are common in the biological brain. Compared to Hopfield and Izhikevich networks in content-addressable memory applications, memcapacitive networks successfully retrieved correct memory patterns with high accuracy, even for distorted inputs of up to 40%. The simulation results illustrate that the novel model of the memcapacitive spiking neuron offers a potential advancement in implementing artificial spiking neurons with high energy efficiency, bringing a step closer to mimicking biological neurons.