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Abstract Memristors for neuromorphic computing have gained prominence over the years for implementing synapses and neurons due to their nano-scale footprint and reduced complexity. Several demonstrations show two-dimensional (2D) materials as a promising platform for the realization of transparent, flexible, ultra-thin memristive synapses. However, unsupervised learning in a spiking neural network (SNN) facilitated by linearity and symmetry in synaptic weight update has not been explored thoroughly using the 2D materials platform. Here, we demonstrate that graphene/MoS₂/SiO_x/Ni synapses exhibit ideal linearity and symmetry when subjected to identical input pulses, which is essential for their role in online training of neural networks. The linearity in weight update holds for a range of pulse width, amplitude and number of applied pulses. Our work illustrates that the mechanism of switching in MoS₂-based synapses is through conductive filaments governed by Poole-Frenkel emission. We demonstrate that the graphene/MoS₂/SiO_x/Ni synapses, when integrated with a MoS₂-based leaky integrate-and-fire neuron, can control the spiking of the neuron efficiently. This work establishes 2D MoS₂as a viable platform for all-memristive SNNs.