It is believed that neurons communicate through neurotransmitters, released from multiple synapses and flow to the axon of the next neuron. But has it been shown if a single neuron communicate with a single neighbour neuron by its electric field? (not involving the chemical synaptic cleft).
Neuronal membranes act as capacitors: there is a charge separation between inside and outside due to the (relatively) impermeable membrane to charged ions. The voltage of a neuron is a charge separated across the membrane. However, the electric field outside a capacitor is negligible. In many physics problems it's treated as simply zero. See some Q&A on Physics.SE:
Because of this, there isn't any good reason from the physics of a neuron that you'd expect a neuron to be affected by the electric field of its neighbor. The field itself is tiny, and changes in the field are tinier still.
Some neurons communicate directly to each other through gap junctions, but that's because these connections cause a gap in the membrane that joins the inside space of one cell with the inside of another, allowing ions to flow freely between them. It's also possible for neighboring neurons to be influenced by changes in extracellular ion concentrations, but this really only happens in bulk circumstances such as cortical spreading depression, ischemia, or seizure, rather than normal neuronal communication.