Quantum sensors using solid state qubits have demonstrated outstanding sensitivity, beyond that possible using classical devices. In particular, those based on colour centres in diamond have demonstrated high sensitivity to magnetic field through exploiting the field-dependent emission of fluorescence under coherent control using microwaves. Given the highly biocompatible nature of diamond, sensing from biological samples is a key interdisciplinary application. In particular, the microscopic-scale study of living systems can be possible through recording of temperature and biomagnetic field. In this work, we use such a quantum sensor to demonstrate such microscopic-scale recording of electrical activity from neurons in fragile living brain tissue. By recording weak magnetic field induced by ionic currents in mouse corpus callosum axons, we accurately recover signals from neuronal action potential propagation while demonstrating in situ pharmacology. Our sensor allows recording of the electrical activity in neural circuits, disruption of which can shed light on the mechanisms of disease emergence. Unlike existing techniques for recording activity, which can require potentially damaging direct interaction, our sensing is entirely passive and remote from the sample. Our results open a promising new avenue for the microscopic recording of neuronal signals, offering the eventual prospect of microscopic imaging of electrical activity in the living mammalian brain.