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The PANOSETI experiment is an all-sky, all-the-time visible search for nanosecond to millisecond time-scale transients. The experiment will deploy observatory domes at several sites, each dome containing ~45 telescopes and covering ~4,440 square degrees. Here we describe the focal-plane electronics for the visible wavelength telescopes, each of which contains a Mother Board and four Quadrant Boards. On each quadrant board, 256 silicon photomultiplier (SiPM) photon detectors are arranged to measure pulse heights to search for nanosecond time-scale pulses. To simultaneously examine pulse widths over a large range of time scales (nanoseconds to milliseconds), the instrument implements both a Continuous Imaging Mode (CI-Mode) and a Pulse Height Mode (PH-Mode). Precise timing is implemented in the gateware with the White Rabbit protocol.