(a) We model phase precession as a von Mises centred at a preferred theta phase which precesses in time. This factor modulates the spatial firing field. It is parameterised by (von Mises width parameter, aka noise) and (fraction of full 2π phase being swept, diagonal line). We showed in a previous figure that biological phase precession parameters are optimal. Any more or less phase precession degrades performance. It is easy to understand why: (b) Consider four place cells on a track (purple, blue, green, yellow) where the first and last just overlap. (c) In the weak phase precession regime, there is no ordering to the spikes and STDP can’t learn the asymmetry in the successor matrix (right) (d) In the medium phase precession regime, spikes are broadly ordered in time (purple then blue then green…) so the symmetry is broken and STDP learns a close approximation the successor matrix (e) In the ‘exaggerated’ phase precession regime, there exist two problems for learning SRs: ‘causal’ bindings (e.g. from presynaptic purple to postsynaptic yellow, which sits in front of purple) are inhibited for anything except the most closely situated cell pairs due to the sharp tuning curves. Secondly, though this is a less important effect, when is too large it is possible for incorrect “acasual” bindings to be formed due to one cell (e.g. yellow) firing late in theta cycle N just before another cell located far behind it on the track fires (e.g. purple) in theta cycle N+1. (f) CA1 cells will phase precess when driven by multiple CA3 place cells. Here, we show phase precession (spike probability for different theta phases against distance travelled through field) for CA3 basis features and CA1 STDP successor features after learning. Although noisier, there is still a clear tendency for CA1 cells to phase precess. Real CA1 cell phase precession can be ‘noisy’; we show for comparison a phase precession plot for CA1 place field taken from Jeewajee et al., 2014, the same data for which we fitted our parameters. The schematic simulation figures showing spiking phase precession data in panels b, c, d, and e were made using an open source hippocampal data generation toolkit (George et al., 2022). Panel f, right has been adapted from Figure 5a from Jeewajee et al., 2014. (g) (Left) A decreasing phase shift is measured between CA3 and CA1, starting from 90º late in the cycle – the phase cells initially spike at as animals enter a field – and ending at 0º early in the cycle, panel adapted from Mizuseki et al., 2012. (Middle) Three phase shifts (0º, 45º and 90º) are simulated and the average of the resulting synaptic weight matrices is taken (right).