Extended Data Fig. 1: Performance of the linear Hebbian/anti-Hebbian network in the PSP task.

(a,b) The PSP error as quantified by \(\left\| {{{{\mathbf{F}}}}_{t}^{{{\bf{ \top }}}}{{{\mathbf{F}}}}_{t} - {{{\mathbf{U}}}}{{{\mathbf{U}}}}^{{{\bf{ \top }}}}} \right\|_F/\left\| {{{{\mathbf{UU}}}}^{{{\bf{ \top }}}}} \right\|_F\), where U is a n × k matrix whose columns are the top k left singular vectors of \({{\mathbf{X}}} \equiv \left[ {{{{\mathbf{x}}}}_1, \cdots ,{{{\mathbf{x}}}}_T} \right]\) and F_t≡M⁻¹_tW_t, drops very quickly during training (a) and maintains the low error in the presence of synaptic noise (b). (c) The relative change of the similarity matrix at time t compared to time point 0, corresponding to the point where the network initially learned the task, defined as \(\left\| {{{{\mathbf{Y}}}}_{t}^ \top {{{\mathbf{Y}}}}_{t} - {{{\mathbf{Y}}}}_0^ \top {{{\mathbf{Y}}}}_0} \right\|_{{{\mathrm{F}}}}/\left\| {{{{\mathbf{Y}}}}_0^ \top {{{\mathbf{Y}}}}_0} \right\|_{{{\mathrm{F}}}}\). (d) Estimating rotational diffusion constant \(D_{{{\mathrm{\varphi }}}}\) from mean squared angular displacement (MSAD). Gray lines are MSAD estimated based on individual representation trajectory \(\mathbf{y}(t)\). The dashed line is a linear fit between \(\left\langle {\left( {{{\Delta }}\varphi } \right)^2} \right\rangle \equiv \left\langle {\left( {{{{{\varphi }}}}\left( {t + {{\Delta }}t} \right) - {{{{\varphi }}}}\left( t \right)} \right)^2} \right\rangle\) and \(\Delta t\) to estimate the rotational diffusion constant. Inset: illustration of Δφ. Parameters are the same as Fig. 2 in the main text.