Example (from one of the 5 highest likelihood solutions) for latent states of a PLRNN with <i>M</i> = 10 estimated from ACC multiple single-unit recordings during working memory (cf. Figs 9 and 10).

<p>Shown are trial averages for left-lever (blue) and right-lever (red) trials with SEM-bands computed across trials. Dashed vertical lines flank the 10 s period of the delay phase used for model estimation. Note that latent variables <i>z</i>₄ and <i>z</i>₅, in particular, differentiate between left and right lever responses throughout most of the delay period.</p

Computational performance of state inference (E-step) and full EM algorithm as the number of latent states is increased.

<p>(A) The number of full mode-search iterations, i.e. the number of constraint-sets <b>Ω</b> visited as defined through constraint vector <b>d</b> (cf. <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005542#pcbi.1005542.e031" target="_blank">Eq 7</a>) within one E-step, increases (sub-)linearly with the number <i>M</i> of latent states included in the model. (B) Likewise, the <i>total</i> number of mode-search steps (evaluated with <i>single-constraint</i> flipping here) summed across all EM iterations increases about linearly with <i>M</i> (single-constraint flipping requires about 10-fold more iterations than full-constraint flipping, but was observed to perform more stably). Note that this measure combines the number of EM iterations with the number of mode-search steps during each EM pass, and in this sense reflects the scaling of the full EM procedure. Performance tests shown were run on the experimental data sets illustrated in Figs <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005542#pcbi.1005542.g009" target="_blank">9</a>–<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005542#pcbi.1005542.g012" target="_blank">12</a>. Means were obtained across 40 different initial conditions (with each, in turn, representing the mean from 3x14 = 42 runs in A, or 14 runs in B, separately for each of 14 trials). Error bars = SEM (across initial conditions).</p

State and parameter estimates for nonlinear cycle example.

<p>(A) True (solid/ open-circle lines) and estimated (dashed-star lines) states over some periods of the simulated limit cycle generated by a 3-state PLRNN when true parameters were provided (for this example, <b>θ</b> ≈ (0.86,0.09,–0.85); all other parameters as in B, see also provided Matlab file ‘PLRNNoscParam.mat’). ‘True states’ refers to the actual states from which the observations <b>X</b> were generated. Inputs of <i>s</i>_<i>it</i> = 1 were provided to units <i>i</i> = 1 and <i>i</i> = 2 on time steps 1 and 10 of each cycle, respectively. Note that true and inferred states are tightly overlapping in this low-noise example (such that the ‘stars’ appear on top of the ‘open circles’). (B) True and estimated model parameters for (from top-left to bottom-right) <b>μ</b>₀,<b>A</b>,<b>W</b>,<b>Σ</b>,<b>B</b>,<b>Γ</b>, when true states (but not their higher-order moments) were provided. Bisectrix lines (black) indicate identity.</p

Initial (gray) and final (black) distributions of maximum (absolute) eigenvalues associated with all fixed points of 400 PLRNNs estimated from the experimental data (cf. Figs 9–11) with different initializations of parameters, including the (fixed) threshold parameters <i>θ</i>_<i>i</i>.

<p>Initial parameter configurations were deliberately chosen to yield a rather uniform distribution of absolute eigenvalues ≤ 3.</p

Log-likelihood of PLRNN and cross-validation performance of linear (LDS) and nonlinear (PLRNN) state space models on the ACC data.

<p>(A) Examples of log-likelihood curves across EM iterations from the 5/36 highest-likelihood runs for a 5-state PLRNN estimated from 19 simultaneously recorded prefrontal neurons on a working memory task (cf. <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005542#pcbi.1005542.g009" target="_blank">Fig 9</a>). State estimation here was performed by inverting only the single constraint corresponding to the largest deviation on each iteration (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005542#sec015" target="_blank">Methods</a>). (B) Cross-validation error (CVE) for the PLRNN (red curve) and the LDS (blue curve) as a function of the number of latent states <i>M</i>. CVE was assessed on each of 14 left-out trials with model parameters estimated from the remaining 14–1 = 13 experimental trials. Shown are squared errors averaged across all units <i>i</i>, time points <i>t</i>, and 40 different initial conditions. (C) Same as A, but with outputs estimated from states predicted Δ<i>t</i> = 1 (solid curves) or Δ<i>t</i> = 3 (dashed curves) time steps ahead. Note that in this case the PLRNN consistently performs better than a LDS for all <i>M</i>, with the PLRNN-LDS difference growing as Δ<i>t</i> increases. Error bars represent SEMs across those of the 40 initial conditions for which stable models were obtained (same for the means).</p

Agreement between simulated (<i>x</i>-axes) and semi-analytical (<i>y</i>-axes) solutions for state expectancies for the model from Fig 1 across all three state variables and <i>T</i> = 750 time steps.

<p>Here, <i>ϕ</i>(<i>z</i>_<i>i</i>) ≔ max{0,<i>z</i>_<i>i</i>−<i>θ</i>_<i>i</i>} is the PL activation function. Simulated state paths and their moments were generated using a bootstrap particle filter with 10⁴ particles. Bisectrix lines in gray indicate identity.</p

Full EM algorithm on working memory model.

<p>(A) Parameter estimates for ML solution from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005542#pcbi.1005542.g005" target="_blank">Fig 5</a>. True parameters (on <i>x</i>-axes or as blue bars, respectively), initial (gray circles or green bars) and final (black circles or yellow bars) parameter estimates for (from left to right) <b>μ</b>₀,<b>A</b>,<b>W</b>,<b>B</b>,<b>Γ</b>. Bisectrix lines in blue. Correlations between true and final estimates are indicated on top (note from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005542#pcbi.1005542.e057" target="_blank">Eq 17C</a> that the estimates for <b>μ</b>₀ are based on just one state, hence will naturally be less precise). (B) Distributions of initial (gray curves), final (black-solid curves), and final after reordering of states (black-dashed curves), deviations between estimated and true parameters across all 240 EM runs from different initial conditions. All final distributions were approximately centered around 0, indicating that final parameter estimates were relatively unbiased. Note that partial information about state assignments was implicitly provided to the network through the unit-specific inputs (and, more generally, may also come from the unit-specific thresholds <i>θ</i>_<i>i</i>, although these were all set to 0 for the present example), and hence state reordering only produced slight improvements in the parameter estimates.</p

Synaptic parameters.

<p>Synaptic parameters.</p

Short-term synaptic plasticity.

<p>Short-term synaptic plasticity.</p

Anatomical and synaptic properties.

<p>(A) Laminar structure of a single network column. Arrow widths represent relative strength of connections (black: excitatory, gray: inhibitory), i.e. the product of connection probability and synaptic peak conductance. (B) Left panel: Distribution of three different short-term plasticity types over different combinations of pre- and postsynaptic neuron types. Arrows from or to one of the shaded blocks (rather than from or to a single neuron type) denote connection types that are identical for all excitatory (PC) or inhibitory (IN) neurons. Where all three types are drawn, they are randomly distributed over all synapses between these two neuron types according to the probabilities given in the figure. Right panel: Illustration of the postsynaptic potential in response to a series of presynaptic spikes for three types of short-term synaptic plasticity for excitatory (E1 to E3) and inhibitory synapses (I1 to I3).</p