Illustration of the interval timing model.

The PFC network (center gray box) with layer 2/3 and layer 5 neurons (excitatory: blue, inhibitory: red, connectivities: gray arrows) is stimulated in the beginning and the end of several inter-stimulus intervals (color-coded from 50 ms to 750 ms). Spike trains are extracted within a window around the second stimulus to compute network states and to train weights of the readout layer via least squares to predict the estimated inter-stimulus intervals ∆ttest.

Psychophysical law and scalar property

A Tuning curves of the readout units (color-coded) with standard deviation for different test intervals (x-axis). B Averaged subjective time (over 50 trials) as a function of test interval durations together with the standard deviations (shaded region), the linear fit (dashed-dotted-line) and the objective time (dashed black line). C Standard deviations of the estimated time as a function of test interval durations, fitted to a linear (red curve) and to a square root function (blue curve). D The trial-averaged Weber fraction as a function of test interval durations. The horizontal line (in green) depicts the average value between (400 - 750 ms).

Assessing the scalar property

Standard deviations of the estimated time as a function of test interval durations averaged across ten different parameter sets. The red line shows the best fit to a piecewise linear function, saturating for ∆ttest above a threshold (see the text for details). The ten different parameter sets were only used for this figure.

Fit parameters for the scalar property over ten parameter sets with ∆ttest = 10 ms.

Effects of dopaminergic modulation

A Subjective duration as a function of test interval durations between 200 ms - 600 ms without modulation (black curve), for agonistic (blue curves) and antagonistic (red curves) D2 modulation. B The slope of the linear psychophysical law for each modulation of synaptic and neuronal parameters together (black curve), for synaptic modulation alone (magenta curve) and for neuronal modulation alone (green curve). The dashed vertical line indicates the point without modulation. C Standard deviations of the estimated time as a function of test interval durations for D2 modulation between −50 % and 20 %.

Retraining after mild dopaminergic modulation.

Estimated times (panel A) and the corresponding standard deviations (panel B) as a function of test interval durations for agonistic (30 %, red) and antagonistic modulation (− 30 %, blue) after retraining. The estimated times and the standard deviations without modulation were re-plotted from Fig. 2 in black for comparison.

Testing for limitations of timing in the model.

Estimated time (panels A, C) and standard deviations (panels B, D) as a function of test interval durations for longer durations up to 2000 ms (panels A, B) and for higher noise levels (panels C, D). Lighter colors in panels C and D represent higher noise levels (see legend in C). The corresponding Poisson weights and the fit parameters are shown in Table 2.

Parameters for different noise levels with 1000 Hz Poisson neurons

Ablation studies.

Estimated durations (panels A-C) and their standard deviations σ (panels D-F) averaged over five parameter sets as a function of test interval durations for different ablations. A, D Removing synaptic processes, namely NMDA (magenta curves) and GABAB (blue curves) currents, short-term plasticity (STP, red curves) and the combination of all three (cyan curves). B, E Removing heterogeneity of the neuronal parameters (red curves), of the synaptic parameters (blue curves), and of both (magenta curves), additional reduction to one type of STP within each pair of neuron types (light blue curves) and removing the heterogeneity of synaptic delays only (green curves). C, F Removing the background current within the PFC model (red curves). In all panels, the original results from Fig. 2B and C are re-plotted in black. The corresponding fit parameters are shown in Suppl. Table A.3 and A.4.

Readout weights and average firing rates within interval-encoding pools

A. Neurons are associated with readout units based on the largest weight w compared to the weights of other readout units. The weight matrix is then first sorted by the association to readout units, and within each group, sorted by their strongest weights to the respective readout unit. The horizontal lines show the borders of the pools of neurons, and the vertical lines separate the encoded intervals. The color scale follows a log10-scale. B Normalized averaged readout weights of neurons within a pool as a function of the normalized encoded interval durations of each pool (color-coded by the readout intervals). The dashed line is the best fit to a sum of two Gaussians of all curves. C The averaged states over 50 trials within a pool normalized for each test interval normalized by the encoded duration of each pool color-coded by the readout intervals, see legend in B. The dashed line represents the best fit of the stereotypic firing rate profiles to a sigmoid function (dashed line).

Synaptic currents within the pools.

Average excitatory (panel A) and inhibitory (panel B) synaptic currents of neurons within each group as a function of simulated time. The three circles and the dashed lines mark three time ranges for which the synaptic currents are compared across pools, namely before stimulation (t = − 500 ms to t =− 5 ms, marked as 1, panel C), during first stimulation (t = − 5 ms to t = 45 ms, marked as 2, panel D), and during the second stimulation (t = − 5 ms + ∆t to t =− 5 ms + ∆t + 50 ms marked as 3, panel E). For each case, average excitatory (blue curves), inhibitory (red curves), and overall currents (black curves) are shown for each pool as a function of the duration encoded in each pool.

Synaptic currents within IEP averaged before stimulation (t = −500 ms to t = −5 ms), after the first stimulation (t = −5 ms to t = 45 ms) and after the second stimulation (t = −5 ms + ∆t to t = −5 ms + ∆t + 20 ms) for all encoding intervals

Excitability of interval-encoding pools

A Averaged rheobase (see Methods for details) and B the difference of the rheobase to the total synaptic current within each pool before stimulation as a function of the duration encoded in each pool, cf. black line in Fig. 9C.

Neuronal and synaptic properties within IEPs, as a function of the duration encoded in each pool

A Averaged synaptic weights onto neurons within IEPs for positive weights only (blue curves), negative weights only (red curves), and all weights (grey curves). B Same as in A for the number of inputs and C for the average synaptic delays. D Averaged time constants of short-term facilitation (τfac) and depression (τrec) for excitatory neurons within IEPs. E Same as D for inhibitory neurons. F Averaged membrane time constants τmem for both neuron types within each pool.

Overview on the derivation of the scalar property

A Translating the activity within interval-encoding pools (bottom, blue: excitatory, and red: inhibitory) into timing estimates and their standard deviations (top). B,C Multiplying the readout units with their corresponding time duration and summing up over all readout units leads to the estimated times and the scalar property. D Tuning curveshaped outputs are generated within the readout neurons with a defined relation between mean and standard deviation (panel E). F Synaptic weights from the pools to the output neurons, forming a stereotypic temporal receptive field. G Firing probabilities peaking at one close to the encoded time of each of the pools. H Relation of mean and standard deviations of the firing rates, following from the binomial distribution.

Binomial distribution of firing rate probabilities.

Standard deviation of the firing probabilities as a function of the mean. The curves are well-fitted by the relation predicted by the binomial distribution (dotted line). For longer intervals, smaller firing rates and higher standard deviations occur. The colors correspond to the intervals that are represented in each pool (see the color bar, cf. Fig. 2A).

Results from the minimal model.

Normalized firing rate as a function of simulated time, shown as a fraction of the encoded interval duration of each pool (color-coded). Each curve was generated by a different threshold for the membrane potential to elicit a spike. Higher thresholds lead to longer durations that need to elapse before that first spike (cf. Fig. 10). When scaling this time by the duration Ti at which the neuron spikes with 95 %, the firing rate curves for all durations largely overlap, as in Fig. 8D for the full model.

Relation between Vierordt’s law and the scalar property.

Comparing the minimal standard deviation of the output curves Oj against the slope of Vierordt’s law for the different cases, results in a strong negative linear trend. The smaller the standard deviation, the better the estimation.

Extracting the states by defining a window.

A. Rasterplot of all 1000 neurons with two stimulations (t0 = 0 ms for 10 ms and at t2 = 100 ms for 10 ms) to reproduce an interval of ∆t2 = 100 ms. The red box identifies the window in which the states from the spike trains are extracted (right: zoomed-in view). B. Result of a grid search to find the optimal parameters for the window with varying window size and end time. The best parameter set was selected by the minimum RMSE value (red box).

Synaptic parameters of GABAB.

Parameter alterations for agonistic D2 modulation.

Matching spike statistics of the PFC and the state-dependent PFC model

The raster plots are shown for the PFC model without GABAB in A and for the statedependent PFC model with GABAB in B. C The numbers of active neurons are depicted as histograms for the averaged ISI for the state-dependent PFC model (green) (μ = 101.1 ms, σ = 93.2 ms and the PFC model (black) (μ = 106.3 ms, σ = 100.6 ms) for neurons with more than 10 spikes. D The CV per neuron for the same neurons (spikes > 10) is calculated for the PFC model in black (μ = 0.4, σ = 0.3) and for the state-dependent PFC model in green (μ = 0.4, σ = 0.4). E The standard deviation of the subthreshold membrane potential over time for non-zero firing neurons is higher for the state-dependent PFC (μ = 3.8 mV, σ = 2.4 mV) than for the PFC model (μ = 2.2 mV, σ = 1.3 mV) as shown with the histograms.

Spike trains for exemplary inter-stimulus intervals ∆t

The raster plots of the main simulation as shown in Fig. 2 are depicted for the training intervals ∆t1 = 50 ms in A, ∆t4 = 200 ms in B, ∆t8 = 400 ms in C and ∆t12 = 600 ms in D.

Ridge regression

Estimated times A and standard deviations B, when using ridge regression instead of the linear least squares method to calculate readout weights. Results are almost identical compared to the linear least squares method; cf. Fig. 2.

Dopamine modulation and the change of the subjective time within the full range.

The estimated time for antagonistic (− 100 % - − 10 %) and agonistic (10 % - 100 %) dopaminergic modulation is presented here. Each color represents the tested interval, such that the evolution of the estimate of one interval can be considered over the modulation. An overestimation and an underestimation of time can only be found within the range of 200 - 600 ms and for percentages below ±50 %.

Outputs of the 400 ms readout neuron with altered DA modulation.

A With increasing antagonistic dopaminergic modulation (from darker to lighter colors), the activation of the 400 ms interval-encoding unit is activated at later time points. B With increasing agonistic modulation the same unit is activated at earlier intervals. The output values decay with increasing modulation.

Standard deviations of the dopaminergic modulations.

The standard deviations with respect to unmodulated Webers law (black) are shown for antagonistic A and agonistic B dopamine modulations of ±10 to ±100 % within the full range and C the slopes of the estimated times for each modulation fitted with a linear regression (red).

Subthreshold membrane potential fluctuations for different cases of Poisson noise

The standard deviation of the sub-threshold membrane potentials over time for non-zero firing neurons for the original simulation with 1 Hz in black and for different levels of Poisson noise with 1000 Hz. The mean μ and variance σ for the distribution of the histogram can be extracted from the legend.

Neuronal and synaptic properties within interval-encoding pools below the readout threshold of 0.01

A The averaged summed synaptic weight B, the connectivities C and the averaged synaptic delays onto an interval-encoding pool for excitatory (blue), for inhibitory (red) and for all weights (black). For the synaptic delays, only inhibitory synaptic delays are significant (slope = −0.0002, R2 = 0.6, p = 0.02). The time constants of STP, τfac and τrec, for excitatory D and inhibitory E neurons within a pool. F The averaged membrane time constants τmem for neurons within IEPs below the threshold for both neuron types.

Parameter adjustments for ablation analysis. The changes were applied in relation to the PFC model from [27].

Linear regression was calculated for parameters (see Fig. 11) within the interval selective pools over the full range (50 - 750 ms).

Ablation studies to identify the components of linear timing

Ablation studies to identify the components of Weber’s law

Assessing the origin of the scalar property

A.Standard deviations of the output neuron activity as a function of their mean activity.

B.Weber fractions (black curve, standard deviation of the duration estimated divided by its mean) and square root of the sum of the variances of all output neurons (red curve) for each of the encoded interval durations.

Interpreting the tuning curves of the output neurons as a probability density function

A Illustration of multiple tuning curves (mean-shifted Gaussians with constant standard deviations) evaluated at a given time t∗ . B The values of the different tuning curves Oj can be interpreted as the values of a single Gaussian O (centered at t∗) at different times that correspond to the shifted means. Note that the area under this Gaussian is one, so O(t) can be interpreted as a probability distribution function for the time t elapsed relative to t∗.