Integrator dynamics in the cortico-basal ganglia loop for flexible motor timing

Experimental model and participant details

Mice

This study is based on both adult male and female mice (aged > P60). We used five mouse lines: C57BL/6 J (JAX# 000664), VGAT-ChR2-eYFP⁶² (JAX #14548), Drd1–cre FK150 (ref. ⁶³), Adora2–cre KG126 (ref. ⁶³) and R26-LNL-GtACR1-Fred-Kv2.1 (ref. ⁴⁷) (JAX #33089). See Supplementary Table 1 for mice used in each experiment.

All procedures were in accordance with protocols approved by the MPFI IACUC committee. We followed the published water restriction protocol⁶⁴. Mice were housed in a 12–12 reverse light–dark cycle and behaviourally tested during the dark phase. Ambient temperature was 74 °F and humidity ranged between 35% and 60%. A typical behavioural session lasts between 1 h and 2 h. Mice obtained all of their water in the behaviour apparatus (approximately 0.6 ml per day). Mice were implanted with a titanium headpost for head fixation⁶⁴ and single housed. For cortical photoinhibition, mice were implanted with a clear skull cap³⁷. For bilateral D1/D2-SPN silencing, tapered fibre optics⁶⁵ (1.0-mm taper, NA 0.37 and core diameter of 200 µm, Doric lenses) were bilaterally implanted during the headpost surgery around the following target coordinates (Bregma): anteroposterior −0.3 mm, mediolateral ±3 mm and dorsoventral 3.5 mm for the VLS; and anteroposterior 0.6 mm, mediolateral ±1.5 mm and dorsoventral 3 mm for the dorsal medial striatum. Craniotomies for recording were made after behavioural training.

Viral injection

To virally express stGtACR1 (ref. ⁶⁶) in the striatum, we followed published protocols⁶⁷ for virus injection. AAV2/5 CamKII-stGtACR1-FusionRed (titre: 9.5 × 10¹²) was injected into anteroposterior −0.3 mm, mediolateral 3 mm, dorsoventral 2.75 and 3.5 mm, 100 nl each depth. The same tapered fibre optics described above were bilaterally implanted at dorsoventral 3.5 mm.

Behaviour

At the beginning of each trial, an auditory cue was presented, which consisted of three repeats of pure tones (3 kHz, 150-ms duration with 100-ms inter-tone intervals, 74 dB). A delay epoch started from the onset of the cue presentation. Licking during the delay epoch aborted the trial without a water reward, followed by a 1.5-s timeout epoch. Licking during the 10-s answer epoch following the delay was considered a ‘correct lick’, and a water reward (approximately 2 µl per drop) was delivered immediately, followed by a 1.5-s consumption epoch. If mice did not lick during the 10-s answer period, the trial would end without a reward. Trials were separated by an ITI randomly sampled from an exponential distribution with a mean of 3 s, with 1-s offset (with a maximum ITI of 7 s). This prevented mice from predicting the trial onset without a cue. Animals had to withhold licking during the full ITI epoch for the next trial to begin (otherwise, the ITI epoch repeated). In approximately 10% of randomly interleaved trials, the auditory cue was omitted to assess spontaneous lick rate (‘no-cue’ trials). No water reward was delivered in no-cue trials.

We followed the protocol described in Majumder et al.⁶⁸ for training. In brief, the delay duration increased from 0.1 s to 1.8 s gradually based on the performance of the animal⁶⁸. Once mice reached 1.8-s delay, we started either the switching delay, the random delay or the constant delay conditions (see Supplementary Fig. 3 for example sessions). In the switching delay condition, we switched the delay between 1 s versus 3 s or 1 versus 1.8 s every 30–70 trials (the number of trials was randomly selected from 30 to 70 and not contingent upon behaviour). Similarly, in the random delay condition, we randomly switched the delay among 0.5, 1.0, 1.5, 2.0, 3.0 or 5.0 s every 30–70 trials. For the constant delay condition, mice were trained with a constant delay of 1.5 s across sessions for at least 2 weeks. For the cue-intensity experiments (Extended Data Fig. 5), we changed the cue intensity (3-kHz auditory cue, ±15 dB, lasting 0.6 s) in randomly interleaved test trials (approximately 20%). Except for this modification, the task structure was identical. Cue intensity stayed constant (74 dB) before the cue-intensity experiments. Otherwise, the task design and reward contingency remained the same. ALM and striatal perturbation experiments (Figs. 4 and 5) were performed under the switching delay condition. To avoid human bias, the behaviour was automatically controlled by Bpod (Sanworks) and custom MATLAB codes.

Optogenetics

Photostimulation was deployed on less than 25% in randomly selected trials. To prevent mice from distinguishing photostimulation trials from control trials using visual cues, a ‘masking flash’ (1-ms pulses at 10 Hz) was delivered using 470-nm LEDs (Luxeon Star) throughout the trial. For both ChR2 and stGtACR1, we used a 488-nm laser (OBIS 488–150C, Coherent).

The ChR2-assisted photoinhibition of the dorsal cortices was performed through clear-skull cap³⁷ (Fig. 2e) or craniotomy (in case of simultaneous recording; Fig. 4). We scanned the 488-nm laser light using Galvo mirrors. We stimulated GABAergic interneurons in Vgat-ChR2-eYFP mice starting at 0.6 s after the cue, lasting for 1.2 s (including 0.2-s ramping down; Fig. 2e) or 0.6-s duration (including 0.3-s ramping down; Fig. 4). Time-averaged laser power was 1.5 mW per spot (or 0.3 mW per spot for Extended Data Fig. 12; 8 spots in total: 4 spots in each hemisphere centred around the target coordinates with 1-mm intervals; we photoinhibited each spot sequentially at the rate of 5 ms per step). For Fig. 2e, the targeted brain area was randomly selected for each photostimulation trial. The target coordinates were anteroposterior 2.5 mm and mediolateral ±1.5 mm for the ALM; anteroposterior 0.5 mm and mediolateral ±1.5 mm for M1B; anteroposterior 0.5 mm and mediolateral ±2.5 mm for S1TJ; anteroposterior −1.0 mm and mediolateral ±1.5 mm for S1TR; anteroposterior −1.0 mm and mediolateral ±3.0 mm for S1B; anteroposterior −2 mm and mediolateral ±1.5 mm for PPC; and anteroposterior −2.5 mm and mediolateral ±3.5 mm for V1, respectively (Bregma).

To silence D1-SPNs using stGtACR1 (Fig. 5), we delivered photostimuli (0.25 mW or 0.5 mW, 488 nm) bilaterally (Fig. 5l–o) or unilaterally (in case of optrode; Fig. 5h–k) in the striatum starting 0.6 s after the cue and lasting for 0.6 s (including 0.3-s ramping down). In precue inhibition trials, photostimuli were delivered 0.81 s, 0.6 s before the cue for the ALM, D1-SPN perturbation, respectively, both lasting for 0.6 s. The light was delivered through implanted fibre optics, and intensity was measured at the fibre tip.

Extracellular electrophysiology

A small craniotomy (diameter of 0.5–1 mm) was made over the recording sites 1 day before the first recording session. Extracellular spikes were recorded acutely using 64-channel two-shank silicon probes (H-2, Cambridge Neurotech) for the ALM and Neuropixels probe 1.0 (ref. ⁶⁹) for the striatum. For the H-2 probes, voltage signals were multiplexed, recorded on a PCI6133 board (National Instruments) and digitized at 400 kHz (14-bit). All recordings were made with the open-source software SpikeGLX (http://billkarsh.github.io/SpikeGLX/). During recordings, the craniotomy was immersed in a cortex buffer (125 mM NaCl, 5 mM KCl, 10 mM glucose, 10 mM HEPES, 2 mM MgSO₄ and 2 mM CaCl₂; adjusted pH to 7.4). Brain tissue was allowed to settle for at least 5 min before recordings.

For the optrode recordings (Fig. 5h–k), we used 64-channel two-shank silicon optrodes with a 1.0-mm taper fibre optic attached adjacently (NA 0.22, core diameter of 200 µm; Cambridge Neurotech). Optrode was acutely inserted in each session and the light delivery protocol was identical to that used for behavioural experiments described in the section ‘Optogenetics’. Neuropixels probe and optrode tracks labelled with CM-DiI were used to determine recording locations⁷⁰.

Histology

Mice were perfused transcardially with PBS, followed by 4% paraformaldehyde/0.1 M PBS. To reconstruct recording tracks, we either generated coronal sections followed by conventional imaging (protocol described in Inagaki et al.⁷¹) or cleared the brain followed by light-sheet microscopy. To clear the brain, we used the EZ Clear method⁷². We followed the previous protocol to map the recording tracks to the Allen Common Coordinate Framework^70,73.

Quantification and statistical analysis

Behavioural analysis

We analysed the time of the first lick after the cue onset in each trial. Lick time was measured by detecting the contact of the tongue with the lick port using an electrical lick detector. For optogenetic experiments, we analysed trials with the first lick occurring after the onset time of photostimulation (0.6 s after the cue) in both control and photostimulated trials to compare the effect of photostimulation on behaviour. The no-lick rate was calculated as the probability of mice not responding within 5 s after the cue. The shift in lick time (Δlick time) was based on the median lick time. The post-stimulation lick rate (Extended Data Fig. 6) was calculated as the probability of mice licking within 0.6 s after the photostimulation offset time in no-cue trials. To analyse behaviour while the mice were engaged in the task, we analysed all trials between the first occurrence of five consecutive cue trials with licks and 20 trials before the last occurrence of three consecutive no-lick trials without photostimulation.

Owing to the attenuation of behavioral effects of optogenetic manipulation (Extended Data Fig. 6), we restricted analyses of both behavioural and physiological data to the first (for striatal manipulation) or the first two (for ALM manipulation) manipulation sessions per mouse. All analyses, including the calculation of confidence intervals and P values, were performed using a hierarchical bootstrap, unless stated otherwise. First, we randomly selected animals with replacements. Second, we randomly selected sessions for each animal with replacement. Third, we randomly selected trials for each session with replacements. Then, we calculated the behavioural metrics described above. This procedure was repeated 1,000 times to estimate the mean, confidence intervals and statistics.

Timer model and hazard rate analyses

To interpret the effects of optogenetic manipulations, we numerically simulated how different operations influence a timer, an accumulator that infers passage of time by integrating a constant input or periodic event, such as a water clock, hourglass, pendulum clock and quartz watch (Fig. 6). We modelled time as a scalar variable representing the temporal integration of a constant inflow signal. Specifically, the internal representation of time T(t) evolves according to the equation:

$$T

In addition to analysing lick-time distributions, we computed the hazard rate, defined as the instantaneous probability of a lick occurring at time t, given that no lick has occurred yet. Mathematically, the hazard rate h(t) is computed as:

$$h

We simulated the effects of two types of transient perturbation to the timer: pause (slowdown), in which the inflow rate r is transiently reduced by the speed coefficient c:

$${r}_{text{during}text{manipulation}}={ctimes r}_{text{before}text{manipulation}}$$

Here c = 0 represents a complete pause, whereas larger values correspond to a slowdown (0.5 was used in Fig. 6b). By contrast, in rewind, the timer state T(t) is transiently decreased as follows:

$$T

In all cases, the manipulation lasted for 600 ms and linearly decayed over the final 300 ms, matching the experimental condition. These manipulations were applied across 10,000 trials to assess their effects on lick timing and hazard rate dynamics.

Trial-history regression analysis

For the linear regression analysis in Fig. 2d, we tested 42 combinations of regressors with 1–6 lags with fivefold cross-validation (see Supplementary Fig. 3 for details). The median absolute deviation of lick time explained by different regression models was calculated as 1 − R1/R2, where R1 is the median of the absolute value of the model residuals, and R2 is the median of the absolute value of the null model residuals.

Extracellular recording analysis

Spike sorting and cell-type classification

JRClust⁷⁴ (https://github.com/JaneliaSciComp/JRCLUST) with manual curations was used for spike sorting. We used quality metrics (described in Majumder et al.⁶⁸) to select single units. Units with a total trial number of less than 75 were excluded from analyses. For the single-session population analysis, units with violated inter-spike interval were included.

For ALM recording, units with a mean spike rate above 0.5 Hz and spike width of 0.5 ms or more³⁷ (putative pyramidal neurons) were analysed. For striatal recording, units within the striatum (regions annotated as ‘striatum’, ‘caudoputamen’, and ‘fundus of striatum’ after registration to the Allen Common Coordinate Framework) with a mean spike rate above 0.1 Hz were analysed. We classified striatal neuron types based on spike features: striatal projection neurons (spike width ≥ 0.4 ms and with post-spike suppression duration ≤ 40 ms), fast-spiking interneurons (spike width < 0.4 ms and with less than 10% chance of having a long interspike interval) and tonically active neurons⁴⁴. For the single-session analyses (decoding and projection to modes), only putative pyramidal neurons were analysed for the ALM recording, whereas all neurons were included for the striatal recording data. See Supplementary Table 1 for the number of recorded neurons in each experiment.

Correlation in neural population activity

To plot the correlation in neural population activity, we calculated the mean spike activity of individual neurons across trials with different lick-time ranges to yield a population activity matrix, with the number of rows equal to the number of neurons and the number of columns equal to the number of time points (200-ms bin). For Fig. 3, we calculated pairwise Pearson’s correlation of these population activity matrices between trials with lick times between 1.40 s and 1.55 s (reference trials) and the trials with other lick-time ranges. For Figs. 4 and 5, we compared the pairwise Pearson’s correlation between unperturbed trials with lick times between 1.4 s and 1.7 s (reference trials) and the photostimulation trials with lick times between 1.7 s and 2.0 s). As a control, we subselected unperturbed trials with lick times closest to the median lick time in the unperturbed condition (the number of trials was matched to the number of trials as in the photostimulation condition). The choice of reference trials did not qualitatively change the results. For each correlation matrix, we identified the points along the y axis with the maximum correlation (above 0.8) for each time point, and repeated this procedure with the hierarchical bootstrap (Figs. 3d,i, 4g,k and 5i,m).

Single-cell analyses

To plot the PSTH of example cells, PSTHs were calculated based on 1-ms time bin and smoothed with a 200-ms causal boxcar filter unless specified otherwise. To temporally warp PSTH for individual cells, we linearly scaled the spike timing after the cue, based on the time from cue to lick. Specifically, Spike time_warped = Spike time_original/(LT_{trial to be warped}/LT_{target warp time}), where LT denotes the first lick time in each trial, and LT_{target warp time} = 1 s.

Across-trial variance (Extended Data Fig. 2e–g) was calculated as the variance of spiking activity across trials for the original or temporally warped data (the across-trial variance was calculated for five 200-ms time windows after the cue and then averaged).

To quantify the number of cells that significantly increase or decrease spike rate before the lick compared with baseline, the trial-averaged spike rate of 0.2–0.5 s before the lick was compared with that of 0–1 s before the cue. Signed-rank tests were performed to determine whether the spike rate difference was significant.

To calculate the proportion of cells affected or unaffected by photostimulation (Figs. 4c and 5e), we analysed the spikes within the time window of 50–250 ms from the photostimulation onset time. To quantify the effect of photostimulation, trials with licks before the photostimulation onset time were excluded from the analysis. For individual cells, the spike rate in control and photostimulation trials was compared using the two-sided rank-sum test. Cells with a mean spike rate above 1 Hz during this window and more than 10 trials per condition were analysed.

In Extended Data Fig. 2, we analysed the partial rank correlation between the spike rate (in specific time windows) and the lick time in previous trials, removing the effect of upcoming lick time, for each cell (we only analysed trials after rewarded trials to avoid confound caused by the representation of rewards; analysis of previous unrewarded trials yielded similar results). Specifically, we calculated the rank correlation between spike rate (R) versus previous lick time (P; (rho {RP})), rank correlation between R versus upcoming lick time (U; (rho {RU})) and rank correlation between P and U ((rho {PU})). Then, the partial correlation between spike rate versus lick time in the previous trial removing the effect of upcoming lick time is as follows:

$$rho {RP}cdot U=frac{rho {RP}-rho {RU}cdot rho {PU}}{sqrt{1-{rho }^{2}{RU}}sqrt{1-{rho }^{2}{PU}}}$$

As controls, we performed a trial shuffle test, which shuffles the trial order and destroys trial history, and a session permutation test to avoid the confound of nonsensical correlations (1,000 iterations)⁴⁵. The proportion of cells with a correlation higher than the chance level estimated by these controls is shown.

Single-cell ramping characterization

In Extended Data Fig. 2a–d, cells with more than 50 trials of lick time between 1.25 s and 1.5 s were used for firing pattern characterization. For each cell, PSTH was smoothed with a 200-ms causal boxcar filter. Trials were randomly split into halves and averaged to generate train and test data. We fit the activity of the train data from cue onset to the first lick time with different orders of polynomial functions (MATLAB polyfit function; tested order 1–8). We then calculated the mean squared error between the fit and the test data. The ‘best-fit order’ is the one with the lowest mean squared error. We repeated this procedure 10 times and defined the final order as the most frequent order among the 10 iterations. The best-fit data were then used to determine the monotonicity and the peak firing time of the cell. Monotonic firing cells are those whose derivative of the best polynomial fit remains consistently positive or negative values from cue onset to lick. Peak firing time was the time point between the cue and lick where the best polynomial fit had the highest firing rate.

Dimensionality reduction

We characterized population activity patterns between the cue and the lick by defining modes that differentiate the baseline activity during the ITI (0–1 s before the cue) from the activity during specific 300-ms time windows after the cue: 0–0.3 s after the cue (cue mode), 0.5–0.8 s before the lick (middle mode), 0.2–0.5 s before the lick (ramp mode) and 0–0.3 s after the lick (execution mode).

Specifically, to calculate ramp mode for a population of n recorded neurons, we looked for an n × 1 unit vector that maximally distinguished the mean activity before the trial onset (0–1 s before cue; r_{before cue}) and the mean activity before the first lick (0.2–0.5 s before the first lick; r_{before lick}) in the n-dimensional activity space. We defined a population ramping vector: w = r_{before lick} – r_{before cue}. Ramp mode is w normalized by its norm. Similarly, we defined cue mode, middle mode and execution mode using different time windows, and middle mode was orthogonalized to ramp mode, and cue mode was orthogonalized to both middle mode and ramp mode using the Gram–Schmidt process. Thus, the upper limit of the sum of square sum of task-modulated spiking activity explained (cue mode + middle mode + ramp mode) in Extended Data Fig. 3 is 1. Execution mode was orthogonalized to ramp mode (Extended Data Fig. 5a₃,b₃).

To define the trial-history mode, we first calculated the predicted lick time in each trial by applying the linear regression model described in the ‘Trial-history regression analysis’ section for each recorded session. Specifically, the model included previous lick times and the interaction between previous lick time and outcome at lags 1 and 2. This predicted value estimates what the lick time would be if it were determined solely by recent behavioural history and reinforcement, according to the fitted regression model, thereby summarizing trial history as a single value for each trial. We then calculated the Spearman rank correlation between the spike rate during the ITI (0–1 s before the cue) and the predicted lick time across trials for each neuron, indicating how strongly the ITI activity for each neuron encodes trial history. We obtained an n × 1 unit vector representing the rank correlation of each neuron and normalized it by its norm to calculate the trial-history mode. Trial-history mode is not orthogonalized to any other modes.

In Fig. 3 and Extended Data Figs. 3 and 5a–c, we have pooled cells recorded across sessions (that is, pseudo-sessions). For each cell, we randomly selected 50 unperturbed control trials to define the mode. These unperturbed trials met the following criteria: the first lick occurred within 1–3 s after the cue, and there were no licks 3 s before the cue onset. Then, we selected a different set of trials to project the activity along these modes. Only neurons with more than 10 trials within all six lick time ranges were included. The six lick time ranges were: 0.80–1.10 s, 1.10–1.25 s, 1.25–1.40 s, 1.40–1.55 s, 1.55–1.70 s and 1.70–2.00 s.

To calculate the square sum of spiking activity explained by individual modes (Extended Data Figs. 3 and 5), we calculated the square sum of the activity along individual modes after subtracting the baseline activity (0–0.2 s before the cue), and then divided that by the square sum of the spike rate across neurons after subtracting the baseline activity. For each lick time range, we averaged across at least 10 trials with lick times within that range, and spiking data for each trial were smoothed using a 200-ms causal boxcar filter. To calculate the square sum of spiking activity explained by the sum of cue mode, middle mode and ramp mode reported in the main text, we calculated the square sum of task-modulated spiking activity explained between 0.2 s from cue (around when task modulation started) to lick for individual lick time ranges and then averaged across them. We calculated the square sum of spiking activity explained by trial-history mode activity similarly but without subtraction of the baseline activity (Extended Data Fig. 5a–c). In Extended Data Fig. 5d, we performed a linear regression analysis between trial-history mode activity during 0–1 s before the cue and the upcoming lick time for each iteration of the hierarchical bootstrap. We then plotted the distribution of the linear regression coefficient (slope) across these iterations as a cumulative distribution function.

To calculate the angle between activity modes (Extended Data Fig. 3m), we computed the cosine similarity between two vectors of interest. Because cosine similarity depends on vector dimensionality (tending towards orthogonality as the number of neurons increases), we assessed statistical significance by shuffling one of the vectors and recalculating cosine similarity. This allowed us to determine whether the observed alignment between modes exceeded chance levels.

Single-session analyses

For analyses based on single sessions (Figs. 4 and 5 and Extended Data Figs. 4, 5, 7, 8, 10 and 12), sessions with more than 300 trials and five neurons were analysed. Spiking activity was binned per 50-ms time window. Activity between 1 s before the cue and the first lick in each trial was analysed (that is, post-lick activity was excluded as we focused on timing dynamics before the first lick). Dimensionality reduction was performed in the same manner as in the pseudo-session analysis, but modes were defined individually for each session. To visualize the time course of activity (for example, Fig. 4i), trials across sessions were pooled based on lick time, and only lick-time ranges that exist in at least two-thirds of the analysed sessions were shown. Therefore, the plotted lick-time ranges vary depending on the manipulation conditions.

To decode the T_{to lick} from simultaneously recorded neural population activity, we conducted a kNN regression analysis. Within each experimental session, trials were partitioned into two sets: a test set comprising randomly selected 100 unperturbed trials and all perturbed trials, and a training set consisting of the remaining trials. For each moment in a test trial (50-ms window), we searched all time points in the training set to identify k data points with the most similar population activity patterns (Mahalanobis distance based on the top principal components explaining 90% of variance). To estimate the T_{to lick} of the test set, we averaged the T_{to lick} in these kNNs. We tested ‘k’ values between 20–50 (which are close to the square root of the number of data points in the training dataset) and found that they yielded similar results and did not change conclusions (data not shown). In the paper, we have reported the results with k = 30. Some sessions showed low decodability due to a small number of recorded neurons, trials and/or lack of task-modulated cells (Extended Data Fig. 4b). We analysed sessions in which the kNN decodability (Pearson’s correlation between decoded lick time at the perturbation onset time, that is, 0.6 s after the cue versus actual lick time) was higher than 0.35.

To analyse the effect of perturbations systematically, we compared unperturbed versus perturbed trials after matching the number of trials and decoded time at the perturbation onset time (Extended Data Figs. 7a,c,d,e,g–i,k–m,o,p and 12e,i). Specifically, we randomly resampled animals, sessions and trials hierarchically (hierarchical bootstrap; 1,000 iterations). For each perturbed trial in each bootstrap iteration, we identified an unperturbed trial within the same session with the closest decoded time at the perturbation onset time (0.6 s after the cue). Then, we pooled these trials. This procedure allowed us to examine how decoded time (and projection along each mode) changed after the perturbation in conditions where their activity patterns were similar before the perturbation.

For the two-dimensional plots (Figs. 4e and 5g) and two-dimensional vector field analysis (Extended Data Fig. 11), we analysed how activity evolves in the two-dimensional space defined by ramp mode and middle mode. Spiking activity was binned in 50-ms time windows, and activity between the cue and the first lick in each trial was analysed. For each session, we projected the activity of ALM neurons along ramp mode and middle mode. The projection was normalized by the standard deviation of activity among control trials, but was not subtracted by the mean so that 0 represents 0 spike activity. For individual activity state (x) in control trials, we calculated the vector ({r}_{x}^{mathrm{control}}) representing the direction that activity evolves in the next time point (50-ms time bin) in the two-dimensional state. Then, we calculated the mean vector for individual states in the two-dimensional space by averaging all vectors within a spatial bin of 0.5 along both the middle mode and ramp mode axes (if the spatial bin contained more than 30 data points): ({r}_{XY}^{mathrm{control}}), where X and Y denote the location of the state along the middle mode and ramp mode axes, respectively. Similarly, we acquired the vector field during inhibition by pooling all time points during inhibition (100–400 ms from the inhibition onset) in photostimulation trials to acquire ({r}_{XY}^{mathrm{stim}}). Then, we calculated the direction between ({r}_{XY}^{mathrm{control}}) and ({{boldsymbol{r}}}_{XY}^{mathrm{stim}}) for all states where both control and stim vectors exist. We excluded points where ({{boldsymbol{r}}}_{XY}^{mathrm{control}}) is within (pi /6) from tanh(Y/X) because if the activity is evolving against the zero point under control conditions, we cannot distinguish between whether the activity is rewinding or moving towards the zero point during the inhibition.

Network models

Using a dynamical systems approach, we considered four variables representing the average membrane currents (h) and spike rates (r = f(h), where f(h) is the neural activation function) of neuronal populations in the ALM and striatum. Conceptually, in these models, the striatum represents both connections within the striatum and the subcortical loop via the thalamus, which is why there are excitatory connections. In these models, the membrane potential of neuron i, ({h}_{i}

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الكاتب: Zidan Yang

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