Graphical representation of TDE-RICA and matrix factorization procedure.

(A and B) Principles of time-delay embedding (TDE) of time-series data are graphically depicted. (C) Principles of reconstruction independent component analysis (RICA), used for a subset of data with no missing values, are graphically depicted. (D) Principles of matrix factorization (MF) used for all neurons in all samples, with missing data, are graphically depicted.</p

Network modeling with gKDR-GMM.

(A) Overview of the gKDR-GMM model. The model learns to predict the target neuron activity (y(t+Δt)) from the previous activity of presynaptic neurons (X) (blue lines represent physical connections). To this end, gKDR reduces the dimension of presynaptic activities X and generates K-dimensional values U representing X, ensuring that sufficient information for predicting y is included in U. GMM models the joint probability of (U, y) as a weighted sum of Gaussian distributions fitted to real data. The conditional probability of y is determined from the GMM model and used for prediction. See Methods and S1 Text—File 23 for details. (B) Neural activities obtained from free run simulation by gKDR-GMM. The time span depicted as “real” is actual activity data, identical to Fig 2A whereas the time span depicted as “Repeat 1/2” displays simulation results. The results of two simulation runs are shown following the same order of neurons as Fig 2A. (C) Cross-correlation of neural activities. The left panel shows the cross-correlation of actual activities as in Fig 2B, shown for comparison. The middle and right panels are cross-correlations of the simulation results in (B). Red and blue color show positive and negative correlations, respectively. See S1 Text—File 11A-C for results of all samples.</p

File information.

The recent advancements in large-scale activity imaging of neuronal ensembles offer valuable opportunities to comprehend the process involved in generating brain activity patterns and understanding how information is transmitted between neurons or neuronal ensembles. However, existing methodologies for extracting the underlying properties that generate overall dynamics are still limited. In this study, we applied previously unexplored methodologies to analyze time-lapse 3D imaging (4D imaging) data of head neurons of the nematode Caenorhabditis elegans. By combining time-delay embedding with the independent component analysis, we successfully decomposed whole-brain activities into a small number of component dynamics. Through the integration of results from multiple samples, we extracted common dynamics from neuronal activities that exhibit apparent divergence across different animals. Notably, while several components show common cooperativity across samples, some component pairs exhibited distinct relationships between individual samples. We further developed time series prediction models of synaptic communications. By combining dimension reduction using the general framework, gradient kernel dimension reduction, and probabilistic modeling, the overall relationships of neural activities were incorporated. By this approach, the stochastic but coordinated dynamics were reproduced in the simulated whole-brain neural network. We found that noise in the nervous system is crucial for generating realistic whole-brain dynamics. Furthermore, by evaluating synaptic interaction properties in the models, strong interactions within the core neural circuit, variable sensory transmission and importance of gap junctions were inferred. Virtual optogenetics can be also performed using the model. These analyses provide a solid foundation for understanding information flow in real neural networks.</div

TDE-RICA captures motifs of neural activity.

(A) TDE-RICA for neurons commonly observed across samples. (A: Top left) Original time series data shown as a heat map as Fig 2A. (A: Top right) Motifs of neural activities obtained by TDE-RICA. Each motif consists of the activity of 94 neurons over 300 time points. The color indicates the relative intensity of each individual neural activity in each motif. Motifs are common between samples. (A: Bottom left) Motif occurrences obtained by TDE-RICA. The occurrences differ between samples. (A: Bottom right) Reconstructed time series. The color range is the same as that of the upper left panel. (B) Fourteen motifs are shown, each consisting of the activity of 177 neurons over 300 time points. The color indicates the relative intensity of each individual neural activity per motif.</p

Data obtained by 4D imaging.

(A) Activity time series of head neurons obtained by 4D imaging. The activity of each neuron in the scaled fluorescence ratio of YFP over CFP is shown in pseudocolor. The top row shows the salt concentration applied to the animal’s nose, and remaining rows indicate the neuronal activity profiles. Each row represents one neuron, whose order was determined by hierarchical clustering based on activity cross-correlations. Note that only a subset of head neurons, which differ between samples, are shown in each panel, because some neurons were unobserved (e.g., too dim) or unannotated. (B) Pairwise cross-correlation of head neuron activities obtained by 4D imaging. Red and blue color show positive and negative correlation, respectively. In (A) and (B), examples of four samples are shown. For all samples, see S1 Text—File 1. (C) Pairwise cross-correlation averaged across samples. Hierarchical clustering based on p-values was performed to arrange the neurons. Red and blue color show positive and negative correlations, respectively. Several large and small correlated groups are observed; two showing prominent negative correlation with each other. Pairs of neurons never co-observed in any sample are filled in black.</p

Experimental setup for 4D imaging.

(A) Left panel shows an overview of the microscope for 4D imaging. The head of the worm was observed by the spinning-disk confocal microscope with 3 cameras for simultaneous multi-color imaging and with the piezo-positioner for z-scanning. Right panel shows that the worm was held in the microfluidic olfactory chip and was stimulated periodically. The inset shows the slight movement of the worm in the chip. CSU: confocal scanning unit. (B) An overview of the image analysis pipeline. The worm expressed 5 nuclear-localized fluorescent proteins and 3 colors were recorded for the annotation movie, and after that 3 colors were recorded for the activity movie. The neuronal nuclei were detected in the high-quality annotation movie. The nuclei were then annotated by labeling them with the names of their respective neurons. Next, the ROI information of the nuclei in the annotation movie was transferred to the activity movie. The nuclei were tracked in the activity movie and neural activity was calculated. ROI: region of interest.</p

Temporal motifs of neural activities obtained by TDE-RICA by matrix factorization.

(A) Motifs corresponding to sensory responses. (B) Motifs corresponding to forward and backward movements. (C) Motifs corresponding to the thermotaxis circuit. (D-F) Common features and individual differences of motif occurrences in the latent space. (D) Left: Occurrence of motif 13 (red) and motif 14 (blue) in sample 16 (upper) and sample 11 (lower). Right: Phase diagram of motif 13 and motif 14 in sample 16 (upper) and sample 11 (lower). (E) Left: Occurrence of motif 1 (red) and motif 2 (blue) in the same samples as D. Right: Phase diagram of motif 1 (red) and motif 2 (blue) in the same samples as D. (F) Left: Occurrence of motif 8 (red) and motif 9 (blue) in the same samples as D. Right: Phase diagram of motif 8 (red) and motif 9 (blue) in the same samples as D.</p

Free-run simulation by deterministic prediction with or without added noise.

(A) The left side in each panel (0–1250 time points) shows the real data. The simulated results follow the zero-activity portion (1250–2500 time points) used as initial conditions for free run simulation. “w/ stim” indicates that periodic salt sensory input was added during the simulation while “w/o stim” indicates omission of the sensory input. “probabilistic” indicates regular gKDR-GMM (Fig 6) while “deterministic” represents prediction by expectation value (which is unique) for GMM. “det + noise” represents deterministic prediction with random independent noise added. (B) Cross correlation of the data obtained in (A). Color codes are as in Figs 2B and 6C.</p

Estimation of synaptic weights from gKDR-GMM models.

(A) Estimated synaptic weights from different models. Each box shows model-estimated synaptic weights from the neuron on the y axis to the neuron on the x axis. In each box, model-estimated weights are shown in 15 x 3 cells as summarized on the right, where 15 rows are results from models using different K for gKDR using offset 0–4 of the data, and 3 columns show results from three parts of split time series. Models that did not show significance in bootstrap cross validation tests at p 10(p), while red colors show positive mean weights and blue colors show negative mean weights. (A) and (B) show part of the table for sample 1 as examples. Full figures are shown in S1 Text—Files 19A and 19B. (C) Estimated synaptic weights from different samples. In this figure, estimated weights from 15 models in each box of (A) are averaged, and shown in 24 x 3 arrangement, as depicted on the right. (D) Consistency across 24 samples in each box in (C) were tested by Wilcoxon’s rank-sum test and p values are shown in color codes as in (B). Darkness of the color shows –log10(p), while red colors show positive mean weights and blue colors show negative mean weights. (C) and (D) show part of the table as examples. Full figures are shown in S1 Text—Files 19D and 19E. In (A)-(D), the order of neurons is the same as that in Fig 2C. Group A and B neurons are shown in magenta and cyan, respectively. (E) Mean synaptic weights from all samples are shown as graph representation. Only the weights that showed consistency at FDR < 0.005 in (D) among direct synaptic connections are shown. (F, G) Graph representation of chemical synapses (F) and gap junctions (G) are shown between neurons shown in (E). (H) Mean (line) and standard deviation (light blue shade, across samples) of estimated synaptic weights at each lag between the neuron classes indicated. (I-L) Mean synaptic weights vs. consistency (-log(p) in (D)) were plotted for each of chemical synapses and gap junctions; for pairs of neurons with only chemical (I), only electrical (J) or both synapses (K), and for all pairs of neurons (L). In (L), gap junctions are shown in red dots and chemical synapses are shown in blue circles.</p

Comparison of real and simulation results.

(A) For comparison of real data and simulated data, cross correlation between activities of neuron pairs are plotted for real (x axis) and simulated (y axis) activities for all pairs of neurons in each sample. Different samples are plotted in different colors. Plot for each animal can be seen in S1 Text—File 12. (B) Correlation coefficients of the real vs simulated relationship in (A) are plotted for each sample. Results of different models (K = 3, 4, 5) are also shown. (C) Lagged cross-correlation of all combinations of neurons. The lags with the best absolute cross-correlation are depicted color coded. Magenta and green indicate positive and negative lags, respectively. (D) Example of time series plot of a pair of neurons, AVA and RIM, showing a lagged correlation in most samples. (E) Periodicity of neural activities. Top row shows the salt stimulus (concentration range, 50–25 mM), which has a regular periodicity. Each of the real and simulation results as Fig 6B (K = 3, 4, 5 for simulation) was split into salt stimulus periods, overlaid and averaged to visualize the periodicity of the activity of each neuron. For visualization purposes, the period averages are shown repeated twice. (F) Fourier power of real AVAL/R activity (circles) and gKDR-GMM simulation results (crosses) with different model parameters A: K = 3, B: K = 4, C: K = 5. Only some of the samples show large periodic components.</p