Table 1.
Experimental protocol for DA receptor antagonists.
Fig 1.
Euclidian distance between scattered two-dimensional data (a) was used for assigning individual points to clusters based on their similarity as measured by the Euclidian distance between data points (b). The vertical axis of the dendrogram (b) gives the distance between different data points (leaves). The closest points #4 and #5 (see panel a) are clustered together in the dendrogram (panel b) with the lowest horizontal linkage (distance). A horizontal dashed line marked with (a) through the dendrogram (panel b) shows two clusters, which are identified in the actual data (panel a) by the corresponding dashed line. Lower maximum distances among data points split the data in three (dashed-dotted line marked with (b)) and five (double dashed line marked with (c)) clusters, respectively.
Fig 2.
The correlation coefficients of a “reference” trial with each of the other 99 trials for a particular animal and condition (black continuous line in panel a1) are relatively low due to the random light stimulus timing relative to the network’s activity. After circularly shifting the trials, we noticed an order of magnitude increase of the correlation coefficient (see a1 dashed line). The vertical error bar corresponds to one standard deviation. Similar analysis was carried out for the sum of the squared differences (SSD) between the arbitrary “reference” trial and each of the other 99 trials for each animal and condition (see panels b1 and b2). The increase in the correlation coefficient due to phase correction also decreases the SSD values. Rectangular shaded areas mark similar conditions, such as scha, schb, etc.
Table 2.
Averages ± standard deviations of the correlation coefficient and the sum of squared differences (SSD) before and after shifting the trials to correct for the phase resetting.
Fig 3.
Dendrogram-based groups of similar LFPs using Euclidian distance among trials.
Twelve similar clusters were formed out of the 100 trials both without (a) and with (b) circular phase shifting to account for network phase resetting due to light stimulus. The distance threshold that generates, for example, six clusters was about 62 arbitrary units without phase correction (a) and decreased to about 42 arbitrary units (b) after phase correction (see the horizontal dashed lines in a1 and b1). Panels a2 and b2 show two representative examples of LFP recordings belonging to two different clusters in the dendrogram. The recordings in panel a2 belong to the largest cluster whereas in b2 they belong to the second largest cluster in the dendrogram shown in panel b1. The LFP pattern classifier correctly separated dissimilar recordings in different clusters and also correctly placed in the same cluster similarly-looking LFPs.
Fig 4.
Percentage overlap between baseline and DA receptor antagonist trials.
The baseline (bc) trials were concatenated with the corresponding DA receptor antagonist trials and classified in either 6 (a) or 12 (b) clusters. Overlapping between bc and DA receptor antagonists trials could be due to the limited number (six) of allowed clusters (a). Some overlapping was removed by allowing the trials to separate in 12 clusters (b). In almost all cases, the mean percentage mixing between bc and DA receptor antagonists trials in the same cluster is within a standard deviation regardless of the number of clusters used by classifier (c). A minimum of 20% overlap between bc and DA receptor antagonists suggests a possible common, invariant, part of the mathematical model that applies to both conditions. The minimum number of clusters that produce a consistent separation of trials may indicate the required number of parameters in a mathematical model that capture trials’ details.
Fig 5.
Correlation-based lag time statistics for all DA receptor antagonist conditions and mice.
For scha condition (a1) the correlation-based lag times have a bimodal distribution with some trials having a very short correlation time around 0.1 s and another group with 0.45 s. Although this and all the other distributions are similar to their corresponding baselines (see Fig 6), they also have significant differences presumably due to the DA receptor antagonist effect. For example, scha condition (a1) significantly shifted towards longer lag times compared to baseline. The weighted average of all lag times were 0.3012 s for scha (panel a1), and 0.2072 s for the baseline (see Fig 6a1), which represents a significant 45% increase in correlation time. Similarly, the weighted average of lag times for sulpa was 0.2864 s (panel a2) compared to 0.2275 s for the baseline (see Fig 7), which is a 26% increase in the correlation time. For botha (panel a3) there was only one viable animal which showed no change in the weighted average lag times compared to the baseline (see Fig 6a3). The weighted averages of lag times were 0.2594 s for the schb (panel b1), 0.2439 s for sulpb (panel b2), and 0.2548 s for bothb (panel b3). In all cased when DA receptor antagonist was applied before cocaine the weighted average of all lag times is the same as the corresponding baseline value.
Fig 6.
Distributions of all baseline lag times.
The animal were grouped according to the DA receptor antagonist treatments shown in Fig 5 for direct comparison. Most of the distributions are unimodal, except a1 and a2. The features of the lag time distributions of the baseline were preserved both under DA receptor antagonist (see Fig 5) and under cocaine alone (see Fig 7).
Fig 7.
Lag time distributions under cocaine alone.
While cocaine preserves the general structure of the lag time distributions noticed in the baseline (Fig 6), there are subtle differences that could be captured, for example, by the weighted average of the lag times. For example, after cocaine the weighted lag time is 0.1937 s (panel a1) and the baseline is at 0.2072 s (7% shorted compared to baseline). For panel b1 the weighed average lag time is 0.2635 s compared to 0.2594 s for baseline (2% longer than baseline). For panel a2 the weighed average lag time is 0.2275 s, which is the same as the baseline. For panel b2 the weighed average lag time is 0.2850 s under cocaine and 0.2439 s for the baseline (17% increase compared to baseline). For panel a3 the weighed average lag time is 0.2423 s after cocaine and 0.1790 s for the baseline (35% increase). For b3 the weighed average lag time is 0.2194 s for cocaine and 0.2548 s for baseline (14% shorter than baseline).
Table 3.
Autocorrelation-based lag time averages (in milliseconds) for baseline, DA receptor antagonists before cocaine, cocaine, and DA receptor antagonists after cocaine.
The first column represents the baseline recording, the second column represents data from DA receptor antagonists applied before cocaine, the third column represents data collected when cocaine alone was applied, and the last column are data recorded when DA receptor antagonists was applied after cocaine. The respective DA receptor antagonists are listed in parentheses. All standards deviations are given with two significant figures.
Fig 8.
Average mutual information-based lag time statistics for all DA receptor antagonist conditions and mice.
For scha condition (a1), the AMI-based lag times have narrow unimodal distributions with an average around 0.017 s (see Table 4 for mean ± standard deviation values). For schb condition (a2), the AMI-based lag times is unimodal with an average of 0.024 s, although for mice 14 and 15 the distributions have long tails that extend past 0.04 s. For sulpa condition (b1), the AMI-based lag times have unimodal distributions with an overall average of about 0.017 s. For sulpb condition (b2), the AMI-based lag times have unimodal distributions with an overall average of about 0.23 s and long tails that extent past 0.04 s. For botha condition (c1), the AMI-based lag time distribution has and average of about 0.023 s and a long tail that extent past 0.04 s. For bothb condition (c1), the AMI-based lag time distribution has and average of about 0.012 s and a long tail up to 0.04 s.
Table 4.
Average mutual information-based lag time averages (in milliseconds) for baseline, DA receptor antagonists before cocaine, cocaine, and DA receptor antagonists after cocaine.
The first column represents the baseline recording, the second column represents data from DA receptor antagonists applied before cocaine, the third column represents data collected when cocaine was applied, and the last column are data recorded when DA receptor antagonists was applied after cocaine. The respective DA receptor antagonists are listed in parentheses. All standards deviations are given with two significant figures.
Fig 9.
Solid black squares indicate lag time distributions belonging to the same class (a1). For example, bc8 and bc9, bc11, bc15, bc20 and bc22 are likely to be drawn from the same distribution. The measure of likelihood is determined by p-value, which is given on a natural logarithmic scale in panel (a2) where the bright orange color covers the range from e0 = 1 to e−10 ≈ 4.5 × 10−5. The maximum difference between any two distributions is shown in panel (a3). Dark colors mean small differences between empirical distributions. Panels a1-a3 refer to baseline distributions and are only provided to help us understand the changes induced by DA receptor antagonists (panels b1-b3). Panel b1 is similar to a1 (baseline), except for a few conditions and mice that seem to be affected by DA receptor antagonists. For example, the mouse # 18 only had similar distribution of lag times with mouse #23 under baseline (see panel a1), whereas when treated with sulpa (sulphiride after cocaine) its lag time distribution became similar to baseline lag time distributions for bc9, bc11, bc12, bc15, bc19, bc20, bc21, and bc22. These new correlations are shown with solid red squares to signal that they are new and unexpected compared to baseline. At the same time, sulpa18 distribution lost its baseline similarity with bc18 and bc23, which is marked with solid blue triangles in panel b1. The corresponding p-values (panel b2) confirm the new membership of the lag time distributions to the set bc9, bc11, bc12, bc15, bc19, bc20, bc21, and bc22. Furthermore, the maximum distance between sulpa18 and bc9, bc11, bc19, bc20, bc21, and bc22 is indeed low (below 0.2), whereas it increases a bit when compared to bc12 and bc15, but remains below 0.4 (panel b3). When cross-correlating the DA receptor antagonists’ effects against themselves (a3) we notice a significant increase similar lag time distributions (solid red squares) compared to the baseline (solid black squares). Some previously similar distributions in the baseline, e.g. bc16, bc22, and bc23, are lost under DA receptor antagonists (marked with solid green squares).
Fig 10.
Three-dimensional reconstructed attractors of neural activity.
Representative reconstructed attractors of neural activity under each condition: scha (a1), schb (b1), sulpa (a2), sulpb (b2), botha (a3), and bothb (b2). For each condition only two traces are shown with thin continuous and dashed lines, respectively. The thick line is the average reconstructed trace over the entire cluster, which is provided only as a visual aid. The trajectories resemble those observed under baseline [102] and cocaine [103].
Fig 11.
Three-dimensional reconstructed attractors both with autocorrelation-based and AMI-based lag times.
Representative reconstructed attractors of neural activity from schb12 trial #5 with a lag time of 1953 sampling times, i.e. 195.3 ms, based on autocorrelation method (a), and with 205 lag times, i.e. 20.5 ms, based on AMI method (b). While the lag times are one order of magnitude apart, they still unfold the attractor without any self-crossings of the phase space trajectory (see also the three-dimensional rotating frame in S1 Fig (for correlation-based lag time reconstruction) and S2 Fig (for AMI-based lag time reconstruction)).
Fig 12.
Fréchet distance between baseline and DA receptor antagonists.
(a) Each white-bordered square contains the color-coded Fréchet distance between the 100 control (bc) trials and the 100 DA receptor antagonist trials. Deep blue colors mean small Fréchet distance and suggest similar phase space trajectories. (b) The average over each 100 × 100 block offers a smoother color-coded overview of topological similarities among conditions. For example, there are a few clusters of similar geometrical structure among the three-dimensional attractors that are below 10% of the maximum observed arbitrary Fréchet distance. Among them, bc8 are topologically similar to sulpb8, sulpa18, sulpa19, and bothb20. Similar clusters are bc19 & bc20 and sulpb19 & sulpb20 and a larger blue cluster with sulpa18, sulpa19, and bothb20 (see upper right side of panel b).
Fig 13.
Synthesized traces from the strongest harmonics of the original signal.
The original LFP recording (black) was first decomposed in Fourier harmonics and then a signal was synthesized from the first 3 (a1 red line), 5 (a1 blue curve), 10 (a1 green curve), and 100 (b1 red curve) frequencies with the largest amplitude in the power spectrum. The insets show that as we include more harmonics, the synthesized signal better approximates the original LFP recording. All distributions of residuals, i.e. the difference between the original and the synthesized signal, have Gaussian shapes (a2 and b2). The amplitude or residuals decreases by an order of magnitude as the number of frequencies for the synthesized signal increased from 3 (a2) to 100 (b2).
Fig 14.
Fourier amplitudes versus frequency for signal synthesis.
The Fourier amplitudes for the first 3 (a1), 5 (a2), and 10 (a3) strongest harmonics over all trials of sulpa for mouse #18. Each frequency has an amplitude range (see shade narrow rectangle) that can be better capture by its distribution (see the corresponding shaded inset). The insets in panels (a) show the percentage contribution (horizontal axis) of each Fourier amplitude (vertical axis) for a particular frequency. Panels (b) show the distribution of Fourier amplitudes of the first four strongest harmonics.
Table 5.
Fourier coefficients for the first four strongest frequencies in Fourier spectrum.
The Fourier amplitudes for sulpa mouse #18 shown in Fig 14 for the 100 trials with three Fourier coefficients (the second and third columns), five coefficients (the fourth and fifth columns), and 10 coefficients (the sixth and seventh columns). The first column represents the integer frequency index k. Although each trial was synthesized using the largest 3, 5, or 10 Fourier amplitudes, different trials have different combinations of frequencies. For each of the three decompositions in Fourier components, we show what fraction of the trials required a particular frequency.