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# Neuronal population activity dynamics reveal a low-dimensional signature of operant learning in Aplysia

## Abstract

Learning engages a high-dimensional neuronal population space spanning multiple brain regions. However, it remains unknown whether it is possible to identify a low-dimensional signature associated with operant conditioning, a ubiquitous form of learning in which animals learn from the consequences of behavior. Using single-neuron resolution voltage imaging, here we identify two low-dimensional motor modules in the neuronal population underlying Aplysia feeding. Our findings point to a temporal shift in module recruitment as the primary signature of operant learning. Our findings can help guide characterization of learning signatures in systems in which only a smaller fraction of the relevant neuronal population can be monitored.

## Introduction

Understanding neuronal population activity spanning multiple brain regions has emerged as central to understanding brain functions. The strategy of building low-dimensional representations of high-dimensional population activity offers insights into the generation of behavior1,2,3, learning1,4,5, and other functions6,7. Low-dimensional representations are useful because high-dimensionality limits the interpretability of the data. Consequently, many dimensionality reduction approaches have been developed, which capture most of the information in the original data using only a reduced number of dimensions. Low-dimensional representations can be used to identify the most distinctive features of a given phenomenon (e.g., behavior, learning, cognitive state, task, event, etc.) in the data. These distinctive features can be referred to as a low-dimensional signature of the phenomenon (e.g., ref. 8).

Despite continued advances, monitoring activity in a substantial proportion of the population that is relevant for a given behavior is still only feasible in a few systems (e.g., refs. 1,3,9,10,11,12,13). As a result, it remains unknown whether it is possible to identify a low-dimensional “learning signature” associated with operant conditioning (OC), a ubiquitous form of learning in which an animal learns from the consequences of its behavior.

We addressed this question by leveraging the advantages of Aplysia feeding behavior. The behavior is well characterized, as is the neuronal circuit underlying its generation. Feeding responses consist of a sequence of movements of the radula, a food-grasping structure. The two primary phases of this sequence are protraction and retraction. During ingestion responses, the radula protracts while open, and subsequently retracts while closed around food. The rate of ingestion responses can be modified in vivo by OC. For example, if seaweed or seaweed juice is contingently presented following ingestions, the rate of these responses increases. This increase is dependent upon the contingency between response and reward, as presenting the same number of rewards irrespective of response occurrence does not increase response rate14. The rewarding effect of seaweed is mediated by the esophageal nerve 2 (En2), and electrical stimulation of this nerve, when contingent upon ingestions, similarly increases the rate of responses15.

Aplysia feeding responses are produced by a central pattern generating circuit located primarily in the buccal ganglia16. Nerves projecting from the buccal ganglia to the musculature of the feeding apparatus control radula movement. Buccal motor patterns (BMPs) of activity in these nerves correspond to protraction, retraction, and closure of the radula during in vivo feeding behavior17. Moreover, buccal ganglia continue to generate such BMPs after excision from the animal, and an increased rate of BMPs can be observed in ganglia isolated from contingently-trained animals14,18. An in vitro analog of OC has been developed by making electrical stimulation of En2 contingent upon BMPs occurring in the isolated ganglia. Similar to in vivo observations, in vitro contingent training leads to an increase in the rate of ingestion-like BMPs (iBMPs)19,20,21. Using in vivo and in vitro OC, some loci of nonsynaptic and synaptic plasticity engaged by OC have been characterized22,23,24.

Given that activity in a substantial proportion of the BMP-generating circuit can be monitored using voltage-sensitive dye (VSD) imaging25,26, we combined VSD imaging with the in vitro analog of OC19 to examine reconfiguration of population activity induced by learning. We identified two low-dimensional motor modules in the population activity dynamics—one active during protraction and the other during retraction. Our findings pointed to an advance in the recruitment of the retraction module as the primary signature of operant learning.

## Results

### In vitro analog of operant conditioning

We used extracellular suction electrodes in buccal nerves 1–3 (Bn1–3) and the radula nerve (Rn) (Fig. 1a, b) to record BMPs before and after OC (Fig. 1c–e). Direct stimulation of the dopamine-rich esophageal nerve 2 (En2), which acts as a reward, was made contingent upon ingestion-like BMPs (iBMPs). Each contingent preparation was paired with a yoked control that received identical stimuli irrespective of its own BMP occurrences (Fig. 1c). Consistent with previous studies, conditioning specifically increased the rate of iBMPs compared to yoked controls, but did not change the rate of other, non-iBMP patterns (Fig. 1f, g).

### VSD imaging captures changes in primary features of population activity

To gain insight into the effects of OC on high-dimensional population dynamics, OC was combined with population-wide, single-neuron resolution VSD imaging (Fig. 2). We monitored ~100 neurons per preparation. About one-third exhibited activity (Fig. 2a–c). Many neurons displayed correlated phasic firing, similar to previous studies25,26.

As a first step toward characterizing population-level features of operant reward learning, we examined overall changes in pairwise firing correlation and firing rate. We computed the mean firing correlation across all pairs of neurons within each preparation and asked whether this correlation matrix differed between contingently-trained and yoked-control preparations. Contingently-trained neuronal populations were significantly more correlated than those with yoked training (Fig. 2d) without a clear difference in overall firing rate (Fig. 2e). Thus, a distinctive feature of operant reward learning was enhanced coincident firing among units, rather than the altered level of activity.

### Contingent training advances recruitment of the retraction, but not the protraction, motor module

The high dimensionality of the neuronal population space hinders further intuitive interpretation. Consequently, we asked whether there is a low-dimensional subspace that captures most of the dynamics, using the dimensionality reduction approach non-negative matrix factorization (NNMF)27. For each preparation, NNMF extracted two modules from the high-dimensional population activity (Fig. 3a and Supplementary Fig. 1c). Each module is defined by: (1) a set of weights or contributions from each neuron, which represents the subset of neurons participating in the module, and (2) a magnitude or level of recruitment of the module at each point in time. Neuron participation in the modules was sparse and largely non-overlapping. Together, these modules explained 74.9 ± 1.8% of the power in the data, suggesting they captured most of the population dynamics while reducing the dimensionality to two.

Given that NNMF can decompose behavior28,29, we asked whether the modules corresponded to specific aspects of BMPs. BMPs consist of two main phases—protraction followed by retraction. Although neither phase timing nor nerve activity were used to obtain the modules, module recruitment overlapped with either protraction or retraction (Fig. 3b). Indeed, each module across all preparations was correlated with the timing of one phase and anti-correlated with the other, forming two clusters (Fig. 3c). Modules were also highly consistent across preparations (Supplementary Fig. 1). These findings suggested the observed population dynamics reflected recruitment of neurons participating in a retraction and a protraction motor module for each animal.

Next, we asked whether a low-dimensional signature for operant reward learning was apparent in these motor modules by examining recruitment during BMPs before and after contingent or yoked training (Fig. 3d, e). The peak recruitment of the retraction module was significantly advanced in contingent preparations compared to yoked controls. In contrast, no significant difference in timing was observed for the protraction module (Fig. 3f). Interestingly, there was no difference between groups in the peak recruitment magnitude of either the retraction or protraction modules (Fig. 3g). These findings suggest the most prominent feature of OC was a change in the relative timing, but not the level, of population activity. Furthermore, that change was specific to a particular subset of neurons within the population (those participating in the retraction module).

### Spatial distribution of neurons contributing to motor modules

Many neurons in the ganglia have been uniquely identified and characterized in previous studies (e.g., refs. 16,30). Thus, one can ask which neurons are consistently part of the cells modulated by OC across animals. Population-wide imaging does not provide information on features typically used to identify neurons, such as synaptic and intrinsic properties. We, therefore, examined the overall localization of neurons contributing to the retraction and protraction modules and compared this distribution to previously characterized neurons (Supplementary Fig. 2). Neurons that strongly contributed to the retraction module were typically localized to the top-left quadrant, whereas neurons contributing to the protraction module tended to be localized more centrally and broadly (Supplementary Fig. 2a). Neurons contributing to both modules stretched from medial to lateral, but neurons contributing to the retraction module were more ventral (Supplementary Fig. 2b). Localization of both modules was consistent with previously described neurons known to be active during either retraction or protraction (Supplementary Fig. 2c). In addition, the region with the most consistent and strong contributions to the retraction module appeared to coincide with the ventral region where several retraction motor neurons are located (e.g., B3, B6, B7, B9, B10, B39, B43, and B44). Determining the extent to which biophysical changes in these neurons are responsible for the shift will require extensive electrophysiological studies focusing on them and on cells that could affect their activity, but that were likely not part of the retraction module due to their localization (e.g., B51 and B64).

## Discussion

Why is activity modified specifically in the retraction module? All outcome-based learning systems must address the credit assignment problem. That is, following an outcome such as a reward, which neurons or synapses should be modified and to what extent? Temporal contiguity is a primary mechanism of credit assignment31,32,33,34. In our training, reward immediately followed retraction termination. Furthermore, the retraction module tended to display peak recruitment toward the end of the retraction phase. Therefore, the specific effect of OC on the retraction module may be explained by the shorter interval between activity and reward compared to neurons in the protraction module.

Neurons participating in the retraction module mediate the retraction phase of feeding behavior. Thus, it is likely that the temporal advance observed at the population level is reflected both on the behavior and on nerve activity. Although this implies that an analysis of the nerve recordings alone could yield similar findings, several advantages of single-neuron resolution population-wide recordings should be noted. First, several interneurons do not send axonal projections through peripheral nerves. Therefore, these neurons would be missed by extracellular nerve recordings, but are captured here through voltage imaging. Second, because several neurons send axons simultaneously through multiple nerves, their contributions would be inappropriately weighted in an analysis of extracellular nerve activity alone. Third, detailed spatial information on individual neurons is obtained through imaging. Thus, the data and findings reported here extend beyond what could be obtained from extracellular recordings alone.

Differences in the retraction phase have been reported between ingestion-like BMPs (iBMPs) and rejection-like BMPs (rBMPs). In particular, the retraction phase tends to be longer during iBMPs20. Given that OC increases the rate of iBMPs19, and given that we included all pattern types in our analyses, an important question is whether the advance we observed in the retraction module could be explained by this difference in retraction duration. This is likely not the case for two reasons. First, we normalized the time scale to the duration of each phase. Thus, simply stretching or shrinking phase duration would have no effect on our analyses. Second, because iBMPs have longer retraction than rBMPs, and given that peak recruitment of the retraction module happens toward the end of the phase, the expected shift, if any, should be in the direction opposite of what we have observed. Therefore, the advance in the retraction module likely reflects a different effect of contingent reinforcement on the retraction phase.

Contingently-trained preparations had a higher overall level of pairwise correlations than yoked preparations (Fig. 2d) without a significant difference between overall firing rates (Fig. 2e). One possibility is that increased electrical coupling between neurons could explain this difference. Operant conditioning is known to increase the coupling between identified neurons B30 and B63, as well as the coupling between B63 and B6518. Given that electrical coupling is a common feature in the feeding circuit, it is also conceivable that coupling is modified among many other neurons that have yet to be examined. Interestingly, both groups tended to display a decrease in correlation relative to the pretest, which may be due to a general nonassociative effect that promotes decorrelation among activities.

Previous studies in vertebrates and invertebrates have similarly found that neuronal activity patterns underlying behavior lie within a low-dimensional subspace of the population space1,2,3,10. However, identifying a low-dimensional signature of OC has been challenging, in part because the number of cells from which one can record is commonly orders of magnitude smaller than the relevant population space. The isolated ganglia preparation allowed recording from a large proportion of the population that participates in consummatory feeding behavior, while removing variability due to sensory stimuli—the only stimulus was the reward. Therefore, we observed the direct effect of the association between fictive behavior and reward on the dynamics of the relevant neuronal population.

One possibility is that a learning signature could be found in changes to the low-dimensional subspace itself. However, primate studies suggest this subspace is resistant to change and imposes important constraints on learning-induced changes5,35,36. Accordingly, we found that neuronal activity before and after learning was consistently captured by a low-dimensional subspace, comprising protraction and retraction modules. The learning signature was not a change in the subspace, but rather in the relative timing of the trajectory within the subspace. Given the importance of timing actions for behavior production, learning may commonly act primarily by modifying neuronal module timing. Our findings can help guide the characterization of learning signatures in systems with many more degrees of freedom, for which it may only be possible to record a small fraction of the relevant neuronal population.

## Methods

### Experimental subjects and preparations

Aplysia californica were obtained either from the National Resource for Aplysia of the University of Miami (Miami, FL) or from South Coast Bio-Marine LLC (San Pedro, CA). Animals (25–50 g) were kept in aerated artificial seawater tanks and fed seaweed three times a week. Animal care was in accordance with guidelines and overseen by the institutional Animal Welfare Committee. Prior to each experiment, animals were food-deprived for 3–5 days (NIH or institutional guidelines for food deprivation protocols do not apply to Aplysia used in the present study because they are marine invertebrates). A small piece of seaweed was used to test the animals’ motivation and feeding behavior prior to dissection. Animals were injected with a volume of isotonic MgCl2 equal in milliliters to 50% of the animals’ mass in grams. The buccal mass with buccal ganglia attached was isolated from the animal and placed in a Sylgard-coated chamber containing artificial seawater with a high concentration of divalent cations (HiDi-ASW; 210 mM NaCl, 10 mM KCl, 33 mM CaCl2, 145 mM MgCl2, 20 mM MgSO4, 10 mM HEPES, pH adjusted to 7.5 with NaOH). HiDi-ASW was used to reduce activity and muscle contractions during dissection. Buccal ganglia with buccal nerves were excised from the buccal mass and pinned down. The sheath of connective tissue around the ganglia was kept intact. Bipolar suction electrodes for recording and stimulation were positioned on each nerve (see below). HiDi-ASW was substituted with normal artificial seawater (ASW; 450 mM NaCl, 10 mM KCl, 10 mM CaCl2, 30 mM MgCl2, 20 mM MgSO4, 10 mM HEPES, pH adjusted to 7.5 with NaOH) prior to recording. Preparations were maintained at 15 °C throughout the experiment.

### Voltage-sensitive dye (VSD) imaging

Isolated buccal ganglia were stained for 7 min with 0.2 mg/ml of the absorbance VSD RH-155 (TRC Canada) in ASW. After staining, the solution was swapped for a lower concentration of RH-155 in ASW (0.01 mg/ml), which was kept for the remainder of the experiment. Using a Gilson Minipuls 3 peristaltic pump, the solution was continuously cycled through a Warner Instruments SC-20 inline cooler/heater under control of a Warner Instruments CL-100 temperature controller, which maintained the preparation at 15 °C. Imaging data were acquired by a Deep Well NeuroCMOS 128 × 128 camera (RedShirtImaging, SciMeasure) sampled at 1 kHz. The camera was fitted to an Olympus BX50WI microscope equipped with an Olympus 20 × 0.95 NA XLUMPlanFL water immersion objective lens. The preparation was illuminated with a 150 W, 24 V Osram halogen light bulb powered by a Kepco ATE 75–8 M power supply. Before reaching the preparation, the light passed through a Brightline 710/40 nm bandpass filter.

Most neurons in the buccal ganglia are bilaterally symmetric, somata are located at the surface of the ganglia, and more neurons can be observed caudally than rostrally. Therefore, the number of recorded cells was maximized by imaging the caudal surface of the left hemiganglion, which accounts for about a quarter of the total surface area of the buccal ganglia. Each preparation was imaged for two 120 s periods, one before and one after either contingent or yoked training (see below).

### Extracellular electrophysiology

In typical ingestion, the animal’s food-grasping radula is first protracted while open, and subsequently retracted back while closed around food. This biphasic protraction-retraction sequence can also be observed in the activity of nerves that control the feeding apparatus, both in intact animals17 and isolated ganglia19,37. Bipolar suction electrodes were used to monitor nerve activity associated with buccal motor patterns (BMPs). Nerve signals were amplified by A-M Systems model 1700 differential AC amplifiers and digitized by Axon Instruments Digidata 1322 A at 20 kHz. The following nerves were recorded: the left and right buccal nerves 2 and 3 (Bn2 and Bn3); right buccal nerve 1 (Bn1); left radula nerve (Rn); and left esophageal nerve 2 (En2). Electrical stimulation of En2, which acts as reinforcement, consisted of a 6-s train of 0.5 ms pulses at 10 Hz. Stimulation was manually triggered by the experimenter using a WPI Pulsemaster A300, and delivered by a WPI 850 A stimulus isolator. Stimulus intensity was calibrated for each preparation by gradually increasing the intensity of a lower rate tonic stimulus (2 Hz, 0.5 ms) until a complete BMP was elicited. The efficacy of the regular 6 s stimulus was then confirmed after a brief rest. This procedure was devised to obtain a physiological threshold for En2 stimulation while minimizing the amount of stimulation prior to training.

### In vitro operant conditioning

After the En2 threshold was determined, experiments followed the timeline depicted in Fig. 1c. Preparations were allowed to rest for 40 min. Preparations that did not emit any spontaneous BMPs during this rest period were allowed to rest for an additional 20 min. The pretest period of 30 min was then initiated. To avoid a potential ceiling effect, and to ensure that preparations had a baseline rate of BMPs that was sufficient for training, preparations were discarded if they displayed <3 or >10 BMPs during the first two-thirds of the pretest. All criteria were identical across groups. During the last third of the pretest, preparations were imaged for 120 s. If during this time no BMPs occurred, additional attempts were made to image during BMP generation. If no BMP could be captured before the end of the pretest, the preparation was discarded. After the pretest, a 30 min training period ensued. For contingent preparations, the occurrence of an iBMP was followed by stimulation of En2. Any BMP elicited by En2 stimulation (i.e., initiated within 20 s after the offset of the stimuli train) was not reinforced. For each yoked preparation, En2 stimuli were identical to those received by its paired contingent preparation, irrespective of the BMPs emitted by the yoked preparation itself. Thus, yoked preparations controlled for any time-dependent or nonassociative effects. Preparations that had <3 effective En2 stimuli during training were discarded. An En2 stimulus was considered effective (for either contingent or yoked preparations) only if it clearly elicited at least one BMP. Upon conclusion of training, a 30 min posttest was performed. As in the pretest, preparations were imaged for 120 s during the last third of the posttest.

### BMP identification

During training, BMPs were manually classified by the experimenter in real-time. For data analysis, BMPs were automatically identified and classified using custom MATLAB code. In both cases, the same criteria were used. Activity in Bn1 was used as an indicator of the timing of each BMP phase. Bn1 is particularly informative because it is active during protraction, inactive during retraction, and active again upon retraction termination. A BMP was identified when three criteria were met: (1) a burst of large-unit activity lasting at least 3 s occurred in Bn1; (2) the Bn1 burst was followed by a burst of large-unit activity in at least one of the two Bn3s; (3) at least one burst of large-unit Rn activity occurred during either the Bn1 burst (protraction) or the subsequent Bn1 suppression (retraction). The large-unit activity was defined as all activity with amplitude above a voltage threshold for each nerve, which was set independently for each experiment. The threshold for Bn1 was set to include all units that were consistently active during protraction and inactive during retraction. The Bn3 threshold was set to include only the largest units, which correspond to B4, a neuron known to be active at the onset of retraction. Similar to previous studies20,21, the Rn threshold was set to include units larger than baseline activity (i.e., activity occurring outside of BMPs). Thresholds were adjusted as necessary to ensure that the algorithm captured all BMPs occurring during the pretest, training, and posttest recordings for each preparation. Bursts were defined as at least 3 spikes with interspike intervals ≤4 s. BMPs were classified as ingestion-like (i.e., iBMPs) if ≥50% of the Rn burst duration occurred during the retraction phase19,20.

### Raw VSD data processing

VSD data were processed in a similar manner to that described elsewhere26. Briefly, regions of interest were drawn around the soma of each neuron, and the light intensity was averaged across pixels within the region for each time point. Traces for each cell were taken as the change in light intensity relative to baseline (i.e., ΔI/I), where the baseline was defined as an average of the first ten frames after shutter opening. Traces were filtered using MATLAB’s elliptic filter (n = 2, Rp = 0.1, Rs = 40, Wp = [0.02 0.2]).

### Denoising

Principal component analysis (PCA) was used to isolate and remove correlated noise, such as that associated with movement. PCA extracts a set of components from the data that can be ordered by the amount of variance-covariance in the data explained by each component. If some components consist primarily of noise, the data can be reconstructed without those components to improve the signal-to-noise ratio. At the millisecond timescale, most of the correlation between the traces of individual neurons was due to noise that affected many neurons simultaneously, such as tissue movement, vibrations, and fluctuations in illumination. This was the case because the amplitude of RH-155 signals is very small (i.e., in the order of 1 × 10−4 ΔI/I), and, importantly, because action potentials in Aplysia are not correlated at the millisecond scale during spontaneous BMP generation. Note that this is different from the correlation between firing rates examined throughout this paper, which is on the scale of seconds. Thus, we performed PCA on the z-scored traces from all neurons in a preparation, removed the top n components accounting for most of the correlation between traces, and reconstructed the data. The number of components to remove was determined by performing reconstructions with n ranging from 0 to all components and selecting the most de-correlated reconstruction. We validated this procedure by adding artificial action potentials of varying amplitudes to a test dataset, performing the full range of reconstructions, and attempting to detect the artificially added action potentials using the algorithm described below. The test data were a set of 100 traces from a real preparation. Because a substantial proportion of the added spikes were buried in real noise, they served as a realistic test of signal improvement by denoising. Consistent with a visual inspection of signal quality, the most de-correlated reconstructions led to the highest true positive rates.

### Spike detection

Spikes were detected in a neuron when a downward deflection in the VSD trace exceeded five times an estimate of the standard deviation of the noise. This estimate was given by

$${\sigma }_{n}=\frac{{median}\left({{{{{\rm{|}}}}}}x{{{{{\rm{|}}}}}}\right)}{b}$$
(1)

where x is the trace of the neuron, and b = 0.6745 is the constant for the relationship between the median absolute deviation estimator and the standard deviation $${\sigma }_{n}$$ for Gaussian distributions38,39,40. A re-arm window of 10 ms was used to prevent a single action potential from being counted more than once. This simple threshold method yielded comparable true positive rates as did a more sophisticated spike detection algorithm41 in a test dataset to which artificial action potentials of varying amplitudes were added.

### Non-negative matrix factorization (NNMF)

NNMF is a dimensionality reduction approach that optimizes an approximation of the data matrix $$X$$ from a predetermined number of components $$k$$, given non-negativity constraints (for a detailed overview, see ref. 27). NNMF is well-suited to identify a low-dimensional signature of learning in the Aplysia feeding circuit for at least three reasons. First, NNMF has been used to identify groups of neurons participating in memory42. Second, NNMF has successfully decomposed behaviors into their motor module building blocks29. Third, the assumptions and constraints adopted by NNMF are compatible with neuronal activity data27. For the activity of a neuronal population over time, NNMF can be described by

$${X}_{n\times t}={W}_{n\times k}{H}_{k\times t}+{E}_{n\times t}$$
(2)

where $${X}_{n\times t}$$ is the activity of $$n$$ neurons over $$t$$ time points, $${W}_{n\times k}$$ is a matrix of the contributions of each of the $$n$$ neurons to each of the $$k$$ components, $${H}_{k\times t}$$ is a matrix of the magnitude of each of the $$k$$ components at each time point $$t$$, and $${E}_{n\times t}$$ is the approximation error or residual term that NNMF minimizes. Note that these $$k$$ components are referred to as modules throughout this manuscript.

NNMF was performed using the seqNMF MATLAB package developed by Mackevicius et al.28 with L = 1, λ = 0, and otherwise default parameters. This set of parameters corresponds to standard NNMF using multiplicative update rules first introduced by Lee and Seung43. Six repetitions were performed for each run to ensure that the algorithm consistently converged to the same modules. Given consistent convergence, an arbitrary repetition was ultimately selected.

Prior to NNMF, data were temporally smoothed by convolving the spike train of each neuron with a Gaussian. Because a difference in pairwise correlations was observed when spikes were binned with 1 s bins (Fig. 2d), the Gaussian was set to have a standard deviation of 1 s. This smoothing is compatible with the duration of BMPs as well as in vivo feeding behaviors, which usually unfold over the course of tens of seconds. After smoothing, pretest and posttest data were concatenated. This held the contribution matrix $$W$$ constant between pretest and posttest, allowing for the definition of identical components (i.e., modules) before and after training. Thus, we were able to compare the time course of recruitment of the same modules before and after training.

Because fixed-duration recordings (i.e., 2 min) were performed, it was possible for BMPs to be only partly captured during the imaging window. To prevent such events from distorting subsequent analysis, all BMPs that were only partially captured were removed from the dataset prior to the steps described below.

Given that overall BMP duration and the duration of each phase vary from pattern to pattern, comparing component recruitment during BMP generation required normalizing these durations. This normalization was performed by using linear interpolation to resample component recruitment during each phase of each BMP to a standard of 180 time points. This yielded equal-length time courses of module recruitment during each individual motor pattern. Finally, this resampled time course of module recruitment was averaged across BMPs to obtain a mean pretest and a mean posttest time course for each preparation. The timing and magnitude of the peaks of these average time courses were then compared (Fig. 3d–g). Furthermore, the average time courses during the pretest (i.e., before any training) were used to examine the consistency of modules across preparations (Supplementary Fig. 1).

Our analyses required comparing BMPs generated before and after two training paradigms (i.e., contingent or yoked), yielding a total of four conditions. Because learning increases the rate of iBMPs, there are inherent differences in the number of patterns captured under each of these conditions with fixed-length windows. Given that this difference in sampling cannot be avoided, we calculated the average recruitment of each module for each preparation before and after training (as described in the preceding paragraph). Thus, whether a preparation received contingent or yoked training, it was represented by one time course of module recruitment for the pretest, and another for the posttest for all comparisons in Fig. 3d–g. Therefore, it is unlikely that the overall number of patterns sampled would directly impact those comparisons.

A critical factor in NNMF and other dimensionality reduction techniques is the selection of the number of modules k used to approximate the data. Various strategies have been used to make this selection6,27,42. Examples include using heuristic solutions (such as a simple threshold on the approximation error or variance-covariance explained by the reconstruction), or using estimates of the tradeoff between information loss and model complexity (e.g., Akaike’s or Bayesian information criteria), among others. Here, priority was given to ensuring that modules were behaviorally relevant and consistent across animals, thereby allowing for direct comparisons of the effects of learning on module recruitment. We assessed this cross-animal consistency by computing the correlation between the average pretest time course of module recruitment during a normalized BMP across preparations (Supplementary Fig. 1a, b). We performed a series of factorizations with $$k$$ ranging from one to ten, each repeated six times. For each repetition, a matching algorithm sorted modules into $$k$$ clusters by matching modules in the order of their correlations across preparations. Within-preparation matches were disallowed, and each module was only allowed to match with one other module per preparation. Note that this is distinct from the hierarchical clustering in Supplementary Fig. 1b, which has no such limitations. This matching algorithm ensured that each preparation had exactly one module in each of the $$k$$ clusters. An overall mean correlation among matched modules was computed for each NNMF run as the final consistency metric (Supplementary Fig. 1c).

The number of modules k = 2 was selected because it yielded the most consistent modules across preparations, and because it did so while explaining most (74.9 ± 1.8%) of the total power in the data. Explained power is a metric of how well a factorization approximates the original data, analogous to the concept of explained variance-covariance in PCA. The explained power was computed as in Mackevicius et al.28, that is

$$\frac{\sum {X}^{2}-\sum {(X-\widetilde{X})}^{2}}{\sum {X}^{2}}$$
(3)

where X is the original data, $$\widetilde{X}$$ is the data reconstruction, and $$\sum {X}^{2}$$ denotes $${\sum }_{n=1}^{N}{\sum }_{t=1}^{T}{x}_{n\times t}^{2}$$.

Hierarchical clustering (Supplementary Fig. 1b) was performed with MATLAB’s linkage function using the average method. Briefly, the function starts with single-module clusters and progressively forms larger clusters by grouping the most similar (i.e., most correlated) clusters, until a single cluster encompasses all modules. Once a module is assigned to a cluster, the module is no longer individually compared to others; rather, a comparison is made between the entire clusters. The dissimilarity between two clusters is equal to the mean dissimilarity among all possible pairs of modules between the two clusters.

### Spatial distribution of motor modules

To allow for comparisons of module localization across animals, all preparations were aligned to a reference ganglion (preparation A in Supplementary Fig. 2a). A regression of the pixel coordinates of the largest neurons (i.e., the upper quartile for size) was used to approximate the orientation and offset of each ganglion26.

### Data acquisition software

Imaging data were acquired using Neuroplex 10.2.2 (RedShirtImaging). Electrophysiological data were acquired simultaneously on Neuroplex 10.2.2 and Clampex 10.4 (Molecular Devices).

### Statistics and reproducibility

All hypothesis tests were performed on SigmaPlot 12. Normality was not assumed and two-tailed Wilcoxon signed-rank tests were used to perform paired comparisons between contingent (N = 7) and yoked (N = 7) groups in Figs. 1f, g, 2d, e, and 3f, g. Alpha was set to 0.05. Data analyses were performed while blinded to the training history of each preparation.

### Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.

## Data availability

Data underlying the study have been deposited to Dryad (https://doi.org/10.5061/dryad.m37pvmd3q). The dataset includes activity for all recorded neurons and source data for the plots in this article.

## Code availability

Data were analyzed using Sigmaplot 12 (Systat Software) and Matlab R2020b (Mathworks). Non-negative matrix factorization was performed using the seqNMF package (https://github.com/FeeLab/seqNMF; Mackevicius et al.28). Custom algorithms are fully described in the Methods section, and corresponding code is available on Dryad/Zenodo (https://doi.org/10.5281/zenodo.5790146) and Github (https://github.com/Byrne-Lab/costa_et_al_OC_VSD).

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## Acknowledgements

We thank Drs. Ryota Homma and Paul Smolen for comments on an earlier draft of the manuscript. This work was supported by National Institutes of Health grant R01 NS101356 (JHB), CNPq Science without Borders Scholarship 203059/2014-0 (RMC), and a Larry Deaven Ph.D. Fellowship in Biomedical Sciences (RMC).

## Author information

Authors

### Contributions

Conceptualization, methodology, and writing—review and editing: RMC, DAB, and JHB. Investigation, visualization, analysis, and writing—original draft: RMC. Funding acquisition and project administration: JHB. Supervision: DAB and JHB.

### Corresponding author

Correspondence to John H. Byrne.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Peer review

### Peer review information

Communications Biology thanks Gyorgy Kemenes, Etsuro Ito and the other, anonymous, reviewers for their contribution to the peer review of this work. Primary Handling Editor: Karli Montague-Cardoso.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Costa, R.M., Baxter, D.A. & Byrne, J.H. Neuronal population activity dynamics reveal a low-dimensional signature of operant learning in Aplysia. Commun Biol 5, 90 (2022). https://doi.org/10.1038/s42003-022-03044-1

• Accepted:

• Published:

• DOI: https://doi.org/10.1038/s42003-022-03044-1


text:
# 2D genetic fractals: a Makers Guide

Next article: plotting fractals

Having plastered pretty pictures all over the internet and even having a genetic fractal in an art gallery (yes…!), the time must have come to out genetic fractals beyond sharing beautiful maths equations. So here it is, a series on constructing genetic fractals in 2 dimensions. “Why only two?”, I hear you groan. Because 3 dimensions are HARD. Not too hard but we’d lose sight of how these fractals work. So, 2D first and if courage remains, we’ll do 3D.

This will be series of posts which I will also paste together in an article. Or a booklet. Hell, let’s go crazy! But as all good stories, in the beginning…

### The Genetic Equation

You may have seen this before on this site and that is not surprising. The creation equation is a mathematical formulation of a genetic fractal. Any genetic fractal.Here it is again.

$C =\int_s D_{r} e^{i\int_s D _{\varphi }}$

Or in trigonometric form:

$C=\left\{\begin{matrix}\int_s D_{r}sin(\int_s D_{\varphi}ds)ds\\ \int_s D_{r}cos(\int_sD_{\varphi}ds)ds\end{matrix}\right..$

In today’s installment, we’re just going to focus on geometry inside this equation. No fancy branches and fractal patterns, just the simple lines (y against x) that this equation will generate.

To try out this equation, we need the driver functions $D_{r}$ and $D_{\varphi}$. We can make them as simple or complex as we want. Since for this first test we can use Excel, we can make these driver functions arbitrary and play around with them. But just to reassure you that there is no slight of hand, if $D_{r}=1$ and $D_{\varphi}=0.3141$ then the creation equation becomes

$C =\int_s e^{i\int_s 0.3141}$

Below is an Excel implementation of this. Source here.  (Go on, click and play)

These simple driver functions generate a circle. No excitement yet. But now you have an Excel model and you can mess around with the drivers. To get you going, I’ve added a few extra sheets in the excel so you get the idea. Have fun – just mess around with the Dr and Dphi columns. (Better not change the other columns). If you make the data list longer in order to get more ” graph”, just change the data selection of the graphs or create new x-y graphs. You know how.

Below are screenshots of the Excel output: “spiral”, “random Dphi”, “specs”, “hexagon”, “hive”.

To be continued: in the next installment we’ll add branches (and make real fractals!!!)

Enjoy.

Next article: plotting fractals

## 7 thoughts on “2D genetic fractals: a Makers Guide”

1. Whooooosh is the sound this makes flying over my head 🙂

Like

2. Phil Boswell says:

“Or in trigonometric form”, formula does not parse.

I can see the raw code in a tooltip but it makes little sense to me, sorry!

Liked by 1 person

• Thanks Phil – somewhere along the line the “\” in latex got stripped. Fixed now. In the meantime I have adopted a better notation. Have a look at the end of this paper: https://arxiv.org/pdf/1703.06307.pdf
Thanks for letting me know. I’ve fixed the code.
Henk

Like

format: plain


text:
# Revision history [back]

### What is the second case of ros::ok() return false?

Here says that

ros::ok() will return false if:

1. a SIGINT is received (Ctrl-C)
2. we have been kicked off the network by another node with the same name
3. ros::shutdown() has been called by another part of the application.

I understand 1 and 3,but what is the second situation?

How to kicked off the network by another node with the same name?

Thank you~


text:
# zbMATH — the first resource for mathematics

## Wang, Yangming

Compute Distance To:
 Author ID: wang.yangming Published as: Wang, Y.; Wang, Y. M.; Wang, Y.-M.; Wang, Yangming
 Documents Indexed: 104 Publications since 1975

#### Co-Authors

 0 single-authored 2 He, Xuanli 2 Li, Shirong

#### Serials

 1 Ukrainian Mathematical Journal 1 Rendiconti del Seminario Matematico della Università di Padova

#### Fields

 2 Group theory and generalizations (20-XX) 1 Combinatorics (05-XX)

#### Citations contained in zbMATH

55 Publications have been cited 233 times in 226 Documents Cited by Year
A reformulation of the strong ellipticity conditions for unconstrained hyperelastic media. Zbl 0876.73030
Wang, Y.; Aron, M.
1996
Characteristic boundary conditions for direct simulations of turbulent counterflow flames. Zbl 1086.80006
Yoo, C. S.; Wang, Y.; Trouvé, A.; Im, H. G.
2005
A note on the numerical solution of high-order differential equations. Zbl 1031.65087
Wang, Y.; Zhao, Y. B.; Wei, G. W.
2003
The influence of inclusion shape on the overall viscoelastic behavior of composites. Zbl 0775.73173
Wang, Y. M.; Weng, G. J.
1992
Implicit-explicit finite-difference lattice Boltzmann method for compressible flows. Zbl 1151.82405
Wang, Y.; He, Y. L.; Zhao, T. S.; Tang, G. H.; Tao, W. Q.
2007
Statistical analysis for stochastic systems including fractional derivatives. Zbl 1183.70062
Huang, Z. L.; Jin, X. L.; Lim, C. W.; Wang, Y.
2010
Three-dimensional non-free-parameter lattice-Boltzmann model and its application to inviscid compressible flows. Zbl 1229.76090
Li, Q.; He, Y. L.; Wang, Y.; Tang, G. H.
2009
The Stroh formalism for anisotropic materials that possess an almost extraordinary degenerate matrix $$\mathbf N$$. Zbl 0943.74511
Wang, Y. M.; Ting, T. C. T.
1997
Mechanics of corrugated surfaces. Zbl 1200.74008
Wang, Y.; Weissmüller, J.; Duan, H. L.
2010
Co-ordinated control design of generator excitation and SVC for transient stability and voltage regulation enhancement of multi-machine power systems. Zbl 1057.93038
Cong, L.; Wang, Y.; Hill, D. J.
2004
Probability-one homotopy algorithms for solving the coupled Lyapunov equations arising in reduced-order $$H^2/H^\infty$$ modeling, estimation, and control. Zbl 1028.93011
Wang, Y.; Bernstein, D. S.; Watson, L. T.
2001
Scheduling projects with labor constraints. Zbl 0984.90012
Cavalcante, C. C. B.; de Souza, C. Carvalho; Savelsbergh, M. W. P.; Wang, Y.; Wolsey, L. A.
2001
Lyapunov function construction for nonlinear stochastic dynamical systems. Zbl 1284.93207
Ling, Quan; Jin, Xiao Ling; Wang, Y.; Li, H. F.; Huang, Zhi Long
2013
Instabilities of core-shell heterostructured cylinders due to diffusions and epitaxy: Spheroidization and blossom of nanowires. Zbl 1162.74317
Duan, H. L.; Weissmüller, J.; Wang, Y.
2008
Impingement of filler dropelts and weld pool dynamics during gas metal arc welding process. Zbl 1064.76624
Wang, Y.; Tsai, Hai Lung
2001
Stochastic vibration model of gear transmission systems considering speed-dependent random errors. Zbl 0946.70505
Wang, Y.; Zhang, W. J.
1998
On deformations with constant modified stretches describing the bending of rectangular blocks. Zbl 0830.73010
Aron, M.; Wang, Y.
1995
Numerical solutions comparison for interval linear programming problems based on coverage and validity rates. Zbl 1427.65094
Lu, H. W.; Cao, M. F.; Wang, Y.; Fan, X.; He, L.
2014
Robustness of non-linear stochastic optimal control for quasi-Hamiltonian systems with parametric uncertainty. Zbl 1287.93110
Wang, Y.; Ying, Z. G.; Zhu, W. Q.
2009
Numerical simulations of gas resonant oscillations in a closed tube using lattice Boltzmann method. Zbl 1143.80332
Wang, Y.; He, Y. L.; Li, Q.; Tang, G. H.
2008
Two-mode ILC with pseudo-downsampled learning in high frequency range. Zbl 1117.93045
Zhang, B.; Wang, D.; Ye, Y.; Wang, Y.; Zhou, K.
2007
Time-dependent Poiseuille flows of visco-elasto-plastic fluids. Zbl 1106.76011
Wang, Y.
2006
Asymptotic finite strip analysis of doubly curved laminated shells. Zbl 1015.74031
Wu, C.-P.; Wang, Y.-M.; Hung, Y.-C.
2001
On a straightening of compressible, nonlinearly elastic, annular cylindrical sectors. Zbl 1001.74531
Aron, M.; Christopher, C.; Wang, Y.
1998
Remarks concerning the flexure of a compressible nonlinearly elastic rectangular block. Zbl 0835.73012
Aron, M.; Wang, Y.
1995
Numerical inverse isoparametric mapping in 3D FEM. Zbl 0638.73037
Murti, V.; Wang, Y.; Valliappan, S.
1988
Theoretical analysis for blow-up behaviors of differential equations with piecewise constant arguments. Zbl 1410.34184
Zhou, Y. C.; Yang, Z. W.; Zhang, H. Y.; Wang, Y.
2016
Chaos and Hopf bifurcation of a finance system with distributed time delay. Zbl 1217.91225
Wang, Y.; Zhai, Y. H.; Wang, J.
2010
Nearly $$s$$-normal subgroups of a finite group. Zbl 1170.20015
Guo, W.; Wang, Y.; Shi, L.
2008
Mass modified outlet boundary for a fully developed flow in the lattice Boltzmann equation. Zbl 1200.76151
Tong, C. Q.; He, Y. L.; Tang, G. H.; Wang, Y.; Liu, Y. W.
2007
Simulation of two-dimensional oscillating flow using the lattice Boltzmann method. Zbl 1107.82370
Wang, Y.; He, Y. L.; Tang, G. H.; Tao, W. Q.
2006
A finite element approach to dynamic modeling of flexible spatial compound bar-gear systems. Zbl 1140.70447
Wang, Y.; Zhang, W. J.; Cheung, H. M. E.
2001
3D dynamic modelling of spatial geared systems. Zbl 1015.70004
Wang, Y.; Cheung, H. M. E.; Zhang, W. J.
2001
Time-dependent absorbing boundary conditions for elastic wave propagation. Zbl 0998.74071
Ju, S. H.; Wang, Y. M.
2001
Transient characterization of flat plate heat pipes during startup and shutdown operations. Zbl 1065.80500
Wang, Y.; Vafai, K.
2000
On the inverse source method of solving inverse scattering problems. Zbl 0805.35147
Chew, W. C.; Wang, Y. M.; Otto, G.; Lesselier, D.; Bolomey, J. Ch.
1994
The influence of the compiler on the cost of mathematical software - in particular on the cost of triangular factorization. Zbl 0312.68017
Parlett, B. N.; Wang, Y.
1975
On BNA-normality and solvability of finite groups. Zbl 1368.20010
He, Xuanli; Li, Shirong; Wang, Yangming
2016
Fluctuating hydrodynamics methods for dynamic coarse-grained implicit-solvent simulations in LAMMPS. Zbl 1364.65172
Wang, Y.; Sigurdsson, J. K.; Atzberger, P. J.
2016
Computationally efficient banding of large covariance matrices for ordered data and connections to banding the inverse Cholesky factor. Zbl 1292.62082
Wang, Y.; Daniels, M. J.
2014
Construction of operator-orthogonalwavelet-based elements for adaptive analysis of thin plate bending problems. Zbl 1356.74127
Wang, Y. M.; Wu, Q.
2013
Bayesian modeling of the dependence in longitudinal data via partial autocorrelations and marginal variances. Zbl 1277.62088
Wang, Y.; Daniels, M. J.
2013
Eikonal equation-based front propagation for arbitrary complex configurations. Zbl 1167.74051
Wang, Y.; Guibault, F.; Camarero, R.
2008
Superlinearly convergent trust-region method without the assumption of positive-definite Hessian. Zbl 1139.90032
Zhang, J. L.; Wang, Y.; Zhang, X. S.
2006
Axisymmetric buckling of transversely isotropic circular and annular plates. Zbl 1097.74025
Xu, R. Q.; Wang, Y.; Chen, W. Q.
2005
Detrended fluctuation analysis of human brain electroencephalogram. Zbl 1209.92032
Pan, C. P.; Zheng, B.; Wu, Y. Z.; Wang, Y.; Tang, X. W.
2004
Sortability of multi-partitions. Zbl 1047.90058
Hwang, F. K.; Wang, Y. M.; Lee, J. S.
2002
Responses of near-optimal, continuous horizontally curved beams to transit loads. Zbl 1235.74201
Wilson, J. F.; Wang, Y.; Threlfall, I.
1999
Subsea vehicle path planning using nonlinear programming and constructive solid geometry. Zbl 0875.93293
Wang, Y.; Lane, D. M.
1997
Radial deformations of cylindrical and spherical shells composed of a generalized Blatz-Ko material. Zbl 0925.73564
Wang, Y.; Aron, Marian
1996
Dynamic crack growth in TDCB specimens. Zbl 0857.73066
Wang, Y.; Williams, J. G.
1996
Uncertainty, variability, and sensitivity analysis in physiological pharmacokinetic models. Zbl 0907.62127
Krewski, D.; Wang, Y.; Bartlett, S.; Krishnan, K.
1995
Parallelizing Strassen’s method for matrix multiplication on distributed-memory MIMD architectures. Zbl 0839.68093
Chou, C.-C.; Deng, Y.-F.; Li, G.; Wang, Y.
1995
Variable structure, bilinear and intelligent control with power system application. Zbl 0774.93017
Mohler, R. R.; Wang, Y.; Zakrzewski, R. R.; Vedam, Rajkumar
1992
The axisymmetric pulsar magnetosphere. Zbl 0573.76124
Mestel, L.; Robertson, J. A.; Wang, Y.-M.; Westfold, K. C.
1985
Theoretical analysis for blow-up behaviors of differential equations with piecewise constant arguments. Zbl 1410.34184
Zhou, Y. C.; Yang, Z. W.; Zhang, H. Y.; Wang, Y.
2016
On BNA-normality and solvability of finite groups. Zbl 1368.20010
He, Xuanli; Li, Shirong; Wang, Yangming
2016
Fluctuating hydrodynamics methods for dynamic coarse-grained implicit-solvent simulations in LAMMPS. Zbl 1364.65172
Wang, Y.; Sigurdsson, J. K.; Atzberger, P. J.
2016
Numerical solutions comparison for interval linear programming problems based on coverage and validity rates. Zbl 1427.65094
Lu, H. W.; Cao, M. F.; Wang, Y.; Fan, X.; He, L.
2014
Computationally efficient banding of large covariance matrices for ordered data and connections to banding the inverse Cholesky factor. Zbl 1292.62082
Wang, Y.; Daniels, M. J.
2014
Lyapunov function construction for nonlinear stochastic dynamical systems. Zbl 1284.93207
Ling, Quan; Jin, Xiao Ling; Wang, Y.; Li, H. F.; Huang, Zhi Long
2013
Construction of operator-orthogonalwavelet-based elements for adaptive analysis of thin plate bending problems. Zbl 1356.74127
Wang, Y. M.; Wu, Q.
2013
Bayesian modeling of the dependence in longitudinal data via partial autocorrelations and marginal variances. Zbl 1277.62088
Wang, Y.; Daniels, M. J.
2013
Statistical analysis for stochastic systems including fractional derivatives. Zbl 1183.70062
Huang, Z. L.; Jin, X. L.; Lim, C. W.; Wang, Y.
2010
Mechanics of corrugated surfaces. Zbl 1200.74008
Wang, Y.; Weissmüller, J.; Duan, H. L.
2010
Chaos and Hopf bifurcation of a finance system with distributed time delay. Zbl 1217.91225
Wang, Y.; Zhai, Y. H.; Wang, J.
2010
Three-dimensional non-free-parameter lattice-Boltzmann model and its application to inviscid compressible flows. Zbl 1229.76090
Li, Q.; He, Y. L.; Wang, Y.; Tang, G. H.
2009
Robustness of non-linear stochastic optimal control for quasi-Hamiltonian systems with parametric uncertainty. Zbl 1287.93110
Wang, Y.; Ying, Z. G.; Zhu, W. Q.
2009
Instabilities of core-shell heterostructured cylinders due to diffusions and epitaxy: Spheroidization and blossom of nanowires. Zbl 1162.74317
Duan, H. L.; Weissmüller, J.; Wang, Y.
2008
Numerical simulations of gas resonant oscillations in a closed tube using lattice Boltzmann method. Zbl 1143.80332
Wang, Y.; He, Y. L.; Li, Q.; Tang, G. H.
2008
Nearly $$s$$-normal subgroups of a finite group. Zbl 1170.20015
Guo, W.; Wang, Y.; Shi, L.
2008
Eikonal equation-based front propagation for arbitrary complex configurations. Zbl 1167.74051
Wang, Y.; Guibault, F.; Camarero, R.
2008
Implicit-explicit finite-difference lattice Boltzmann method for compressible flows. Zbl 1151.82405
Wang, Y.; He, Y. L.; Zhao, T. S.; Tang, G. H.; Tao, W. Q.
2007
Two-mode ILC with pseudo-downsampled learning in high frequency range. Zbl 1117.93045
Zhang, B.; Wang, D.; Ye, Y.; Wang, Y.; Zhou, K.
2007
Mass modified outlet boundary for a fully developed flow in the lattice Boltzmann equation. Zbl 1200.76151
Tong, C. Q.; He, Y. L.; Tang, G. H.; Wang, Y.; Liu, Y. W.
2007
Time-dependent Poiseuille flows of visco-elasto-plastic fluids. Zbl 1106.76011
Wang, Y.
2006
Simulation of two-dimensional oscillating flow using the lattice Boltzmann method. Zbl 1107.82370
Wang, Y.; He, Y. L.; Tang, G. H.; Tao, W. Q.
2006
Superlinearly convergent trust-region method without the assumption of positive-definite Hessian. Zbl 1139.90032
Zhang, J. L.; Wang, Y.; Zhang, X. S.
2006
Characteristic boundary conditions for direct simulations of turbulent counterflow flames. Zbl 1086.80006
Yoo, C. S.; Wang, Y.; Trouvé, A.; Im, H. G.
2005
Axisymmetric buckling of transversely isotropic circular and annular plates. Zbl 1097.74025
Xu, R. Q.; Wang, Y.; Chen, W. Q.
2005
Co-ordinated control design of generator excitation and SVC for transient stability and voltage regulation enhancement of multi-machine power systems. Zbl 1057.93038
Cong, L.; Wang, Y.; Hill, D. J.
2004
Detrended fluctuation analysis of human brain electroencephalogram. Zbl 1209.92032
Pan, C. P.; Zheng, B.; Wu, Y. Z.; Wang, Y.; Tang, X. W.
2004
A note on the numerical solution of high-order differential equations. Zbl 1031.65087
Wang, Y.; Zhao, Y. B.; Wei, G. W.
2003
Sortability of multi-partitions. Zbl 1047.90058
Hwang, F. K.; Wang, Y. M.; Lee, J. S.
2002
Probability-one homotopy algorithms for solving the coupled Lyapunov equations arising in reduced-order $$H^2/H^\infty$$ modeling, estimation, and control. Zbl 1028.93011
Wang, Y.; Bernstein, D. S.; Watson, L. T.
2001
Scheduling projects with labor constraints. Zbl 0984.90012
Cavalcante, C. C. B.; de Souza, C. Carvalho; Savelsbergh, M. W. P.; Wang, Y.; Wolsey, L. A.
2001
Impingement of filler dropelts and weld pool dynamics during gas metal arc welding process. Zbl 1064.76624
Wang, Y.; Tsai, Hai Lung
2001
Asymptotic finite strip analysis of doubly curved laminated shells. Zbl 1015.74031
Wu, C.-P.; Wang, Y.-M.; Hung, Y.-C.
2001
A finite element approach to dynamic modeling of flexible spatial compound bar-gear systems. Zbl 1140.70447
Wang, Y.; Zhang, W. J.; Cheung, H. M. E.
2001
3D dynamic modelling of spatial geared systems. Zbl 1015.70004
Wang, Y.; Cheung, H. M. E.; Zhang, W. J.
2001
Time-dependent absorbing boundary conditions for elastic wave propagation. Zbl 0998.74071
Ju, S. H.; Wang, Y. M.
2001
Transient characterization of flat plate heat pipes during startup and shutdown operations. Zbl 1065.80500
Wang, Y.; Vafai, K.
2000
Responses of near-optimal, continuous horizontally curved beams to transit loads. Zbl 1235.74201
Wilson, J. F.; Wang, Y.; Threlfall, I.
1999
Stochastic vibration model of gear transmission systems considering speed-dependent random errors. Zbl 0946.70505
Wang, Y.; Zhang, W. J.
1998
On a straightening of compressible, nonlinearly elastic, annular cylindrical sectors. Zbl 1001.74531
Aron, M.; Christopher, C.; Wang, Y.
1998
The Stroh formalism for anisotropic materials that possess an almost extraordinary degenerate matrix $$\mathbf N$$. Zbl 0943.74511
Wang, Y. M.; Ting, T. C. T.
1997
Subsea vehicle path planning using nonlinear programming and constructive solid geometry. Zbl 0875.93293
Wang, Y.; Lane, D. M.
1997
A reformulation of the strong ellipticity conditions for unconstrained hyperelastic media. Zbl 0876.73030
Wang, Y.; Aron, M.
1996
Radial deformations of cylindrical and spherical shells composed of a generalized Blatz-Ko material. Zbl 0925.73564
Wang, Y.; Aron, Marian
1996
Dynamic crack growth in TDCB specimens. Zbl 0857.73066
Wang, Y.; Williams, J. G.
1996
On deformations with constant modified stretches describing the bending of rectangular blocks. Zbl 0830.73010
Aron, M.; Wang, Y.
1995
Remarks concerning the flexure of a compressible nonlinearly elastic rectangular block. Zbl 0835.73012
Aron, M.; Wang, Y.
1995
Uncertainty, variability, and sensitivity analysis in physiological pharmacokinetic models. Zbl 0907.62127
Krewski, D.; Wang, Y.; Bartlett, S.; Krishnan, K.
1995
Parallelizing Strassen’s method for matrix multiplication on distributed-memory MIMD architectures. Zbl 0839.68093
Chou, C.-C.; Deng, Y.-F.; Li, G.; Wang, Y.
1995
On the inverse source method of solving inverse scattering problems. Zbl 0805.35147
Chew, W. C.; Wang, Y. M.; Otto, G.; Lesselier, D.; Bolomey, J. Ch.
1994
The influence of inclusion shape on the overall viscoelastic behavior of composites. Zbl 0775.73173
Wang, Y. M.; Weng, G. J.
1992
Variable structure, bilinear and intelligent control with power system application. Zbl 0774.93017
Mohler, R. R.; Wang, Y.; Zakrzewski, R. R.; Vedam, Rajkumar
1992
Numerical inverse isoparametric mapping in 3D FEM. Zbl 0638.73037
Murti, V.; Wang, Y.; Valliappan, S.
1988
The axisymmetric pulsar magnetosphere. Zbl 0573.76124
Mestel, L.; Robertson, J. A.; Wang, Y.-M.; Westfold, K. C.
1985
The influence of the compiler on the cost of mathematical software - in particular on the cost of triangular factorization. Zbl 0312.68017
Parlett, B. N.; Wang, Y.
1975
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#### Cited by 513 Authors

 11 Qi, Liqun 5 Molabahrami, Ahmad 4 He, Yaling 4 Ling, Chen 4 Zhao, Jianxing 3 Bu, Changjiang 3 Guo, Lixin 3 Javili, Ali 3 Sang, Caili 3 Shidfar, Abdollah 3 Vervisch, Luc 3 Wang, Haotian 3 Wang, Jingyue 3 Wang, Xinwei 3 Wang, Yong 3 Wei, Guowei 3 Yao, Hongmei 3 Yoo, Chun Sang 3 Zhang, Xinzhen 3 Zhu, Weiqiu 2 Abd-Elhameed, Waleed Mohamed 2 Aron, Marian 2 Aron, Miles 2 Babaei, Afshin 2 Bonnet, Guy 2 Cen, Kefa 2 Char, Ming-I 2 Chen, Rongqian 2 Coussement, Axel 2 Dai, Hui-Hui 2 Degrez, Gérard 2 Domingo, Pascale 2 Duong, Pham Luu Trung 2 Fang, Chung 2 Fiorina, Benoît 2 Fu, Yiming 2 Gicquel, Olivier 2 Guo, Wenbin 2 Han, Deren 2 Hawkes, Evatt R. 2 He, Pei 2 He, Yilin 2 Im, Hong G. 2 Ishaquddin, Md. 2 Khani, Farzad 2 Lanzoni, Luca 2 Lee, Moonyong 2 Lévesque, Martin 2 Li, Chaoqian 2 Li, Jackie Ji 2 Liu, Qing 2 Liu, Weiyan 2 Lodato, Guido 2 Lv, Yu 2 Mackerle, Jaroslav 2 Pian, Theodore Hsueh-huang 2 Poinsot, Thierry J. 2 Qiu, Ruofan 2 Shu, Chang 2 Tai, Bo-Chen 2 Tang, Guanhua 2 Tao, Wen-Qing 2 Ting, Thomas C. T. 2 Tsai, Hai-Lung 2 Tsai, Hsien-Lin 2 Vielma, Juan Pablo 2 Wang, Yuer 2 Wang, Zhihua 2 Weng, George J. 2 Wu, Chih-Ping 2 Xu, Wei 2 Yang, Qingzhi 2 Yang, Yuning 2 Yazdanian, A. R. 2 You, Yancheng 2 Yuan, Kuan-Ya 2 Zhao, Shan 2 Zhou, Guanglu 2 Zhou, Jiang 2 Zhou, Junhu 2 Zhu, Chengxiang 1 Abdollahi, Vahid 1 Agrawal, Rashmi 1 Al-Mdallal, Qasem M. 1 Albin, Eric 1 Alipour, Mir Mohammad 1 Alizad, Karim 1 Allahdadi, Mehdi 1 Alonso, Andrés M. 1 Araujo, Janniele A. S. 1 Atzberger, Paul J. 1 Avagyan, Vahe 1 Ayatollahi, Mojtaba 1 Azrar, Lahcen 1 Bader, Philipp 1 Bagheri, Rasul 1 Bahei-El-Din, Yehia A. 1 Bakkali, A. 1 Baptiste, Didier 1 Barello, Romina B. ...and 413 more Authors
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# What is the 'right' slope formula of a regression? deltas or Pearson?

this may be a silly question, but still:

• I've been told that the slope formula equals the rise/run ratio, like this:

$$m = \frac{rise}{run} = \frac{y_2 - y_1}{x_2 - x_1}$$

in which rise equals detaY and run equals deltaX

• however, I've also seen it using Pearson's correlation coefficient, such as in many posts here, like this one or this one :

$$b = r_{x,y} \frac{\sigma_y}{\sigma_x}$$

• Probably they're the same thing and I'm not able to see it... but, how are these two equations the same? I'd really want to understand this.
• Hi Larissa. If you are still not aware of, Cross Validated is MathJax enabled; for a quick guide, check: Instructions on how to use LaTeX on CrossValidated. Sep 24 at 20:34
• thank you for the link! and thanks @Tim for the answer and the edit, the post looks much nicer now! I'll work on the answer! Sep 24 at 23:52

For only two points they are the same.

The slope of simple linear regression is

$$\hat \beta = \frac{\sum_i (x_i - \bar x) (y_i - \bar y)}{\sum_i (x_i - \bar x)^2}$$

that is the same form you mentioned in the second bullet. Notice that when we have two points, then things like $$x_1 - \bar x$$ can be written as

$$x_1 - \frac{x_1 + x_2}{2} = \frac{x_1 - x_2}{2}$$

So we can re-write

\require{cancel} \begin{align} \hat \beta &= \frac{\frac{x_1 - x_2}{2}\frac{y_1 - y_2}{2} + \frac{x_2 - x_1}{2} \frac{y_2 - y_1}{2}}{ \frac{x_1 - x_2}{2}\frac{x_1 - x_2}{2} + \frac{x_2 - x_1}{2}\frac{x_2 - x_1}{2} } \\ &= \frac{ (\frac{x_1y_1}{4} - \frac{x_1y_2}{4} - \frac{x_2y_1}{4} + \frac{x_2y_2}{4}) + (\frac{x_2y_2}{4} - \frac{x_2y_1}{4} - \frac{x_1y_2}{4} + \frac{x_1y_1}{4}) }{\cancel{2} \frac{x_1 - x_2}{\cancel 2}\frac{x_1 - x_2}{2}} \\ &= \frac{\frac{x_1 y_1 - x_1 y_2 - x_2 y_1 + x_2 y_2}{\cancel 2}}{ (x_1 - x_2)\frac{x_1 - x_2}{\cancel 2}} \\ &= \frac{\cancel{(x_1 - x_2)}( y_1 - y_2)}{\cancel{(x_1 - x_2)}(x_1 - x_2)} \\ &= \frac{y_1 - y_2}{x_1 - x_2} = \frac{y_2 - y_1}{x_2 - x_1} \end{align}

They must have been the same because the regression line calculated for two points needs to pass through them as there's no "noise". Linear regression is a linear model, so they need to be algebraically the same.

If you have more than two points, as Dave mentioned, you cannot use the rise/run formula anymore.

These are fairly unrelated concepts.

In the first equation, that is the slope of the line connecting two points.

In the second, you are finding a line that fits multiple points best (according to a particular and common definition of best). This line is called the ordinary least squares regression line. This is what is produced by “add trendline” in Excel. Unlike in the first case, your “trendline” might not hit any of the points.

The two will coincide if you have two points and fit a “trendline” to those points (aside from a technicality if the points have the same $$x$$- or $$y$$-coordinate), but I’d say that the similarities mostly end there.

• huhhh, very interesting, thank you. If you don't mind, could you go over the part "The two will coincide if you have two points and fit a “trendline” to those points, but I’d say that the similarities mostly end there" ? @Dave Sep 24 at 16:37
• What about it? The math works out that way. It might be a useful exercise to do the algebra. @LarissaCury
– Dave
Sep 24 at 16:45


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Graphene Nanoplatelets Screen-Printed on Woven and Knitted Fabrics: History
Please note this is an old version of this entry, which may differ significantly from the current revision.

Although the force/pressure applied onto a textile substrate through a uniaxial compression is constant and independent of the yarn direction, it should be noted that such mechanical action causes a geometric change in the substrate, which can be identified by the reduction in its lateral thickness. Therefore, researchers investigate the influence of the fabric orientation on both knitted and woven pressure sensors, in order to generate knowledge for a better design process during textile piezoresistive sensor development.

• smart textiles
• piezoresistive sensors
• graphene nanoplatelets

## 1. Introduction

In addition to the role as a social communication tool [1] and the function of protection against environmental factors [2], textiles, by wrapping the body, can be the “territory” to detect internal (corporeal) and external (environmental) stimuli. This is the concept of Smart Textiles, in other words, fabrics that can interact with the environment/user by detecting and, sometimes, reacting and adapting to mechanical, thermal, chemical or electrical stimuli [3]. One of these stimuli is the pressure that can be exerted by the body itself, through the bones and blood pressure, or through external objects, such as the use of wearable medical devices. In turn, the measurement and action upon said body pressure can, in some cases, prevent the development of skin diseases, like pressure ulcers [4], and cardiovascular diseases, through pulse pressure monitoring [5], as well as being inserted into haptic feedback systems to generate tactile information for amputees [6].
Given their wide applicability, pressure sensors in a wearable textile form can, non-invasively, measure mechanical action on the body that can be used, e.g., to quantify performance or identify possible harm. To meet this sensing requirement, flexible sensors seem promising, as they can mechanically deform, granting them the ability to sense the stimuli at its origin, where the signal is most accurate [7]. Regarding the perspective of comfort, they can also meet the requirements of negligible weight, tailorability to regular daily garments and conformability.
The transduction mechanisms that present electrical responses from mechanical deformations are piezoresistive, piezoelectric, capacitive, triboelectric and transistive [8]. These transduction methods, in general, rely on an active material whose electrical property changes upon mechanical stress. The presence of two electrodes captures this property on the said active material. Among the mechanoelectrical ones, piezoresistive sensors are the most prevalent in the literature to be applied as electronic skin for their simple mechanism, compact structure, low cost, energy consumption and ease of signal acquisition [9].
Sensitivity and detection range are two important parameters to evaluate pressure sensors’ performance. In this context, sensitivity is defined as the rate of change of the electrical property with the change of the measured variable. In bulk piezoresistive sensors, the sensitivity is, generally, measured by the polymer matrix compression modulus and the filler direction change due to decreased percolation pathways caused by disconnections and the change in the compatibility between the conductive materials and the matrix. Furthermore, it should be noted that under uniaxial compression conditions, the load applied to the material will be uniformly distributed over it. However, when referring to an intrinsically anisotropic material, such as a textile substrate, the contact between the fibres in different directions can lead to peculiar deformation performance and, consequently, alter the path of current conduction. In line with this, a recent study carried out by Xie et al. indicated that pressure sensors made of knitted fabric have different sensitivities in the walewise and coursewise directions [10]. This study emphasized that the way the yarns are in contact with each other will influence their compressibility property and, therefore, the electrical response of the sensor. The mechanical performance of a textile substrate will also be influenced by the yarn properties (such as the diameter, the coefficient of friction and the initial Young’s modulus) [11] and by its weave or knit structure.

## 2. Piezoresistive Behaviour in Polymer Composites

The term piezo comes from the Greek word “piezen”, which means to press and to compress. Through this etymological meaning, it is understood that a piezoresistive sensor transduces an applied pressure into an electrical resistance variation [12].
A simple piezoresistive sensor model consists of an active part positioned between two electrodes. The resistance value of the sensor is defined by being the resistance of the electrodes (Re) added to the resistance of the active material (Ra). It is noteworthy that the change in Ra is the reason for the piezoresistive behaviour, considering that Re is a constant value. Through this discursive reasoning, it is worth mentioning the material resistance equation:
where $\mathsf{\rho }$ is the material resistivity, L is the length and A is the cross-sectional area.
According to the equation above, and in line with previous studies, the resistance variation in piezoresistive sensors is influenced by two main factors: geometric deformation of the elastomeric composite, where the parameters L and A change according to the material deformation, and the resistivity of fillers in band structure and interparticle separation [9]. The conductive polymer composites, generally composed of conductive particles, i.e., fillers, dispersed in an insulating matrix, rely on the percolation theory. From phase a, known for presenting very low electrical conductivity values, up to zone c, where the filler particles come into direct contact to form perfect networks, the conductivity values increase significantly. At some point, though, the percolation threshold is exceeded, and conductivity does not increase substantially. Hereupon, particle geometries and properties, insulating matrix properties and polymer–particle interaction will directly influence the tunnelling conduction mechanism under a mechanical deformation [9,13].

## 3. Materials (Active, Electrodes and Substrates)

Given the above, the materials that compose a sensor have a direct relationship with its piezoresistive response. In its simplest form, the sensor is composed of the sensing material, also known as active material, signal transfer components and a flexible substrate. Each component comprises different requirements.
Active materials are those responsible for detecting the mechanical stimulus. In this sense, they need to possess reliable electrical conduction paths, exceptional chemical stability, good mechanical compliance and compatibility with large-area processing techniques [9]. Semiconducting polymers such as polydimethylsiloxane (PDMS) [14], polypropylene (PP) [15], semiconducting nanowires and carbon-based materials are some of these. However, carbon and its allotropes, such as graphene and nanotubes, are the most used for application in polymer-based composites. In addition, graphene, the single sp2-hybridized carbon atom-thick 2D material, has shown to be highly promising due to its excellent mechanical, thermal and electrical properties provided by its internal honeycomb-like structure [16]. Among its variations, graphene nanoplatelets (GNPs) have gained evidence due to their ease of scalability production and low price when compared to others [17]. Furthermore, GNPs have few graphite layers, are lightweight, and have a higher aspect ratio with a planar shape [18]. However, the key challenge in the synthesis and processing of bulk-quantity graphene sheets is aggregation, which tends to occur through van der Waals interactions. In this sense, many synthesis and modification methods have been used, which involve either preparing “amphiphilic” graphene nanoplatelets (a strategy that involves exfoliation followed by Graphene oxide (GO) in situ reductions [19]) or a dry-low-cost method for obtaining N-rich graphene, through gamma irradiation with the use of ethylamine [20]. Furthermore, when dispersed in biopolymers and doped onto textile substrates, such as linen, GNPs’ ecocomposites may have their properties potentiated, namely, improved hydrophobic capacity and UV protection, in addition to their piezoresistive response [21].
Therefore, graphene nanoplatelets were chosen in this study due to their aforementioned superior mechanical, thermal, electrical and scalability properties, beyond the already proven applicability as a filler in polymer matrices with a piezoresistive response.
Regarding signal transfer materials, better known as electrodes, it is expected that they are stretchable materials that can maintain high conductivity under large strains and possess excellent stability [9]. For this purpose, metal electrode materials with a specific design are suitable for stretchable electronics, such as Au, Ag or Cu [22]. Furthermore, in this sense, silver ink was chosen since it has an appropriate electrical conductivity value for electrodes.
For flexible sensors, the substrate can be understood as the base, known as the insulating matrix, in which the active material will be dispersed. To this end, the substrate should have excellent chemical stability, low surface roughness and flexible mechanical properties [7]. Elastomers, self-healing materials, polyurethane (PU) and textile fabrics are presented as some of the materials that meet these requirements.

This entry is adapted from 10.3390/ma15155185

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# Difference between revisions of "2015 USAMO Problems"

## Day 1

### Problem 1

Solve in integers the equation $$x^2+xy+y^2 = \left(\frac{x+y}{3}+1\right)^3.$$

### Problem 2

Quadrilateral $APBQ$ is inscribed in circle $\omega$ with $\angle P = \angle Q = 90^{\circ}$ and $AP = AQ < BP$. Let $X$ be a variable point on segment $\overline{PQ}$. Line $AX$ meets $\omega$ again at $S$ (other than $A$). Point $T$ lies on arc $AQB$ of $\omega$ such that $\overline{XT}$ is perpendicular to $\overline{AX}$. Let $M$ denote the midpoint of chord $\overline{ST}$. As $X$ varies on segment $\overline{PQ}$, show that $M$ moves along a circle.

## Day 2

### Problem 4

Find all functions $f:\mathbb{Q}\rightarrow\mathbb{Q}$ such that$$f(x)+f(t)=f(y)+f(z)$$for all rational numbers $x that form an arithmetic progression. ($\mathbb{Q}$ is the set of all rational numbers.)

### Problem 5

Let $ABCD$ be a cyclic quadrilateral. Prove that there exists a point $X$ on segment $\overline{BD}$ such that $\angle BAC=\angle XAD$ and $\angle BCA=\angle XCD$ if and only if there exists a point $Y$ on segment $\overline{AC}$ such that $\angle CBD=\angle YBA$ and $\angle CDB=\angle YDA$.

### Problem 6

Steve is piling $m\geq 1$ indistinguishable stones on the squares of an $n\times n$ grid. Each square can have an arbitrarily high pile of stones. After he finished piling his stones in some manner, he can then perform stone moves, defined as follows. Consider any four grid squares, which are corners of a rectangle, i.e. in positions $(i, k), (i, l), (j, k), (j, l)$ for some $1\leq i, j, k, l\leq n$, such that $i and $k. A stone move consists of either removing one stone from each of $(i, k)$ and $(j, l)$ and moving them to $(i, l)$ and $(j, k)$ respectively,j or removing one stone from each of $(i, l)$ and $(j, k)$ and moving them to $(i, k)$ and $(j, l)$ respectively.

Two ways of piling the stones are equivalent if they can be obtained from one another by a sequence of stone moves.

How many different non-equivalent ways can Steve pile the stones on the grid?


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# Intersection triangle of Steiner System

How to use the intersection triangle of the Steiner System $S(5,6,12)$ to show the completement of any of its block is also a block?

My argument: Since the parameter $m_{60}$ in the intersection triangle is 1, so each block $B$ has a unique block $B'$ which is disjoint to the former, thus $B' \subseteq X\diagdown B$. Moreover, since $|X\diagdown B|=12-6=6=|B'|$, so $X\diagdown B=B'$ which implies $X\diagdown B$ is a block. Is my argument valid based on my interpretation of $m_{60}$?

For the convention of the subscripts please Google the book “Topics on Steiner Systems” By Charles C. Lindner, A. Rosa, on page 49

-
Since there are 12 points, two disjoint blocks of six points each must clearly be the complement of each other. Assuming the $m_{60}$ means what you think it does (I cannot see that page in googlebooks) then what you wrote is correct. –  Mariano Suárez-Alvarez May 24 '12 at 6:50
"No eBook available", so hard to tell. Otherwise obviously agree with Mariano –  Jyrki Lahtonen May 24 '12 at 7:04
Along with a googlebooks link, it would be helpful if you always provided a bibliographic reference to the book —title, authors, edition— and a location in the book, if needed: it may well happened that someone has the book but not access to that page in googlebooks! –  Mariano Suárez-Alvarez May 24 '12 at 7:20
I don't have any copy of this book. Can you please give me the meaning of the notations $m_{60}$ and definition of intersection triangle ..So that I can understand your argument. –  users31526 May 24 '12 at 7:57