s2fieldsofstudy: Biology Medicine

text:
Personalized analysis of pathway aberrance induced by sevoflurane and propofol

Anesthetic agents are used in surgical operations to reversibly reduce consciousness and pain. Sevoflurane is an inhalational anesthetic. Propofol is a short-acting intravenous general anesthetic. The mechanism of anesthetic agents at pathway level on individual patients has not been reported to date. In the present study, pathway aberrance in the human atrial tissue in response to anesthetics was examined. Microarray data of anesthesia-treated samples were downloaded from the Array Express database. Pathway information was obtained from the Reactome Pathway Database. The individual pathway aberrance score (iPAS) was introduced to identify dysregulated pathways in individual patients. The present data demonstrated 157 dysregulated pathways in the sevoflurane group, and 44 pathways were identified with the least P-values. A subset of 49 differentially expressed genes (DEGs) that were shared between the expression profiling results and the dysregulated pathways results were constructed into a co-expression network. The top 5 ranked DEGs, nuclear receptor subfamily 4 group A member 3 (NR4A3), JUNB proto-oncogene, MYC proto-oncogene, tachykinin precursor 1 and nicotinamide phosphoribosyltransferase, were identified as important in the topology analysis. In the propofol group, 87 dysregulated pathways were identified and 44 pathways had the least P-values. In total 28 DEGs were constructed into a co-expression network, of which 5 DEGs were important in the topology analysis, NR4A3, suppressor of cytokine signaling 3, cyclin dependent kinase inhibitor 1A, C-C motif chemokine ligand 2 and C-X-C motif chemokine ligand 1. A total of 72 dysregulated pathways were identified in common in the two groups. In conclusion, the two types of anesthetics induced partially similar mechanisms. The pathways enriched by DEGs, particularly those that were unique to sevoflurane and propofol, may affect surgical outcomes and aid the prevention of complications from anesthetics.

Introduction
Anesthetic reagents are used in surgical operations to reduce consciousness and pain reversibly. They have been used for ~150 years due to their efficacy and safety (1). Sevoflurane is an inhalational anesthetic used for general anesthesia. It dissolves poorly in blood, and can be breathed out quickly by the lungs, providing rapid recovery. Propofol is a short-acting, most commonly used, intravenous general anesthetic and it is administered as an alkylphenol formulated in a lipid emulsion (2). It dissolves from the blood to the surrounding tissues and central nerve systems quickly. In addition, propofol exhibits rapid clearance from the body (3). Since these two types of anesthetic are widely used in surgery, a better understanding of their effect in biological processes is warranted.
Although the study of Lucchinetti et al (4) provided insights into the gene expression effects mediated by anesthetics administration inhuman atrial tissues, pathway aberrance in individual patients remains unknown.
Gene-based pathway analysis is becoming an important method for understanding disease mechanisms. Especially pathway analysis based on differentially expressed genes (DEGs) has become an important approach to study pathogenesis mechanisms in an individual sample (5). The analysis of altered pathways in an individual patient may help to understand the disease status and suggest customized therapies. Although the personalized interpretation of pathways can be demanding, most current pathway analyses have been developed to investigate deregulated pathways between two phenotype groups. Three methods were described by Khatri et al (6), overrepresentation analysis, functional class scoring and pathway topology-based approach. They were mainly focused on pathway aberrance between normal samples and cancer samples, but not suitable for an individual sample. The method of individual pathway aberrance score (iPAS) proposed by Ahn et al (5), which is based on overrepresentation analysis and functional class scoring, provides a series of analyses steps, containing four parts: Data processing, gene-level statistics, iPAS and a significance test, with the clinical importance of this method being to provide pathway interpretation of a single cancer sample.
With the development of molecular biology and bioinformatics, network analysis has become a more informative and powerful approach to study disease mechanisms (7). The relationship of disease to a specific network, especially a co-expression network, provides valuable information to analyze gene regulation (8). The topological characteristics may reflect the properties of the network and may be used to investigate the underlying implications of the network.
To test the hypothesis that sevoflurane and propofol may induce pathway aberrance, microarray data of human atrial tissue treated with sevoflurane and propofol were examined separately. DEGs and individual pathway aberrance were identified using individual pathway analysis and co-expression network analysis.

Datasets
Gene expression data. Microarray data of E-GEOD-4386 (4), and its annotation file were downloaded from the Array Express database (http://www.ebi.ac.uk/arrayexpress/). The dataset includes 10 sevoflurane-treated atrial tissues, 10 propofol-treated atrial tissues and 10 matched non-anesthetic-treated atrial tissues. The tissues were obtained from patients who underwent coronary artery bypass graft (CABG) surgery and were given the corresponding anesthetics. The sample platform that was used in the study was the A-AFFY-44-Affymetrix Gene Chip Human Genome U133 Plus 2.0 [HG-U133_Plus_2]. According to platform annotation files, probes were mapped to gene symbols by filtering with the function of the feature Filter (9). The model description was as follows: We used (w, b) as the coefficients and intercept of L1 logistic regression. (1) where L was the loss function, λ was a regularization parameter which had the ability to dispose high dimensional data, xn ϵRN was the n-th feature and yn ϵ RN was the label of the n-th sample. After performing ID transformation with the get SYMBOL function (10), 20,102 genes were obtained.
Pathway data. Information from gene sets representing biological pathways was obtained from the Reactome Pathway Database (http://www.reactome.org) (11). Pathways with large number of genes are more difficult to understand for human experts. Therefore, pathways with gene set size >100 were filtered out (5). Considering that information from different platform may vary, we filtered out the pathways that had no transformational gene symbols. Eventually 1,005 pathways were obtained, including 4,852 genes.
DEG analysis DEG screening. At present, many methods are routinely used to screen for DEGs, including Linear Models for Microarray Data (Limma) (12), significant analysis of microarrays (13), statistical methods for ranking (14), and R package siggenes (15) In the present study, the Limma package was selected to screen for DEGs. With this method, we obtained a P-value for all the genes. The genes with log |fold change (FC)|≥2 and P≤0.01 were selected for further analysis. In total, 269 DEGs were obtained from the sevoflurane samples, and 129 DEGs were obtained from the propofol samples.

Co-expression gene analysis.
To explore the interactive relationship between genes under different conditions, differentially co-expressed genes were constructed using the DEGs, with the following functions in the weighted gene co-expression network analysis (WGCNA) package (16).
where sij is the absolute value of the correlation coefficient between the profiles of nodes i and j, aij encodes the network connection strength between nodes i and j, and τ is the 'hard' threshold parameter. A total of 180 co-expression relationships (813 gene pairs) were identified when samples were treated with sevoflurane, and 119 co-expression relationships (1,216 gene pairs) with propofol. Then a network was constructed by linking atrial genes with the selected gene signatures using Cytoscape (17), a free software package for visualizing, modeling and analyzing the integration of biomolecular interaction networks with high-throughput expression data and other molecular states for further analysis in the topological characteristics.
Pathway analysis. Pathway analysis has become important in capturing clinical information. An iPAS model was introduced to identify altered pathways in an individual sample (18). Normal tissue data were represented as nRef. The process of this analysis was described as follows. Gene level statistics. Gene level statistics was described by Irizarry et al (19,20) by making use of multichip averages. Genes from normal samples were normalized by the function: (5) where represents the unit diagonal, and qk=(qk1,...,qkn ) for k=1,..., p represents the vector of the k-th quantiles for all n arrays For normal controls, the mean and standard deviation of gene expression level were calculated. For individual tumor cases, quantile normalization was performed after combining the single tumor microarray with all nRef samples. The gene-level statistics of individual tumor samples were standardized using the mean and standard deviation of the reference. The formula was: (6) where Zi indicates the standardized expression value of i-th gene and n represents the number of genes belonging to the pathway (5).
Pathway statistics. For each pathway, the pathway statistics is the average of all the gene level statistics which was extracted from the same pathway, calculated as: (7) Where zi symbolizes the standardized expression value of i-th gene belonging to the pathway, and n symbolizes the number of genes belonging to the pathway. The clinical characteristics of a single anesthesia sample can be obtained by measuring its difference against common characteristic of normal samples, which was represented as nRef in our study.
Dysregulated pathways. Significance test was performed to assess altered pathways associated with anesthetics administration. To assess the statistical significance, pathway statistics, were tested by the two-group comparison method Wilcoxon (21), with the function of: where n is the number of samples. The significance levels were revised by false discovery rate (FDR) (22). Finally, each pathway yielded a P-value. A pathway with P<0.01 was considered to be significantly different between the anesthetics group and the normal group.
Individual altered pathway. Significance can be obtained against the null distribution generated from normal samples. The statistic of the null distribution was obtained by comparing every normal sample with all nRef samples. The P-value of each pathway was obtained by comparing each anesthetic sample with the null distribution. P<0.05 was considered to be significant difference.
Analysis of DEGs in dysregulated pathways. DEGs in dysregulated pathways were mapped to co-expression networks to analyze the co-expression relationships among these DEGs. In the sevoflurane group, 49 DEGs were constructed into a network. In the propofol group, 28 DEGs were constructed into a network. The procedure for analysis of topological characteristics of the networks was performed with the following steps.
The degree for counting the number of interactions of a given node was measured as described (23). Stress centrality (24) for the number of nodes in the shortest path between two other nodes was calculated as follows: (11) Closeness centrality (24) for measuring the average length of the shortest paths to access all other proteins in the network was calculated with the function of (12) Betweenness centrality (24) for determining how the neighbors of a node are inter-connected was calculated with the function of (13) Where σst is the total number of shortest paths from node s to node t and σst (v) is the number of those paths that pass through v. dG (s, t) represents the length of the shortest path between two nodes s and t in graph G, which is the sum of the weights of all edges on this shortest path.
Statistical analysis. All data were expressed as mean ± standard deviation. The Student's t-test was used for comparisons between groups unless otherwise noted. P<0.05 was considered to indicate a statistically significant difference.

DEG data analysis.
In the sevoflurane group, 269 DEGs were identified with logFC≥2 and P≤0.01, and they were mapped to 189 pathways. In the propofol group, 129 DEGs were identified, and they were overlapped to 140 pathways.
Dysregulated pathway analysis. Pathway statistics of anesthetic-treated samples and normal samples were obtained by using the method of iPAS. The pathway statistics of the pathways in which most DEGs were participating were displayed in Figs. 1 and 2 to illustrate the difference between the two groups. Fig. 1 presents pathway statistics for each of the samples and controls for the 'chemokine receptors bind chemokines' pathway, as this was the pathway that most DEGs were associated with in the sevoflurane group, the P-value for this pathway was P=0.002838. For the propofol group, the pathway that the most DEGs were associated with was the 'signaling by TGF-β receptor complex', and the P-value for this pathway was P=0.01213. However, although these two pathways had the highest number of DEGs that were associated with them, they did not exhibit the lowest P-values in their respective treatment groups. In total, 157 dysregulated pathways with P<0.01 were identified in the sevoflurane group. Following ranking of the pathways by P-value, the top 44 pathways with the lowest P-value (all P=0.000247) were listed in Table I. In the propofol group, 82 dysregulated pathways were obtained. Following ranking of the pathways by P-value, the top 29 pathways with the lowest P-value (all P=0.000375) were listed in Table II. Among these dysregulated pathways, 72 dysregulated pathways were common between the two groups.
The dysregulated pathways were converted into a heatmap to help understand pathway aberrance in different samples. Heatmaps are commonly used in pathway analysis and are constructed as a visual aid to analyze dysregulated pathways. The heatmap of the dysregulated pathways in the sevoflurane group is illustrated in Fig. 3 and the heatmap of the dysregulated pathways in the propofol group is illustrated in Fig. 4, and two

Analysis of DEGs in the dysregulated pathways.
To analyze the DEGs involved in the dysregulated pathways, the genes were mapped into the corresponding co-expression network, and then the interactive relationships and the topological characteristics of the DEGs were obtained. The interactive relationship networks are illustrated in Figs. 5 and 6 for the sevoflurane and the propofol groups, respectively.
For the sevoflurane group, 49 DEGs were constructed into a network (Fig. 5). In the network, only two genes were downregulated, Fanconi Anemia complementation group F and cingulin, but the two genes didn't relate with other DEGs (Fig. 5). All the other genes were upregulated at different extents. Amongst these, six genes, nuclear receptor subfamily 4 group A member 3 (NR4A3), MYC proto-oncogene (MYC), tachykinin precursor 1 (TAC1), JUNB proto-oncogene (JUNB), nicotinamide phosphoribosyltransferase (NAMPT) and interleukin 1 receptor antagonist (IL1RN), were central nodes of interactions, displaying the most complex relationships with the other DEGs.    The topology properties of all the DEGs in the network were analyzed, including betweenness centrality, closeness centrality, degree and stress. Following ranking DEGs by their topology characteristics, the top five ranked genes were NR4A3, JUNB, MYC, TAC1 and NAMPT (Table III).
For the propofol group, 28 DEGs were constructed into the network, all of which were upregulated (Fig. 6). The genes encoding interleukin (IL) 6 and IL8 were the most highly upregulated compared with the other DEGs. The top five ranked genes by topology characteristics were NR4A3, suppressor of cytokine signaling 3 (SOCS3), cyclin dependent kinase inhibitor 1A (CDKN1A), C-C motif chemokine ligand 2 (CCL2) and C-X-C motif chemokine ligand 1 (CXCL1) ( Table IV).

Discussion
In the present study, pathway aberrance induced by anesthetic treatment in patients who underwent cardiac surgery was analyzed. The microarray data used in the present study were based on a previously published study by Lucchinetti et al (4), and the aim was to identify altered pathways in individuals.
Sevoflurane and propofol were the experimental anesthetics involved in the data, as these are the most commonly used anesthetic agents in everyday clinical practice in several countries. Sevoflurane is a volatile anesthetic agent that is widely used by mask induction (25). Propofol is the intravenous anesthetic agent of choice in surgery, because of its favorable operating conditions and association with rapid recovery (26).
Pathway regulation patterns of samples treated with general anesthetic were distinct from those in the normal groups. With the iPAS method, anesthetic-treated samples were analyzed for pathway aberrance. A total of 157 dysregulated pathways were obtained from the sevoflurane samples. There were 44 pathways significantly influenced (all P= 0.000247) by sevoflurane treatment, such as the 'activation of C3 and C5' pathway, which participates in regulation of the immune system. Three pathways involved in vitamin metabolism were also significantly altered, which was consistent with the fact that vitamin C supplementation offers significant protection against anesthetic-induced neurotoxicity and behavioral alterations, as proposed by Xu et al (27). Seven pathways involving the Toll like receptor (TLR) were significantly altered, which is consistent with the report by Sun et al which demonstrated that sevoflurane exerts direct relaxant and anti-inflammatory effects by inhibiting the TLR4/ nuclear factor-κB pathway (28). The present results indicate that these identified pathways were closely related to sevoflurane treatment, and suggest that special attention should be given to patients with correlated diseases when administering sevoflurane anesthesia. In the propofol samples, there were 82 dysregulated pathways, of which 29 pathways were significantly influenced (all P=0.000375), such as the 'nicotinamide salvaging' pathway, which is involved in nicotinamide utilization. Propofol administration reduces the heart rate and the baseline cerebral blood flow. These 29 identified pathways in the present study were demonstrated to be closely related to propofol treatment. In the original study by Lucchinetti et al (4), the peroxisome proliferator-activated receptor γ coactivator 1-α pathway and the granulocyte-colony stimulator factor survival pathway, integral components of the protective program in the heart, were proposed as potential therapeutic targets in perioperative cardioprotection. A total of 72 pathways were common in both groups, indicating that, in some aspects, sevoflurane and propofol treatments caused similar effects. Previous studies have provided evidence that both sevoflurane and propofol enhance the function of the γ-aminobutyric acid type A receptor (GABA A R) in neurons and in recombinant α 1 β 2 γ 2 GABA A R systems (29,30). The difference between sevoflurane and propofol is that they bind to separate sites and modulate receptor function via distinctly different mechanisms (31).
To visualize dysregulated pathways and provide a sensitive measurement of a patient's clinical features, heatmaps of the two anesthetic groups were constructed. In the heatmap, intuitive information of the comparison of pathways between anesthetic-treated samples and normal samples was  obtained. The differences between the normal samples and anesthetic-treated samples are evident, and dysregulated pathways varied among the different anesthetic agents.
Networks were constructed for the DEGs involved in the dysregulated pathways. In the sevoflurane group, two genes were downregulated and 47 genes were upregulated. Six DEGs in the center of the interaction network were closely linked with other genes, so the corresponding pathways, nuclear receptor transcription and circadian clock pathways, may have been influenced by sevoflurane. In the propofol group, all the 28 DEGs were upregulated, and several genes in the center of the interaction network were significantly altered, including IL6 and IL8, which were also identified by Lucchinetti et al (4). As IL6 and IL8 participate in immune response, it is probable that administration of the anesthetic propofol may influence the immune system.
Following ranking genes by their topological characteristics, the top ranked gene in the sevoflurane group was NR4A3. In the interaction network, NR4A3 was a central node with a complex relationship with the other genes. NR4A3 is important in the regulation of cell proliferation, differentiation, metabolism (32) and apoptosis (33). It is involved in the 'nuclear receptor transcription' pathway. These genes may potentially be targets for therapies to prevent anesthetic-associated side effects in patients undergoing CABG surgery, and may contribute to the clinical action of sevoflurane and propofol. In the propofol network, the top ranked gene was SOCS3, which functions in the negative regulation of fetal liver hematopoiesis and placental development. SOCS3was involved in the pathway of 'growth hormone receptor signaling', 'interleukin-6 signaling', and 'regulation of interferon γ signaling'.
In summary, the present study identified 157 dysregulated pathways in the sevoflurane group and 87 dysregulated pathways in the propofol group. Of these, 72 dysregulated pathways were common in both groups. Following ranking DEGs by topology analysis, the top 5 ranked DEGs in the sevoflurane group were NR4A3, JUNB, MYC, TAC1 and NAMPT, and in the propofol group, NR4A3, SOCS3, CDKN1A, CCL2 and CXCL1. These genes and pathways may be important in regulating the effects of sevoflurane and propofol administration. Among them, several have been reported previously, while others were reported in the present study for the first time. In conclusion, both types of anesthetic caused gene and pathway aberrance, and the present results may provide indicators of physical condition in patients that may aid in preventing complications from anesthetics. Further prospective studies will be required to validate the present results.


extfieldsofstudy: Chemistry Medicine

text:
The nuclear factor-erythroid 2-related factor/heme oxygenase-1 axis is critical for the inflammatory features of type 2 diabetes–associated osteoarthritis

Epidemiological findings support the hypothesis that type 2 diabetes mellitus (T2DM) is a risk factor for osteoarthritis (OA). Moreover, OA cartilage from patients with T2DM exhibits a greater response to inflammatory stress, but the molecular mechanism is unclear. To investigate whether the antioxidant defense system participates in this response, we examined here the expression of nuclear factor-erythroid 2-related factor (Nrf-2), a master antioxidant transcription factor, and of heme oxygenase-1 (HO-1), one of its main target genes, in OA cartilage from T2DM and non-T2DM patients as well as in murine chondrocytes exposed to high glucose (HG). Ex vivo experiments indicated that Nrf-2 and HO-1 expression is reduced in T2DM versus non-T2DM OA cartilage (0.57-fold Nrf-2 and 0.34-fold HO-1), and prostaglandin E2 (PGE2) release was increased in samples with low HO-1 expression. HG-exposed, IL-1β-stimulated chondrocytes had lower Nrf-2 levels in vitro, particularly in the nuclear fraction, than chondrocytes exposed to normal glucose (NG). Accordingly, HO-1 levels were also decreased (0.49-fold) in these cells. The HO-1 inducer cobalt protoporphyrin IX more efficiently attenuated PGE2 and IL-6 release in HG+IL-1β-treated cells than in NG+IL-1β-treated cells. Greater reductions in HO-1 expression and increase in PGE2/IL-6 production were observed in HG+IL-1β-stimulated chondrocytes from Nrf-2−/− mice than in chondrocytes from wild-type mice. We conclude that the Nrf-2/HO-1 axis is a critical pathway in the hyperglucidic-mediated dysregulation of chondrocytes. Impairments in this antioxidant system may explain the greater inflammatory responsiveness of OA cartilage from T2DM patients and may inform treatments of such patients.

explain the greater inflammatory responsiveness of OA cartilage from T2DM patients and may inform treatments of such patients.
Type 2 diabetes mellitus (T2DM) 5 is a chronic metabolic disorder associated with many adverse complications. Accumulating epidemiological and experimental findings support the hypothesis that T2DM is an independent risk factor for osteoarthritis (OA), the most frequent joint disease (1), or for its severity (2)(3)(4). However, the mechanisms underlying the connection between both diseases remain unclear.
Together, oxidative stress and pro-inflammatory mediators, particularly interleukin 1␤ (IL-1␤), actively induce the changes in articular cartilage that predispose this tissue to the development of OA (5,6) as well as to T2DM and T2DM complications (7)(8)(9)(10). Elevated intracellular glucose levels primarily generate oxidative damage as a consequence of glycolytic pathway saturation in the cell and the subsequent production of advanced glycation end products (11)(12)(13). This oxidative stress mediated by excess glucose-induced impairments in the antioxidant defense system may also contribute to diabetic complications. Moreover, similar oxidative stress disturbances could occur in OA (14,15).
cro ARTICLE As shown in recent studies, Nrf-2 is a pivotal target for the prevention and attenuation of diabetes mellitus (17,21) and for controlling bone and cartilage destruction induced by oxidative stress (22,23). Under physiological conditions, Nrf-2 is generally located in the cytoplasm and binds to its inhibitor, Kelchlike ECH-associated protein 1 (Keap1), leading to its degradation. However, in response to oxidative or electrophilic stress, Nrf-2 dissociates from Keap1 and translocates to the nucleus to bind antioxidant-responsive elements in the promoter regions of its downstream antioxidant genes, including heme oxygenase-1 (HO-1) (24).
HO-1 is a crucial antioxidant enzyme that catalyzes the degradation of heme into iron, carbon monoxide, and biliverdin (25). This enzyme regulates catabolic and anabolic processes in OA chondrocytes (26). HO-1 overexpression in cartilage prevents the pro-inflammatory mediator-induced activation of catabolic, apoptotic, or senescence pathways (27)(28)(29). Thus, HO-1 represents an important part of the cellular response to inflammatory and oxidative stress in joints (30). Interestingly, the expression and activity of this enzyme are down-regulated in the vasculature and retinal tissue from patients with diabetes (31)(32)(33) as well as in in vitro and in vivo cellular models of high glucose exposure (34,35). However, the expression and involvement of the Nrf-2/HO-1 antioxidant axis in type 2 diabetes-related OA has not yet been studied.
For a better understanding of the link between OA and T2DM, we recently reported the higher responsiveness of OA cartilage from patients with T2DM to IL-1␤-induced inflammatory stress (36). Similarly, the exposure of IL-1␤-stimulated chondrocytes to high glucose exacerbates the activation of pathological pathways, which is blocked by ROS scavengers (36). These findings further support the critical role of oxidative stress generated under hyperglucidic conditions in the activation of catabolic responses in cartilage. However, the pathological mechanisms triggered by a high glucose environment that ultimately establishes the redox imbalance in cartilage remain evasive. We studied the Nrf-2/HO-1 axis in human OA cartilage from patients with or without diabetes and in chondrocytes in a high-glucose environment to determine whether the antioxidant defense system is impaired in this excess glucose-and low-grade inflammation-induced phenomenon in cartilage.

Nrf-2/HO-1 expression was decreased in OA cartilage from patients with diabetes and was inversely correlated with the production of inflammatory mediators
We recently observed a more pronounced inflammatory phenotype of OA cartilage from patients with diabetes, based on IL-6 and prostaglandin E 2 (PGE 2 ) release (36). We evaluated the expression of the Nrf-2 and HO-1 proteins in 8 cartilage explants from patients with T2DM and 8 explants from nondiabetic patients matched for age and body mass index (BMI) to elucidate whether impairments to the antioxidant system underlie this altered phenotype (age: 64.8 Ϯ 11.1 versus 68.0 Ϯ 8.7; BMI: 30.1 Ϯ 4.9 versus 29.9 Ϯ 4.7; gender: 6 versus 3 females for T2DM and non-T2DM patients, respectively). Additionally, the other clinical characteristics of all patients were similar.
The Nrf-2 and HO-1 expression levels were reduced in OA cartilage from patients with T2DM compared with the expression levels in non-diabetic patients (0.57-fold and 0.34-fold for Nrf-2 and HO-1, respectively, p Ͻ 0.05) (Fig. 1, A and B). Likewise, the HO-1 and Nrf-2 expression levels varied similarly; Nrf-2 expression was higher in patients with HO-1 expression levels above the median (Fig. 1C). Conversely, PGE 2 release and HO-1 expression varied in opposite ways; PGE 2 release was higher in patients with HO-1 expression levels below the median (Fig. 1D).

A high-glucose environment reduced Nrf-2/HO-1 expression in IL-1␤-stimulated murine chondrocytes
Glucose has been shown to down-regulate HO-1 promoter activity and HO-1 levels (37). Consequently, a high glucose environment in T2DM may be responsible for the reduced Nrf-2/HO-1 expression, as observed in OA cartilage from patients with T2DM. Murine chondrocytes were stimulated with or without IL-1␤ (5 ng/ml) for the indicated times in the presence of normal glucose (5.5 mM) or high glucose (25 mM) to explore this possibility. As shown in Fig. 2 and Fig. 3, reduced HO-1 and Nrf-2 mRNA expression levels were observed at 72 h of incubation with high glucose compared with cells treated with normal glucose; however, these differences in expression were not significant at the protein level. The IL-1␤ treatment further enhanced the impact of high glucose on the Nrf-2 and HO-1 expression levels at both the mRNA ( Fig. 2A and Fig. 3A) and protein levels ( Fig. 2D and Fig. 3B). This effect was significant at 48 h and particularly at 72 h (0.81-fold Nrf-2 and 0.48-fold HO-1 reductions at the protein level, p Ͻ 0.05). Additionally, we did not observe a simultaneous regulation of Kelch-like ECH-associated protein 1 (Keap-1) expression (Nrf-2 cytoplasmic inhibitor) that counteracted the variations in the Nrf-2 levels (Fig. 2B). Subsequently, the Nrf-2/Keap-1 ratio confirmed the modulation of Nrf-2 expression by high glucose (Fig. 2C).
We assayed the variations in the nuclear levels of this transcription factor 30 min after treatment to determine whether Nrf-2 nuclear translocation was impaired (Fig. 2E). Nuclear levels of Nrf-2 were reduced in chondrocytes grown under highglucose conditions in both the presence and absence of IL-1␤ compared with that of cells grown under normal glucose conditions (0.73-fold and 0.61-fold, respectively). Consistent with these findings and similar to our observations with human OA cartilage, we observed a significant dependence between the Nrf-2 and HO-1 expression levels in cultured murine chondrocytes, as the levels of these proteins varied in the same way (Fig. 3C).

Reduced HO-1 expression favored the increased responsiveness of chondrocytes to IL-1␤ in a high-glucose environment
Consistent with our observations in OA cartilage, HO-1 expression was inversely correlated with PGE 2 production in IL-1␤-stimulated murine chondrocytes (Fig. 3D). Based on these findings, HO-1 expression may regulate chondrocyte activation by pro-inflammatory cytokines. Chondrocytes were co-incubated with cobalt protoporphyrin IX (CoPP), an inducer of HO-1 activity, for 72 h, and IL-6 and PGE 2 release Nrf-2/HO-1 axis in type 2 diabetes-associated osteoarthritis was measured to further confirm this hypothesis (Fig. 4, A and B). CoPP attenuated both IL-1␤-induced IL-6 and PGE 2 production. Interestingly, this reduction was significantly stronger in cells grown in high glucose, with a 49% decrease in IL-6 release and a 98% decrease in PGE 2 release compared with a 25 and 91% release, respectively, from cells grown in normal glucose. Moreover, because HO-1 is an antioxidant enzyme, we also determined whether CoPP reduced the previously observed high glucose-stimulated increase in IL-1␤-induced ROS production (36). As expected, CoPP, through HO-1, strongly reduced the ROS levels, returning them to the control values (Fig. 4C).

Nrf-2 knock-out chondrocytes showed reduced HO-1 expression and higher responsiveness to IL-1␤ in a high glucose environment
Nrf-2 is recognized as the main inducer of HO-1 gene expression (38). Here, Nrf-2 and HO-1 expression were shown to vary in the same way in OA cartilage and in murine chondrocytes exposed to a high glucose environment. Additionally, sulforaphane, a known natural inducer of Nrf-2, increased HO-1 expression, reduced ROS accumulation, and attenuated the production of IL-6 in chondrocytes exposed to IL-1␤ (Fig. 5), suggesting a role for Nrf-2 in controlling the catabolic response. Therefore, we conducted studies in chondrocytes from Nrf-2 knock-out (Nrf-2 Ϫ/Ϫ ) mice using the same experimental approach to further address the dependence of IL-1␤-induced HO-1 expression in response to high glucose on Nrf-2 ( Fig. 6). As expected, HO-1 expression was drastically reduced in Nrf-2 Ϫ/Ϫ chondrocytes at 72 h compared with that in the wild-type cells (Fig. 6A). IL-6 and PGE 2 production were measured to determine whether the Nrf-2 knock-out also had an impact on the pro-inflammatory phenotype of chondrocytes. We observed a strengthening of PGE 2 production induced by IL-1␤ in chondrocytes from Nrf-2 Ϫ/Ϫ mice compared with wildtype mice, achieving significant differences in those cells incubated in high glucose (Fig. 6B); moreover, similar results were obtained for IL-6 levels.

Discussion
T2DM is currently considered an additional risk factor for OA occurrence and OA severity, delineating the T2DM-associated OA phenotype (2,3,39). In addition to insulin resistance, chronic hyperglycemia is one of the main biological features involved in diabetic complications (10,13). Moreover, we recently reported a more pronounced inflammatory phenotype in OA cartilage from patients with diabetes and enhanced IL-1␤-induced inflammation in cultured chondrocytes exposed to excess glucose (36). To further elucidate these findings, we show in this study the expression of Nrf-2, the master transcrip-

Nrf-2/HO-1 axis in type 2 diabetes-associated osteoarthritis
tional regulator of antioxidant responses, and one of its main target genes, HO-1, is reduced in OA cartilage from patients with T2DM. These observations are mimicked in vitro by exposing IL-1␤-stimulated chondrocytes to high glucose. Linking our findings, impaired Nrf-2/HO-1 signaling is responsible for the increased responsiveness to IL-1␤, as shown by the increased IL-6 and PGE 2 release and ROS production by chondrocytes.
The Nrf-2/HO-1 axis is a crucial cell survival mechanism that counteracts oxidative stress and inflammation (24,40). Deficiencies in this axis have been identified in some systemic diabetic complications, such as retinopathy or cardiopathy (33)(34)(35)41). In the present study we investigated whether the Nrf-2/HO-1 pathway was also impaired in OA joints from patients with T2DM and whether it participates in the pathological mechanisms predisposing cartilage to diabetes-associated OA. For this purpose, we selected osteoarthritic cartilage explants from patients with and without diabetes and matched them based on age, BMI, gender, and other metabolic co-morbidities, similar to our previous report (36). Both Nrf-2 and HO-1 expression were decreased in diabetic OA cartilage. The HO-1 promoter (HMOX1) contains binding sites for several transcription factors, notably the antioxidant-responsive element site for Nrf-2; however, activator protein-1 (AP-1), cAMP

Nrf-2/HO-1 axis in type 2 diabetes-associated osteoarthritis
response element-binding protein (CREB), and nuclear factor-B (NF-B) can also activate its expression (38). In our study HO-1 expression in OA cartilage at least partially depended on the Nrf-2 levels. Nrf-2/HO-1 signaling protects the joint against the activation of pathological pathways (42)(43)(44). Here, we revealed a negative correlation between PGE 2 production and the HO-1 levels, suggesting a deleterious effect of a reduction in Nrf-2/HO-1 expression on the inflammatory profile of cartilage.
A decrease in Nrf-2/HO-1 signaling and a subsequent increase in ROS release are associated with exposure to a high glucose environment in different cell types. Retinal endothelial cells incubated with high glucose exhibit reduced Nrf-2 transcriptional activity (45). Similar results were observed in human microvessel endothelial cells (46). Interestingly, decreased HO-1 levels were also detected in these in vitro models (46) as well as in animal models of diabetes mellitus (31,35). Therefore, we used an in vitro approach to evaluate whether a diabetesrelated high glucose environment may participate in reducing Nrf-2/HO-1 signaling in T2DM OA cartilage. Murine chon-drocytes were stimulated with IL-1␤, a cytokine known to be involved in the pathophysiology of OA and T2DM, and were exposed to high glucose. The hyperglucidic environment reduced the early nuclear translocation of Nrf-2 as well as its total protein level. As expected, HO-1 expression was also reduced and was positively correlated with the Nrf-2 levels. This correlation between the expression levels of both proteins was further confirmed in chondrocytes from Nrf-2 Ϫ/Ϫ mice. The genetic invalidation of Nrf-2 expression drastically reduced HO-1 expression.
The transcriptional activity of Nrf-2 is mainly regulated by its cytoplasmic repressor, Keap-1, although other inhibitors also block its activity in the nucleus (24). We failed to detect any modulation of Keap-1 expression under high glucose conditions and/or IL-1␤ stimulation, suggesting that repressors other than Keap-1 participate in the alterations in the Nrf-2 pathway observed in our model. For instance, the glycogen synthase kinase 3␤ (GSK3␤)/Fyn pathway and BTB domain and CNC homolog 1 (bach1) (47) impair Nrf-2 signaling in diabetic complications and have been involved in pathological processes . HO-1 expression was reduced in murine chondrocytes incubated with high glucose and is associated with Nrf-2 levels and negatively associated with the production of inflammatory mediators. Chondrocytes were incubated with NG or HG in the presence or absence of IL-1␤ for the indicated times. Expression of the HO-1 gene (A) or protein (B) was evaluated by qRT-PCR or Western blotting, respectively. The values were normalized to the HPRT levels for gene expression or to the ␤-actin levels for protein quantification (n ϭ 6 per condition). Each symbol represents an experiment from one litter of mice in normal glucose (E) and high glucose (•) conditions. The bars represent the means Ϯ S.D. for each condition. *, p Յ 0.05. Dependence analysis between Nrf-2 expression (C) or IL-1␤-induced PGE 2 production (D) and HO-1 expression was assayed by dichotomizing Nrf-2/PGE 2 values as a function of the median HO-1 levels. Each symbol (‚) represents the Nrf-2/PGE 2 value obtained from chondrocytes incubated in both normal and high glucose and whose HO-1 expression is lower or higher than the median level (n ϭ 72 for the Nrf-2/HO-1 analysis; n ϭ 10 for the PGE2/HO-1 analysis). The bars represent the means Ϯ S.D. for each condition. *, p Յ 0.05; **, p Յ 0.01.

Nrf-2/HO-1 axis in type 2 diabetes-associated osteoarthritis
in joints (48 -52). Based on these findings, hyperglycemia/diabetes modulates the Nrf-2/HO-1 axis through different repressors; however, additional studies are required to elucidate the specific pathways activated in chondrocytes.
The beneficial effect of activation of the Nrf-2/HO-1 axis on diabetic conditions is widely accepted (17,35). Moreover, accumulating evidence has revealed a protective role for the Nrf-2/ HO-1 axis in joint diseases (22,23,26,43,53). As shown in the present study, the reduction of the Nrf-2/HO-1 levels contributes to a pro-inflammatory imbalance in IL-1␤-stimulated chondrocytes exposed to a high glucose environment. HO-1 expression was negatively correlated with the production of inflammatory mediators. Likewise, treatment with sulforaphane, a known natural inductor of Nrf-2, and subsequent HO-1 up-regulation protected chondrocytes against ROS accumulation and inflammatory production. Accordingly, the Release of the IL-6 (A) and PGE 2 (B) was assayed. C, ROS production was evaluated using a fluorometric assay with DCFDA. Represented data are -fold induction compared with the control condition without IL-1␤ and normalized to intracellular protein quantity. Each symbol represents an experiment from one litter of mice in normal glucose (E) and high glucose (•) conditions. The bars represent the means Ϯ S.D. for each condition (n ϭ 6). *, p Յ 0.05.

Nrf-2/HO-1 axis in type 2 diabetes-associated osteoarthritis
chemical induction of HO-1 activity reduced IL-6 and PGE 2 release in a more significant manner in those chondrocytes exposed to high glucose. Finally, HO-1 expression was reduced in chondrocytes from Nrf-2 knock-out mice incubated with high glucose and the inflammatory response to IL-1␤ was further exacerbated. Our results are corroborated by Cai et al. (43), who observed that an Nrf-2 deletion results in increased disease severity in different animal models of OA. Additionally, the recovery of Nrf-2 activity inducing HO-1 expression decreased OA pathogenesis (43). Accordingly, deficiency of bach1 favoring Nrf-2 transcriptional activity protects against development of two different types of OA: aging-and post-traumatic-associated OA pathogenesis (51). Thereby, these findings suggest anti-catabolic roles of Nrf-2 in different OA subsets. Here, we provide for the first time strong evidence that Nrf2 is also pivotal to counteract the pathological pathways activated by highglucose stress in a diabetes-related OA context. However, HO-1 activation independent of Nrf-2 may also protect against the development of this disease (51,54). In this sense, Nrf-2 knock-out mice failed to completely block the effect of high glucose on HO-1 expression, suggesting that pathways other than Nrf-2 signaling may also be involved in regulating HO-1 expression.
In conclusion, the Nrf-2/HO-1 axis is dysfunctional in diabetic osteoarthritic cartilage and represents a critical pathway involved in the hyperglucidic-mediated dysregulation of articular chondrocytes (Fig. 7). The impairment of this antioxidant system may explain the greater inflammatory responsiveness of cartilage from patients with T2DM and may provide new targeted therapeutic avenues for treating patients with the diabetes-related OA phenotype.

Collection of OA human cartilage
Human knee explants were obtained from patients with OA who were undergoing total joint replacement at Saint-Antoine Hospital (Paris, France). The diagnosis of OA of the knee was based on criteria from the American College of Rheumatology (55). Patients were screened for diabetes using their medical files, drug prescriptions, and patient interviews. For each patient with diabetes who was included in the study, we matched a non-diabetic patient undergoing total knee joint replacement due to OA by age and BMI to avoid confounding factors. The explants from each patient were manually dissected from all remaining cartilage zones (i.e. tibial plateaus and femoral condyles), mixed to obtain homogeneous isolated cartilage samples and managed using previously described methods (36). Briefly, the cartilage explants were cut into small pieces (ϳ5 mm 3 ), washed several times with PBS, and incubated in DMEM (25 mM) supplemented with 100 units/ml pen- Figure 6. Nrf-2 knock-out (Nrf-2 ؊/؊ ) chondrocytes exposed to high glucose exhibited lower HO-1 expression and increased responsiveness to IL-1␤. A, the expression of the HO-1 protein was evaluated in chondrocytes from wild-type (WT) or Nrf-2 Ϫ/Ϫ mice that were stimulated as described above for 72 h. The values were normalized to ␤-actin expression. Represented data are -fold induction compared with the control condition without IL-1␤. B, PGE 2 production was also measured. Each symbol represents an experiment from one litter of wild-type (E) and Nrf-2 Ϫ/Ϫ (•) mice. The bars represent the means Ϯ S.D. for each condition. *, p Յ 0.05.(n ϭ 4).

Figure 7.
Hypothetical mechanism by which impairments in the Nrf-2/ HO-1 axis favors catabolic responsiveness to IL-1␤ in chondrocytes exposed to a high glucose environment. The Nrf-2/HO-1 axis is one of the most important anti-inflammatory and antioxidant protective systems. Under stress conditions Nrf-2 is translocated to the nucleus to activate antioxidant gene expression, including one of its main target HO-1. Next, HO-1 counteracts the catabolic pathways induced by inflammatory stimuli such as IL-1␤. However, Nrf-2 signaling is impaired in a high glucose environment, which is manifested as dysfunction in Nrf-2 synthesis and translocation and a subsequent reduction in the HO-1 levels. Thus, chondrocytes lost the capacity to control ROS production and inflammation (i.e. PGE 2 and IL-6 release). This event further favors the activation of pathological pathways, leading to an increase in the susceptibility to OA and disease severity. SFN, a Nrf-2 activator, and CoPP, an inducer of HO-1, attenuate this phenomenon.

Nrf-2/HO-1 axis in type 2 diabetes-associated osteoarthritis
icillin, 100 g/ml streptomycin, and 4 mM glutamine for 24 h at 37°C. After incubation, the explants were frozen, and protein extracts were obtained using previously established methods (56). Briefly, the tissues were ground in liquid nitrogen using a mortar and pestle, after which the proteins were extracted with lysis buffer and used for the Western blotting experiments. In parallel, conditioned media (CM) were also collected, centrifuged (1600 ϫ g for 6 min), and stored at Ϫ20°C. Each volume of medium was normalized to the wet weight of the explants (6 ml/g tissue) (36). Informed consent was obtained from each patient for the use of their tissues and clinical data. All experiments using human samples were approved by a French Institutional Review Board (Comité de Protection des Personnes, Paris Ile de France 5, April 2012).

Primary culture and treatment of murine articular chondrocytes
Mouse primary chondrocytes were isolated from the articular cartilage of 5-6-day-old newborn C57BL/6 mice from Janvier (St. Berthevin, France) and were seeded at a density of 8 ϫ 10 3 cells per cm as previously described (57). Articular chondrocytes obtained from newborn mice using this protocol were validated as cells presenting characteristics similar to fully mature murine chondrocytes (58). Chondrocytes were cultured in complete DMEM for a week and were then incubated in serum-free DMEM containing 0.1% bovine serum albumin (BSA) for 24 h before treatment (basal medium). Subsequently, the murine chondrocytes were incubated with normal glucose (5.5 mM) or high glucose (25 mM) in the presence or absence of IL-1␤ (5 ng/ml) (PeproTech, Rocky Hill, NJ) for the indicated times. Cell lysates were collected for mRNA or protein extraction, and supernatants were collected for the assays.
For mechanistic studies, chondrocytes that had been cultured with normal or high glucose in the presence or absence of IL-1␤ (5 ng/ml) for 72 h were co-treated with CoPP, an inducer of HO-1 activity (10 M) (Enzo Life Sciences, Villeurbanne, France) whose non-toxic effect at this concentration had been checked using an LDH assay and its specificity described in previous publications (29,59,60). In additional experiments, cells were pretreated for 30 min with the Nrf2 activator compound sulforaphane (SFN; 5 M) (Sigma). Moreover, experiments were also performed using Nrf-2 knock-out (Nrf-2 Ϫ/Ϫ ) (61) mice generated from inbred Nrf-2 heterozygous mice on a C57BL/6J background, as described by El Ali et al. (62). The mice were housed in a pathogen-free facility and were handled in accordance with the principles and procedures outlined in Council Directive 86/609/EEC.
The efficiency of the IL-1␤ dose employed was assayed previously, and the cytotoxic effects of the treatments and an osmotic effect of high glucose were excluded (36). The CoPP dose was chosen based on dose-effect experiments (data not shown) and litera-ture data. All experiments with murine articular chondrocytes were performed using protocols approved by the French and European ethics committees (Comité Régional d'Ethique en Expérimentation Animale N°3 de la région Ile de France).

RNA extraction and quantitative RT-PCR (qRT-PCR)
Total RNA was extracted from the chondrocytes using the ReliaPrep RNA Cell Miniprep System kit (Promega, Madison, WI), and the concentrations were determined by spectrophotometry (Eppendorf, Le Pecq, France). Reverse transcription utilized 500 ng of total RNA and the Omniscript RT kit (Qiagen). The levels of the IL-6, HO-1, Nrf-2, and Keap-1 mRNAs were quantified using a Light Cycler LC480 (Roche Diagnostics). The PCR amplification conditions were: initial denaturation for 5 min at 95°C followed by 40 cycles consisting of 10 s at 95°C, 15 s at 60°C, and 10 s at 72°C. Product formation was detected at 72°C in the fluorescein isothiocyanate channel. The relative mRNA expression levels were calculated and normalized to the levels of the murine hypoxanthine guanine phosphoribosyltransferase (HPRT) mRNA using the 2 ⌬⌬CT method (specific mouse primer sequences are shown in Table 1). All measurements were performed in duplicate.

IL-6 and PGE 2 assessments
The IL-6 concentrations in the murine cell supernatants were measured using the Quantikine enzyme-linked immunosorbent assay (ELISA) kit (R&D Systems, Lille, France). The concentrations of human IL-6 were measured in human CM using the Pelikine compact kit (Sanquin, Amsterdam, The Netherlands). The concentrations of the murine and human pro-inflammatory bioactive lipid PGE 2 in the cell supernatants and CM were measured using the enzymatic immunoassay (EIA) kit (Cayman Chemical, Ann Arbor, MI). The limits of detection were 7.8, 0.6, and 7.8 pg/ml for the murine/human IL-6 and PGE 2 assessments, respectively. Duplicate measurements were performed.

Protein extraction and Western blotting
Protein extracts from human cartilage explants were prepared using previously described methods (56). Murine chondrocytes were cultured in 12-well plates (4 ϫ 10 4 cells/well) in duplicate and were treated as indicated. Subsequently, the total intracellular proteins were obtained as previously described (63). In some experiments, cytosolic and nuclear fractions were obtained from the murine chondrocytes using commercially available NE-PER Nuclear and Cytoplasmic Extraction Reagents (Thermo Scientific, Waltham, MA) according to the manufacturer's recommendations. Protein extracts from human cartilage (30 g) and total proteins (30 g) or nuclear fractions (20 g) from murine chondrocytes were resolved by SDS-PAGE and immunoblotted with the indicated antibodies: Table 1 Primer sequences used for real-time PCR

Cellular ROS production
Chondrocytes were seeded and cultured in 96-well plates at a density of 1 ϫ 10 4 cells per well, as described above. After 72-h of treatment, ROS production was measured using a fluorometric assay with dichlorodihydrofluorescein diacetate (DCFDA) (Molecular Probes, Life Technologies, Saint Aubin, France). Briefly, chondrocytes were incubated with 17 M DCFDA diluted in the fasting medium for 60 min at 37°C in the dark. Subsequently, the chondrocytes were washed with PBS, and fluorescence was measured using the Fluostar Galaxy reader (BMG Labtech, Ortenberg, Germany) at an excitation wavelength of 485 nm and an emission wavelength of 520 nm and then analyzed using the Biolise system (Labsystems, Helsinki, Finland). The intracellular proteins were collected with NaOH (0.5 M), and the concentrations were measured using a spectrophotometer and a protein assay kit (Bio-Rad) to normalize the results. The ROS production is represented as -fold induction from that of the control and by micrograms of protein. All measurements were performed in triplicate.

Statistical analysis
All data are reported as points representing one single experiment from one litter of mice or one patient with standard deviation to represent error. All tests were analyzed using GraphPad Prism 5 (GraphPad Software, San Diego, CA) with the Wilcoxon test for paired variables and the Mann-Whitney test for unpaired variables. Additionally, analysis of data from Nrf-2 Ϫ/Ϫ experiments was performed by paired t test. p Յ 0.05 was considered statistically significant.
Author contributions-C. V.-G., A. C., M.-C. L., X. H., S. K.-R., R. M., F. B., and J. S. were responsible for the study design, manuscript preparation, and data interpretation. A. S. organized and collected the human tissue samples and participated in designing the experiments with human tissue and in data interpretation. C. V.-G., A. C., and A. P. performed the experiments. S. K.-R. was responsible for generating the Nrf-2 Ϫ/Ϫ mice and was involved in data interpretation. All authors reviewed and approved the final manuscript.


extfieldsofstudy: Biology Medicine

text:
Quantitative magnetic resonance imaging of brain atrophy in a mouse model of Niemann-Pick type C disease

In vivo magnetic resonance imaging (MRI) was used to investigate regional and global brain atrophy in the neurodegenerative Niemann Pick Type C1 (NPC1) disease mouse model. Imaging experiments were conducted with the most commonly studied mouse model of NPC1 disease at early and late disease states. High-resolution in vivo images were acquired at early and late stages of the disease and analyzed with atlas-based registration to obtain measurements of twenty brain region volumes. A two-way ANOVA analysis indicated eighteen of these regions were different due to genotype and thirteen showed a significant interaction with age and genotype. The ability to measure in vivo neurodegeneration evidenced by brain atrophy adds to the ability to monitor disease progression and treatment response in the mouse model.

Introduction
Niemann Pick Type C (NPC) is a rare genetic neurodegenerative disease which currently lacks effective treatments, and is universally fatal with death occurring prior to adulthood in the majority of patients [1] [2]. NPC disease is most commonly diagnosed in early childhood with symptoms including ataxia, dysarthria, dysphagia, vertical supranuclear gaze palsy, and progressive neurological decline. The primary cause of the disease is mutation of the NPC1 gene, resulting in a lack of functional NPC1 protein. The precise functions of the NPC1 protein are a topic of recent studies [3][4] [5], and are known to include cholesterol transport within cells throughout the body. The dysfunction of NPC1 protein in NPC disease causes impaired cholesterol trafficking leading to a buildup of cholesterol and glycolipids in cells. A small percentage of NPC cases are caused by defects in NPC2 protein, which has been found to work with a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 NPC1 to transport cholesterol [6] [7]. The current study focuses on NPC disease caused by NPC1 gene mutations (NPC1 disease).
A number of MRI studies of NPC patients have reported abnormalities in white matter tracts throughout the brain, gray matter atrophy, enlargement of ventricles and reduced cerebellar volume [8] [9][10][11][12] [13] [14] [15]. A study of adult patients reported a pattern of cortical frontal lobe atrophy associated with psychiatric or cognitive symptoms, while patients with gait and movement disorders had more pronounced brainstem and cerebellar atrophy, and at late stages of the disease, diffuse atrophy was found throughout the brain [16]. Another study of adult NPC patients with MRI morphometry techniques described widespread alterations in white matter tracts but focal rather than widespread reductions in gray matter volumes [17]. Variability in clinical reports of MRI-visible brain atrophy is likely due to differences in disease severity associated with the over 240 separate mutations of NPC1 which are currently known to cause disease [18]. Due to the low incidence of NPC disease, estimated at approximately 1 in 150,000 live births [19], large clinical treatment studies are prohibitively difficult, making the use of animal models a valuable component of NPC disease research.
A commonly studied mouse model of NPC1 disease, Npc1 -/has a mutation in the Npc1 gene which causes truncation of 11 of 13 transmembrane domains of the Npc1 protein and consequently, a complete lack of functional Npc1 protein [20]. The Npc1 -/mouse model exhibits disease symptoms mimicking a severe infantile form of NPC disease, and the mice live to approximately 10 weeks of age [2] on the BALB/cNctr genetic background. The Npc1 -/model has been used in many studies of NPC disease, including recent studies of promising therapies using miglustat and cyclodextrins [21][22][23]. The phenotype of the Npc1 -/brain has been described with white matter abnormalities including hypomyelination and myelin degeneration [24] and atrophy including a progressive loss of Purkinje cell neurons in the cerebellum [25]. MRI studies of the Npc1 -/mouse model in vivo have shown abnormal myelination and atrophy of brain structures including the cerebellum, but have not examined brain abnormalities across the entire brain [26][27][28][29].
In vivo MRI-based mouse brain atlases and templates have made it possible to determine volumes of multiple brain regions from 3-dimensional (3D) MRI without the need for manual image segmentation. Currently available preclinical MRI systems and 3D imaging techniques have allowed us to use MRI atlas-based volumetry techniques in the preclinical disease model of the Npc1 -/mouse. The current study utilized high-resolution in vivo 3D MRI in combination with atlas-based segmentation to investigate whole brain and region specific brain atrophy in the Npc1 -/mouse model of NPC1 disease.

Materials & methods
All animal procedures were carried out under approval of the University of Arizona Institutional Animal Care and Use Committee (IACUC). Npc1 -/and age matched wild-type control (WT) mice were studied at 3 and 9 weeks of age, corresponding to an early pre-symptomatic age and an end-stage near-death time point, respectively. The numbers of animals in each group are indicated in Table 1. Imaging was carried out on a 7T Bruker Biospec MRI system (Bruker Biospin Corp., Billerica MA), using a 72mm ID birdcage coil for excitation and a four element phased array surface coil for signal reception. Mice were anesthetized with 2% isoflurane gas and positioned with an animal holding system including bite bar and ear bars for head fixation within the imaging coils. Breathing rate was monitored throughout all experiments and body temperature was maintained at 37˚C with a heated circulating water system and monitored with a rectal fiber optic probe. Images were collected with a 3D fast spinecho sequence with the following parameters: TR = 1800 ms, ETL = 8, Echo Spacing = 10 ms, TE eff = 40 ms, FOV = 30 x 17 x 9.6 mm 3 , 100 μm isotropic resolution, and scan time: 60:08 (min:sec). Mice were euthanized at the end of the study via CO 2 exposure.
In the resulting images, brain tissue was semi-automatically segmented from non-brain tissue. An initial brain surface demarcation was made with an intensity based 3D region of interest (ROI) selection tool included in the MRIcron software package (http://people.cas.sc.edu/ rorden/mricron/index.html). The brain edges were manually examined and adjusted in the coronal orientation for each slice of the 3D image datasets, with reference to a mouse brain atlas [30]. The brainstem was manually trimmed at the base of the cerebellum for each dataset. Heterogeneous signal intensity due to the use of a four element phased array surface coil was corrected by the use of the N4ITK bias correction method [31], as implemented in the Advanced Normalization Tools (ANTS) toolbox [32].
Following this, images were registered to an in vivo mouse brain MRI atlas [33], using the Symmetric Normalization (SyN) registration algorithm as implemented in the ANTS toolbox [32] using parameters appropriate for mouse brain registration [34]. Registration with the in vivo MRI brain atlas allowed for quantization of volumes from twenty brain regions to be calculated for each dataset without the need for manual drawings of ROIs.
Two-way ANOVA was performed on each brain region defined by the atlas in order to evaluate changes in volume due to both age and disease status. For all statistical analyses, a Holm-Bonferroni correction was used to correct for multiple comparisons with a significance level of 0.05.

Results
Representative images of late stage (9 weeks) and early presymptomatic (3 weeks) Npc1 -/mice and WT controls are shown in Fig 1. Differences in contrast between gray and white matter are visible in the white matter regions of the corpus callosum, external capsule, fimbria, and internal capsule. The cerebellum of the Npc1 -/mouse appears reduced in size relative to the WT mouse, especially apparent in the sagittal section at 9 weeks of age. The overall size of the brain at 9 weeks of age also appears reduced in the Npc1 -/mouse relative to WT. Fig 2 demonstrates the steps used in the processing of the high resolution T2-weighted datasets for analysis. Orthogonal sections from an individual WT mouse at 9 weeks of age are shown in Fig 2a. Fig 2b displays the result of the semi-automated masking of non-brain tissue from the 3D volume. The estimated bias field from the heterogeneous sensitivity of the 4-channel surface coil, as determined by the N4 technique, is shown in Fig 2c. The high signal intensities in the upper regions of the cortex and cerebellum due to surface coil sensitivity are visible as bright regions in the bias field. The segmented and bias field intensity-corrected images are shown in Fig 2d and are used for volumetric analyses. Example high resolution T2-weighted in vivo images of Npc1 -/-(a-c, g-i) and WT (d-f, j-l) mice at 3 weeks (a-f) and 9 weeks (g-l) of age. At three weeks of age there are only slight differences between in the brains of Npc1 -/and WT mice, but can be seen in the white matter regions of the brain. These differences become more pronounced at 9 weeks of age. The arrow in panel j points to the region of the corpus callosum and external capsule in a WT mouse, which demonstrates a dark band of intensity compared to the surrounding gray matter. This is reversed in the Npc1 -/mouse. The cerebellum is circled in panel k, which is visibly reduced in size in the Npc1 -/mouse at 9 weeks. The arrow in panel l indicates the bright signal of the CSF in the lateral ventricles, which are increased in size in the Npc1 -/mouse at 9 weeks. Volumes of twenty brain regions and whole brain volume generated from atlas-based registration of WT and Npc1 -/mice are listed in Table 1 and brain region volumes are plotted in Fig 3. Sixteen of the twenty brain regions identified showed a significant effect of disease status, i.e. genotype, on brain region volume. Two of the regions showed a significant effect of age on volume and ten of the regions showed a significant interaction between age and genotype. For most brain regions, volumes in Npc1 -/mice at 3 weeks of age were smaller than the regions in WT mice. Notable exceptions to this are the ventricles and corpus callosum and external capsule. At 9 weeks of age, every brain region in Npc1 -/mice was smaller than those in WT mice, except the ventricles, which remain larger in Npc1 -/mice. In order to evaluate the change in brain region volume with age specifically, the percent change in brain region volume from 3 to 9 weeks of age is plotted in Fig 4. In WT mice, every brain region, except for the anterior commissure, increased in size from 3 to 9 weeks of age. The entire brain was increased from 458.3 to 477.2 mm 3 and the cerebellum from 58.4 to 63.1 mm 3 . In Npc1 -/mice, thirteen of the brain regions identified decreased in volume. The entire brain decreased from 424.8 to 400.6 mm 3 and the cerebellum from 54.7 to 48.1 mm 3 . In the brain regions which increased with age in Npc1 -/mice, the increase was always less than that of WT mice for the same region.

Discussion
The high-resolution 3D MRI and semi-automated atlas-based registration carried out in the Npc1 -/mouse model demonstrated differences in the volume of several brain regions in Npc1 -/mice compared to WT mice at an early, pre-symptomatic stage of the disease and more pronounced differences at a late, end-stage of the disease. Significant effects of the disease were seen in several brain regions including the hippocampus, caudate and putamen, thalamus cerebellum, superior colliculi and neocortex. These brain regions have previously been reported to be affected by neurodegeneration in clinical NPC cases as well as in ex vivo studies of the Npc1 -/mouse model and are the regions of highest levels of Npc1 mRNA [35].
Clinical case studies of NPC disease have described widespread but variable amounts of MRI visible brain atrophy associated with neurodegeneration. A study of adult NPC patients quantified brain atrophy with morphometric methods [17], and reported focal areas of grey matter reductions in the areas of the hippocampus, cerebellum, and thalamus. Similar regions were found to be significantly reduced in volume at 9 weeks of age in the current study of the Npc1 -/mouse. Recent studies of NPC patients across a wide spectrum of ages and disease severities were able to correlated corpus callosal and cerebellar volume to disease severity[13] [14]. Alterations in microstructural properties of white matter in these two regions were also measured. Neuroimaging case studies of adult NPC patients have also been carried out with 18 F-flourodeoxyglucose positron emission tomography (FDG-PET), which provides a measure of tissue metabolic activity that can be associated with neurodegeneration. Reported FDG-PET findings include severe hypometabolism in the frontal cortex, thalamus, and cerebellum [36], which would be expected with neurodegenerative atrophy in those regions, similar to the current findings in the Npc1 -/mouse. A second FDG-PET study found hypometabolism in the prefrontal cortex and thalamus, but hypermetabolism in the cerebellum hypothesized to be related to symptomatic dystonia [37].
An ex vivo MRI study reported reduced whole-brain and cerebellar volumes at 6 weeks of age [29]. In vivo MRI studies of the Npc1 -/mouse to date have identified, but not quantified, atrophy in the brain [26][27][28]38]. We found a trend towards decreased cerebellar volume at 3 weeks which was highly significant at 9 weeks. Purkinje cell degeneration is detectable at 4 weeks [39] and is quite advanced by 6 weeks [40], but the volume of the cerebellum is more dependent on the number of granule cells [41] which do not undergo the last round of division [42]. Thus, the marked volume changes we found reflect failure of development of granule cells as well. This defective initial development of the cerebellum lead to delayed motor skill acquisition [43]and eventually results in motor in-coordination and tremors which have been abundantly described [44].
The decreased volume of the thalamus likely reflects the previously described neuronal loss in the thalamus, especially in the ventral posterior lateral and medial nuclei [45] while the relative lack of volumetric change in the brainstem may reflect the gliosis which is abundant there [46]. The neocortical volume loss we found correlates with the findings of a quantitative ex vivo study of regional brain volumes in 11 week old Npc1 -/mice which used a stereological cell counting method and found significantly decreased numbers of neurons in the prefrontal cortex (as well as the thalamus) [47]. Memory loss [48] is a consequence of this neocortical and, also, the hippocampal volume losses. Change in the volume of mouse brain regions with age. Percent change in whole brain and brain region volumes between 3 and 9 months of age for WT (blue) and Npc1 -/-(red) mice. Errors bars indicate the standard deviation of the percent change. Significant effects of genotype (*) and age (*) and significant interaction between age and genotype (*) were determined from a two-way ANOVA and p<0.05 after Holm-Bonferroni correction. https://doi.org/10.1371/journal.pone.0178179.g004 The current work, while able to semi automatically measure in vivo brain volume differences in the Npc1 -/model compared to WT, has limitations. With all atlas-based registration methodology the results will necessarily depend on how well the registration process worked and whether it worked the same on images from both experimental groups. Consistent image registration is assumed for analysis used in this work. While the ANTS program has been shown to be well suited to MRI brain registration tasks [32], and used often in mouse brain MRI evaluation [34][49] [50], registrations of small structures with complex shapes such as white matter tracts and ventricular spaces could be subject to error. A major question is whether the difference in volume reported by the atlas-based registration is truly due to differences in volume, or to differences in the performance of the registration. While visual inspection indicates a qualitatively similar registration this does not guarantee that registration differences do not impact the volumes reported. This uncertainty, however, does not diminish the confidence in the statistical results that show brain regions in Npc1 -/and WT mice are different. Because of the small error in the volumes reported in each group and at each age, we can be confident that differences between the brains are real, but these differences may be from more than a simple volume change, e.g. results could be impacted by changes in tissue signal from brain regions which might affect registration.
To our knowledge, this is the first study to utilize high-resolution volumetric MRI in conjunction with semi-automated atlas-based registration to evaluate changes in the brain of the Npc1 -/mouse model at early and late disease stages of the disease. A significant effect of genotype on volume was observed in almost every brain structure analyzed. These regions correspond to findings reported in ex vivo studies of the Npc1 -/model as well as clinical reports of NPC disease. The ability to measure in vivo neurodegeneration evidenced by brain atrophy provides an additional means to monitor disease status and could prove useful in future studies of treatments in the Npc1 -/mouse model of NPC1 disease.
The Npc1 -/mouse model studied in this work has been utilized extensively in NPC research but represents a severe, infantile form of the disease. It would be very interesting to compare the results of this study with similar studies of the Npc1 nmf164 mouse model, which contains a point mutation similar to many human forms of the disease and has some NPC1 activity (37). The slower progression of the disease in the Npc1 nmf164 mouse could result in greater "temporal separation" of the changes in differing brain regions which could allow delimitation of possibly differing pathological processes in them. Such changes could also serve as biomarkers for the success or failure of current and future therapeutic interventions.


extfieldsofstudy:
Mathematics
Computer Science
Medicine

text:
A Low-Rank Method for Characterizing High-Level Neural Computations

The signal transformations that take place in high-level sensory regions of the brain remain enigmatic because of the many nonlinear transformations that separate responses of these neurons from the input stimuli. One would like to have dimensionality reduction methods that can describe responses of such neurons in terms of operations on a large but still manageable set of relevant input features. A number of methods have been developed for this purpose, but often these methods rely on the expansion of the input space to capture as many relevant stimulus components as statistically possible. This expansion leads to a lower effective sampling thereby reducing the accuracy of the estimated components. Alternatively, so-called low-rank methods explicitly search for a small number of components in the hope of achieving higher estimation accuracy. Even with these methods, however, noise in the neural responses can force the models to estimate more components than necessary, again reducing the methods' accuracy. Here we describe how a flexible regularization procedure, together with an explicit rank constraint, can strongly improve the estimation accuracy compared to previous methods suitable for characterizing neural responses to natural stimuli. Applying the proposed low-rank method to responses of auditory neurons in the songbird brain, we find multiple relevant components making up the receptive field for each neuron and characterize their computations in terms of logical OR and AND computations. The results highlight potential differences in how invariances are constructed in visual and auditory systems.

Alternative constraint formulations
The advantage of the linear equality constraints (Eq 6) is that they provide strong optimality guarantees while scaling as a function of rD but come with the disadvantage of requiring the choice of π k for each pair U •,k and V •,k . There are several other possible equality constraints that satisfy rank(J sym ) = rank(J) and do not require choosing π k but they each are accompanied by their own disadvantages. A few possible alternatives are summarized below.
Two simple choices are the bilinear constraints UV T = VU T which enforce symmetry and the quadratic constraints the latter of which can be proven to satisfy rank(J sym ) = rank(J) by showing P N (U)V = 0 and P N (V)U = 0 (Fazel et al., 2003). The trouble with the bilinear and quadratic equality constraints is that they require optimizing D(D − 1)/2 and D(D + 1)/2 unique constraints making optimization with interior-point methods only tenable when D is small. Since the number of constraints does not depend on r, it means that block coordinate descent is not an option to decrease dimensionality in these cases. Furthermore, Hessian approximation techniques like L-BFGS usually do not attempt to approximate the Jacobian of the constraints and therefore will also not lead to a sufficient decrease in the dimensionality of the problem (Nocedal and Wright, 2006;Wächter and Biegler, 2006). With present memory and speed limitations, the best option in this case would be to use a constrained optimization technique that does not require second-order information; for instance, optimizing an augmented Lagrangian with conjugate gradient descent. In our experience, however, first-order methods may require more problem-specific tweaking of the optimization parameters (e.g. update parameters for the barrier function in an augmented Lagrangian method) to attain reasonable convergence speed compared to second-order methods. This may make first-order methods less attractive for large datasets.
Another possible alternative to the linear constraints is to instead insist that each U •,k and V •,k pair is parallel via the equality constraint which comes from setting the square of the dot product of normalized U •,k and V •,k to 1. The advantages of this form for the equality constraints is that it scales as r and is thus lower dimensional than the linear equality constraints while not requiring a choice of π k . Compared to the other constraints, however, this constraint is quite nonlinear as the Jacobians in the Hessian will depend explicitly on U and V. This constraint may still behave well in practice, but it is not clear if convergence to a solution can be guaranteed. Indeed, experiments with this formulation often got stuck at non-stationary points which would appear to indicate that it does not behave well at least with interior-point methods. There are other variations one can try, including discontinuous variations of Eq 24 where the power-2 is replaced by the absolute value, but we will not explore these further.
Lastly, an alternative form of the constraints that may be attempted is a modification of the linear equality constraints (Eq 6) where the discretization of the π k parameters is relaxed such that π k can take on any real number. In this case, π k would ensure that U •,k and V •,k are equal up to a signed scaling factor.
The downside of the relaxation is that one either needs to insert U •,k = −π k V •,k leading to a nonlinear optimization over a third-order polynomial or use the Lagrangian method in which case the Hessian of f (Eq 10) would be rank-deficient.

Optimality conditions
Here we show the conditions under which a stationary point of L (Eq 9) is a feasible local minimum of f (Eq 10) and derive some useful quantities for the subsequent discussion. In order to certify that a weight vector, is a feasible local minimizer of f (x), the first-order necessary conditions, also known as the Karush-Kuhn-Tucker (KKT) conditions, and the second-order sufficient conditions must be satisfied (Nocedal and Wright, 2006). The KKT conditions state that the gradient of L(x, Λ) with respect to x is zero and x must be a feasible point in weight space. The second-order sufficient conditions require that the Hessian of f (x) is positive semidefinite along feasible descent directions near x.
The necessary conditions for x to be a feasible local minimizer of f appear in Prop 1 where ∇ a , ∇ h , and ∇ Q •,k are the gradient operators with respect to a, h, and Q •,k , respectively, and is a quadratic feature matrix.
PROPOSITION 1. Karush-Kuhn-Tucker (KKT) conditions: the first-order necessary conditions for a feasible local minimum of f (x) are is the Jacobian of the constraint vector with respect to Q •,k and are the feasibility conditions.
With the constraints being linear and the Jacobian being full row-rank, all w k satisfy the linear independence constraint qualification regularity condition and therefore the KKT conditions are guaranteed to be satisfied at a feasible stationary point of L (Nocedal and Wright, 2006). Notably, when the KKT conditions are satisfied Q •,k is complementary to (i.e. is a null space component of) Eq 29 because Q T •,k A T k,k = w T k = 0 (Eq 8) and therefore where the right-hand-side of the arrow follows from the fact that the bracketed term is a symmetric matrix. In other words, Q •,k is complementary to the quadratic gradient found in the bracketed term. This also implies that the Lagrange multipliers, Λ •,k , are components of the null column-space of the partial Jacobian matrix, A k,k , for each k. This result will become important in the following discussion of locally/globally optimal regularization domains.
For nonconvex problems, the KKT conditions in Prop 1 are insufficient to guarantee that a stationary point is a feasible local minimum of f . Instead, we turn to the second-order sufficient conditions. PROPOSITION 2. Second-order sufficient conditions: if the KKT conditions in Prop 1 are satisfied at point x * in weight space and where A is the full rD × (1 + D + 2rD) Jacobian matrix of the constraints and N (A) returns the null space of A, then x * is a feasible local minimum of f .
Intuitively, the second-order sufficient conditions mean that the Hessian must be positive semidefinite along feasible descent directions arbitrarily close to the stationary point x * .
In terms of z t (Eq 1), the Hessian with respect to x may be written as where the Hessian operator is defined as is a symmetric block diagonal indefinite matrix and is a strictly positive semidefinite diagonal matrix. Due to the indefinite matrix, M (see definition in Eq 33), f is a nonconvex function.

Locally and globally optimal regularization domains
Generally speaking, the fact that the low-rank MNE optimization problem is nonconvex invites the possibility that there are multiple local minima, many of which may be suboptimal. However, since the nuclear-norm regularization penalty is convex, it can be shown that there is a regularization domain for which any solution to the low-rank MNE problem (Eq 8) is guaranteed to be globally optimal. We first observe that there is some value of the regularization parameters for which the Hessian of f (Eq 33) becomes positive semidefinite at a given x.
PROPOSITION 3. For a given x, there is a threshold value of k that satisfies k < λ max (M) where λ max (M) is the largest eigenvalue of M such that if all k are greater than or equal to this threshold then ∇ 2 xx f evaluated at x is guaranteed to be positive semidefinite.
PROOF. Under the assumption of fixed x, the characteristic polynomial of M + ∇ 2 xx * (Eq 33) is (using the block LDU decomposition) where λ is an eigenvalue of the Hessian and I is the identity matrix. For any λ that is a solution, there is a corresponding eigenvalue, λ = k ± |λ − k |, symmetric across k that is also a solution. Since all k ≥ 0, the minimum possible eigenvalue of M + ∇ 2 xx * at a given x occurs when any k = 0 and is equal to the smallest eigenvalue of M, λ min (M) ≡ −λ max (M). Therefore, if k ≥ λ max (M) for all k at a given x, then the Hessian is guaranteed to be positive semidefinite at x. Since λ min (∇ 2 xx f ) ≥ λ min (RR T ) + λ min (M + ∇ 2 xx * ), k = λ max (M) is an upper bound on the value of k above which the Hessian becomes positive semidefinite for a given x. Now, suppose we design a (nonlinear) semidefinite program (SDP) equivalent to the low-rank MNE problem (Eq 8), min a,h,X 1 ,··· ,Xr f (a, h, X 1 , · · · , X r ) where X k ≡ Q •,k Q T •,k is a 2D ×2D rank-one matrix with regularization parameter k , J k is the upper-right D × D block of X k , and X k ≥ 0 indicates X k is constrained to be positive semidefinite. In this form, the SDP is nonconvex with potentially many local minima. If, however, we relax the SDP by eliminating the rank constraint (or equivalently modifying it to rank(X k ) ≤ 2D for all k), then the problem becomes convex.
solved at x * then x * is a feasible global minimizer of the equivalent SDP (note that ∇ X k L is a 2D × 2D gradient matrix instead of a gradient vector).
PROOF. According to Prop 3, there is some value for each k above which the eigenvalue spectrum of 2∇ X k f is strictly positive for a given x. However, since ∇ X j L = ∇ X k L for any j and k, the threshold value of k such that the Hessian is positive semidefinite is the same for all k. Thus, if the minimal assigned value of k across all k satisfies Eq 38, the Hessian is positive semidefinite at x. If x * is a solution to the low-rank MNE minimization problem (Eq 8) and Prop 4 is true, then all ∇ X k f are positive semidefinite matrices and ∇ X k f X * k = 0 (Eq 31) at x * . It follows that x * is then a solution to the relaxed SDP and a global minimizer of f (Burer and Monteiro, 2003;Bach et al., 2008;Haeffele et al., 2014). This is true because the corresponding SDP weights for a, h, and all X k are a feasible local minimizer of the relaxed SDP along feasible descent directions shown via the first-order Taylor series expansion of f with respect to all X k about solution a * , h * , X * = k X * k : No feasible X k can locally decrease f because the trace of a product of two positive semidefinite matrices is greater than or equal to zero. Furthermore, since f is a convex function of a, h, and X k , this local information is sufficient to conclude that x * is at a feasible global minimum of f . Therefore, solutions to the low-rank MNE minimization problem (Eq 8) and the rank-constrained SDP (Eq 37) are globally optimal for a given rank and set of regularization parameters provided Prop 4 is satisfied.
Low-rank MNE models optimized to satisfy Prop 4 constitute a regularization domain of globally optimal solutions to the low-rank MNE minimization problem. Low-rank models within this globally optimal domain can be a consistent, good approximation to J of a given maximum rank r provided the training data is sufficiently representative of the underlying ground truth value of J. This can be particularly helpful when attempting to extract low-rank J from exceptionally high-dimensional problems that are impractical to solve at full-rank.
A secondary consequence of the above analysis is that, in the locally optimal regularization domain, solutions are unique along a given unit matrix,Q = Q Q F , where · F is the Frobenius norm.
PROPOSITION 5. Given a solution x * to the low-rank MNE problem, the associated quadratic weights, Q * , are a unique solution along the unit matrixQ * up to a change in sign of the columns (i.e. ±Q * •,k are equivalent solutions).
PROOF. Unlike in the globally optimal regularization domain where ∇ X k f is a positive semidefinite matrix at solution x * , in the locally optimal regularization domain ∇ X k f may be indefinite at a solution x * . However, f is still a convex function of the SDP weights, a, h, and X k . Furthermore, X * k is still complementary to ∇ X k f (Eq 39) at the solution. Therefore, X * k is a unique solution along unit matrixX * k . This unit direction matrix is represented in factorized space byX * k = (±Q * •,k )(±Q * T •,k ) =Q * •,kQ * T •,k . This factorization of the unit matrix,X * k , defines a unique direction in factorized space,Q * •,k , and thus Q * •,k from solution x * is also a unique solution up to a change in sign.
Intuitively, this means that a local minimum constrained to the unit directionQ is a global minimum alonĝ Q. In the case of degeneracy in Q, arbitrary rotations of degenerate vectors are globally optimal along any unit direction in this degenerate subspace. This result does not find use in this paper but we provide it here for completeness in case future advancements in numerical optimization techniques are able to exploit this structure to efficiently obtain certifiable global minima of problems in the locally optimal domain. At the moment, this property would theoretically decrease the number of iterations necessary to reach the global minimum using a branch and bound algorithm but the algorithm would still be at worst EXPTIME and impractical to implement for large D.
To investigate the existence of suboptimal local minima in the locally optimal domain, we took an empirical approach where we searched for a counterexample to the hypothesis that, despite nonconvexity, there are no suboptimal local minima. This was done by drawing random weights, x, from a normal distribution with D = 2 and r opt = 2. For each problem, N = 100 stimulus feature vectors were also drawn from a normal distribution. The response was generated using Eq 1. Then r = 1 low-rank MNE models were fit 10 times for each randomly drawn problem with different random initializations of the weights. If two models for a given random problem differed in negative log-likelihood by 1 · 10 −4 at their respective minima, then that problem qualified as a potential counterexample and was stored. To ensure that the difference in negative log-likelihood was not due to imprecise fitting, the detected counterexamples were verified by plotting f (Eq 10) in U space and observing the existence of spatially separated minima. We found that there are indeed counterexamples to the hypothesis and it is therefore possible that suboptimal local minima will exist in a given problem. However, we remark that suboptimal local minima are seemingly rare among problems drawn from a normal distribution, often requiring the generation of on the order of ∼ 10 3 random problems to find one with a suboptimal local minimum.

Convergence of the block coordinate descent algorithm
Since the interior-point algorithm (Nocedal and Wright, 2006) used to solve each block problem already guarantees global convergence to a feasible local minimizer of a given block of weights, x k , with fixed regularization parameter, k , each cycle through the r blocks (Eq 13) of the block coordinate descent algorithm leads to a monotonically decreasing series f (x r ) for the jth cycle. Because f ≥ 0 is bounded from below on the domain of the weights, the cost function value cannot decrease indefinitely and the series saturates as j → ∞ to a stationary point. At this stationary point, the gradient of the Lagrangian with respect to each block is simultaneously zero and all ∇ 2 x k x k f are simultaneously positive semidefinite when projected into the null space of the constraints, However, satisfying these conditions alone does not guarantee that this stationary point is a feasible local minimizer of the low-rank MNE problem (Eq 8). It can be shown that this stationary point is, in fact, a feasible local minimum of L by verifying that the necessary and sufficient conditions of the full problem (Prop 1 & 2 in appendix) are satisfied when the block KKT conditions (right-hand-side of Eq 14) and block second-order sufficient conditions (Eq 40) are satisfied.
In the following discussion, we will assume that the data has been prepared sufficiently according to Assumption 1, which may be done without loss of generality. We formally state the necessary and sufficient conditions under which a stationary point of the block coordinate descent is a feasible local minimizer of the low-rank MNE problem (Eq 8) in Prop 6. ASSUMPTION 1. The feature space satisfies 41) or has been transformed in such a way that this is true.
Intuitively, this assumption means that the covariance of the stimulus feature space must be full-rank. If the stimulus space does not satisfy this assumption, one can, for example, project the stimuli into the subset of principal components with non-zero variance without loss of generality.
PROPOSITION 6. If, for all k, x k are feasible local minima of the block subproblems (Eq 13) where the block KKT conditions and block second-order sufficient conditions S k ≥ 0 (Eq 40) are satisfied, then the equivalent weight vector, x, of the full problem is a feasible local minimizer of the low-rank MNE problem (Eq 8).
PROOF. (Part 1: necessary conditions) The KKT conditions in Eq 27, 28, & 29 are trivially satisfied when the block gradients are zero. Also, each constraint vector w k is only dependent on the kth block and therefore the feasibility conditions in Eq 30 are satisfied as well.
Showing that the second-order sufficient conditions are satisfied is a bit more involved. To make the proof less cumbersome, we define the abbreviations for 0 < i ≤ j ≤ r where i and j can be thought of as slicing indices (the same conditions apply to i and j ) that define a submatrix of a larger matrix. Under Assumption 1, R i:j,• R T i:j,• is positive definite provided all Q •,i , · · · , Q •,j = 0. If any Q •,k = 0 for i ≤ k ≤ j, then R i:j,• R T i:j,• is strictly positive semidefinite. The matrix Z i,j is generally an indefinite matrix and has rank-deficiency rank N (Z i,j ) ≥ rank ([Q •,i , Q •,i+1 , · · · , Q •,j ]) (recall from Eq 31 that Q •,k is complementary to Z k,k ). The matrix B is always positive definite.
In terms of these abbreviations, we rewrite S k (Eq 40) as and take the Shur complement over B which is positive semidefinite and therefore is positive semidefinite. Provided Q •,k = 0, it is much more likely that Θ k,k will be positive definite than strictly positive semidefinite given the structural dissimilarity of R k,• R T k,• and Z k,k . Even if Θ k,k is singular, an infinitesimal increase to k would theoretically produce a full-rank approximation to Θ k,k with negligible impact on the weights. From here forward, it is assumed that Θ k,k is positive definite unless Q •,k = 0.
We will now use the result in Eq 50 (and the surrounding discussion) and recursive application of the Schur complement to show that the second-order sufficient conditions (Prop 2) of the full problem are satisfied.