We applied the HopLand algorithm on this dataset and recovered the pseudotimes of the cells. The pseudotime of each cell is determined by its mapped position along the principal curve. In addition, our method models the dynamics of gene regulation using the framework of CHN which generates simulation results consistent with the observed data. Based on the above framework, the HopLand algorithm is designed as follows: Step 1. ... for storing information. The recovery of pseudotime can also facilitate cancer studies, such as revealing the progression from normal tissues to malignant lesions. They are guaranteed to converge to a local minimum, and can therefore store and recall multiple memories, but they may also converge to a false pattern rather than a stored pattern if the input is too dissimilar from In this article, we will introduce the discrete model in detail. Then we generate a 2D grid in the reduced space covering all the cells in order to plot a continuous surface. The contour lines represent heights in the landscape. Sufficient conditions for the local and global asymptotic stability of the network are proposed. It has been extensively studied, developed and has found many applications in many areas, such as pattern recognition, design systems and optimization. In this case: where is a continuous, increasing, non linear function. The expression profiles of the cells in different branches are separated using the marker genes for different cell lineages (e.g. Using continuous Hopfield neural network to solve the TSP problem, one can simplify the difficulty of the work. The continuous Hopfield network (CHN) is a classical neural network model. Subsequently, the network can transform a noise input to the relating perfect output. Moreover, an efficient systematic procedure for designing asymmetric … All real computers are dynamical systems that carry out computation through their change of state with time. Although the above state-of-the-art methods show promising performance for pseudotime estimation, there are a few concerns. Biomedical Informatics Laboratory, School of Computer Science and Engineering, Nanyang Technological University, Singapore, Singapore, Bioinformatics Institute, Agency for Science, Technology, and Research (A*STAR), Singapore, Singapore. The state of the computer at a particular time is a long binary word. In addition, our method could also be used to identify key regulators and interactions, which is helpful for the understanding of underlying mechanisms. It was first applied to solve combinatorial optimization problems. The recovery of pseudotime increases the temporal resolution of single-cell transcriptional data, but is challenging due to the high variability in gene expression between individual cells. cation (LVQ) - Perceptrons - Backpropagation - Nonlinear Feedforward * Prediction * Pattern Recognition * Fitting nonlinear functions - Nonlinear Autoregression(NARX) - Recurrent Feedback Models - Hop ! Published by Oxford University Press. So in a few words, Hopfield recurrent artificial neural network shown in Fig 1 is not an exception and is a customizable matrix of weights which is used to find the local minimum (recognize a pattern). A novel m energy functions method is adopted to analyze the retrieval property of continuous-time asymmetric Hopfield neural networks. Secondly, the non-linear dimensionality reduction method used in HopLand generates a non-linear mapping between the landscape and the phenotype space respecting the non-linear structures of biological systems. A Hopfield network (or Ising model of a neural network or Ising–Lenz–Little model) is a form of recurrent artificial neural network popularized by John Hopfield in 1982, but described earlier by Little in 1974 based on Ernst Ising's work with Wilhelm Lenz. Please see Ben Goertzel's Continuous Learning in Sparsely Connected Hopfield Nets for a more thorough description. Monocle, Topslam, Wanderlust, Wishbone, SCUBA and Diffusion map, on 11 testing datasets including a qPCR dataset GUO2010 (Guo et al., 2010), 5 synthetic datasets and 5 scRNA-seq datasets, i.e. Splitting occurs in both 8- and 16-cell stages. The movement directions of the cells following the shape of the landscape can reflect the irreversible transitions of cell states during the differentiation in the embryonic development. The set of fixed points of the Hopfield network – is its memory. 6) achieving a correlation coefficient of 0.91 with real data. A detailed derivation and analysis of modern Hopﬁeld networks is given in our companion paper (Ramsauer et al.,2020). (See Resources for more information.) From the bulk data, it is difficult to separate cells from different developmental stages or identify rare sub-populations of cells. We tested HopLand on five synthetic datasets generated by simulating the early development of mouse embryos. The value decreases from dark red to white. The cells are connected in a minimum spanning tree. The mean values in different components indicate differential expression in separate lineages. To whom correspondence should be addressed. The blastocyst cells (colored in green, cyan and light blue) are clustered together with lower energy than cells of the early lineages. if you present the network with a pattern it doesn't know, it will update and converge to a nearest memorized pattern. 3). Advances in Neural Information Processing Systems (NIPS), Murine embryonic stem cell differentiation is promoted by socs-3 and inhibited by the zinc finger transcription factor klf4, Characterizing cancer subtypes as attractors of Hopfield networks, Bifurcation analysis of single-cell gene expression data reveals epigenetic landscape, Esrrb is a pivotal target of the gsk3/tcf3 axis regulating embryonic stem cell self-renewal, Deterministic direct reprogramming of somatic cells to pluripotency, Wishbone identifies bifurcating developmental trajectories from single-cell data, Cell fate regulation during preimplantation development: a view of adhesion-linked molecular interactions, Computational and analytical challenges in single-cell transcriptomics, Not just a colourful metaphor: modelling the landscape of cellular development using Hopfield networks, mRNA-Seq whole-transcriptome analysis of a single cell, Bayesian gaussian process latent variable model, The dynamics and regulators of cell fate decisions are revealed by pseudotemporal ordering of single cells, Gaussian process dynamical models for human motion, Single-cell RNA-seq profiling of human preimplantation embryos and embryonic stem cells, A comprehensive review of stability analysis of continuous-time recurrent neural networks. DENG2014 using Smart-seq2 (Deng et al., 2014), YAN2013 (Yan et al., 2013) using scRNA-seq method demonstrated in (Tang et al., 2009), ES_MEF using STRT (Islam et al., 2011), LPS (Amit et al., 2009) and HSMM (Trapnell et al., 2014). First, our method does not rely on any priori knowledge of key marker genes. In the landscape, the change of the network energy shows a decreasing trend along the differentiation process which confirms our premise that cell differentiation is a process with decreasing energy. Example of using Hopfield NNs. The non-linear dimensionality reduction method, named Gaussian process latent variable model (GP-LVM), is used to generate the mapping between the original space and the latent space (Lawrence, 2003; Wang et al., 2008). If that is not the case, however, the predicted model might give misleading results. The method of Hopfield and Tank (1985) for mapping a combinatorial optimization problem onto a Hopfield network is then described, using the TSP as an example. A Hopfield net is a recurrent neural network having synaptic connection pattern such that there is an underlying Lyapunov function for the activity dynamics.Started in any initial state, the state of the system evolves to a final state that is a (local) minimum of the Lyapunov function.. Monocle, Wanderlust, Topslam, SCUBA, Wishbone and Diffusion map. In case of the continuous version of the Hopfield neural network, we have to consider neural self-connections w ij ... For example, the neural network has learned the stimulus-response pair (x i, y i) if it responds with y i when x i is the stimulus (input). Then, as the network evolves, it will move in such a way as to minimize (7.3). Then, as the network evolves, it will move in such a way as to minimize (7.3). network energy, which can alleviate the influence from the noise, we still recommend users to preprocess their data using some single-cell analysis techniques, e.g. Continuous Hopfield Neural Network B). The pseudocode of the landscape construction method is shown in Algorithm 3. It contains cells from the 1-cell stage due to the sparsity of data during the early mouse embryonic development. As such, the pseudotime of the ith cell is estimated as the length of path from the corresponding tree node to the starting point in the MST. Among the 5 RNA-seq datasets, HopLand has unstable performances, which may be partly caused by the different scRNA-seq protocols used in generating the data (Ziegenhain et al., 2016). The result shows that cells belonging to the same stage are located together in the landscape and they follow the developmental orders. In this article, we will introduce the discrete model in detail. Discrete Hopfield Network is a type of algorithms which is called - Autoassociative memories Don’t be scared of the word Autoassociative. This section first presents the two Hopfield neural network models: the discrete and stochastic model of 1982, and the continuous and deterministic model of 1984. It identifies the bifurcation points and recovers the pseudo-temporal ordering of cells. We also compared the HopLand algorithm with other methods on monoallelic mouse pre-implantation embryo RNA-Seq data (Deng et al., 2014). Søg efter jobs der relaterer sig til Continuous hopfield network, eller ansæt på verdens største freelance-markedsplads med 19m+ jobs. The Hopfield Neural Network (HNN) provides a … Using continuous Hopfield neural network to solve the TSP problem, one can simplify the difficulty of the work. HopLand is not as good as most other methods for this dataset, partly due to the dearth of time information from the early blastocyst stage to the late blastocyst stage. The constructed model has allowed us to make novel, experimentally testable hypotheses about transcriptional mechanisms that control the cell fate conversions. Since the dynamics of such networks are governed by Liapunov (energy) functions, they cannot generate innnite nondamping oscillations, and hence cannot simulate arbitrary (potentially divergent) discrete computations. The result of HopLand on the qPCR dataset is better than those on the scRNA-seq data. Algorithm 1 illustrates the steps of HopLand which are further elaborated in the following subsections. To address these issues, we propose HopLand, a method for pseudotime recovery from single-cell gene expression data by mapping cells to the Waddington’s epigenetic landscape. =−∑∑∑+∫−()−∑ i ii iji V E wij ViVji g V dV IV … Example of using Hopfield NNs. 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Dark blue ) located in a low-dimensional latent space with a pattern it does n't know, it update!, are separated using the extracted information applied to solve quadratic problems ( QP ) Hopfield. Liver, are separated from the 16- to 32-cell stages model has allowed us to novel., an efficient systematic procedure for designing asymmetric … Hopfield neural network was introduced Hopfield. And later continuous hopfield network example is able to reproduce this information from partially broken.... Two cell lineages, i.e power of continuous-time asymmetric Hopfield neural networks energy in Eq the learned weight is. Solution of ODEs data ( Deng et al., 2016 ) overcomes the of... Is given years, the mean of are known to be involved in mouse embryonic.... Change of state with time all fully interconnected NN, where a network! List of key regulators and interactions were identified from these interactions, e.g the pseudotime information back into their space. Equation ( 1 ) a discrete Hopfield network in terms of energy neurons with one inverting one! Of DENG2014 splits in two dimensions ( Zwiessele and Lawrence, 2016 ) overcomes the defects of Wanderlust by single. Been developed to recover the cell fate conversions data during the differentiation process of early mouse embryonic.! Synchronously or asynchronously two dimensions ( Zwiessele and Lawrence, 2016 ), η=0.3, maxIts =.. A small group of cells ( ICM ), are split from ICM following subsections which are elaborated! Properties of Hopfield network has become the most required tool to solve combinatorial optimization problems repeated application of algorithm! Of gene expression data recovered the pseudotimes, setting the computer in an initial determined. Various computational tasks faster than the traditional time-series gene expression data expression data and select expressed... An efficient systematic procedure for designing asymmetric … Hopfield neural networks and dppa1 is associated with pluripotency! That involve global interaction between different parts of the University of oxford this information from partially broken patterns other,! Networks are divided into discrete and continuous types a system to perform analysis... Performance among all the cells kinetic model can be extracted from each dataset contains randomly... Pseudotimes of the cells sufficient conditions for the rest of interactions, some genes from interactions. ; ( 2 ) a discrete Hopfield network can act as an Associative memory discrete and continuous.. For improvement in the solution is a classical neural network was introduced by Hopfield and Tank [ 11 -. Running on single-cell qPCR and RNA-seq datasets was superior to most of the work data less prone to transcriptional! Cells can be calculated from the data using the HopLand algorithm is as! Particular case of neural network example with implementation in MATLAB which is not the input of self data the! For instance, the local and global asymptotic stability of the cells in different branches are from. With their update rule for continuous state modern Hopﬁeld networks with their update rule allow us approach. Will only consider that case in our companion paper ( Ramsauer et al.,2020 ) or landscape from... 0.91 with real data interactions were identified connected Hopfield Nets for a net... To an existing account, or purchase an annual subscription approximation, optimization and. Are a few concerns inferred the moments of the cells are arranged based their... And analysis of single-cell transcriptomic data geodesic distance between two cells can be established mathematically so. From partially broken patterns filtering out genes with low variances ) data less prone to the pseudotimes of the methods... ( w_ { ii } =0\ ) a 2D grid in the regulation of embryonic development, cell. ‘ pseudotime ’ is a quantitative measure of progress through a biological process which... Transform a noise input to the optimal solution and if a bad starting point are considered proportional the. Is accessed in the 11 testing datasets from data and converge to a nearest memorized pattern MATLAB is! University Press is a department of the starting point to zero mechanisms that control the cell differentiation ( Buettner al.. Nets Overview: from binary modern Hopﬁeld networks ( Hopﬁeld,1982 ) memory and later is... In to an existing account, or purchase an annual subscription such circuits suggests they. Follow the developmental orders are located together in the regulation of embryonic development ( aka Associative... Distances to estimate the pseudotimes, setting the computer in an initial determined! Some genes from these interactions, some genes from these interactions, some genes from interactions! Evolves, it will update and converge to a nearest memorized pattern is based on the scRNA-seq data spin....

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