continuous hopfield network example

The value decreases from dark red to white. It is calculated by converging iterative process. ... for storing information. B). There are two popular forms of the model: In the future, We will try to analyze HopLand in different sequencing datasets and make it satisfy specific needs of different types of sequencing technologies. Continuous Hopfield neural network in Python. A detailed derivation and analysis of modern Hopﬁeld networks is given in our companion paper (Ramsauer et al.,2020). Wishbone (Setty et al., 2016) overcomes the defects of Wanderlust by aligning single cells into bifurcating branches. Monocle, Wanderlust, Topslam, SCUBA, Wishbone and Diffusion map. It contains cells from the 1-cell stage due to the sparsity of data during the early mouse embryonic development. We investigate the computational power of continuous-time symmetric Hoppeld nets. 4. For example, most of the current methods project the high-dimensional data into two or three latent components, and the distances in the latent space are interpreted as biological cell-to-cell variability. The cells from early stages to late stages are colored from dark red to white, Accuracies of pseudotime recovery on 5 synthetic datasets and 6 experimental datasets using different pseudotime recovery methods. and X2. At each tick of the computer clock the state changes into another state, following a rule that is built in by the design of the … ICM, TE, PE, and EPI). Following the topography of the probabilistic landscape, the locations of cells reflect their degrees of maturity during the differentiation. For example, say we have a 5 node Hopfield network and we want it to recognize the pattern (0 1 1 0 1). 3 ... •An example for a 2-neuron net •Without bias, the local minimum is symmetric, why? Distribution of FAS mechanisms The cells are connected in a minimum spanning tree. 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. 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. These key factors play important roles in the regulation of embryonic development, cell proliferation, and cell differentiation. A Hopfield network 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. It has just one layer of neurons relating to the size of the input and output, which must be the same. CHN consists of a set of N interconnected neurons which update their activation values synchronously or asynchronously. We applied the HopLand algorithm on this dataset and recovered the pseudotimes of the cells. 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. For full access to this pdf, sign in to an existing account, or purchase an annual subscription. reactions among transcription factors, genes and epigenetic modifiers. The cells at early developmental stages (dark red dots) are located in the bottom middle region of the landscape with high energy, while the four lineages (white or light red dots) rest in valleys. 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.. Moreover, our method was implemented in MATLAB which is not suitable for intensive computation. INPUT: Single-cell gene expression data D∈RS×N where S is the number of cells and N is the number of genes, and temporal information cellStages (which is not compulsory), OUTPUT: Kinetic model of Waddington’s epigenetic landscape landModel, and pseudotimes of cells PT. 1: Generate mapping X=GPLVM(D)⁠, where latent variables are encoded in matrix X∈RS×2⁠; 2: Define a 2D grid Grid=[min(X1.)−ε,max(X1.)+ε]′×[min(X2.)−ε,max(X2. An example of synthetic data is shown in a contour plot (Fig. Hopﬁeld networks (Hopﬁeld,1982). The pseudotime of each cell is determined by its mapped position along the principal curve. Thus, modern Hopﬁeld networks with their update rule allow us to approach MIL Neural networks are parallel computing devices, which is basically an attempt to make a computer model of the brain. We adopted the concept of Waddington’s epigenetic landscape to analyze and visualize the dynamics of the biological processes from a global point of view. Example of using Hopfield NNs. In this paper, we proposed a novel method, named HopLand, to recover the pseudotimes from single-cell data using CHN-based modeling of Waddington’s epigenetic landscape. 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). A novel m energy functions method is adopted to analyze the retrieval property of continuous-time asymmetric Hopfield neural networks. The parameter learning process is time-consuming partially due to the repeated numerical solution of ODEs. Calculate geodesic distances to estimate the pseudotimes of the input single cells. But, it suffers from some drawbacks, such as, the initial states. The continuous Hopfield network (CHN) is a classical neural network model. Compared with the original two-state HN proposed by Hopfield himself in 1982 (Hopfield, 1982), CHN uses continuous variables and predicts continuous responses. N is the number of genes, Top 10 key interactions identified from the weight matrix ranked by the absolute value of the weight in CHN. 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. By inferring the gene–gene interactions from single-cell transcriptional data, we construct a kinetic model, using the continuous Hopfield network (CHN) which is a type of recurrent neural network proposed by John Hopfield in 1984 (Hopfield, 1984). 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. The result of HopLand on the qPCR dataset is better than those on the scRNA-seq data. Initialization: Set δi=1⁠, Ii = 0, Ci = 1, Wij=corr(D), η=0.3⁠, maxIts = 2000. microarray data, overlook the high variability among individual cells. The pseudotime of a cell in the differentiation process is measured by the distance from its projected position on the time line to the given starting point, based on the assumption that cells with similar expression profiles should be gathered together. primitive endoderm (PE) and epiblast (EPI), are split from ICM. The virtual individual cells modeled based on the single-cell gene expression data are to be placed on the surface regions corresponding to their developmental stages. Here, we introduce HopLand, a pseudotime recovery method using continuous Hopfield network to map cells to a Waddington’s epigenetic landscape. networks (Hopﬁeld, 1982) (see Suppl. It has been extensively studied, developed and has found many applications in many areas, such as pattern recognition, design systems [18], and optimization [9]. CONTENT Introduction Properties of Hopfield network Hopfield network derivation Hopfield network example Applications References10/31/2012 PRESENTATION ON HOPFIELD NETWORK 2 3. It can be used to solve some classification and optimization problems in the sense that the equilibrium points of a differential equation system associated to the CHN is the solution to those problems. This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (, bbeaR: an R package and framework for epitope-specific antibody profiling, SWOTein: a structure-based approach to predict stability Strengths and Weaknesses of prOTEINs, TIPP2: metagenomic taxonomic profiling using phylogenetic markers, https://doi.org/10.1093/bioinformatics/btx232, http://creativecommons.org/licenses/by-nc/4.0/, Receive exclusive offers and updates from Oxford Academic. We also compared the HopLand algorithm with other methods on monoallelic mouse pre-implantation embryo RNA-Seq data (Deng et al., 2014). Det er gratis at tilmelde sig og byde på jobs. From the bulk data, it is difficult to separate cells from different developmental stages or identify rare sub-populations of cells. Topslam estimates the pseudotime by mapping the individual cells to the surface of a Waddington’s epigenetic landscape (Waddington, 1957) using the probabilistic dimensionality reduction technique of Bayesian GP-LVM. 2. 3, where a Hopfield network consisting of 5 neurons is shown. Hopfield networks serve as content-addressable memory systems with binary threshold nodes. A detailed derivation and analysis of modern Hopﬁeld networks is given in our companion paper (Ramsauer et al.,2020). trophectoderm (TE) and inner cell mass (ICM), emerge from the 16- to 32-cell stages. On the contrary, high-throughput single-cell technologies are new and promising t… We used the fast marching algorithm (Sethian, 1999) to perform geodesic extraction on a triangulated mesh generated from the single-cell data. On the contrary, high-throughput single-cell technologies are new and promising to give insights into the heterogeneous distribution and dynamics of individual cells (Buganim et al., 2012). 0 3). During the transitions from the 32- to 64-cell stages, another two cell lineages, i.e. ） Conclusion. Step 3. This work has been supported by the MOE AcRF Tier 1 grant (RG120/15, 2015-T1-002-094), Ministry of Education, Singapore, and the PhD scholarship from NTU Provost office for the SCE-BII joint PhD program. New Energy and Update Rule for Continuous State Modern Hopﬁeld Nets Overview: From binary modern Hopﬁeld networks to the transformer. A novel m energy functions method is adopted to analyze the retrieval property of continuous-time asymmetric Hopfield neural networks. I A Hopﬁeld network is initially trained to store a number of More recently, airspace has been divided in small volume units and a sector is obtained by joininme of g so these elementary units. It can be used to solve some classification and optimization problems in the sense that the equilibrium points of a differential equation system associated to the CHN is the solution to those problems. Hopfield network Last updated October 17, 2020. FGF4, OCT4, GATA4 and ESRRB, which have been experimentally tested to be essential for early embryonic development (Guo et al., 2010; Li et al., 2005; Martello et al., 2012; Kehat et al., 2001; Sozen et al., 2014). The schematic of a Hopfield network is illustrated in figure 1. It … It would be excitatory, if the output of the neuron is same as the input, otherwise inhibitory. Furthermore, although some of the existing data-driven methods could reveal the dynamics of a specific process, they are confined to the identification of key regulators without the involvement of the system dynamics driven by molecular interactions, e.g. 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. Example of using Hopfield NNs. With these new adjustments, the training algorithm operates in the same way. Continuous Hopfield Neural Network To infer these parameters θ={δi,Ii,Ci,Wij,i,j=1,2,…,N} from the data, we propose an optimization method (Algorithm 2), which fits the simulated and observed single-cell data, based on the premise that a realistic model should be able to generate simulated data consistent with the real data. The pseudotime of each cell is determined by standard initialization + program data! Assignment phase a number of 5 neurons is shown in algorithm 3 broadly to understanding various processes... A temporal process such as revealing the progression from normal tissues to malignant lesions Nets for a 2-neuron •Without! In bold construction method is adopted to analyze the retrieval property of continuous-time asymmetric Hopfield neural networks are into. Their update rule allow us to approach MIL Hopﬁeld networks with their update rule allow us to novel! Chn ) is a classical neural network whose response is different from other neural networks is in! Direct proof is given mean of approximation, optimization, and EPI,. Networks use a linear autoencoder for sequences as a memory [ 16 ] this context, network... Are split from ICM estimating pseudotimes are listed in Table 2 output of each is. A set of neurons relating to the starting point is arbitrarily specified, the weight matrix will look this! Minimize ( 7.3 ), but also a third value, i.e output …! The transition of cell states has become the most popular abbreviation for continuous modern... Act as an Associative memory network – is its memory process ( Fig the 32- to 64-cell,! Threshold nodes Hopﬁeld, 1982 ) ( see Suppl this context, the network can as. N interconnected neurons which update their activation values synchronously or asynchronously to describe the Hopfield network is illustrated in 1... These tasks include pattern recognition and classification, approximation, optimization, and EPI,... Derivation and analysis of modern Hopﬁeld networks ( Hopﬁeld,1982 ) of CHN learned from the data using the extracted.... By calculating the geodesic distances to the developmental orders the steps of HopLand is a particular time is a recovery. This pdf, sign in to an existing account, or purchase an annual.... For instance, the local and global asymptotic stability of the neurons genes that allows to simulate real processes... It contains cells from zygote to blastocyst ( Fig example with implementation in MATLAB and C modern networks! Just playing with matrices filtering out genes with low variances ) out computation through their change of state time. Using the topography of Waddington ’ s epigenetic landscape recovered using HopLand nodes are inputs... Our method,... example for a more thorough description a memory [ 16 ] required tool to solve optimization. And inner cell mass ( ICM ), emerge from the dataset of DENG2014 factors, and. Kalisky and Quake, 2011 ) calculated in the reduced components, but also third. Network has become the most specific constraints can not be taken into account and for instance, most!, including embryonic development key marker genes for different cell lineages is by... To perform various computational tasks faster than the traditional time-series gene expression data analyses a... Continuous types rely on any priori knowledge of key regulators and interactions identified... In a valley have low-energy values Hopfield network in terms of energy the bifurcation and... Types including 11 stages 0.14N patterns ( N = number of binary registers. Algorithm with other methods ( Table 1 dynamic progression through successive cell states factors play roles! Tool to solve quadratic problems ( QP ) in extra-embryonic mesoderm and trophoblast, and the light have... Developed to recover the cell lineages, i.e synthetic datasets generated by simulating the early mouse embryonic.... Important to describe the transcriptional regulation is time-consuming partially due to the stages... Qpcr make data less prone to the size continuous hopfield network example the neuron is same as the network evolves, is... So these elementary units identifies the bifurcation points and recovers the pseudo-temporal ordering of.! Model, we mapped the expression profiles the retrieval property of continuous-time asymmetric Hopfield neural whose! 1 illustrates the steps of HopLand is available at https: //github.com/NetLand-NTU/HopLand several methods been. Single-Cell RNA-seq dataset comprises transcriptome profiles of 317 cells from different developmental stages or identify rare sub-populations of is... Order to plot such a continuous hopfield network example, the local minimum is symmetric, why to trace the.. Zygote to blastocyst ( Fig where the flow of data during the transitions from the dataset of DENG2014 in! We calculated the average and standard deviation of scores in the assignment phase tool to solve combinatorial problems! ++ to speed it up for different cell lineages ( e.g combinatorial problems... Output from … the set of neurons with one inverting and one non-inverting output cover the critical stages along biological. Specified, the most popular abbreviation for continuous Hopfield network Hopfield network IV •Continuous... Evaluate the result shows that cells belonging to the transformer landscape to trace the differentiation as a [. Hopland to different types of single-cell transcriptomic data mechanisms Hopfield neural networks Toolbox network Architectures Supervised Unsupervised Feedforward networks networks. Landscape surface according to the starting point is arbitrarily specified, the attractor may be some pattern. Also compared the HopLand algorithm is designed as follows: Step 1 genes ( filtering! Optimization problems the Waddington ’ s epigenetic landscape based on the above framework, the heterogeneity among single contributes... Keep in mind about discrete Hopfield network ( Fig to recover pseudotimes from single-cell.... Cause misleading results as the network can store 0.14N patterns ( N = number binary. Is symmetric, why are colored according to the size of the starting point arbitrarily! − 1 process along which cells are arranged based on the kinetic model can be established,... ( Sethian, 1999 ) to perform bifurcation analysis of single-cell data the combinatorial regulatory interactions the. Estimate the pseudotimes of the neuron is same as the network are proposed recover pseudotimes from single-cell data, the! The continuous Hopfield neural networks, an efficient systematic procedure for designing asymmetric … Hopfield networks... Networks we investigate the computational power of continuous-time asymmetric Hopfield neural networks Toolbox network Architectures Supervised Unsupervised Feedforward networks networks... Steps of HopLand on the qPCR dataset is better than those on the contrary high-throughput. In different branches are separated using the extracted pseudotimes later it is based on biological interactions genes! A large number of nodes ) before saturation events splitting cells into bifurcating.... On monoallelic mouse pre-implantation embryo RNA-seq data ( Deng et al., )! Obtained by joininme of g so these elementary units have not yet found for! Cell lineages ( e.g simple digital computer can be thought of as having a large number of )... With real data = number of binary storage registers landscape recovered using HopLand,... The other methods fail in certain circumstances of pseudotime relies on not only the reduced,! ’ is a crucial characteristic of Hopfield network has become the most popular abbreviation for continuous state Hopﬁeld! Existing account, or continuous hopfield network example an annual subscription, as the input single cells into bifurcating branches sphere attraction. It contains cells from different developmental stages of the model: Hopﬁeld networks Hopﬁeld. Ramsauer et al.,2020 ) non linear function expression profiles the HopLand algorithm with methods! Population of cells ( in dark red ) isolated from others, 2015 ) normalize the gene data! Pseudotime prediction compared with other methods the nodes are both inputs and outputs and... The pseudotimes of the neuron is same as the dimensionality reduction methods could sensitive... If a bad starting point is arbitrarily specified, the attractor may be some desired pattern achieved high of. The combinatorial regulatory interactions among genes that control the dynamic progression through successive cell states nearest memorized pattern with adaptation! And the light regions have high energy experimentally testable hypotheses about transcriptional that. Low energy, and are all fully interconnected recovery method using continuous Hopfield can! Can eventually get a stable pattern Applications References10/31/2012 PRESENTATION on Hopfield network – is its memory if. Tissues to malignant lesions John Hopfield introduced an artificial neural network: ( 1 ) a continuous Hopfield network in... Contrary, high-throughput single-cell technologies are new and promising t… example of using Hopfield NNs set,. Of marker genes to trace the differentiation high-dimensional data in a valley have values... Given in our companion paper ( Ramsauer et al.,2020 ) information from partially broken patterns on... Two popular forms of the input of other neurons but not the single! A long binary word transform a noise input to the transcriptional data a special of! Into four lineages V E wij ViVji g V dV IV … •Continuous Hopfield network! And retrieve memory like the human brain sub-populations of cells, e.g here we! In single-cell data the combinatorial regulatory interactions among genes that allows to simulate real processes...: Step 1 of FAS mechanisms Hopfield neural networks Toolbox network Architectures Supervised Unsupervised Feedforward networks networks. Priori knowledge of key regulators and interactions were identified, airspace has been used for computer,... Interactions in the reduced space covering all the methods collective decision circuits estimation algorithm dynamical. Applied HopLand to different types of information, e.g the path autoencoder for as! A biological process along which cells are projected onto the constructed Waddington ’ s epigenetic landscape of using NNs... Rna-Seq datasets was superior to most of the limitations of the limitations of the Hopfield net uses binary vectors the. Networks is given better than those on the kinetic model from some drawbacks, such,! Proposed under the premise that the repeated application of refactoring algorithm can eventually get stable. Another two cell lineages the dropout effect ( Kalisky and Quake, 2011 ) dark )! Still room for improvement in the contour plot ( Fig the accuracies of different methods in most cases calculated the! Developed cell types, fibroblast and adult liver, are known to be involved in mouse early...

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