Màster Oficial - Física dels Sistemes Complexos i Biofísica
URI permanent per a aquesta col·leccióhttps://hdl.handle.net/2445/189184
Treballs Finals del Màster en Física dels Sistemes Complexos i Biofísica de la Facultat de Física de la Universitat de Barcelona.
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Treball de fi de màsterMaximum entropy filtering of weighted networks(2025-06) Ramoneda Perales, Shanti; Cardillo, AlessioThe increasing availability of data of large-scale complex systems has resulted in noisy and extremely dense complex networks, reducing their sparsity that constitutes one of the hallmarks of complex networks, making them more difficult to interpret, visualize andZ study from a computational perspective. Network filtering responds to the necessity to continue using the toolbox of network science to analyze larger networks, and several methods and approaches exist deal with this problem, focusing on extracting the so-called backbone of a given network, which is a sparsesubgraph retaining some properties of the original network. The properties a backbone retains from its ancestor network will depend on the method used to retrieve it. We study backbone extraction with the Enhanced Configuration Model (ECM) filter, a maximum-entropy approach that shares the advantages of some existing filtering techniques, while solving some of their problems. Remarkably, the ECM-filter is an unbiased approach that preserves the degree and strength distribution. First, we have performed a systematic benchmark of the ECM-filter using synthetic networks topologies with a known backbone. We have found that an optimal acceptance level can be identified for which the extracted backbone is the most alike possible to the expected, synthetic, one. Also, we notice that such acceptance level scales with the network size as a power-law. Then, we have studied how the properties of real complex networks vary as they are filtered with the ECM-filter. We have found that even if there is no universal behavior shared by the empirical networks considered, the ECMfilter consistently extracts non trivial and well connected backbones comprising a major part of the nodes even when a significant fraction of the edges are already pruned. We discuss the different criteria that can be used to choose a suitable acceptance value for the ECM-filter depending on the type of structure and properties we want the extracted backbone to have- Treball de fi de màsterEntropy Production of Nonequilibrium Steady State Systems(2025-06) Palau Leonés, David; Ritort Farran, FèlixEntropy production is a fundamental feature in nonequilibrium physics that characterizes many aspects in physical and biological systems. Within this work, the framework of stochastic thermodynamics is utilized to reproduce the derivation of a variance sum rule (VSR) for displacements and force variances that allows to obtain an expression for the entropy production rate σ in nonequilibrium steady states. I also provide a generalization of this VSR to take into account inertial effects. Furthermore, these methodologies are applied to different cases of stochastic switching trap (SST) systems, where stochastic jumps in the potential energy perform a mechanical work into the system, leading it to a nonequilibrium state with a net production of entropy. To compute σ analytically, I demonstrate that those problems can be easily solved in the Laplace space, yielding to results equivalent to those obtained via stochastic energetics.
- Treball de fi de màsterShape-sensing motility of cell clusters and collective durotaxis(2025-06) Jiménez Becerra, Àlex; Casademunt i Viader, JaumeCollective cell migration plays a central role in many biological processes, such as embryonic development, wound healing and cancer progression. Recent studies have predicted that non-circular epithelial cell clusters could collectively migrate through a shape-sensing mechanism, independently of any guiding external cues. This phenomenon has been called ‘shape-sensing spontaneous motility’ and has been demonstrated in numerical simulations based on the equations for active polar gels adapted to a free-boundary problem, and partially in experiments. On the other hand, collective durotaxis is a well-established phenomenon that manifests in the migration of cell clusters towards stiffer regions when the environment exhibits a stiffness gradient. In this study, we aim to observe and characterise the behaviour of simulated clusters when these two mechanisms are combined and may compete. To address this problem at a quantitative level and in different parameter regimes, one must rely on numerical simulations. Using a code based on the finite elements method, we have modelled the dynamics of cell clusters of different shapes and sizes with or without the effect of durotaxis. Results can then be analysed to draw meaningful conclusions about the motility of these cell clusters and the interplay between the two guiding effects. We find that most cases are highly non-trivial and we find some unexpected behaviours. In addition, we perform a theoretical analysis of the model for the circular case, where the radial symmetry allows us to obtain explicit expressions for most of the magnitudes in the problem. Finally, we study some particular questions arising from simulations, related to the mixed effect of the shape of the clusters and the durotaxis. After our studies, we see that cell clusters are very complex systems where the balance between forces gives rise to intricate and surprising dynamical behaviours. Many cases are not intuitive, but a certain control over the motility of these clusters seems to be possible.
- Treball de fi de màsterImpact of autistic-derived metabolites on the dynamics and connectivity of neuronal cultures(2025-06) Gómez García, Sarah; Soriano i Fradera, JordiAutism spectrum disorder (ASD) is a neurodevelopmental condition with complex and largely unknown causes. While genetic factors are widely studied, growing evidence suggests that environmental influences, including gut microbiome alterations, may play a significant role in ASD pathophysiology. Recent hypotheses propose that disruptions in gut microbiota composition could impact brain function and contribute to the development of ASD. This study investigates the effects of ASD-associated microbiota metabolites on the dynamics and connectivity of neuronal cultures, which provide an accessible model to explore brain-like processes at the cellular level, offering insights into how neuronal abnormalities may lead to dysfunction and disease. To examine this, three types of cultures are analyzed: cultures treated with fecal-derived metabolites from individuals with ASD or neurotypical individuals, and control cultures. All samples contain metabolites that differentially influence neuronal behavior. Experimental data is combined with an in silico model to help understanding the relationship between neuron-level connectivity and network-wide collective behavior. Results show that ASD cultures exhibit alterations in spontaneous activity and effective connectivity as compared to the neurotypical condition, although these alterations are not preserved along development, an aspect that hints at the activation of plasticity mechanisms that counterbalance possible damage By elucidating the effects of ASD-associated microbiota metabolites on neuronal function, this study helps advance our understanding of gut-brain interactions in ASD and their role in neurodevelopmental disorders
- Treball de fi de màsterMixing in Active Turbulence: from fully developed turbulence to labyrinthine arrest(2025-06) Dejuán Bitriá, Eduardo; Casademunt i Viader, JaumeActive fluids represent a new paradigm in non-equilibrium physics. Even at low Reynolds numbers, they can develop spontaneous flows that resemble inertial turbulence in chaotic behaviour, also with a universal scaling law. However, unlike classical turbulence, active turbulence lacks an energy cascade. Instead, it self-organizes the scales at which energy is injected and dissipated. This suggests we may be dealing with a new class of turbulence—one we are still learning how to describe. This thesis is the first to study mixing in this rich scenario, in which two qualitatively different turbulence limits emerge: one characterized by fully developed turbulence and another where labyrinth-like patterns form and the system becomes dynamically arrested. We study the mixing of a passive scalar and the movement of tracer particles within these simulated flows. We aim to characterize each flow regime using our mixing analysis. We find that tracer particles provide a more effective way to probe the transport properties of each regime, particularly through quantities like the ballistic time and the excess kurtosis, which clearly distinguish the two types of flow and point towards a non-diffusive stationary state of the labyrinths. Interestingly, also in the labyrinthine regime, tracer particles show a heavy-tailed distribution in concentration, indicating clustering and the possible formation of particle caustics. Although this study is still in progress, our results already point toward labyrinthine flows as a distinct form of quantitative and qualitative turbulence, with unique transport properties.
- Treball de fi de màsterModeling Public Opinion Dynamics: The Spiral of silence in clustered homophilic networks(2025-06) Castillo Sanz, Juan; Cozzo, EmanuelePublic discourses are altered by multiple social factors that affect both the willingness of the individuals to express their opinion and the structure of social relations. In this study, we explore the combined role of choice homophily and triadic closure in the formation of a public opinion in societies. This is achieved by combining mean-field models for network formation and opinion sharing, which allows us to derive the willingness of individuals to express their opinions from their fundamental relational dynamics. The theoretical framework is built upon a multigroup majority game and a dynamical system model. This approach is also able to determine bifurcations in the macroscopic behaviour of the population. Monte Carlo simulations are used to validate the analytical results and to explore how finite-size effects and non-mean field network features influence the system’s behaviour. These findings contribute to a deeper understanding of how the network structure can suppress or support minority opinions in digital and offline communities
- Treball de fi de màsterNumerical simulations of neuronal cultures with modular organization: uncovering the mechanisms that shape a rich dynamics(2025-04) Alberic i Torrent, Júlia; Soriano i Fradera, JordiThe study of modular neuronal cultures has proven useful to provide insights into the mechanisms that shape a rich dynamics in the human brain, specifically the capacity to alternate between small-scale computation (segregation) and whole-network information exchange (integration). Motivated by experimental data, in this work we investigated the dynamics of simulated neuronal networks using the Izhikevich model, with the key ingredient that neurons were organized in modules coupled to one another with short- and long-range connections. We considered both cross-correlation and transfer entropy to analyze activity data and extract major metrics of network’s functional behavior, such as modularity and functional richness, and observed that these quantities were highly sensitive to the coupling among modules and noise, which is the main driver of spontaneous activity. We also assessed whether the resulting functional and effective networks retained information from the underlying structural connectivity. Our simulations, despite their simplicity and refinements left behind, may help experimentalists in the design of new in vitro systems to explore open questions of functionality and information processing in rich neuronal assemblies
- Treball de fi de màsterDynamical mean-field theory for non-reciprocal spin-glasses(2024-06) Garcés Ortiz, Ot; Levis, DemianIn out-of-equilibrium systems, the lack of reciprocity in interactions is more the rule than the exception. Non-reciprocal interactions arise generically in out-of-equilibrium systems, such as metamaterials, neural networks, or ecosystems. In the context of glassy systems, it is known that they are crucial in the process of learning in neural networks but their role in glassy dynamics is still widely debated. In this work, we develop a generalization of a dynamical mean-field theory of spinglass models which includes non-reciprocal interactions among spins, with full analytical detail. Furthermore, we show how the dynamics of mean-field spin-glasses are quantitatively and qualitatively modified when considering non-reciprocal interactions, focusing on the high-temperature relaxational dynamics. Our theory predicts critical slowing down of the dynamics and glass melting when considering weakly non-reciprocal interactions, although we suspect that new physics can be further explored beyond that limit
- Treball de fi de màsterSherrington-Kirkpatrick model analysis of fMRI-BOLD data from Alzheimer’s patients and healthy controls(2024-07) Damiani, Giada; Ruffini, Giulio; Levis, Demian; Vohryzek, JakubThis thesis aimed to advance the understanding of Alzheimer’s disease (AD) using computational modelling tools rooted in statistical physics. Specifically, pseudo-likelihood maximisation of a spinglass model was employed to extract coupling matrices J from fMRI-BOLD data of elderly subjects diagnosed with AD and healthy controls (HC). Data was sourced from participants in the European Project Neurotwin’s clinical trial, where AD patients undergo brain stimulation as a potential treatment, and from the AD Neurological Initiative (ADNI) database for the healthy controls. The derived coupling matrices were then compared between conditions to identify differences in brain connectivity. The research also explored the criticality of these systems using Metropolis simulations to assess phase transitions and critical temperatures. First, the focus was on extracting and analysing the J matrices. It was found that the J homotopic connectivity decreased in the AD subjects compared to the healthy ones with weak statistical significance (p = 0.0496), a finding consistent with other studies on inter-hemispheric connectivity disruption in AD. Moreover, the J matrices’ standard deviation significantly differed between the HC and AD groups (p = 0.0039). Additionally, brain areas with the highest change in J across conditions aligned with regions previously identified in functional connectivity studies of AD. Then, the spin-glass systems — defined by the condition-specific J’s — were simulated with the Metropolis algorithm. The critical temperature was found to be lower in the AD spin lattice compared to the HC spin lattice, suggesting that the AD state is closer to a disordered (paramagnetic) phase, which aligns with the hypothesis that weaker inter-parcel connections in AD may lead to a state nearer to the paramagnetic phase transition. The research highlighted the potential of the J coupling matrix to capture structural features and homotopic connections, which can serve as a synthetic brain connectome when dMRI is unavailable. Future work will include using longer data and other type of data (e.g. new healthy controls, and pre- and post-stimulation data), and the optimisation of the sparsity value in the extraction of J. Also, the criticality analysis may be improved by building a theoretical phase diagram based on J characteristics.
- Treball de fi de màsterFloquet stability analysis of a wall-bounded oscillatory flow of a viscoelastic fluid(2024-09) Codina Vilanova, Arnau; Ortín, Jordi, 1959-Oscillatory fluid flows play an important role in fluid mechanics for their long history and numerous applications. In this work we will start off from Stokes’ second problem of the boundary layer adjacent to an oscillatory wall in order to study the wall-bounded zero-mean oscillatory flow of a viscoelastic fluid placed between two parallel plates oscillating synchronously. From related experiments on the oscillatory flow of a viscoelastic solution in a vertical tube we know that the rectilinear flow at small forcing amplitudes gives rise to a secondary flow with toroidal vortices at larger amplitudes. Our purpose is to provide a theoretical understanding of this instability in a simpler setup. We analytically solve the governing equations of the periodic base flow, and carry out a Floquet linear stability analysis of the stress and velocity fields. We apply the Galerkin spectral method to numerically solve the corresponding generalized eigenvalue problem, and provide instability thresholds in forcing amplitude for both resonant and non-resonant forcing frequencies
- Treball de fi de màsterInverse inference of a quantum spin glass(2023-06) Torres Hugas, Lluís; Palassini, MatteoIn statistical physics, inverse problems arise when we need to design a manybody system with particular desired properties. Rather than calculating observables based on known model parameters, inverse problems involve inferring the parameters of a model based on observations. In my final degree project, we studied the inverse problem for the Viana-Bray spin glass model with a discrete distribution of the couplings, and with simulated annealing, we tried to infer the couplings of the system with the maximum pseudolikelihood method. In this project, we studied the inverse problem for the quantum Viana-Bray spin glass model with the same coupling distribution. The goal was to extend the pseudolikelihood maximization approach to a quantum spin glass with a transverse field since most research efforts have focused on the classical version and hardly anything is known about its quantum counterpart. To achieve this, we generated equilibrium configurations from the quantum partition function using quantum Monte Carlo techniques, and then we employed simulated annealing to maximize the pseudolikelihood function and infer the couplings of the system. We derived a modified version of the pseudolikelihood function from the initial proposal after closely following the approach used in the classical case. We found that when introducing the transverse field in the system, the algorithm was still able to infer the couplings; however, because of how the quantum system is treated, certain modifications had to be made to the pseudolikelihood method. Moreover, as in the classical case, we found that the algorithm performed the best around the phase transition boundaries.
- Treball de fi de màsterSelf-regulation of a network of Kuramoto oscillators(2023-05) Pirker Díaz, Paula; Díaz Guilera, Albert; Soriano i Fradera, JordiPersistent global synchronization of a neuronal network is considered a pathological, undesired state. Such as synchronization is often caused by the loss of neurons that regulate network dynamics, or cells that assist these neurons such as glial cells. Here we propose a self-regulation model in the framework of complex networks in which we assume that, for sake of simplicity, glial cells prevent the over synchronization of the neuronal network. We have considered a brain-like network characterized by a modular organization combined with a dynamic description of the nodes as Kuramoto oscillators. We have applied a self-regulation mechanism to keep local synchronization while avoiding global synchronization at the same time. To do so, we have added self-regulation to the system by switching off for a certain period of time a selection of edges that link nodes showing a synchronization above a certain threshold. Despite the simplicity of the approximation, our results show that it is possible to maintain a high local synchronization (module level) while keeping low the global one. In addition, characteristic dynamic patterns have been observed when analysing synchronization between modules in large modular networks. Our work could help to understand the effects of localized regulatory actions on modular systems with synchronous phenomena, such as neuroscience and other fields.
- Treball de fi de màsterDynamics and critical behaviour of neuronal cultures grown on topographical patterns with fractal structure(2023-06) Olives Verger, Mireia; Soriano i Fradera, JordiNeuronal cultures are an excellent experimental tool to study the collective behaviour of neuronal ensembles, providing information on the principles of synaptic functioning and propagation. However, neurons cultured on flat surfaces present limitations in terms of their functionality, as they exhibit a synchronous dynamic behaviour that differs from the much richer repertoire of activity of the brain. In order to address this limitation and help developing better in vitro tools to model the brain, here we studied the capacity to break off synchrony by modulating the spatial arrangement of neurons in the substrate they grow. For that, we designed polydimethylsiloxane (PDMS) topographical patterns with fractal geometry and used them as the substrate to grow neurons, with the goal to break the isotropy in connectivity and enrich dynamics. Neuronal activity was recorded with calcium fluorescence imaging and data analysed in the context of criticality, which was inspired by recent findings suggesting that a rich structural connectivity in the brain is behind its functioning at the edge of criticality. We observed that, first, neurons cultured on fractal patterns exhibited richer and more complex dynamics as compared to standard cultures; and, second, that an analysis of the data using the renormalisation group approach, revealed the presence of scale invariance and typical features of systems poised at criticality. Our study is a multidisciplinary endeavour that combined experimental, theoretical and data analysis aspects to validate the hypothesis of the existence of a self-organised criticality in living neuronal networks, from cultures up to the brain.
- Treball de fi de màsterValidation and Refinement of a Laminar Neural Mass Model Using in vivo Mice Data(2023-06) Moreno Fina, Martina; Clusella Corberó, Pau; Sánchez-Todo, Roser; Soriano i Fradera, JordiGamma oscillations (30-80 Hz) play a crucial role in cognitive functions and are associated with neurological disorders, including Alzheimer’s disease. Non-invasive brain stimulation techniques, such as 40 Hz transcranial alternating current stimulation (tACS), offer potential in modulating these oscillations and impact cognitive functions. The complexity of the brain, however, necessitates the use of advanced models for effective understanding and the development of therapies. This study aims to validate a framework combining Neural Mass Models (NMMs) with volume conduction physics that takes into account the brain’s physical properties and the distribution of synapses across cortical layers. The validation involves predicting a synaptic distribution across various neuronal groups and employing a Genetic Algorithm (GA) to iteratively refine the model to match experimental data. Key findings include the ability of the NMM to achieve greater similarity with experimental results by varying stochastic noise and the dominance of gamma and alpha oscillations in experimental data aligning well with model predictions. The GA also shows robustness in fitting the model to experimental data, and the predicted synaptic distribution is evaluated against existing literature for physiological accuracy. Despite limitations, our enhanced NMM provides valuable insights into cortical layer interactions, contributing to the understanding of human brain function and the development of treatments for neurological disorders.
- Treball de fi de màsterNon-reciprocal interactions in the XY Model(2023-06) Mazzanti Tarancón, David; Levis, DemianThis work focuses on the investigation of non-reciprocal interactions in the XY Model using the Kuramoto model of synchronization in the overdamped limit. Initially, we provide partial results of the reciprocal XY Model by examining the spatial correlation function and the transition temperature. Through a comparison of simulation and theoretical results, we gain insights into the critical behavior of the model. To extend the analysis, we introduce non-reciprocal interactions using the Kuramoto model in the overdamped regime, which offers a nonlinear mathematical framework for understanding the dynamics of the system. This is particularly relevant as the reciprocal XY Model lacks a Hamiltonian description. By incorporating non-reciprocal interactions, we observe that the system does not undergo a topological phase transition. Instead, a dynamic analysis reveals, under certain initial distribution and conditions, the emergence of waves and their characteristic propagation. We explore these phenomena in both one-dimensional and two-dimensional scenarios, demonstrating that the waves propagate with a linear velocity and exhibit a linear dispersion relation
- Treball de fi de màsterViscosity of blood at the microscale infected by Plasmodium falciparum(2023-06) Masó Castro, Nil; Hernández Machado, Aurora; Portillo Obando, Hernando A. delThe viscosity of blood infected by malaria is studied using a microfluidic-based rheometer. Malaria is an infectious disease that causes numerous infectious worldwide, being one of the most prevalent diseases. The findings provide insights into the physical properties of the parasite, with potential implications for malaria diagnostics. The microrheometer allows to perform experiments and obtain the viscosity curve as a function of the shear rate using low volumes of blood, contrary to commercial viscosimeters. The viscosity of blood is well characterized using a power-law model, which only needs two parameters to be defined. Blood samples in culture media (RPMI) and at a physiological temperature were analysed. Moreover, blood samples infected with Plasmodium falciparum with different percentage of infected red blood cells were analyzed. Results show a shift towards a Newtonian behaviour as parasitemia increased, losing the characteristic shear-thinning behaviour of blood. This study also examines the influence of mature parasite stages on blood viscosity, revealing their contribution to the observed changes. These findings have two implications. Firstly, we enhance our understanding of the rheological alterations caused by malaria infection, which can aid in developing improved diagnostic tools based on rheological markers. Secondly, the surprising results obtained contribute to the knowledge about the effects of the parasite as the sample ages at 37◦C, which ultimately could aid the development of new malaria treatment techniques.
- Treball de fi de màsterAssembly and dynamics of propelling ferromagnetic colloids(2023-06) Manzano González, Andrés Javier; Junot, Gaspard; Tierno, PietroActive colloids is a growing field of research that studies the emergent phenomena exhibited in systems of self-propelling particles as a result of the interactions between them that could not be observed if these components were isolated. These phenomena can be observed in a wide variety of systems. In this work, I show how a collection of oscillating colloidal rotors, or simply, shakers, immersed in a viscoelastic fluid, specifically, a solution of polyacrylamide (PAAM), can display this behaviour, forming complex zig-zag dynamic bands that grow linearly in time. I have investigated the dynamics of this band growth for two different PAAM concentrations, as well as the hydrodynamic flow around these shakers for both cases. This helps understanding the complex dynamics of these fluids, having implications in soft matter physics and microfluidics and in biophysics
- Treball de fi de màsterDeveloping modified Lotka-Volterra models to simulate in vitro clonal dynamics(2023-06) Jurado Rodríguez, Imanol; Ibañes Miguez, Marta; Alemany i Arias, AnnaDuring the last decades, novel technological approaches have allowed uniquely labelling cells by integrating random barcodes in their DNAs. Such barcodes are permanent and are inherited by the cellular offspring. Thus, DNA sequencing permits their reading in order to identify those cells with common ancestors. This process, known as lineage tracing, enables the study of complex biological processes such as embryonic development, tissue homeostasis or even cancer metastasis with clonal, and even single cell, resolution. Here we aim to reveal the foundations behind the experimental results of clonal dynamics of colon cancer organoids through in silico simulations. We formulated three modified Lotka-Volterra models that allow us to investigate the role of clonal carrying capacity, proliferation rates and inter-clonal interaction network to achieve our purpose. The results show the vital role of partial interactions among clones and the importance of implementing nonequilibrium networks, i.e. architectures of interactions that vary in time. Furthermore, our results reveal a direct relationship between the harvesting time and the average number of surviving species at the end of the experiment, suggesting that external perturbations to the system can have big effects to clonal dynamics
- Treball de fi de màsterThe community structure of the geometric soft configuration model(2023-08) González Gea, Viktor; Serrano Moral, Ma. Ángeles (María Ángeles)Network models serve as an approach to explain the properties of real networks. The geometric soft configuration model, also known as the S1/H2 model, can be used to generate synthetic networks that replicate many features of real complex networks —sparsity, a heterogeneous degree distribution, the small world property, a high level of clustering, and more— while randomizing others. In this work, a range of parameters of the S1/H2 model has been explored, satisfactorily manipulating the level of heterogeneity of the degree distribution with the parameter γ and the level of clustering with the parameter β, in order to probe the level of control that is possible to attain in the generation of random networks. Recent theoretical evidence supports that hyperbolic networks like this one possess topological community structure, up to being maximally modular in the thermodynamic limit, even if the model is not purposefully equipped with geometric communities. The community structure of the S1/H2 model was put under scrutiny using computational simulations, revealing that synthetic networks generated according to it could be consistently partitioned with a high modularity. The modularity of equally sized angular partitions of the generated random networks was evaluated, confirming that this model tends to maximal modularity in the limit of large network size and in a regime of high clustering. The Louvain method for community detection in the topology of complex networks using modularity maximization was employed as well, giving rise to no significantly better results in comparison with the initial approach. With the S1/H2 model, it was also explored how much of the community structure of real networks can be attributed to the effect of clustering in combination with their heterogeneous degree distribution —networks with these two features are called hierarchical—. The results suggest that the communities detected in some real networks are, in part or totally, a byproduct of their hierarchicity.
- Treball de fi de màsterStatistical Mechanics of non−reciprocally interacting Ising spins(2023-07) Garcés Ortiz, Adrià; Levis, DemianNon−reciprocal interactions are present in a large number of out−of−equilibrium systems such as active matter, social, ecological and non−Hermitian quantum systems. They are believed to be responsible for non−equilibrium phase transitions and are, still, an open topic of major interest in recent research. In this work, we present a generalization of the Ising model that includes non−reciprocal interactions among spins and analytically characterize the mean field stationary behaviour of two proposed models that incorporate non−reciprocal interactions. We show how the models exhibit a first order phase transition and how their mean field solutions are no longer spin−inversion symmetric. Furthermore, we also study d = 1 spin chains with nearest neighbours interactions, and derive the evolution equations for the first two moments. Finally, we discuss the dynamical equations for the proposed models. The derived dynamic equations signal the presence of steady currents, e.g. traveling states, in non−reciprocally interacting spin chains.